## wp17105 ## Source details **Canonical URL:** [wp17105](https://www.imf.org/-/media/files/publications/wp/2017/wp17105.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2017/wp17105.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2017/wp17105.pdf.json) --- ### Firms and Production Technology - Firms: continuum of perfectly competitive firms produce y_t with Cobb-Douglas technology: - y_t = A_y [ z_{i,t-1} ]^{Â} (k_{t-1})^{–} (e_t^‰ l_t)^{1––} - Parameters: – ∈ (0,1);  ∈ (0,1); ‰ > 0; A_y > 0. - Factor demands (first-order conditions): - – y_t / k_{t-1} = r_k_t - (1––) y_t / l_t = w_t e_t^‰ - Wage normalization: wage per unit of raw labor grows at rate g. ### Households, Preferences, and Time Allocation - Representative household utility: - max Σ_{t=0}^∞ —^t Q_c [ c_t (1–n_t)^{’} ]^{1–1/Ÿ} –1 / (1–1/Ÿ) - — = (1+Í)^{–1} (1+g)^{1–1/Ÿ} ∈ (0,1); Ÿ > 0; ’ > 0. - Time allocation: - n_t = l_t + u_t - l_t: time producing goods (labor supplied) - u_t: time to accumulate human capital (schooling) ### Household Budget and Capital Accumulation - Intertemporal budget constraint (normalized): - (1+·_t) c_t + I_t + b^d_t = w_t e_t^‰ l_t + r_k_t k_{t-1} + (1/(1+r^d_{t-1})) b^d_{t-1} / (1+g) + T_t + Π_t - Private capital law of motion: - (1+g) k_t = (1–”_k) k_{t-1} + I_t ### Human Capital Accumulation (Two-stage with Inertia) - Schooling production: - A_e [ z^e_{t-1} ]^{„} (e_t^‰ u_t)^{‹} - Parameters: A_e > 0; „ > 0; ‹ > 0. - Intermediate stock in schools ›_t: - (1+g) ›_t = (1–Ê) ›_{t-1} + A_e [ z^e_{t-1} ]^{„} (e_t^‰ u_t)^{‹} - Fraction Ê moves from schools to labor force each period; average delay = 1/Ê periods. - Productive human capital e_t: - e_t = (1–”_e) e_{t-1} / (1+g) + Ê ›_{t-1} ### Household First-Order Conditions and Shadow Values (selected) - Consumption: - [ c_t (1–l_t–u_t)^{’} ]^{1–1/Ÿ} c_t = ⁄_{1,t} (1+·_t) - Labor supply: - ’ [ c_t (1–l_t–u_t)^{’} ]^{1–1/Ÿ} / (1–l_t–u_t) = ⁄_{1,t} w_t e_t^‰ - Schooling time: - ’ [ c_t (1–l_t–u_t)^{’} ]^{1–1/Ÿ} / (1–l_t–u_t) = ⁄_{2,t} ‹ A_e [ z^e_{t-1} ]^{„} (e_t^‰ u_t)^{‹} / u_t - Euler for private capital: - (1+g) ⁄_{1,t} = — ⁄_{1,t+1} 1/(1+r_k_{t+1}–”_k) - Asset return relation: - r^d_t = r_k_{t+1} – ”_k - Human capital value equations (selected): - (1+g) ⁄_{2,t} = — [ (1–Ê) ⁄_{2,t+1} + Ê ⁄_{3,t+1} ] - ⁄_{3,t} = ⁄_{1,t} ‰ w_t e_t^‰ l_t e_t + ⁄_{2,t} ‰ ‹ A_e [ z^e_{t-1} ]^{„} (e_t^‰ u_t)^{‹} e_t + — ⁄_{3,t+1} (1–”_e)/(1+g) ### Economic Interpretations - Equation (17): value of human capital in school equals discounted value of units remaining in school plus those that become productive next period. - Equation (18): value of one unit of human capital equals sum of (1) higher current wage income benefit, (2) marginal value in producing new human capital, and (3) discounted undepreciated social capital next period. ### Key quantitative insights from Sections 1–2 (summary) - Roads increase firms’ productivity relatively quickly; schools raise workers’ productivity mostly in the long run and have larger current expenditures (operations and maintenance). - Baseline calibration yields: - Investing exclusively in roads yields faster growth for around 15 years. - About 24 years for output from investing in schools to overtake that from roads. - Schools cause a threefold peak increase in government debt relative to roads. - With a 15 percent return differential favoring schools, government limits the fraction of the investment increase to schools to about three fourths; share falls to about a half with a “big push.” - Political leaders with horizon < 30 years would not invest in schools at all; about twice as large a horizon needed for investment in schools to match benevolent planner. --- ### Government: Investment, Maintenance, and Fiscal Framework - Infrastructure law of motion (j = e, i): - (1+g) z_j_t = (1−”_j_z) z_j_{t−1} + g_j_t - ”_j_z ∈ (0,1) is depreciation rate. - Operation and maintenance costs: - m_j_t = “_j_z z_j_{t−1}; “_j_z > 0. - Government budget constraint (real terms): - Δb_x_t + Δb_d_t = m_z_t + g_z_t + T_t + 1/(1+g) r_d_{t−1} − g^2 b_d_{t−1} + 1/(1+g) r_x_{t−1} − g^2 b_x_{t−1} − ·_t c_t − G_t - m_z_t ≡ m_e_t + m_i_t; g_z_t ≡ g_e_t + g_i_t. - Government issues either new domestic or foreign debt, but not both at the same time: Δb_x_t = 0 or Δb_d_t = 0. - External real interest rate: - r_x_t = r_f + ‚_g ÷_g ( (b_x_t / y_t) − (b_x_o / y_o) ) - ‚_g > 0, ÷_g > 0, r_f risk-free world real rate (implies upward-sloping supply of foreign funds). - Fiscal gap before adjustment: - Gap_t = g_z_t + m_z_t + 1/(1+g) r_d_{t−1} − g^2 b_d_{t−1} + 1/(1+g) r_x_{t−1} − g^2 b_x_{t−1} + T_t − ·_t c_t − G_t - Debt-stabilizing targets: - ·_target_t = ·_t + (1 − ⁄) Gap_t / c_t - T_target_t = T_t − ⁄ Gap_t - ⁄ ∈ [0,1] divides adjustment between taxes and transfers. - Fiscal reaction functions: - ·_t = ·_{t−1} + ⁄_{·,1} (·_target_t − ·_{t−1}) + ⁄_{·,2} ( (b_x_{t−1} + b_d_{t−1}) − (b_x + b_d)_{target} ) / y_t - T_t analogous with ⁄_{T,1}, ⁄_{T,2} - Allocation between schools and roads: - Share to schools È_e; to roads È_i = (1 − È_e). - g_e_t + m_e_t = È_e (g_z_t + m_z_t) - g_e_t = È_e (g_z_t + m_z_t) − m_e_t - Market clearing and external balance: - −[ (b_x_t − b_x_{t−1}) / (1+g) ] = y_t + G_t − 1/(1+g) r_x_{t−1} b_x_{t−1} − (m_z_t + g_z_t + c_t + I_t) - Goods market clearing: y_t = c_t + I_t + g_z_t + m_z_t + nx_t - Current account: ca_t = nx_t + G_t − 1/(1+g) r_x_{t−1} b_x_{t−1} - Competitive markets imply Π_t = 0. - System: 35 equations in 35 variables given exogenous (b_x + b_d)_{target} and paths for g_z_t and G_t. ### Calibration Highlights (annual frequency; average low-income country) - Ÿ = 0.34 - n_o = 0.36 → ’ = 1.1648; implied Frisch elasticity = 1.0051 - – = 0.475 - R_i z,o = 0.25 →  = 0.1123 - Ê = 0.08 (average delay 1/Ê ≈ 12 periods) - R_e z,o = 0.40; implied: ‰ = 0.6911, „ = 0.5467, ‹ = 0.5838 with 1/u_o n_o = 0.10 - Depreciation rates: ”_k, ”_e_z, ”_i_z, ”_e = 0.0500 - g = 0.015 - r_d_o = 0.1000; r_k_o = 0.1500 - r_f = 0.0400; r_x_o = 0.0600; implied ‚_g = 0.0200; ÷_g = 0.0000 - b_d_o = 0.2000 of GDP; b_x_o = 0.0000 - G_o = 0.0400 of GDP - Initial total public infrastructure expenditure = 0.0600 of GDP; O+M ≈ 0.0340 of GDP for LIDCs in SSA reference - Initial shares and O+M ratios: - Two-thirds of investment in roads and one-third in schools. - Recurrent expenditure as fraction of installed capital: 70% for schools, 50% for roads → average 56.7%. - “_i_z = 0.0650 and “_e_z = 0.1517. - È_i = 0.7692 in initial equilibrium. - ·_o = 0.1500; T_o = 7.9376 of GDP - ⁄ = 0.0000 (simulations: only taxes bear adjustment) - fl_· = fl_T = 0.9900 - Policy reaction parameters: ⁄_{·,1} = ⁄_{T,1} = 0.2500; ⁄_{·,2} = ⁄_{T,2} = 0.0200 --- ### Simulation Design and Main Experiment - Numerical simulations: annual, global nonlinear saddle path, perfect foresight, no uncertainty. - Experiments: at least a one-time permanent change in policy; economy converges to different steady state. - Main experiment: permanent increase in combined public investment and current expenditure equal to 0.0100 (1%) of initial GDP (a 16.7% real scaling up from initial infrastructure expenditures). --- ### Permanent Scale-Up of Public Investment (Section 4.1) - Scenario: total government infrastructure expenditures rise from 0.0600 to 0.0700 of GDP. - Roads scenario: - Total expenditure on roads rises permanently from 0.0400 to 0.0500 of GDP. - Capital expenditures in roads rise from 0.0200 to 0.0300 of GDP on impact. - Long-run split: investment and current expenditures on roads evenly split the 0.0100 increase (both rise from 0.0200 to 0.0250 of GDP). - Schools scenario: - Long-run split between capital and current expenditures is 30%-70%: - Investment increases from 0.0060 to 0.0090 of GDP. - Current expenditures jump from 0.0140 to 0.0210 of GDP. - Long-run output effects: - Investment in schools: long-run increase in output of 24%. - Investment in roads: long-run increase in output of 5%. - Transition dynamics: - First 13 years: economy grows faster with roads investment (above g) than with schools investment. - With schools, growth dips below trend for about 9 years; with roads, growth stays above trend. - It takes 24 years for additional output from schools to overtake roads. - Fiscal outcomes: - Roads: public debt rises by about 0.0200 of GDP; increased debt burden lasts ~30 years. - Schools: public debt rises by about 0.0600 of GDP (almost threefold relative to roads); increase lasts ~60 years. - Key qualitative insight: schools yield larger long-run gains but slower realization, requiring more debt financing during transition and raising debt-sustainability concerns relative to roads. ### Permanent Scale-Up with a “Big Push” (Section 4.2) - Big push specification: short-run increase nearly doubles public expenditure from 0.0600 to almost 0.1300 in the first 2 years and remains over 0.0800 for about 14 years (short-run increase, as a percent of GDP, over and above the long-run 0.0100 rise, is specified as 14(e ≠.2t ≠2e ≠.9t ) in model formulation). - Transition speed: - For schools, transition almost complete in 60 years with big push versus >100 years without it. - Medium-run growth: - Big push raises growth well above roads in the medium run. - All-schools with big push: output, private consumption, and private investment overtake all-roads counterparts about 4-5 years earlier (in ~20 years) than without big push. - Fiscal dynamics: - Big push raises tax and debt burdens initially but shortens duration taxes and debt remain elevated. - Public debt returns to original level, and below, within 20 years because GDP rises faster. - Fiscal differences between roads and schools largely vanish with big push. - Welfare and short-run costs: - Short-run costs in private consumption and output are much higher with big push. - Big push strengthens intertemporal labor substitution, causing medium-run labor supply decline and raising welfare concerns. --- ### Optimal Composition and Key Determinants (Section 5 and 5.1) - Welfare-maximizing constant split of the 0.0100 scale-up: - Without big push: optimal share to education ≈ 76% - With big push: optimal share to education ≈ 51% - Explanation: big push accelerates benefits of schools but raises short-run consumption costs; in baseline calibration short-run costs outweigh acceleration benefits, lowering optimal education share despite higher returns. - Return differential (baseline): schools 0.4000; roads 0.2500 → 15 percent annual differential. - If school return falls to 0.2500 (equal to roads): - Without big push: optimal share of schools drops from ~76% to 12%. - With big push: optimal share drops from ~51% to ~25%. - If return to schools declines below the 15–20 percent range (i.e., below roads), optimal share of schools → 0%. - Debt aversion (parameter λ τ,1 in [0,1]): - Greater debt aversion reduces optimal share of schools because schools produce pronounced spike in government debt. - Quantitative illustration (baseline returns): - Without big push: optimal share of schools falls from almost 90% when government relies on debt to under 70% when government uses taxes only (λ τ,1 = 1). - With big push: optimal share less sensitive to debt aversion. - Policy implication: concessional financing and grants can mitigate debt-intolerance and support higher social spending. - Political myopia (finite planner horizon): - Without big push: - Planner horizon < 30 years → no investment in schools. - Horizon 60–70 years needed to approach altruistic planner outcome. - With big push: - Threshold for no investment falls to 20 years. - 45–55 years needed to reach share comparable to