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doc405 08rt26-delivering-public-investment-efficiently-_conefrey-lozej-o'reilly-and-walsh_.pdf_sfvrsn_52df701a_3.pdf
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"text": "Public investment can be used to affect the economy over the business cycle, but also to boost its long-term potential. However, public investment is often subject to delays related to planning, construction, or both. We show that these delays can materially affect the usefulness of public investment for managing the economy. In particular in the case of a downturn, a delay in delivering an announced public investment can worsen the downturn if public investment is not delivered quickly. If public investment that has been planned during the recession is delayed so that it occurs when the recession is over, it risks overheating the economy. Delays in the delivery of public investment shift the benefits from higher public capital further into the future and reduce welfare. These findings hold in the standard model as well as in the model with search frictions, but with search frictions planning delays become less concerning, because the business cycle induced by the delay becomes smoother.",
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"text": "We are grateful to Christian Merkl and Valeria Patella for discussions and suggestions, as well as to the participants of the 29th International Conference on Macroeconomic Analysis and International Finance, workshop Fiscal policy and the role of cohesion funds in the European Union, 2nd annual Irish public economics workshop, and the participants of the Central Bank of Ireland economics seminar. The opinions expressed are those of the authors and do not reflect views of the Central Bank of Ireland or of the European System of Central Banks.",
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"text": "Public investment has been increasingly highlighted as an important policy tool because it stimulates the economy in the short run by increasing aggregate demand, while at the same time also improves the productive capacity of the economy in the long run, which increases aggregate supply. However, delivering public investment is often slow and subject to delays. These delays can be of a more procedural nature due to the delays in planning, but also technical, as it takes a long time to construct large projects that are typically done by governments.",
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"text": "We investigate the effects of these two frictions that cause delays in delivering public investment. The first friction, which we call a planning delay or a time-to-plan, is the period of time between the moment a public investment is announced and the time it starts being carried out. No material spending occurs during this time and therefore also no material stimulus to the economy other than information that something will be constructed and put to productive use in the future. The second friction, which we call construction delay or time-to-build, is the time that passes from the moment construction starts to when the project is delivered and put to use. During this time, there is an active demand stimulus to the economy provided by the government purchasing goods and services needed for government investment. Similarly as with the planning delay, this is accompanied by the information that, when finished, the investment project will be put to productive use in the future.",
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"text": "To assess the effects of delays related to public investment, we conduct a comparative analysis in two classes of models of a small open economy in a monetary union, calibrated to Ireland. One model features a standard labour market with sticky wages, while the other model is otherwise identical, but features search frictions and unemployment. We find that in both classes of models a permanent increase in (productive) public investment always leads to an increase in output, consumption, and exports in the long run. Delays in delivering public investment always result in lower welfare, even when the long-run increase in consumption and other variables is the same.",
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"text": "Moreover, we find that planing delays can make the stabilisation of business cycles more difficult. Planning delays can result in output decreasing in the short run, so that during a recession an announcement of an increase in public investment, if not followed quickly by implementation, can exacerbate the downturn. Moreover, if the eventual stimulus in the future coincides with a future boom in the business cycle, this may exacerbate the boom and overheat the economy.",
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"text": "Public investment is an important policy tool (Ramey, 2021), because it has two distinct properties. First, it has short-term demand-side effects similar to those of government consumption, and can therefore be used to stimulate the economy in the short run. Second, public investment increases long-lasting public capital that has long-term supply effects similar as the productivity increase. Both properties have been increasingly recognised by policymakers as useful instruments to manage the economy, especially after the Great Recession (see, e.g., Coenen et al. (2012) and Coenen et al. (2013)), but also in the more strategic longer run. A very prominent recent example is the report of Mario Draghi (Draghi, 2024), which among other things calls for a structural reform in Europe with a permanent increase in public investment. However, the effect of public investment on the economy depends not only on the level of investment, but also on the efficiency with which it is delivered. Typically, delays happen (Leeper et al., 2010), so benefits from investment accrue further in the future.",
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"text": "We investigate the effects of two frictions related to public investment that cause delays. The first friction, which we call time-to-plan, is the time that passes between the moment a public investment is announced and the time it starts being carried out. The second friction, called time-to-build, is the time that passes from the moment construction starts to when the project is delivered and put to use. There is an important distinction between them. Time-to-plan implies an announcement of public investment in the future, but no (or minimal) spending and hence no immediate stimulus until investment commences. Time-to-build implies that the stimulus from the demand-side related to public investment is already taking place, but the project has not yet been finished and put to use, so that there are no supply-side effects from higher public capital yet.",
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"text": "Delays can happen in either phase. During the planning phase the frictions causing delays are typically of a more bureaucratic nature: there are zoning issues, various permissions must be obtained, there may be objections from the public, litigations, etc. These can be sizeable. For instance, in Ireland the time between a permission is obtained and construction commences can come close to two years in some regions (NCPC, 2024). Frictions causing delays in time-to-build are typically more technical and related to the organisation of work, supply lines, availability of workers, etc.",
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"text": "Regardless of the reason, both types of delays are a common occurrence when public investment is concerned (Leeper et al., 2010), and at the same time important for the short-term economic fluctuations that are caused by government investment (Ramey, 2021). While these authors focused on the closed economy, we analyse these issues in an open economy that is a member of a monetary union. This has two implications. First, monetary policy reaction is (almost) absent, as the economy is small relative to the rest of the monetary union. Second, the effects of domestic prices on external competitiveness, i.e., on exports and imports, play an important role.",
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"text": "Delays in delivering public investment, however, are particularly interesting because of its productive property, in the sense that it increases the stock of public capital, which in turn increases productivity of the private sector. The notion has a long tradition starting at least from Baxter and King (1993) and continuing in Leeper et al. (2010) and Ramey (2021). This increase in productivity of the private sector occurs in the future (when public capital is constructed and put in productive use). Therefore, it has very similar properties to a positive news shock about an increase in future productivity,",
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Page 5 JSON
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"text": "which has an extensive literature on its own, starting with Beaudry and Portier (2006). The key issue in that literature was whether a standard model can generate a typical economic boom, with an increase in output, consumption, investment, and employment immediately, even if productivity will only increase in the future. Den Haan and Kaltenbrunner (2009) have pointed out that this can happen in a closed economy if there are search frictions. Den Haan and Lozej (2011) showed that this can also be the case in an open economy. The reason is that in a standard model with search frictions, employing a worker is essentially an investment decision on behalf of a firm. When deciding to post a job vacancy, a firm looks at the present value of future profits that will be obtained by employing an additional worker.",
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"text": "This intertemporal dimension of hiring is what makes delays in the delivery of public investment particularly interesting, because while delays shift the productivity into the future, firms may still hire workers now in anticipation of future profits. If this mechanism is strong, it may make delays in public investment less costly. This interaction of search frictions and public investment has, to the best of our knowledge, only been investigated in the context of a closed real economy model by Matusche (2025). The importance of analysing this in an open economy stems from the fact that a productive element of the public capital lowers marginal costs of domestic producers and therefore improves the external competitiveness of the economy. This channel is particularly important in small open economies.",
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"text": "We investigate the interaction of the two types of delays described above with the search frictions in the context of a large policy model of a small open economy in a monetary union.[p5.3.4] To do so, we simulate a structural reform that leads to a permanent increase in public investment in a small open economy in Europe, Ireland, using a global model. We do so once in the standard setting, with no search frictions on the labour market, and once in the otherwise identical model, but where the standard labour market with staggered wage setting has been replaced by search frictions with staggered wage setting, closely following Gomes et al. (2023).",
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"text": "We find that a permanent increase in (productive) public investment always leads to an increase in output, consumption, and exports in the long run. Delays in delivering public investment always result in the benefits of higher output, exports and consumption occurring further in the future. In the case of planning delays, long delays can even result in output decreasing in the short run, implying that announcement of public investment in a recession, if not followed quickly by implementation, does not stimulate the economy. Moreover, it can lead to a fluctuation in the future that looks very much like a small business cycle. For example, if it coincides with a boom in the business cycle that is not related to public investment, this may exacerbate the boom and overheat the economy. In addition, the demand stimulus during the implementation phase of public investment worsens the external competitiveness of the economy.",
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"text": "To conduct the analysis in a reasonably realistic modelling environment as used in policy institutions, we require a few features: (1) we need a small open economy that is a member of the monetary union; (2) we need a reasonable trade structure that approximates a modern global economy, where countries trade globally and not just within their currency area or a region; (3) we need a sufficient degree of real and nominal frictions, in particular sticky prices and sticky wages; (4) we need the fiscal policy that allows for public investment.",
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"text": "To satisfy these requirements, we use as the basic framework the model developed by Gomes et al. (2010) and Gomes et al. (2012), which features a small open economy in a monetary union, both integrated in a global economy.[p6.3.8] To model the high degree of openness of the Irish economy as an export platform, we use the augmented framework of this model that features import-content of exports (Brzoza-Brzezina et al. (2014)), as this allows us to calibrate a realistic share of trade relative to GDP. The core model features four blocs, each of which is similar to a standard Smets-Wouters model with sticky prices and wages. Two of the blocs are in a monetary union, so that they share the same monetary policy rate and have the nominal exchange rate fixed to unity. The remaining two blocs are the US and the rest of the world. Risk-sharing is imperfect. Households can save in an intra-euro-area bond that is traded between the two blocs constituting the euro area. In addition, households can also hold an international bond (denominated in USD). These features of the core model satisfy conditions (1)-(2) and partly the condition (3), as the framework has the New Keynesian price and wage rigidities.",
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Page 7 JSON
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"mathml": "<math aria-label=\"K sub G\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>K</mi><mrow><mi>G</mi></mrow></msub></mrow></math>"
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{
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"type": "formula",
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"r": 379.0956,
"b": 639.0972
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"text": "K_{G,t}=(1-\\delta_{G})K_{G,t-1}+I_{G,t-h},",
"attrs": {
"block_label": "display_formula",
"latex": "K_{G,t}=(1-\\delta_{G})K_{G,t-1}+I_{G,t-h},",
"notation": "latex",
"alt": "K sub G, t equals 1 minus delta sub G K sub G, t minus 1 plus I sub G, t minus h,",
"mathml": "<math aria-label=\"K sub G, t equals 1 minus delta sub G K sub G, t minus 1 plus I sub G, t minus h,\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>K</mi><mrow><mi>G</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>=</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>G</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>K</mi><mrow><mi>G</mi><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>I</mi><mrow><mi>G</mi><mo>,</mo><mi>t</mi><mo>−</mo><mi>h</mi></mrow></msub><mo>,</mo></mrow></math>"
},
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{
"id": "p7.4",
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"type": "text",
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{
"id": "p7.5",
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"type": "text",
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"text": "where[p7.5.1] is the depreciation rate of public capital, and $h$ measures the time-to-build (in quarters) that is needed before public investment becomes part of public capital that is used in production (for instance, for 2-year time-to-build,[p7.5.2]).",
"children": [
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"alt": "h equals 8",
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"text": "Government investment is decided by the government and follows an autoregressive process:",
"children": []
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{
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"type": "formula",
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"text": "I_{G,t}=(1-\\rho)\\overline{{I_{G}}}+\\rho*I_{G,t-1}+\\varepsilon_{I_{G},t-j},",
"attrs": {
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"latex": "I_{G,t}=(1-\\rho)\\overline{{I_{G}}}+\\rho*I_{G,t-1}+\\varepsilon_{I_{G},t-j},",
"notation": "latex",
"alt": "I sub G, t equals 1 minus rho overline I sub G plus rho *I sub G, t minus 1 plus varepsilon sub I sub G, t minus j,",
"mathml": "<math aria-label=\"I sub G, t equals 1 minus rho overline I sub G plus rho *I sub G, t minus 1 plus varepsilon sub I sub G, t minus j,\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>I</mi><mrow><mi>G</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>=</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><mi>ρ</mi><mo stretchy=\"false\">)</mo><mover><mrow><mrow><msub><mi>I</mi><mrow><mi>G</mi></mrow></msub></mrow></mrow><mo accent=\"true\">―</mo></mover><mo>+</mo><mi>ρ</mi><mo>*</mo><msub><mi>I</mi><mrow><mi>G</mi><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>ε</mi><mrow><msub><mi>I</mi><mrow><mi>G</mi></mrow></msub><mo>,</mo><mi>t</mi><mo>−</mo><mi>j</mi></mrow></msub><mo>,</mo></mrow></math>"
},
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{
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"type": "text",
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{
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"text": "where[p7.9.1] measures the persistence of public investment,[p7.9.2] is the steady state level of public investment, and[p7.9.3] is the change in public investment that is decided by the government. Index $j$ measures the time to plan, i.e., the time from the announcement to the implementation of public investment (e.g., for 2-year time-to-plan,[p7.9.4]). Throughout the paper, we assume that the announcement is fully credible.",
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"mathml": "<math aria-label=\"I sub G bar\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mover><mrow><msub><mi>I</mi><mrow><mi>G</mi></mrow></msub></mrow><mo accent=\"true\">―</mo></mover></mrow></math>"
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"mathml": "<math aria-label=\"varepsilon sub I sub G\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>ε</mi><mrow><msub><mi>I</mi><mrow><mi>G</mi></mrow></msub></mrow></msub></mrow></math>"
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"type": "formula",
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"l": 67.9013,
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"text": "j = 8",
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"inline": true,
"text_offset": 347,
"text_span": 8,
"bbox_estimated": true,
"alt": "j equals 8",
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{
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"text": "Importantly, public capital improves the productivity of the private sector. In the context of both models, it enters in the production functions of tradable goods,[p7.10.5], and non-tradable goods,[p7.10.6], following the specifications that are standard in the literature (Baxter and King (1993) or Leeper et al. (2010)):",
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"type": "link",
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"text": "(2010)):",
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"text": "\\begin{array}{ccl}{Y_{N,t}}&{=}&{z_{N,t}K_{G,t}^{\\alpha_{G}}K_{N,t}^{\\alpha_{N}}N_{N,t}^{1-\\alpha_{N}}-\\psi_{N}}\\\\ \\end{array}",
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{
"id": "p7.15",
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"type": "text",
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"text": "where[p7.15.2] and[p7.15.3] are sector-specific productivities,[p7.15.4] and[p7.15.5] is capital used in each sector, and[p7.15.6] and[p7.15.7] is labour used in each sector. These equations show that an increase in public capital is the same as the increase in productivity ([p7.15.8] enters production function in the same way as productivity z). The only difference is that productivity increase happens exogenously and is \"free\" in the sense that nobody has to pay for it, while the increase in public capital is a decision of the government and comes at a cost - the government must raise the required funds either by borrowing or by raising taxes. The analysis in this paper will assume that the government borrows the required funds. [p7.15.9]",
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"alt": "to the power of 5",
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"text": "To understand the importance of productivity of public capital for the external competitiveness of the economy, recall that price is a markup over the marginal cost and that the marginal cost equation with public capital is (reproduced here for the tradable sector): [p7.16.2]",
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"text": "[p7.17.1]The borrowed funds are repaid over time by levying lump-sum taxes on households.",
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"text": "[p7.18.1]The equation for marginal costs in the non-tradable sector is analogous.",
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footer (1)
footnote (1)
formula (14)
link (6)
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1 · FORMULA
2 · TEXT
3 · TEXT
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8 · TEXT
9
10 · TEXT
11
12
13 · TEXT
14
15 · FORMULA
16 · TEXT
17 · TEXT
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25 · FOOTNOTE
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29 · FOOTER
Page 8 JSON
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"text": "MC_{T,t}=\\frac{1}{z_{T,t}K_{G,t}^{\\alpha_{G}}\\alpha_{T}^{\\alpha_{T}}(1-\\alpha_{T})^{1-\\alpha_{T}}}r_{k,t}^{\\alpha_{T}}w_{t}^{1-\\alpha_{T}},",
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"text": "where[p8.3.1] is the return on private capital and[p8.3.2] is firms' cost of labour. Note that[p8.3.3] appears in the denominator of marginal costs, so that an increase in public capital, if it is productive (which is the case when[p8.3.4]), leads to a reduction in marginal costs and as a consequence to a reduction in prices and a depreciation in the real exchange rate.,",
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"text": "A delay in either planning or in construction of public investment goods will shift the increase in[p8.4.1] into the future and with it the decrease in marginal costs. Forward-looking agents in the model know this and react accordingly. However, their reaction must also take into account other frictions. In the standard model, this is the friction due to sticky wages, while in the search model, this is, in addition to sticky wages, also the search friction. To see how these frictions work, consider the following intuition for each of the models considered.",
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"text": "In the standard model, the wage setting is forward-looking, as unions set wages as a markdown over the marginal disutility of work (this is the standard Erceg et al. (2000) mechanism). Unions know that labour demand will increase when the demand for labour will increase, which will happen when government will start spending on investment. If wages were flexible, unions would increase wages then. However, when this increase in labour demand happens in the future (which is the case with planning delays), unions start increasing wages already now, because wages are sticky and it takes time to increase them. Faced with higher wages and no increase in goods demand before the stimulus begins, firms will reduce the number of workers.",
"children": [
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"type": "link",
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"text": "Erceg et al.",
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"text": "(2000)",
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"text": "In the search model, there is the aforementioned friction due to sticky wages, but also an additional search friction. Firms know that hiring takes time (unlike in the standard model, where hiring is instantaneous), so in order to satisfy the need for more workers in the future, they will have to start hiring already now. To see this, consider the optimality condition of the firm in the search model, which we reproduce below (see Appendix B for the details):",
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"text": "\\psi_{s}=p_{t}^{F}\\beta\\frac{u^{\\prime}(c_{s,t+1})}{u^{\\prime}(c_{s,t})}\\left[(1-\\kappa_{w,s})\\mathcal{A}_{t}^{F}(w_{s,t+1}^{*})+\\kappa_{w,s}\\mathcal{A}_{t}^{F}(w_{s,t+1})\\right].",
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"text": "The condition states that a firm that is posting a vacancy for household type s equalises the per-period constant cost[p8.9.3] for having a vacancy open with the expected value of getting a worker. This expected value depends on several factors. The first is the probability that the firm will find a worker,[p8.9.4]. The remainder of the right-hand side of the equation 6 is the discounted value of the benefits that this firm will have from finding a worker. This depends on whether the firm will be able to renegotiate the wage with the worker or not. If[p8.9.5] denotes the probability that a firm cannot renegotiate the wage for a newly hired worker from household type s, then the value of employing a new worker is, in monetary terms, equal to the weighted average of the value of a worker at a newly-renegotiated job[p8.9.6] and the value of a worker hired at average wage[p8.9.7].7 Because these values are forward-looking, they will increase immediately upon announcement, and firms will immediately post more vacancies, even if wages increase temporarily and if there is no immediate need for additional workers.",
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1 · HEADING
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Page 9 JSON
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"text": "We calibrate both models to Ireland, the rest of the euro area (REA), the rest of the world (RW) and the US. First, we use the some standard values from the literature, but where available, we augment these with the empirical estimates. These are mostly based on the estimates for the euro area in Christoffel et al. (2008), and the values used in Gomes et al. (2012). Second, in addition to these standard values from the literature, we account for the fact that the neutral rate has followed a downward trend throughout and therefore adjust its level to be 1% in the steady sate (as in Gomes et al. (2023)). Finally, we update the trade matrix to account for the fact that trade flows have increased markedly from the levels before the Great Financial Crisis, and to account for the increased role of the global value chains, we also target the empirical levels of import-content of exports for all blocs in the model. Below we report the full set of values used in the calibration (please see A for the equations that correspond to the reported parameters).",
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"text": "Table 1 reports parameters used to calibrate utility and production functions. Parameters governing the utility function and intermediate goods production are mostly standard values from the literature. The productivity of public capital,[p9.3.7], was calibrated to be a conservative estimate from the literature (in the lower end of the range considered by Leeper et al. (2010), and lower than the estimates in Bom and Ligthart (2014), as Ireland is a developed economy and additional investment in infrastructure would likely be less productive than investment in infrastructure in countries where infrastructure is missing). Note also that the value of this parameter captures the average productivity of public investment, and there are arguably types of public investment that are more productive than the others. In such settings, the composition of public investment bundle matters. Our choice of relatively conservative value for this parameter can also be viewed as reflecting the view that that most productive types of public investment have been made first, and that the additional public investment - while still productive - is less productive that past public investment. [p9.3.8]",
