{ "document_id": 395, "page": 36, "width_pt": 595.276, "height_pt": 841.89, "engine": "paddleocr_vl", "source_run_id": 520, "item_count": 39, "counts_by_type": { "footer": 1, "formula": 20, "heading": 1, "link": 3, "text": 14 }, "build_ms": 3, "items": [ { "id": "p36.1", "order": 1, "type": "formula", "bbox": { "l": 191.9471, "t": 760.0529, "r": 402.1292, "b": 728.374 }, "text": "K_{t+1}=(1-\\delta)K_{t}+\\left(1-\\Gamma_{I,t}\\left(\\frac{I_{t}}{I_{t-1}}\\right)\\right),", "attrs": { "block_label": "display_formula", "latex": "K_{t+1}=(1-\\delta)K_{t}+\\left(1-\\Gamma_{I,t}\\left(\\frac{I_{t}}{I_{t-1}}\\right)\\right),", "notation": "latex", "alt": "K sub t plus 1 equals 1 minus delta K sub t plus 1 minus capital gamma sub I, t of I sub t over I sub t minus 1,", "mathml": "Kt+1=(1−δ)Kt+(1−ΓI,t(ItIt−1))," } }, { "id": "p36.2", "order": 2, "type": "text", "bbox": { "l": 67.4214, "t": 725.0142, "r": 417.245, "b": 710.8547 }, "text": "where[p36.2.1] is the depreciation rate and investment-adjustment cost is", "children": [ { "id": "p36.2.1", "order": 3, "type": "formula", "bbox": { "l": 67.4214, "t": 725.0142, "r": 417.245, "b": 710.8547 }, "text": "δ", "attrs": { "latex": "\\delta", "inline": true, "text_offset": 5, "text_span": 9, "bbox_estimated": true, "alt": "delta", "mathml": "δ" } } ] }, { "id": "p36.3", "order": 4, "type": "formula", "bbox": { "l": 215.7006, "t": 700.0551, "r": 377.416, "b": 667.1762 }, "text": "\\Gamma_{I,t}\\left(\\frac{I_{t}}{I_{t-1}}\\right)\\equiv\\frac{\\gamma}{2}\\left(\\frac{I_{t}}{I_{t-1}}-1\\right)^{2}.", "attrs": { "block_label": "display_formula", "latex": "\\Gamma_{I,t}\\left(\\frac{I_{t}}{I_{t-1}}\\right)\\equiv\\frac{\\gamma}{2}\\left(\\frac{I_{t}}{I_{t-1}}-1\\right)^{2}.", "notation": "latex", "alt": "capital gamma sub I, t of I sub t over I sub t minus 1 equiv gamma over 2 times I sub t over I sub t minus 1 minus 1 squared .", "mathml": "ΓI,t(ItIt−1)≡γ2(ItIt−1−1)2." } }, { "id": "p36.4", "order": 5, "type": "text", "bbox": { "l": 68.3812, "t": 663.5764, "r": 343.3454, "b": 648.9369 }, "text": "The first-order condition for investment in capital is:" }, { "id": "p36.5", "order": 6, "type": "text", "bbox": { "l": 501.9417, "t": 752.1332, "r": 524.7354, "b": 737.7337 }, "text": "(14)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" } }, { "id": "p36.6", "order": 7, "type": "text", "bbox": { "l": 502.4216, "t": 690.4554, "r": 524.4955, "b": 676.0559 }, "text": "(15)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" } }, { "id": "p36.7", "order": 8, "type": "formula", "bbox": { "l": 77.7386, "t": 629.4976, "r": 493.3041, "b": 596.1388 }, "text": "p_{I,t}=q_{i,t}\\left(1-\\Gamma_{I,t}\\left(\\frac{I_{t}}{I_{t-1}}\\right)-\\Gamma_{I,t}^{\\prime}\\left(\\frac{I_{t}}{I_{t-1}}\\right)I_{t}\\right)+\\beta\\frac{u^{\\prime}(c_{I,t+1})}{u^{\\prime}(c_{I,t})}q_{I,t+1}\\Gamma_{I,t}^{\\prime}\\left(\\frac{I_{t}}{I_{t-1}}\\right)\\frac{I_{t+1}^{2}}{I_{t}},", "attrs": { "block_label": "display_formula", "latex": "p_{I,t}=q_{i,t}\\left(1-\\Gamma_{I,t}\\left(\\frac{I_{t}}{I_{t-1}}\\right)-\\Gamma_{I,t}^{\\prime}\\left(\\frac{I_{t}}{I_{t-1}}\\right)I_{t}\\right)+\\beta\\frac{u^{\\prime}(c_{I,t+1})}{u^{\\prime}(c_{I,t})}q_{I,t+1}\\Gamma_{I,t}^{\\prime}\\left(\\frac{I_{t}}{I_{t-1}}\\right)\\frac{I_{t+1}^{2}}{I_{t}},", "notation": "latex", "alt": "p