## _wp15272

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---

### 1. Introduction — research question, efficiency concept, and high-level intuition
- Key research question:
  - What is the growth impact of an increase in the rate of public investment spending, and how does this depend on the efficiency of public investment spending?
- Core concept of efficiency (ε):
  - Efficiency ε is the ratio between the actual increment of public capital and the amount spent (the fraction of spending converted into productive capital).
  - Four formalized manifestations of inefficiency:
    - Corruption: a fraction ε of spending is redirected or misclassified.
    - Waste: project costs are higher than necessary.
    - Poorly designed projects: low flow of capital services for the same spending.
    - Poor allocation across sectors/types of investment.
  - ε treated as time-invariant in the basic analysis (0 ≤ ε ≤ 1).
- High-level invariance intuition (Cobb-Douglas case):
  - Marginal contribution of an extra dollar of investment = (capital installed per dollar spent, ∝ ε) × (marginal productivity of that capital, decreasing in ε).
  - For a Cobb-Douglas production function with standard decreasing returns, these two effects exactly offset: growth impact of additional public investment spending does not depend on a permanent level of ε.
- Key model primitives and measured variables (notation preserved):
  - Public capital accumulation: G_t = G_{t−1}(1−δ) + ε I^g_t
  - Measured public capital stock (perpetual inventory): G^m_t ≡ Σ_{j=0}^{∞} (1−δ)^j I^g_{t−j} = G_t / ε
  - Simplest production function used: Y_t = A_t G_{t−1}^ψ
- Empirical note:
  - Regressed log GDP per capita on public capital stock controlling for private capital; fitted line in scatter: beta = 0.19 - 0.16*PIMI (t-stat = 0.64).
  - No significant correlation reported between PIMI (public investment management index) and estimated growth impact.

### 2. The basic invariance result (formal statement and steady-state)
- Growth relationship (logs/differences) for Cobb-Douglas:
  - Δy_t = a_t + ψ Δg_{t−1}
  - Using Δg_t = Δg^m_t (percent change in measured capital equals percent change in true capital) implies ε drops out.
- Steady-state relations (notation preserved):
  - G = ε I^g / δ
  - Marginal return to a permanent extra dollar of investment:
    - dY/dI^g = MPK * (dG/dI^g)
    - MPK = ψ (Y/G)
    - dG/dI^g = ε / δ
    - Combining: dY/dI^g = ψ Y / I^g
    - In growth-rate terms: (dY/Y) / (dI^g/I^g) = ψ
- Main formal result:
  - The effect of additional investment spending on the growth rate of output does not depend on the (constant) level of ε in the Cobb-Douglas specification.

### 3. Thought experiments, comparisons, and key subtleties
- Three thought experiments comparing countries with different permanent ε yield the same invariance:
  1. Keep investment-to-GDP share identical across countries — lower ε implies lower output level; invariance of growth impact holds.
     - Empirical support cited: (Pub Inv)/GDP = 9.79 - 1.25*PIMI (t-stat = 1.38; coefficient not statistically significant; p-value = 0.17).
  2. Keep nominal investment spending I^g identical across countries — dY/dI^g rises with ε in level terms, but percent growth effect invariant.
  3. Reclassify ε given observed Y and I^g — TFP (A) must adjust; marginal growth effect still invariant.
- Distinction emphasized:
  - Invariance applies to permanent cross-country differences in ε.
  - Time-varying ε within a country can have strong effects (analyzed in later sections).

