Bridging micro and macro production functions: The fiscal multiplier of infrastructure investment
What this paper finds — and why it matters
Layer 1 — Overview
Research Question
This paper investigates the fiscal multiplier of infrastructure investment, specifically by incorporating firm-level investment decisions — a dimension absent from prior literature. The central analytical challenge is bridging the micro (firm-level) and macro (state-level) production functions for infrastructure, given that public capital is non-rivalrous: it can be used simultaneously by all firms without being depleted. The paper demonstrates that this non-rivalry generates a systematic discrepancy between firm-level and aggregate-level estimates of the elasticity of substitution between private and public capital, and it shows how this discrepancy shapes the magnitude of the fiscal multiplier.
Data and Methodology
The authors build and estimate a heterogeneous-firm general equilibrium model. Firms operate a constant-elasticity-of-substitution (CES) production function using private capital, non-rivalrous public capital (infrastructure), and labor. Firms are subject to idiosyncratic productivity shocks and make lumpy investment decisions subject to both fixed and convex capital adjustment costs, following Cooper and Haltiwanger (2006) and Winberry (2021). The economy has two regions — one with poor infrastructure and one with good infrastructure — motivated by the near-invariant cross-state distribution of infrastructure spending observed in U.S. data.
The model is estimated via an extended Simulated Method of Moments (SMM) that treats market clearing prices as additional parameters estimated simultaneously with structural parameters, reducing computational cost relative to standard GE estimation. Estimation uses a multi-block Metropolis-Hastings algorithm. Target moments include lumpy investment fraction (0.14, from Zwick and Mahon 2017), average investment-to-capital ratio (0.10), standard deviation of i/k (0.16), private-to-infrastructure capital ratio (0.75, from BEA), high-infrastructure region’s private capital share (0.83, from Census BDS), and total working hours (0.33).
The identification of the key parameter — the firm-level elasticity of substitution between private and public capital (λ) — comes from the relative size of private capital stocks across the two infrastructure groups: under greater complementarity, regions with more infrastructure should hold relatively more private capital.
External validation is provided by estimating the state-level elasticity from the model’s simulated data using a nonlinear least squares method following An et al. (2019), and comparing it to empirical state-level estimates from actual U.S. state data.
Main Findings with Quantitative Magnitudes
Firm-level vs. aggregate-level elasticity gap. The estimated firm-level elasticity of substitution is λ = 1.185, implying gross substitutability between private and public capital at the firm level. The state-level elasticity implied by the same model is 0.48 (or 0.35 in a decreasing-returns-to-scale specification), implying gross complementarity. The empirical state-level counterpart estimated from actual U.S. data is 0.445. The paper proves theoretically (Proposition 1) that, given non-rivalry and under mild conditions, firm-level gross substitutability implies aggregate-level gross complementarity. Proposition 2 further shows that this same mechanism micro-founds the increasing-returns-to-scale assumption in Baxter and King’s (1993) Cobb-Douglas aggregate production function.
Fiscal multiplier (baseline, 2-year horizon). The aggregate output multiplier over a 2-year horizon in the heterogeneous-firm general equilibrium model is 1.088 in response to a one-time unexpected infrastructure spending shock equal to 1% of steady-state GDP, financed by a lump-sum tax. The corresponding partial-equilibrium output multiplier (holding prices fixed at steady state) is 1.858; the gap reflects crowding out of private investment induced by the general equilibrium interest rate response. In the baseline, the interest rate rises by 0.39% after the shock; the investment multiplier is -0.043.
Comparison with representative-agent model. When the same implied returns-to-scale parameters are used in a representative-agent model (following Baxter and King 1993), the output multiplier is 0.991 and the investment multiplier is -0.157, both substantially lower than the heterogeneous-firm baseline. The key mechanism: under convex adjustment costs, the Jensen’s inequality effect implies that heterogeneous firms face a greater average adjustment burden than the representative firm, making their investment less responsive to the general equilibrium crowding-out pressure.
Sensitivity to elasticity of substitution. Across the heterogeneous-firm model: at λ = 3 (high substitutability), the output multiplier falls to 0.672; at λ = 0.5 (complementarity), it rises to 1.364. The multiplier is significantly more sensitive to λ in the heterogeneous-firm model than in the representative-agent model, because non-rivalry amplifies the effect of any given elasticity value through each firm’s production function.