sub I, t equals q sub i, t of 1 minus capital gamma sub I, t of I sub t over I sub t minus 1 minus capital gamma sub I, t to the power of prime of I sub t over I sub t minus 1 I sub t plus beta u to the power of prime of c sub I, t plus 1 over u to the power of prime of c sub I, t q sub I, t plus 1 capital gamma sub I, t to the power of prime of I sub t over I sub t minus 1 I sub t plus 1 squared over I sub t,", "mathml": "pI,t=qi,t(1−ΓI,t(ItIt−1)−ΓI,t′(ItIt−1)It)+βu′(cI,t+1)u′(cI,t)qI,t+1ΓI,t′(ItIt−1)It+12It," } }, { "id": "p36.8", "order": 9, "type": "text", "bbox": { "l": 502.4216, "t": 620.8579, "r": 524.2556, "b": 606.4584 }, "text": "(16)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" } }, { "id": "p36.9", "order": 10, "type": "text", "bbox": { "l": 67.1815, "t": 587.0191, "r": 526.175, "b": 555.3402 }, "text": "where primes indicate derivatives,[p36.9.1] is Tobin's q, and[p36.9.2] is the relative price of investment goods.", "children": [ { "id": "p36.9.1", "order": 11, "type": "formula", "bbox": { "l": 67.1815, "t": 587.0191, "r": 526.175, "b": 555.3402 }, "text": "q_{I,t}", "attrs": { "latex": "q_{I,t}", "inline": true, "text_offset": 34, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "q sub I, t", "mathml": "qI,t" } }, { "id": "p36.9.2", "order": 12, "type": "formula", "bbox": { "l": 67.1815, "t": 587.0191, "r": 526.175, "b": 555.3402 }, "text": "p_{I,t} \\equiv \\frac{P_{I,t}}{P_{C,t}}", "attrs": { "latex": "p_{I,t} \\equiv \\frac{P_{I,t}}{P_{C,t}}", "inline": true, "text_offset": 61, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "p sub I, t equiv P sub I, t over P sub C, t", "mathml": "pI,t≡PI,tPC,t" } } ] }, { "id": "p36.10", "order": 13, "type": "formula", "bbox": { "l": 67.4214, "t": 535.1809, "r": 526.8948, "b": 500.8622 }, "text": "q_{I,t}=\\beta\\frac{u^{\\prime}(c_{I,t+1})}{u^{\\prime}(c_{I,t})}\\left[(1-\\delta)q_{I,t+1}+(1-\\tau_{t+1}^{k})r_{K,t+1}u_{t+1}+\\left(\\tau_{t+1}^{k}\\delta-(1-\\tau_{t+1}^{k})\\Gamma(u_{t+1})\\right)p_{I,t+1}\\right].", "attrs": { "block_label": "display_formula", "latex": "q_{I,t}=\\beta\\frac{u^{\\prime}(c_{I,t+1})}{u^{\\prime}(c_{I,t})}\\left[(1-\\delta)q_{I,t+1}+(1-\\tau_{t+1}^{k})r_{K,t+1}u_{t+1}+\\left(\\tau_{t+1}^{k}\\delta-(1-\\tau_{t+1}^{k})\\Gamma(u_{t+1})\\right)p_{I,t+1}\\right].", "notation": "latex", "alt": "q sub I, t equals beta u to the power of prime of c sub I, t plus 1 over u to the power of prime of c sub I, t 1 minus delta q sub I, t plus 1 plus 1 minus tau sub t plus 1 to the power of k r sub K, t plus 1 u sub t plus 1 plus tau sub t plus 1 to the power of k delta minus 1 minus tau sub t plus 1 to the power of k capital gamma of u sub t plus 1 p sub I, t plus 1 .", "mathml": "qI,t=βu′(cI,t+1)u′(cI,t)[(1−δ)qI,t+1+(1−τt+1k)rK,t+1ut+1+(τt+1kδ−(1−τt+1k)Γ(ut+1))pI,t+1]." } }, { "id": "p36.11", "order": 