{
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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": "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": "where[p41.5.1] is the number of new matches formed in a period, and[p41.5.2] is the fraction of existing employment relationships that have (exogenously) separated. The number of matches[p41.5.3] is:",
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"text": "M_{s,t}=\\phi_{s,M}(un_{s,t})^{\\mu_{s}}(vac_{s,t})^{1-\\mu_{s}}=p_{s,t}^{W}un_{s,t}=p_{s,t}^{F}vac_{s,t},",
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"latex": "M_{s,t}=\\phi_{s,M}(u n_{s,t})^{\\mu_{s}}(v a c_{s,t})^{1-\\mu_{s}}=p_{s,t}^{W}u n_{s,t}=p_{s,t}^{F}v a c_{s,t},",
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"alt": "M sub s, t equals phi sub s, M of un sub s, t to the power of mu sub s of vac sub s, t to the power of 1 minus mu sub s equals p sub s, t to the power of W un sub s, t equals p sub s, t to the power of F vac sub s, t,",
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}
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"text": "where[p41.8.1] is matching efficiency,[p41.8.2] is the number of searching workers in each segment,[p41.8.3] is the number of vacancies,[p41.8.4] is the matching probability for workers of each type,[p41.8.5] is the matching probability for firms, and[p41.8.6] is the elasticity of the matching function with respect to unemployment.",
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"text": "p_{s,t}^{F}",
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"text": "The probability for a searching worker to find a job is"
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{
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"t": 502.0621,
"r": 376.6962,
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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}}",
"attrs": {
"block_label": "display_formula",
"latex": "p_{s,t}^{W}=\\frac{M_{t}}{u n_{s,t}}=\\phi_{s,M}(\\frac{v a c_{s,t}}{u n_{s,t}})^{1-\\mu_{s}}",
"notation": "latex",
"alt": "p sub s, t to the power of W equals M sub t over un sub s, t equals phi sub s, M of vac sub s, t over un sub s, t to the power of 1 minus mu sub s",
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}
},
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"text": "and the probability of a firm finding a worker is"
},
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"type": "text",
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"text": "(49)",
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},
{
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"type": "formula",
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"r": 375.4965,
"b": 411.3453
},
"text": "p_{s,t}^{F}=\\frac{M_{s,t}}{vac_{s,t}}=\\phi_{s,M}(\\frac{vac_{s,t}}{un_{s,t}})^{-\\mu_{s}}",
"attrs": {
"block_label": "display_formula",
"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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"id": "p41.14",
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"type": "text",
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"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:"
},
{
"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,",
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}
},
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"order": 34,
"type": "text",
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"text": "(51)",
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},
{
"id": "p41.18",
"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].",
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"text": "ω",
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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,
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"text_span": 10,
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"notation": "latex",
"alt": "une sub t equals omega une sub j, t plus 1 minus omega une sub i, t",
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"id": "p41.19",
"order": 38,
"type": "heading",
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"text": "B.1 Value functions",
"attrs": {
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}
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{
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"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": [
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"text": "Bodart",
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},
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"text": "et al.",
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},
{
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"text": "(2006),",
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"text": "A^{F}(w_{s,t}^{*})",
"attrs": {
"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 *",
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{
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"type": "formula",
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"text": "s \\in [i, j]",
"attrs": {
"latex": "s \\in [i, j]",
"inline": true,
"text_offset": 123,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
"alt": "s in i, j",
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},
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"type": "formula",
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"text": "w_{s,j}^{*}",
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"inline": true,
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"text": "\\mathcal{A}^{F}(w_{s,t}^{*}) \\equiv u'(c_{s,t})A^{F}(w_{s,t}^{*})",
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"latex": "\\mathcal{A}^{F}(w_{s,t}^{*}) \\equiv u'(c_{s,t})A^{F}(w_{s,t}^{*})",
"inline": true,
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"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 *",
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}
},
{
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"type": "formula",
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"inline": true,
"text_offset": 371,
"text_span": 10,
"bbox_estimated": true,
"notation": "latex",
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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}",
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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 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": ""
}
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"text": "(52)",
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