### 4. Main corollaries and implications (Section 2.2)
- Observable measures and ε:
  - Replacing unobserved G_t by measured G_m_t = ε G_t removes ε from the growth-effect expression:
    - ∆y_t = a_t − δ + ψ (Y/G_m)_t (I_g/Y)_t. (Equation 14)
  - Controlling for output level and history of investment spending, the output effect depends on:
    - production parameter ψ, and
    - output/capital ratio (with capital measured as if ε = 1).
- Interpretation of empirical rates of return:
  - Empirical estimates typically measure the marginal product of public investment spending, not an increment to public capital.
  - Estimated gross marginal product ψ Y/G_m embeds inefficiency; an analyst computing ψ Y/G_m will effectively calculate ε ψ Y/G — so applying an additional inefficiency discount to published rate-of-return ranges is incorrect.
- Inferring ε and TFP:
  - Inferring ε from GDP growth and an assumed production function is misleading: time-invariant ε cannot explain negative TFP growth via the identity a_t = ∆y_t − ψ∆g_m_t.
  - Alternative causes for apparent negative TFP: very low ψ, time-varying ε, mismeasurement, conflict, supply shocks, policy reversals.
  - Conditional on output level and history of public investment, TFP level/change does not affect the effect of public investment on growth.
- Efficiency-adjusted capital stocks and level decompositions:
  - Time-invariant efficiency adjustments to capital stocks change levels but not growth rates; such adjusted series are uninformative for growth analysis.
  - Level decompositions of output into public capital and TFP do depend on ε; growth decompositions between periods do not.
- Time variation in efficiency matters:
  - Holding historical efficiency constant, an improvement in ε increases the output effect of public investment.
  - Efficiency declines (e.g., from investment surges overwhelming capacity) reduce the output effect.
- CES production function extension (qualitative and calibrated implications):
  - For CES between private capital K and public capital G with elasticity of substitution η, marginal returns depend on η:
    - dY/dI_g = ψ (A/δ)^{(η−1)/η} (1 / (I_g/Y))^{1/η} (ε)^{(η−1)/η}. (Equation 22)
    - Growth term: dY/Y ÷ dI_g/I_g = ψ (A I_g / δ Y)^{(η−1)/η} (ε)^{(η−1)/η}. (Equation 23)
  - Implications by η:
    - η = 1 (Cobb-Douglas): growth effect independent of ε (invariance).
    - η > 1 (substitutes): growth effect increasing in ε; as η → ∞ (“AK”), growth effect ∝ ε.
    - η < 1 (complements): growth effect decreasing in ε.
  - Illustration parameters: A = 1, ψ = 0.1, δ = 0.05, I_g/Y = 0.06, efficiencies ε_H = 0.9 and ε_L = 0.5.
  - Authors use η = 1 baseline: empirical evidence supports approximately constant elasticity of output w.r.t. public capital.
- Continuous-time dynamics (invariance holds dynamically):
  - With Y_t = A G_t^ψ and ̇G_t = ε I_g_t − δ G_t, an investment increase at t = 0 yields G_t = (G_0 − G_1) e^{−δ t} + G_1 with G_0 = ε I_g_0 / δ and G_1 = ε I_g_1 / δ.
  - Growth rate expression (Equation 27) is independent of ε for every t — dynamic invariance holds.

### 5. The role of private capital (Section 3.3)
- Model setup highlights (closed economy, log preferences, one-sector production):
  - Production: Y_t = A(G_{t−1})^ψ (K_{t−1})^α (L_t)^{1−α}. (equation (28))
  - Private capital accumulation: K_t = (1−δ)K_{t−1} + I_t. (equation (29))
  - Euler equation: (1/β) (C_{t+1} / C_t) = α (Y_{t+1} / K_t) + 1−δ. (equation (30))
  - Public capital accumulation: G_t = (1−δ)G_{t−1} + ε I^g_t. (equation (31))
  - Resource constraint: Y_t = C_t + I_t + I^g_t. (equation (32))
- Steady-state crowding-in and invariance:
  - K = Φ G^{ψ/(1−α)}  (equation (33)), Y = Γ G^{ψ/(1−α)}.
  - Invariance of long-run impact: dY/Y ÷ dI^g/I^g = ψ/(1−α), independent of ε.
- Dynamic simulations and calibration (parameters preserved):
  - Investment path: hump-shaped from 6 percent of initial GDP → almost 11 percent by year 3 → permanent new level 9.3 percent.
  - Calibration: time unit year; β = 0.94; initial infrastructure investment = 6 percent of GDP; α = 0.5; δ = 0.05; ψ set to match a net-of-depreciation rate of return of 25 percent for the low-efficient country.
  - Efficiency cases: low-efficiency ε = 0.5; high-efficiency ε = 1.
- Simulation findings:
  - For the same percentage increase in public investment, high- and low-efficiency countries attain the same growth benefits (invariance confirmed numerically); impulse responses are indistinguishable.
  - Public investment scaling-up raises GDP by 0.5percent, on average, in the first 10 years.
  - Private consumption and private investment are crowded out when scaling-up is tax-financed.
- Financing and private capital market frictions:
  - External financing (e.g., aid) dampens crowding-out and magnifies growth effects.
  - Private capital adjustment costs specified as AC_t = ν/2 ( (I_t / K_{t−1} − δ)^2 ) K_{t−1}; ν = 0 recovers no adjustment costs.
  - If low- and high-efficiency countries share ν, growth effects coincide; cross-country differences in ν affect transition dynamics (higher ν delays crowding-in and reduces transitional growth).
  - Example simulations with ε = 0.5: ν = 10 (severe adjustment costs) vs ν = 0 show delayed crowding-in and slower transition for high ν; convergence eventually occurs.
- Institutional interpretation:
  - PIMI and ICRG correlation ≈ 30 percent (sample of 50 countries; ICRG averaged 2007–2010).
  - Observed weaker output responses in low-efficiency countries may reflect private capital market frictions (higher ν) rather than lower marginal productivity of public capital per se.
- Policy implications:
  - Long-run output gains from scaling up public investment can be similar across ε levels, but transitional dynamics depend on financing and private capital market frictions.
  - External financing and strengthening private capital markets (reducing ν) can speed crowding-in and improve transitional growth responses.

