{ "document_id": 395, "page": 7, "width_pt": 595.276, "height_pt": 841.89, "engine": "paddleocr_vl", "source_run_id": 520, "item_count": 46, "counts_by_type": { "footer": 1, "footnote": 2, "formula": 25, "link": 6, "text": 12 }, "build_ms": 4, "items": [ { "id": "p7.1", "order": 1, "type": "text", "bbox": { "l": 67.6614, "t": 770.3725, "r": 526.6549, "b": 684.2156 }, "text": "Here we reproduce the key equations related to public investment and the associated delays that are common to both versions of the model and that illustrate how the intertemporal dimension of public investment feeds into the public capital stock and into production and marginal costs. The latter feature is particularly important for small open economies, as it determines the effect of public capital on the external competitiveness of the economy." }, { "id": "p7.2", "order": 2, "type": "text", "bbox": { "l": 85.6564, "t": 683.0157, "r": 388.9328, "b": 668.8562 }, "text": "Public investment,[p7.2.1], forms public capital,[p7.2.2], as follows:", "children": [ { "id": "p7.2.1", "order": 3, "type": "formula", "bbox": { "l": 85.6564, "t": 683.0157, "r": 388.9328, "b": 668.8562 }, "text": "I_{G}", "attrs": { "latex": "I_{G}", "inline": true, "text_offset": 18, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "I sub G", "mathml": "IG" } }, { "id": "p7.2.2", "order": 4, "type": "formula", "bbox": { "l": 85.6564, "t": 683.0157, "r": 388.9328, "b": 668.8562 }, "text": "K_{G}", "attrs": { "latex": "K_{G}", "inline": true, "text_offset": 49, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "K sub G", "mathml": "KG" } } ] }, { "id": "p7.3", "order": 5, "type": "formula", "bbox": { "l": 215.4606, "t": 654.2167, "r": 379.0956, "b": 639.0972 }, "text": "K_{G,t}=(1-\\delta_{G})K_{G,t-1}+I_{G,t-h},", "attrs": { "block_label": "display_formula", "latex": "K_{G,t}=(1-\\delta_{G})K_{G,t-1}+I_{G,t-h},", "notation": "latex", "alt": "K sub G, t equals 1 minus delta sub G K sub G, t minus 1 plus I sub G, t minus h,", "mathml": "KG,t=(1−δG)KG,t−1+IG,t−h," } }, { "id": "p7.4", "order": 6, "type": "text", "bbox": { "l": 509.6196, "t": 654.6967, "r": 524.4955, "b": 640.5372 }, "text": "(1)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" } }, { "id": "p7.5", "order": 7, "type": "text", "bbox": { "l": 67.4214, "t": 633.5774, "r": 526.175, "b": 590.139 }, "text": "where[p7.5.1] is the depreciation rate of public capital, and $h$ measures the time-to-build (in quarters) that is needed before public investment becomes part of public capital that is used in production (for instance, for 2-year time-to-build,[p7.5.2]).", "children": [ { "id": "p7.5.1", "order": 8, "type": "formula", "bbox": { "l": 67.4214, "t": 633.5774, "r": 526.175, "b": 590.139 }, "text": "\\delta_{G}", "attrs": { "latex": "\\delta_{G}", "inline": true, "text_offset": 5, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "delta sub G", "mathml": "δG" } }, { "id": "p7.5.2", "order": 9, "type": "formula", "bbox": { "l": 67.4214, "t": 633.5774, "r": 526.175, "b": 590.139 }, "text": "h = 8", "attrs": { "latex": "h=8", "inline": true, "text_offset": 245, "text_span": 8, "bbox_estimated": true, "alt": "h equals 8", "mathml": "h=8" } } ] }, { "id": "p7.6", "order": 10, "type": "text", "bbox": { "l": 68.1412, "t": 589.899, "r": 526.175, "b": 561.58 }, "text": "Government investment is decided by the government and follows an autoregressive process:" }, { "id": "p7.7", "order": 11, "type": "formula", "bbox": { "l": 199.1451, "t": 546.9405, "r": 394.2114, "b": 531.3411 }, "text": "I_{G,t}=(1-\\rho)\\overline{{I_{G}}}+\\rho*I_{G,t-1}+\\varepsilon_{I_{G},t-j},", "attrs": { "block_label": "display_formula", "latex": "I_{G,t}=(1-\\rho)\\overline{{I_{G}}}+\\rho*I_{G,t-1}+\\varepsilon_{I_{G},t-j},", "notation": "latex", "alt": "I