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"text": "Values for the calibration of final consumption good firms, final investment good firms, and re-exported goods are obtained as follows. First, the substitution elasticities are based on the values used in similar models (Christoffel et al. (2008), Gomes et al. (2012), and Clancy et al. (2016)). Second, the shares of goods are obtained by matching the openness of all blocs in terms of bilateral imports and import content of exports, with targets reported in Tables 2 and 3. In addition, net foreign assets are calibrated such that they match the trade balance. For the euro area, we assume a roughly balanced trade balance, for the US a 5% deficit, for Ireland, we target about 15% surplus in the steady state. The latter is a difficult statistic to match, as trade balance has fluctuated between almost zero and as much as 40% in recent years (2000-2024). To obtain a reasonable number that is approximately in line with the trade balance to GNI[p9.4.9] ratio over the recent years, we opt for the relatively large surplus. Note, however, that due to large foreign direct investment to Ireland, the large negative value of net foreign assets and the corresponding need for a large trade surplus are not unexpected. A similar issue as with the trade balance is with the level of imports and exports, which often exceed GDP. Since 2000, Irish imports have fluctuated between 65% and 124% of GDP. We have opted for the calibration that targets Irish import-to-GDP ratio of about 75%,",
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"text": "which corresponds to the period before the one-off GDP increase in 2015.[p10.1.2] The main implication of the calibration of large trade surplus and low net foreign assets for Ireland is that we are able to obtain a relatively low consumption-to-GDP ratio, which is in line with the Irish data.",
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"text": "[p10.2.1]In 2015, Irish GDP increased by 25% due to several reasons, but the main factor was that intangible assets of several multinationals have been transferred to Ireland, and with them also the revenues pertaining to these assets. For the dynamic results reported in this paper, the exact value of the great ratios does not make a material difference, but it does help the credibility of the model.",
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Table integrity:
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r11c0 (29%): “Bilateral bias towards imported goods, Home ( $ \nu_{MC} $)”
r12c0 (29%): “Bilateral bias towards imported goods, REA ( $ \nu_{MC} $)”
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r14c0 (29%): “Bilateral bias towards imported goods, RW ( $ \nu_{MC} $)”
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caption (2)
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footnote (2)
link (8)
table (2)
table_cell (110)
text (1)
1 · CAPTION
2 · TABLE
3
4
5
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8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28 · FOOTNOTE
29 · CAPTION
30 · TABLE
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
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71
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114
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116 · FOOTNOTE
117 · TEXT
118
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125
126 · FOOTER
Page 12 JSON
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caption (3)
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1 · CAPTION
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"text": "To calibrate the model with labour market frictions, we use the following strategy. First, we use identical parameters as in the model without labour market frictions (discussed above) wherever possible, in order to keep the two model versions harmonised to the greatest extent possible so that they are comparable. We can do this only for those parameters that are common to both models, such as the Frisch elasticity of labour supply and the frequency of resetting prices (see Table 5 for nominal price rigidities that are common across models) and wages. Second, for parameters that are specific to the model with search frictions, we use the values based either on the literature or empirical estimates. The calibration parameters are reported in Table 7 and we discuss both groups of parameters below in more detail.",
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"text": "In the first group of parameters that we can harmonise across both models are wage rigidities and Frisch labour supply elasticities. The Frisch labour supply elasticity is set to 0.5 as is typical in the literature (its inverse,[p13.11.3]). Even though the model with labour market frictions can distinguish between wage rigidities of new hires and wage rigidities of existing workers (measured by the probabilities that they start working at existing average wage), we set both to equal value (and to the value equal to that in the model with the standard labour market), so that[p13.11.4]. These values are based on the estimates in De Walque et al. (2009).",
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Page 14 JSON
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"text": "with search frictions as harmonised as possible with the standard model, which does not distinguish the labour market for different groups of households. The parameter governing the production of labour services is in line with Bodart et al. (2006).",
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"text": "Matching efficiency and vacancy posting costs are obtained by targeting matching probabilities of workers and firms. Matching probabilities of workers come from the OECD and Eurostat data for unemployment duration by educational attainment for 2017 and 2018 using the method of Shimer (2012).[p14.2.9] The matching probability for firms is based on the estimates for the US in Den Haan et al. (2000) and Stähler and Thomas (2012) for Europe. These two probabilities determine matching efficiency and (indirectly) vacancy posting costs (the latter being lower for HtM households).",
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"text": "Separation rates are obtained by matching unemployment rates, which are OECD data averages from 2004-2019. Consistently with job finding probabilities, we assume that the unemployment rate for non-Ricardian households is the unemployment rate for persons with educational attainment below upper-secondary. Matching the unemployment rates results in break-up rates for HtM households exceeding those of Ricardian households. Bargaining power has been set to 0.5, for both groups of households, in line with the literature.",
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"text": "[p14.5.5]As described in Gomes et al. (2023) and Herman and Lozej (2023), we assume that non-Ricardian households have below upper-secondary education and Ricardians have above upper secondary education. The estimates are population-weighted across countries.",
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"text": "[p14.6.1]Concretely, replacement rates for one-year horizon of unemployment duration for Ricardian households and two-year horizon of unemployment duration for non-Ricardian households, given that they are more likely to be long-term unemployed.",
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caption (1)
footer (1)
footnote (3)
formula (4)
heading (1)
link (4)
table (1)
table_cell (35)
text (2)
1 · CAPTION
2 · TABLE
3
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38 · FOOTNOTE
39 · HEADING
40 · TEXT
41
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45 · TEXT
46 · FOOTNOTE
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48 · FOOTNOTE
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52 · FOOTER
Page 15 JSON
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"text": "4 Simulation design and results",
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"text": "To analyse the effects of delays in the delivery of public investment, we simulate a gradual, but permanent debt-financed increase in public investment. To clearly separate the effects of different types of delays, we simulate the increase in public investment without any delays, and then compare this with 2- and 5-year planning delays (time-to-plan). We then repeat the same simulation, but with 2- and 5-year construction delays (time-to-build).[p15.5.3] We conduct the same set of simulations across two models, first in the model without search frictions on the labour market, and second in the otherwise identical model, except that this time the standard labour market is replaced by search frictions. [p15.5.4]",
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"text": "It is important to clarify in detail the information structure of the experiments conducted. Before the announcement, the economy is in the steady state, and the",
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"text": "[p15.7.1]Note that while the model has a steady-state 2% inflation, we assume that government investment spending is in real terms, as this allows us to compare shocks of the same size across time. In all cases we assume the government borrows when it invests, i.e., there is no front-loading in government borrowing.",
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"text": "announcement of public investment is a surprise. The announcement is fully credible, the time path of public investment is known at the time of the announcement, and this includes the delays (planning or construction). Note that in this setting a surprise in the sense of the delay being longer than initially announced could be inferred from the charts shown below as the difference between the impulse responses in the charts. For instance, the effect of a 2-year delay, when the initial expectation is immediate implementation, could be computed as the difference between two corresponding impulse responses (this would be exact in a linear model, but is only approximate in our nonlinear setting). [p16.1.2]",
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"text": "4.1 Delays in delivering public investment in a standard model",
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"text": "Figure 1 shows the effects of the delays in planning, and Figure 2 shows the effects of the delays in construction (time-to-build) in the standard model without search frictions. In each figure, the top-left chart shows the path of public investment, and the adjacent chart shows the path of the public capital stock. The path of public capital is the same in both figures, as the source of the delay in public investment does not matter for the stock of the public capital (only the delay itself matters). The key difference, therefore, between planning and construction delays is in the timing when the outlays of funds and the associated payments for public investment goods occur.",
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"text": "This timing of events matters. In Figure 1, the planning delay leads to a period of time where agents in the economy know that public investment will happen in the future, but there is no demand or supply stimulus from public investment yet, because the expenditure has not yet occurred due to the planning delay. Such a planning delay is equivalent to an announcement effect of future outlays for public investment (and future increase in productivity due to higher stock of productive public capital). [p16.4.3]",
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"text": "Figure 1 shows three cases. The black line shows the case when public investment increases without any delay. This is the typical way public investment is analysed in standard cases that do not deal with delays, and therefore serves as a useful benchmark. The dashed red line shows the 2-year delay in planning, and the dotted blue line shows the 5-year delay in planning. Compared to the benchmark case when the stimulus occurs at the announcement, the delays in planning result in output falling slightly during the planning period, with the decrease being larger, the longer is the planning delay. Inflation increases upon announcement and also later in the future (when the demand stimulus occurs), and it remains elevated during the planning period. This is because during this period private consumption increases, and so does (mildly) private investment. The decline in output in the case of delays is due to the fall in net foreign demand, which is primarily caused by the decrease in exports and an increase in imports. Output starts to increase only after the stimulus begins and afterwards, and inflation falls further out in the future, when the demand stimulus eases and the supply effects from higher public capital prevail (recall that public capital is productive). The fluctuations in prices affect",
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"text": "the external competitiveness of the economy, which is reflected in exports and imports, with exports decreasing when prices increase. Exports recover only in the medium to long run when the supply-side effects of productive public capital reduce marginal costs and hence inflation, which restores and improves external competitiveness. Private investment initially increases, then decreases in the medium term, and recovers in the longer run.[p17.1.3] Labour falls if there is a planing delay, and the fall is more pronounced the longer the delay is. This happens because households anticipate the increase in resources in the future and increase their consumption. The wealth effect reduced the supply of labour and this leads to an increase in wages. Firms, observing the increase in wages, reduce the demand for labour (and increase the demand for capital, causing an increase in investment), as labour becomes less attractive due to higher wages. [p17.1.4]",
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"text": "In the long run, shown in Figure 1 as the dot on the extreme right-hand side of the plots, the benefits of higher public investment are obvious. Output, consumption, private investment and wages all increase. There is a permanent decrease in labour services, caused by a reduction in labour supply by households. This happens despite the increase in public debt (recall that the increase in public investment is debt-financed), because higher public capital implies larger productivity and more available resources, which compensates for higher taxes needed to finance higher level of public investment. [p17.2.3]",
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"text": "Note that the paths of inflation and to some extent output look very much like a standard business cycle that was caused by first a negative and then a positive supply shock. However, this behaviour is caused by only one (demand) shock and the effect of its announcement. Moreover, the longer the delay, the more pronounced the fluctuation is, and the cycle is longer (the increase is shifted further into the future when planning delays are longer). From the policy perspective, this can be problematic, because the state of the future business cycle is unknown when the public investment is announced. For instance, suppose an economy is in a recession and the government announces an increase in investment. Because of planning delays, this investment occurs in the future, when the economy may already be out of the recession and in a boom, which risks overheating the economy. Moreover, our results imply that announcing an increase in public investment during the recession will not help stimulate the economy in a countercyclical manner if there are long planning delays - note that output decreases during the planning phase. This implies that an increase in public investment during a downturn will help only if public investment is increased without delay.",
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"text": "[p17.5.1]Note that this mechanism is different than it would be in an RBC model, where labour would decrease because of the decrease in labour supply. Here, labour decreases because of the decrease in labour demand caused by higher wages (which are in turn caused by lower labour supply).",
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"text": "Figure 2 shows delays due to construction (time-to-build). As before, the benchmark case, where there are no construction-related delays, is shown in black line and serves as the benchmark against which to compare the two cases with delays. Dashed red line",
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"text": "shows 2-year construction delay, and the dotted blue line shows the 5-year construction delay. Delays related to construction are less problematic than planning from the perspective of short-term business cycle management, because the demand stimulus occurs immediately in all cases (top left panel) and output starts increasing already on impact. However, the strength of the stimulus and the increase in output are slower the longer it takes to complete the project, because the supply-side benefits from higher public capital kick in later. In particular, inflation stays higher for longer when the construction delays are long, which exacerbates and prolongs the real exchange rate appreciation (a fall in the real effective exchange rate means appreciation) and is detrimental for (net) exports.",
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"text": "Unlike in the case of planning delays, labour increases despite the increase in wages. This is because there is an immediate stimulus from higher demand for public investment goods, which have to be produced, and this increases the demand for labour despite higher wages (and pushes wages up slightly more in the initial phases than in the case of planning delays). Private investment increases in the short run because there are more resources available (from higher output) and because firms wish to supplement the increase in labour with more capital. The longer the delay, the stronger is the initial increase in labour. This is because private consumption increases by less initially, and therefore labour supply decreases by less. This leads to a lower increase in wages, which stimulates the demand for labour by firms. In the long run (shown as dots on the right axis of each plot), all responses are identical to the responses after a planning delay.",
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"text": "As explained above, due to the fact that the decision to hire a worker is forward-looking in a model with search frictions, one would expect that delays in the delivery of public investment are less detrimental in the short run. Forward-looking firms anticipate the increase in future productivity regardless of the delay (it is just shifted farther in the future), and because it takes time to hire workers, they start hiring workers already now. This brings some of the benefits of higher future productivity forward. Figure 3 indicates that there is indeed some evidence of such a channel. Labour increases during the planning phase, if the planning phase is not too long, and decreases by less than in the standard model if the planning phase is long. The outcomes for output are similar, with output mildly increasing during the delay phase for short delays, and remaining practically unchanged for longer delays.",
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"text": "Private consumption increases for the same reason as in the standard model, i.e., an expected long-run increase in available resources due to the higher level of productive public capital. Private investment increases by less than in the standard model during the delay phase because labour does not decrease as much and a greater share of aggregate demand can be met in production using labour. Wages increase, but less than in the standard model because labour increases (or falls by less than in the standard model). As a result, output increases during the planning phase.",
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"text": "Inflation also increases already on impact and tends to stay above zero until the outlays for public investment begin, when it increases again. The initial increase in inflation is due to higher demand that has not been met by the increase in quantities supplied. The initial increase in inflation quickly disappears as supply increases, which is in part due to new hiring being brought forward. As explained in Section 2, new hiring in the search model does not depend only on wages, but also on the probability that a firm will be able to find a worker, which takes time. Firms therefore start hiring immediately when there is news about the needs for workers in the future, as long as that future is not too far away. This can clearly be seen if a 2-year delay and a 5-year delay are compared. When the delay is short, firms increase hiring, while if the delay is longer, firms still decrease hiring, although not as much as in the standard model. We further explore this issue in the next section.",
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"text": "Note also that the long-run effects of increasing public investment in the search model are somewhat larger than in the standard model. This happens because in the long run, labour increases (note that labour fell in the long run in the standard model), which happens because labour demand effect from higher level of public capital dominates the negative labour supply effect. Higher stock of productive public capital increases the value of each worker for the firm, and this increase is permanent. In the long run, firms increase the level of vacancies, and this implies more hiring of workers (lower long-run unemployment). While households still reduce the per-capita hours worked, the effect of hiring (extensive margin) dominates the effect of hours (see also next section).",
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"text": "is the consequence of the stronger wage increase, as outlays for public investment (and therefore the increase in labour demand and employment) happen immediately, regardless of the time needed to construct public capital. High inflation worsens the external competitiveness of the economy through the real effective exchange rate appreciation and the resulting fall in exports. This is worse, the longer the delay lasts, because the positive supply-side effect of public capital occurs later in the future.",
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"text": "The reason why this experiment is interesting is because a delayed increase in (productive) public investment is akin to a news shock about future productivity, which has been a subject of a number of papers in the past, both in terms of empirics (Beaudry and Portier (2006)) and theory (Jaimovich and Rebelo (2009), Den Haan and Kaltenbrunner (2009), Den Haan and Lozej (2011)). One of the main issues in that literature was whether a news shock about future productivity can cause a so-called Pigou cycle, where output, consumption, investment, and labour jointly increase in the period before the shock materialises. In our setting, this question is particularly interesting when we have planning delays, because such a delay is closest to a news shock: households and firms find out today that productive public capital will be higher in the future, but only after some time. During this time there will be no demand stimulus because of the planning delay (construction delay is different, as there is a demand stimulus during this period; we consider this case in Appendix C).",
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"text": "The results of planning delays with varying degrees of search frictions are shown in Figures 5 and 6. Figure 5 shows the main macroeconomic aggregates and Figure 6 shows the main labour market variables. In each figure, the first column shows the benchmark results and the remaining two columns show the results of alternatives. First, note that in the benchmark case, there is no \"Pigou cycle\", as output and labour services (the latter due to the fall in hours worked) do not increase during the delay period, and the increase in private investment is meager. If the labour market is more volatile than in the benchmark case, then we do observe a small increase in output and labour services (the",
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"text": "[p25.6.8]Note that our model features the choice of hours worked (see Appendix B, in particular equation 70). Because in equation 70[p25.6.9] is less than 1, the hours production function is concave, and the marginal revenue for the firm from an additional hour supplied is[p25.6.10]. This nonlinearity matters, see e.g. Kudoh and Sasaki (2011) or Mangin and Julien (2021). In the model used here, this nonlinearity is not very strong, as[p25.6.11] is already close to 1.",
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"text": "[p25.7.3]Recall, Shimer (2005) pointed out that a standard search model has difficulties generating realistic volatility in unemployment and vacancies.",
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"text": "number of hires dominates the fall in hours), while investment response lingers slightly above zero during the delay phase. Consumption increases in all cases. [p26.1.2]",
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"text": "Inflation fluctuations are also lower when the labour market is more volatile. The reason for this is that a more volatile labour market implies more hiring, which dominates the reduction in hours worked per capita (see Figure 6). More responsive labour supply during the delay reduces wage increase and therefore dampens the increase in marginal cost of firms and inflation.",
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"text": "FIGURE 5. Varying degree of search frictions: macroeconomic variables",
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"embedded_text": "Benchmark Low surplus Very low surplus\n4 4 4\n3 3 3\nOutput\n2 2 2\n1 1 1\n0 0 0\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\nconsumption\n2 2 2\n1 1 1\nPrivate\n0 0 0\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\n1 1 1\ninvestment\n0 0 0\nPrivate\n-1 -1 -1\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\n0.15 0.15 0.15\n0.1 0.1 0.1\n0.05 0.05 0.05\n0 0 0\n-0.05 -0.05 -0.05\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT"
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"text": "Horizontal axes: quarters; vertical axes: percent deviations from the steady state, except inflation, which is in annualised p.p. deviations from the steady state. All variables are in real terms.",
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"embedded_text": "Benchmark Low surplus Very low surplus\n0.4 0.4 0.4\nLabour\n0.2 0.2 0.2\n0 0 0\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\n1 1 1\nEmployment\n0.5 0.5 0.5\n0 0 0\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\n0 0 0\nHours\n-0.5 -0.5 -0.5\n-1 -1 -1\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\n1 1 1\nWages\n0.5 0.5 0.5\n0 0 0\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\nGov. investment increase 2-year time to plan 5-year time to plan Long-term effect"
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"text": "4.4 Welfare",
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"text": "In the model with the standard labour market and in the model with search (and in each case for both time-to-build and time-to-plan), most of the output increase occurs later in time if there are delays. However, in some cases, in particular in the case of planning",
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"text": "delays with the standard labour market, labour falls during the planning phase. This could, in principle, be to some extent beneficial in terms of welfare, as households prefer leisure to work, despite the fact that consumption falls. Similarly, in the model with search, an early increase in labour that is disliked by the households could reduce the utility of households even though consumption increases. To assess these issues, we compute welfare of households, which is computed as the present value of household's utility, which is in turn defined by the utility of consumption and disutility of labour, as follows:",
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"text": "U_{s,t}=\\left(\\frac{1}{1-\\sigma}\\left(\\frac{c_{s,t+k}-\\kappa c_{s,t+k-1}}{1-\\kappa}\\right)^{1-\\sigma}-\\frac{1}{1+\\zeta}h_{s,t+k}^{1+\\zeta}\\right)",
