{ "document_id": 405, "page": 8, "width_pt": 595.276, "height_pt": 841.89, "engine": "paddleocr_vl", "source_run_id": 530, "item_count": 29, "counts_by_type": { "footer": 1, "footnote": 1, "formula": 14, "link": 6, "text": 7 }, "build_ms": 4, "items": [ { "id": "p8.1", "order": 1, "type": "formula", "bbox": { "l": 180.6702, "t": 758.853, "r": 413.886, "b": 729.574 }, "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}},", "attrs": { "block_label": "display_formula", "latex": "M C_{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}},", "notation": "latex", "alt": "MC sub T, t equals frac 1 z sub T, t K sub G, t to the power of alpha sub G alpha sub T to the power of alpha sub T of 1 minus alpha sub T to the power of 1 minus alpha sub T r sub k, t to the power of alpha sub T w sub t to the power of 1 minus alpha sub T,", "mathml": "MCT,t=1zT,tKG,tαGαTαT(1−αT)1−αTrk,tαTwt1−αT," }, "children": [] }, { "id": "p8.2", "order": 2, "type": "text", "bbox": { "l": 509.3797, "t": 753.0932, "r": 524.4955, "b": 739.4137 }, "text": "(5)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p8.3", "order": 3, "type": "text", "bbox": { "l": 67.1815, "t": 723.3342, "r": 525.9351, "b": 665.9763 }, "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.,", "children": [ { "id": "p8.3.1", "order": 4, "type": "formula", "bbox": { "l": 67.1815, "t": 723.3342, "r": 525.9351, "b": 665.9763 }, "text": "r_{k,t}", "attrs": { "latex": "r_{k,t}", "inline": true, "text_offset": 5, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "r sub k, t", "mathml": "rk,t" }, "children": [] }, { "id": "p8.3.2", "order": 5, "type": "formula", "bbox": { "l": 67.1815, "t": 723.3342, "r": 525.9351, "b": 665.9763 }, "text": "w_t", "attrs": { "latex": "w_t", "inline": true, "text_offset": 50, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "w sub t", "mathml": "wt" }, "children": [] }, { "id": "p8.3.3", "order": 6, "type": "formula", "bbox": { "l": 67.1815, "t": 723.3342, "r": 525.9351, "b": 665.9763 }, "text": "K_{G,t}^{\\alpha_G}", "attrs": { "latex": "K_{G,t}^{\\alpha_G}", "inline": true, "text_offset": 94, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "K sub G, t to the power of alpha sub G", "mathml": "KG,tαG" }, "children": [] }, { "id": "p8.3.4", "order": 7, "type": "formula", "bbox": { "l": 67.1815, "t": 723.3342, "r": 525.9351, "b": 665.9763 }, "text": "\\alpha_G > 0", "attrs": { "latex": "\\alpha_G > 0", "inline": true, "text_offset": 231, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "alpha sub G is greater than 0", "mathml": "αG>0" }, "children": [] } ] }, { "id": "p8.4", "order": 8, "type": "text", "bbox": { "l": 67.4214, "t": 665.2563, "r": 526.6549, "b": 565.1799 }, "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.", "children": [ { "id": "p8.4.1", "order": 9, "type": "formula", "bbox": { "l": 67.4214, "t": 665.2563, "r": 526.6549, "b": 565.1799 }, "text": "K_{G,t}", "attrs": { "latex": "K_{G,t}", "inline": true, "text_offset": 99, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "K sub G, t", "mathml": "KG,t" }, "children": [] } ] }, { "id": "p8.5", "order": 10, "type": "text", "bbox": { "l": 67.9013, "t": 563.9799, "r": 526.6549, "b": 435.1045 }, "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": [ { "id": "p8.5.1", "order": 11, "type": "link", "bbox": { "l": 462.869, "t": 549.463, "r": 525.406, "b": 536.238 }, "text": "Erceg et al.", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p8.5.2", "order": 12, "type": "link", "bbox": { "l": 73.456, "t": 535.017, "r": 103.185, "b": 522.169 }, "text": "(2000)", "attrs": { "action_type": "/Named" }, "children": [] } ] }, { "id": "p8.6", "order": 13, "type": "text", "bbox": { "l": 67.6614, "t": 433.9045, "r": 526.8948, "b": 347.7476 }, "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):", "children": [ { "id": "p8.6.1", "order": 14, "type": "link", "bbox": { "l": 498.136, "t": 376.113, "r": 507.864, "b": 362.889 }, "text": "B", "attrs": { "action_type": "/Named" }, "children": [] } ] }, { "id": "p8.7", "order": 15, "type": "formula", "bbox": { "l": 141.8009, "t": 336.708, "r": 450.8358, "b": 307.669 }, "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].", "attrs": { "block_label": "display_formula", "latex": "\\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].", "notation": "latex", "alt": "psi