{ "document_id": 405, "page": 42, "width_pt": 595.276, "height_pt": 841.89, "engine": "paddleocr_vl", "source_run_id": 530, "item_count": 33, "counts_by_type": { "footer": 1, "formula": 16, "link": 5, "text": 11 }, "build_ms": 7, "items": [ { "id": "p42.1", "order": 1, "type": "text", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "The term[p42.1.2] denotes the effective hours that a labour firm produces from hours[p42.1.3] supplied by the worker from household s.[p42.1.4] is what the labour packer pays for such unit of labour. The first term in equation 52 therefore measures earnings of the labour firm from selling hours worked. But for these hours it has to pay to the household hourly wage, which is in this case newly renegotiated,[p42.1.5]. Because we assume that labour firms pay some labour taxes (social security contributions), the cost for the labour firm is increased by taxes paid, at the rate[p42.1.6]. In the next period, if the firm and the worker do not separate, which occurs with probability[p42.1.7], two cases can arise. In the first case, which occurs with the probability[p42.1.8], wages are renegotiated and the value of the worker for the labour firm is again the value of a worker with a renegotiated wage, just in the next period,[p42.1.9]. With probability[p42.1.10] wages are not renegotiated and the firm is in next period stuck with the worker value at the current wage,[p42.1.11].", "children": [ { "id": "p42.1.1", "order": 2, "type": "link", "bbox": { "l": 284.855, "t": 739.805, "r": 300.716, "b": 727.064 }, "text": "52", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p42.1.2", "order": 3, "type": "formula", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "h_{s,t}^{\\alpha_{H}}", "attrs": { "latex": "h_{s,t}^{\\alpha_{H}}", "inline": true, "text_offset": 8, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "h sub s, t to the power of alpha sub H", "mathml": "hs,tαH" }, "children": [] }, { "id": "p42.1.3", "order": 4, "type": "formula", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "h_{s,t}", "attrs": { "latex": "h_{s,t}", "inline": true, "text_offset": 84, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "h sub s, t", "mathml": "hs,t" }, "children": [] }, { "id": "p42.1.4", "order": 5, "type": "formula", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "x_{s,t}", "attrs": { "latex": "x_{s,t}", "inline": true, "text_offset": 134, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "x sub s, t", "mathml": "xs,t" }, "children": [] }, { "id": "p42.1.5", "order": 6, "type": "formula", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "w_{s,t}^{*}", "attrs": { "latex": "w_{s,t}^{*}", "inline": true, "text_offset": 409, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "w sub s, t to the power of *", "mathml": "ws,t*" }, "children": [] }, { "id": "p42.1.6", "order": 7, "type": "formula", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "\\tau_{t}^{wf}", "attrs": { "latex": "\\tau_{t}^{wf}", "inline": true, "text_offset": 579, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "tau sub t to the power of wf", "mathml": "τtwf" }, "children": [] }, { "id": "p42.1.7", "order": 8, "type": "formula", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "(1 - \\delta_{x,s})", "attrs": { "latex": "(1 - \\delta_{x,s})", "inline": true, "text_offset": 683, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "1 minus delta sub x, s", "mathml": "(1−δx,s)" }, "children": [] }, { "id": "p42.1.8", "order": 9, "type": "formula", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "(1 - \\xi_{w,s})", "attrs": { "latex": "(1 - \\xi_{w,s})", "inline": true, "text_offset": 767, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "1 minus xi sub w, s", "mathml": "(1−ξw,s)" }, "children": [] }, { "id": "p42.1.9", "order": 10, "type": "formula", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "\\mathcal{A}_{t+1}^{F}(w_{s,t+1}^{*})", "attrs": { "latex": "\\mathcal{A}_{t+1}^{F}(w_{s,t+1}^{*})", "inline": true, "text_offset": 930, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "mathcal A sub t plus 1 to the power of F of w sub s, t plus 1 to the power of *", "mathml": "𝒜t+1F(ws,t+1*)" }, "children": [] }, { "id": "p42.1.10", "order": 11, "type": "formula", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "\\xi_{w,s}", "attrs": { "latex": "\\xi_{w,s}", "inline": true, "text_offset": 957, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "xi sub w, s", "mathml": "ξw,s" }, "children": [] }, { "id": "p42.1.11", "order": 12, "type": "formula", "bbox": { "l": 66.9416, "t": 769.1726, "r": 526.6549, "b": 595.8988 }, "text": "\\mathcal{A}_{t+1}^{F}(w_{s,t}^{*})", "attrs": { "latex": "\\mathcal{A}_{t+1}^{F}(w_{s,t}^{*})", "inline": true, "text_offset": 1074, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "mathcal A sub t plus 1 to the power of F of w sub s, t to the power of *", "mathml": "𝒜t+1F(ws,t*)" }, "children": [] } ] }, { "id": "p42.2", "order": 13, "type": "text", "bbox": { "l": 85.8963, "t": 595.6588, "r": 449.6361, "b": 582.9392 }, "text": "The value at $ t+1 $ of the worker with renegotiated wage from time t is", "children": [] }, { "id": "p42.3", "order": 14, "type": "formula", "bbox": { "l": 122.6062, "t": 561.82, "r": 470.7503, "b": 511.9018 }, "text": "\\begin{aligned}\\mathcal{A}_{t+1}^{F}(w_{s,t}^{*})&=u^{\\prime}(c_{s,t+1})\\left(h_{s,t+1}^{\\alpha_{H}}x_{s,t+1}-h_{s,t+1}w_{s,t}^{*}\\frac{(1+\\overline{\\pi})P_{t}}{P_{t+1}}(1+\\tau_{t+1}^{wf})\\right)\\\\&+\\beta(1-\\delta_{x,s})\\left[(1-\\xi_{w,s})\\mathcal{A}_{t+2}^{F}(w_{s,t+2}^{*})+\\xi_{w,s}\\mathcal{A}_{t+2}^{F}(w_{s,t}^{*})\\right]\\end{aligned}", "attrs": { "block_label": "display_formula", "latex": "\\begin{aligned}\\mathcal{A}_{t+1}^{F}(w_{s,t}^{*})&=u^{\\prime}(c_{s,t+1})\\left(h_{s,t+1}^{\\alpha_{H}}x_{s,t+1}-h_{s,t+1}w_{s,t}^{*}\\frac{(1+\\overline{\\pi})P_{t}}{P_{t+1}}(1+\\tau_{t+1}^{wf})\\right)\\\\&+\\beta(1-\\delta_{x,s})\\left[(1-\\xi_{w,s})\\mathcal{A}_{t+2}^{F}(w_{s,t+2}^{*})+\\xi_{w,s}\\mathcal{A}_{t+2}^{F}(w_{s,t}^{*})\\right]\\end{aligned}", "notation": "latex", "alt": "begin aligned mathcal A sub t plus 1 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 plus 1 times h sub s, t plus 1 to the power of alpha sub H x sub s, t plus 1 minus h sub s, t plus 1 w sub s, t to the power of * 1 plus pi bar P sub t over P sub t plus 1 times 1 plus tau sub t plus 1 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 2 to the power of F of w sub s, t plus 2 to the power of * plus xi sub w, s mathcal A sub t plus 2 to the power of F of w sub s, t to the power of * end aligned", "mathml": "𝒜t+1F(ws,t*)&=u′(cs,t+1)(hs,t+1αHxs,t+1−hs,t+1ws,t*(1+π―)PtPt+1(1+τt+1wf))&+β(1−δx,s)[(1−ξw,s)𝒜t+2F(ws,t+2*)+ξw,s𝒜t+2F(ws,t*)]" }, "children": [] }, { "id": "p42.4", "order": 15, "type": "text", "bbox": { "l": 502.4216, "t": 545.0206, "r": 524.0156, "b": 531.1011 }, "text": "(53)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p42.5", "order": 16, "type": "text", "bbox": { "l": 67.9013, "t": 503.982, "r": 525.4552, "b": 475.4231 }, "text": "The wage from the previous period has been indexed by the ratio of trend inflation[p42.5.1] and