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"text": "where[p29.4.4] stands for Ricardian (i) and hand-to-mouth (j) households,[p29.4.5] is consumption,[p29.4.6] is the discount rate,[p29.4.7] is the inverse of the intertemporal elasticity of substitution and[p29.4.8] is the (inverse of) the Frisch labour supply elasticity.[p29.4.9] is the degree of habit formation in consumption.²³ We include the fact that wage dispersion creates inefficiencies in allocation of labour and drives a wedge between labour demand from firms and labour supply by households, following Sims and Wolff (2018). Therefore, our measure of hours worked for welfare computation,[p29.4.10], includes the additional hours that occurs because of wage dispersion. Welfare is just the discounted present value of future utilities,",
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"text": "The results for welfare are shown in Figure 7, where the first row shows the results from the standard model and the second row shows the results from the search model. In both cases, welfare increases instantaneously upon announcement of the increase in public investment. This is because welfare is the present value of all future utilities, and the increase is mostly driven by the long-term effects (see equation 8). The long-run effects in turn are driven by productive public capital, and the stock of this capital increases when public investment goes up.",
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"text": "Delays of either type make relatively little difference in the short run (and no difference in the long run, as the final steady state is identical regardless of the length or the type of the delay). However, construction delays (time to build) generate somewhat larger differences between the lines showing welfare for different length of the delay, and this is the case in both models. The main reason is that in the case of time to build, construction commences immediately, and this implies an immediate increase in labour. Hours worked generate disutility, and this disutility is larger, the longer the time that is needed to build an investment project. In addition, the increase in labour occurs early",
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1 · TEXT
2 · CAPTION
3 · FIGURE
4 · FOOTNOTE
5 · FOOTER
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"text": "and the disutility of working more is therefore not discounted as much as the future benefits.",
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"text": "FIGURE 7. Welfare",
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"embedded_text": "Standard model, time to plan Standard model, time to build\n8 8\n6 6\n4 4\n2 2\n0 0\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\nSearch model, time to plan Search model, time to build\n10 10\n8 8\n6 6\n4 4\n2 2\n0 0\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\nGov. investment increase 2-year delay 5-year delay Long-term effect"
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2 · TEXT
3 · TEXT
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5 · TEXT
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"text": "We analyse planning and construction delays in a small open economy in a monetary union. We do so in two models, one that features a standard labour market with sticky wages, and one with the addition of search frictions on the labour market (with sticky nominal wages), but is otherwise identical. The comparison is related to the literature on news shocks about the future productivity, where it has been shown that the modelling choice can have different outcomes during the period of anticipation of the future productivity increase. That literature focused on productivity, but when public capital is productive, a delay related to public investment is akin to a news shock about future productivity. The novelty of our paper is that we focus on a small open economy in a monetary union, whereas the literature has mostly looked at real models of closed economies.",
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"text": "We find that delays in public investment, either due to planning or due to time needed for construction, result in the benefits from higher public capital occurring later in the future. From the business cycles perspective, planning and construction delays cause fluctuations not only immediately upon announcement, but also in the future, which may complicate the management of business cycles. This is particularly the case for planning delays, where future fluctuations tend to be stronger.",
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"text": "A Outline of the standard model (without search frictions)",
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"text": "This section reports the equations of the core model. It closely follows the description of the EAGLE model Gomes et al. (2012), and the reader should refer for all the details to that paper.",
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"text": "Gomes et al.",
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"text": "(2012),",
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{
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"text": "A.1 Households",
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"text": "There are two types of households in the model, Ricardian (unconstrained) and non-Ricardian (hand-to-mouth, HtM, or unconstrained). Their utilities are shown in equation 10:",
"children": [
{
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"text": "U_{s,t}=\\left(\\frac{1}{1-\\sigma}\\left(\\frac{c_{s,t+k}-\\kappa c_{s,t+k-1}}{1-\\kappa}\\right)^{1-\\sigma}-\\frac{1}{1+\\zeta}h_{s,t+k}^{1+\\zeta}\\right)",
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"latex": "U_{s,t}=\\left(\\frac{1}{1-\\sigma}\\left(\\frac{c_{s,t+k}-\\kappa c_{s,t+k-1}}{1-\\kappa}\\right)^{1-\\sigma}-\\frac{1}{1+\\zeta}h_{s,t+k}^{1+\\zeta}\\right)",
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{
"id": "p35.6",
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"text": "(10)",
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{
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"text": "where[p35.7.1] is consumption of type $s$ households,[p35.7.2] is their hours worked,[p35.7.3] is the (inverse of) intertemporal elasticity of substitution,[p35.7.4] is the degree of habit formation and[p35.7.5] is the (inverse of) the Frisch labour supply elasticity.",
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"children": []
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{
"id": "p35.7.3",
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"type": "formula",
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"text": "σ",
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"text_offset": 85,
"text_span": 9,
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"alt": "sigma",
"mathml": "<math aria-label=\"sigma\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>σ</mi></mrow></math>"
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"children": []
},
{
"id": "p35.7.4",
"order": 14,
"type": "formula",
"bbox": {
"l": 67.6614,
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"r": 525.9351,
"b": 510.2218
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"text": "κ",
"attrs": {
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"text_offset": 156,
"text_span": 9,
"bbox_estimated": true,
"alt": "kappa",
"mathml": "<math aria-label=\"kappa\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>κ</mi></mrow></math>"
},
"children": []
},
{
"id": "p35.7.5",
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"type": "formula",
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"text": "ζ",
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"text_offset": 202,
"text_span": 9,
"bbox_estimated": true,
"alt": "zeta",
"mathml": "<math aria-label=\"zeta\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>ζ</mi></mrow></math>"
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},
{
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"text": "Ricardian households hold capital, invest in internationally traded bonds denominated in US dollars[p35.8.1], that pay the interest rate[p35.8.2] set by the Federal Reserve. They also hold domestic government bonds[p35.8.3] that pay the rate[p35.8.4], which is determined by the European Central Bank. Investment in capital is subject to investment-adjustment costs that give rise to the standard Tobin's marginal q. The Euler equation with respect to domestic government bonds is",
"children": [
{
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"text": "B_{t}^{*}",
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"mathml": "<math aria-label=\"B sub t to the power of *\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>B</mi><mrow><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup></mrow></math>"
},
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{
"id": "p35.8.2",
"order": 18,
"type": "formula",
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"text": "R_{t}^{*}",
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"text_offset": 136,
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"notation": "latex",
"alt": "R sub t to the power of *",
"mathml": "<math aria-label=\"R sub t to the power of *\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>R</mi><mrow><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup></mrow></math>"
},
"children": []
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{
"id": "p35.8.3",
"order": 19,
"type": "formula",
"bbox": {
"l": 67.6614,
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"text": "B_{t}",
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"latex": "B_{t}",
"inline": true,
"text_offset": 214,
"text_span": 9,
"bbox_estimated": true,
"notation": "latex",
"alt": "B sub t",
"mathml": "<math aria-label=\"B sub t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>B</mi><mrow><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
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{
"id": "p35.8.4",
"order": 20,
"type": "formula",
"bbox": {
"l": 67.6614,
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"b": 422.8649
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"text": "R_{t}",
"attrs": {
"latex": "R_{t}",
"inline": true,
"text_offset": 241,
"text_span": 9,
"bbox_estimated": true,
"notation": "latex",
"alt": "R sub t",
"mathml": "<math aria-label=\"R sub t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>R</mi><mrow><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
}
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},
{
"id": "p35.9",
"order": 21,
"type": "formula",
"bbox": {
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"text": "u^{\\prime}(c_{s,t})=\\beta\\left(u^{\\prime}(c_{s,t+1})\\frac{R_{t}(1-\\Gamma_{B,t+1})}{\\pi_{t+1}}\\right),",
"attrs": {
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"latex": "u^{\\prime}(c_{s,t})=\\beta\\left(u^{\\prime}(c_{s,t+1})\\frac{R_{t}(1-\\Gamma_{B,t+1})}{\\pi_{t+1}}\\right),",
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"alt": "u to the power of prime of c sub s, t equals beta of u to the power of prime of c sub s, t plus 1 R sub t of 1 minus capital gamma sub B, t plus 1 over pi sub t plus 1,",
"mathml": "<math aria-label=\"u to the power of prime of c sub s, t equals beta of u to the power of prime of c sub s, t plus 1 R sub t of 1 minus capital gamma sub B, t plus 1 over pi sub t plus 1,\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>u</mi><mrow><mi>′</mi></mrow></msup><mo stretchy=\"false\">(</mo><msub><mi>c</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo>=</mo><mi>β</mi><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msup><mi>u</mi><mrow><mi>′</mi></mrow></msup><mo stretchy=\"false\">(</mo><msub><mi>c</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo><mfrac><mrow><msub><mi>R</mi><mrow><mi>t</mi></mrow></msub><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>Γ</mi><mrow><mi>B</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></mrow><mrow><msub><mi>π</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mo>,</mo></mrow></math>"
},
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{
"id": "p35.10",
"order": 22,
"type": "text",
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"text": "(11)",
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{
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"type": "text",
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"text": "where[p35.11.2] is the transaction cost for the euro area traded bonds, which depends on the intra-EA bond holdings.²⁴ There is an analogous Euler equation for internationally traded bonds:",
"children": [
{
"id": "p35.11.1",
"order": 24,
"type": "link",
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{
"id": "p35.11.2",
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{
"id": "p35.12",
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"type": "formula",
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"text": "u^{\\prime}(c_{s,t})=\\beta\\left(u^{\\prime}(c_{s,t+1})\\frac{R_{t}^{*}(1-\\Gamma_{B^{*},t+1})}{\\pi_{t+1}^{*}}\\frac{rer_{t+1}}{rer_{t}}\\right),",
"attrs": {
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"latex": "u^{\\prime}(c_{s,t})=\\beta\\left(u^{\\prime}(c_{s,t+1})\\frac{R_{t}^{*}(1-\\Gamma_{B^{*},t+1})}{\\pi_{t+1}^{*}}\\frac{r e r_{t+1}}{r e r_{t}}\\right),",
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"alt": "u to the power of prime of c sub s, t equals beta of u to the power of prime of c sub s, t plus 1 R sub t to the power of * 1 minus capital gamma sub B to the power of *, t plus 1 over pi sub t plus 1 to the power of * rer sub t plus 1 over rer sub t,",
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},
{
"id": "p35.13",
"order": 27,
"type": "text",
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"r": 524.2556,
"b": 316.3087
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"text": "(12)",
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"block_label": "formula_number"
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"children": []
},
{
"id": "p35.14",
"order": 28,
"type": "text",
"bbox": {
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"text": "where[p35.14.1] is the real exchange rate and[p35.14.2] is the international transaction premium for US-denominated bonds, where[p35.14.3] is the nominal exchange rate, expressed in terms of units of Home currency per unit of US dollars:",
"children": [
{
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"text": "rer_{t}",
"attrs": {
"latex": "rer_{t}",
"inline": true,
"text_offset": 5,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
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{
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"type": "formula",
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"text": "\\Gamma_{B^{*}}",
"attrs": {
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"inline": true,
"text_offset": 45,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "capital gamma sub B to the power of *",
"mathml": "<math aria-label=\"capital gamma sub B to the power of *\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>Γ</mi><mrow><msup><mi>B</mi><mrow><mo>*</mo></mrow></msup></mrow></msub></mrow></math>"
},
"children": []
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{
"id": "p35.14.3",
"order": 31,
"type": "formula",
"bbox": {
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"text": "S_{t}^{H,US}",
"attrs": {
"latex": "S_{t}^{H,US}",
"inline": true,
"text_offset": 128,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "S sub t to the power of H, US",
"mathml": "<math aria-label=\"S sub t to the power of H, US\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>S</mi><mrow><mi>t</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>U</mi><mi>S</mi></mrow></msubsup></mrow></math>"
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"text": "\\Gamma_{B^{*}}\\left(\\frac{S_{t}^{H,US}B_{t+1}^{*}}{P_{Y,t}Y_{t}};rp_{t}\\right)\\equiv\\gamma_{B^{*}}\\left(\\exp\\left(\\frac{S_{t}^{H,US}B_{t+1}^{*}}{P_{Y,t}Y_{t}}-\\overline{B_{Y}^{*}}\\right)-1\\right)-rp_{t}",
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"notation": "latex",
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},
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{
"id": "p35.16",
"order": 33,
"type": "text",
"bbox": {
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"text": "(13)",
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{
"id": "p35.17",
"order": 34,
"type": "text",
"bbox": {
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"text": "where[p35.17.3] is a parameter,[p35.17.4] is the steady-state net foreign asset position,[p35.17.5] is a risk premium shock,[p35.17.6] is the GDP deflator,[p35.17.7] is the GDP in real terms. The term[p35.17.8] in the Euler equation for euro area bonds is analogous and depends on the target for intra-EA bond holdings. For further details see the Appendix in Gomes et al. (2010).",
"children": [
{
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"notation": "latex",
"alt": "capital gamma sub B, t plus 1",
"mathml": "<math aria-label=\"capital gamma sub B, t plus 1\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>Γ</mi><mrow><mi>B</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></math>"
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"text": "Ricardian households also invest[p35.18.1] in physical capital[p35.18.2], which involves as:",
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{
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"order": 44,
"type": "formula",
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"mathml": "<math aria-label=\"I sub t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>I</mi><mrow><mi>t</mi></mrow></msub></mrow></math>"
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{
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"type": "formula",
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"mathml": "<math aria-label=\"K sub t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>K</mi><mrow><mi>t</mi></mrow></msub></mrow></math>"
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"type": "footnote",
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"text": "[p35.19.1]The bond traded in the union is in euros and the transaction cost is needed to stabilise the model.",
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},
"children": [
{
"id": "p35.19.1",
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"type": "formula",
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"text": "²⁴",
"attrs": {
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"inline": true,
"text_offset": 0,
"text_span": 10,
"bbox_estimated": true,
"alt": "to the power of 24",
"mathml": "<math aria-label=\"to the power of 24\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi /><mrow><mn>24</mn></mrow></msup></mrow></math>"
},
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footer (1)
formula (20)
heading (1)
link (3)
text (14)
1 · FORMULA
2 · TEXT
3 · TEXT
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5 · FORMULA
6 · TEXT
7 · TEXT
8 · FORMULA
9 · TEXT
10 · TEXT
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13 · FORMULA
14 · TEXT
15 · TEXT
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19 · TEXT
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22 · FORMULA
23 · TEXT
24 · HEADING
25 · TEXT
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34 · FORMULA
35 · TEXT
36 · TEXT
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39 · FOOTER
Page 36 JSON
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"text": "K_{t+1}=(1-\\delta)K_{t}+\\left(1-\\Gamma_{I,t}\\left(\\frac{I_{t}}{I_{t-1}}\\right)\\right),",
"attrs": {
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"notation": "latex",
"alt": "K sub t plus 1 equals 1 minus delta K sub t plus 1 minus capital gamma sub I, t of I sub t over I sub t minus 1,",
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"text": "where[p36.3.1] is the depreciation rate and investment-adjustment cost is",
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"text": "\\Gamma_{I,t}\\left(\\frac{I_{t}}{I_{t-1}}\\right)\\equiv\\frac{\\gamma}{2}\\left(\\frac{I_{t}}{I_{t-1}}-1\\right)^{2}.",
"attrs": {
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"latex": "\\Gamma_{I,t}\\left(\\frac{I_{t}}{I_{t-1}}\\right)\\equiv\\frac{\\gamma}{2}\\left(\\frac{I_{t}}{I_{t-1}}-1\\right)^{2}.",
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"text": "The first-order condition for investment in capital is:",
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"text": "p_{I,t}=q_{i,t}\\left(1-\\Gamma_{I,t}\\left(\\frac{I_{t}}{I_{t-1}}\\right)-\\Gamma_{I,t}^{\\prime}\\left(\\frac{I_{t}}{I_{t-1}}\\right)I_{t}\\right)+\\beta\\frac{u^{\\prime}(c_{I,t+1})}{u^{\\prime}(c_{I,t})}q_{I,t+1}\\Gamma_{I,t}^{\\prime}\\left(\\frac{I_{t}}{I_{t-1}}\\right)\\frac{I_{t+1}^{2}}{I_{t}},",
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"notation": "latex",
"alt": "p sub I, t equals q sub i, t of 1 minus capital gamma sub I, t of I sub t over I sub t minus 1 minus capital gamma sub I, t to the power of prime of I sub t over I sub t minus 1 I sub t plus beta u to the power of prime of c sub I, t plus 1 over u to the power of prime of c sub I, t q sub I, t plus 1 capital gamma sub I, t to the power of prime of I sub t over I sub t minus 1 I sub t plus 1 squared over I sub t,",
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},
"children": []
},
{
"id": "p36.8",
"order": 9,
"type": "text",
"bbox": {
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"b": 606.4584
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"text": "(16)",
"attrs": {
"raw_type": "FORMULA_NUMBER",
"block_label": "formula_number"
},
"children": []
},
{
"id": "p36.9",
"order": 10,
"type": "text",
"bbox": {
"l": 67.1815,
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"r": 526.175,
"b": 555.3402
},
"text": "where primes indicate derivatives,[p36.9.1] is Tobin's q, and[p36.9.2] is the relative price of investment goods.",
"children": [
{
"id": "p36.9.1",
"order": 11,
"type": "formula",
"bbox": {
"l": 67.1815,
"t": 587.0191,
"r": 526.175,
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"text": "q_{I,t}",
"attrs": {
"latex": "q_{I,t}",
"inline": true,
"text_offset": 34,
"text_span": 9,
"bbox_estimated": true,
"notation": "latex",
"alt": "q sub I, t",
"mathml": "<math aria-label=\"q sub I, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>q</mi><mrow><mi>I</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p36.9.2",
"order": 12,
"type": "formula",
"bbox": {
"l": 67.1815,
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},
"text": "p_{I,t} \\equiv \\frac{P_{I,t}}{P_{C,t}}",
"attrs": {
"latex": "p_{I,t} \\equiv \\frac{P_{I,t}}{P_{C,t}}",
"inline": true,
"text_offset": 61,
"text_span": 9,
"bbox_estimated": true,
"notation": "latex",
"alt": "p sub I, t equiv P sub I, t over P sub C, t",
"mathml": "<math aria-label=\"p sub I, t equiv P sub I, t over P sub C, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>p</mi><mrow><mi>I</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>≡</mo><mfrac><mrow><msub><mi>P</mi><mrow><mi>I</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><mrow><msub><mi>P</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mfrac></mrow></math>"
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},
{
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"type": "formula",
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"text": "q_{I,t}=\\beta\\frac{u^{\\prime}(c_{I,t+1})}{u^{\\prime}(c_{I,t})}\\left[(1-\\delta)q_{I,t+1}+(1-\\tau_{t+1}^{k})r_{K,t+1}u_{t+1}+\\left(\\tau_{t+1}^{k}\\delta-(1-\\tau_{t+1}^{k})\\Gamma(u_{t+1})\\right)p_{I,t+1}\\right],",
"attrs": {
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"latex": "q_{I,t}=\\beta\\frac{u^{\\prime}(c_{I,t+1})}{u^{\\prime}(c_{I,t})}\\left[(1-\\delta)q_{I,t+1}+(1-\\tau_{t+1}^{k})r_{K,t+1}u_{t+1}+\\left(\\tau_{t+1}^{k}\\delta-(1-\\tau_{t+1}^{k})\\Gamma(u_{t+1})\\right)p_{I,t+1}\\right].",
"notation": "latex",
"alt": "q sub I, t equals beta u to the power of prime of c sub I, t plus 1 over u to the power of prime of c sub I, t 1 minus delta q sub I, t plus 1 plus 1 minus tau sub t plus 1 to the power of k r sub K, t plus 1 u sub t plus 1 plus tau sub t plus 1 to the power of k delta minus 1 minus tau sub t plus 1 to the power of k capital gamma of u sub t plus 1 p sub I, t plus 1 .",
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},
"children": []
},
{
"id": "p36.11",
"order": 14,
"type": "text",
"bbox": {
"l": 502.4216,
"t": 505.662,
"r": 524.0156,
"b": 492.4625
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"text": "(17)",
"attrs": {
"raw_type": "FORMULA_NUMBER",
"block_label": "formula_number"
},
"children": []
},
{
"id": "p36.12",
"order": 15,
"type": "text",
"bbox": {
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"r": 525.2153,
"b": 462.9435
},
"text": "where[p36.12.1] is the return on physical capital,[p36.12.2] is capital utilisation, and[p36.12.3] is capital utilisation adjustment cost.",
"children": [
{
"id": "p36.12.1",
"order": 16,
"type": "formula",
"bbox": {
"l": 67.9013,
"t": 491.2625,
"r": 525.2153,
"b": 462.9435
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"text": "r_{K,t}",
"attrs": {
"latex": "r_{K,t}",
"inline": true,
"text_offset": 5,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "r sub K, t",
"mathml": "<math aria-label=\"r sub K, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>r</mi><mrow><mi>K</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p36.12.2",
"order": 17,
"type": "formula",
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"b": 462.9435
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"text": "u_{t}",
"attrs": {
"latex": "u_{t}",
"inline": true,
"text_offset": 50,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "u sub t",
"mathml": "<math aria-label=\"u sub t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>u</mi><mrow><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p36.12.3",
"order": 18,
"type": "formula",
"bbox": {
"l": 67.9013,
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"b": 462.9435
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"text": "\\Gamma(u_{t})",
"attrs": {
"latex": "\\Gamma(u_{t})",
"inline": true,
"text_offset": 88,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "capital gamma of u sub t",
"mathml": "<math aria-label=\"capital gamma of u sub t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>Γ</mi><mo stretchy=\"false\">(</mo><msub><mi>u</mi><mrow><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo></mrow></math>"
},
"children": []
}
]
},
{
"id": "p36.13",
"order": 19,
"type": "text",
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"text": "Total capital services in the economy are the product of the capital stock and its utilisation, and are used by tradable[p36.13.1] and non-tradable[p36.13.2] sectors of intermediate goods firms:",
"children": [
{
"id": "p36.13.1",
"order": 20,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 462.2235,
"r": 525.9351,
"b": 418.7851
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"text": "K_{T,t}",
"attrs": {
"latex": "K_{T,t}",
"inline": true,
"text_offset": 120,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "K sub T, t",
"mathml": "<math aria-label=\"K sub T, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>K</mi><mrow><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p36.13.2",
"order": 21,
"type": "formula",
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"text": "K_{N,t}",
"attrs": {
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"inline": true,
"text_offset": 147,
"text_span": 10,
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"notation": "latex",
"alt": "K sub N, t",
"mathml": "<math aria-label=\"K sub N, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>K</mi><mrow><mi>N</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
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}
]
},
{
"id": "p36.14",
"order": 22,
"type": "formula",
"bbox": {
"l": 243.5329,
"t": 404.1456,
"r": 350.0636,
"b": 389.0261
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"text": "u_{t}K_{t}=K_{N,t}+K_{T,t}.",
"attrs": {
"block_label": "display_formula",
"latex": "u_{t}K_{t}=K_{N,t}+K_{T,t}.",
"notation": "latex",
"alt": "u sub t K sub t equals K sub N, t plus K sub T, t .",
"mathml": "<math aria-label=\"u sub t K sub t equals K sub N, t plus K sub T, t .\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>u</mi><mrow><mi>t</mi></mrow></msub><msub><mi>K</mi><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mi>K</mi><mrow><mi>N</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>+</mo><msub><mi>K</mi><mrow><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>.</mo></mrow></math>"
},
"children": []
},
{
"id": "p36.15",
"order": 23,
"type": "text",
"bbox": {
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"r": 524.4955,
"b": 390.9461
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"text": "(18)",
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"block_label": "formula_number"
},
"children": []
},
{
"id": "p36.16",
"order": 24,
"type": "heading",
"bbox": {
"l": 68.1412,
"t": 369.3468,
"r": 202.7441,
"b": 353.5074
},
"text": "A.2 Labour market",
"attrs": {
"block_label": "paragraph_title"