### 6. Conclusion, appendices, and policy cautions
- Main result restated:
  - In the benchmark model (Cobb-Douglas), cross-country differences in the level of public investment efficiency do not matter for the growth impact of increases in public investment spending.
  - Result hinges on Cobb-Douglas specification but is approximately true when empirical evidence supports ψ constant.
- Complementarity caveat:
  - If public capital is highly complementary to private inputs (η < 1 or other non-Cobb-Douglas cases), intuition can reverse; with η > 1 the growth effect increases with ε.
- Robustness:
  - Inclusion of private capital, adjustment costs, and alternative inefficiency definitions does not overturn the broad invariance conclusion.
- Policy cautions (case-specific factors to consider before scaling up public investment):
  - marginal product of resulting capital,
  - efficiency,
  - cost of financing,
  - available "fiscal space",
  - discretionary taxation effects to finance investment,
  - prospects for operations and maintenance,
  - risks of debt distress.
- Measurement and timing notes:
  - PIMI cannot be readily mapped into the model’s ε; PIMI is cross-sectional and assumed monotonically related to ε.
  - Changes in public investment efficiency may be slow; waiting for efficiency improvements may not be viable in policy practice.
- Appendix A (project selection and four conceptions of efficiency):
  - The four conceptions (corruption, waste, low service flow, misallocation) are algebraically equivalent in lowering levels of output but not changing proportional growth impacts of additional public investment.
  - Example two-type production Y = A G^ψ H^φ with accumulation and allocation equations shows dY/Y ÷ dI/I = ψ + φ, independent of allocation θ_G and efficiencies ϵ.
- Appendix B (consumption with corruption — steady-state comparisons):
  - Two countries a (inefficient) and b (efficient) with equal initial output:
    - Y_a0 = A_a (ϵ G_a0)^ψ = Y_b0 = A_b (G_b0)^ψ ≡ 1. (B-1)
    - For higher common investment level I_1: Y_a1 / Y_a0 = Y_b1 / Y_b0 = (I_1 / I_0)^ψ ≡ Y_1. (B-4) — invariance of output change.
  - Consumption responses differ: for any ϵ < 1, the increase in consumption is larger in the inefficient (corrupt) country because less realized investment is needed to achieve the same output increase (see equations (B-5) and (B-6)).
  - Conclusion: inefficient countries can display larger long-run consumption gains from higher investment spending because a given spending increase translates into a smaller realized addition to public capital, freeing resources for consumption while output increases by the same proportion.

*Source: _wp15272 (IMF working paper PDF content provided).*

### 1. Introduction ........................................................................................................

### 1    Introduction

### Key research question and context
- What is the growth impact of an increase in the rate of public investment spending, and how does this depend on the efficiency of public investment spending?
- Builds on Pritchett (2000): one (real) dollar spent on public investment typically yields only a fraction in actual public capital; growth impact of additional investment spending may be lower in countries that are inefficient.
- The paper uncovers a theoretical invariance result: the growth impact of public investment spending is not higher in countries with a (permanently) higher level of public investment efficiency.

### Core concept of efficiency
- Efficiency (ε) is defined as the ratio between the actual increment of public capital and the amount spent (the fraction of spending converted into productive capital).
- Four ways inefficiency can manifest (and can be formalized in the model):
  - Corruption: a fraction ε of spending is redirected or misclassified (e.g., transfers to civil servants).
  - Waste: project costs are higher than necessary.
  - Poorly designed projects: low flow of capital services for the same spending.
  - Poor investment allocation across sectors or types of investment.
- ε is treated as time-invariant in the basic analysis (0 ≤ ε ≤ 1).