altruistic planner. - Interpretation: big push helps overcome political myopia by bringing benefits forward, but trade-offs remain. - Central mechanism: front-loaded fiscal costs and slow accrual of schooling benefits versus quicker road gains interact with distortionary taxation and debt intolerance to limit optimal schooling shares. - Role for multilateral agencies: - Providing tied concessional financing and grants can (1) mitigate debt-intolerance concerns and (2) address political myopia by tying aid to investment in schools. --- ### Key Parameters and Initial Stocks (selected exact values) - Initial proportion of time used for non-leisure activities: 0.3600 - Preference parameter for leisure: ’ = 1.1648 - Capital’s share in value added: – = 0.4750 - Initial return on infrastructure: 0.2500 - Elasticity of output w.r.t. infrastructure:  = 0.1123 - Speed of transition of human capital from schools to labor force: Ê = 0.0800 - Initial return on schools: 0.4000 - Elasticities in human capital accumulation: „ = 0.5467; ‹ = 0.5838; ‰ = 0.6911 - Depreciation rates: private and human capital = 0.0500; public infrastructure and schools = 0.0500 - Trend growth rate: g = 0.015 - Initial real interest rates: r_d = 0.1000; r_k = 0.1500; r_f = 0.0400; r_x = 0.0600 - Public debt risk premium: ‚_g = 0.0200; ÷_g = 0.0000 - Initial fiscal stocks and flows (ratios to GDP): - b_d_o / y_o = 0.2000 - b_x_o / y_o = 0.0000 - G_o / y_o = 0.0400 - g_i_o / y_o = 0.0200 - g_e_o / y_o = 0.0060 - m_i_o / y_o = 0.0200 - m_e_o / y_o = 0.0140 - “_i_z = 0.0650 - “_e_z = 0.1517 - s_i = 0.7692 - ·_o = 0.1500 - T_o = 7.9376 - ⁄ = 0.0000 - fl_· , fl_T = 0.9900 - ⁄_{·,1} , ⁄_{T,1} = 0.2500 - ⁄_{·,2} , ⁄_{T,2} = 0.0200 *Source: wp17105 - IMF Working Paper.* ### 2.1 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 ### 2.1 Firms ### Summary: major themes from Sections 1–2 - Paper distinguishes “economic” (roads) and “social” (schools) infrastructure and studies macro-fiscal implications of their composition in a single-good, small-open dynamic general equilibrium model. - Roads (economic infrastructure) increase firms’ productivity relatively quickly; schools (social infrastructure) raise workers’ productivity mostly in the long run and entail larger current expenditures (operations and maintenance). - Baseline calibration to an average developing economy yields these key quantitative insights: - Investing exclusively in roads yields faster growth for a prolonged time (around 15 years). - It takes about a generation (almost 24 years) for output obtained by investing in schools to overtake that delivered by investing in roads. - Schools cause a threefold peak increase in government debt relative to roads. - With a large (15 percent) return differential in favor of schools, the government chooses to limit the fraction of the investment increase dedicated to schools to about three fourths; this share falls to about a half with a “big push.” - If political leaders have a planning horizon of less than 30 years, they would not invest in schools at all; an approximately twice as large a time horizon is needed for investment in schools to be comparable to a benevolent social planner. - A “big push” (front-loading of investment expenditures) shrinks the delay for schools’ output to overtake roads and results in more similar (but amplified) public debt paths; however, it amplifies the medium-run drop in output and private consumption by strengthening intertemporal labor substitution associated with schooling. ### Production technology and firm behavior - Firms: continuum of perfectly competitive firms producing good y_t by combining private capital k_{t-1}, effective labor e_t^‰ l_t, and government-supplied infrastructure z_{i,t-1}, with Cobb-Douglas technology: - y_t = A_y [ z_{i,t-1} ]^{Â} (k_{t-1})^{–} (e_t^‰ l_t)^{1––} (equation (1)) - Parameters: – ∈ (0,1);  ∈ (0,1); ‰ > 0; A_y > 0. - First-order conditions / factor demands: - – y_t / k_{t-1} = r_k_t (equation (2)) - (1––) y_t / l_t = w_t e_t^‰ (equation (3)) - Wage normalization: wage per unit of effective labor has been normalized by dividing by (1+g)^{(1–‰) t}; the wage rate per unit of raw labor grows at rate g. ### 2.2 Households ### Preferences and time allocation - Representative household: - Utility: max Σ_{t=0}^∞ —^t Q_c [ c_t (1–n_t)^{’} ]^{1–1/Ÿ} –1 / (1–1/Ÿ) (equation (4)) - Discount factor — = (1+Í)^{–1} (1+g)^{1–1/Ÿ} ∈ (0,1); Í is the pure rate of time preference. - Elasticity of intertemporal substitution: Ÿ > 0. - Preference parameter for consumption–leisure substitution: ’ > 0 (Frisch elasticity). - Time allocation: - n_t = l_t + u_t (equation (5)) - l_t: time producing goods (labor supplied) - u_t: time to accumulate human capital (schooling) ### Budget constraint and capital accumulation - Household income from labor and capital, firm profits Π_t, government transfers T_t. - Household saves via private capital investment I_t (depreciation ”_k ∈ (0,1)) and domestic bonds b^d_t with real return r^d_t. - Intertemporal budget constraint (after normalization): - (1+·_t) c_t + I_t + b^d_t = w_t e_t^‰ l_t + r_k_t k_{t-1} + (1/(1+r^d_{t-1})) b^d_{t-1} / (1+g) + T_t + Π_t (equation (6)) - Private capital law of motion: - (1+g) k_t = (1–”_k) k_{t-1} + I_t (equation (7)) ### 2.2.1 Human Capital Accumulation ### Technology and timing - Schooling combines government-provided schools z^e_{t-1} and effective schooling time e_t^‰ u_t: - A_e [ z^e_{t-1} ]^{„} (e_t^‰ u_t)^{‹} (equation (8)) - Parameters: A_e > 0; „ > 0 (elasticity wrt schools); ‹ > 0 (elasticity wrt schooling time). - Two-stage accumulation with inertia: - Intermediate stock trapped in schools: ›_t evolves as - (1+g) ›_t = (1–Ê) ›_{t-1} + A_e [ z^e_{t-1} ]^{„} (e_t^‰ u_t)^{‹} (equation (9)) - Fraction Ê moves from schools to labor