},
"children": []
},
{
"id": "p36.17",
"order": 25,
"type": "text",
"bbox": {
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"text": "The labour market in the standard model is identical to what is typical in the literature - a standard staggered wage setting. This is the part where the main difference between the model with and without search frictions lies, namely, the standard part described briefly below is replaced by the fully-fledged search frictions, described in detail in Appendix B.",
"children": [
{
"id": "p36.17.1",
"order": 26,
"type": "link",
"bbox": {
"l": 512.282,
"t": 299.162,
"r": 522.01,
"b": 286.422
},
"text": "B.",
"attrs": {
"action_type": "/Named"
},
"children": []
}
]
},
{
"id": "p36.18",
"order": 27,
"type": "text",
"bbox": {
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"text": "Households supply differentiated labour services in monopolistically competitive market for each type of labour service. Wages are determined by staggered wage setting in terms of nominal wages, where the probability that a wage contract is reset in a given period is[p36.18.1], where $s$ stands for Ricardian or hand-to-mouth households. Households that can reset wage choose the same wage[p36.18.2], while wages that are not reset are indexed to past wages by a combination of the past CPI inflation,[p36.18.3], where[p36.18.4] is the consumer price index, and inflation target,[p36.18.5], where the indexation weight is[p36.18.6]. The first order condition for the optimal reset wage is",
"children": [
{
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"order": 28,
"type": "formula",
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"text": "1 - \\xi_{s}",
"attrs": {
"latex": "1 - \\xi_{s}",
"inline": true,
"text_offset": 267,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "1 minus xi sub s",
"mathml": "<math aria-label=\"1 minus xi sub s\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mn>1</mn><mo>−</mo><msub><mi>ξ</mi><mrow><mi>s</mi></mrow></msub></mrow></math>"
},
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{
"id": "p36.18.2",
"order": 29,
"type": "formula",
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"text": "\\tilde{W}_{s,t}",
"attrs": {
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"text_offset": 390,
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"notation": "latex",
"alt": "tilde W sub s, t",
"mathml": "<math aria-label=\"tilde W sub s, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mover><mrow><mi>W</mi></mrow><mo stretchy=\"false\">~</mo></mover><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
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{
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"type": "formula",
"bbox": {
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"text": "\\Pi_{C,t-1} = P_{C,t-1}/P_{C,t-2}",
"attrs": {
"latex": "\\Pi_{C,t-1} = P_{C,t-1}/P_{C,t-2}",
"inline": true,
"text_offset": 502,
"text_span": 10,
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"notation": "latex",
"alt": "capital pi sub C, t minus 1 equals P sub C, t minus 1 over P sub C, t minus 2",
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{
"id": "p36.18.4",
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"type": "formula",
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"text": "P_{C,t}",
"attrs": {
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"inline": true,
"text_offset": 519,
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"notation": "latex",
"alt": "P sub C, t",
"mathml": "<math aria-label=\"P sub C, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>P</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
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{
"id": "p36.18.5",
"order": 32,
"type": "formula",
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"text": "\\overline{\\Pi}",
"attrs": {
"latex": "\\overline{\\Pi}",
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"text_offset": 580,
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"notation": "latex",
"alt": "capital pi bar",
"mathml": "<math aria-label=\"capital pi bar\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mover><mrow><mi>Π</mi></mrow><mo accent=\"true\">―</mo></mover></mrow></math>"
},
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"text": "\\chi_{s} \\colon W_{t,i} = \\Pi_{C,t-1}^{\\chi_{s}} \\overline{\\Pi}^{1-\\chi_{s}} W_{t-1,i}",
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"latex": "\\chi_{s} \\colon W_{t,i} = \\Pi_{C,t-1}^{\\chi_{s}} \\overline{\\Pi}^{1-\\chi_{s}} W_{t-1,i}",
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"alt": "chi sub s colon W sub t, i equals capital pi sub C, t minus 1 to the power of chi sub s capital pi bar to the power of 1 minus chi sub s W sub t minus 1, i",
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"text": "E_{t}\\left[\\sum_{k=0}^{\\infty}(\\beta\\xi_{s})^{k}\\left(u^{\\prime}(c)_{s,t+k}(1-\\tau_{t+k}^{N}-\\tau_{t+k}^{W_{h}})\\frac{\\tilde{W}_{s,t}}{P_{C,t+k}}\\left(\\frac{P_{C,t+k-1}}{P_{C,t-1}}\\right)^{\\chi_{s}}\\overline{\\Pi}^{(1-\\chi_{s})k}-\\frac{\\eta_{s}}{\\eta_{s}-1}N_{t+k}^{\\zeta}\\right)N_{t+k}\\right]=0",
"attrs": {
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"latex": "E_{t}\\left[\\sum_{k=0}^{\\infty}(\\beta\\xi_{s})^{k}\\left(u^{\\prime}(c)_{s,t+k}(1-\\tau_{t+k}^{N}-\\tau_{t+k}^{W_{h}})\\frac{\\tilde{W}_{s,t}}{P_{C,t+k}}\\left(\\frac{P_{C,t+k-1}}{P_{C,t-1}}\\right)^{\\chi_{s}}\\overline{\\Pi}^{(1-\\chi_{s})k}-\\frac{\\eta_{s}}{\\eta_{s}-1}N_{t+k}^{\\zeta}\\right)N_{t+k}\\right]=0",
"notation": "latex",
"alt": "E sub t the sum of sub k equals 0 to the power of infinity of beta xi sub s to the power of k of u to the power of prime of c sub s, t plus k of 1 minus tau sub t plus k to the power of N minus tau sub t plus k to the power of W sub h frac tilde W sub s, t P sub C, t plus k of P sub C, t plus k minus 1 over P sub C, t minus 1 to the power of chi sub s capital pi bar to the power of of 1 minus chi sub s k minus eta sub s over eta sub s minus 1 N sub t plus k to the power of zeta N sub t plus k equals 0",
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"text": "(19)",
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{
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"text": "For details see the Appendix in Gomes et al. (2010).",
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{
"id": "p36.21.1",
"order": 37,
"type": "link",
"bbox": {
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"text": "Gomes et al.",
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},
{
"id": "p36.21.2",
"order": 38,
"type": "link",
"bbox": {
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"text": "(2010).",
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}
]
},
{
"id": "p36.22",
"order": 39,
"type": "footer",
"bbox": {
"l": 290.0801,
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"r": 305.4359,
"b": 39.3586
},
"text": "35",
"attrs": {
"block_label": "number"
},
"children": []
}
],
"ai_page_review": {
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}
footer (1)
formula (26)
heading (2)
link (2)
text (12)
1 · HEADING
2 · TEXT
3
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7 · FORMULA
8 · TEXT
9 · HEADING
10 · FORMULA
11 · TEXT
12 · TEXT
13
14
15
16 · FORMULA
17 · TEXT
18 · TEXT
19
20
21
22
23
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28 · FORMULA
29 · TEXT
30 · TEXT
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34 · FORMULA
35 · TEXT
36 · FORMULA
37 · TEXT
38 · FORMULA
39 · TEXT
40 · FORMULA
41 · TEXT
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43 · FOOTER
Page 37 JSON
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{
"id": "p37.1",
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"type": "heading",
"bbox": {
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"text": "A.3 Final goods",
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{
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"type": "text",
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"text": "Final goods are consumption goods[p37.2.1] and investment goods[p37.2.2] (to save space, we list equations only for consumption goods, as the equations for investment goods are analogous). They are assembled from tradable goods,[p37.2.3] and non-tradable goods,[p37.2.4] using the constant elasticity of substitution technology:",
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{
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"type": "formula",
"bbox": {
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"text": "Q_{C,t}",
"attrs": {
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"notation": "latex",
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{
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"text": "Q_{I,t}",
"attrs": {
"latex": "Q_{I,t}",
"inline": true,
"text_offset": 63,
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"notation": "latex",
"alt": "Q sub I, t",
"mathml": "<math aria-label=\"Q sub I, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>Q</mi><mrow><mi>I</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
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{
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"type": "formula",
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"text": "TT_{C,t}",
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"inline": true,
"text_offset": 228,
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"bbox_estimated": true,
"notation": "latex",
"alt": "TT sub C, t",
"mathml": "<math aria-label=\"TT sub C, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>T</mi><msub><mi>T</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
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{
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"type": "formula",
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"text": "NT_{C,t}",
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"notation": "latex",
"alt": "NT sub C, t",
"mathml": "<math aria-label=\"NT sub C, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>N</mi><msub><mi>T</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
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{
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"type": "formula",
"bbox": {
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"t": 677.0159,
"r": 425.6427,
"b": 643.6571
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"text": "Q_{C,t}=\\left[\\nu_{C}^{\\frac{1}{\\mu_{C}}}TT_{C,t}^{\\frac{\\mu_{C}-1}{\\mu_{C}}}+(1-\\nu_{C})^{\\frac{1}{\\mu_{C}}}NT_{C,t}^{\\frac{\\mu_{C}-1}{\\mu_{C}}}\\right]^{\\frac{\\mu_{C}}{\\mu_{C-1}}},",
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"latex": "Q_{C,t}=\\left[\\nu_{C}^{\\frac{1}{\\mu_{C}}}T T_{C,t}^{\\frac{\\mu_{C}-1}{\\mu_{C}}}+(1-\\nu_{C})^{\\frac{1}{\\mu_{C}}}N T_{C,t}^{\\frac{\\mu_{C}-1}{\\mu_{C}}}\\right]^{\\frac{\\mu_{C}}{\\mu_{C-1}}},",
"notation": "latex",
"alt": "Q sub C, t equals nu sub C to the power of 1 over mu sub C TT sub C, t to the power of mu sub C minus 1 over mu sub C plus 1 minus nu sub C to the power of 1 over mu sub C NT sub C, t to the power of mu sub C minus 1 over mu sub C to the power of mu sub C over mu sub C minus 1,",
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"text": "(20)",
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"text": "where:",
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"type": "formula",
"bbox": {
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"text": "TT_{C,t}=\\left[\\nu_{TC}^{\\frac{1}{\\mu_{TC}}}HT_{C,t}^{\\frac{\\mu_{TC}-1}{\\mu_{TC}}}+(1-\\nu_{TC})^{\\frac{1}{\\mu_{TC}}}IM_{C,t}^{\\frac{\\mu_{TC}-1}{\\mu_{TC}}}\\right]^{\\frac{\\mu_{TC}}{\\mu_{TC}-1}}.",
"attrs": {
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"latex": "T T_{C,t}=\\left[\\nu_{T C}^{\\frac{1}{\\mu_{T C}}}H T_{C,t}^{\\frac{\\mu_{T C}-1}{\\mu_{T C}}}+(1-\\nu_{T C})^{\\frac{1}{\\mu_{T C}}}I M_{C,t}^{\\frac{\\mu_{T C}-1}{\\mu_{T C}}}\\right]^{\\frac{\\mu_{T C}}{\\mu_{T C}-1}}.",
"notation": "latex",
"alt": "TT sub C, t equals nu sub TC to the power of 1 over mu sub TC HT sub C, t to the power of mu sub TC minus 1 over mu sub TC plus 1 minus nu sub TC to the power of 1 over mu sub TC IM sub C, t to the power of mu sub TC minus 1 over mu sub TC to the power of mu sub TC over mu sub TC minus 1 .",
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"text": "(21)",
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{
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"text": "Tradable goods are a composite bundle of domestic tradable goods,[p37.8.1], and imports,[p37.8.2]. Imports, in turn, are also a composite of imports from other regions CO, denoted by[p37.8.3], and H stands for the home country:",
"children": [
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"text": "HT_{C,t}",
"attrs": {
"latex": "HT_{C,t}",
"inline": true,
"text_offset": 65,
"text_span": 9,
"bbox_estimated": true,
"notation": "latex",
"alt": "HT sub C, t",
"mathml": "<math aria-label=\"HT sub C, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>H</mi><msub><mi>T</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p37.8.2",
"order": 14,
"type": "formula",
"bbox": {
"l": 67.9013,
"t": 588.459,
"r": 525.2153,
"b": 544.5406
},
"text": "IM_{C,t}",
"attrs": {
"latex": "IM_{C,t}",
"inline": true,
"text_offset": 88,
"text_span": 9,
"bbox_estimated": true,
"notation": "latex",
"alt": "IM sub C, t",
"mathml": "<math aria-label=\"IM sub C, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p37.8.3",
"order": 15,
"type": "formula",
"bbox": {
"l": 67.9013,
"t": 588.459,
"r": 525.2153,
"b": 544.5406
},
"text": "IM_{C,CO,t}",
"attrs": {
"latex": "IM_{C,CO,t}",
"inline": true,
"text_offset": 182,
"text_span": 9,
"bbox_estimated": true,
"notation": "latex",
"alt": "IM sub C, CO, t",
"mathml": "<math aria-label=\"IM sub C, CO, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
}
]
},
{
"id": "p37.9",
"order": 16,
"type": "formula",
"bbox": {
"l": 78.9383,
"t": 521.7414,
"r": 512.4988,
"b": 472.5432
},
"text": "IM_{C,t}=\\left[\\sum_{CO\\neq H}\\left(\\nu_{IM_{C}}^{H,CO}\\right)^{\\frac{1}{\\mu_{IMC}}}\\left(IM_{C,CO,t}\\left(1-\\Gamma_{IM^{C}}^{H,CO}\\left(\\frac{IM_{C,CO,t}}{Q_{C,t}}\\right)\\right)\\right)^{\\frac{\\mu_{IMC}-1}{\\mu_{IMC}}}\\right]^{\\frac{\\mu_{IMC}}{\\mu_{IMC}-1}}",
"attrs": {
"block_label": "display_formula",
"latex": "I M_{C,t}=\\left[\\sum_{C O\\neq H}\\left(\\nu_{I M_{C}}^{H,C O}\\right)^{\\frac{1}{\\mu_{I M C}}}\\left(I M_{C,C O,t}\\left(1-\\Gamma_{I M^{C}}^{H,C O}\\left(\\frac{I M_{C,C O,t}}{Q_{C,t}}\\right)\\right)\\right)^{\\frac{\\mu_{I M C}-1}{\\mu_{I M C}}}\\right]^{\\frac{\\mu_{I M C}}{\\mu_{I M C}-1}}",
"notation": "latex",
"alt": "IM sub C, t equals the sum of sub CO is not equal to H of nu sub IM sub C to the power of H, CO to the power of 1 over mu sub IMC of IM sub C, CO, t of 1 minus capital gamma sub IM to the power of C to the power of H, CO of IM sub C, CO, t over Q sub C, t to the power of mu sub IMC minus 1 over mu sub IMC to the power of mu sub IMC over mu sub IMC minus 1",
"mathml": "<math aria-label=\"IM sub C, t equals the sum of sub CO is not equal to H of nu sub IM sub C to the power of H, CO to the power of 1 over mu sub IMC of IM sub C, CO, t of 1 minus capital gamma sub IM to the power of C to the power of H, CO of IM sub C, CO, t over Q sub C, t to the power of mu sub IMC minus 1 over mu sub IMC to the power of mu sub IMC over mu sub IMC minus 1\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>=</mo><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">[</mo><msub><mo>∑</mo><mrow><mi>C</mi><mi>O</mi><mo>≠</mo><mi>H</mi></mrow></msub><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>ν</mi><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi></mrow></msub></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><mi>O</mi></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>μ</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow></msub></mrow></mfrac></mrow></msup><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi></mrow></msub><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>Γ</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>C</mi></mrow></msup></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><mi>O</mi></mrow></msubsup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mfrac><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><mrow><msub><mi>Q</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><mfrac><mrow><msub><mi>μ</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow></msub><mo>−</mo><mn>1</mn></mrow><mrow><msub><mi>μ</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow></msub></mrow></mfrac></mrow></msup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">]</mo></mrow><mrow><mfrac><mrow><msub><mi>μ</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow></msub></mrow><mrow><msub><mi>μ</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow></msub><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup></mrow></math>"
},
"children": []
},
{
"id": "p37.10",
"order": 17,
"type": "text",
"bbox": {
"l": 502.4216,
"t": 474.4631,
"r": 524.2556,
"b": 461.0236
},
"text": "(22)",
"attrs": {
"raw_type": "FORMULA_NUMBER",
"block_label": "formula_number"
},
"children": []
},
{
"id": "p37.11",
"order": 18,
"type": "text",
"bbox": {
"l": 67.4214,
"t": 459.5836,
"r": 526.175,
"b": 378.2265
},
"text": "Parameters[p37.11.1],[p37.11.2], and[p37.11.3] are intratemporal elasticities of substitution between the inputs in the production functions, while[p37.11.4],[p37.11.5], and[p37.11.6] are weights (quasi-shares) of the inputs into the production functions, with[p37.11.7]. The function[p37.11.8] is the adjustment cost for bilateral consumption imports of country H from country CO, where[p37.11.9] determines the magnitude of the cost:",
"children": [
{
"id": "p37.11.1",
"order": 19,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 459.5836,
"r": 526.175,
"b": 378.2265
},
"text": "\\mu_{C}",
"attrs": {
"latex": "\\mu_{C}",
"inline": true,
"text_offset": 10,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "mu sub C",
"mathml": "<math aria-label=\"mu sub C\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>μ</mi><mrow><mi>C</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p37.11.2",
"order": 20,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 459.5836,
"r": 526.175,
"b": 378.2265
},
"text": "\\mu_{TC}",
"attrs": {
"latex": "\\mu_{TC}",
"inline": true,
"text_offset": 21,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "mu sub TC",
"mathml": "<math aria-label=\"mu sub TC\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>μ</mi><mrow><mi>T</mi><mi>C</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p37.11.3",
"order": 21,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 459.5836,
"r": 526.175,
"b": 378.2265
},
"text": "\\mu_{IMC}",
"attrs": {
"latex": "\\mu_{IMC}",
"inline": true,
"text_offset": 36,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "mu sub IMC",
"mathml": "<math aria-label=\"mu sub IMC\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>μ</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p37.11.4",
"order": 22,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 459.5836,
"r": 526.175,
"b": 378.2265
},
"text": "\\nu_{C}",
"attrs": {
"latex": "\\nu_{C}",
"inline": true,
"text_offset": 147,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "nu sub C",
"mathml": "<math aria-label=\"nu sub C\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>ν</mi><mrow><mi>C</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p37.11.5",
"order": 23,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 459.5836,
"r": 526.175,
"b": 378.2265
},
"text": "\\nu_{TC}",
"attrs": {
"latex": "\\nu_{TC}",
"inline": true,
"text_offset": 158,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "nu sub TC",
"mathml": "<math aria-label=\"nu sub TC\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>ν</mi><mrow><mi>T</mi><mi>C</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p37.11.6",
"order": 24,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 459.5836,
"r": 526.175,
"b": 378.2265
},
"text": "\\nu_{IMC}",
"attrs": {
"latex": "\\nu_{IMC}",
"inline": true,
"text_offset": 173,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "nu sub IMC",
"mathml": "<math aria-label=\"nu sub IMC\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>ν</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p37.11.7",
"order": 25,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 459.5836,
"r": 526.175,
"b": 378.2265
},
"text": "\\sum_{CO\\neq H} v_{IMC}^{H,CO} = 1",
"attrs": {
"latex": "\\sum_{CO\\neq H} v_{IMC}^{H,CO} = 1",
"inline": true,
"text_offset": 260,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "the sum of sub CO is not equal to H v sub IMC to the power of H, CO equals 1",
"mathml": "<math aria-label=\"the sum of sub CO is not equal to H v sub IMC to the power of H, CO equals 1\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mo>∑</mo><mrow><mi>C</mi><mi>O</mi><mo>≠</mo><mi>H</mi></mrow></msub><msubsup><mi>v</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><mi>O</mi></mrow></msubsup><mo>=</mo><mn>1</mn></mrow></math>"
},
"children": []
},
{
"id": "p37.11.8",
"order": 26,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 459.5836,
"r": 526.175,
"b": 378.2265
},
"text": "\\Gamma_{IMC}^{H,CO} \\left( \\frac{IM_{C,CO,t}}{Q_{C,t}^{C}} \\right)",
"attrs": {
"latex": "\\Gamma_{IMC}^{H,CO} \\left( \\frac{IM_{C,CO,t}}{Q_{C,t}^{C}} \\right)",
"inline": true,
"text_offset": 284,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "capital gamma sub IMC to the power of H, CO of IM sub C, CO, t over Q sub C, t to the power of C",
"mathml": "<math aria-label=\"capital gamma sub IMC to the power of H, CO of IM sub C, CO, t over Q sub C, t to the power of C\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>Γ</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><mi>O</mi></mrow></msubsup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mfrac><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><mrow><msubsup><mi>Q</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>C</mi></mrow></msubsup></mrow></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow></mrow></math>"
},
"children": []
},
{
"id": "p37.11.9",
"order": 27,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 459.5836,
"r": 526.175,
"b": 378.2265
},
"text": "\\gamma_{IMC}",
"attrs": {
"latex": "\\gamma_{IMC}",
"inline": true,
"text_offset": 387,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "gamma sub IMC",
"mathml": "<math aria-label=\"gamma sub IMC\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>γ</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow></msub></mrow></math>"
},
"children": []
}
]
},
{
"id": "p37.12",
"order": 28,
"type": "formula",
"bbox": {
"l": 149.7188,
"t": 367.9069,
"r": 442.4381,
"b": 334.7881
},
"text": "\\Gamma_{IM^{C}}^{H,CO}\\left(\\frac{IM_{C,CO,t}}{Q_{C,t}}\\right)\\equiv\\frac{\\gamma_{IM^{C}}}{2}\\left(\\frac{IM_{C,CO,t}/Q_{C,t}}{IM_{C,CO,t-1}/Q_{C,t-1}}-1\\right)^{2}.",
"attrs": {
"block_label": "display_formula",
"latex": "\\Gamma_{I M^{C}}^{H,C O}\\left(\\frac{I M_{C,C O,t}}{Q_{C,t}}\\right)\\equiv\\frac{\\gamma_{I M^{C}}}{2}\\left(\\frac{I M_{C,C O,t}/Q_{C,t}}{I M_{C,C O,t-1}/Q_{C,t-1}}-1\\right)^{2}.",
"notation": "latex",
"alt": "capital gamma sub IM to the power of C to the power of H, CO of IM sub C, CO, t over Q sub C, t equiv gamma sub IM to the power of C over 2 times IM sub C, CO, t over Q sub C, t over IM sub C, CO, t minus 1 over Q sub C, t minus 1 minus 1 squared .",
"mathml": "<math aria-label=\"capital gamma sub IM to the power of C to the power of H, CO of IM sub C, CO, t over Q sub C, t equiv gamma sub IM to the power of C over 2 times IM sub C, CO, t over Q sub C, t over IM sub C, CO, t minus 1 over Q sub C, t minus 1 minus 1 squared .\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>Γ</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>C</mi></mrow></msup></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><mi>O</mi></mrow></msubsup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mfrac><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><mrow><msub><mi>Q</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mo>≡</mo><mfrac><mrow><msub><mi>γ</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>C</mi></mrow></msup></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mfrac><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>/</mo><msub><mi>Q</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mi>Q</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac><mo>−</mo><mn>1</mn><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>.</mo></mrow></math>"
},
"children": []
},
{
"id": "p37.13",
"order": 29,
"type": "text",
"bbox": {
"l": 502.4216,
"t": 359.0272,
"r": 524.2556,
"b": 344.8677
},
"text": "(23)",
"attrs": {
"raw_type": "FORMULA_NUMBER",
"block_label": "formula_number"
},
"children": []
},
{
"id": "p37.14",
"order": 30,
"type": "text",
"bbox": {
"l": 67.9013,
"t": 326.1484,
"r": 526.6549,
"b": 267.3505
},
"text": "A final good's firm chooses the combination of the tradable and nontradable bundles that minimizes the expenditure[p37.14.3] subject to technology constraints (20) and (21), taking the input prices as given, which yields the following demand functions:",
"children": [
{
"id": "p37.14.1",
"order": 31,
"type": "link",
"bbox": {
"l": 196.752,
"t": 296.139,
"r": 212.612,
"b": 282.914
},
"text": "(20)",
"attrs": {
"action_type": "/Named"
},
"children": []
},
{
"id": "p37.14.2",
"order": 32,
"type": "link",
"bbox": {
"l": 243.761,
"t": 296.139,
"r": 259.622,
"b": 282.914
},
"text": "",
"attrs": {
"action_type": "/Named"
},
"children": []
},
{
"id": "p37.14.3",
"order": 33,
"type": "formula",
"bbox": {
"l": 67.9013,
"t": 326.1484,
"r": 526.6549,
"b": 267.3505
},
"text": "P_{HT,t}HT_{C,t} + P_{IMC,t}IM_{C,t} + P_{NT,t}NT_{C,t}",
"attrs": {
"latex": "P_{HT,t}HT_{C,t} + P_{IMC,t}IM_{C,t} + P_{NT,t}NT_{C,t}",
"inline": true,
"text_offset": 114,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "P sub HT, t HT sub C, t plus P sub IMC, t IM sub C, t plus P sub NT, t NT sub C, t",
"mathml": "<math aria-label=\"P sub HT, t HT sub C, t plus P sub IMC, t IM sub C, t plus P sub NT, t NT sub C, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>P</mi><mrow><mi>H</mi><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub><mi>H</mi><msub><mi>T</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>+</mo><msub><mi>P</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>+</mo><msub><mi>P</mi><mrow><mi>N</mi><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub><mi>N</mi><msub><mi>T</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
}
]
},
{
"id": "p37.15",
"order": 34,
"type": "formula",
"bbox": {
"l": 87.5759,
"t": 258.4708,
"r": 355.582,
"b": 225.112
},
"text": "\\begin{array}{rlr}{HT_{C,t}}&{=}&{\\nu_{TC}\\nu_{C}\\left(\\displaystyle\\frac{P_{HT,t}}{P_{TT^{C},t}}\\right)^{-\\mu_{TC}}\\left(\\displaystyle\\frac{P_{TT^{C},t}}{P_{C,t}}\\right)^{-\\mu_{C}}Q_{C,t}}\\end{array}",
"attrs": {
"block_label": "display_formula",
"latex": "\\begin{array}{r l r}{H T_{C,t}}&{=}&{\\nu_{T C}\\nu_{C}\\left(\\displaystyle\\frac{P_{H T,t}}{P_{T T^{C},t}}\\right)^{-\\mu_{T C}}\\left(\\displaystyle\\frac{P_{T T^{C},t}}{P_{C,t}}\\right)^{-\\mu_{C}}Q_{C,t}}\\end{array}",
"notation": "latex",
"alt": "begin array rlr HT sub C, t & equals & nu sub TC nu sub C of displaystyle P sub HT, t over P sub TT to the power of C, t to the power of minus mu sub TC of displaystyle P sub TT to the power of C, t over P sub C, t to the power of minus mu sub C Q sub C, t end array",
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{
"id": "p37.16",
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{
"id": "p37.17",
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"l": 88.0557,
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"text": "\\begin{array}{rcl}{IM_{C,t}}&{=}&{\\left(1-\\nu_{TC}\\right)\\nu_{C}\\left(\\cfrac{P_{IM^{C},t}}{P_{TT^{C},t}}\\right)^{-\\mu_{TC}}\\left(\\cfrac{P_{TT^{C},t}}{P_{C,t}}\\right)^{-\\mu_{C}}Q_{C,t}}\\end{array}",
"attrs": {
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"latex": "\\begin{array}{r c l}{I M_{C,t}}&{=}&{\\left(1-\\nu_{T C}\\right)\\nu_{C}\\left(\\cfrac{P_{I M^{C},t}}{P_{T T^{C},t}}\\right)^{-\\mu_{T C}}\\left(\\cfrac{P_{T T^{C},t}}{P_{C,t}}\\right)^{-\\mu_{C}}Q_{C,t}}\\end{array}",
"notation": "latex",
"alt": "begin array rcl IM sub C, t & equals & 1 minus nu sub TC nu sub C of cfrac P sub IM to the power of C, t P sub TT to the power of C, t to the power of minus mu sub TC of cfrac P sub TT to the power of C, t P sub C, t to the power of minus mu sub C Q sub C, t end array",
