{ "document_id": 405, "page": 40, "width_pt": 595.276, "height_pt": 841.89, "engine": "paddleocr_vl", "source_run_id": 530, "item_count": 39, "counts_by_type": { "footer": 1, "formula": 20, "heading": 1, "link": 5, "text": 12 }, "build_ms": 4, "items": [ { "id": "p40.1", "order": 1, "type": "text", "bbox": { "l": 66.2218, "t": 769.8926, "r": 526.415, "b": 636.9373 }, "text": "Price setting for non-tradable goods. All non-tradable goods firms that are able to reset their prices, which happens with the probability[p40.1.1], choose the same price[p40.1.2]. Firms that do not reset their prices index their prices according to[p40.1.3], i.e., to a geometric average of past sector-specific inflation,[p40.1.4] and the (constant) inflation target,[p40.1.5]. The parameter[p40.1.6] measures the degree of indexation. Reoptimising firm maximizes the discounted sum of its expected nominal profits subject to the price-indexation scheme and taking as given the demand for its brand. The marginal costs[p40.1.7] are symmetric across producers.[p40.1.8] is the stochastic discount factor of i-type households, who own firms.", "children": [ { "id": "p40.1.1", "order": 2, "type": "formula", "bbox": { "l": 66.2218, "t": 769.8926, "r": 526.415, "b": 636.9373 }, "text": "1 - \\xi_{N}", "attrs": { "latex": "1 - \\xi_{N}", "inline": true, "text_offset": 138, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "1 minus xi sub N", "mathml": "1−ξN" }, "children": [] }, { "id": "p40.1.2", "order": 3, "type": "formula", "bbox": { "l": 66.2218, "t": 769.8926, "r": 526.415, "b": 636.9373 }, "text": "\\widetilde{P}_{NT,t}", "attrs": { "latex": "\\widetilde{P}_{NT,t}", "inline": true, "text_offset": 170, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "widetilde P sub NT, t", "mathml": "P~NT,t" }, "children": [] }, { "id": "p40.1.3", "order": 4, "type": "formula", "bbox": { "l": 66.2218, "t": 769.8926, "r": 526.415, "b": 636.9373 }, "text": "P_{t} = (\\Pi_{NT,t-1})^{\\chi_{N}} \\overline{\\Pi}^{1-\\chi_{N}} P_{t-1}", "attrs": { "latex": "P_{t} = (\\Pi_{NT,t-1})^{\\chi_{N}} \\overline{\\Pi}^{1-\\chi_{N}} P_{t-1}", "inline": true, "text_offset": 249, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "P sub t equals capital pi sub NT, t minus 1 to the power of chi sub N capital pi bar to the power of 1 minus chi sub N P sub t minus 1", "mathml": "Pt=(ΠNT,t−1)χNΠ―1−χNPt−1" }, "children": [] }, { "id": "p40.1.4", "order": 5, "type": "formula", "bbox": { "l": 66.2218, "t": 769.8926, "r": 526.415, "b": 636.9373 }, "text": "\\Pi_{NT,t-1} \\equiv P_{NT,t-1} / P_{NT,t-2}", "attrs": { "latex": "\\Pi_{NT,t-1} \\equiv P_{NT,t-1} / P_{NT,t-2}", "inline": true, "text_offset": 323, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "capital pi sub NT, t minus 1 equiv P sub NT, t minus 1 over P sub NT, t minus 2", "mathml": "ΠNT,t−1≡PNT,t−1/PNT,t−2" }, "children": [] }, { "id": "p40.1.5", "order": 6, "type": "formula", "bbox": { "l": 66.2218, "t": 769.8926, "r": 526.415, "b": 636.9373 }, "text": "\\overline{\\Pi}", "attrs": { "latex": "\\overline{\\Pi}", "inline": true, "text_offset": 369, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "capital pi bar", "mathml": "Π―" }, "children": [] }, { "id": "p40.1.6", "order": 7, "type": "formula", "bbox": { "l": 66.2218, "t": 769.8926, "r": 526.415, "b": 636.9373 }, "text": "\\chi_{N}", "attrs": { "latex": "\\chi_{N}", "inline": true, "text_offset": 393, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "chi sub N", "mathml": "χN" }, "children": [] }, { "id": "p40.1.7", "order": 8, "type": "formula", "bbox": { "l": 66.2218, "t": 769.8926, "r": 526.415, "b": 636.9373 }, "text": "MC_{N,t}", "attrs": { "latex": "MC_{N,t}", "inline": true, "text_offset": 620, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "MC sub N, t", "mathml": "MCN,t" }, "children": [] }, { "id": "p40.1.8", "order": 9, "type": "formula", "bbox": { "l": 66.2218, "t": 769.8926, "r": 526.415, "b": 636.9373 }, "text": "\\Lambda_{i,t,t+k}", "attrs": { "latex": "\\Lambda_{i,t,t+k}", "inline": true, "text_offset": 661, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "capital lambda sub i, t, t plus k", "mathml": "Λi,t,t+k" }, "children": [] } ] }, { "id": "p40.2", "order": 10, "type": "text", "bbox": { "l": 86.6161, "t": 637.1773, "r": 274.2444, "b": 623.7378 }, "text": "The implied first-order condition is:", "children": [] }, { "id": "p40.3", "order": 11, "type": "formula", "bbox": { "l": 80.1379, "t": 612.9382, "r": 494.0239, "b": 573.5796 }, "text": "E_{t}\\left[\\sum_{k=0}^{\\infty}\\left(\\xi_{N}\\right)^{k}\\Lambda_{i,t,t+k}\\left(\\prod_{s=1}^{k}\\Pi_{NT,t+s-1}^{\\chi_{N}}\\overline{\\Pi}^{1-\\chi_{N}}\\widetilde{P}_{NT,t}-\\frac{\\theta_{N}}{\\theta_{N}-1}MC_{N,t+k}\\right)NT_{t+k}\\right]=0", "attrs": { "block_label": "display_formula", "latex": "E_{t}\\left[\\sum_{k=0}^{\\infty}\\left(\\xi_{N}\\right)^{k}\\Lambda_{i,t,t+k}\\left(\\prod_{s=1}^{k}\\Pi_{NT,t+s-1}^{\\chi_{N}}\\overline{\\Pi}^{1-\\chi_{N}}\\widetilde{P}_{NT,t}-\\frac{\\theta_{N}}{\\theta_{N}-1}MC_{N,t+k}\\right)NT_{t+k}\\right]=0", "notation": "latex", "alt": "E sub t the sum of sub k equals 0 to the power of infinity of xi sub N to the power of k capital lambda sub i, t, t plus k of the product of sub s equals 1 to the power of k capital pi sub NT, t plus s minus 1 to the power of chi sub N capital pi bar to the power of 1 minus chi sub N widetilde P sub NT, t minus theta sub N over theta sub N minus 1 MC sub N, t plus k NT sub t plus k equals 0", "mathml": "Et[∑k=0∞(ξN)kΛi,t,t+k(∏s=1kΠNT,t+s−1χNΠ―1−χNP~NT,t−θNθN−1MCN,t+k)NTt+k]=0" }, "children": [] }, { "id": "p40.4", "order": 12, "type": "text", "bbox": { "l": 502.6615, "t": 600.9386, "r": 524.2556, "b": 587.0191 }, "text": "(43)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p40.5", "order": 13, "type": "text", "bbox": { "l": 68.1412, "t": 563.9799, "r": 526.415, "b": 535.6609 }, "text": "Firms whose price contracts are re-optimized set prices to equate the discounted sum of expected revenues to the discounted sum of expected marginal costs.", "children": [] }, { "id": "p40.6", "order": 14, "type": "text", "bbox": { "l": 84.9366, "t": 534.461, "r": 487.3057, "b": 520.0615 }, "text": "With price setting as described above, the sector-specific price index[p40.6.1] is:", "children": [ { "id": "p40.6.1", "order": 15, "type": "formula", "bbox": { "l": 84.9366, "t": 534.461, "r": 487.3057, "b": 520.0615 }, "text": "P_{NT,t}", "attrs": { "latex": "P_{NT,t}", "inline": true, "text_offset": 70, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "P sub NT, t", "mathml": "PNT,t" }, "children": [] } ] }, { "id": "p40.7", "order": 16, "type": "formula", "bbox": { "l": 113.9686, "t": 507.1019, "r": 478.4282, "b": 474.9431 }, "text": "P_{NT,t}=\\left[\\xi_{N}\\left(\\Pi_{NT,t-1}^{\\chi_{N}}\\overline{\\Pi}^{1-\\chi_{N}}P_{NT,t-1}\\right)^{1-\\theta_{N}}+\\left(1-\\xi_{N}\\right)\\left(\\widetilde{P}_{NT,t}\\right)^{1-\\theta_{N}}\\right]^{\\frac{1}{1-\\theta_{N}}}", "attrs": { "block_label": "display_formula", "latex": "P_{N T,t}=\\left[\\xi_{N}\\left(\\Pi_{N T,t-1}^{\\chi_{N}}\\overline{\\Pi}^{1-\\chi_{N}}P_{N T,t-1}\\right)^{1-\\theta_{N}}+\\left(1-\\xi_{N}\\right)\\left(\\widetilde{P}_{N T,t}\\right)^{1-\\theta_{N}}\\right]^{\\frac{1}{1-\\theta_{N}}}", "notation": "latex", "alt": "P sub NT, t equals xi sub N of capital pi sub NT, t minus 1 to the power of chi sub N capital pi bar to the power of 1 minus chi sub N P sub NT, t minus 1 to the power of 1 minus theta sub N plus 1 minus xi sub N times widetilde P sub NT, t to the power of 1 minus theta sub N to the power of 1 over 1 minus theta sub N", "mathml": "PNT,t=[ξN(ΠNT,t−1χNΠ―1−χNPNT,t−1)1−θN+(1−ξN)(P~NT,t)1−θN]11−θN" }, "children": [] }, { "id": "p40.8", "order": 17, "type": "text", "bbox": { "l": 502.4216, "t": 497.9823, "r": 524.2556, "b": 483.5828 }, "text": "(44)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p40.9", "order": 18, "type": "text", "bbox": { "l": 67.6614, "t": 456.7037, "r": 524.4955, "b": 412.5453 }, "text": "Price setting for home tradable goods. Price setting for home tradable goods is analogous to that of non-tradable goods. The probability of resetting the price is[p40.9.1]. The sector-specific price index[p40.9.2] evolves according to:", "children": [ { "id": "p40.9.1", "order": 19, "type": "formula", "bbox": { "l": 67.6614, "t": 456.7037, "r": 524.4955, "b": 412.5453 }, "text": "(1 - \\xi_{H})", "attrs": { "latex": "(1 - \\xi_{H})", "inline": true, "text_offset": 162, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "1 minus xi sub H", "mathml": "(1−ξH)" }, "children": [] }, { "id": "p40.9.2", "order": 20, "type": "formula", "bbox": { "l": 67.6614, "t": 456.7037, "r": 524.4955, "b": 412.5453 }, "text": "P_{HT,t}", "attrs": { "latex": "P_{HT,t}", "inline": true, "text_offset": 204, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "P sub HT, t", "mathml": "PHT,t" }, "children": [] } ] }, { "id": "p40.10", "order": 21, "type": "formula", "bbox": { "l": 115.8881, "t": 400.3057, "r": 475.3091, "b": 367.9069 }, "text": "P_{HT,t}=\\left[\\xi_{H}\\left(\\Pi_{HT,t-1}^{\\chi_{H}}\\overline{{\\Pi}}^{1-\\chi_{H}}P_{HT,t-1}\\right)^{1-\\theta_{T}}+\\left(1-\\xi_{H}\\right)\\left(\\widetilde{P}_{HT,t}\\right)^{1-\\theta_{T}}\\right]^{\\frac{1}{1-\\theta_{T}}}", "attrs": { "block_label": "display_formula", "latex": "P_{H T,t}=\\left[\\xi_{H}\\left(\\Pi_{H T,t-1}^{\\chi_{H}}\\overline{{\\Pi}}^{1-\\chi_{H}}P_{H T,t-1}\\right)^{1-\\theta_{T}}+\\left(1-\\xi_{H}\\right)\\left(\\widetilde{P}_{H T,t}\\right)^{1-\\theta_{T}}\\right]^{\\frac{1}{1-\\theta_{T}}}", "notation": "latex", "alt": "P sub HT, t equals xi sub H of capital pi sub HT, t minus 1 to the power of chi sub H overline capital pi to the power of 1 minus chi sub H P sub HT, t minus 1 to the power of 1 minus theta sub T plus 1 minus xi sub H times widetilde P sub HT, t to the power of 1 minus theta sub T to the power of 1 over 1 minus theta sub T", "mathml": "PHT,t=[ξH(ΠHT,t−1χHΠ―1−χHPHT,t−1)1−θT+(1−ξH)(P~HT,t)1−θT]11−θT" }, "children": [] }, { "id": "p40.11", "order": 22, "type": "text", "bbox": { "l": 502.4216, "t": 390.4661, "r": 524.2556, "b": 376.3066 }, "text": "(45)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p40.12", "order": 23, "type": "text", "bbox": { "l": 66.2218, "t": 349.4275, "r": 526.6549, "b": 238.5515 }, "text": "Price setting for export goods. Firms discriminates across countries, by invoicing and setting the price in the currency of the generic destination market CO (local currency pricing assumption). The probability of optimally resetting prices[p40.12.1], and all firms that reset their prices choose the same price,[p40.12.2], while the others index their prices to[p40.12.3], where[p40.12.4] is the sector-specific inflation. Foreign inflation target is time invariant and equal to the Home inflation target,[p40.12.5].", "children": [ { "id": "p40.12.1", "order": 24, "type": "formula", "bbox": { "l": 66.2218, "t": 349.4275, "r": 526.6549, "b": 238.5515 }, "text": "(1 - \\xi_X)", "attrs": { "latex": "(1 - \\xi_X)", "inline": true, "text_offset": 240, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "1 minus xi sub X", "mathml": "(1−ξX)" }, "children": [] }, { "id": "p40.12.2", "order": 25, "type": "formula", "bbox": { "l": 66.2218, "t": 349.4275, "r": 526.6549, "b": 238.5515 }, "text": "\\widetilde{P}_{X,t}^{H,CO}", "attrs": { "latex": "\\widetilde{P}_{X,t}^{H,CO}", "inline": true, "text_offset": 312, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "widetilde P sub X, t to the power of H, CO", "mathml": "P~X,tH,CO" }, "children": [] }, { "id": "p40.12.3", "order": 26, "type": "formula", "bbox": { "l": 66.2218, "t": 349.4275, "r": 526.6549, "b": 238.5515 }, "text": "P_{X,t}^{CO} = \\left(\\Pi_{X,t-1}^{H,CO}\\right)^{\\chi_X} \\overline{\\Pi}^{1-\\chi_X} P_{X,t-1}^{CO}", "attrs": { "latex": "P_{X,t}^{CO} = \\left(\\prod_{X,t-1}^{H,CO}\\right)^{\\chi_X} \\overline{\\Pi}^{1-\\chi_X} P_{X,t-1}^{CO}", "inline": true, "text_offset": 362, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "P sub X, t to the power of CO equals the product of sub X, t minus 1 to the power of H, CO to the power of chi sub X capital pi bar to the power of 1 minus chi sub X P sub X, t minus 1 to the power of CO", "mathml": "PX,tCO=(∏X,t−1H,CO)χXΠ―1−χXPX,t−1CO" }, "children": [] }, { "id": "p40.12.4", "order": 27, "type": "formula", "bbox": { "l": 66.2218, "t": 349.4275, "r": 526.6549, "b": 238.5515 }, "text": "\\Pi_{X,t-1}^{H,CO} \\equiv P_{X,t-1}^{H,CO} / P_{X,t-2}^{H,CO}", "attrs": { "latex": "\\prod_{X,t-1}^{H,CO} \\equiv P_{X,t-1}^{H,CO} / P_{X,t-2}^{H,CO}", "inline": true, "text_offset": 379, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "the product of sub X, t minus 1 to the power of H, CO equiv P sub X, t minus 1 to the power of H, CO over P sub X, t minus 2 to the power of H, CO", "mathml": "∏X,t−1H,CO≡PX,t−1H,CO/PX,t−2H,CO" }, "children": [] }, { "id": "p40.12.5", "order": 28, "type": "formula", "bbox": { "l": 66.2218, "t": 349.4275, "r": 526.6549, "b": 238.5515 }, "text": "\\overline{\\Pi}^{CO} = \\overline{\\Pi}", "attrs": { "latex": "\\overline{\\Pi}^{CO} = \\overline{\\Pi}", "inline": true, "text_offset": 506, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "capital pi bar to the power of CO equals capital pi bar", "mathml": "Π―CO=Π―" }, "children": [] } ] }, { "id": "p40.13", "order": 29, "type": "text", "bbox": { "l": 85.4165, "t": 237.3515, "r": 485.3863, "b": 222.7121 }, "text": "The bilateral exports price index (of country H to the generic country CO) is:", "children": [] }, { "id": "p40.14", "order": 