force each period. Average delay until newly accumulated human capital becomes productive: 1/Ê periods. - Productive human capital e_t evolves as - e_t = (1–”_e) e_{t-1} / (1+g) + Ê ›_{t-1} (equation (10)) - ”_e is human capital depreciation rate. ### Interpretation - Delay captures that human capital acquired via schooling becomes productive only over time as cohorts enter the labor force. - This delay is distinct from time-to-build for public physical capital, which the authors abstract from for clarity. ### 2.2.2 Household’s Optimization ### First-order conditions and shadow values - Household chooses c_t, l_t, u_t, e_t, ›_t, b^d_t, k_t to maximize (4) subject to (9), (10), and (11) (budget constraint with I_t eliminated via (7)): - (1+·_t) c_t + (1+g) k_t + b^d_t = w_t e_t^‰ l_t + (1+r_k_t–”_k) k_{t-1} + (1/(1+r^d_{t-1})) b^d_{t-1} / (1+g) + T_t + Π_t (equation (11)) - Let ⁄_{1,t}, ⁄_{2,t}, ⁄_{3,t} be Lagrange multipliers for constraints (11), (9), (10). First-order conditions: - c_t: - [ c_t (1–l_t–u_t)^{’} ]^{1–1/Ÿ} c_t = ⁄_{1,t} (1+·_t) (equation (12)) - l_t: - ’ [ c_t (1–l_t–u_t)^{’} ]^{1–1/Ÿ} / (1–l_t–u_t) = ⁄_{1,t} w_t e_t^‰ (equation (13)) - u_t: - ’ [ c_t (1–l_t–u_t)^{’} ]^{1–1/Ÿ} / (1–l_t–u_t) = ⁄_{2,t} ‹ A_e [ z^e_{t-1} ]^{„} (e_t^‰ u_t)^{‹} / u_t (equation (14)) - k_t: - (1+g) ⁄_{1,t} = — ⁄_{1,t+1} 1/(1+r_k_{t+1}–”_k) (equation (15)) - b_t: - r^d_t = r_k_{t+1} – ”_k (equation (16)) - ›_t: - (1+g) ⁄_{2,t} = — [ (1–Ê) ⁄_{2,t+1} + Ê ⁄_{3,t+1} ] (equation (17)) - e_t: - ⁄_{3,t} = ⁄_{1,t} ‰ w_t e_t^‰ l_t e_t + ⁄_{2,t} ‰ ‹ A_e [ z^e_{t-1} ]^{„} (e_t^‰ u_t)^{‹} e_t + — ⁄_{3,t+1} (1–”_e)/(1+g) (equation (18)) ### Economic interpretations of optimality conditions - Equation (17): value of one unit of human capital in school equals present discounted value of (1–Ê) units remaining in school plus Ê units that become available to transform raw labor into effective labor next period. - Equation (18): value of one unit of human capital equals sum of: - its benefit in higher current wage income, - its marginal value in production of new human capital, - the present discounted value of undepreciated social capital next period. *Source: wp17105 - 2.1 Firms (IMF working paper pdf chapter content).* ### 2.3 Government ### 2.3 Government ### Government investment and maintenance framework - Public investment augments stocks of economic infrastructure (roads) and social infrastructure (schools) according to (1 +g)z_j_t = (1−”_j_z) z_j_{t−1} + g_j_t, for j = e, i, where ”_j_z ∈ (0,1) is the depreciation rate of the corresponding stock of infrastructure. - Operation and maintenance costs m_j_t are modeled as a constant proportion of the stock: m_j_t = “_j_z z_j_{t−1}, where “_j_z > 0. This captures different sizes of O+M costs for roads versus schools and implications for time profiles of costs and benefits. ### Government budget constraint, borrowing, and interest on external debt - Revenues: VAT on consumption ·_t c_t and grants/other revenues G_t. Government also makes transfers T_t to households. - Deficit financing: domestic borrowing Δb_d_t = b_d_t − b_d_{t−1} and external concessional borrowing Δb_x_t = b_x_t − b_x_{t−1}. - Government budget constraint (real terms): - Δb_x_t + Δb_d_t = m_z_t + g_z_t + T_t + 1/(1+g) r_d_{t−1} − g^2 b_d_{t−1} + 1/(1+g) r_x_{t−1} − g^2 b_x_{t−1} − ·_t c_t − G_t, - where m_z_t ≡ m_e_t + m_i_t and g_z_t ≡ g_e_t + g_i_t are total O+M (current) and investment (capital) expenditures. - Government issues either new domestic or foreign debt, but not both at the same time: Δb_x_t = 0 or Δb_d_t = 0. - External (real) interest rate r_x_t = r_f + ‚_g ÷_g ( (b_x_t / y_t) − (b_x_o / y_o) ), with ‚_g > 0, ÷_g > 0, and r_f the risk-free world real interest rate. This implies an upward-sloping supply of foreign funds. ### Fiscal gap and fiscal adjustment mechanism - Fiscal gap before policy adjustment: - Gap_t = g_z_t + m_z_t + 1/(1+g) r_d_{t−1} − g^2 b_d_{t−1} + 1/(1+g) r_x_{t−1} − g^2 b_x_{t−1} + T_t − ·_t c_t − G_t. - Gap_t is the excess of expenditures (including interest payments) over revenues, holding transfers and taxes at reference values. - Reference values for taxes and transfers evolve as x_t = x_f + fl_x (x_{t−1} − x_f), for x = ·, T, with x_{−1} = x_o and x_o, x_f initial and final steady-state values. ·_f is determined endogenously; T_f = T_o × (y_f / y_o). - Financing and adjustment equation: - Gap_t = −Δb_x_t − Δb_d_t + (·_t − ·_t) c_t − 1/2 (T_t − T_t). - The gap can be covered by borrowing short/medium term, but ultimately VAT and transfers must adjust to cover the entire gap. - Debt-stabilizing targets: - ·_target_t = ·_t + (1 − ⁄) Gap_t / c_t, - T_target_t = T_t − ⁄ Gap_t, - where ⁄ ∈ [0,1] controls the division of adjustment between taxes and transfers (⁄ = 0 → adjustment fully on taxes; ⁄ = 1 → fully on transfers). - Fiscal reaction functions (transition dynamics): - ·_t = ·_{t−1} + ⁄_{·,1} (·_target_t − ·_{t−1}) + ⁄_{·,2} ( (b_x_{t−1} + b_d_{t−1}) − (b_x + b_d)_{target} ) / y_t, - T_t = T_{t−1} + ⁄_{T,1} (T_target_t − T_{t−1}) + ⁄_{T,2} ( (b_x_{t−1} + b_d_{t−1}) − (b_x + b_d)_{target} ) / y_t, - where (b_x + b_d)_{target} is the exogenously specified steady-state government debt level. - Transition behavior: government smooths policy changes; in response to shocks it typically reaches fiscal targets over time while using borrowing in the interim. Later transition characterized by smaller transfers and higher taxes than targets to generate surpluses to pay down debt. ### Allocation between schools and roads - Total expenditure on infrastructure g_z_t + m_z_t is split with share È_e to schools and È_i = (1 − È_e) to roads. - Total spending on schools: g_e_t + m_e_t = È_e (g_z_t + m_z_t). - Since m_e_t is a fraction of the stock of social infrastructure, capital expenditures on