"mathml": "<math aria-label=\"begin array rcl IM sub C, t & equals & 1 minus nu sub TC nu sub C of cfrac P sub IM to the power of C, t P sub TT to the power of C, t to the power of minus mu sub TC of cfrac P sub TT to the power of C, t P sub C, t to the power of minus mu sub C Q sub C, t end array\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mtable><mtr><mtd columnalign=\"right\"><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mtd><mtd columnalign=\"center\"><mrow><mo>=</mo></mrow></mtd><mtd columnalign=\"left\"><mrow><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ν</mi><mrow><mi>T</mi><mi>C</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><msub><mi>ν</mi><mrow><mi>C</mi></mrow></msub><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mfrac><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mrow><msub><mi>P</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>C</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></msub></mrow></mstyle><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mrow><msub><mi>P</mi><mrow><mi>T</mi><msup><mi>T</mi><mrow><mi>C</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></msub></mrow></mstyle></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><mo>−</mo><msub><mi>μ</mi><mrow><mi>T</mi><mi>C</mi></mrow></msub></mrow></msup><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mfrac><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mrow><msub><mi>P</mi><mrow><mi>T</mi><msup><mi>T</mi><mrow><mi>C</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></msub></mrow></mstyle><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mrow><msub><mi>P</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mstyle></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><mo>−</mo><msub><mi>μ</mi><mrow><mi>C</mi></mrow></msub></mrow></msup><msub><mi>Q</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow></math>"
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{
"id": "p37.18",
"order": 37,
"type": "text",
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"b": 200.1529
},
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},
{
"id": "p37.19",
"order": 38,
"type": "formula",
"bbox": {
"l": 88.0557,
"t": 187.6733,
"r": 283.8418,
"b": 156.2344
},
"text": "\\begin{array}{rcl}{NT_{C,t}}&{=}&{\\left(1-\\nu_{C}\\right)\\left(\\displaystyle\\frac{P_{NT,t}}{P_{C,t}}\\right)^{-\\mu_{C}}Q_{C,t}}\\end{array}",
"attrs": {
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"latex": "\\begin{array}{r c l}{N T_{C,t}}&{=}&{\\left(1-\\nu_{C}\\right)\\left(\\displaystyle\\frac{P_{N T,t}}{P_{C,t}}\\right)^{-\\mu_{C}}Q_{C,t}}\\end{array}",
"notation": "latex",
"alt": "begin array rcl NT sub C, t & equals & 1 minus nu sub C times displaystyle P sub NT, t over P sub C, t to the power of minus mu sub C Q sub C, t end array",
"mathml": "<math aria-label=\"begin array rcl NT sub C, t & equals & 1 minus nu sub C times displaystyle P sub NT, t over P sub C, t to the power of minus mu sub C Q sub C, t end array\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mtable><mtr><mtd columnalign=\"right\"><mrow><mi>N</mi><msub><mi>T</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mtd><mtd columnalign=\"center\"><mrow><mo>=</mo></mrow></mtd><mtd columnalign=\"left\"><mrow><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ν</mi><mrow><mi>C</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mrow><msub><mi>P</mi><mrow><mi>N</mi><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><mrow><msub><mi>P</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mstyle></mrow><mrow><mo>−</mo><msub><mi>μ</mi><mrow><mi>C</mi></mrow></msub></mrow></msup><msub><mi>Q</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow></math>"
},
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},
{
"id": "p37.20",
"order": 39,
"type": "text",
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"l": 502.4216,
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},
"text": "(26)",
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},
{
"id": "p37.21",
"order": 40,
"type": "formula",
"bbox": {
"l": 71.2604,
"t": 152.1546,
"r": 524.4955,
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},
"text": "\\begin{array}{rlr}{IM_{C,CO,t}}&{=}&{\\nu_{IM^{C}}^{H,CO}\\left(\\frac{P_{IM,t}^{H,CO}}{P_{IM^{C},t}\\Gamma_{IM^{C}}^{H,CO\\dagger}\\left(IM_{C,CO,t}/Q_{C,t}\\right)}\\right)^{-\\mu_{IMC}}\\frac{IM_{C,t}}{1-\\Gamma_{IM^{C}}^{H,CO}\\left(IM_{C,CO,t}/Q_{C,t}\\right)}\\mathrm{(27)}}\\end{array}",
"attrs": {
"block_label": "display_formula",
"latex": "\\begin{array}{r l r}{I M_{C,C O,t}}&{=}&{\\nu_{I M^{C}}^{H,C O}\\left(\\frac{P_{I M,t}^{H,C O}}{P_{I M^{C},t}\\Gamma_{I M^{C}}^{H,C O\\dagger}\\left(I M_{C,C O,t}/Q_{C,t}\\right)}\\right)^{-\\mu_{I M C}}\\frac{I M_{C,t}}{1-\\Gamma_{I M^{C}}^{H,C O}\\left(I M_{C,C O,t}/Q_{C,t}\\right)}\\mathrm{(27)}}\\end{array}",
"notation": "latex",
"alt": "begin array rlr IM sub C, CO, t & equals & nu sub IM to the power of C to the power of H, CO of P sub IM, t to the power of H, CO over P sub IM to the power of C, t capital gamma sub IM to the power of C to the power of H, CO dagger of IM sub C, CO, t over Q sub C, t to the power of minus mu sub IMC IM sub C, t over 1 minus capital gamma sub IM to the power of C to the power of H, CO of IM sub C, CO, t over Q sub C, t times 27 end array",
"mathml": "<math aria-label=\"begin array rlr IM sub C, CO, t & equals & nu sub IM to the power of C to the power of H, CO of P sub IM, t to the power of H, CO over P sub IM to the power of C, t capital gamma sub IM to the power of C to the power of H, CO dagger of IM sub C, CO, t over Q sub C, t to the power of minus mu sub IMC IM sub C, t over 1 minus capital gamma sub IM to the power of C to the power of H, CO of IM sub C, CO, t over Q sub C, t times 27 end array\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mtable><mtr><mtd columnalign=\"right\"><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mtd><mtd columnalign=\"left\"><mrow><mo>=</mo></mrow></mtd><mtd columnalign=\"right\"><mrow><msubsup><mi>ν</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>C</mi></mrow></msup></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><mi>O</mi></mrow></msubsup><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mfrac><mrow><msubsup><mi>P</mi><mrow><mi>I</mi><mi>M</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><mi>O</mi></mrow></msubsup></mrow><mrow><msub><mi>P</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>C</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></msub><msubsup><mi>Γ</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>C</mi></mrow></msup></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><mi>O</mi><mi>†</mi></mrow></msubsup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>/</mo><msub><mi>Q</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow></mrow></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><mo>−</mo><msub><mi>μ</mi><mrow><mi>I</mi><mi>M</mi><mi>C</mi></mrow></msub></mrow></msup><mfrac><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><mrow><mn>1</mn><mo>−</mo><msubsup><mi>Γ</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>C</mi></mrow></msup></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><mi>O</mi></mrow></msubsup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>/</mo><msub><mi>Q</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow></mrow></mfrac><mrow><mo stretchy=\"false\">(</mo><mn>27</mn><mo stretchy=\"false\">)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math>"
},
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{
"id": "p37.22",
"order": 41,
"type": "text",
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"text": "The term[p37.22.1] in the bilateral import bundle is the derivative of the adjustment cost.",
"children": [
{
"id": "p37.22.1",
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"type": "formula",
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"text": "\\Gamma_{IM^{C}}^{H,CO^{\\dagger}}\\left(\\frac{IM_{C,CO,t}}{Q_{C,t}}\\right)",
"attrs": {
"latex": "\\Gamma_{IM^{C}}^{H,CO^{\\dagger}}\\left(\\frac{IM_{C,CO,t}}{Q_{C,t}}\\right)",
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"alt": "capital gamma sub IM to the power of C to the power of H, CO to the power of dagger of IM sub C, CO, t over Q sub C, t",
"mathml": "<math aria-label=\"capital gamma sub IM to the power of C to the power of H, CO to the power of dagger of IM sub C, CO, t over Q sub C, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>Γ</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>C</mi></mrow></msup></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><msup><mi>O</mi><mrow><mi>†</mi></mrow></msup></mrow></msubsup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mfrac><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mo>,</mo><mi>C</mi><mi>O</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><mrow><msub><mi>Q</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow></mrow></math>"
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}
]
},
{
"id": "p37.23",
"order": 43,
"type": "footer",
"bbox": {
"l": 290.0801,
"t": 51.1182,
"r": 305.1959,
"b": 39.3586
},
"text": "36",
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}
],
"ai_page_review": {
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footer (1)
formula (14)
heading (1)
link (3)
text (15)
1 · TEXT
2 · FORMULA
3 · TEXT
4 · FORMULA
5 · TEXT
6 · FORMULA
7 · TEXT
8 · HEADING
9 · TEXT
10
11
12
13
14
15 · FORMULA
16 · TEXT
17 · TEXT
18 · FORMULA
19 · TEXT
20 · FORMULA
21 · TEXT
22 · TEXT
23 · FORMULA
24 · TEXT
25 · TEXT
26 · FORMULA
27 · TEXT
28 · TEXT
29
30
31
32 · FORMULA
33 · TEXT
34 · FOOTER
Page 38 JSON
{
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{
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"bbox": {
"l": 85.8963,
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"r": 329.4292,
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"text": "The corresponding cost-minimizing prices are:",
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},
{
"id": "p38.2",
"order": 2,
"type": "formula",
"bbox": {
"l": 128.1247,
"t": 743.2535,
"r": 350.0636,
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"text": "\\begin{array}{rcl}{P_{C,t}}&{=}&{\\left[\\nu_{C}P_{TT^{C},t}^{1-\\mu_{C}}+\\left(1-\\nu_{C}\\right)P_{NT,t}^{1-\\mu_{C}}\\right]^{\\frac{1}{1-\\mu_{C}}}}\\\\ \\end{array}",
"attrs": {
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"latex": "\\begin{array}{r c l}{P_{C,t}}&{=}&{\\left[\\nu_{C}P_{T T^{C},t}^{1-\\mu_{C}}+\\left(1-\\nu_{C}\\right)P_{N T,t}^{1-\\mu_{C}}\\right]^{\\frac{1}{1-\\mu_{C}}}}\\\\ \\end{array}",
"notation": "latex",
"alt": "begin array rcl P sub C, t & equals & nu sub C P sub TT to the power of C, t to the power of 1 minus mu sub C plus 1 minus nu sub C P sub NT, t to the power of 1 minus mu sub C to the power of 1 over 1 minus mu sub C end array",
"mathml": "<math aria-label=\"begin array rcl P sub C, t & equals & nu sub C P sub TT to the power of C, t to the power of 1 minus mu sub C plus 1 minus nu sub C P sub NT, t to the power of 1 minus mu sub C to the power of 1 over 1 minus mu sub C end array\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mtable><mtr><mtd columnalign=\"right\"><mrow><msub><mi>P</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mtd><mtd columnalign=\"center\"><mrow><mo>=</mo></mrow></mtd><mtd columnalign=\"left\"><mrow><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">[</mo><msub><mi>ν</mi><mrow><mi>C</mi></mrow></msub><msubsup><mi>P</mi><mrow><mi>T</mi><msup><mi>T</mi><mrow><mi>C</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>μ</mi><mrow><mi>C</mi></mrow></msub></mrow></msubsup><mo>+</mo><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ν</mi><mrow><mi>C</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><msubsup><mi>P</mi><mrow><mi>N</mi><mi>T</mi><mo>,</mo><mi>t</mi></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>μ</mi><mrow><mi>C</mi></mrow></msub></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">]</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>μ</mi><mrow><mi>C</mi></mrow></msub></mrow></mfrac></mrow></msup></mrow></mtd></mtr></mtable></mrow></math>"
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{
"id": "p38.3",
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{
"id": "p38.4",
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"bbox": {
"l": 116.128,
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"text": "\\begin{array}{rcl}{P_{TT^{C},t}}&{=}&{\\left[\\nu_{TC}P_{HT,t}^{1-\\mu_{TC}}+\\left(1-\\nu_{TC}\\right)P_{IM^{C},t}^{1-\\mu_{TC}}\\right]^{\\frac{1}{1-\\mu_{TC}}}}\\\\ \\end{array}",
"attrs": {
"block_label": "display_formula",
"latex": "\\begin{array}{r c l}{P_{T T^{C},t}}&{=}&{\\left[\\nu_{T C}P_{H T,t}^{1-\\mu_{T C}}+\\left(1-\\nu_{T C}\\right)P_{I M^{C},t}^{1-\\mu_{T C}}\\right]^{\\frac{1}{1-\\mu_{T C}}}}\\\\ \\end{array}",
"notation": "latex",
"alt": "begin array rcl P sub TT to the power of C, t & equals & nu sub TC P sub HT, t to the power of 1 minus mu sub TC plus 1 minus nu sub TC P sub IM to the power of C, t to the power of 1 minus mu sub TC to the power of 1 over 1 minus mu sub TC end array",
"mathml": "<math aria-label=\"begin array rcl P sub TT to the power of C, t & equals & nu sub TC P sub HT, t to the power of 1 minus mu sub TC plus 1 minus nu sub TC P sub IM to the power of C, t to the power of 1 minus mu sub TC to the power of 1 over 1 minus mu sub TC end array\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mtable><mtr><mtd columnalign=\"right\"><mrow><msub><mi>P</mi><mrow><mi>T</mi><msup><mi>T</mi><mrow><mi>C</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></msub></mrow></mtd><mtd columnalign=\"center\"><mrow><mo>=</mo></mrow></mtd><mtd columnalign=\"left\"><mrow><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">[</mo><msub><mi>ν</mi><mrow><mi>T</mi><mi>C</mi></mrow></msub><msubsup><mi>P</mi><mrow><mi>H</mi><mi>T</mi><mo>,</mo><mi>t</mi></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>μ</mi><mrow><mi>T</mi><mi>C</mi></mrow></msub></mrow></msubsup><mo>+</mo><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ν</mi><mrow><mi>T</mi><mi>C</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><msubsup><mi>P</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>C</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>μ</mi><mrow><mi>T</mi><mi>C</mi></mrow></msub></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">]</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>μ</mi><mrow><mi>T</mi><mi>C</mi></mrow></msub></mrow></mfrac></mrow></msup></mrow></mtd></mtr></mtable></mrow></math>"
},
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{
"id": "p38.5",
"order": 5,
"type": "text",
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"text": "(29)",
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{
"id": "p38.6",
"order": 6,
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"bbox": {
"l": 114.6884,
"t": 683.0157,
"r": 478.1883,
"b": 634.2974
},
"text": "\\begin{array}{lll}{P_{{IM}^{C},t}}&{=}&{\\displaystyle\\left(\\sum_{CO\\neq H}\\nu_{{IM}^{C}}^{H,CO}\\left(\\frac{P_{{IM},t}^{H,CO}}{\\Gamma_{{IM}^{C}}^{H,CO\\dagger}\\left(IM_{t}^{C,CO}/Q_{C,t}\\right)}\\right)^{1-\\mu_{{IMC}}}\\right)^{\\frac{1}{1-\\mu_{{IMC}}}}.}\\\\ \\end{array}",
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"latex": "\\begin{array}{l l l}{P_{{I M}^{C},t}}&{=}&{\\displaystyle\\left(\\sum_{C O\\neq H}\\nu_{{I M}^{C}}^{H,C O}\\left(\\frac{P_{{I M},t}^{H,C O}}{\\Gamma_{{I M}^{C}}^{H,C O\\dagger}\\left(I M_{t}^{C,C O}/Q_{C,t}\\right)}\\right)^{1-\\mu_{{I M C}}}\\right)^{\\frac{1}{1-\\mu_{{I M C}}}}.}\\\\ \\end{array}",
"notation": "latex",
"alt": "begin array lll P sub IM to the power of C, t & equals & displaystyle of the sum of sub CO is not equal to H nu sub IM to the power of C to the power of H, CO of frac P sub IM, t to the power of H, CO capital gamma sub IM to the power of C to the power of H, CO dagger of IM sub t to the power of C, CO over Q sub C, t to the power of 1 minus mu sub IMC to the power of frac 1 1 minus mu sub IMC . end array",
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},
"children": []
},
{
"id": "p38.7",
"order": 7,
"type": "text",
"bbox": {
"l": 502.4216,
"t": 665.7363,
"r": 524.0156,
"b": 651.5768
},
"text": "(30)",
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"raw_type": "FORMULA_NUMBER",
"block_label": "formula_number"
},
"children": []
},
{
"id": "p38.8",
"order": 8,
"type": "heading",
"bbox": {
"l": 67.9013,
"t": 615.8181,
"r": 192.9069,
"b": 599.4986
},
"text": "A.4 Export goods",
"attrs": {
"block_label": "paragraph_title"
},
"children": []
},
{
"id": "p38.9",
"order": 9,
"type": "text",
"bbox": {
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"t": 590.619,
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"b": 532.781
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"text": "The model features import-content of exports, which is modelled as in Brzoza-Brzezina et al. (2014), where export firms create a bundle of home-produced tradable goods for export,[p38.9.4] and goods imported for the purpose of reexporting,[p38.9.5], and then export the bundle of both goods. The export bundle is:",
"children": [
{
"id": "p38.9.1",
"order": 10,
"type": "link",
"bbox": {
"l": 437.324,
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"b": 576.662
},
"text": "Brzoza-Brzezina",
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},
{
"id": "p38.9.2",
"order": 11,
"type": "link",
"bbox": {
"l": 69.87,
"t": 575.44,
"r": 97.246,
"b": 562.216
},
"text": "et al.",
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"action_type": "/Named"
},
"children": []
},
{
"id": "p38.9.3",
"order": 12,
"type": "link",
"bbox": {
"l": 101.844,
"t": 575.44,
"r": 131.573,
"b": 562.216
},
"text": "(2014),",
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"action_type": "/Named"
},
"children": []
},
{
"id": "p38.9.4",
"order": 13,
"type": "formula",
"bbox": {
"l": 67.9013,
"t": 590.619,
"r": 526.415,
"b": 532.781
},
"text": "HT_{X,t}",
"attrs": {
"latex": "HT_{X,t}",
"inline": true,
"text_offset": 179,
"text_span": 9,
"bbox_estimated": true,
"notation": "latex",
"alt": "HT sub X, t",
"mathml": "<math aria-label=\"HT sub X, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>H</mi><msub><mi>T</mi><mrow><mi>X</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
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},
{
"id": "p38.9.5",
"order": 14,
"type": "formula",
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"r": 526.415,
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"text": "IM_{X,t}",
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"latex": "IM_{X,t}",
"inline": true,
"text_offset": 239,
"text_span": 9,
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"notation": "latex",
"alt": "IM sub X, t",
"mathml": "<math aria-label=\"IM sub X, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>X</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
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"children": []
}
]
},
{
"id": "p38.10",
"order": 15,
"type": "formula",
"bbox": {
"l": 166.994,
"t": 521.2614,
"r": 426.1226,
"b": 488.8626
},
"text": "EX_{t}=\\left\\lceil\\nu_{X}^{\\frac{1}{\\mu_{X}}}HT_{X,t}^{\\frac{\\mu_{X}-1}{\\mu_{X}}}+(1-\\nu_{X})^{\\frac{1}{\\mu_{X}}}IM_{X,t}^{\\frac{\\mu_{x}-1}{\\mu_{X}}}\\right\\rceil^{\\frac{\\mu_{X}}{\\mu_{X}-1}}.",
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"latex": "E X_{t}=\\left\\lceil\\nu_{X}^{\\frac{1}{\\mu_{X}}}H T_{X,t}^{\\frac{\\mu_{X}-1}{\\mu_{X}}}+(1-\\nu_{X})^{\\frac{1}{\\mu_{X}}}I M_{X,t}^{\\frac{\\mu_{x}-1}{\\mu_{X}}}\\right\\rceil^{\\frac{\\mu_{X}}{\\mu_{X}-1}}.",
"notation": "latex",
"alt": "EX sub t equals lceil nu sub X to the power of 1 over mu sub X HT sub X, t to the power of mu sub X minus 1 over mu sub X plus 1 minus nu sub X to the power of 1 over mu sub X IM sub X, t to the power of mu sub x minus 1 over mu sub X rceil to the power of mu sub X over mu sub X minus 1 .",
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},
"children": []
},
{
"id": "p38.11",
"order": 16,
"type": "text",
"bbox": {
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"b": 496.3023
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"text": "(31)",
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{
"id": "p38.12",
"order": 17,
"type": "text",
"bbox": {
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"text": "Analogously to demands for goods in the consumption bundle, we have the demands for home tradable goods for exports and imported goods for reexports:",
"children": []
},
{
"id": "p38.13",
"order": 18,
"type": "formula",
"bbox": {
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"text": "\\begin{array}{ccc}{HT_{X,t}}&{=}&{\\nu_{X}\\left(\\displaystyle\\frac{MC_{T,t}}{MC_{X,t}}\\right)^{-\\mu_{X}}EX_{t}}\\\\ \\end{array}",
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"latex": "\\begin{array}{c c c}{H T_{X,t}}&{=}&{\\nu_{X}\\left(\\displaystyle\\frac{M C_{T,t}}{M C_{X,t}}\\right)^{-\\mu_{X}}E X_{t}}\\\\ \\end{array}",
"notation": "latex",
"alt": "begin array ccc HT sub X, t & equals & nu sub X of displaystyle MC sub T, t over MC sub X, t to the power of minus mu sub X EX sub t end array",
"mathml": "<math aria-label=\"begin array ccc HT sub X, t & equals & nu sub X of displaystyle MC sub T, t over MC sub X, t to the power of minus mu sub X EX sub t end array\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mtable><mtr><mtd columnalign=\"center\"><mrow><mi>H</mi><msub><mi>T</mi><mrow><mi>X</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mtd><mtd columnalign=\"center\"><mrow><mo>=</mo></mrow></mtd><mtd columnalign=\"center\"><mrow><msub><mi>ν</mi><mrow><mi>X</mi></mrow></msub><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mrow><mi>M</mi><msub><mi>C</mi><mrow><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><mrow><mi>M</mi><msub><mi>C</mi><mrow><mi>X</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mstyle></mrow><mrow><mo>−</mo><msub><mi>μ</mi><mrow><mi>X</mi></mrow></msub></mrow></msup><mi>E</mi><msub><mi>X</mi><mrow><mi>t</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow></math>"
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"children": []
},
{
"id": "p38.14",
"order": 19,
"type": "text",
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"b": 408.2255
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"text": "(32)",
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},
{
"id": "p38.15",
"order": 20,
"type": "formula",
"bbox": {
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"r": 402.3691,
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"text": "\\begin{array}{lcl}{IM_{X,t}}&{=}&{\\left(1-\\nu_{X}\\right)\\left(\\cfrac{P_{IM^{X},t}}{MC_{X,t}}\\right)^{-\\mu_{X}}EX_{t}.}\\\\ \\end{array}",
"attrs": {
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"latex": "\\begin{array}{l c l}{I M_{X,t}}&{=}&{\\left(1-\\nu_{X}\\right)\\left(\\cfrac{P_{I M^{X},t}}{M C_{X,t}}\\right)^{-\\mu_{X}}E X_{t}.}\\\\ \\end{array}",
"notation": "latex",
"alt": "begin array lcl IM sub X, t & equals & 1 minus nu sub X times cfrac P sub IM to the power of X, t MC sub X, t to the power of minus mu sub X EX sub t . end array",
"mathml": "<math aria-label=\"begin array lcl IM sub X, t & equals & 1 minus nu sub X times cfrac P sub IM to the power of X, t MC sub X, t to the power of minus mu sub X EX sub t . end array\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mtable><mtr><mtd columnalign=\"left\"><mrow><mi>I</mi><msub><mi>M</mi><mrow><mi>X</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mtd><mtd columnalign=\"center\"><mrow><mo>=</mo></mrow></mtd><mtd columnalign=\"left\"><mrow><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ν</mi><mrow><mi>X</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mfrac><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mrow><msub><mi>P</mi><mrow><mi>I</mi><msup><mi>M</mi><mrow><mi>X</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></msub></mrow></mstyle><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mrow><mi>M</mi><msub><mi>C</mi><mrow><mi>X</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mstyle></mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><mo>−</mo><msub><mi>μ</mi><mrow><mi>X</mi></mrow></msub></mrow></msup><mi>E</mi><msub><mi>X</mi><mrow><mi>t</mi></mrow></msub><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math>"
},
"children": []
},
{
"id": "p38.16",
"order": 21,
"type": "text",
"bbox": {
"l": 502.6615,
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"b": 373.6667
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"text": "(33)",
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"block_label": "formula_number"
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"children": []
},
{
"id": "p38.17",
"order": 22,
"type": "text",
"bbox": {
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"b": 324.9484
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"text": "Imported goods for the purpose of reexports are a bundle of imported goods from all other regions, subject to adjustment costs:",
"children": []
},
{
"id": "p38.18",
"order": 23,
"type": "formula",
"bbox": {
"l": 73.8996,
"t": 300.4693,
"r": 517.7774,
"b": 252.951
},
"text": "IM_{X,t}=\\left[\\sum_{CO\\neq H}\\left(\\nu_{IM_{X}}^{H,CO}\\right)^{\\frac{1}{\\mu_{IMX}}}\\left(IM_{X,CO,t}\\left(1-\\Gamma_{IM^{X}}^{H,CO}\\left(\\frac{IM_{X,CO,t}}{EX_{t}}\\right)\\right)\\right)^{\\frac{\\mu_{IMX}-1}{\\mu_{IMX}}}\\right]^{\\frac{\\mu_{IMX}}{\\mu_{IMX}-1}}.",
"attrs": {
"block_label": "display_formula",
"latex": "I M_{X,t}=\\left[\\sum_{C O\\neq H}\\left(\\nu_{I M_{X}}^{H,C O}\\right)^{\\frac{1}{\\mu_{I M X}}}\\left(I M_{X,C O,t}\\left(1-\\Gamma_{I M^{X}}^{H,C O}\\left(\\frac{I M_{X,C O,t}}{E X_{t}}\\right)\\right)\\right)^{\\frac{\\mu_{I M X}-1}{\\mu_{I M X}}}\\right]^{\\frac{\\mu_{I M X}}{\\mu_{I M X}-1}}.",
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"text": "These give rise to bilateral demands for reexport goods:",
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"text": "The real price (in terms of consumption goods) of the export good is just its marginal cost,[p38.23.1], which is the weighted average of (real) marginal costs of home tradable goods,[p38.23.2], and the relative price of imported goods for reexport,[p38.23.3]:",
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"text": "\\begin{array}{r}{MC_{X,t}=\\left[\\nu_{X}MC_{T,t}^{1-\\mu_{X}}+(1-\\nu_{X})P_{IM^{X},t}^{1-\\mu_{X}}\\right]^{\\frac{1}{1-\\mu_{X}}},}\\end{array}",
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footer (1)
formula (24)
heading (2)
link (2)
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1 · TEXT
2 · FORMULA
3 · TEXT
4 · HEADING
5 · TEXT
6
7
8 · FORMULA
9 · TEXT
10 · FORMULA
11 · TEXT
12 · TEXT
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21 · TEXT
22 · FORMULA
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24 · TEXT
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33 · FORMULA
34 · TEXT
35 · FORMULA
36 · TEXT
37 · HEADING
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41 · FOOTER
Page 39 JSON
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"text": "and the price of reexport good is the weighted average of the import prices from the regions where its components came from:",
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"text": "P_{IM^{X},t}=\\left(\\sum_{CO\\neq H}\\nu_{IM^{X}}^{H,CO}\\left(\\frac{P_{IM,t}^{H,CO}}{\\Gamma_{IM^{X}}^{H,CO\\dagger}\\left(IM_{t}^{X,CO}/EX_{t}\\right)}\\right)^{1-\\mu_{IMX}}\\right)^{\\frac{1}{1-\\mu_{IMX}}}.",
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"text": "A.5 Intermediate goods",
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"text": "Non-tradable[p39.5.1] and tradable[p39.5.2] intermediate goods are produced using a Cobb-Douglas technologies:",
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},
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}
]
},
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"alt": "begin array ccl Y sub N, t & equals & z sub N, t K sub G, t to the power of alpha sub G K sub N, t to the power of alpha sub N N sub N, t to the power of 1 minus alpha sub N minus psi sub N end array",
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},
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{
"id": "p39.7",
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"type": "text",
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"text": "(38)",
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{
"id": "p39.8",
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"text": "\\begin{array}{ccl}{Y_{T,t}}&{=}&{z_{T,t}K_{G,t}^{\\alpha_{G}}K_{T,t}^{\\alpha_{T}}N_{T,t}^{1-\\alpha_{T}}-\\psi_{T}}\\\\ \\end{array}",
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"notation": "latex",
"alt": "begin array ccl Y sub T, t & equals & z sub T, t K sub G, t to the power of alpha sub G K sub T, t to the power of alpha sub T N sub T, t to the power of 1 minus alpha sub T minus psi sub T end array",