the price level growth,[p42.5.2].", "children": [ { "id": "p42.5.1", "order": 17, "type": "formula", "bbox": { "l": 67.9013, "t": 503.982, "r": 525.4552, "b": 475.4231 }, "text": "\\bar{\\pi}", "attrs": { "latex": "\\bar{\\pi}", "inline": true, "text_offset": 82, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "bar pi", "mathml": "π¯" }, "children": [] }, { "id": "p42.5.2", "order": 18, "type": "formula", "bbox": { "l": 67.9013, "t": 503.982, "r": 525.4552, "b": 475.4231 }, "text": "P_{t}/P_{t+1} = (1 + \\pi_{t+1})", "attrs": { "latex": "P_{t}/P_{t+1} = (1 + \\pi_{t+1})", "inline": true, "text_offset": 119, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "P sub t over P sub t plus 1 equals 1 plus pi sub t plus 1", "mathml": "Pt/Pt+1=(1+πt+1)" }, "children": [] } ] }, { "id": "p42.6", "order": 19, "type": "text", "bbox": { "l": 68.1412, "t": 474.7031, "r": 525.9351, "b": 446.3841 }, "text": "If we substitute equation $ 53 $ into equation $ 52 $, and do this for every future period, we arrive at the following expression:", "children": [ { "id": "p42.6.1", "order": 20, "type": "link", "bbox": { "l": 219.15, "t": 474.01, "r": 235.01, "b": 461.27 }, "text": "53", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p42.6.2", "order": 21, "type": "link", "bbox": { "l": 307.106, "t": 474.01, "r": 322.967, "b": 461.27 }, "text": "52,", "attrs": { "action_type": "/Named" }, "children": [] } ] }, { "id": "p42.7", "order": 22, "type": "formula", "bbox": { "l": 113.9686, "t": 425.5048, "r": 479.3879, "b": 312.2289 }, "text": "\\begin{aligned}\\mathcal{A}_{t}^{F}(w_{s,t}^{*})&=\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}u^{\\prime}(c_{s,t+k})h_{s,t+k}^{\\alpha_{H}}x_{s,t+k}\\\\&-w_{s,t}^{*}\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}u^{\\prime}(c_{s,t+k})\\frac{(1+\\overline{\\pi})^{k}P_{t}}{P_{t+k}}h_{s,t+k}(1+\\tau_{t+k}^{wf})\\\\&+\\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}^{*})\\end{aligned}", "attrs": { "block_label": "display_formula", "latex": "\\begin{aligned}\\mathcal{A}_{t}^{F}(w_{s,t}^{*})&=\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}u^{\\prime}(c_{s,t+k})h_{s,t+k}^{\\alpha_{H}}x_{s,t+k}\\\\&-w_{s,t}^{*}\\sum_{k=0}^{\\infty}\\left[\\beta(1-\\delta_{x,s})\\xi_{w,s}\\right]^{k}u^{\\prime}(c_{s,t+k})\\frac{(1+\\overline{\\pi})^{k}P_{t}}{P_{t+k}}h_{s,t+k}(1+\\tau_{t+k}^{wf})\\\\&+\\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}^{*})\\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 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 plus k to the power of alpha sub H x sub s, t plus k & minus w sub s, t to the power of * 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 times 1 plus pi bar to the power of k P sub t over P sub t plus k h sub s, t plus k of 1 plus tau sub t plus k to the power of wf & 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 * end aligned", "mathml": "𝒜tF(ws,t*)&=∑k=0∞[β(1−δx,s)ξw,s]ku′(cs,t+k)hs,t+kαHxs,t+k&−ws,t*∑k=0∞[β(1−δx,s)ξw,s]ku′(cs,t+k)(1+π―)kPtPt+khs,t+k(1+τt+kwf)&+∑k=0∞β(1−δx,s)(1−ξw,s)[β(1−δx,s)ξw,s]k𝒜t+k+1F(ws,t+k+1*)" }, "children": [] }, { "id": "p42.8", "order": 23, "type": "text", "bbox": { "l": 502.4216, "t": 376.5466, "r": 524.2556, "b": 362.3871 }, "text": "(54)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p42.9", "order": 24, "type": "text", "bbox": { "l": 67.6614, "t": 303.1092, "r": 525.6952, "b": 273.8302 }, "text": "One can define auxiliary variables and write the infinite sums in recursive form. For the first line in equation 54, define", "children": [ { "id": "p42.9.1", "order": 