{ "document_id": 405, "page": 43, "width_pt": 595.276, "height_pt": 841.89, "engine": "paddleocr_vl", "source_run_id": 530, "item_count": 20, "counts_by_type": { "footer": 1, "formula": 8, "text": 11 }, "build_ms": 4, "items": [ { "id": "p43.1", "order": 1, "type": "formula", "bbox": { "l": 68.6211, "t": 744.4535, "r": 547.2892, "b": 684.9356 }, "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*}", "attrs": { "block_label": "display_formula", "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*}", "notation": "latex", "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": "𝒜tF(ws,t*)=(Ss,tx−Ss,twfws,t*)+∑k=0∞β(1−δx,s)(1−ξw,s)[β(1−δx,s)ξw,s]k𝒜t+k+1F(ws,t+k+1*)=(Ss,tx−Ss,twfws,t*)−β(1−δx,s)ξw,s(Ss,t+1x−Ss,t+1wfws,t+1*)+β(1−δx,s)𝒜t+1F(ws,t+1*)" }, "children": [] }, { "id": "p43.2", "order": 2, "type": "text", "bbox": { "l": 502.4216, "t": 685.1756, "r": 524.4955, "b": 671.9761 }, "text": "(57)", "attrs": { "raw_type": "FORMULA_NUMBER", "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 }, "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": { "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 & 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": "𝒜tF(ws,t)&=u′(cs,t)(hs,tαHxs,t−hs,tws,t(1+τtwf))&+β(1−δx,s)[(1−ξw,s)𝒜t+1F(ws,t+1*)+ξw,s𝒜t+1F(ws,t)]" }, "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": "𝒜tF(ws,t)=(Ss,tx−Ss,twfws,t)−β(1−δx,s)ξw,s(Ss,t+1x−Ss,t+1wfws,t+1)+β(1−δx,s)𝒜t+1F(ws,t+1)" }, "children": [] }, { "id": "p43.9", "order": 9, "type": "text", "bbox": { "l": 502.4216, "t": 481.4228, "r": 524.4955, "b": 467.7433 }, "text": "(59)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p43.10", "order": 10, "type": "text", "bbox": { "l": 66.4617, "t": 447.5841, "r": 526.6549, "b": 345.3477 }, "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": "ψs" }, "children": [] }, { "id": "p43.10.2", "order": 12, "type": "formula", "bbox": { "l": 66.4617, "t": 447.5841, "r": 526.6549, "b": 345.3477 }, "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": "κw,s" }, "children": [] }, { "id": "p43.10.3", "order": 13, "type": "formula", "bbox": { "l": 66.4617, "t": 447.5841, "r": 526.6549, "b": 345.3477 }, "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": "𝒜F(ws,t*)≡u′(cs,t)AF(ws,t*)" }, "children": [] } ] }, { "id": "p43.11", "order": 14, "type": "formula", "bbox": { "l": 142.0409, "t": 335.2681, "r": 451.0757, "b": 305.2691 }, "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": "p43.12", "order": 15, "type": "text", "bbox": { "l": 502.4216, "t": 329.0283, "r": 524.2556, "b": 314.3888 }, "text": "(60)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p43.13", "order": 16, "type": "text", "bbox": { "l": 66.7016, "t": 287.2698, "r": 526.415, "b": 230.1518 }, "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", "children": [] }, { "id": "p43.14", "order": 17, "type": "formula", "bbox": { "l": 135.0828, "t": 207.1126, "r": 457.314, "b": 136.5551 }, "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}", "attrs": { "block_label": "display_formula", "latex": "\\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}", "notation": "latex", "alt": "begin aligned mathcal A sub t to the power of H 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 w sub s, t to the power of * 1 minus tau sub t to the power of wh minus b sub s, t minus 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 1 minus xi sub w, s mathcal A sub t plus 1 to the power of H 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 H of w sub s, t to the power of * & minus beta p sub s, t to the power of W 1 minus kappa sub w, s mathcal A sub t plus 1 to the power of H of w sub s, t plus 1 to the power of * plus kappa sub w, s mathcal A sub t plus 1 to the power of H of w sub s, t plus 1 end aligned", "mathml": "𝒜tH(ws,t*)&=u′(cs,t)(hs,tws,t*(1−τtwh)−bs,t)−χhs,t1+φ1+φ&+β(1−δx,s)[(1−ξw,s)𝒜t+1H(ws,t+1*)+ξw,s𝒜t+1H(ws,t*)]&−βps,tW[(1−κw,s)𝒜t+1H(ws,t+1*)+κw,s𝒜t+1H(ws,t+1)]" }, "children": [] }, { "id": "p43.15", "order": 18, "type": "text", "bbox": { "l": 502.4216, "t": 180.2336, "r": 524.2556, "b": 165.5941 }, "text": "(61)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p43.16", "order": 19, "type": "text", "bbox": { "l": 68.1412, "t": 127.1955, "r": 526.175, "b": 97.6765 }, "text": "We can define additional auxiliary variables to sum the utility obtained from wages, unemployment benefits, and the disutility of labour terms,", "children": [] }, { "id": "p43.17", "order": 20, "type": "footer", "bbox": { "l": 289.3603, "t": 51.1182, "r": 305.6758, "b": 39.3586 }, "text": "42", "attrs": { "block_label": "number" }, "children": [] } ], "ai_page_review": { "model": "claude-opus-4-8", "applied": true, "changed": true, "reviewed_at": "2026-08-21T10:46:51+00:00" } }