### High-level intuition for the invariance result
- The marginal contribution of an additional dollar of investment spending to output is the product of:
  - The amount of capital actually installed per dollar spent (increasing in ε).
  - The marginal productivity of that capital (decreasing in ε because lower steady-state scarcity raises capital stock).
- For a Cobb-Douglas production function with standard decreasing returns to a single reproducible factor, these two effects exactly offset: high- and low-efficiency countries have the same growth impact from additional public investment spending.

### Model primitives and empirical note
- Public capital accumulation equation (notation preserved):
  - G_t = G_{t−1}(1−δ) + ε I^g_t
- Measured public capital stock via perpetual inventory method:
  - G^m_t ≡ Σ_{j=0}^{∞} (1−δ)^j I^g_{t−j} = G_t / ε
- Simplest production function used:
  - Y_t = A_t G_{t−1}^ψ
- Empirical exploration:
  - Regressed log GDP per capita on public capital stock controlling for private capital, obtaining country-specific coefficients (betas) as supply-side measures of growth effect of public investment.
  - Figure 1 reports the scatter of estimated coefficients against the public investment management index (PIMI). The fitted line: beta = 0.19 - 0.16*PIMI (t-stat = 0.64).
  - No significant correlation between the efficiency measure (PIMI) and the size of the growth impact in the presented scatter.

### Relevance and positioning
- Results derived in the context of standard exogenous growth models (Solow, 1956; Mankiw, Romer and Weil, 1992) because:
  - These models are workhorses of growth theory and empirics explaining features like conditional convergence.
  - Pritchett’s assessment is also based on exogenous growth models.
- Paper acknowledges other frameworks and approaches to modeling efficiency (e.g., network effects, rent-seeking bureaucracies) but focuses on the ε-concept consistent with Pritchett (2000), Caselli (2005), Gupta et al. (2014), and related macro models.

---

### 2    The basic invariance result

### Formal statement (Cobb-Douglas case)
- Growth rate relation (logs and differences):
  - Δy_t = a_t + ψ Δg_{t−1}
  - Using Δg_t = Δg^m_t, the percent change in measured capital stock equals percent change in true capital stock, implying ε does not affect growth.
- Steady-state expressions:
  - G = ε I^g / δ
  - Marginal return to a permanent extra dollar of investment:
    - dY/dI^g = MPK * (dG/dI^g)
    - MPK = ψ (Y/G)
    - dG/dI^g = ε / δ
    - Combining gives dY/dI^g = ψ Y / I^g
    - In growth-rate terms: (dY/Y) / (dI^g/I^g) = ψ
- Main result: the effect of additional investment spending on the growth rate of output does not depend on the (constant) level of ε in the Cobb-Douglas specification.

### Intuition recap
- Two opposing forces as ε rises:
  - More capital per unit of spending (raises direct impact).
  - Larger capital stock reduces marginal productivity (lowers direct impact).
- Under Cobb-Douglas, offset is exact; growth elasticity of output to investment is ψ irrespective of ε.

---

### 3    Some subtleties and thought experiments

### Comparing countries with different permanent ε
- Three natural thought experiments yield same invariance result, though other variables adjust:
  1. Keep investment-to-GDP share the same across countries (most natural approach in paper). Then lower ε implies lower level of output; invariance holds.
     - Empirical support: the ratio of investment spending to output is not correlated with PIMI across countries (Figure 2). The linear fit: (Pub Inv)/GDP = 9.79 - 1.25*PIMI (t-stat = 1.38) (coefficient not statistically significant; p-value = 0.17).
  2. Keep nominal investment spending I^g identical across countries (not the investment share). Then dY/dI^g rises with ε (MPK inversely proportional to ε^{1−ψ}), but the growth effect (percent change) remains invariant because output changes proportionally.
  3. Revise assessment of a country’s ε given observed Y and I^g (i.e., reclassifying measured capital stock). Then A (TFP) must adjust in the model to reconcile observed Y and I^g; invariance of marginal growth effect still holds.

### Important distinction: permanent cross-country differences vs. time-varying ε
- The invariance result applies to comparing countries with different permanent levels of ε.
- Changes in ε over time within a country can have strong effects (these are analyzed in later sections of the paper).

---

### 4    Implications emphasized in the introduction
- The trade-off between scarcity and efficiency matters for policy: ignoring that lower efficiency raises marginal productivity through scarcity can lead to incorrect policy conclusions.
- Both need (scarcity) and efficiency matter for public investment policy; policy guidance should take into account the invariance insight when using exogenous growth frameworks.