14, "type": "text", "bbox": { "l": 502.4216, "t": 505.662, "r": 524.0156, "b": 492.4625 }, "text": "(17)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" } }, { "id": "p36.12", "order": 15, "type": "text", "bbox": { "l": 67.9013, "t": 491.2625, "r": 525.2153, "b": 462.9435 }, "text": "where[p36.12.1] is the return on physical capital,[p36.12.2] is capital utilisation, and[p36.12.3] is capital utilisation adjustment cost.", "children": [ { "id": "p36.12.1", "order": 16, "type": "formula", "bbox": { "l": 67.9013, "t": 491.2625, "r": 525.2153, "b": 462.9435 }, "text": "r_{K,t}", "attrs": { "latex": "r_{K,t}", "inline": true, "text_offset": 5, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "r sub K, t", "mathml": "rK,t" } }, { "id": "p36.12.2", "order": 17, "type": "formula", "bbox": { "l": 67.9013, "t": 491.2625, "r": 525.2153, "b": 462.9435 }, "text": "u_{t}", "attrs": { "latex": "u_{t}", "inline": true, "text_offset": 50, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "u sub t", "mathml": "ut" } }, { "id": "p36.12.3", "order": 18, "type": "formula", "bbox": { "l": 67.9013, "t": 491.2625, "r": 525.2153, "b": 462.9435 }, "text": "\\Gamma(u_{t})", "attrs": { "latex": "\\Gamma(u_{t})", "inline": true, "text_offset": 88, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "capital gamma of u sub t", "mathml": "Γ(ut)" } } ] }, { "id": "p36.13", "order": 19, "type": "text", "bbox": { "l": 67.4214, "t": 462.2235, "r": 525.9351, "b": 418.7851 }, "text": "Total capital services in the economy are the product of the capital stock and its utilisation, and are used by tradable[p36.13.1] and non-tradable[p36.13.2] sectors of intermediate goods firms:", "children": [ { "id": "p36.13.1", "order": 20, "type": "formula", "bbox": { "l": 67.4214, "t": 462.2235, "r": 525.9351, "b": 418.7851 }, "text": "K_{T,t}", "attrs": { "latex": "K_{T,t}", "inline": true, "text_offset": 120, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "K sub T, t", "mathml": "KT,t" } }, { "id": "p36.13.2", "order": 21, "type": "formula", "bbox": { "l": 67.4214, "t": 462.2235, "r": 525.9351, "b": 418.7851 }, "text": "K_{N,t}", "attrs": { "latex": "K_{N,t}", "inline": true, "text_offset": 147, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "K sub N, t", "mathml": "KN,t" } } ] }, { "id": "p36.14", "order": 22, "type": "formula", "bbox": { "l": 243.5329, "t": 404.1456, "r": 350.0636, "b": 389.0261 }, "text": "u_{t}K_{t}=K_{N,t}+K_{T,t}.", "attrs": { "block_label": "display_formula", "latex": "u_{t}K_{t}=K_{N,t}+K_{T,t}.", "notation": "latex", "alt": "u sub t K sub t equals K sub N, t plus K sub T, t .", "mathml": "utKt=KN,t+KT,t." } }, { "id": "p36.15", "order": 23, "type": "text", "bbox": { "l": 502.4216, "t": 404.8656, "r": 524.4955, "b": 390.9461 }, "text": "(18)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" } }, { "id": "p36.16", "order": 24, "type": "heading", "bbox": { "l": 68.1412, "t": 369.3468, "r": 202.7441, "b": 353.5074 }, "text": "A.2 Labour market", "attrs": { "block_label": "paragraph_title" } }, { "id": "p36.17", "order": 25, "type": "text", "bbox": { "l": 