"mathml": "<math aria-label=\"begin array ccl Y sub T, t & equals & z sub T, t K sub G, t to the power of alpha sub G K sub T, t to the power of alpha sub T N sub T, t to the power of 1 minus alpha sub T minus psi sub T end array\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mtable><mtr><mtd columnalign=\"center\"><mrow><msub><mi>Y</mi><mrow><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></mtd><mtd columnalign=\"center\"><mrow><mo>=</mo></mrow></mtd><mtd columnalign=\"left\"><mrow><msub><mi>z</mi><mrow><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub><msubsup><mi>K</mi><mrow><mi>G</mi><mo>,</mo><mi>t</mi></mrow><mrow><msub><mi>α</mi><mrow><mi>G</mi></mrow></msub></mrow></msubsup><msubsup><mi>K</mi><mrow><mi>T</mi><mo>,</mo><mi>t</mi></mrow><mrow><msub><mi>α</mi><mrow><mi>T</mi></mrow></msub></mrow></msubsup><msubsup><mi>N</mi><mrow><mi>T</mi><mo>,</mo><mi>t</mi></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>α</mi><mrow><mi>T</mi></mrow></msub></mrow></msubsup><mo>−</mo><msub><mi>ψ</mi><mrow><mi>T</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow></math>"
},
"children": []
},
{
"id": "p39.9",
"order": 11,
"type": "text",
"bbox": {
"l": 502.6615,
"t": 560.62,
"r": 524.0156,
"b": 546.4605
},
"text": "(39)",
"attrs": {
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},
{
"id": "p39.10",
"order": 12,
"type": "text",
"bbox": {
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"t": 533.501,
"r": 526.175,
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"text": "where[p39.10.1] and[p39.10.2] are fixed costs. The inputs are homogenous capital services,[p39.10.3] and[p39.10.4], and labour services,[p39.10.5] and[p39.10.6]. Capital services are supplied by domestic households under perfect competition, yielding that demand for capital is determined by its marginal product being equal to the rental rate, while labour demand is determined by the marginal product of labour at intermediate goods firm being equal to the cost of labour determined by labour packers, as described in the main text.[p39.10.7] and[p39.10.8] are sector-specific productivity shocks.",
"children": [
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},
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},
{
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"order": 16,
"type": "formula",
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"text": "K_{NTt}",
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"notation": "latex",
"alt": "K sub NTt",
"mathml": "<math aria-label=\"K sub NTt\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>K</mi><mrow><mi>N</mi><mi>T</mi><mi>t</mi></mrow></msub></mrow></math>"
},
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"text": "N_{N,t}",
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"notation": "latex",
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"mathml": "<math aria-label=\"N sub N, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>N</mi><mrow><mi>N</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
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{
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"text": "N_{N,t}",
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"inline": true,
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"notation": "latex",
"alt": "N sub N, t",
"mathml": "<math aria-label=\"N sub N, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>N</mi><mrow><mi>N</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p39.10.7",
"order": 19,
"type": "formula",
"bbox": {
"l": 67.9013,
"t": 533.501,
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"b": 433.6645
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"text": "z_{N,t}",
"attrs": {
"latex": "z_{N,t}",
"inline": true,
"text_offset": 534,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "z sub N, t",
"mathml": "<math aria-label=\"z sub N, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>z</mi><mrow><mi>N</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p39.10.8",
"order": 20,
"type": "formula",
"bbox": {
"l": 67.9013,
"t": 533.501,
"r": 526.175,
"b": 433.6645
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"text": "z_{T,t}",
"attrs": {
"latex": "z_{T,t}",
"inline": true,
"text_offset": 548,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "z sub T, t",
"mathml": "<math aria-label=\"z sub T, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>z</mi><mrow><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
}
]
},
{
"id": "p39.11",
"order": 21,
"type": "text",
"bbox": {
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"r": 525.9351,
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},
"text": "Note that the labour market clearing implies that the total labour services provided must equal to the labour demanded by intermediate goods firms:",
"children": []
},
{
"id": "p39.12",
"order": 22,
"type": "formula",
"bbox": {
"l": 250.7309,
"t": 387.8262,
"r": 342.6256,
"b": 374.6267
},
"text": "n_{t}=N_{N,t}+N_{T,t}.",
"attrs": {
"block_label": "display_formula",
"latex": "n_{t}=N_{N,t}+N_{T,t}.",
"notation": "latex",
"alt": "n sub t equals N sub N, t plus N sub T, t .",
"mathml": "<math aria-label=\"n sub t equals N sub N, t plus N sub T, t .\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>n</mi><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mi>N</mi><mrow><mi>N</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>+</mo><msub><mi>N</mi><mrow><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>.</mo></mrow></math>"
},
"children": []
},
{
"id": "p39.13",
"order": 23,
"type": "text",
"bbox": {
"l": 502.4216,
"t": 389.2661,
"r": 524.2556,
"b": 375.3466
},
"text": "(40)",
"attrs": {
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"block_label": "formula_number"
},
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},
{
"id": "p39.14",
"order": 24,
"type": "text",
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"text": "Goods markets for non-tradable and tradable goods clear by equating supply and demand. Demand for non-tradable goods comes from demands for non-tradable consumption and investment goods,[p39.14.1] and[p39.14.2]), and from government[p39.14.3], which is fully biased towards non-tradable goods. Demand for tradable goods comes from demands for tradable consumption and investment goods[p39.14.4] and from trading partners' imports[p39.14.5].[p39.14.6],[p39.14.7], and[p39.14.8] are price dispersions in non-tradable, tradable, and export sectors:",
"children": [
{
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"type": "formula",
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},
"text": "(NT_{C,t}",
"attrs": {
"latex": "(NT_{C,t}",
"inline": true,
"text_offset": 186,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "NT sub C, t",
"mathml": "<math aria-label=\"NT sub C, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo stretchy=\"false\">(</mo><mi>N</mi><msub><mi>T</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
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{
"id": "p39.14.2",
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"type": "formula",
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"text": "NT_{I,t}",
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"text_offset": 200,
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"notation": "latex",
"alt": "NT sub I, t",
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"type": "formula",
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"text": "(G_t)",
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"inline": true,
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"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "G sub t",
"mathml": "<math aria-label=\"G sub t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo stretchy=\"false\">(</mo><msub><mi>G</mi><mi>t</mi></msub><mo stretchy=\"false\">)</mo></mrow></math>"
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"type": "formula",
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"text": "(HT_{C,t}, HT_{I,t})",
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"inline": true,
"text_offset": 384,
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"alt": "HT sub C, t, HT sub I, t",
"mathml": "<math aria-label=\"HT sub C, t, HT sub I, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo stretchy=\"false\">(</mo><mi>H</mi><msub><mi>T</mi><mrow><mi>C</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>,</mo><mi>H</mi><msub><mi>T</mi><mrow><mi>I</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo></mrow></math>"
},
"children": []
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{
"id": "p39.14.5",
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"type": "formula",
"bbox": {
"l": 67.9013,
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"text": "(IM_{CO,H,t})",
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"text_offset": 429,
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"bbox_estimated": true,
"notation": "latex",
"alt": "IM sub CO, H, t",
"mathml": "<math aria-label=\"IM sub CO, H, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo stretchy=\"false\">(</mo><mi>I</mi><msub><mi>M</mi><mrow><mi>C</mi><mi>O</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo></mrow></math>"
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"notation": "latex",
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},
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"type": "formula",
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"bbox_estimated": true,
"notation": "latex",
"alt": "s sub HT, t",
"mathml": "<math aria-label=\"s sub HT, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>s</mi><mrow><mi>H</mi><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
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"type": "formula",
"bbox": {
"l": 67.9013,
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"r": 526.6549,
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},
"text": "s_{X,t}^{H,CO}",
"attrs": {
"latex": "s_{X,t}^{H,CO}",
"inline": true,
"text_offset": 466,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "s sub X, t to the power of H, CO",
"mathml": "<math aria-label=\"s sub X, t to the power of H, CO\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>s</mi><mrow><mi>X</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>H</mi><mo>,</mo><mi>C</mi><mi>O</mi></mrow></msubsup></mrow></math>"
},
"children": []
}
]
},
{
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},
"text": "\\begin{array}{rlr}{Y_{N,t}}&{=}&{s_{N,t}\\left(NT_{C,t}+NT_{I,t}+G_{t}\\right)}\\end{array}",
"attrs": {
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"latex": "\\begin{array}{r l r}{Y_{N,t}}&{=}&{s_{N,t}\\left(N T_{C,t}+N T_{I,t}+G_{t}\\right)}\\end{array}",
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"text": "A.6 Price setting",
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"text": "Price setting follows the standard Calvo (1983) framework. Intermediate goods firms, however, set prices differently, depending on the market to which they are pricing their goods. This applies to non-tradable, home tradable and to export goods, in the latter case also for each export market separately (the so-called local currency pricing framework). Prices that are not reset are indexed to a composite of past inflation and inflation target.",
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footer (1)
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1 · TEXT
2
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10 · TEXT
11 · FORMULA
12 · TEXT
13 · TEXT
14 · TEXT
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16 · FORMULA
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21 · FORMULA
22 · TEXT
23 · TEXT
24
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30 · FORMULA
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32 · HEADING
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39 · FOOTER
Page 40 JSON
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"alt": "P sub NT, t equals xi sub N of capital pi sub NT, t minus 1 to the power of chi sub N capital pi bar to the power of 1 minus chi sub N P sub NT, t minus 1 to the power of 1 minus theta sub N plus 1 minus xi sub N times widetilde P sub NT, t to the power of 1 minus theta sub N to the power of 1 over 1 minus theta sub N",
"mathml": "<math aria-label=\"P sub NT, t equals xi sub N of capital pi sub NT, t minus 1 to the power of chi sub N capital pi bar to the power of 1 minus chi sub N P sub NT, t minus 1 to the power of 1 minus theta sub N plus 1 minus xi sub N times widetilde P sub NT, t to the power of 1 minus theta sub N to the power of 1 over 1 minus theta sub N\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>P</mi><mrow><mi>N</mi><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>=</mo><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">[</mo><msub><mi>ξ</mi><mrow><mi>N</mi></mrow></msub><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>Π</mi><mrow><mi>N</mi><mi>T</mi><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow><mrow><msub><mi>χ</mi><mrow><mi>N</mi></mrow></msub></mrow></msubsup><msup><mover><mrow><mi>Π</mi></mrow><mo accent=\"true\">―</mo></mover><mrow><mn>1</mn><mo>−</mo><msub><mi>χ</mi><mrow><mi>N</mi></mrow></msub></mrow></msup><msub><mi>P</mi><mrow><mi>N</mi><mi>T</mi><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>θ</mi><mrow><mi>N</mi></mrow></msub></mrow></msup><mo>+</mo><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ξ</mi><mrow><mi>N</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msub><mover><mrow><mi>P</mi></mrow><mo>~</mo></mover><mrow><mi>N</mi><mi>T</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>θ</mi><mrow><mi>N</mi></mrow></msub></mrow></msup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">]</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>θ</mi><mrow><mi>N</mi></mrow></msub></mrow></mfrac></mrow></msup></mrow></math>"
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{
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"text": "Price setting for home tradable goods. Price setting for home tradable goods is analogous to that of non-tradable goods. The probability of resetting the price is[p40.9.1]. The sector-specific price index[p40.9.2] evolves according to:",
"children": [
{
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"text": "P_{HT,t}=\\left[\\xi_{H}\\left(\\Pi_{HT,t-1}^{\\chi_{H}}\\overline{{\\Pi}}^{1-\\chi_{H}}P_{HT,t-1}\\right)^{1-\\theta_{T}}+\\left(1-\\xi_{H}\\right)\\left(\\widetilde{P}_{HT,t}\\right)^{1-\\theta_{T}}\\right]^{\\frac{1}{1-\\theta_{T}}}",
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"latex": "P_{H T,t}=\\left[\\xi_{H}\\left(\\Pi_{H T,t-1}^{\\chi_{H}}\\overline{{\\Pi}}^{1-\\chi_{H}}P_{H T,t-1}\\right)^{1-\\theta_{T}}+\\left(1-\\xi_{H}\\right)\\left(\\widetilde{P}_{H T,t}\\right)^{1-\\theta_{T}}\\right]^{\\frac{1}{1-\\theta_{T}}}",
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{
"id": "p40.11",
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{
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"text": "Price setting for export goods. Firms discriminates across countries, by invoicing and setting the price in the currency of the generic destination market CO (local currency pricing assumption). The probability of optimally resetting prices[p40.12.1], and all firms that reset their prices choose the same price,[p40.12.2], while the others index their prices to[p40.12.3], where[p40.12.4] is the sector-specific inflation. Foreign inflation target is time invariant and equal to the Home inflation target,[p40.12.5].",
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"mathml": "<math aria-label=\"1 minus xi sub X\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ξ</mi><mi>X</mi></msub><mo stretchy=\"false\">)</mo></mrow></math>"
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{
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"text": "P_{X,t}^{CO} = \\left(\\Pi_{X,t-1}^{H,CO}\\right)^{\\chi_X} \\overline{\\Pi}^{1-\\chi_X} P_{X,t-1}^{CO}",
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"alt": "P sub X, t to the power of CO equals the product of sub X, t minus 1 to the power of H, CO to the power of chi sub X capital pi bar to the power of 1 minus chi sub X P sub X, t minus 1 to the power of CO",
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{
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"text": "\\Pi_{X,t-1}^{H,CO} \\equiv P_{X,t-1}^{H,CO} / P_{X,t-2}^{H,CO}",
"attrs": {
"latex": "\\prod_{X,t-1}^{H,CO} \\equiv P_{X,t-1}^{H,CO} / P_{X,t-2}^{H,CO}",
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"text": "\\overline{\\Pi}^{CO} = \\overline{\\Pi}",
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"alt": "capital pi bar to the power of CO equals capital pi bar",
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"type": "text",
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"text": "The bilateral exports price index (of country H to the generic country CO) is:",
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{
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"order": 30,
"type": "formula",
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"text": "P_{X,t}^{H,CO}=\\left[\\xi_{X}\\left(\\left(\\Pi_{X,t-1}^{H,CO}\\right)^{\\chi_{X}}\\overline{\\Pi}^{1-\\chi_{X}}P_{X,t-1}^{H,CO}\\right)^{1-\\theta_{T}}+\\left(1-\\xi_{X}\\right)\\left(\\widetilde{P}_{X,t}^{H,CO}\\right)^{1-\\theta_{T}}\\right]^{\\frac{1}{1-\\theta_{T}}}",
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"latex": "P_{X,t}^{H,CO}=\\left[\\xi_{X}\\left(\\left(\\Pi_{X,t-1}^{H,CO}\\right)^{\\chi_{X}}\\overline{\\Pi}^{1-\\chi_{X}}P_{X,t-1}^{H,CO}\\right)^{1-\\theta_{T}}+\\left(1-\\xi_{X}\\right)\\left(\\widetilde{P}_{X,t}^{H,CO}\\right)^{1-\\theta_{T}}\\right]^{\\frac{1}{1-\\theta_{T}}}",
"notation": "latex",
"alt": "P sub X, t to the power of H, CO equals xi sub X of capital pi sub X, t minus 1 to the power of H, CO to the power of chi sub X capital pi bar to the power of 1 minus chi sub X P sub X, t minus 1 to the power of H, CO to the power of 1 minus theta sub T plus 1 minus xi sub X times widetilde P sub X, t to the power of H, CO to the power of 1 minus theta sub T to the power of 1 over 1 minus theta sub T",
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},
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},
{
"id": "p40.15",
"order": 31,
"type": "text",
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"b": 186.9533
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"text": "(46)",
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{
"id": "p40.16",
"order": 32,
"type": "heading",
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"r": 418.9246,
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"text": "B Outline of the model with search frictions",
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{
"id": "p40.17",
"order": 33,
"type": "text",
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"text": "This appendix if based on, and follows closely the model description in Gomes et al. (2023), and the reader should refer to this paper for all the details. The labour market structure with search frictions follows the standard search model of Mortensen and Pissarides (1999), but adds sticky wages by means of staggered wage setting, and a",
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{
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"order": 34,
"type": "link",
"bbox": {
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"b": 112.463
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"text": "Gomes et al.",
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footer (1)
formula (25)
heading (1)
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1 · TEXT
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11 · FORMULA
12 · TEXT
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38 · HEADING
39 · TEXT
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49 · TEXT
50 · FOOTER
Page 41 JSON
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"text": "potential to distinguish wage stickiness of new hires and existing workers, as in Bodart et al. (2006) and De Walque et al. (2009). In addition, it distinguishes between labour market segments for Ricardian and non-Ricardian households.",
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"text": "et al.",
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"text": "The number of workers in segment $s$ ([p41.2.1] for Ricardian and[p41.2.2] for non-Ricardian households) that are employed after the matching process has been completed,[p41.2.3], is:",
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"text": "The probability for a searching worker to find a job is",
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"text": "p_{s,t}^{W}=\\frac{M_{t}}{un_{s,t}}=\\phi_{s,M}(\\frac{vac_{s,t}}{un_{s,t}})^{1-\\mu_{s}}",
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"type": "text",
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"text": "and the probability of a firm finding a worker is",
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{
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"text": "p_{s,t}^{F}=\\frac{M_{s,t}}{vac_{s,t}}=\\phi_{s,M}(\\frac{vac_{s,t}}{un_{s,t}})^{-\\mu_{s}}",
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"latex": "p_{s,t}^{F}=\\frac{M_{s,t}}{v a c_{s,t}}=\\phi_{s,M}(\\frac{v a c_{s,t}}{u n_{s,t}})^{-\\mu_{s}}",
"notation": "latex",
"alt": "p sub s, t to the power of F equals M sub s, t over vac sub s, t equals phi sub s, M of vac sub s, t over un sub s, t to the power of minus mu sub s",
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},
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{
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"type": "text",
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"text": "(50)",
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{
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"order": 32,
"type": "text",
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"text": "The number of unemployed workers who search for work at the beginning of period t is equal to those who were unemployed at the end of period t-1 after the t-1 matching has been completed plus the newly separated workers:",
"children": []
},
{
"id": "p41.16",
"order": 33,
"type": "formula",
"bbox": {
"l": 218.3398,
"t": 348.4676,
"r": 375.7365,
"b": 334.0681
},
"text": "un_{s,t}=une_{s,t-1}+\\delta_{x,s}nde_{s,t-1},",
"attrs": {
"block_label": "display_formula",
"latex": "u n_{s,t}=u n e_{s,t-1}+\\delta_{x,s}n d e_{s,t-1},",
"notation": "latex",
"alt": "un sub s, t equals une sub s, t minus 1 plus delta sub x, s nde sub s, t minus 1,",
"mathml": "<math aria-label=\"un sub s, t equals une sub s, t minus 1 plus delta sub x, s nde sub s, t minus 1,\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>u</mi><msub><mi>n</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>=</mo><mi>u</mi><mi>n</mi><msub><mi>e</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mi>n</mi><mi>d</mi><msub><mi>e</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo></mrow></math>"
},
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"type": "text",
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"text": "(51)",
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},
{
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"order": 35,
"type": "text",
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"text": "Total unemployment in a bloc is a weighted unemployment across labour market segments, where[p41.18.1] is the share of non-Ricardian households:[p41.18.2].",
"children": [
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"id": "p41.18.1",
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"type": "formula",
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"text": "ω",
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"bbox_estimated": true,
"alt": "omega",
"mathml": "<math aria-label=\"omega\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>ω</mi></mrow></math>"
},
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},
{
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"type": "formula",
"bbox": {
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},
"text": "une_{t} = \\omega \\ une_{j,t} + (1 - \\omega)une_{i,t}",
"attrs": {
"latex": "une_{t} = \\omega \\ une_{j,t} + (1 - \\omega)une_{i,t}",
"inline": true,
"text_offset": 144,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "une sub t equals omega une sub j, t plus 1 minus omega une sub i, t",
"mathml": "<math aria-label=\"une sub t equals omega une sub j, t plus 1 minus omega une sub i, t\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>u</mi><mi>n</mi><msub><mi>e</mi><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>ω</mi><mtext> </mtext><mi>u</mi><mi>n</mi><msub><mi>e</mi><mrow><mi>j</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>+</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><mi>ω</mi><mo stretchy=\"false\">)</mo><mi>u</mi><mi>n</mi><msub><mi>e</mi><mrow><mi>i</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></math>"
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},
{
"id": "p41.19",
"order": 38,
"type": "heading",
"bbox": {
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"b": 248.1512
},
"text": "B.1 Value functions",
"attrs": {
"block_label": "paragraph_title"
},
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},
{
"id": "p41.20",
"order": 39,
"type": "text",
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"text": "Value functions for a labour firm Let[p41.20.4] denote the value of a job for a firm employing a worker from household type[p41.20.5], where[p41.20.6] is the renegotiated wage, and i stands for a Ricardian and j for a non-Ricardian household. Following Bodart et al. (2006), it will be convenient to use this value in marginal utility terms, so we define[p41.20.7], where[p41.20.8] is the marginal utility of consumption of household s. The value of a job with a renegotiated wage for a labour firm can then be written as",
"children": [
{
"id": "p41.20.1",
"order": 40,
"type": "link",
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"text": "Bodart",
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{
"id": "p41.20.2",
"order": 41,
"type": "link",
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},
"text": "et al.",
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},
{
"id": "p41.20.3",
"order": 42,
"type": "link",
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"text": "(2006),",
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{
"id": "p41.20.4",
"order": 43,
"type": "formula",
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},
"text": "A^{F}(w_{s,t}^{*})",
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"latex": "A^{F}(w_{s,t}^{*})",
"inline": true,
"text_offset": 37,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "A to the power of F of w sub s, t to the power of *",
"mathml": "<math aria-label=\"A to the power of F of w sub s, t to the power of *\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>A</mi><mrow><mi>F</mi></mrow></msup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo></mrow></math>"
},
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{
"id": "p41.20.5",
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"type": "formula",
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"text": "s \\in [i, j]",
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"latex": "s \\in [i, j]",
"inline": true,
"text_offset": 123,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "s in i, j",
"mathml": "<math aria-label=\"s in i, j\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mi>s</mi><mo>∈</mo><mo stretchy=\"false\">[</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy=\"false\">]</mo></mrow></math>"
},
"children": []
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{
"id": "p41.20.6",
"order": 45,
"type": "formula",
"bbox": {
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"text": "w_{s,j}^{*}",
"attrs": {
"latex": "w_{s,j}^{*}",
"inline": true,
"text_offset": 140,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "w sub s, j to the power of *",
"mathml": "<math aria-label=\"w sub s, j to the power of *\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>j</mi></mrow><mrow><mo>*</mo></mrow></msubsup></mrow></math>"
},
"children": []
},
{
"id": "p41.20.7",
"order": 46,
"type": "formula",
"bbox": {
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"r": 526.6549,
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},
"text": "\\mathcal{A}^{F}(w_{s,t}^{*}) \\equiv u'(c_{s,t})A^{F}(w_{s,t}^{*})",
"attrs": {
"latex": "\\mathcal{A}^{F}(w_{s,t}^{*}) \\equiv u'(c_{s,t})A^{F}(w_{s,t}^{*})",
"inline": true,
"text_offset": 354,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "mathcal A to the power of F of w sub s, t to the power of * equiv u' c sub s, t A to the power of F of w sub s, t to the power of *",
"mathml": "<math aria-label=\"mathcal A to the power of F of w sub s, t to the power of * equiv u' c sub s, t A to the power of F of w sub s, t to the power of *\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>𝒜</mi><mrow><mi>F</mi></mrow></msup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mo>≡</mo><msup><mi>u</mi><mi>′</mi></msup><mo stretchy=\"false\">(</mo><msub><mi>c</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo><msup><mi>A</mi><mrow><mi>F</mi></mrow></msup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo></mrow></math>"