*Source: _wp15272 - 1. Introduction (IMF working paper PDF content provided).*

### 2.2    Implications

### 2.2    Implications

### Main corollaries of the invariance result
- The assertion that the growth impact of public investment spending depends on  is incorrect when observable measures are used. Replacing unobserved G_t by measured G_m_t = G_t removes  from the expression for the growth effect:
  - ∆y_t = a_t − δ + ψ (Y/G_m)_t (I_g/Y)_t. (Equation 14)
- Controlling for the level of output and the history of investment spending, the output effect of additional investment spending depends on:
  - the production function parameter ψ, and
  - the output/capital ratio (with capital measured as if efficiency is 1).

### Interpretation of empirical rates of return
- Empirical estimates of the rate of return to public investment generally measure the marginal product of public investment spending, not an increment to public capital.
- Although estimations often implicitly assume  = 1, the estimated rate of return includes the effects of inefficiency. An analyst computing the gross marginal product as ψ Y/G_m will actually calculate ψ Y/G, i.e., the marginal product of public investment spending including inefficiency.
- Analysts who take published rate-of-return ranges (for example, the 15-30 percent range proposed in Dalgaard and Hansen (2015)) and then apply an additional inefficiency discount would be mistaken.

### Inferring efficiency and TFP
- Efforts to infer efficiency  from data on GDP growth and an assumed production function are misguided. Using Pritchett (2000)’s logic, setting  = 1 and solving a_t = ∆y_t − ψ∆g_m_t does not yield a valid measure of  because equation (4) shows time-invariant efficiency cannot explain negative TFP growth.
- Alternative explanations for apparently negative TFP growth include:
  - very low values of ψ,
  - time-varying ,
  - various mismeasurement issues,
  - conflict, supply shocks, or policy reversals.
- TFP does not matter for the output effect of public investment once spending/GDP is controlled for: the level and change in TFP do not matter for the effect of public investment on growth, conditional on the level of output and the history of public investment spending.

### On “efficiency-adjusted” capital stocks and level decompositions
- “Efficiency-adjusted” capital stocks are generally uninformative for growth analysis when the efficiency proxy is time-invariant:
  - Adjusting public capital stocks by a time-invariant quality index (e.g., GCR ‘quality of roads’ index) produces series that differ only by a constant percentage; their growth rates are identical and thus irrelevant for growth issues.
- Level decompositions of output into contributions of public capital and TFP do depend on :
  - Example: Y_b_t / Y_a_t = A_b_t (_b G_m_b)^ψ / [ A_a_t (_a G_m_a)^ψ ]. (Equation 15)
  - If Y_b_t / Y_a_t = 2^{1/ψ} and G_m are equal, then:
    - assuming  = 1 (“full efficiency case”) implies A_b_t / A_a_t = 2^{1/ψ} (output difference due to TFP),
    - but if _b = 1 and _a = 1/2 (“different efficiency case”) then A_b_t = A_a_t and the output difference is due to efficiency-adjusted capital stocks.
  - While level decompositions differ with assumptions about time-invariant , decompositions of growth between periods do not depend on .

### Time variation in efficiency matters
- Changes in efficiency over time affect the output effect of public investment:
  - Holding historical efficiency constant, an increase in efficiency increases the output effect of public investment (e.g., via structural reforms or “investing in investing”).
  - Decreases in efficiency can arise from investment surges that overwhelm administrative and implementation capacity.
- Time-varying efficiency will alter growth effects; this issue is explored further in Section 3.5.

### CES production function extension (qualitative and calibrated implications)
- Generalizing to a CES production function combining private capital K and public capital G shows the role of the elasticity of substitution η:
  - With K = 1 and production Y = A [ ψ (G)^{(η−1)/η} + 1−ψ ]^{η/(η−1)}, the marginal product of public capital is:
    - dY/dG = ψ A^{(η−1)/η} (Y/G)^{1/η}. (Equation 18)
  - Given G =  I_g / δ (Equation 19) and dG/dI_g = /δ (Equation 21), the return on public investment is:
    - dY/dI_g = ψ (A/δ)^{(η−1)/η} (1 / (I_g/Y))^{1/η} ()^{(η−1)/η}. (Equation 22)
  - Rewriting in growth terms:
    - dY/Y ÷ dI_g/I_g = ψ (A I_g / δ Y)^{(η−1)/η} ()^{(η−1)/η}. (Equation 23)
- Implications by η:
  - If η = 1 (Cobb-Douglas), the growth effect is independent of efficiency (invariance result).
  - If η > 1 (inputs substitutes), the growth effect is increasing in efficiency ; in the limit η → ∞ (“AK” technology), the growth effect is directly proportional to .
  - If η < 1 (inputs complements), the growth effect is decreasing in efficiency .
- Illustration parameters used in the example:
  - A = 1, ψ = 0.1, δ = 0.05, I_g/Y = 0.06, with efficiencies _H = 0.9 and _L = 0.5.
- The authors maintain η = 1 (Cobb-Douglas) as baseline because:
  - empirical evidence suggests the elasticity of output with respect to public capital is constant (Calderón, Moral-Benito and Servén, 2015), and
  - when η ≈ 1 results are approximately Cobb-Douglas.