schools: g_e_t = È_e (g_z_t + m_z_t) − m_e_t. ### Market clearing and external balance - External balance (balance of payments) combining household and government constraints: - −[ (b_x_t − b_x_{t−1}) / (1+g) ] = y_t + G_t − 1/(1+g) r_x_{t−1} b_x_{t−1} − (m_z_t + g_z_t + c_t + I_t). - Left-hand side is negative of capital/financial account; right-hand side is the current account. - Goods market clearing: y_t = c_t + I_t + g_z_t + m_z_t + nx_t, where nx_t is net exports. - Current account: ca_t = nx_t + G_t − 1/(1+g) r_x_{t−1} b_x_{t−1}. - Competitive markets and constant returns in private factors imply zero firms’ profits: Π_t = 0. - System summary: model consists of 35 equations in 35 variables given exogenous (b_x + b_d)_{target} and exogenous paths for g_z_t and G_t. ### Calibration highlights (annual frequency; average low-income country) - Elasticity of intertemporal substitution (Ÿ): base case 0.34. - Proportion of non-leisure time n_o = 0.36 in initial equilibrium → ’ = 1.1648; implied Frisch elasticity = 1.0051. - Capital share (–): 0.475. - Return to economic infrastructure R_i z,o = 0.25 at initial equilibrium → implied  = 0.1123. - Speed of transition of human capital from schools to labor force Ê = 0.08 (K-12 assumption → average delay 1/Ê ≈ 12 periods). - Return to schools R_e z,o = 0.40 at initial equilibrium; human capital parameters implied: ‰ = 0.6911, „ = 0.5467, ‹ = 0.5838 with 1/u_o n_o = 0.10 in initial equilibrium. - Depreciation rates: ”_k, ”_e_z, ”_i_z, ”_e set to 5%. - Trend growth rate g = 1.5%. - Domestic real interest rate r_d_o = 10% in initial steady state; real return on private capital r_k_o is a markup over r_d_o equivalent to capital depreciation (15%). - Risk-free world real interest rate r_f = 4%; initial real interest rate on foreign borrowing r_x_o = 6%; implied ‚_g = 0.02; ÷_g = 0 (risk premium constant). - Public domestic debt b_d_o = 20% of GDP. - Public external debt b_x_o = 0. - Grants/other revenues G_o = 4% of GDP. - Initial total public expenditure on infrastructure (current and capital) = 6% of GDP; O+M ≈ 3.4% of GDP for LIDCs in SSA reference. - Initial shares and O+M ratios: - Two-thirds of investment in roads and one-third in schools. - Recurrent expenditure as fraction of installed capital: 70% for schools, 50% for roads → average 56.7%. - Derived values: “_i_z = 0.0650 and “_e_z = 0.1517. - Fraction of government expenditures on roads È_i = 76.92% in initial equilibrium. - Consumption VAT ·_o = 15% at initial steady state. - Net transfers T_o = 7.94% of GDP at initial steady state. - Division of fiscal adjustment parameter ⁄ = 0 for simulations (only taxes bear adjustment). - Speed of adjustment of reference values fl_· = fl_T = 0.99. - Policy reaction parameters: ⁄_{·,1} = ⁄_{T,1} = 0.25; ⁄_{·,2} = ⁄_{T,2} = 0.02. Sensitivity analysis done for range of tax reactivity. ### Simulation design and experiments - Numerical simulations are annual, global nonlinear saddle path, perfect foresight, no uncertainty. - Experiments involve at least a one-time permanent change in policy; economy converges to a different steady state. - Main experiment: permanent increase in combined public investment and current expenditure equal to 1% of initial GDP (a 16.7% real scaling up from initial infrastructure expenditures). This is modest relative to a 50% scaling up (6% to 9% of GDP) in related literature. - Paper compares effects of scale-up exclusively in roads versus exclusively in schools; later sections analyze “big-push” and optimal composition. *Source: IMF Working Paper, wp17105 — Section 2.3 Government (extracted from provided content).* ### 4.1 A Permanent Scale-Up of Public Investment ### 4.1 A Permanent Scale-Up of Public Investment ### Comparative scenarios: scale-up entirely in roads vs entirely in schools - Total government infrastructure expenditures rise from 6% to 7% of GDP. - Roads scenario: - Total expenditure on roads rises permanently from 4% to 5% of GDP. - Capital expenditures in roads rise from 2% to 3% of GDP on impact. - Long-run split of the 1% increase: investment and current expenditures on roads evenly split the 1% increase (both rise from 2% to 2.5% of GDP). - Schools scenario: - Long-run split between capital and current expenditures is 30%-70% (investment increases from 0.6% to 0.9% of GDP; current expenditures jump from 1.4% to 2.1% of GDP). - The model calibration assumes investment in schools has higher returns, implying higher long-run growth from school investment but sharper transition trade-offs. ### Macroeconomic dynamics and transition trade-offs - Long-run output: - Investment in schools results in a long-run increase in output (above the underlying trend) of 24%. - Investment in roads yields a much smaller long-run increase of 5%. - Short- and medium-run growth dynamics: - For the first 13 years, the economy grows faster when public investment is in roads rather than in schools (growth over and above the exogenous growth rate of g). - With investment in schools, growth dips below its trend for about 9 years; with roads investment, growth stays above trend. - It takes 24 years for the additional output from investing in schools to overtake that delivered by investing in roads. - Private sector and consumption effects: - In the initial ~20 years, differences in private consumption across scenarios are moderate because: - Larger future productivity increases from school investment generate a stronger wealth effect and intertemporal labor substitution toward the future. - Agents cannot borrow from abroad, so private investment falls in the short run when output rises only slowly. - Fiscal and external implications: - As government ramps up investment and consumption (its tax base) and revenues respond little on impact, the fiscal deficit