66.9416, "t": 343.9077, "r": 526.175, "b": 286.0698 }, "text": "The labour market in the standard model is identical to what is typical in the literature - a standard staggered wage setting. This is the part where the main difference between the model with and without search frictions lies, namely, the standard part described briefly below is replaced by the fully-fledged search frictions, described in detail in Appendix B.", "children": [ { "id": "p36.17.1", "order": 26, "type": "link", "bbox": { "l": 512.282, "t": 299.162, "r": 522.01, "b": 286.422 }, "text": "B.", "attrs": { "action_type": "/Named" } } ] }, { "id": "p36.18", "order": 27, "type": "text", "bbox": { "l": 66.4617, "t": 285.1098, "r": 526.415, "b": 152.6346 }, "text": "Households supply differentiated labour services in monopolistically competitive market for each type of labour service. Wages are determined by staggered wage setting in terms of nominal wages, where the probability that a wage contract is reset in a given period is[p36.18.1], where $s$ stands for Ricardian or hand-to-mouth households. Households that can reset wage choose the same wage[p36.18.2], while wages that are not reset are indexed to past wages by a combination of the past CPI inflation,[p36.18.3], where[p36.18.4] is the consumer price index, and inflation target,[p36.18.5], where the indexation weight is[p36.18.6]. The first order condition for the optimal reset wage is", "children": [ { "id": "p36.18.1", "order": 28, "type": "formula", "bbox": { "l": 66.4617, "t": 285.1098, "r": 526.415, "b": 152.6346 }, "text": "1 - \\xi_{s}", "attrs": { "latex": "1 - \\xi_{s}", "inline": true, "text_offset": 267, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "1 minus xi sub s", "mathml": "1−ξs" } }, { "id": "p36.18.2", "order": 29, "type": "formula", "bbox": { "l": 66.4617, "t": 285.1098, "r": 526.415, "b": 152.6346 }, "text": "\\tilde{W}_{s,t}", "attrs": { "latex": "\\tilde{W}_{s,t}", "inline": true, "text_offset": 390, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "tilde W sub s, t", "mathml": "W~s,t" } }, { "id": "p36.18.3", "order": 30, "type": "formula", "bbox": { "l": 66.4617, "t": 285.1098, "r": 526.415, "b": 152.6346 }, "text": "\\Pi_{C,t-1} = P_{C,t-1}/P_{C,t-2}", "attrs": { "latex": "\\Pi_{C,t-1} = P_{C,t-1}/P_{C,t-2}", "inline": true, "text_offset": 502, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "capital pi sub C, t minus 1 equals P sub C, t minus 1 over P sub C, t minus 2", "mathml": "ΠC,t−1=PC,t−1/PC,t−2" } }, { "id": "p36.18.4", "order": 31, "type": "formula", "bbox": { "l": 66.4617, "t": 285.1098, "r": 526.415, "b": 152.6346 }, "text": "P_{C,t}", "attrs": { "latex": "P_{C,t}", "inline": true, "text_offset": 519, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "P sub C, t", "mathml": "PC,t" } }, { "id": "p36.18.5", "order": 32, "type": "formula", "bbox": { "l": 66.4617, "t": 285.1098, "r": 526.415, "b": 152.6346 }, "text": "\\overline{\\Pi}", "attrs": { "latex": "\\overline{\\Pi}", "inline": true, "text_offset": 580, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "capital