sub s equals p sub t to the power of F beta u to the power of prime of c sub s, t plus 1 over u to the power of prime of c sub s, t 1 minus kappa sub w, s mathcal A sub t to the power of F of w sub s, t plus 1 to the power of * plus kappa sub w, s mathcal A sub t to the power of F of w sub s, t plus 1 .", "mathml": "ψs=ptFβu′(cs,t+1)u′(cs,t)[(1−κw,s)𝒜tF(ws,t+1*)+κw,s𝒜tF(ws,t+1)]." }, "children": [] }, { "id": "p8.8", "order": 16, "type": "text", "bbox": { "l": 509.6196, "t": 330.2282, "r": 524.2556, "b": 316.0687 }, "text": "(6)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p8.9", "order": 17, "type": "text", "bbox": { "l": 66.9416, "t": 301.6693, "r": 526.415, "b": 111.356 }, "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.", "children": [ { "id": "p8.9.1", "order": 18, "type": "link", "bbox": { "l": 181.427, "t": 243.256, "r": 190.353, "b": 230.516 }, "text": "6", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p8.9.2", "order": 19, "type": "link", "bbox": { "l": 202.643, "t": 156.894, "r": 209.757, "b": 141.465 }, "text": "7", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p8.9.3", "order": 20, "type": "formula", "bbox": { "l": 66.9416, "t": 301.6693, "r": 526.415, "b": 111.356 }, "text": "\\psi_{s}", "attrs": { "latex": "\\psi_{s}", "inline": true, "text_offset": 118, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "psi sub s", "mathml": "ψs" }, "children": [] }, { "id": "p8.9.4", "order": 21, "type": "formula", "bbox": { "l": 66.9416, "t": 301.6693, "r": 526.415, "b": 111.356 }, "text": "p^{F}", "attrs": { "latex": "p^{F}", "inline": true, "text_offset": 308, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "p to the power of F", "mathml": "pF" }, "children": [] }, { "id": "p8.9.5", "order": 22, "type": "formula", "bbox": { "l": 66.9416, "t": 301.6693, "r": 526.415, "b": 111.356 }, "text": "\\kappa_{w,s}", "attrs": { "latex": "\\kappa_{w,s}", "inline": true, "text_offset": 557, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "kappa sub w, s", "mathml": "κw,s" }, "children": [] }, { "id": "p8.9.6", "order": 23, "type": "formula", "bbox": { "l": 66.9416, "t": 301.6693, "r": 526.415, "b": 111.356 }, "text": "\\mathcal{A}^{F}(w_{s,t}^{*})", "attrs": { "latex": "\\mathcal{A}^{F}(w_{s,t}^{*})", "inline": true, "text_offset": 824, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "mathcal A to the power of F of w sub s, t to the power of *", "mathml": "𝒜F(ws,t*)" }, "children": [] }, { "id": "p8.9.7", "order": 24, "type": "formula", "bbox": { "l": 66.9416, "t": 301.6693, "r": 526.415, "b": 111.356 }, "text": "\\mathcal{A}^{F}(w_{s,t})", "attrs": { "latex": "\\mathcal{A}^{F}(w_{s,t})", "inline": true, "text_offset": 880, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "mathcal A to the power of F of w sub s, t", "mathml": "𝒜F(ws,t)" }, "children": [] } ] }, { "id": "p8.10", "order": 25, "type": "footnote", "bbox": { "l": 68.1412, "t": 100.5564, "r": 526.175, "b": 72.2374 }, "text": "[p8.10.2]Note that[p8.10.3], and analogous for the value at average wage, see Appendix B.", "attrs": { "block_label": "footnote" }, "children": [ { "id": "p8.10.1", "order": 26, "type": "link", "bbox": { "l": 117.608, "t": 84.915, "r": 126.659, "b": 70.981 }, "text": "B.", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p8.10.2", "order": 27, "type": "formula", "bbox": { "l": 68.1412, "t": 100.5564, "r": 526.175, "b": 72.2374 }, "text": "⁷", "attrs": { "latex": "^{7}", "inline": true, "text_offset": 0, "text_span": 9, "bbox_estimated": true, "alt": "to the power of 7", "mathml": "7" }, "children": [] }, { "id": "p8.10.3", "order": 28, "type": "formula", "bbox": { "l": 68.1412, "t": 100.5564, "r": 526.175, "b": 72.2374 }, "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": 18, "text_span": 9, "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": "𝒜F(ws,t*)≡u′(cs,t)AF(ws,t*)" }, "children": [] } ] }, { "id": "p8.11", "order": 29, "type": "footer", "bbox": { "l": 293.1992, "t": 51.1182, "r": 302.0768, "b": 39.8386 }, "text": "7", "attrs": { "block_label": "number" }, "children": [] } ], "ai_page_review": { "model": "claude-opus-4-8", "applied": true, "changed": true, "reviewed_at": "2026-08-21T10:44:48+00:00" } }