sub G, t equals 1 minus rho overline I sub G plus rho *I sub G, t minus 1 plus varepsilon sub I sub G, t minus j,", "mathml": "IG,t=(1−ρ)IG―+ρ*IG,t−1+εIG,t−j," } }, { "id": "p7.8", "order": 12, "type": "text", "bbox": { "l": 509.6196, "t": 546.9405, "r": 524.4955, "b": 533.261 }, "text": "(2)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" } }, { "id": "p7.9", "order": 13, "type": "text", "bbox": { "l": 67.9013, "t": 525.5813, "r": 526.175, "b": 453.8238 }, "text": "where[p7.9.1] measures the persistence of public investment,[p7.9.2] is the steady state level of public investment, and[p7.9.3] is the change in public investment that is decided by the government. Index $j$ measures the time to plan, i.e., the time from the announcement to the implementation of public investment (e.g., for 2-year time-to-plan,[p7.9.4]). Throughout the paper, we assume that the announcement is fully credible.", "children": [ { "id": "p7.9.1", "order": 14, "type": "formula", "bbox": { "l": 67.9013, "t": 525.5813, "r": 526.175, "b": 453.8238 }, "text": "ρ", "attrs": { "latex": "\\rho", "inline": true, "text_offset": 5, "text_span": 8, "bbox_estimated": true, "alt": "rho", "mathml": "ρ" } }, { "id": "p7.9.2", "order": 15, "type": "formula", "bbox": { "l": 67.9013, "t": 525.5813, "r": 526.175, "b": 453.8238 }, "text": "\\overline{I_{G}}", "attrs": { "latex": "\\overline{I_{G}}", "inline": true, "text_offset": 60, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "I sub G bar", "mathml": "IG―" } }, { "id": "p7.9.3", "order": 16, "type": "formula", "bbox": { "l": 67.9013, "t": 525.5813, "r": 526.175, "b": 453.8238 }, "text": "\\varepsilon_{I_{G}}", "attrs": { "latex": "\\varepsilon_{I_{G}}", "inline": true, "text_offset": 120, "text_span": 8, "bbox_estimated": true, "notation": "latex", "alt": "varepsilon sub I sub G", "mathml": "εIG" } }, { "id": "p7.9.4", "order": 17, "type": "formula", "bbox": { "l": 67.9013, "t": 525.5813, "r": 526.175, "b": 453.8238 }, "text": "j = 8", "attrs": { "latex": "j=8", "inline": true, "text_offset": 347, "text_span": 8, "bbox_estimated": true, "alt": "j equals 8", "mathml": "j=8" } } ] }, { "id": "p7.10", "order": 18, "type": "text", "bbox": { "l": 67.9013, "t": 453.1039, "r": 526.415, "b": 395.0259 }, "text": "Importantly, public capital improves the productivity of the private sector. In the context of both models, it enters in the production functions of tradable goods,[p7.10.5], and non-tradable goods,[p7.10.6], following the specifications that are standard in the literature (Baxter and King (1993) or Leeper et al. (2010)):", "children": [ { "id": "p7.10.1", "order": 19, "type": "link", "bbox": { "l": 73.456, "t": 409.529, "r": 158.776, "b": 396.305 }, "text": "(Baxter and King", "attrs": { "action_type": "/Named" } }, { "id": "p7.10.2", "order": 20, "type": "link", "bbox": { "l": 162.678, "t": 409.529, "r": 192.406, "b": 396.305 }, "text": "(1993)", "attrs": { "action_type": "/Named" } }, { "id": "p7.10.3", "order": 21, "type": "link", "bbox": { "l": 210.08, "t": 409.529, "r": 275.208, "b": 396.305 }, "text": "Leeper et al.", "attrs": { "action_type": "/Named" } }, { "id": "p7.10.4", "order": 22, "type": "link", "bbox": { "l": 279.109, "t": 409.529, "r": 308.837, "b": 396.305 }, "text": "(2010)):", "attrs": { "action_type": "/Named" } }, { "id": "p7.10.5", "order": 23, "type": "formula", "bbox": { "l": 67.9013, "t": 453.1039, "r": 526.415, "b": 395.0259 }, "text": "Y_{T,t}", "attrs": { "latex": "Y_{T,t}", "inline": true, "text_offset": 164, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "Y sub T, t", "mathml": "YT,t" } }, { "id": "p7.10.6", "order": 24, "type": "formula", "bbox": { "l": 67.9013, "t": 453.1039, "r": 526.415, "b": 395.0259 }, "text": "Y_{N,t}", "attrs": { "latex": "Y_{N,t}", "inline": true, "text_offset": 198, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "Y sub N, t", "mathml": "YN,t" } } ] }, { "id": "p7.11", "order": 25, "type": "formula", "bbox": { "l": 204.9035, "t": 370.3068, "r": 387.7332, "b": 352.7874 }, "text": "\\begin{array}{ccl}{Y_{N,t}}&{=}&{z_{N,t}K_{G,t}^{\\alpha_{G}}K_{N,t}^{\\alpha_{N}}N_{N,t}^{1-\\alpha_{N}}-\\psi_{N}}\\\\ \\end{array}", "attrs": { "block_label": "display_formula", "latex": "\\begin{array}{c c l}{Y_{N,t}}&{=}&{z_{N,t}K_{G,t}^{\\alpha_{G}}K_{N,t}^{\\alpha_{N}}N_{N,t}^{1-\\alpha_{N}}-\\psi_{N}}\\\\ \\end{array}", "notation": "latex", "alt": "begin array ccl Y sub N, t & equals & z sub N, t K sub G, t to the power of alpha sub G K sub N, t to the power of alpha sub N N sub N, t to the power of 1 minus alpha sub N minus psi sub N end array", "mathml": "YN,t=zN,tKG,tαGKN,tαNNN,t1−αN−ψN" } }, { "id": "p7.12", "order": 26, "type": "formula", "bbox": { "l": 206.1032, "t": 350.3875, "r": 387.0134, "b": 333.3481 }, "text": "\\begin{array}{lcl}{Y_{T,t}}&{=}&{z_{T,t}K_{G,t}^{\\alpha_{G}}K_{T,t}^{\\alpha_{T}}N_{T,t}^{1-\\alpha_{T}}-\\psi_{T},}\\\\ \\end{array}", "attrs": { "block_label": "display_formula", "latex": "\\begin{array}{l c l}{Y_{T,t}}&{=}&{z_{T,t}K_{G,t}^{\\alpha_{G}}K_{T,t}^{\\alpha_{T}}N_{T,t}^{1-\\alpha_{T}}-\\psi_{T},}\\\\ \\end{array}", "notation": "latex", "alt": "begin array lcl Y sub T, t & equals & z sub T, t K sub G, t to the power of alpha sub G K sub T, t to the power of alpha sub T N sub T, t to the power of 1 minus alpha sub T minus psi sub T, end array", "mathml": "YT,t=zT,tKG,tαGKT,tαTNT,t1−αT−ψT," } }, { "id": "p7.13", "order": 27, "type": "text", "bbox": { "l": 510.0995, "t": 369.8268, "r": 524.2556, "b": 356.3873 }, "text": "(3)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" } }, { "id": "p7.14", "order": 28, "type": "text", "bbox": { "l": 510.0995, "t": 350.1475, "r": 524.0156, "b": 336.468 }, "text": "(4)", "attrs": { "raw_type": "FORMULA_NUMBER", "block_label": "formula_number" } }, { "id": "p7.15", "order": 29, "type": "text", "bbox": { "l": 67.9013, "t": 323.7485, "r": 526.6549, "b": 194.1531 }, "text": "where[p7.15.2] and[p7.15.3] are sector-specific productivities,[p7.15.4] and[p7.15.5] is capital used in each sector, and[p7.15.6] and[p7.15.7] is labour used in each sector. These equations show that an increase in public capital is the same as the increase in productivity ([p7.15.8] enters production function in the same way as productivity z). The only difference is that productivity increase happens exogenously and is \"free\" in the sense that nobody has to pay for it, while the increase in public capital is a decision of the government and comes at a cost - the government must raise the required funds either by borrowing or by raising taxes. The analysis in this paper will assume that the government borrows the required funds. [p7.15.9]", "children": [ { "id": "p7.15.1", "order": 30, "type": "link", "bbox": { "l": 101.599, "t": 208.322, "r": 108.712, "b": 196.141 }, "text": "5", "attrs": { "action_type": "/Named" } }, { "id": "p7.15.2", "order": 31, "type": "formula", "bbox": { "l": 67.9013, "t": 323.7485, "r": 526.6549, "b": 194.1531 }, "text": "z_{N,t}", "attrs": { "latex": "z_{N,t}", "inline": true, "text_offset": 5, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "z sub N, t", "mathml": "zN,t" } }, { "id": "p7.15.3", "order": 32, "type": "formula", "bbox": { "l": 67.9013, "t": 323.7485, "r": 526.6549, "b": 194.1531 }, "text": "z_{T,t}", "attrs": { "latex": "z_{T,t}", "inline": true, "text_offset": 18, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "z sub T, t", "mathml": "zT,t" } }, { "id": "p7.15.4", "order": 33, "type": "formula", "bbox": { "l": 67.9013, "t": 323.7485, "r": 526.6549, "b": 194.1531 }, "text": "K_{N,t}", "attrs": { "latex": "K_{N,t}", "inline": true, "text_offset": 63, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "K sub N, t", "mathml": "KN,t" } }, { "id": "p7.15.5", "order": 34, "type": "formula", "bbox": { "l": 67.9013, "t": 323.7485, "r": 526.6549, "b": 194.1531 }, "text": "K_{T,t}", "attrs": { "latex": "K_{T,t}", "inline": true, "text_offset": 76, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "K sub T, t", "mathml": "KT,t" } }, { "id": "p7.15.6", "order": 35, "type": "formula", "bbox": { "l": 67.9013, "t": 323.7485, "r": 526.6549, "b": 194.1531 }, "text": "N_{N,t}", "attrs": { "latex": "N_{N,t}", "inline": true, "text_offset": 121, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "N sub N, t", "mathml": "NN,t" } }, { "id": "p7.15.7", "order": 36, "type": "formula", "bbox": { "l": 67.9013, "t": 323.7485, "r": 526.6549, "b": 194.1531 }, "text": "N_{T,t}", "attrs": { "latex": "N_{T,t}", "inline": true, "text_offset": 134, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "N sub T, t", "mathml": "NT,t" } }, { "id": "p7.15.8", "order": 37, "type": "formula", "bbox": { "l": 67.9013, "t": 323.7485, "r": 526.6549, "b": 194.1531 }, "text": "K_{G}", "attrs": { "latex": "K_{G}", "inline": true, "text_offset": 276, "text_span": 9, "bbox_estimated": true, "notation": "latex", "alt": "K sub G", "mathml": "KG" } }, { "id": "p7.15.9", "order": 38, "type": "formula", "bbox": { "l": 67.9013, "t": 323.7485, "r": 526.6549, "b": 194.1531 }, "text": "⁵", "attrs": { "latex": "^{5}", "inline": true, "text_offset": 741, "text_span": 9, "bbox_estimated": true, "alt": "to the power of 5", "mathml": "5" } } ] }, { "id": "p7.16", "order": 39, "type": "text", "bbox": { "l": 67.6614, "t": 193.1931, "r": 526.6549, "b": 134.8752 }, "text": "To understand the importance of productivity of public capital for the external competitiveness of the economy, recall that price is a markup over the marginal cost and that the marginal cost equation with public capital is (reproduced here for the tradable sector): [p7.16.2]", "children": [ { "id": "p7.16.1", "order": 40, "type": "link", "bbox": { "l": 109.429, "t": 150.539, "r": 116.543, "b": 136.714 }, "text": "6", "attrs": { "action_type": "/Named" } }, { "id": "p7.16.2", "order": 41, "type": "formula", "bbox": { "l": 67.6614, "t": 193.1931, "r": 526.6549, "b": 134.8752 }, "text": "⁶", "attrs": { "latex": "^{6}", "inline": true, "text_offset": 267, "text_span": 9, "bbox_estimated": true, "alt": "to the power of 6", "mathml": "6" } } ] }, { "id": "p7.17", "order": 42, "type": "footnote", "bbox": { "l": 83.0171, "t": 123.8356, "r": 487.3057, "b": 108.4761 }, "text": "[p7.17.1]The borrowed funds are repaid over time by levying lump-sum taxes on households.", "attrs": { "block_label": "footnote" }, "children": [ { "id": "p7.17.1", "order": 43, "type": "formula", "bbox": { "l": 83.0171, "t": 123.8356, "r": 487.3057, "b": 108.4761 }, "text": "⁵", "attrs": { "latex": "^{5}", "inline": true, "text_offset": 0, "text_span": 9, "bbox_estimated": true, "alt": "to the power of 5", "mathml": "5" } } ] }, { "id": "p7.18", "order": 44, "type": "footnote", "bbox": { "l": 83.2571, "t": 104.6363, "r": 431.6411, "b": 90.2368 }, "text": "[p7.18.1]The equation for marginal costs in the non-tradable sector is analogous.", "attrs": { "block_label": "footnote" }, "children": [ { "id": "p7.18.1", "order": 45, "type": "formula", "bbox": { "l": 83.2571, "t": 104.6363, "r": 431.6411, "b": 90.2368 }, "text": "⁶", "attrs": { "latex": "^{6}", "inline": true, "text_offset": 0, "text_span": 9, "bbox_estimated": true, "alt": "to the power of 6", "mathml": "6" } } ] }, { "id": "p7.19", "order": 46, "type": "footer", "bbox": { "l": 292.9593, "t": 50.6382, "r": 302.0768, "b": 39.1186 }, "text": "6", "attrs": { "block_label": "number" } } ] }