30, "type": "formula", "bbox": { "l": 92.1346, "t": 211.4325, "r": 481.3074, "b": 178.0737 }, "text": "P_{X,t}^{H,CO}=\\left[\\xi_{X}\\left(\\left(\\Pi_{X,t-1}^{H,CO}\\right)^{\\chi_{X}}\\overline{\\Pi}^{1-\\chi_{X}}P_{X,t-1}^{H,CO}\\right)^{1-\\theta_{T}}+\\left(1-\\xi_{X}\\right)\\left(\\widetilde{P}_{X,t}^{H,CO}\\right)^{1-\\theta_{T}}\\right]^{\\frac{1}{1-\\theta_{T}}}", "attrs": { "block_label": "display_formula", "latex": "P_{X,t}^{H,CO}=\\left[\\xi_{X}\\left(\\left(\\Pi_{X,t-1}^{H,CO}\\right)^{\\chi_{X}}\\overline{\\Pi}^{1-\\chi_{X}}P_{X,t-1}^{H,CO}\\right)^{1-\\theta_{T}}+\\left(1-\\xi_{X}\\right)\\left(\\widetilde{P}_{X,t}^{H,CO}\\right)^{1-\\theta_{T}}\\right]^{\\frac{1}{1-\\theta_{T}}}", "notation": "latex", "alt": "P sub X, t to the power of H, CO equals xi sub X of capital pi sub X, t minus 1 to the power of H, CO to the power of chi sub X capital pi bar to the power of 1 minus chi sub X P sub X, t minus 1 to the power of H, CO to the power of 1 minus theta sub T plus 1 minus xi sub X times widetilde P sub X, t to the power of H, CO to the power of 1 minus theta sub T to the power of 1 over 1 minus theta sub T", "mathml": "PX,tH,CO=[ξX((ΠX,t−1H,CO)χXΠ―1−χXPX,t−1H,CO)1−θT+(1−ξX)(P~X,tH,CO)1−θT]11−θT" }, "children": [] }, { "id": "p40.15", "order": 31, "type": "text", "bbox": { "l": 502.4216, "t": 201.1128, "r": 524.2556, "b": 186.9533 }, "text": "(46)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p40.16", "order": 32, "type": "heading", "bbox": { "l": 68.1412, "t": 159.1143, "r": 418.9246, "b": 140.635 }, "text": "B Outline of the model with search frictions", "attrs": { "block_label": "paragraph_title" }, "children": [] }, { "id": "p40.17", "order": 33, "type": "text", "bbox": { "l": 67.6614, "t": 126.4755, "r": 526.415, "b": 68.3976 }, "text": "This appendix if based on, and follows closely the model description in Gomes et al. (2023), and the reader should refer to this paper for all the details. The labour market structure with search frictions follows the standard search model of Mortensen and Pissarides (1999), but adds sticky wages by means of staggered wage setting, and a", "children": [ { "id": "p40.17.1", "order": 34, "type": "link", "bbox": { "l": 455.964, "t": 125.203, "r": 525.406, "b": 112.463 }, "text": "Gomes et al.", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p40.17.2", "order": 35, "type": "link", "bbox": { "l": 73.456, "t": 111.241, "r": 103.185, "b": 98.017 }, "text": "(2023),", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p40.17.3", "order": 36, "type": "link", "bbox": { "l": 441.862, "t": 96.311, "r": 525.406, "b": 85.591 }, "text": "Mortensen and", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p40.17.4", "order": 37, "type": "link", "bbox": { "l": 69.87, "t": 82.35, "r": 124.68, "b": 69.125 }, "text": "Pissarides", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p40.17.5", "order": 38, "type": "link", "bbox": { "l": 130.537, "t": 82.35, "r": 160.265, "b": 69.125 }, "text": "(1999),", "attrs": { "action_type": "/Named" }, "children": [] } ] }, { "id": "p40.18", "order": 39, "type": "footer", "bbox": { "l": 290.0801, "t": 51.1182, "r": 305.4359, "b": 39.5986 }, "text": "39", "attrs": { "block_label": "number" }, "children": [] } ], "ai_page_review": { "model": "claude-opus-4-8", "applied": true, "changed": true, "reviewed_at": "2026-08-21T10:46:33+00:00" } }