increases public debt (public external debt in this model). - Public debt rises by about 2% of GDP for investment in roads. - Investment in schools results in an almost threefold increase of 6% of GDP in public debt. - Persistence of debt increase: - Roads: increases the debt burden for a period of 30 years. - Schools: leads to an increase in public debt that lasts around 60 years. ### Key qualitative insight - Investment in schools produces larger long-run gains but slower realization of those gains, which forces more debt financing during the transition and exacerbates debt-sustainability concerns relative to investment in roads. *Italic source: IMF Working Paper (wp17105), section 4.1 — "A Permanent Scale-Up of Public Investment" (source PDF: wp17105 - 4.1 A Permanent Scale-Up of Public Investment)* ### 4.2 A Permanent Scale-Up with a “Big Push” ### Modeling the short-run “big push” - The short-run increase, as a percent of GDP, over and above the long-run 1% rise, is specified as 14(e ≠.2t ≠2e ≠.9t ). - The “big push” nearly doubles public expenditure from 6% to almost 13% in the first 2 years and it remains over 8% for about 14 years. ### Effects on transition speed, growth, and fiscal outcomes - Transition speed: - For investment in schools, the transition is almost complete in 60 years with the “big push” compared to more than 100 years in its absence. - Medium-run growth: - The “big push” takes growth well above the long-run outcome for investment in roads in the medium run. - With the “big push” under all-investment-in-schools, output, private consumption, and private investment overtake their all-roads counterparts about 4-5 years earlier (in about 20 years) than in scenarios without the “big push.” - Fiscal and tax dynamics: - The “big push” increases tax and debt burdens in initial years but shortens the duration for which taxes and debt remain elevated. - Public debt (as a fraction of GDP) comes back to the original level, and below, within 20 years because GDP rises much faster. - Pronounced differences between consumption taxes and public debt paths across roads vs schools (present without the “big push”) vanish with the “big push”; the fiscal handicap of schools vis-à-vis roads almost vanishes. - Welfare and short-run costs: - Although the “big push” accelerates educational investment benefits, short-run costs in terms of private consumption (and output) are much higher. - The “big push” strengthens intertemporal labor substitution effects, causing labor supply to decline sharply in the medium run relative to a permanent policy shift alone, reducing output and private consumption and raising welfare concerns. *Italic source: IMF Working Paper (wp17105), section 4.2 — "A Permanent Scale-Up with a “Big Push”" (source PDF: wp17105 - 4.1 A Permanent Scale-Up of Public Investment)* ### 5 Roads or Schools: “Optimal” Composition and Its Key Determinants ### Welfare-maximizing split of the scale-up - Consider a government choosing a constant split of the 1% long-run scale-up between roads and schools to maximize household welfare. - Without the “big push” (permanent 1% scale-up only): - The optimal share of expenditure allocated to education is 76%. - With the “big push” included: - The optimal share of education falls to 51%. ### Explanation of the change in optimal composition - The “big push” has two opposing effects: - Positive: school investment catches up faster with roads in consumption, output, tax, and debt terms. - Negative: short-run costs in consumption are much higher with the “big push.” - In the baseline calibration, the short-run costs outweigh the acceleration benefits, lowering the optimal share of schools despite their higher returns. ### Equilibrium paths and policy implication - Equilibrium macro paths for the optimal composition lie between the two extreme scenarios (all-roads and all-schools). - Without the “big push,” optimal paths are closer to the all-schools scenario (reflecting high optimal education share). - With the “big push,” optimal paths are more clearly between the two extremes. - Policy implication: - An intermediate, welfare-maximizing composition that blends roads and schools can improve welfare vis-a-vis all investment in schools by trading some future welfare gains for present gains. - The optimal composition also modestly reduces distortionary taxation and risks to debt sustainability, effects that are implicitly accounted for in the welfare comparison. *Italic source: IMF Working Paper (wp17105), section 5 — "Roads or Schools: “Optimal” Composition and Its Key Determinants" (source PDF: wp17105 - 4.1 A Permanent Scale-Up of Public Investment)* ### 5.1 Key Determinants of the Optimal Investment Composition ### 5.1 Key Determinants of the Optimal Investment Composition ### Return Differential - Baseline assumed differential: 15 percent in annual terms between returns to schools and roads (return is 25 percent for roads and 40 percent for schools). - Alternative extreme calibration analyzed: zero differential (return of 25 percent for both schools and roads) with the same aggregate scale-up in public investment of one percent of initial GDP. - Qualitative effects when school returns are lower or equal to roads: - Investing in schools gives rise to a stronger increase in public debt (differences stronger without the “big push”). - Higher taxes are required to stabilize debt. - More severe crowding-out of private consumption and investment in the short and medium run (more pronounced with a “big push”). - More pronounced intertemporal substitution of labor. - Higher GDP in the long run when investing in schools, offsetting lower initial-period GDP under equal-return assumption. - Quantitative implications for the welfare-optimal share of schools (keeping roads return at 