Cross-state distribution of gains. Under the baseline spending allocation (81% to Good states, 19% to Poor states), per $1 of infrastructure spending, Good states receive $1.072 of the $1.088 total output gain, while Poor states receive only $0.016. In a counterfactual with equal spending across states, the total output multiplier falls to 0.873, Good states’ output multiplier falls to 0.810, and Poor states’ output multiplier rises to approximately 0.062 (about four times the baseline level of 0.016). This quantifies a sharp efficiency-equality trade-off in the allocation of infrastructure investment.
Employment and earnings effects. Compared to steady state, the baseline fiscal shock produces an average annual increase of 0.304% in employment and 0.389% in wages, yielding a $0.713 increase in earnings and a $0.148 increase in consumption per $1 of fiscal spending in general equilibrium. In partial equilibrium (no price changes), earnings increase by $1.294 and consumption by $0.605 per $1 spent.
Scope Conditions
Results are conditional on: (i) lump-sum tax financing of the fiscal shock; (ii) a one-time unexpected (MIT) shock with no persistence; (iii) a closed-economy framework with endogenous real interest rate; (iv) the estimated two-region structure calibrated to U.S. state-level infrastructure data; (v) firm-level investment dynamics calibrated to Compustat and BDS moments. The authors note that incorporating time-to-build assumptions (tested in an appendix) reduces the aggregate fiscal multiplier, consistent with Ramey (2020).
Layer 2 — Q&A
Q1: What is the core theoretical result connecting firm-level and aggregate-level elasticities, and what is the intuition?
A: Proposition 1 proves that, given non-rivalrous public capital and mild data conditions (at least one firm has private capital below total infrastructure, and aggregate private capital exceeds total infrastructure), if the firm-level elasticity of substitution λ ≥ 1 (gross substitutes), then the aggregate-level elasticity ξ < 1 (gross complements). The intuition is that a marginal increase in public capital raises the marginal product of private capital for every firm simultaneously due to non-rivalry; the sum of these MPK gains across all firms exceeds any single firm’s gain. To represent this amplified benefit within an aggregate production function, a stronger complementarity is required than what any single firm faces. Put differently, non-rivalry means aggregate private and public capital “look” more complementary than they truly are at the firm level.
Q2: How does non-rivalry micro-found the Baxter-King aggregate production function?
A: Proposition 2 shows that if firms use a CES production function with gross substitutability (λ ≥ 1) and non-rivalrous public capital, then fitting aggregate output with a Cobb-Douglas production function (as in Baxter and King 1993, H(K,N,L) = zK^α L^{1-α} N^ζ) yields ζ > 0, implying increasing returns to scale (IRS). This is the paper’s micro-foundation for a widely-used but previously ad hoc assumption in the macro-fiscal literature. The corollary states that both gross complementarity in the aggregate CES function and IRS in the aggregate Cobb-Douglas follow from the same non-rivalry mechanism at the firm level.
Q3: Why does the heterogeneous-firm model produce a higher output multiplier than the representative-agent model?
A: Two mechanisms drive the difference. First, due to Jensen’s inequality and the convexity of adjustment costs, heterogeneous firms face a higher average adjustment burden than the representative (average) firm; this means heterogeneous firms are less responsive to interest rate changes that crowd out investment. The investment multiplier is -0.043 in the heterogeneous-agent baseline versus -0.157 in the representative-agent model. Second, the fixed adjustment cost (present in the baseline but absent from the representative-agent model) further dampens investment sensitivity via the extensive margin. Because less private investment is crowded out, more of the direct output boost from infrastructure spending survives into the aggregate multiplier, yielding 1.088 versus 0.991.
Q4: What is the novel estimation procedure and why is it necessary?
A: Standard SMM applied to GE models requires solving for market-clearing prices for every candidate parameter vector, creating a nested optimization loop that is computationally prohibitive. The authors extend SMM by treating market-clearing prices (wage w and marginal utility of consumption p) as additional parameters and appending market-clearing conditions as additional target moments — effectively requiring those moments to equal zero. A multi-block Metropolis-Hastings algorithm jointly draws from the price block and the parameter block. This approach generates posterior draws that simultaneously satisfy market clearing and fit empirical moments, without the inner loop. The resulting market-clearing accuracy is e^{-4} at the posterior mean.