### Dynamics: invariance in continuous time
- In a continuous-time dynamic setup with:
  - Y_t = A G_t^ψ (Equation 24),
  -  ̇G_t =  I_g_t − δ G_t (Equation 25),
  - suppose at t = 0 investment increases from I_g_0 to I_g_1 and stays there (I_g_1 > I_g_0).
- Solving yields G_t = (G_0 − G_1) e^{−δ t} + G_1 with G_0 =  I_g_0 / δ and G_1 =  I_g_1 / δ.
- The growth rate is:
  - ̇Y_t / Y_t = ψ [ δ (I_g_1 / I_g_0) (I_g_1 / I_g_0) (1 − e^{−δ t}) + e^{−δ t} − δ ], (Equation 27)
  - which is independent of efficiency  for every point in time — the invariance result holds dynamically as well.

*Source: _wp15272 - 2.2    Implications*

### 3.3    The Role of Private Capital

### _wp15272 - 3.3    The Role of Private Capital

### Model setup and key equations
- One-sector neoclassical growth model (closed economy) with perfectly competitive markets; long-run exogenous growth rate is zero.
- Representative consumer maximizes intertemporal utility from consumption Ct with logarithmic preferences (intertemporal elasticity of substitution = 1).
- Labor supply Lt is inelastic and equal to 1.
- Firms produce Yt using private capital Kt, labor Lt, and public capital Gt.
- Government invests Ig_t units in public capital and finances it with lump-sum taxes Tt.
- Core reduced-form equations:
  - Production: Yt = A(Gt−1)^ψ (Kt−1)^α (Lt)^{1−α}  (equation (28))
  - Private capital accumulation: Kt = (1−δ)Kt−1 + It  (equation (29))
  - Euler equation: (1/β) (Ct+1 / Ct) = α (Yt+1 / Kt) + 1−δ  (equation (30))
  - Public capital accumulation with inefficiencies: Gt = (1−δ)Gt−1 + ϵ Ig_t  (equation (31))
  - Resource constraint: Yt = Ct + It + Ig_t  (equation (32))

### Long-run (steady-state) results and invariance
- Steady-state crowding-in of private capital:
  - K = Φ G^{ψ/(1−α)}  (equation (33)), where Φ>0 depends on β, α, A, and δ.
  - Y = Γ G^{ψ/(1−α)}, with Γ depending on β, α, A, and δ.
- Invariance result (impact of public investment on output accounting for crowding-in):
  - dY/Y  divided by  dIg/Ig  = ψ/(1−α)
  - This impact is independent of the level of efficiency ϵ (i.e., the invariance result holds in steady state even when private capital and crowding-in are included).

### Dynamic analysis and calibrated simulations
- Simulation scenario for public investment path (percent of initial GDP): hump-shaped path starting at 6 percent, almost 11 percent by year 3, then tapering to a permanent new level of 9.3 percent (Figure 5).
- Calibration notes (broadly in line with Buffie et al. (2012) for an average low-income country):
  - Time unit: year; discount factor β = 0.94.
  - Initial infrastructure investment set to 6 percent of GDP.
  - Capital’s share α = 0.5.
  - Depreciation rate δ = 0.05.
  - Elasticity of output w.r.t. public capital ψ set to match a rate of return on public capital (net of depreciation) of 25 percent for the low-efficient country.
- Efficiency parameter choices for comparison:
  - Low-efficiency case: ϵ = 0.5 (in line with estimates by Arestoff and Hurlin (2010)).
  - High-efficiency case: ϵ = 1.
- Main simulation findings:
  - For the same percentage increase in public investment, high-efficiency and low-efficiency countries enjoy the same growth benefits (invariance result confirmed numerically); impulse responses for low and high-efficiency cases are indistinguishable in Figure 5.
  - The public investment scaling-up translates into more public capital and raises GDP by 0.5percent, on average, in the first 10 years.
  - Private consumption and private investment are crowded out (tax financing of the scaling-up shifts resources away from the private sector).