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{
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"text": "u'(c_{s,t})",
"attrs": {
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"inline": true,
"text_offset": 371,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "u' c sub s, t",
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{
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"type": "formula",
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"text": "\\begin{aligned}\\mathcal{A}_{t}^{F}(w_{s,t}^{*})&=u^{\\prime}(c_{s,t})\\left(h_{s,t}^{\\alpha_{H}}x_{s,t}-h_{s,t}w_{s,t}^{*}(1+\\tau_{t}^{wf})\\right)\\\\&+\\beta(1-\\delta_{x,s})\\left[(1-\\xi_{w,s})\\mathcal{A}_{t+1}^{F}(w_{s,t+1}^{*})+\\xi_{w,s}\\mathcal{A}_{t+1}^{F}(w_{s,t}^{*})\\right]\\end{aligned}",
"attrs": {
"block_label": "display_formula",
"latex": "\\begin{aligned}\\mathcal{A}_{t}^{F}(w_{s,t}^{*})&=u^{\\prime}(c_{s,t})\\left(h_{s,t}^{\\alpha_{H}}x_{s,t}-h_{s,t}w_{s,t}^{*}(1+\\tau_{t}^{wf})\\right)\\\\&+\\beta(1-\\delta_{x,s})\\left[(1-\\xi_{w,s})\\mathcal{A}_{t+1}^{F}(w_{s,t+1}^{*})+\\xi_{w,s}\\mathcal{A}_{t+1}^{F}(w_{s,t}^{*})\\right]\\end{aligned}",
"notation": "latex",
"alt": "begin aligned mathcal A sub t to the power of F of w sub s, t to the power of * & equals u to the power of prime of c sub s, t times h sub s, t to the power of alpha sub H x sub s, t minus h sub s, t w sub s, t to the power of * 1 plus tau sub t to the power of wf & plus beta of 1 minus delta sub x, s 1 minus xi sub w, s mathcal A sub t plus 1 to the power of F of w sub s, t plus 1 to the power of * plus xi sub w, s mathcal A sub t plus 1 to the power of F of w sub s, t to the power of * end aligned",
"mathml": "<math aria-label=\"begin aligned mathcal A sub t to the power of F of w sub s, t to the power of * & equals u to the power of prime of c sub s, t times h sub s, t to the power of alpha sub H x sub s, t minus h sub s, t w sub s, t to the power of * 1 plus tau sub t to the power of wf & plus beta of 1 minus delta sub x, s 1 minus xi sub w, s mathcal A sub t plus 1 to the power of F of w sub s, t plus 1 to the power of * plus xi sub w, s mathcal A sub t plus 1 to the power of F of w sub s, t to the power of * end aligned\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mrow><msubsup><mi>𝒜</mi><mrow><mi>t</mi></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mi>&</mi><mo>=</mo><msup><mi>u</mi><mrow><mi>′</mi></mrow></msup><mo stretchy=\"false\">(</mo><msub><mi>c</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>h</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><msub><mi>α</mi><mrow><mi>H</mi></mrow></msub></mrow></msubsup><msub><mi>x</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>−</mo><msub><mi>h</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">(</mo><mn>1</mn><mo>+</mo><msubsup><mi>τ</mi><mrow><mi>t</mi></mrow><mrow><mi>w</mi><mi>f</mi></mrow></msubsup><mo stretchy=\"false\">)</mo><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mspace linebreak=\"newline\" /><mi>&</mi><mo>+</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">[</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mo>+</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">]</mo></mrow></mrow></mrow></math>"
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footer (1)
formula (16)
link (5)
text (11)
1 · TEXT
2
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13 · TEXT
14 · FORMULA
15 · TEXT
16 · TEXT
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33 · FOOTER
Page 42 JSON
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"text": "The term[p42.1.2] denotes the effective hours that a labour firm produces from hours[p42.1.3] supplied by the worker from household s.[p42.1.4] is what the labour packer pays for such unit of labour. The first term in equation 52 therefore measures earnings of the labour firm from selling hours worked. But for these hours it has to pay to the household hourly wage, which is in this case newly renegotiated,[p42.1.5]. Because we assume that labour firms pay some labour taxes (social security contributions), the cost for the labour firm is increased by taxes paid, at the rate[p42.1.6]. In the next period, if the firm and the worker do not separate, which occurs with probability[p42.1.7], two cases can arise. In the first case, which occurs with the probability[p42.1.8], wages are renegotiated and the value of the worker for the labour firm is again the value of a worker with a renegotiated wage, just in the next period,[p42.1.9]. With probability[p42.1.10] wages are not renegotiated and the firm is in next period stuck with the worker value at the current wage,[p42.1.11].",
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{
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{
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"alt": "1 minus xi sub w, s",
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{
"id": "p42.1.10",
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{
"id": "p42.1.11",
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"text": "\\mathcal{A}_{t+1}^{F}(w_{s,t}^{*})",
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},
{
"id": "p42.2",
"order": 13,
"type": "text",
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"text": "The value at $ t+1 $ of the worker with renegotiated wage from time t is",
"children": []
},
{
"id": "p42.3",
"order": 14,
"type": "formula",
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"text": "\\begin{aligned}\\mathcal{A}_{t+1}^{F}(w_{s,t}^{*})&=u^{\\prime}(c_{s,t+1})\\left(h_{s,t+1}^{\\alpha_{H}}x_{s,t+1}-h_{s,t+1}w_{s,t}^{*}\\frac{(1+\\overline{\\pi})P_{t}}{P_{t+1}}(1+\\tau_{t+1}^{wf})\\right)\\\\&+\\beta(1-\\delta_{x,s})\\left[(1-\\xi_{w,s})\\mathcal{A}_{t+2}^{F}(w_{s,t+2}^{*})+\\xi_{w,s}\\mathcal{A}_{t+2}^{F}(w_{s,t}^{*})\\right]\\end{aligned}",
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"notation": "latex",
"alt": "begin aligned mathcal A sub t plus 1 to the power of F of w sub s, t to the power of * & equals u to the power of prime of c sub s, t plus 1 times h sub s, t plus 1 to the power of alpha sub H x sub s, t plus 1 minus h sub s, t plus 1 w sub s, t to the power of * 1 plus pi bar P sub t over P sub t plus 1 times 1 plus tau sub t plus 1 to the power of wf & plus beta of 1 minus delta sub x, s 1 minus xi sub w, s mathcal A sub t plus 2 to the power of F of w sub s, t plus 2 to the power of * plus xi sub w, s mathcal A sub t plus 2 to the power of F of w sub s, t to the power of * end aligned",
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},
"children": []
},
{
"id": "p42.4",
"order": 15,
"type": "text",
"bbox": {
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"r": 524.0156,
"b": 531.1011
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"text": "(53)",
"attrs": {
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{
"id": "p42.5",
"order": 16,
"type": "text",
"bbox": {
"l": 67.9013,
"t": 503.982,
"r": 525.4552,
"b": 475.4231
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"text": "The wage from the previous period has been indexed by the ratio of trend inflation[p42.5.1] and the price level growth,[p42.5.2].",
"children": [
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"id": "p42.5.1",
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"text": "\\bar{\\pi}",
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"notation": "latex",
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"children": []
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{
"id": "p42.5.2",
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"type": "formula",
"bbox": {
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"text": "P_{t}/P_{t+1} = (1 + \\pi_{t+1})",
"attrs": {
"latex": "P_{t}/P_{t+1} = (1 + \\pi_{t+1})",
"inline": true,
"text_offset": 119,
"text_span": 9,
"bbox_estimated": true,
"notation": "latex",
"alt": "P sub t over P sub t plus 1 equals 1 plus pi sub t plus 1",
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"children": []
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},
{
"id": "p42.6",
"order": 19,
"type": "text",
"bbox": {
"l": 68.1412,
"t": 474.7031,
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"b": 446.3841
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"text": "If we substitute equation $ 53 $ into equation $ 52 $, and do this for every future period, we arrive at the following expression:",
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{
"id": "p42.6.1",
"order": 20,
"type": "link",
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"text": "53",
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"id": "p42.6.2",
"order": 21,
"type": "link",
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"t": 474.01,
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"text": "52,",
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"children": []
}
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},
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"order": 22,
"type": "formula",
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"text": "\\begin{aligned}\\mathcal{A}_{t}^{F}(w_{s,t}^{*})&=\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}u^{\\prime}(c_{s,t+k})h_{s,t+k}^{\\alpha_{H}}x_{s,t+k}\\\\&-w_{s,t}^{*}\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}u^{\\prime}(c_{s,t+k})\\frac{(1+\\overline{\\pi})^{k}P_{t}}{P_{t+k}}h_{s,t+k}(1+\\tau_{t+k}^{wf})\\\\&+\\sum_{k=0}^{\\infty}\\beta(1-\\delta_{x,s})(1-\\xi_{w,s})\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}\\mathcal{A}_{t+k+1}^{F}(w_{s,t+k+1}^{*})\\end{aligned}",
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"latex": "\\begin{aligned}\\mathcal{A}_{t}^{F}(w_{s,t}^{*})&=\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}u^{\\prime}(c_{s,t+k})h_{s,t+k}^{\\alpha_{H}}x_{s,t+k}\\\\&-w_{s,t}^{*}\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}u^{\\prime}(c_{s,t+k})\\frac{(1+\\overline{\\pi})^{k}P_{t}}{P_{t+k}}h_{s,t+k}(1+\\tau_{t+k}^{wf})\\\\&+\\sum_{k=0}^{\\infty}\\beta(1-\\delta_{x,s})(1-\\xi_{w,s})\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}\\mathcal{A}_{t+k+1}^{F}(w_{s,t+k+1}^{*})\\end{aligned}",
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"alt": "begin aligned mathcal A sub t to the power of F of w sub s, t to the power of * & equals the sum of sub k equals 0 to the power of infinity beta of 1 minus delta sub x, s xi sub w, s to the power of k u to the power of prime of c sub s, t plus k h sub s, t plus k to the power of alpha sub H x sub s, t plus k & minus w sub s, t to the power of * the sum of sub k equals 0 to the power of infinity beta of 1 minus delta sub x, s xi sub w, s to the power of k u to the power of prime of c sub s, t plus k times 1 plus pi bar to the power of k P sub t over P sub t plus k h sub s, t plus k of 1 plus tau sub t plus k to the power of wf & plus the sum of sub k equals 0 to the power of infinity beta of 1 minus delta sub x, s times 1 minus xi sub w, s beta of 1 minus delta sub x, s xi sub w, s to the power of k mathcal A sub t plus k plus 1 to the power of F of w sub s, t plus k plus 1 to the power of * end aligned",
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"text": "(54)",
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"text": "One can define auxiliary variables and write the infinite sums in recursive form. For the first line in equation 54, define",
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"text": "S_{s,t}^{x}\\equiv\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}u^{\\prime}(c_{s,t+k})h_{s,t+k}^{\\alpha_{H}}x_{s,t+k}=u^{\\prime}(c_{s,t})h_{s,t}^{\\alpha_{H}}x_{s,t}+\\beta(1-\\delta_{x,s})\\xi_{w,s}S_{s,t+1}^{x}",
"attrs": {
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"latex": "S_{s,t}^{x}\\equiv\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}u^{\\prime}(c_{s,t+k})h_{s,t+k}^{\\alpha_{H}}x_{s,t+k}=u^{\\prime}(c_{s,t})h_{s,t}^{\\alpha_{H}}x_{s,t}+\\beta(1-\\delta_{x,s})\\xi_{w,s}S_{s,t+1}^{x}",
"notation": "latex",
"alt": "S sub s, t to the power of x equiv the sum of sub k equals 0 to the power of infinity beta of 1 minus delta sub x, s xi sub w, s to the power of k u to the power of prime of c sub s, t plus k h sub s, t plus k to the power of alpha sub H x sub s, t plus k equals u to the power of prime of c sub s, t h sub s, t to the power of alpha sub H x sub s, t plus beta of 1 minus delta sub x, s xi sub w, s S sub s, t plus 1 to the power of x",
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{
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"text": "(55)",
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"text": "For the second line, define",
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"text": "\\begin{align*}S_{s,t}^{wf}&\\equiv\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k} u^{\\prime}(c_{s,t+k})h_{s,t}\\frac{(1+\\overline{\\pi})^{k} P_{t}}{P_{t+k}}(1+\\tau_{t}^{wf})\\\\&=u^{\\prime}(c_{s,t})h_{s,t}(1+\\tau_{t}^{wf})+\\beta(1-\\delta_{x,s})\\frac{(1+\\overline{\\pi})}{(1+\\pi_{t+1})}\\xi_{w,s}S_{s,t+1}^{wf}\\end{align*}",
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"latex": "\\begin{align*}S_{s,t}^{wf}&\\equiv\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k} u^{\\prime}(c_{s,t+k})h_{s,t}\\frac{(1+\\overline{\\pi})^{k} P_{t}}{P_{t+k}}(1+\\tau_{t}^{wf})\\\\&=u^{\\prime}(c_{s,t})h_{s,t}(1+\\tau_{t}^{wf})+\\beta(1-\\delta_{x,s})\\frac{(1+\\overline{\\pi})}{(1+\\pi_{t+1})}\\xi_{w,s}S_{s,t+1}^{wf}\\end{align*}",
"notation": "latex",
"alt": "begin align* S sub s, t to the power of wf & equiv the sum of sub k equals 0 to the power of infinity beta of 1 minus delta sub x, s xi sub w, s to the power of k u to the power of prime of c sub s, t plus k h sub s, t of 1 plus pi bar to the power of k P sub t over P sub t plus k of 1 plus tau sub t to the power of wf & equals u to the power of prime of c sub s, t h sub s, t of 1 plus tau sub t to the power of wf plus beta of 1 minus delta sub x, s times 1 plus pi bar over of 1 plus pi sub t plus 1 xi sub w, s S sub s, t plus 1 to the power of wf end align*",
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"text": "Using these definitions, we can simplify equation 54 to",
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footer (1)
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1 · FORMULA
2 · TEXT
3 · TEXT
4 · TEXT
5 · FORMULA
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8 · FORMULA
9 · TEXT
10 · TEXT
11
12
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14 · FORMULA
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17 · FORMULA
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20 · FOOTER
Page 43 JSON
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"text": "\\begin{align*}\\mathcal{A}_{t}^{F}(w_{s,t}^{*})&=\\left(S_{s,t}^{x}-S_{s,t}^{wf}w_{s,t}^{*}\\right)+\\sum_{k=0}^{\\infty}\\beta(1-\\delta_{x,s})(1-\\xi_{w,s})\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}\\mathcal{A}_{t+k+1}^{F}(w_{s,t+k+1}^{*})\\\\&=\\left(S_{s,t}^{x}-S_{s,t}^{wf}w_{s,t}^{*}\\right)-\\beta(1-\\delta_{x,s})\\xi_{w,s}\\left(S_{s,t+1}^{x}-S_{s,t+1}^{wf}w_{s,t+1}^{*}\\right)+\\beta(1-\\delta_{x,s})\\mathcal{A}_{t+1}^{F}(w_{s,t+1}^{*})\\end{align*}",
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"latex": "\\begin{align*}\\mathcal{A}_{t}^{F}(w_{s,t}^{*})&=\\left(S_{s,t}^{x}-S_{s,t}^{wf}w_{s,t}^{*}\\right)+\\sum_{k=0}^{\\infty}\\beta(1-\\delta_{x,s})(1-\\xi_{w,s})\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}\\mathcal{A}_{t+k+1}^{F}(w_{s,t+k+1}^{*})\\\\&=\\left(S_{s,t}^{x}-S_{s,t}^{wf}w_{s,t}^{*}\\right)-\\beta(1-\\delta_{x,s})\\xi_{w,s}\\left(S_{s,t+1}^{x}-S_{s,t+1}^{wf}w_{s,t+1}^{*}\\right)+\\beta(1-\\delta_{x,s})\\mathcal{A}_{t+1}^{F}(w_{s,t+1}^{*})\\end{align*}",
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"alt": "begin align* mathcal A sub t to the power of F of w sub s, t to the power of * & equals S sub s, t to the power of x minus S sub s, t to the power of wf w sub s, t to the power of * plus the sum of sub k equals 0 to the power of infinity beta of 1 minus delta sub x, s times 1 minus xi sub w, s beta of 1 minus delta sub x, s xi sub w, s to the power of k mathcal A sub t plus k plus 1 to the power of F of w sub s, t plus k plus 1 to the power of * & equals S sub s, t to the power of x minus S sub s, t to the power of wf w sub s, t to the power of * minus beta of 1 minus delta sub x, s xi sub w, s of S sub s, t plus 1 to the power of x minus S sub s, t plus 1 to the power of wf w sub s, t plus 1 to the power of * plus beta of 1 minus delta sub x, s mathcal A sub t plus 1 to the power of F of w sub s, t plus 1 to the power of * end align*",
"mathml": "<math aria-label=\"begin align* mathcal A sub t to the power of F of w sub s, t to the power of * & equals S sub s, t to the power of x minus S sub s, t to the power of wf w sub s, t to the power of * plus the sum of sub k equals 0 to the power of infinity beta of 1 minus delta sub x, s times 1 minus xi sub w, s beta of 1 minus delta sub x, s xi sub w, s to the power of k mathcal A sub t plus k plus 1 to the power of F of w sub s, t plus k plus 1 to the power of * & equals S sub s, t to the power of x minus S sub s, t to the power of wf w sub s, t to the power of * minus beta of 1 minus delta sub x, s xi sub w, s of S sub s, t plus 1 to the power of x minus S sub s, t plus 1 to the power of wf w sub s, t plus 1 to the power of * plus beta of 1 minus delta sub x, s mathcal A sub t plus 1 to the power of F of w sub s, t plus 1 to the power of * end align*\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mtable displaystyle=\"true\" rowspacing=\"3pt\" columnspacing=\"0em 2em\"><mtr><mtd columnalign=\"right\"><msubsup><mi>𝒜</mi><mrow><mi>t</mi></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo></mtd><mtd columnalign=\"left\"><mi /><mo>=</mo><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>x</mi></mrow></msubsup><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>w</mi><mi>f</mi></mrow></msubsup><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">[</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">]</mo></mrow><mrow><mi>k</mi></mrow></msup><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo></mtd></mtr><mtr><mtd columnalign=\"right\" /><mtd columnalign=\"left\"><mi /><mo>=</mo><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>x</mi></mrow></msubsup><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>w</mi><mi>f</mi></mrow></msubsup><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mo>−</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi></mrow></msubsup><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>w</mi><mi>f</mi></mrow></msubsup><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mo>+</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo></mtd></mtr></mtable></mrow></math>"
},
"children": []
},
{
"id": "p43.2",
"order": 2,
"type": "text",
"bbox": {
"l": 502.4216,
"t": 685.1756,
"r": 524.4955,
"b": 671.9761
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"text": "(57)",
"attrs": {
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"block_label": "formula_number"
},
"children": []
},
{
"id": "p43.3",
"order": 3,
"type": "text",
"bbox": {
"l": 67.9013,
"t": 659.4965,
"r": 302.7966,
"b": 646.297
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"text": "After rearranging, we obtain the second line.",
"children": []
},
{
"id": "p43.4",
"order": 4,
"type": "text",
"bbox": {
"l": 66.7016,
"t": 644.857,
"r": 525.9351,
"b": 616.058
},
"text": "We can then similarly define the value of a worker with an average wage for a labour firm:",
"children": []
},
{
"id": "p43.5",
"order": 5,
"type": "formula",
"bbox": {
"l": 135.5627,
"t": 593.0189,
"r": 457.314,
"b": 551.0204
},
"text": "\\begin{aligned}\\mathcal{A}_{t}^{F}(w_{s,t})&=u^{\\prime}(c_{s,t})\\left(h_{s,t}^{\\alpha_{H}}x_{s,t}-h_{s,t}w_{s,t}(1+\\tau_{t}^{wf})\\right)\\\\&+\\beta(1-\\delta_{x,s})\\left[(1-\\xi_{w,s})\\mathcal{A}_{t+1}^{F}(w_{s,t+1}^{*})+\\xi_{w,s}\\mathcal{A}_{t+1}^{F}(w_{s,t})\\right]\\end{aligned}",
"attrs": {
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"latex": "\\begin{aligned}\\mathcal{A}_{t}^{F}(w_{s,t})&=u^{\\prime}(c_{s,t})\\left(h_{s,t}^{\\alpha_{H}}x_{s,t}-h_{s,t}w_{s,t}(1+\\tau_{t}^{wf})\\right)\\\\&+\\beta(1-\\delta_{x,s})\\left[(1-\\xi_{w,s})\\mathcal{A}_{t+1}^{F}(w_{s,t+1}^{*})+\\xi_{w,s}\\mathcal{A}_{t+1}^{F}(w_{s,t})\\right]\\end{aligned}",
"notation": "latex",
"alt": "begin aligned mathcal A sub t to the power of F of w sub s, t & equals u to the power of prime of c sub s, t times h sub s, t to the power of alpha sub H x sub s, t minus h sub s, t w sub s, t of 1 plus tau sub t to the power of wf & plus beta of 1 minus delta sub x, s 1 minus xi sub w, s mathcal A sub t plus 1 to the power of F of w sub s, t plus 1 to the power of * plus xi sub w, s mathcal A sub t plus 1 to the power of F of w sub s, t end aligned",
"mathml": "<math aria-label=\"begin aligned mathcal A sub t to the power of F of w sub s, t & equals u to the power of prime of c sub s, t times h sub s, t to the power of alpha sub H x sub s, t minus h sub s, t w sub s, t of 1 plus tau sub t to the power of wf & plus beta of 1 minus delta sub x, s 1 minus xi sub w, s mathcal A sub t plus 1 to the power of F of w sub s, t plus 1 to the power of * plus xi sub w, s mathcal A sub t plus 1 to the power of F of w sub s, t end aligned\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mrow><msubsup><mi>𝒜</mi><mrow><mi>t</mi></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo><mi>&</mi><mo>=</mo><msup><mi>u</mi><mrow><mi>′</mi></mrow></msup><mo stretchy=\"false\">(</mo><msub><mi>c</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>h</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><msub><mi>α</mi><mrow><mi>H</mi></mrow></msub></mrow></msubsup><msub><mi>x</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>−</mo><msub><mi>h</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">(</mo><mn>1</mn><mo>+</mo><msubsup><mi>τ</mi><mrow><mi>t</mi></mrow><mrow><mi>w</mi><mi>f</mi></mrow></msubsup><mo stretchy=\"false\">)</mo><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mspace linebreak=\"newline\" /><mi>&</mi><mo>+</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">[</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mo>+</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">]</mo></mrow></mrow></mrow></math>"
},
"children": []
},
{
"id": "p43.6",
"order": 6,
"type": "text",
"bbox": {
"l": 502.1816,
"t": 580.2993,
"r": 524.4955,
"b": 565.8998
},
"text": "(58)",
"attrs": {
"raw_type": "FORMULA_NUMBER",
"block_label": "formula_number"
},
"children": []
},
{
"id": "p43.7",
"order": 7,
"type": "text",
"bbox": {
"l": 85.6564,
"t": 541.4207,
"r": 425.8827,
"b": 526.3012
},
"text": "Following the same steps as above, we obtain, after some algebra",
"children": []
},
{
"id": "p43.8",
"order": 8,
"type": "formula",
"bbox": {
"l": 68.3812,
"t": 505.422,
"r": 547.0493,
"b": 478.7829
},
"text": "\\mathcal{A}_{t}^{F}(w_{s,t})=\\left(S_{s,t}^{x}-S_{s,t}^{wf}w_{s,t}\\right)-\\beta(1-\\delta_{x,s})\\xi_{w,s}\\left(S_{s,t+1}^{x}-S_{s,t+1}^{wf}w_{s,t+1}\\right)+\\beta(1-\\delta_{x,s})\\mathcal{A}_{t+1}^{F}(w_{s,t+1})",
"attrs": {
"block_label": "display_formula",
"latex": "\\mathcal{A}_{t}^{F}(w_{s,t})=\\left(S_{s,t}^{x}-S_{s,t}^{wf}w_{s,t}\\right)-\\beta(1-\\delta_{x,s})\\xi_{w,s}\\left(S_{s,t+1}^{x}-S_{s,t+1}^{wf}w_{s,t+1}\\right)+\\beta(1-\\delta_{x,s})\\mathcal{A}_{t+1}^{F}(w_{s,t+1})",
"notation": "latex",
"alt": "mathcal A sub t to the power of F of w sub s, t equals S sub s, t to the power of x minus S sub s, t to the power of wf w sub s, t minus beta of 1 minus delta sub x, s xi sub w, s of S sub s, t plus 1 to the power of x minus S sub s, t plus 1 to the power of wf w sub s, t plus 1 plus beta of 1 minus delta sub x, s mathcal A sub t plus 1 to the power of F of w sub s, t plus 1",
"mathml": "<math aria-label=\"mathcal A sub t to the power of F of w sub s, t equals S sub s, t to the power of x minus S sub s, t to the power of wf w sub s, t minus beta of 1 minus delta sub x, s xi sub w, s of S sub s, t plus 1 to the power of x minus S sub s, t plus 1 to the power of wf w sub s, t plus 1 plus beta of 1 minus delta sub x, s mathcal A sub t plus 1 to the power of F of w sub s, t plus 1\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>𝒜</mi><mrow><mi>t</mi></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo>=</mo><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>x</mi></mrow></msubsup><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>w</mi><mi>f</mi></mrow></msubsup><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mo>−</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi></mrow></msubsup><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>w</mi><mi>f</mi></mrow></msubsup><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mo>+</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></mrow></math>"
},
"children": []
},
{
"id": "p43.9",
"order": 9,
"type": "text",
"bbox": {
"l": 502.4216,
"t": 481.4228,
"r": 524.4955,
"b": 467.7433
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"text": "(59)",
"attrs": {
"raw_type": "FORMULA_NUMBER",
"block_label": "formula_number"
},
"children": []
},
{
"id": "p43.10",
"order": 10,
"type": "text",
"bbox": {
"l": 66.4617,
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"text": "Free entry condition A firm posting a vacancy for household type s must pay a per-period constant cost[p43.10.1] for having a vacancy open. If[p43.10.2] denotes the probability that a firm cannot renegotiate the wage for a newly hired worker from household type s, then the value of employing a new worker is, in monetary terms (recall,[p43.10.3], and the same for the value at average wage), equal to the weighted average of the value of a worker at a newly-renegotiated job and the value of a worker hired at average wage. The free-entry condition is:",
"children": [
{
"id": "p43.10.1",
"order": 11,
"type": "formula",
"bbox": {
"l": 66.4617,
"t": 447.5841,
"r": 526.6549,
"b": 345.3477
},
"text": "\\psi_{s}",
"attrs": {
"latex": "\\psi_{s}",
"inline": true,
"text_offset": 102,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "psi sub s",
"mathml": "<math aria-label=\"psi sub s\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>ψ</mi><mrow><mi>s</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p43.10.2",
"order": 12,
"type": "formula",
"bbox": {
"l": 66.4617,
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"b": 345.3477
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"text": "\\kappa_{w,s}",
"attrs": {
"latex": "\\kappa_{w,s}",
"inline": true,
"text_offset": 142,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "kappa sub w, s",
"mathml": "<math aria-label=\"kappa sub w, s\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>κ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub></mrow></math>"
},
"children": []
},
{
"id": "p43.10.3",
"order": 13,
"type": "formula",
"bbox": {
"l": 66.4617,
"t": 447.5841,
"r": 526.6549,
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"text": "\\mathcal{A}^{F}(w_{s,t}^{*}) \\equiv u'(c_{s,t})A^{F}(w_{s,t}^{*})",
"attrs": {
"latex": "\\mathcal{A}^{F}(w_{s,t}^{*}) \\equiv u'(c_{s,t})A^{F}(w_{s,t}^{*})",
"inline": true,
"text_offset": 336,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "mathcal A to the power of F of w sub s, t to the power of * equiv u' c sub s, t A to the power of F of w sub s, t to the power of *",
"mathml": "<math aria-label=\"mathcal A to the power of F of w sub s, t to the power of * equiv u' c sub s, t A to the power of F of w sub s, t to the power of *\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msup><mi>𝒜</mi><mrow><mi>F</mi></mrow></msup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mo>≡</mo><msup><mi>u</mi><mi>′</mi></msup><mo stretchy=\"false\">(</mo><msub><mi>c</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo><msup><mi>A</mi><mrow><mi>F</mi></mrow></msup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo></mrow></math>"
},
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}
]
},
{
"id": "p43.11",
"order": 14,
"type": "formula",
"bbox": {
"l": 142.0409,
"t": 335.2681,
"r": 451.0757,
"b": 305.2691
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"text": "\\psi_{s}=p_{t}^{F}\\beta\\frac{u^{\\prime}(c_{s,t+1})}{u^{\\prime}(c_{s,t})}\\left[(1-\\kappa_{w,s})\\mathcal{A}_{t}^{F}(w_{s,t+1}^{*})+\\kappa_{w,s}\\mathcal{A}_{t}^{F}(w_{s,t+1})\\right].",
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"text": "Value functions for a worker We have two types of value functions, one for a newly-renegotiated wage and one for the average wage, for each type of household. The value of a job, net of the value of unemployment, for a worker with newly-renegotiated wage is",
