{ "document_id": 405, "page": 35, "width_pt": 595.276, "height_pt": 841.89, "engine": "paddleocr_vl", "source_run_id": 530, "item_count": 48, "counts_by_type": { "footer": 1, "footnote": 1, "formula": 26, "heading": 2, "link": 6, "text": 12 }, "build_ms": 4, "items": [ { "id": "p35.1", "order": 1, "type": "heading", "bbox": { "l": 67.9013, "t": 772.7725, "r": 525.6952, "b": 753.8131 }, "text": "A Outline of the standard model (without search frictions)", "attrs": { "block_label": "paragraph_title" }, "children": [] }, { "id": "p35.2", "order": 2, "type": "text", "bbox": { "l": 67.6614, "t": 740.8536, "r": 526.6549, "b": 696.9352 }, "text": "This section reports the equations of the core model. It closely follows the description of the EAGLE model Gomes et al. (2012), and the reader should refer for all the details to that paper.", "children": [ { "id": "p35.2.1", "order": 3, "type": "link", "bbox": { "l": 165.153, "t": 725.066, "r": 232.05, "b": 712.218 }, "text": "Gomes et al.", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p35.2.2", "order": 4, "type": "link", "bbox": { "l": 236.562, "t": 725.066, "r": 266.29, "b": 712.218 }, "text": "(2012),", "attrs": { "action_type": "/Named" }, "children": [] } ] }, { "id": "p35.3", "order": 5, "type": "heading", "bbox": { "l": 67.9013, "t": 675.8159, "r": 184.2692, "b": 659.9765 }, "text": "A.1 Households", "attrs": { "block_label": "paragraph_title" }, "children": [] }, { "id": "p35.4", "order": 6, "type": "text", "bbox": { "l": 67.9013, "t": 650.6168, "r": 526.415, "b": 607.8983 }, "text": "There are two types of households in the model, Ricardian (unconstrained) and non-Ricardian (hand-to-mouth, HtM, or unconstrained). Their utilities are shown in equation 10:", "children": [ { "id": "p35.4.1", "order": 7, "type": "link", "bbox": { "l": 69.87, "t": 620.132, "r": 85.73, "b": 609.412 }, "text": "10:", "attrs": { "action_type": "/Named" }, "children": [] } ] }, { "id": "p35.5", "order": 8, "type": "formula", "bbox": { "l": 155.4772, "t": 596.1388, "r": 435.9599, "b": 560.86 }, "text": "U_{s,t}=\\left(\\frac{1}{1-\\sigma}\\left(\\frac{c_{s,t+k}-\\kappa c_{s,t+k-1}}{1-\\kappa}\\right)^{1-\\sigma}-\\frac{1}{1+\\zeta}h_{s,t+k}^{1+\\zeta}\\right)", "attrs": { "block_label": "display_formula", "latex": "U_{s,t}=\\left(\\frac{1}{1-\\sigma}\\left(\\frac{c_{s,t+k}-\\kappa c_{s,t+k-1}}{1-\\kappa}\\right)^{1-\\sigma}-\\frac{1}{1+\\zeta}h_{s,t+k}^{1+\\zeta}\\right)", "notation": "latex", "alt": "U sub s, t equals 1 over 1 minus sigma of c sub s, t plus k minus kappa c sub s, t plus k minus 1 over 1 minus kappa to the power of 1 minus sigma minus 1 over 1 plus zeta h sub s, t plus k to the power of 1 plus zeta", "mathml": "Us,t=(11−σ(cs,t+k−κcs,t+k−11−κ)1−σ−11+ζhs,t+k1+ζ)" }, "children": [] }, { "id": "p35.6", "order": 9, "type": "text", "bbox": { "l": 502.4216, "t": 585.0992, "r": 524.2556, "b": 571.1796 }, "text": "(10)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p35.7", "order": 10, "type": "text", "bbox": { "l": 67.6614, "t": 553.1803, "r": 525.9351, "b": 510.2218 }, "text": "where[p35.7.1] is consumption of type $s$ households,[p35.7.2] is their hours worked,[p35.7.3] is the (inverse of) intertemporal elasticity of substitution,[p35.7.4] is the degree of habit formation and[p35.7.5] is the (inverse of) the Frisch labour supply elasticity.", "children": [ { "id": "p35.7.1", "order": 11, "type": "formula", "bbox": { "l": 67.6614, "t": 553.1803, "r": 525.9351, "b": 510.2218 }, "text": "c_{s,t}", "attrs": { "latex": "c_{s,t}", "inline": true, "text_offset": 5, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "c sub s, t", "mathml": "cs,t" }, "children": [] }, { "id": "p35.7.2", "order": 12, "type": "formula", "bbox": { "l": 67.6614, "t": 553.1803, "r": 525.9351, "b": 510.2218 }, "text": "h_{s,t}", "attrs": { "latex": "h_{s,t}", "inline": true, "text_offset": 53, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "h sub s, t", "mathml": "hs,t" }, "children": [] }, { "id": "p35.7.3", "order": 13, "type": "formula", "bbox": { "l": 67.6614, "t": 553.1803, "r": 525.9351, "b": 510.2218 }, "text": "σ", "attrs": { "latex": "\\sigma", "inline": true, "text_offset": 85, "text_span": 9, "bbox_estimated": true, "alt": "sigma", "mathml": "σ" }, "children": [] }, { "id": "p35.7.4", "order": 14, "type": "formula", "bbox": { "l": 67.6614, "t": 553.1803, "r": 525.9351, "b": 510.2218 }, "text": "κ", "attrs": { "latex": "\\kappa", "inline": true, "text_offset": 156, "text_span": 9, "bbox_estimated": true, "alt": "kappa", "mathml": "κ" }, "children": [] }, { "id": "p35.7.5", "order": 15, "type": "formula", "bbox": { "l": 67.6614, "t": 553.1803, "r": 525.9351, "b": 510.2218 }, "text": "ζ", "attrs": { "latex": "\\zeta", "inline": true, "text_offset": 202, "text_span": 9, "bbox_estimated": true, "alt": "zeta", "mathml": "ζ" }, "children": [] } ] }, { "id": "p35.8", "order": 16, "type": "text", "bbox": { "l": 67.6614, "t": 509.7418, "r": 526.415, "b": 422.8649 }, "text": "Ricardian households hold capital, invest in internationally traded bonds denominated in US dollars[p35.8.1], that pay the interest rate[p35.8.2] set by the Federal Reserve. They also hold domestic government bonds[p35.8.3] that pay the rate[p35.8.4], which is determined by the European Central Bank. Investment in capital is subject to investment-adjustment costs that give rise to the standard Tobin's marginal q. The Euler equation with respect to domestic government bonds is", "children": [ { "id": "p35.8.1", "order": 17, "type": "formula", "bbox": { "l": 67.6614, "t": 509.7418, "r": 526.415, "b": 422.8649 }, "text": "B_{t}^{*}", "attrs": { "latex": "B_{t}^{*}", "inline": true, "text_offset": 99, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "B sub t to the power of *", "mathml": "Bt*" }, "children": [] }, { "id": "p35.8.2", "order": 18, "type": "formula", "bbox": { "l": 67.6614, "t": 509.7418, "r": 526.415, "b": 422.8649 }, "text": "R_{t}^{*}", "attrs": { "latex": "R_{t}^{*}", "inline": true, "text_offset": 136, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "R sub t to the power of *", "mathml": "Rt*" }, "children": [] }, { "id": "p35.8.3", "order": 19, "type": "formula", "bbox": { "l": 67.6614, "t": 509.7418, "r": 526.415, "b": 422.8649 }, "text": "B_{t}", "attrs": { "latex": "B_{t}", "inline": true, "text_offset": 214, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "B sub t", "mathml": "Bt" }, "children": [] }, { "id": "p35.8.4", "order": 20, "type": "formula", "bbox": { "l": 67.6614, "t": 509.7418, "r": 526.415, "b": 422.8649 }, "text": "R_{t}", "attrs": { "latex": "R_{t}", "inline": true, "text_offset": 241, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "R sub