### Role of financing and private capital markets (transition dynamics)
- Financing matters for the magnitude/timing of crowding-out and crowding-in:
  - If the public investment increase were financed with external resources (e.g., aid), crowding-out effects would be dampened and growth effects magnified.
- Distortions that slow private capital responsiveness modeled as private capital adjustment costs:
  - Adjustment cost specification: AC_t = ν/2 ( (It / Kt−1 − δ)^2 ) Kt−1, where ν measures severity of costs; ν = 0 reproduces previous model without adjustment costs.
  - With adjustment costs, part of output is allocated to cover these costs; Tobin’s Q and Euler equations are modified accordingly (as described in the text).
- If low- and high-efficiency countries share the same ν, growth effects of public investment increases still coincide.
- Cross-country differences in ν matter for dynamic (but not long-run) effects:
  - Simulation comparison (Figure 6) with ϵ = 0.5:
    - Severe private capital adjustment costs: ν = 10.
    - No adjustment costs: ν = 0.
  - Finding: higher ν (severe adjustment costs) delays crowding-in of private capital and reduces growth in the transitional dynamics; differences can persist for decades though lines eventually converge.

### Institutional interpretation and implications
- Measures of “efficiency” are correlated with broader measures of institutions:
  - PIMI and ICRG measures of “institutions” have a correlation coefficient of 30 percent on a sample of 50 countries (ICRG scores averaged over 2007–2010 period for which PIMI is calculated).
- Economies with poorly functioning private capital markets (higher effective ν) may show weaker crowding-in dynamics; cross-country empirical associations between low measured “efficiency” and smaller output responses may reflect these private capital market frictions rather than lower marginal productivity of public capital per se.
- Policy implications (implied by model and simulations):
  - Scaling up public investment can raise output similarly across efficiency levels in the long run, but transition dynamics depend on private capital market frictions and financing modalities.
  - External financing for public investment scaling-up can mitigate private crowding-out and amplify growth effects.
  - Strengthening private capital markets and lowering adjustment costs (reducing ν) can speed crowding-in and improve transitional growth responses to public investment.

*Italic: Source — _wp15272 - 3.3    The Role of Private Capital (authors’ calculations and text from the source PDF).*

### 0.62 might look like “slow progress”, but an investment that earns 15.5 percent

### _wp15272 - 0.62 might look like “slow progress”, but an investment that earns 15.5 percent

### Conclusion — main results and policy implications
- Main result: In a simple benchmark model, cross-country differences in the level of public investment efficiency do not matter for the growth impact of increases in public investment spending.
- This invariance hinges on the Cobb-Douglas specification for the production function, but is "approximately true" if the production function is Cobb-Douglas, consistent with empirical evidence cited.
- If public capital is highly complementary to private factors of production, the standard intuition is overturned: investment spending in relatively low-efficiency countries would have a relatively large effect on real output growth, if they have the same investment-spending/output ratios.
- Robustness: Careful treatments of private capital, adjustment costs, and different definitions of inefficiency do not change the broad conclusion.
- Key caution: Whether low-efficiency countries should increase public investment depends on multiple case-specific factors, including:
  - marginal product of the resulting capital,
  - efficiency,
  - cost of financing,
  - available "fiscal space",
  - discretionary taxation effects needed to finance investment,
  - prospects for and costs of operations and maintenance,
  - risks of debt distress.
- Measurement challenge: Public Investment Management Index (PIMI) cannot readily be mapped into a quantitative efficiency parameter (denoted  in the text); it is only available in cross-section and is only assumed to be monotonically related to .
- Policy note on timing: Evidence suggests changes in public investment efficiency may be slow, so waiting for efficiency improvements may not always be viable.