Q5: How is the firm-level elasticity of substitution (λ) identified from the data?
A: λ is identified from the cross-state difference in private capital stocks between high- and low-infrastructure regions. Under the model, if private and public capital are more complementary (lower λ), high-infrastructure regions should attract relatively more private capital. The data moment used is the Good region’s share of aggregate private capital (0.83 from Census BDS data). This identification strategy is analogous to Bartik-instrument approaches in the empirical literature, where a parameter governing cross-state sensitivity to aggregate shocks is identified from cross-sectional variation.
Q6: How is the model validated externally?
A: The authors compute the state-level elasticity from the estimated model by fixing firm-level parameters and re-estimating only the elasticity and regional productivity from the model’s simulated state-level data, using the same NLLS estimator as An et al. (2019). The model-implied state-level elasticity is 0.349 (DRS specification) or 0.482 (CRS specification). The empirical estimate from actual U.S. state-level data following the same estimator is 0.445. Both indicate gross complementarity at the state level, consistent with the theoretical prediction. This external validation is not used in the estimation itself, providing an independent check.
Q7: What are the roles of extensive vs. intensive investment margins in the crowding-out effect?
A: Table 9 decomposes the investment multiplier of -0.043 by investment margin. When only the extensive margin (the discrete decision of whether to invest) is allowed to respond, the investment multiplier is -0.032 — approximately 74% of the baseline crowding-out effect. When only the intensive margin (investment size conditional on adjusting) responds, the multiplier is -0.011 — about 25% of the total. Thus the extensive margin is the dominant channel through which higher interest rates crowd out private investment. When both margins are held fixed, the output multiplier rises to 1.139, confirming that investment crowding-out reduces the output multiplier by about 0.05.
Q8: How does the elasticity of substitution affect the fiscal multiplier quantitatively, and why does this matter more in the heterogeneous-firm model?
A: In the heterogeneous-firm GE model: λ = 3 gives an output multiplier of 0.672, λ = 1.185 (baseline) gives 1.088, and λ = 0.5 gives 1.364 — a range of 0.692. In the representative-agent model, the comparable range across the implied ζ values is much narrower (0.970 to 0.998). The amplification in the heterogeneous-firm model occurs because non-rivalry means each firm’s production function directly incorporates the public capital stock, so the elasticity parameter has first-order consequences for every firm’s investment incentive response to a fiscal shock. This heightened sensitivity underscores why accurately estimating λ at the firm level — rather than importing a state-level estimate — is critical for quantifying infrastructure multipliers.
Q9: What is the efficiency-equality trade-off in cross-state infrastructure allocation?
A: Under the baseline allocation (81% of infrastructure spending to Good states, 19% to Poor states), per $1 of infrastructure spending, the Good states receive $1.072 of output gains and Poor states receive only $0.016. In the equal-spending counterfactual, the total output multiplier falls from 1.088 to 0.873. The Poor states’ output multiplier rises from $0.016 to $0.062 (approximately fourfold), while the Good states’ falls from $1.072 to $0.810. The Poor states also see earnings multipliers more than double (from $0.017 to $0.042). This trade-off arises because Good states have both more private capital (benefiting from non-rivalry) and higher estimated TFP — so each dollar of infrastructure is more productive there. Equal allocation reduces aggregate efficiency while partially mitigating regional inequality.
Q10: How do the paper’s multiplier estimates compare to the existing literature?
A: In partial equilibrium (no GE adjustment), the authors find an output multiplier of 1.858, consistent with Chodorow-Reich’s (2019) cross-sectional multiplier of approximately 1.8. Once the general equilibrium interest rate effect is included, the multiplier falls to 1.09, which falls within the 0.6-1.2 range from Ramey (2011). Literature using representative-agent models without non-rivalry (e.g., Ramey 2020) typically reports multipliers of 0.3 to 0.8 using returns-to-scale parameters of 0.07-0.12; the paper shows these correspond to fiscal multipliers of 0.847-0.882 in the representative-agent framework. The heterogeneous-firm model, once it incorporates the non-rivalry-corrected elasticities, yields a meaningfully higher multiplier of 1.088.