25 percent): - When school return drops from 40 percent to 25 percent: - In the absence of a “big push”, optimal share of schools drops from around 76 percent to 12 percent. - With a “big push”, optimal share drops from around 51 percent to about 25 percent. - If return to schools declines below the 15-20 percent range (i.e., below that of roads), schools’ optimal share goes to zero. - Policy interpretation: - A relatively high expected return differential is required for a non-trivial share of the investment scale-up to be optimally allocated to schools. - Narrower differentials make the dynamic trade-offs more stringent and shift optimal composition toward roads. ### Debt Aversion - Context: developing-country constraints on government borrowing (debt intolerance) can stem from political instability, poor debt track record, high macro volatility, inability to mobilize tax revenues, or market-imposed limits. - Model representation: debt aversion measured by parameter ⁄·,1 œ(0,1]: a higher value corresponds to a larger share of the fiscal gap being covered by tax increases as opposed to bond issuance; in the limit, when ⁄·,1 = 1, the government runs a balanced budget and no new debt is issued. - Main result: greater debt aversion reduces the optimal share of schools in the investment scale-up because investing in schools produces a pronounced spike in government debt. - Quantitative illustration under baseline returns: - In the absence of a “big push”: - Optimal share of schools goes from almost 90 percent when the government resorts almost entirely to debt (taxes used minimally) to under 70 percent when the government resorts entirely to taxes. - In the presence of a “big push”, optimal share of schools is less sensitive to debt aversion because debt-path differences between schools and roads are smaller. - Policy implications: - Lack of access to external financing and higher distortionary taxation make it optimal to allocate a smaller fraction of the investment scale-up to schools. - International cooperation via concessional financing and grants can mitigate this financial friction and help governments achieve higher social spending targets. ### Political Myopia - Modeling approach: incumbents’ selfish re-election motives modeled as myopia — incumbents disregard benefits of policies after a finite time horizon. A fully altruistic planner has an infinite time horizon; greater selfishness implies shorter horizon. Rankings still based on agents’ discounted utility but summed over limited horizons. - Welfare-maximizing share of schools as a function of planner’s time horizon: - No “big push” case: - A planner with a horizon of less than 30 years would not invest at all in schools. - A time horizon of 60-70 years is needed to bring a myopic leader’s desired share close to that of an altruistic planner with infinite horizon. - “Big push” case: - Threshold for no investment in schools falls to 20 years. - It takes 45-55 years to attain a share comparable to that under an altruistic planner. - Interpretation: - The cost of political myopia falls with a “big push” because a “big push” brings forward the benefits of investing in schools. - Political myopia helps justify policymakers’ preference for quicker gains from roads in low-income developing countries; investing in schools requires a vision extending beyond one generation. - Policy implication: - Incorporating a “big push” into infrastructure scale-up can mitigate adverse effects of political myopia, but trade-offs with fiscal and debt sustainability remain. ### Key Conclusions and Policy Recommendations - Even with a 15 percent return differential favoring schools (25 percent for roads; 40 percent for schools), the model’s baseline optimal allocation limits the fraction of the investment increase dedicated to schools to about three fourths. - Central mechanism: front-loaded fiscal costs and slow accrual of growth benefits from schools versus quicker gains from roads, interacting with distortionary taxation and debt intolerance; political myopia further reduces optimal schooling shares. - Effects of a “big push”: - Mitigates policy-maker myopia by bringing benefits forward. - Decreases the fraction of the investment scale-up dedicated to schools to about one half, exacerbating threats to fiscal and debt sustainability. - Role for multilateral agencies: - Providing tied concessional financing and grants can (1) mitigate debt-intolerance concerns and (2) address political myopia by tying aid to investment in schools. - Extension note: - Similar trade-offs would arise when considering health infrastructure; analysis of hospitals versus roads is left for future research. *Source: wp17105 - 5.1 Key Determinants of the Optimal Investment Composition* ### 0.3600 Initial proportion of time used for non-leisure activities ### wp17105 - 0.3600 Initial proportion of time used for non-leisure activities ### Model parameters and structural elasticities - 0.3600 Initial proportion of time used for non-leisure activities - ’1.1648 Preference parameter for leisure - –0.4750 Capital’s share in value added - R i z (label present; no numeric value) - 0.2500 Initial return on infrastructure - Â0.1123 Elasticity of output with respect to infrastructure - Ê0.0800 Speed of transition of human capital from schools to the labor force - R e z (label present; no numeric value) - 0.4000 Initial return on schools - „0.5467 Elasticity of human capital accumulation with respect to schools - ‹0.5838 Elasticity of human capital accumulation with respect to private effort - ‰0.6911 Elasticity of effective units of labor with respect to human capital - ” k , ” e (labels present; no numeric values) - 0.0500 Depreciation rate of private economic and human capital - ” i z , ” e z (labels present; no