t", "mathml": "Rt" }, "children": [] } ] }, { "id": "p35.9", "order": 21, "type": "formula", "bbox": { "l": 192.9069, "t": 413.9852, "r": 398.5302, "b": 381.8264 }, "text": "u^{\\prime}(c_{s,t})=\\beta\\left(u^{\\prime}(c_{s,t+1})\\frac{R_{t}(1-\\Gamma_{B,t+1})}{\\pi_{t+1}}\\right),", "attrs": { "block_label": "display_formula", "latex": "u^{\\prime}(c_{s,t})=\\beta\\left(u^{\\prime}(c_{s,t+1})\\frac{R_{t}(1-\\Gamma_{B,t+1})}{\\pi_{t+1}}\\right),", "notation": "latex", "alt": "u to the power of prime of c sub s, t equals beta of u to the power of prime of c sub s, t plus 1 R sub t of 1 minus capital gamma sub B, t plus 1 over pi sub t plus 1,", "mathml": "u′(cs,t)=β(u′(cs,t+1)Rt(1−ΓB,t+1)πt+1)," }, "children": [] }, { "id": "p35.10", "order": 22, "type": "text", "bbox": { "l": 502.1816, "t": 405.3456, "r": 524.2556, "b": 390.9461 }, "text": "(11)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p35.11", "order": 23, "type": "text", "bbox": { "l": 67.4214, "t": 377.2666, "r": 525.6952, "b": 335.2681 }, "text": "where[p35.11.2] is the transaction cost for the euro area traded bonds, which depends on the intra-EA bond holdings.²⁴ There is an analogous Euler equation for internationally traded bonds:", "children": [ { "id": "p35.11.1", "order": 24, "type": "link", "bbox": { "l": 214.066, "t": 363.446, "r": 225.802, "b": 349.245 }, "text": "24", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p35.11.2", "order": 25, "type": "formula", "bbox": { "l": 67.4214, "t": 377.2666, "r": 525.6952, "b": 335.2681 }, "text": "\\Gamma_{B,t+1}", "attrs": { "latex": "\\Gamma_{B,t+1}", "inline": true, "text_offset": 5, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "capital gamma sub B, t plus 1", "mathml": "ΓB,t+1" }, "children": [] } ] }, { "id": "p35.12", "order": 26, "type": "formula", "bbox": { "l": 174.4319, "t": 337.428, "r": 418.2048, "b": 306.4691 }, "text": "u^{\\prime}(c_{s,t})=\\beta\\left(u^{\\prime}(c_{s,t+1})\\frac{R_{t}^{*}(1-\\Gamma_{B^{*},t+1})}{\\pi_{t+1}^{*}}\\frac{rer_{t+1}}{rer_{t}}\\right),", "attrs": { "block_label": "display_formula", "latex": "u^{\\prime}(c_{s,t})=\\beta\\left(u^{\\prime}(c_{s,t+1})\\frac{R_{t}^{*}(1-\\Gamma_{B^{*},t+1})}{\\pi_{t+1}^{*}}\\frac{r e r_{t+1}}{r e r_{t}}\\right),", "notation": "latex", "alt": "u to the power of prime of c sub s, t equals beta of u to the power of prime of c sub s, t plus 1 R sub t to the power of * 1 minus capital gamma sub B to the power of *, t plus 1 over pi sub t plus 1 to the power of * rer sub t plus 1 over rer sub t,", "mathml": "u′(cs,t)=β(u′(cs,t+1)Rt*(1−ΓB*,t+1)πt+1*rert+1rert)," }, "children": [] }, { "id": "p35.13", "order": 27, "type": "text", "bbox": { "l": 502.4216, "t": 330.4682, "r": 524.2556, "b": 316.3087 }, "text": "(12)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p35.14", "order": 28, "type": "text", "bbox": { "l": 67.4214, "t": 301.9092, "r": 526.415, "b": 259.1908 }, "text": "where[p35.14.1] is the real exchange rate and[p35.14.2] is the international transaction premium for US-denominated bonds, where[p35.14.3] is the nominal exchange rate, expressed in terms of units of Home currency per unit of US dollars:", "children": [ { "id": "p35.14.1", "order": 29, "type": "formula", "bbox": { "l": 67.4214, "t": 301.9092, "r": 526.415, "b": 259.1908 }, "text": "rer_{t}", "attrs": { "latex": "rer_{t}", "inline": true, "text_offset": 5, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "rer sub t", "mathml": "rert" }, "children": [] }, { "id": "p35.14.2", "order": 30, "type": "formula", "bbox": { "l": 67.4214, "t": 301.9092, "r": 526.415, "b": 259.1908 }, "text": "\\Gamma_{B^{*}}", "attrs": { "latex": "\\Gamma_{B^{*}}", "inline": true, "text_offset": 45, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "capital gamma sub B to the power of *", "mathml": "ΓB*" }, "children": [] }, { "id": "p35.14.3", "order": 31, "type": "formula", "bbox": { "l": 67.4214, "t": 301.9092, "r": 526.415, "b": 259.1908 }, "text": "S_{t}^{H,US}", "attrs": { "latex": "S_{t}^{H,US}", "inline": true, "text_offset": 128, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "S sub t to the power of H, US", "mathml": "StH,US" }, "children": [] } ] }, { "id": "p35.15", "order": 32, "type": "formula", "bbox": { "l": 120.6868, "t": 249.8311, "r": 472.9097, "b": 210.7125 }, "text": "\\Gamma_{B^{*}}\\left(\\frac{S_{t}^{H,US}B_{t+1}^{*}}{P_{Y,t}Y_{t}};rp_{t}\\right)\\equiv\\gamma_{B^{*}}\\left(\\exp\\left(\\frac{S_{t}^{H,US}B_{t+1}^{*}}{P_{Y,t}Y_{t}}-\\overline{B_{Y}^{*}}\\right)-1\\right)-rp_{t}", "attrs": { "block_label": "display_formula", "latex": "\\Gamma_{B^{*}}\\left(\\frac{S_{t}^{H,U S}B_{t+1}^{*}}{P_{Y,t}Y_{t}};r p_{t}\\right)\\equiv\\gamma_{B^{*}}\\left(\\exp\\left(\\frac{S_{t}^{H,U S}B_{t+1}^{*}}{P_{Y,t}Y_{t}}-\\overline{B_{Y}^{*}}\\right)-1\\right)-r p_{t}", "notation": "latex", "alt": "capital gamma sub B to the power of * S sub t to the power of H, US B sub t plus 1 to the power of * over P sub Y, t Y sub t ;rp sub t equiv gamma sub B to the power of * exp of S sub t to the power of H, US B sub t plus 1 to the power of * over P sub Y, t Y sub t minus B sub Y to the power of * bar minus 1 minus rp sub t", "mathml": "ΓB*(StH,USBt+1*PY,tYt;rpt)≡γB*(exp(StH,USBt+1*PY,tYt−BY*―)−1)−rpt" }, "children": [] }, { "id": "p35.16", "order": 33, "type": "text", "bbox": { "l": 502.4216, "t": 237.8315, "r": 524.0156, "b": 223.672 }, "text": "(13)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" }, "children": [] }, { "id": "p35.17", "order": 34, "type": "text", "bbox": { "l": 67.6614, "t": 198.4729, "r": 525.2153, "b": 142.0749 }, "text": "where[p35.17.3] is a parameter,[p35.17.4] is the steady-state net foreign asset position,[p35.17.5] is a risk premium shock,[p35.17.6] is the GDP deflator,[p35.17.7] is the GDP in real terms. The term[p35.17.8] in the Euler equation for euro area bonds is analogous and depends on the target for intra-EA bond holdings. For further details see the Appendix in Gomes et al. (2010).", "children": [ { "id": "p35.17.1", "order": 35, "type": "link", "bbox": { "l": 366.884, "t": 154.857, "r": 432.561, "b": 141.632 }, "text": "Gomes et al.", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p35.17.2", "order": 36, "type": "link", "bbox": { "l": 436.463, "t": 154.857, "r": 466.191, "b": 141.632 }, "text": "(2010).", "attrs": { "action_type": "/Named" }, "children": [] }, { "id": "p35.17.3", "order": 37, "type": "formula", "bbox": { "l": 67.6614, "t": 198.4729, "r": 525.2153, "b": 142.0749 }, "text": "\\gamma_{B^{*}}", "attrs": { "latex": "\\gamma_{B^{*}}", "inline": true, "text_offset": 5, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "gamma sub B to the power of *", "mathml": "γB*" }, "children": [] }, { "id": "p35.17.4", "order": 38, "type": "formula", "bbox": { "l": 