Q11: What role does time-to-build play, and how does the paper handle it?
A: The baseline model assumes a time-to-build period s = 1 year (one-year lag before new infrastructure is productive). The paper notes in Appendix H that incorporating extended time-to-build reduces the aggregate fiscal multiplier, operating through two channels: a news effect (agents adjust behavior upon anticipating future infrastructure) and a general equilibrium effect endogenous to the news effect. This finding is consistent with Ramey (2020). The baseline results are therefore reported under the minimal one-year time-to-build assumption, with longer lags serving as a robustness check.
Q12: What is the role of region-specific TFP heterogeneity in the model?
A: The model includes two regions that differ both in infrastructure levels and in region-specific productivity (TFP) levels. The TFP of the Good region is estimated to be approximately double that of the Poor region (x = 2.064 for Good vs. 1 for Poor). This productivity difference is estimated to partially capture heterogeneous congestion effects (which are not separately modeled) and is estimated jointly with the infrastructure elasticity. The productivity differential is identified from the Good region’s share of aggregate output (0.849 in the data). The large TFP gap is also the reason why equal spending on Poor states generates a much smaller output gain than spending on Good states: not only is infrastructure utilization lower (fewer firms), but underlying productivity is also lower.
Key Concepts
Non-rivalry of public capital: The property by which infrastructure stock (Nj,t) enters each firm’s production function at the full regional level, not divided among firms. Formally, a single marginal unit of public capital raises every firm’s marginal product of private capital simultaneously, so the aggregate marginal product gain summed across firms exceeds any single firm’s gain. This is the central mechanism driving the micro-macro elasticity discrepancy in the paper.
Firm-level elasticity of substitution (λ): The elasticity governing the degree of substitutability between private capital (k) and public infrastructure (N) in the firm’s CES production function. At λ = 1 the production function is Cobb-Douglas; λ > 1 is gross substitutability; λ < 1 is gross complementarity. In the paper’s estimation, λ = 1.185, meaning private and public capital are gross substitutes at the firm level.
Gross substitutability vs. gross complementarity: Two inputs are gross substitutes (complements) if an increase in the quantity of one raises (lowers) the demand for the other, holding output price fixed. In the paper’s framework, private and public capital are gross substitutes at the firm level (λ = 1.185 > 1) but gross complements at the state level (ξ ≈ 0.48 < 1), with non-rivalry explaining the inversion upon aggregation.
Convex adjustment cost: A cost C(I,k) = (µ/2)(I/k)² · k that scales quadratically with the investment rate. In the heterogeneous-firm model, this cost plays a critical role: by Jensen’s inequality, heterogeneous firms’ average adjustment burden under a convex cost exceeds that of the representative (average) firm, making aggregate investment less sensitive to interest rate changes and thereby dampening crowding out.
Fixed adjustment cost (ξ): A one-time overhead cost drawn from a uniform distribution [0, ξ̄], paid only when a firm makes a large-scale investment outside the “inaction band” [−νk, νk]. This cost generates lumpy investment at the firm level, with about 14% of firms making lumpy investments in any given year. It also creates an extensive margin of investment adjustment that accounts for approximately 74% of the baseline crowding-out effect.
Fiscal multiplier (as defined in this paper): The ratio of the present value of aggregate output deviations from steady state to the present value of the fiscal spending shock, both summed over a T-year horizon. For the short run, T = 2 years; for the long run, T = 5 years. This is computed as a perfect-foresight transition path response to a one-time MIT shock equal to 1% of steady-state GDP.
MIT shock (one-time unexpected shock): An unanticipated, non-persistent one-period deviation in infrastructure spending. The term “MIT shock” refers to a deterministic transition experiment where agents have perfect foresight about all future values after the initial shock occurs. This contrasts with persistent policy rules and allows isolating the dynamic effects of a one-time fiscal impulse.
Extended SMM with market-clearing moments: The paper’s estimation innovation. Rather than solving for market-clearing prices at each parameter candidate (the standard costly inner loop), wages (w) and marginal utility of consumption (p) are treated as parameters with associated moments being the market-clearing conditions set to zero. A multi-block Metropolis-Hastings algorithm draws from the price block and the parameter block separately, generating posterior draws that jointly satisfy market clearing and empirical moment conditions.