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"text": "\\begin{aligned}\\mathcal{A}_{t}^{H}(w_{s,t}^{*})&=u^{\\prime}(c_{s,t})\\left(h_{s,t}w_{s,t}^{*}(1-\\tau_{t}^{wh})-b_{s,t}\\right)-\\chi\\frac{h_{s,t}^{1+\\varphi}}{1+\\varphi}\\\\&+\\beta(1-\\delta_{x,s})\\left[(1-\\xi_{w,s})\\mathcal{A}_{t+1}^{H}(w_{s,t+1}^{*})+\\xi_{w,s}\\mathcal{A}_{t+1}^{H}(w_{s,t}^{*})\\right]\\\\&-\\beta p_{s,t}^{W}\\left[(1-\\kappa_{w,s})\\mathcal{A}_{t+1}^{H}(w_{s,t+1}^{*})+\\kappa_{w,s}\\mathcal{A}_{t+1}^{H}(w_{s,t+1})\\right]\\end{aligned}",
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"text": "We can define additional auxiliary variables to sum the utility obtained from wages, unemployment benefits, and the disutility of labour terms,",
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footer (1)
formula (11)
heading (1)
text (15)
1 · FORMULA
2 · TEXT
3 · FORMULA
4 · TEXT
5 · FORMULA
6 · TEXT
7 · TEXT
8 · FORMULA
9 · TEXT
10 · TEXT
11 · FORMULA
12 · TEXT
13 · HEADING
14 · TEXT
15
16 · FORMULA
17 · TEXT
18 · TEXT
19 · FORMULA
20 · TEXT
21 · TEXT
22
23 · TEXT
24 · FORMULA
25 · TEXT
26 · TEXT
27
28 · FOOTER
Page 44 JSON
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"text": "\\begin{align*}S_{s,t}^{wh}&\\equiv\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k} u^{\\prime}(c_{s,t+k})h_{s,t}\\frac{(1+\\overline{\\pi})^{k} P_{t}}{P_{t+k}}(1-\\tau_{t}^{wh})\\\\&=u^{\\prime}(c_{s,t})h_{s,t}(1-\\tau_{t}^{wh})+\\beta(1-\\delta_{x,s})\\frac{(1+\\overline{\\pi})}{(1+\\pi_{t+1})}\\xi_{w,s}S_{s,t+1}^{wh},\\end{align*}",
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"text": "S_{s,t}^{b}=u^{\\prime}(c_{s,t+k})b_{s,t}+\\beta(1-\\delta_{x,s})\\xi_{w,s}\\frac{(1+\\overline{{\\pi}})}{(1+\\pi_{t+1})}S_{s,t+1}^{b},",
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{
"id": "p44.5",
"order": 5,
"type": "formula",
"bbox": {
"l": 201.5445,
"t": 633.5774,
"r": 393.2517,
"b": 601.4186
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"text": "S_{s,t}^{h}=\\chi\\frac{h_{s,t}^{1+\\varphi}}{1+\\varphi}+\\beta(1-\\delta_{x,s})\\xi_{w,s}S_{s,t+1}^{h},",
"attrs": {
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"latex": "S_{s,t}^{h}=\\chi\\frac{h_{s,t}^{1+\\varphi}}{1+\\varphi}+\\beta(1-\\delta_{x,s})\\xi_{w,s}S_{s,t+1}^{h},",
"notation": "latex",
"alt": "S sub s, t to the power of h equals chi h sub s, t to the power of 1 plus varphi over 1 plus varphi plus beta of 1 minus delta sub x, s xi sub w, s S sub s, t plus 1 to the power of h,",
"mathml": "<math aria-label=\"S sub s, t to the power of h equals chi h sub s, t to the power of 1 plus varphi over 1 plus varphi plus beta of 1 minus delta sub x, s xi sub w, s S sub s, t plus 1 to the power of h,\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>h</mi></mrow></msubsup><mo>=</mo><mi>χ</mi><mfrac><mrow><msubsup><mi>h</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>φ</mi></mrow></msubsup></mrow><mrow><mn>1</mn><mo>+</mo><mi>φ</mi></mrow></mfrac><mo>+</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>h</mi></mrow></msubsup><mo>,</mo></mrow></math>"
},
"children": []
},
{
"id": "p44.6",
"order": 6,
"type": "text",
"bbox": {
"l": 502.6615,
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"r": 524.2556,
"b": 609.8183
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"text": "(64)",
"attrs": {
"raw_type": "FORMULA_NUMBER",
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"children": []
},
{
"id": "p44.7",
"order": 7,
"type": "text",
"bbox": {
"l": 68.6211,
"t": 597.3387,
"r": 394.6913,
"b": 583.6592
},
"text": "then the net value of a job for a new wage for the household is:",
"children": []
},
{
"id": "p44.8",
"order": 8,
"type": "formula",
"bbox": {
"l": 94.294,
"t": 563.7399,
"r": 478.1883,
"b": 508.0619
},
"text": "\\begin{aligned}\\mathcal{A}_{t}^{H}(w_{s,t}^{*})&=S_{s,t}^{w}w_{s,t}^{*}-S_{s,t}^{b}-\\beta(1-\\delta_{x,s})\\xi_{w,s}\\left(S_{t+1}^{w}w_{s,t+1}^{*}-S_{s,t+1}^{b}\\right)\\\\&-S_{s,t}^{h}+\\beta(1-\\delta_{x,s})\\xi_{w,s}S_{s,t+1}^{h}\\\\&+\\beta\\left[1-\\delta_{x,s}-(1-\\kappa_{w,s})p_{s,t}^{W}\\right]\\mathcal{A}_{t+1}^{H}(w_{s,t+1}^{*})-\\beta\\kappa_{w,s}p_{s,t}^{W}\\mathcal{A}_{t+1}^{H}(w_{s,t+1})\\end{aligned}",
"attrs": {
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"latex": "\\begin{aligned}\\mathcal{A}_{t}^{H}(w_{s,t}^{*})&=S_{s,t}^{w}w_{s,t}^{*}-S_{s,t}^{b}-\\beta(1-\\delta_{x,s})\\xi_{w,s}\\left(S_{t+1}^{w}w_{s,t+1}^{*}-S_{s,t+1}^{b}\\right)\\\\&-S_{s,t}^{h}+\\beta(1-\\delta_{x,s})\\xi_{w,s}S_{s,t+1}^{h}\\\\&+\\beta\\left[1-\\delta_{x,s}-(1-\\kappa_{w,s})p_{s,t}^{W}\\right]\\mathcal{A}_{t+1}^{H}(w_{s,t+1}^{*})-\\beta\\kappa_{w,s}p_{s,t}^{W}\\mathcal{A}_{t+1}^{H}(w_{s,t+1})\\end{aligned}",
"notation": "latex",
"alt": "begin aligned mathcal A sub t to the power of H of w sub s, t to the power of * & equals S sub s, t to the power of w w sub s, t to the power of * minus S sub s, t to the power of b minus beta of 1 minus delta sub x, s xi sub w, s of S sub t plus 1 to the power of w w sub s, t plus 1 to the power of * minus S sub s, t plus 1 to the power of b & minus S sub s, t to the power of h plus beta of 1 minus delta sub x, s xi sub w, s S sub s, t plus 1 to the power of h & plus beta 1 minus delta sub x, s minus 1 minus kappa sub w, s p sub s, t to the power of W mathcal A sub t plus 1 to the power of H of w sub s, t plus 1 to the power of * minus beta kappa sub w, s p sub s, t to the power of W mathcal A sub t plus 1 to the power of H of w sub s, t plus 1 end aligned",
"mathml": "<math aria-label=\"begin aligned mathcal A sub t to the power of H of w sub s, t to the power of * & equals S sub s, t to the power of w w sub s, t to the power of * minus S sub s, t to the power of b minus beta of 1 minus delta sub x, s xi sub w, s of S sub t plus 1 to the power of w w sub s, t plus 1 to the power of * minus S sub s, t plus 1 to the power of b & minus S sub s, t to the power of h plus beta of 1 minus delta sub x, s xi sub w, s S sub s, t plus 1 to the power of h & plus beta 1 minus delta sub x, s minus 1 minus kappa sub w, s p sub s, t to the power of W mathcal A sub t plus 1 to the power of H of w sub s, t plus 1 to the power of * minus beta kappa sub w, s p sub s, t to the power of W mathcal A sub t plus 1 to the power of H of w sub s, t plus 1 end aligned\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mrow><msubsup><mi>𝒜</mi><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mi>&</mi><mo>=</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>w</mi></mrow></msubsup><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>b</mi></mrow></msubsup><mo>−</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>S</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>w</mi></mrow></msubsup><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>b</mi></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mspace linebreak=\"newline\" /><mi>&</mi><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>h</mi></mrow></msubsup><mo>+</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>h</mi></mrow></msubsup><mspace linebreak=\"newline\" /><mi>&</mi><mo>+</mo><mi>β</mi><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">[</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo>−</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>κ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msubsup><mi>p</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>W</mi></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">]</mo></mrow><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>H</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mo>−</mo><mi>β</mi><msub><mi>κ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><msubsup><mi>p</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>W</mi></mrow></msubsup><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>H</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></mrow></mrow></math>"
},
"children": []
},
{
"id": "p44.9",
"order": 9,
"type": "text",
"bbox": {
"l": 502.6615,
"t": 543.8206,
"r": 524.4955,
"b": 529.9011
},
"text": "(65)",
"attrs": {
"raw_type": "FORMULA_NUMBER",
"block_label": "formula_number"
},
"children": []
},
{
"id": "p44.10",
"order": 10,
"type": "text",
"bbox": {
"l": 86.3762,
"t": 500.8622,
"r": 394.9312,
"b": 486.9426
},
"text": "The value of a job for an average wage for the household is:",
"children": []
},
{
"id": "p44.11",
"order": 11,
"type": "formula",
"bbox": {
"l": 131.0039,
"t": 467.5033,
"r": 461.3929,
"b": 392.866
},
"text": "\\begin{aligned}\\mathcal{A}_{t}^{H}(w_{s,t})&=S_{s,t}^{w}w_{s,t}-S_{s,t}^{b}-\\beta(1-\\delta_{x,s})\\xi_{w,s}\\left(S_{s,t+1}^{w}w_{s,t+1}-S_{s,t+1}^{b}\\right)\\\\&-S_{s,t}^{h}+\\beta(1-\\delta_{x,s})\\xi_{w,s}S_{s,t+1}^{h}\\\\&+\\beta\\left[(1-\\delta_{x,s})(1-\\xi_{w,s})-(1-\\kappa_{w,s})p_{s,t}^{W}\\right]\\mathcal{A}_{t+1}^{H}(w_{s,t+1}^{*})\\\\&+\\beta\\left[(1-\\delta_{x,s})\\xi_{w,s}-\\kappa_{w,s}p_{s,t}^{W}\\right]\\mathcal{A}_{t+1}^{H}(w_{s,t+1})\\\\ \\end{aligned}",
"attrs": {
"block_label": "display_formula",
"latex": "\\begin{aligned}\\mathcal{A}_{t}^{H}(w_{s,t})&=S_{s,t}^{w}w_{s,t}-S_{s,t}^{b}-\\beta(1-\\delta_{x,s})\\xi_{w,s}\\left(S_{s,t+1}^{w}w_{s,t+1}-S_{s,t+1}^{b}\\right)\\\\&-S_{s,t}^{h}+\\beta(1-\\delta_{x,s})\\xi_{w,s}S_{s,t+1}^{h}\\\\&+\\beta\\left[(1-\\delta_{x,s})(1-\\xi_{w,s})-(1-\\kappa_{w,s})p_{s,t}^{W}\\right]\\mathcal{A}_{t+1}^{H}(w_{s,t+1}^{*})\\\\&+\\beta\\left[(1-\\delta_{x,s})\\xi_{w,s}-\\kappa_{w,s}p_{s,t}^{W}\\right]\\mathcal{A}_{t+1}^{H}(w_{s,t+1})\\\\ \\end{aligned}",
"notation": "latex",
"alt": "begin aligned mathcal A sub t to the power of H of w sub s, t & equals S sub s, t to the power of w w sub s, t minus S sub s, t to the power of b minus beta of 1 minus delta sub x, s xi sub w, s of S sub s, t plus 1 to the power of w w sub s, t plus 1 minus S sub s, t plus 1 to the power of b & minus S sub s, t to the power of h plus beta of 1 minus delta sub x, s xi sub w, s S sub s, t plus 1 to the power of h & plus beta 1 minus delta sub x, s times 1 minus xi sub w, s minus 1 minus kappa sub w, s p sub s, t to the power of W mathcal A sub t plus 1 to the power of H of w sub s, t plus 1 to the power of * & plus beta 1 minus delta sub x, s xi sub w, s minus kappa sub w, s p sub s, t to the power of W mathcal A sub t plus 1 to the power of H of w sub s, t plus 1 end aligned",
"mathml": "<math aria-label=\"begin aligned mathcal A sub t to the power of H of w sub s, t & equals S sub s, t to the power of w w sub s, t minus S sub s, t to the power of b minus beta of 1 minus delta sub x, s xi sub w, s of S sub s, t plus 1 to the power of w w sub s, t plus 1 minus S sub s, t plus 1 to the power of b & minus S sub s, t to the power of h plus beta of 1 minus delta sub x, s xi sub w, s S sub s, t plus 1 to the power of h & plus beta 1 minus delta sub x, s times 1 minus xi sub w, s minus 1 minus kappa sub w, s p sub s, t to the power of W mathcal A sub t plus 1 to the power of H of w sub s, t plus 1 to the power of * & plus beta 1 minus delta sub x, s xi sub w, s minus kappa sub w, s p sub s, t to the power of W mathcal A sub t plus 1 to the power of H of w sub s, t plus 1 end aligned\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><mrow><msubsup><mi>𝒜</mi><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo><mi>&</mi><mo>=</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>w</mi></mrow></msubsup><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>b</mi></mrow></msubsup><mo>−</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>w</mi></mrow></msubsup><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>b</mi></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mspace linebreak=\"newline\" /><mi>&</mi><mo>−</mo><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>h</mi></mrow></msubsup><mo>+</mo><mi>β</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><msubsup><mi>S</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>h</mi></mrow></msubsup><mspace linebreak=\"newline\" /><mi>&</mi><mo>+</mo><mi>β</mi><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">[</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo>−</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>κ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msubsup><mi>p</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>W</mi></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">]</mo></mrow><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>H</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mspace linebreak=\"newline\" /><mi>&</mi><mo>+</mo><mi>β</mi><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">[</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>δ</mi><mrow><mi>x</mi><mo>,</mo><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>ξ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><mo>−</mo><msub><mi>κ</mi><mrow><mi>w</mi><mo>,</mo><mi>s</mi></mrow></msub><msubsup><mi>p</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mi>W</mi></mrow></msubsup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">]</mo></mrow><msubsup><mi>𝒜</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>H</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo><mspace linebreak=\"newline\" /></mrow></mrow></math>"
},
"children": []
},
{
"id": "p44.12",
"order": 12,
"type": "text",
"bbox": {
"l": 502.4216,
"t": 437.2644,
"r": 524.2556,
"b": 423.5849
},
"text": "(66)",
"attrs": {
"raw_type": "FORMULA_NUMBER",
"block_label": "formula_number"
},
"children": []
},
{
"id": "p44.13",
"order": 13,
"type": "heading",
"bbox": {
"l": 68.3812,
"t": 375.5866,
"r": 216.4204,
"b": 359.5072
},
"text": "B.2 Wages and hours",
"attrs": {
"block_label": "paragraph_title"
},
"children": []
},
{
"id": "p44.14",
"order": 14,
"type": "text",
"bbox": {
"l": 67.4214,
"t": 350.3875,
"r": 526.6549,
"b": 306.4691
},
"text": "Wages and hours are determined using efficient Nash bargaining, so that every period, wages and hours worked are determined by maximising the following expression, where[p44.14.1] is the bargaining power of workers of type s:",
"children": [
{
"id": "p44.14.1",
"order": 15,
"type": "formula",
"bbox": {
"l": 67.4214,
"t": 350.3875,
"r": 526.6549,
"b": 306.4691
},
"text": "\\eta_{s}",
"attrs": {
"latex": "\\eta_{s}",
"inline": true,
"text_offset": 169,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "eta sub s",
"mathml": "<math aria-label=\"eta sub s\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>η</mi><mrow><mi>s</mi></mrow></msub></mrow></math>"
},
"children": []
}
]
},
{
"id": "p44.15",
"order": 16,
"type": "formula",
"bbox": {
"l": 207.3029,
"t": 295.4295,
"r": 388.453,
"b": 270.4704
},
"text": "\\max_{w_{s,t}^{*},h_{s,t}}\\left(A_{t}^{H}(w_{s,t}^{*})\\right)^{\\eta_{s}}\\left(A_{t}^{F}(w_{s,t}^{*})\\right)^{1-\\eta_{s}}.",
"attrs": {
"block_label": "display_formula",
"latex": "\\max_{w_{s,t}^{*},h_{s,t}}\\left(A_{t}^{H}(w_{s,t}^{*})\\right)^{\\eta_{s}}\\left(A_{t}^{F}(w_{s,t}^{*})\\right)^{1-\\eta_{s}}.",
"notation": "latex",
"alt": "max sub w sub s, t to the power of *, h sub s, t of A sub t to the power of H of w sub s, t to the power of * to the power of eta sub s of A sub t to the power of F of w sub s, t to the power of * to the power of 1 minus eta sub s .",
"mathml": "<math aria-label=\"max sub w sub s, t to the power of *, h sub s, t of A sub t to the power of H of w sub s, t to the power of * to the power of eta sub s of A sub t to the power of F of w sub s, t to the power of * to the power of 1 minus eta sub s .\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mo>max</mo><mrow><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo>,</mo><msub><mi>h</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></msub><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>A</mi><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><msub><mi>η</mi><mrow><mi>s</mi></mrow></msub></mrow></msup><msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><msubsup><mi>A</mi><mrow><mi>t</mi></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>η</mi><mrow><mi>s</mi></mrow></msub></mrow></msup><mo>.</mo></mrow></math>"
},
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{
"id": "p44.16",
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"text": "(67)",
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{
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"text": "The Nash sharing rule is:",
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},
{
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},
"text": "\\eta_{s}(1-\\tau_{t}^{wh})A_{t}^{F}(w_{s,t}^{*})=(1-\\eta_{s})(1+\\tau_{t}^{wf})A_{t}^{H}(w_{s,t}^{*}).",
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"notation": "latex",
"alt": "eta sub s of 1 minus tau sub t to the power of wh A sub t to the power of F of w sub s, t to the power of * equals 1 minus eta sub s times 1 plus tau sub t to the power of wf A sub t to the power of H of w sub s, t to the power of * .",
"mathml": "<math aria-label=\"eta sub s of 1 minus tau sub t to the power of wh A sub t to the power of F of w sub s, t to the power of * equals 1 minus eta sub s times 1 plus tau sub t to the power of wf A sub t to the power of H of w sub s, t to the power of * .\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>η</mi><mrow><mi>s</mi></mrow></msub><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>τ</mi><mrow><mi>t</mi></mrow><mrow><mi>w</mi><mi>h</mi></mrow></msubsup><mo stretchy=\"false\">)</mo><msubsup><mi>A</mi><mrow><mi>t</mi></mrow><mrow><mi>F</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msub><mi>η</mi><mrow><mi>s</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">(</mo><mn>1</mn><mo>+</mo><msubsup><mi>τ</mi><mrow><mi>t</mi></mrow><mrow><mi>w</mi><mi>f</mi></mrow></msubsup><mo stretchy=\"false\">)</mo><msubsup><mi>A</mi><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msubsup><mi>w</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy=\"false\">)</mo><mo>.</mo></mrow></math>"
},
"children": []
},
{
"id": "p44.19",
"order": 20,
"type": "text",
"bbox": {
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"text": "(68)",
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},
{
"id": "p44.20",
"order": 21,
"type": "text",
"bbox": {
"l": 68.861,
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"r": 251.9306,
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},
"text": "This equation thus determines[p44.20.1].",
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{
"id": "p44.20.1",
"order": 22,
"type": "formula",
"bbox": {
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},
"text": "w_{t,s}^{*}",
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"text_offset": 29,
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"bbox_estimated": true,
"notation": "latex",
"alt": "w sub t, s to the power of *",
"mathml": "<math aria-label=\"w sub t, s to the power of *\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msubsup><mi>w</mi><mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow><mrow><mo>*</mo></mrow></msubsup></mrow></math>"
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{
"id": "p44.21",
"order": 23,
"type": "text",
"bbox": {
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"text": "The average wage is determined by using the law of motion of labour as follows:",
"children": []
},
{
"id": "p44.22",
"order": 24,
"type": "formula",
"bbox": {
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"text": "\\begin{align*}nde_{s,t}h_{s,t}w_{s,t}&=(1-\\delta_{x,s})nde_{s,t-1}\\left[\\xi_{w,s}\\frac{1+\\overline{\\pi}}{1+\\pi_{t-1}}w_{s,t-1}h_{s,t}+(1-\\xi_{w,s})w^{*}_{s,t}h_{s,t}\\right]\\\\&+M_{s,t}\\left[\\kappa_{w,s}w_{s,t}h_{s,t}+(1-\\kappa_{w,s})w^{*}_{s,t}h_{s,t}\\right],\\end{align*}",
"attrs": {
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"latex": "\\begin{align*}nde_{s,t}h_{s,t}w_{s,t}&=(1-\\delta_{x,s})nde_{s,t-1}\\left[\\xi_{w,s}\\frac{1+\\overline{\\pi}}{1+\\pi_{t-1}}w_{s,t-1}h_{s,t}+(1-\\xi_{w,s})w^{*}_{s,t}h_{s,t}\\right]\\\\&+M_{s,t}\\left[\\kappa_{w,s}w_{s,t}h_{s,t}+(1-\\kappa_{w,s})w^{*}_{s,t}h_{s,t}\\right],\\end{align*}",
"notation": "latex",
"alt": "begin align* nde sub s, t h sub s, t w sub s, t & equals 1 minus delta sub x, s nde sub s, t minus 1 xi sub w, s 1 plus pi bar over 1 plus pi sub t minus 1 w sub s, t minus 1 h sub s, t plus 1 minus xi sub w, s w to the power of * sub s, t h sub s, t & plus M sub s, t kappa sub w, s w sub s, t h sub s, t plus 1 minus kappa sub w, s w to the power of * sub s, t h sub s, t, end align*",
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},
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},
{
"id": "p44.23",
"order": 25,
"type": "text",
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"text": "(69)",
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},
{
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"bbox": {
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"text": "where the first row is the average wage of existing workers and the second row is the average wage of new hires, with the number of new matches[p44.24.1] counting the number of new hires.",
"children": [
{
"id": "p44.24.1",
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},
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},
{
"id": "p44.25",
"order": 28,
"type": "footer",
"bbox": {
"l": 289.3603,
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"text": "43",
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}
footer (1)
formula (5)
heading (2)
link (3)
text (7)
1 · TEXT
2
3 · FORMULA
4 · TEXT
5 · HEADING
6 · TEXT
7 · FORMULA
8 · TEXT
9 · TEXT
10
11
12
13 · HEADING
14 · TEXT
15 · TEXT
16
17
18 · FOOTER
Page 45 JSON
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"page": 45,
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"items": [
{
"id": "p45.1",
"order": 1,
"type": "text",
"bbox": {
"l": 68.1412,
"t": 769.8926,
"r": 526.415,
"b": 740.8536
},
"text": "Hours worked are determined by per-period Nash bargaining, and after taking into account the wage decision (equation 68), hours worked expression is:",
"children": [
{
"id": "p45.1.1",
"order": 2,
"type": "link",
"bbox": {
"l": 260.997,
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"text": "68),",
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}
]
},
{
"id": "p45.2",
"order": 3,
"type": "formula",
"bbox": {
"l": 185.4689,
"t": 730.054,
"r": 407.1678,
"b": 697.1752
},
"text": "\\alpha_{H}x_{s,t}(h_{s,t})^{\\alpha_{H-1}}=\\frac{\\chi}{u^{\\prime}(c_{s,t})}\\frac{(1+\\tau_{t}^{wf})}{(1-\\tau_{t}^{wh})}(h_{s,t})^{\\varphi}.",
"attrs": {
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"latex": "\\alpha_{H}x_{s,t}(h_{s,t})^{\\alpha_{H-1}}=\\frac{\\chi}{u^{\\prime}(c_{s,t})}\\frac{(1+\\tau_{t}^{w f})}{(1-\\tau_{t}^{w h})}(h_{s,t})^{\\varphi}.",
"notation": "latex",
"alt": "alpha sub H x sub s, t of h sub s, t to the power of alpha sub H minus 1 equals chi over u to the power of prime of c sub s, t times 1 plus tau sub t to the power of wf over of 1 minus tau sub t to the power of wh times h sub s, t to the power of varphi .",
"mathml": "<math aria-label=\"alpha sub H x sub s, t of h sub s, t to the power of alpha sub H minus 1 equals chi over u to the power of prime of c sub s, t times 1 plus tau sub t to the power of wf over of 1 minus tau sub t to the power of wh times h sub s, t to the power of varphi .\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mrow><msub><mi>α</mi><mrow><mi>H</mi></mrow></msub><msub><mi>x</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mi>h</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><msup><mo stretchy=\"false\">)</mo><mrow><msub><mi>α</mi><mrow><mi>H</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msup><mo>=</mo><mfrac><mrow><mi>χ</mi></mrow><mrow><msup><mi>u</mi><mrow><mi>′</mi></mrow></msup><mo stretchy=\"false\">(</mo><msub><mi>c</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo></mrow></mfrac><mfrac><mrow><mo stretchy=\"false\">(</mo><mn>1</mn><mo>+</mo><msubsup><mi>τ</mi><mrow><mi>t</mi></mrow><mrow><mi>w</mi><mi>f</mi></mrow></msubsup><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>τ</mi><mrow><mi>t</mi></mrow><mrow><mi>w</mi><mi>h</mi></mrow></msubsup><mo stretchy=\"false\">)</mo></mrow></mfrac><mo stretchy=\"false\">(</mo><msub><mi>h</mi><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><msup><mo stretchy=\"false\">)</mo><mrow><mi>φ</mi></mrow></msup><mo>.</mo></mrow></math>"
},
"children": []
},
{
"id": "p45.3",
"order": 4,
"type": "text",
"bbox": {
"l": 502.1816,
"t": 720.9343,
"r": 524.4955,
"b": 706.5348
},
"text": "(70)",
"attrs": {
"raw_type": "FORMULA_NUMBER",
"block_label": "formula_number"
},
"children": []
},
{
"id": "p45.4",
"order": 5,
"type": "heading",
"bbox": {
"l": 68.1412,
"t": 679.8958,
"r": 199.625,
"b": 663.8163
},
"text": "B.3 Labour packer",
"attrs": {
"block_label": "paragraph_title"
},
"children": []
},
{
"id": "p45.5",
"order": 6,
"type": "text",
"bbox": {
"l": 67.4214,
"t": 654.6967,
"r": 525.9351,
"b": 624.9377
},
"text": "Labour from Ricardian and non-Ricardian households is aggregated by a labour packer using a CES technology, as follows:",
"children": []
},
{
"id": "p45.6",
"order": 7,
"type": "formula",
"bbox": {
"l": 150.9184,
"t": 614.8581,
"r": 442.4381,
"b": 589.899
},
"text": "n_{t}^{1-\\frac{1}{\\eta}}=\\left[\\left(1-\\omega\\right)^{\\frac{1}{\\eta}}\\left(nde_{i,t}h_{i,t}^{\\alpha_{H}}\\right)^{1-\\frac{1}{\\eta}}+\\omega^{\\frac{1}{\\eta}}\\left(nde_{j,t}h_{j,t}^{\\alpha_{H}}\\right)^{1-\\frac{1}{\\eta}}\\right],",
"attrs": {
"block_label": "display_formula",
"latex": "n_{t}^{1-\\frac{1}{\\eta}}=\\left[\\left(1-\\omega\\right)^{\\frac{1}{\\eta}}\\left(n d e_{i,t}h_{i,t}^{\\alpha_{H}}\\right)^{1-\\frac{1}{\\eta}}+\\omega^{\\frac{1}{\\eta}}\\left(n d e_{j,t}h_{j,t}^{\\alpha_{H}}\\right)^{1-\\frac{1}{\\eta}}\\right],",
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"alt": "n sub t to the power of 1 minus 1 over eta equals 1 minus omega to the power of 1 over eta of nde sub i, t h sub i, t to the power of alpha sub H to the power of 1 minus 1 over eta plus omega to the power of 1 over eta of nde sub j, t h sub j, t to the power of alpha sub H to the power of 1 minus 1 over eta,",
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"text": "where[p45.8.1] are aggregate labour services and[p45.8.2] are total labour services provided by labour firms in household segment $s$. Parameter[p45.8.3] measures the share of non-Ricardian households in the economy.",
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"text": "C Search frictions and time to build",
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"text": "Here we briefly discuss the implications of more volatile labour market when the delays are related to construction rather than planning. As discussed in the main text, this case is less interesting from the perspective of the news shocks literature than a planning delay, because there is an immediate stimulus of the economy from government spending on public investment (rather than no immediate stimulus in the case of the planning delay). The news of the future productivity of higher level of public capital is, however, identical in the case of planning and construction delays.",
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"text": "The results of the time to build delay are shown in Figure 8 for the main macroeconomic variables and in Figure 9 for the labour market variables. The main findings are similar than for planning delays: the increases in consumption, investment, and labour during the construction phase are somewhat larger if the labour market is more volatile. On the other hand, the increase in inflation is more dampened, which happens for the same reason as in the case of planning delays, i.e., an early increase in hiring causes and increase in labour, which dampens the increase in firms' marginal costs and therefore inflation. Long-run results are identical in the case of planning or construction delays, as the final steady state does not depend on the delay.",
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figure (1)
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1 · HEADER
2 · FIGURE
3 · FOOTNOTE
4 · FOOTNOTE
5 · FOOTER
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"embedded_text": "Benchmark Low surplus Very low surplus\n4 4 4\n3 3 3\nOutput\n2 2 2\n1 1 1\n0 0 0\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\nconsumption\n2 2 2\n1 1 1\nPrivate\n0 0 0\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\n1 1 1\ninvestment\n0 0 0\n-1 -1 -1\nPrivate\n-2 -2 -2\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT\n0.15 0.15 0.15\n0.1 0.1 0.1\n0.05 0.05 0.05\n0 0 0\n-0.05 -0.05 -0.05\n0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT 0 10 20 30 40 50 60 LT"
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caption (1)
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1 · HEADER
2 · FIGURE
3 · CAPTION
4 · FOOTER
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text (1)
1 · TEXT
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"text": "T: +353 (0)1 224 6000\nwww.centralbank.ie\npublications@centralbank.ie\n\nBosca PO 559, Baile Átha Cliath 1, Éire\nPO Box 559, Dublin 1, Ireland",
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