pi bar", "mathml": "Π―" } }, { "id": "p36.18.6", "order": 33, "type": "formula", "bbox": { "l": 66.4617, "t": 285.1098, "r": 526.415, "b": 152.6346 }, "text": "\\chi_{s} \\colon W_{t,i} = \\Pi_{C,t-1}^{\\chi_{s}} \\overline{\\Pi}^{1-\\chi_{s}} W_{t-1,i}", "attrs": { "latex": "\\chi_{s} \\colon W_{t,i} = \\Pi_{C,t-1}^{\\chi_{s}} \\overline{\\Pi}^{1-\\chi_{s}} W_{t-1,i}", "inline": true, "text_offset": 622, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "chi sub s colon W sub t, i equals capital pi sub C, t minus 1 to the power of chi sub s capital pi bar to the power of 1 minus chi sub s W sub t minus 1, i", "mathml": "χs:Wt,i=ΠC,t−1χsΠ―1−χsWt−1,i" } } ] }, { "id": "p36.19", "order": 34, "type": "formula", "bbox": { "l": 68.6211, "t": 133.9152, "r": 567.6836, "b": 92.1567 }, "text": "E_{t}\\left[\\sum_{k=0}^{\\infty}(\\beta\\xi_{s})^{k}\\left(u^{\\prime}(c)_{s,t+k}(1-\\tau_{t+k}^{N}-\\tau_{t+k}^{W_{h}})\\frac{\\tilde{W}_{s,t}}{P_{C,t+k}}\\left(\\frac{P_{C,t+k-1}}{P_{C,t-1}}\\right)^{\\chi_{s}}\\overline{\\Pi}^{(1-\\chi_{s})k}-\\frac{\\eta_{s}}{\\eta_{s}-1}N_{t+k}^{\\zeta}\\right)N_{t+k}\\right]=0", "attrs": { "block_label": "display_formula", "latex": "E_{t}\\left[\\sum_{k=0}^{\\infty}(\\beta\\xi_{s})^{k}\\left(u^{\\prime}(c)_{s,t+k}(1-\\tau_{t+k}^{N}-\\tau_{t+k}^{W_{h}})\\frac{\\tilde{W}_{s,t}}{P_{C,t+k}}\\left(\\frac{P_{C,t+k-1}}{P_{C,t-1}}\\right)^{\\chi_{s}}\\overline{\\Pi}^{(1-\\chi_{s})k}-\\frac{\\eta_{s}}{\\eta_{s}-1}N_{t+k}^{\\zeta}\\right)N_{t+k}\\right]=0", "notation": "latex", "alt": "E sub t the sum of sub k equals 0 to the power of infinity of beta xi sub s to the power of k of u to the power of prime of c sub s, t plus k of 1 minus tau sub t plus k to the power of N minus tau sub t plus k to the power of W sub h frac tilde W sub s, t P sub C, t plus k of P sub C, t plus k minus 1 over P sub C, t minus 1 to the power of chi sub s capital pi bar to the power of of 1 minus chi sub s k minus eta sub s over eta sub s minus 1 N sub t plus k to the power of zeta N sub t plus k equals 0", "mathml": "Et[∑k=0∞(βξs)k(u′(c)s,t+k(1−τt+kN−τt+kWh)W~s,tPC,t+k(PC,t+k−1PC,t−1)χsΠ―(1−χs)k−ηsηs−1Nt+kζ)Nt+k]=0" } }, { "id": "p36.20", "order": 35, "type": "text", "bbox": { "l": 502.1816, "t": 95.7566, "r": 524.7354, "b": 82.797 }, "text": "(19)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" } }, { "id": "p36.21", "order": 36, "type": "text", "bbox": { "l": 85.6564, "t": 82.3171, "r": 354.6223, "b": 67.4376 }, "text": "For details see the Appendix in Gomes et al. (2010).", "children": [ { "id": "p36.21.1", "order": 37, "type": "link", "bbox": { "l": 248.668, "t": 80.978, "r": 314.346, "b": 67.754 }, "text": "Gomes et al.", "attrs": { "action_type": "/Named" } }, { "id": "p36.21.2", "order": 38, "type": "link", "bbox": { "l": 318.247, "t": 80.978, "r": 347.976, "b": 67.754 }, "text": "(2010).", "attrs": { "action_type": "/Named" } } ] }, { "id": "p36.22", "order": 39, "type": "footer", "bbox": { "l": 290.0801, "t": 51.1182, "r": 305.4359, "b": 39.3586 }, "text": "35", "attrs": { "block_label": "number" } } ] }