25, "type": "link", "bbox": { "l": 195.268, "t": 287.661, "r": 211.128, "b": 274.921 }, "text": "54,", "attrs": { "action_type": "/Named" }, "children": [] } ] }, { "id": "p42.10", "order": 26, "type": "formula", "bbox": { "l": 75.0993, "t": 254.3909, "r": 515.378, "b": 216.4723 }, "text": "S_{s,t}^{x}\\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+k}^{\\alpha_{H}}x_{s,t+k}=u^{\\prime}(c_{s,t})h_{s,t}^{\\alpha_{H}}x_{s,t}+\\beta(1-\\delta_{x,s})\\xi_{w,s}S_{s,t+1}^{x}", "attrs": { "block_label": "display_formula", "latex": "S_{s,t}^{x}\\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+k}^{\\alpha_{H}}x_{s,t+k}=u^{\\prime}(c_{s,t})h_{s,t}^{\\alpha_{H}}x_{s,t}+\\beta(1-\\delta_{x,s})\\xi_{w,s}S_{s,t+1}^{x}", "notation": "latex", "alt": "S sub s, t to the power of x 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 plus k to the power of alpha sub H x sub s, t plus k equals u to the power of prime of c sub s, t h sub s, t to the power of alpha sub H x sub s, t plus beta of 1 minus delta sub x, s xi sub w, s S sub s, t plus 1 to the power of x", "mathml": "Ss,tx≡∑k=0∞[β(1−δx,s)ξw,s]ku′(cs,t+k)hs,t+kαHxs,t+k=u′(cs,t)hs,tαHxs,t+β(1−δx,s)ξw,sSs,t+1x" }, "children": [] }, { "id": "p42.11", "order": 27, "type": "text", "bbox": { "l": 502.4216, "t": 218.1522, "r": 524.2556, "b": 203.5127 }, "text": "(55)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p42.12", "order": 28, "type": "text", "bbox": { "l": 85.8963, "t": 193.4331, "r": 226.0177, "b": 179.5136 }, "text": "For the second line, define", "children": [] }, { "id": "p42.13", "order": 29, "type": "formula", "bbox": { "l": 142.2808, "t": 159.1143, "r": 451.0757, "b": 90.2368 }, "text": "\\begin{align*}S_{s,t}^{wf}&\\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}^{wf})\\\\&=u^{\\prime}(c_{s,t})h_{s,t}(1+\\tau_{t}^{wf})+\\beta(1-\\delta_{x,s})\\frac{(1+\\overline{\\pi})}{(1+\\pi_{t+1})}\\xi_{w,s}S_{s,t+1}^{wf}\\end{align*}", "attrs": { "block_label": "display_formula", "latex": "\\begin{align*}S_{s,t}^{wf}&\\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}^{wf})\\\\&=u^{\\prime}(c_{s,t})h_{s,t}(1+\\tau_{t}^{wf})+\\beta(1-\\delta_{x,s})\\frac{(1+\\overline{\\pi})}{(1+\\pi_{t+1})}\\xi_{w,s}S_{s,t+1}^{wf}\\end{align*}", "notation": "latex", "alt": "begin align* S sub s, t to the power of wf & 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 plus tau sub t to the power of wf & equals u to the power of prime of c sub s, t h sub s, t of 1 plus tau sub t to the power of wf 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 wf end align*", "mathml": "Ss,twf≡∑k=0∞[β(1−δx,s)ξw,s]ku′(cs,t+k)hs,t(1+π―)kPtPt+k(1+τtwf)=u′(cs,t)hs,t(1+τtwf)+β(1−δx,s)(1+π―)(1+πt+1)ξw,sSs,t+1wf" }, "children": [] }, { "id": "p42.14", "order": 30, "type": "text", "bbox": { "l": 502.1816, "t": 132.2353, "r": 524.2556, "b": 118.3158 }, "text": "(56)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p42.15", "order": 31, "type": "text", "bbox": { "l": 85.8963, "t": 82.0771, "r": 372.1375, "b": 67.1976 }, "text": "Using these definitions, we can simplify equation 54 to", "children": [ { "id": "p42.15.1", "order": 32, "type": "link", "bbox": { "l": 341.261, "t": 80.494, "r": 357.121, "b": 67.754 }, "text": "54", "attrs": { "action_type": "/Named" }, "children": [] } ] }, { "id": "p42.16", "order": 33, "type": "footer", "bbox": { "l": 289.3603, "t": 50.8782, "r": 305.1959, "b": 39.3586 }, "text": "41", "attrs": { "block_label": "number" }, "children": [] } ], "ai_page_review": { "model": "claude-opus-4-8", "applied": true, "changed": true, "reviewed_at": "2026-08-21T10:46:48+00:00" } }