### Appendix A — Project selection and four conceptions of efficiency
- Four ways to think about public investment efficiency:
  1. A fraction  of spending is literally wasted (“corruption”).
  2. Project costs are higher than needed because of inefficiently high use of inputs (“waste”).
  3. Government may choose projects that yield a greater or lesser flow of capital services for the same investment spending (“poorly designed projects”).
  4. Governments may misallocate public investment spending across sectors or categories (“poor investment allocation”).
- Common implication: All four conceptions have similar implications for the public investment/growth relationship.
- Static ranking metaphor: Available projects can be ranked from highest to lowest rate of return; marginal product of public investment equals the return of the best available project (Figure A-1).
- Interpretation of inefficiency definitions in model equations:
  - Definitions (1) and (2): only a fraction  of spending makes its way into public capital G; 1− portion is not converted into capital. The basic accumulation equations of section (2), equations (1) and (3), apply.
  - Definition (3): different projects produce the same stock but different service flows; flow of infrastructure services equals G_m and output is Y = A(G_m)^ψ. This is algebraically equivalent to discounting investment by  and using effective capital G in the production function.
  - Definition (4): sectoral/types heterogeneity where different project types are not perfect substitutes. Example two-type production function:
    - Y = A G^ψ H^φ . (A-1)
    - Capital accumulation:
      - G_t = (1−δ_G) G_{t−1} + _G I_G (A-2)
      - H_t = (1−δ_H) H_{t−1} + _H I_H (A-3)
    - Investment allocation: I_G = θ_G I and I_H = (1−θ_G) I. (A-4)
    - Steady-state rewritten:
      - Y = A (_G θ_G I / δ_G)^ψ (_H (1−θ_G) I / δ_H)^φ . (A-5)
    - Optimal allocation equalizes marginal products:
      - θ^*_G = ψ / (φ + ψ) . (A-6)
    - Growth impact of additional investment spending is independent of θ_G and :
      - dY/Y ÷ dI/I = ψ + φ.
- Synthesis: A country may choose the wrong mix of types of projects and, within types, choose especially wasteful or low-service-flow projects. All such forms of inefficiency are consistent with the main-text results: they lower levels of output but do not change the proportional growth impact of additional public investment.

### Appendix B — Consumption with corruption (steady-state comparisons)
- Setup and key identities:
  - Consider two countries a and b at initial steady state with equal output but different efficiencies: country b fully efficient, country a inefficient (corrupt); assume no private capital.
  - Initial identities:
    - Y_a0 = A_a ( G_a0)^ψ = Y_b0 = A_b (G_b0)^ψ ≡ 1. (B-1)
    - I_b0 = I_a0 ≡ I_0, G_a0 =  I_0 / δ, G_m0_a = G_m0 / , G_b0 = G_m0_b = I_0 / δ.
    - Equivalence of output implies A_b = A_a ^ψ.
    - Budget constraints:
      - C_a0 = Y_0 −  I_0 (B-2)
      - C_b0 = Y_0 − I_0 (B-3)
- Invariance of output change to efficiency:
  - For a higher common investment level I_1:
    - Y_a1 / Y_a0 = Y_b1 / Y_b0 = (I_1 / I_0)^ψ ≡ Y_1. (B-4)
- Consumption responses differ by efficiency:
  - Country b (efficient): C_b1 / C_b0 = (I_1 / I_0)^ψ − I_1 (1 − I_0). (B-5)
  - Country a (inefficient): C_a1 / C_a0 = (I_1 / I_0)^ψ − I_1 (1 −  I_0). (B-6)
  - For any  < 1, the increase in consumption is larger in the inefficient (corrupt) country because less actual realized investment is needed to achieve the same output increase, leaving more for consumption.
- Alternate assumption (same TFP, different efficiency):
  - If initial Y_a0 = A ( I_a / δ)^ψ = ^ψ Y_b0 (B-7), invariance still holds:
    - Y_a1 / Y_a0 = Y_b1 / Y_b0 = (I_1 / I_0)^ψ ≡ Y_1. (B-8)
  - Consumption ratios:
    - C_b1 / C_b0 = (I_1 / I_0)^ψ − I_1 (1 − I_0). (B-9)
    - C_a1 / C_a0 = ^ψ (I_1 / I_0)^ψ −  I_1 / (^ψ −  I_0) = (I_1 / I_0)^ψ − I_1 ^{1−ψ} (1−I_0) / (^{1−ψ}). (B-10 as presented)
  - Conclusion: When 0 ≤  < 1, ψ < 1, and I_1 > I_0, consumption growth in the inefficient country a is higher than in the efficient country b.
- Economic intuition: Inefficient/corrupt countries may exhibit larger long-run consumption gains from higher investment spending because a given increase in spending translates into a smaller realized addition to public capital, freeing resources for consumption even while output increases by the same proportion.

*Source: _wp15272 - 0.62 might look like “slow progress”, but an investment that earns 15.5 percent*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15272.pdf_