{ "document_id": 405, "page": 44, "width_pt": 595.276, "height_pt": 841.89, "engine": "paddleocr_vl", "source_run_id": 530, "item_count": 28, "counts_by_type": { "footer": 1, "formula": 11, "heading": 1, "text": 15 }, "build_ms": 3, "items": [ { "id": "p44.1", "order": 1, "type": "formula", "bbox": { "l": 143.0006, "t": 749.9733, "r": 451.0757, "b": 680.3758 }, "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*}", "attrs": { "block_label": "display_formula", "latex": "\\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*}", "notation": "latex", "alt": "begin align* S sub s, t to the power of wh & 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 minus tau sub t to the power of wh & equals u to the power of prime of c sub s, t h sub s, t of 1 minus tau sub t to the power of wh 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 wh, end align*", "mathml": "Ss,twh≡∑k=0∞[β(1−δx,s)ξw,s]ku′(cs,t+k)hs,t(1+π―)kPtPt+k(1−τtwh)=u′(cs,t)hs,t(1−τtwh)+β(1−δx,s)(1+π―)(1+πt+1)ξw,sSs,t+1wh," }, "children": [] }, { "id": "p44.2", "order": 2, "type": "text", "bbox": { "l": 502.1816, "t": 723.3342, "r": 524.4955, "b": 708.9347 }, "text": "(62)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p44.3", "order": 3, "type": "formula", "bbox": { "l": 163.1551, "t": 671.9761, "r": 430.4414, "b": 640.7772 }, "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},", "attrs": { "block_label": "display_formula", "latex": "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},", "notation": "latex", "alt": "S sub s, t to the power of b equals u to the power of prime of c sub s, t plus k b sub s, t plus beta of 1 minus delta sub x, s xi sub w, s frac of 1 plus overline pi times 1 plus pi sub t plus 1 S sub s, t plus 1 to the power of b,", "mathml": "Ss,tb=u′(cs,t+k)bs,t+β(1−δx,s)ξw,s(1+π―)(1+πt+1)Ss,t+1b," }, "children": [] }, { "id": "p44.4", "order": 4, "type": "text", "bbox": { "l": 502.4216, "t": 664.2963, "r": 524.2556, "b": 650.3768 }, "text": "(63)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p44.5", "order": 5, "type": "formula", "bbox": { "l": 201.5445, "t": 633.5774, "r": 393.2517, "b": 601.4186 }, "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": { "block_label": "display_formula", "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": "Ss,th=χhs,t1+φ1+φ+β(1−δx,s)ξw,sSs,t+1h," }, "children": [] }, { "id": "p44.6", "order": 6, "type": "text", "bbox": { "l": 502.6615, "t": 623.7378, "r": 524.2556, "b": 609.8183 }, "text": "(64)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "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": { "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_{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": "𝒜tH(ws,t*)&=Ss,twws,t*−Ss,tb−β(1−δx,s)ξw,s(St+1wws,t+1*−Ss,t+1b)&−Ss,th+β(1−δx,s)ξw,sSs,t+1h&+β[1−δx,s−(1−κw,s)ps,tW]𝒜t+1H(ws,t+1*)−βκw,sps,tW𝒜t+1H(ws,t+1)" }, "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": "𝒜tH(ws,t)&=Ss,twws,t−Ss,tb−β(1−δx,s)ξw,s(Ss,t+1wws,t+1−Ss,t+1b)&−Ss,th+β(1−δx,s)ξw,sSs,t+1h&+β[(1−δx,s)(1−ξw,s)−(1−κw,s)ps,tW]𝒜t+1H(ws,t+1*)&+β[(1−δx,s)ξw,s−κw,sps,tW]𝒜t+1H(ws,t+1)" }, "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": "ηs" }, "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": "maxws,t*,hs,t(AtH(ws,t*))ηs(AtF(ws,t*))1−ηs." }, "children": [] }, { "id": "p44.16", "order": 17, "type": "text", "bbox": { "l": 502.4216, "t": 292.7896, "r": 524.2556, "b": 278.8701 }, "text": "(67)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p44.17", "order": 18, "type": "text", "bbox": { "l": 86.1363, "t": 264.9506, "r": 216.9002, "b": 251.511 }, "text": "The Nash sharing rule is:", "children": [] }, { "id": "p44.18", "order": 19, "type": "formula", "bbox": { "l": 168.9135, "t": 237.3515, "r": 423.7233, "b": 220.3121 }, "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}^{*}).", "attrs": { "block_label": "display_formula", "latex": "\\eta_{s}(1-\\tau_{t}^{w h})A_{t}^{F}(w_{s,t}^{*})=(1-\\eta_{s})(1+\\tau_{t}^{w f})A_{t}^{H}(w_{s,t}^{*}).", "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": "ηs(1−τtwh)AtF(ws,t*)=(1−ηs)(1+τtwf)AtH(ws,t*)." }, "children": [] }, { "id": "p44.19", "order": 20, "type": "text", "bbox": { "l": 502.6615, "t": 236.3916, "r": 524.2556, "b": 222.7121 }, "text": "(68)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p44.20", "order": 21, "type": "text", "bbox": { "l": 68.861, "t": 215.9923, "r": 251.9306, "b": 201.8328 }, "text": "This equation thus determines[p44.20.1].", "children": [ { "id": "p44.20.1", "order": 22, "type": "formula", "bbox": { "l": 68.861, "t": 215.9923, "r": 251.9306, "b": 201.8328 }, "text": "w_{t,s}^{*}", "attrs": { "latex": "w_{t,s}^{*}", "inline": true, "text_offset": 29, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "w sub t, s to the power of *", "mathml": "wt,s*" }, "children": [] } ] }, { "id": "p44.21", "order": 23, "type": "text", "bbox": { "l": 86.1363, "t": 201.5928, "r": 499.7823, "b": 187.9133 }, "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": { "l": 92.1346, "t": 167.994, "r": 478.9081, "b": 118.5558 }, "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": { "block_label": "display_formula", "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*", "mathml": "ndes,ths,tws,t=(1−δx,s)ndes,t−1[ξw,s1+π―1+πt−1ws,t−1hs,t+(1−ξw,s)ws,t*hs,t]+Ms,t[κw,sws,ths,t+(1−κw,s)ws,t*hs,t]," }, "children": [] }, { "id": "p44.23", "order": 25, "type": "text", "bbox": { "l": 502.6615, "t": 150.7146, "r": 524.2556, "b": 136.5551 }, "text": "(69)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p44.24", "order": 26, "type": "text", "bbox": { "l": 67.9013, "t": 110.3961, "r": 526.175, "b": 67.6776 }, "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", "order": 27, "type": "formula", "bbox": { "l": 67.9013, "t": 110.3961, "r": 526.175, "b": 67.6776 }, "text": "M_{s,t}", "attrs": { "latex": "M_{s,t}", "inline": true, "text_offset": 143, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "M sub s, t", "mathml": "Ms,t" }, "children": [] } ] }, { "id": "p44.25", "order": 28, "type": "footer", "bbox": { "l": 289.3603, "t": 50.8782, "r": 305.6758, "b": 39.5986 }, "text": "43", "attrs": { "block_label": "number" }, "children": [] } ], "ai_page_review": { "model": "claude-opus-4-8", "applied": true, "changed": true, "reviewed_at": "2026-08-21T10:46:55+00:00" } }