67.6614, "t": 198.4729, "r": 525.2153, "b": 142.0749 }, "text": "\\overline{B}_{Y}^{*}", "attrs": { "latex": "\\overline{B}_{Y}^{*}", "inline": true, "text_offset": 31, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "B bar sub Y to the power of *", "mathml": "B―Y*" }, "children": [] }, { "id": "p35.17.5", "order": 39, "type": "formula", "bbox": { "l": 67.6614, "t": 198.4729, "r": 525.2153, "b": 142.0749 }, "text": "rp_{t}", "attrs": { "latex": "rp_{t}", "inline": true, "text_offset": 89, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "rp sub t", "mathml": "rpt" }, "children": [] }, { "id": "p35.17.6", "order": 40, "type": "formula", "bbox": { "l": 67.6614, "t": 198.4729, "r": 525.2153, "b": 142.0749 }, "text": "P_{Y,t}", "attrs": { "latex": "P_{Y,t}", "inline": true, "text_offset": 124, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "P sub Y, t", "mathml": "PY,t" }, "children": [] }, { "id": "p35.17.7", "order": 41, "type": "formula", "bbox": { "l": 67.6614, "t": 198.4729, "r": 525.2153, "b": 142.0749 }, "text": "Y_{t}", "attrs": { "latex": "Y_{t}", "inline": true, "text_offset": 155, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "Y sub t", "mathml": "Yt" }, "children": [] }, { "id": "p35.17.8", "order": 42, "type": "formula", "bbox": { "l": 67.6614, "t": 198.4729, "r": 525.2153, "b": 142.0749 }, "text": "\\Gamma_{B,t+1}", "attrs": { "latex": "\\Gamma_{B,t+1}", "inline": true, "text_offset": 200, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "capital gamma sub B, t plus 1", "mathml": "ΓB,t+1" }, "children": [] } ] }, { "id": "p35.18", "order": 43, "type": "text", "bbox": { "l": 84.6967, "t": 140.635, "r": 477.9483, "b": 126.9555 }, "text": "Ricardian households also invest[p35.18.1] in physical capital[p35.18.2], which involves as:", "children": [ { "id": "p35.18.1", "order": 44, "type": "formula", "bbox": { "l": 84.6967, "t": 140.635, "r": 477.9483, "b": 126.9555 }, "text": "I_{t}", "attrs": { "latex": "I_{t}", "inline": true, "text_offset": 32, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "I sub t", "mathml": "It" }, "children": [] }, { "id": "p35.18.2", "order": 45, "type": "formula", "bbox": { "l": 84.6967, "t": 140.635, "r": 477.9483, "b": 126.9555 }, "text": "K_{t}", "attrs": { "latex": "K_{t}", "inline": true, "text_offset": 62, "text_span": 10, "bbox_estimated": true, "notation": "latex", "alt": "K sub t", "mathml": "Kt" }, "children": [] } ] }, { "id": "p35.19", "order": 46, "type": "footnote", "bbox": { "l": 68.1412, "t": 113.9959, "r": 526.175, "b": 86.6369 }, "text": "[p35.19.1]The bond traded in the union is in euros and the transaction cost is needed to stabilise the model.", "attrs": { "block_label": "footnote" }, "children": [ { "id": "p35.19.1", "order": 47, "type": "formula", "bbox": { "l": 68.1412, "t": 113.9959, "r": 526.175, "b": 86.6369 }, "text": "²⁴", "attrs": { "latex": "^{24}", "inline": true, "text_offset": 0, "text_span": 10, "bbox_estimated": true, "alt": "to the power of 24", "mathml": "24" }, "children": [] } ] }, { "id": "p35.20", "order": 48, "type": "footer", "bbox": { "l": 290.0801, "t": 51.1182, "r": 305.6758, "b": 39.5986 }, "text": "34", "attrs": { "block_label": "number" }, "children": [] } ], "ai_page_review": { "model": "claude-opus-4-8", "applied": true, "changed": true, "reviewed_at": "2026-08-21T10:46:17+00:00" } }