numeric values) - 0.0500 Depreciation rate of public infrastructure and schools - g0.015 Trend growth rate - r d (label present; no numeric value) - 0.1000 Initial real interest rate on domestic debt - r k (label present; no numeric value) - 0.1500 Initial gross return on capital - r f (label present; no numeric value) - 0.0400 Risk-free real world interest rate - r x (label present; no numeric value) - 0.0600 Initial real interest rate on public external borrowing - ‚ g (label present; no numeric value) - 0.0200 Public debt risk premium - ÷ g (label present; no numeric value) - 0.0000 Public debt risk premium sensitivity parameter ### Initial fiscal stocks and flows (ratios to GDP) - b d o / y o = 0.2000 Initial public domestic debt to GDP ratio - b x o / y o = 0.0000 Initial public external debt to GDP ratio - G o / y o = 0.0400 Initial grants and other revenues to GDP ratio - g i o / y o = 0.0200 Initial capital investment in infrastructure to GDP ratio - g e o / y o = 0.0060 Initial capital investment in schools to GDP ratio - m i o / y o = 0.0200 Initial current expenditure on infrastructure to GDP ratio - m e o / y o = 0.0140 Initial current expenditure on schools to GDP ratio - “ i z = 0.0650 Current expenditure on infrastructure as fraction of infrastructure stock - “ e z = 0.1517 Current expenditure on schools as fraction of school stock - s i = 0.7692 Fraction of government capital expenditure going to infrastructure - · o = 0.1500 Initial consumption VAT rate - T o = 7.9376 Initial transfers to GDP ratio - ⁄0.0000 Share of fiscal adjustment borne by transfers - fl · , fl T = 0.9900 Speed of adjustment of reference values for tax and transfers - ⁄ ·,1 , ⁄ T,1 = 0.2500 Fiscal policy reaction parameters for policy instruments - ⁄ ·,2 , ⁄ T,2 = 0.0200 Fiscal policy reaction parameters for debt ### Stylized empirical relationships (Figure 1) - Public investment regression: pubinv = 13.06 - .917 rgdp_pc (R2: 0.34) - Social spending regression: soc_spend = -1.48 + 1.02 rgdp_pc (R2: 0.20) - Notes: Social spending is the sum of government spending on education and health (includes both current and capital expenditures). ### Simulation experiments and comparative outcomes (figures 2–9) - Aggregate shock size used across experiments: 1% of initial steady-state GDP (noted for multiple figures). - Figures compare two extreme allocations of a public investment scale-up: "All roads" versus "All schools"; later figures include an "Optimal composition" policy path. - Figure 2 (expenditures over time for a permanent increase in public infrastructure—All roads vs All schools): - Charts shown for total infrastructure expenditures, total expenditures on roads, total expenditures on schools, total capital expenditures, capital expenditures on roads, capital expenditures on schools, total current expenditures, current expenditures on roads, current expenditures on schools. - X-axes in years (0–100); Y-axes in percent of initial steady-state GDP. - Figure 3 (effects on key macro variables without a "big push"): - Variables plotted as percent deviations from initial steady state (unless otherwise indicated): Real GDP (%∆from SS), Real GDP growth (% YoY), Private consumption (%∆from SS), Private investment (%∆from SS), Labor supply (%∆from SS), Education effort (%∆from SS), Current account deficit (% of GDP), Consumption tax (%), Total public debt (% of GDP). - Figure 4 (effects with a "big push" short-run increase): - Long-run aggregate shock size: 1% of initial steady-state GDP, with a big push. - Variables include total infrastructure expenditures (% of GDP), Real GDP (%∆from SS), Real GDP growth (% YoY), Private consumption, Private investment, Labor supply, Education effort, Consumption tax (%), Total public debt (% of GDP). - Figures 6–9 (effects of the optimal composition of the public investment scale-up): - Compare "All roads", "All schools", and "Optimal composition" outcomes under both "Without the 'Big Push'" (Figures 6 and 8) and "With the 'Big Push'" (Figures 7 and 9). - Key macro variables reported: Real GDP (%∆from SS), Private consumption (%∆from SS), Private investment (%∆from SS), Labor supply (%∆from SS), Consumption tax (%), Total public debt (% of GDP). - Figures 8 and 9 present the same comparisons under the assumption of the same returns in roads and schools. ### Welfare and optimal composition results (Figures 5, 10–12) - Figure 5: Share of schools in the public investment scale-up and associated welfare - Panel (a) Without "Big Push": welfare plotted versus share of schools (%) with values on the welfare axis between -3.88×10−3 and -3.86×10−3 (tick values shown). - Panel (b) With "Big Push": welfare plotted versus share of schools (%) with welfare axis values from -3.82×10−3 to -3.75×10−3 (tick values shown). - Figure 10: Optimal share of schools in the public investment scale-up as a function of the return on schools (Return on schools (%) from 10 to 45): - Panel (a) Without "Big Push": Share of schools (%) plotted from 0 to 100. - Panel (b) With "Big Push": Share of schools (%) plotted from 0 to 100. - Figure 11: Effect of government’s debt aversion on optimal share of schools - Government’s debt aversion (λ τ,1 ) ranges shown from 0 to 1 in increments (00.10.20.30.40.50.60.70.80.91). - Panels for Without and With "Big Push" both plot Share of schools (%) from 0 to 100. - Figure 12: Effect of government’s myopia (time horizon in years) on optimal share of schools - Government’s time horizon (in years) shown: 10 20 30 40 50 60 70 80 90 100 +∞ - Panels for Without and With "Big Push" both plot Share of schools (%) from 0 to 100. *Source: wp17105 - 0.3600 Initial proportion of time used for non-leisure activities (PDF chapter/section).* --- _Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17105.pdf_