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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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"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)",
"notation": "latex",
"alt": "U sub s, t equals 1 over 1 minus sigma of c sub s, t plus k minus kappa c sub s, t plus k minus 1 over 1 minus kappa to the power of 1 minus sigma minus 1 over 1 plus zeta h sub s, t plus k to the power of 1 plus zeta",
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"text": "(7)",
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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,",
"children": [
{
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"order": 5,
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"text": "23",
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"text": "Sims and Wolff",
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"text": "(2018).",
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"text": "\\text{where } s \\in [i, j] \\text{ stands for Ricardian (}i\\text{) and hand-to-mouth (}j\\text{) households, } c_{s,t} \\text{ is consumption, } \\beta \\text{ is the discount rate, } \\sigma \\text{ is the inverse of the intertemporal elasticity of substitution and } \\zeta \\text{ is the (inverse of) the Frisch labour supply elasticity. } \\kappa \\text{ is the degree of habit formation in consumption.}^{23} \\text{ 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, } h_{s,t}, \\text{ includes the additional hours that occurs because of wage dispersion. Welfare is just the discounted present value of future utilities,}",
"attrs": {
"latex": "\\text{where } s \\in [i, j] \\text{ stands for Ricardian (}i\\text{) and hand-to-mouth (}j\\text{) households, } c_{s,t} \\text{ is consumption, } \\beta \\text{ is the discount rate, } \\sigma \\text{ is the inverse of the intertemporal elasticity of substitution and } \\zeta \\text{ is the (inverse of) the Frisch labour supply elasticity. } \\kappa \\text{ is the degree of habit formation in consumption.}^{23} \\text{ 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, } h_{s,t}, \\text{ includes the additional hours that occurs because of wage dispersion. Welfare is just the discounted present value of future utilities,}",
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"alt": "where s in i, j stands for Ricardian of i and hand minus to minus mouth of j households, c sub s, t is consumption, beta is the discount rate, sigma is the inverse of the intertemporal elasticity of substitution and zeta is the of inverse of the Frisch labour supply elasticity. kappa is the degree of habit formation in consumption. to the power of 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 of 2018 . Therefore, our measure of hours worked for welfare computation, h sub s, t, 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": "c_{s,t}",
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"text": "σ",
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"text": "κ",
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"text": "h_{s,t}",
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"text": "\\mathcal{W}_{s,t}=\\sum_{k=0}^{\\infty}\\beta^{k}U_{s,t+k}.",
"attrs": {
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"latex": "\\mathcal{W}_{s,t}=\\sum_{k=0}^{\\infty}\\beta^{k}U_{s,t+k}.",
"notation": "latex",
"alt": "mathcal W sub s, t equals the sum of sub k equals 0 to the power of infinity beta to the power of k U sub s, t plus k .",
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}
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{
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"text": "(8)",
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"text": "The aggregate welfare measure is just the present value of all future utilities, which we aggregate over the Ricardian and hand-to-mouth households using their respective shares,[p29.7.1]:",
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"text": "\\mathcal{W}_{t}=\\omega\\mathcal{W}_{j,t}+(1-\\omega)\\mathcal{W}_{i,t}",
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"latex": "\\mathcal{W}_{t}=\\omega\\mathcal{W}_{j,t}+(1-\\omega)\\mathcal{W}_{i,t}",
"notation": "latex",
"alt": "mathcal W sub t equals omega mathcal W sub j, t plus 1 minus omega mathcal W sub i, t",
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}
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{
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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.",
"children": [
{
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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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"text": "[p29.12.1]To prevent that the value of[p29.12.2] affects the steady state, we divide the first term in the utility function with[p29.12.3]. This does not affect model dynamics.",
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"text": "²³",
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"text": "κ",
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"text": "1 - κ",
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