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Public investment, debt, and welfare: A quantitative analysis
Accepted Manuscript
Public Investment, Debt, and Welfare: A Quantitative Analysis Santanu Chatterjee, John Gibson, Felix Rioja PII: DOI: Reference:
S0164-0704(17)30452-4 10.1016/j.jmacro.2018.01.007 JMACRO 3007
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Journal of Macroeconomics
Received date: Revised date: Accepted date:
1 November 2017 28 January 2018 29 January 2018
Please cite this article as: Santanu Chatterjee, John Gibson, Felix Rioja, Public Investment, Debt, and Welfare: A Quantitative Analysis, Journal of Macroeconomics (2018), doi: 10.1016/j.jmacro.2018.01.007
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Public Investment, Debt, and Welfare: A Quantitative Analysis ∗ John Gibson‡ Georgia State University
Felix Rioja§ Georgia State University February 13, 2018
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Abstract
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Santanu Chatterjee† University of Georgia
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In this paper, we examine the relationship between infrastructure investment and economic welfare in the context of a heterogeneous agent, incomplete-markets economy. Using a quantitative model to match the key aggregate and distributional features of the U.S. economy over the period 1990-2015, we show that the welfare-maximizing share of public investment in GDP depends critically on whether one internalizes the transition path between stationary equilibria or not. When welfare changes are evaluated by only comparing long-run stationary equilibria, the model implies that the government should increase infrastructure investment above its average share of 4 percent of GDP in the data. However, once the transition path and short-run dynamics are internalized, welfare-maximization generates an intertemporal trade-off in the path of infrastructure spending: a short-run increase significantly above its observed share in the data, but a long-run decline below this share to satisfy the government’s budget constraint.
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Keywords: Infrastructure, public investment, heterogeneous agents, public debt, welfare, transitional dynamics. JEL Classification: E2, E6, H3, H4, H6
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We would like to thank the Editor, William Lastrapes, an anonymous referee, and B. Ravikumar for helpful comments that helped improve this paper significantly. † Department of Economics, University of Georgia, Athens, GA 30602, USA. Telephone: 706-542-1709. Email: [email protected] ‡ Department of Economics, Georgia State University, Atlanta, GA 30303, USA. Telephone: 404-413-0202. Email: [email protected] § Department of Economics, Georgia State University, Atlanta, GA 30303, USA. Telephone: 404-413-0164. Email: [email protected]
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Introduction
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Adequate provision of public infrastructure is a critical component of supporting productive economic activity within a country. Firms need reliable road networks, airports, port facilities, and electrical and water networks to produce goods and deliver them efficiently to markets. However, the rate at which countries invest in infrastructure is not constant over time. For example, while the United States experienced a period of rising infrastructure spending as a share of GDP between between 1947-1960, this share has steadily declined since then (see Figure 1). The declining share of infrastructure investment in the United States since the 1970s has raised several concerns on its consequences for private productivity and economic growth. Moreover, the American Society for Civil Engineers (ASCE) have assigned a failing grade of D+ to the stock of U.S. infrastructure for the past several years (ASCE, 2017). These concerns have, subsequently, led recent administrations to request the U.S. Congress for legislation on an “infrastructure bill” that would significantly expand funding for new projects as well as maintenance and repair of the existing stock. Therefore, it is important to know, from society’s point of view, what level of infrastructure investment yields the highest welfare gains? This is the key question we try to answer in the paper. The literature on the link between public infrastructure spending and economic growth is voluminous. On the empirical side, the findings of a positive association go back to the work of Aschauer (1989). The theoretical foundations for analyzing public investment in dynamic macro models go back to Barro (1990) and Glomm and Ravikumar (1994), among others.1 Using an “AK” endogenous growth model, Barro (1990) demonstrated that economic welfare is maximized when the share of infrastructure investment in GDP is set equal to its output elasticity in the aggregate production function. Subsequent papers, however, have refined this result in other contexts, such as the presence of transition dynamics, installation costs for infrastructure and aggregate uncertainty, as in Cassou and Lansing (1998), Turnovsky (1997) and, more recently, Chatterjee and Turnovsky (2012). The majority of the theoretical literature on the link between public infrastructure and welfare has relied on the representative agent complete-markets modeling framework. However, infrastructure investment can play a key role in determining economic activity in a heterogeneous agent incomplete markets framework. Specifically, when agents are ex-post heterogenous, government investment in infrastructure can affect not only aggregate productivity, but also the precautionary savings motives for individual households. This, in 1
See Gramlich (1994), Romp and den Haan (2007), and Bom and Ligthart (2014a) for surveys of the empirical literature. Calderon and Serven (2014) provide a comprehensive review of the theoretical literature.
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turn, generates a differential impact on household labor supply and savings decisions and, ultimately, economic welfare. Moreover, these effects also change the economy’s wealth and income distributions over time. A representative agent framework is not equipped to analyze these transmission mechanisms. While Chatterjee et al. (2017) and Gibson and Rioja (2017a, b) introduce public infrastructure into the incomplete-markets framework in other contexts, the present paper is the first to characterize the welfare-maximizing rate of infrastructure investment, both in the short-run as well as the long run. 2 We extend the incomplete-markets framework of Aiyagari and McGrattan (1998) by introducing a stock of government-provided infrastructure into the aggregate production function and solve for a baseline stationary equilibrium that is calibrated to match key features of the U.S. economy, both at the aggregate and distributional levels, over the sample period 1990-2015. We then compute the welfare implications of moving from this equilibrium to various counterfactual stationary equilibria with alternative infrastructure investment rates. These welfare results are then compared to those where the full transition path between the stationary equilibria has been incorporated into the welfare calculations.3 Since new investment in infrastructure is typically financed by issuing public debt, we consider scenarios where changes in the economy’s infrastructure investment rates are financed by an appropriate adjustment of the share of public debt in GDP. In conducting our numerical experiments, we focus on how the implied welfare-maximizing share of public infrastructure spending obtained from our model compares with its average share in the data during the sample period of 1990-2015. Our results suggest that when we compute welfare effects by comparing stationary equilibria (thereby ignoring the transition paths between these equilibria), the long-run welfaremaximizing share of infrastructure investment in GDP is (i) larger than its corresponding share of 4 percent observed in the data during our sample period, and (ii) depends on the assumed output elasticity of infrastructure in the model. Specifically, when we set this parameter to a low value of 0.05, we find that it is optimal for the government to allocate approximately 5 percent of GDP to infrastructure investment in the long run. As we in-
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Bom and Ligthart (2014b) also try to characterize the welfare-maximizing level of public infrastructure in a representative agent framework. They use a small open economy overlapping generations Blanchard-Yaari model and find that OECD countries should invest about 6% of GDP in infrastructure. The governments source of funding in their model is labor income taxes. Our paper differs in that we utilize a heterogenous agents framework and that we allow the government to use debt financing. 3 Several recent papers, including Desbonnet and Weitzenblum (2012), Rohrs and Winter (2017), and Chatterjee et al. (2017) have demonstrated that welfare implications derived from comparing stationary equilibria can be very different from those derived from internalizing the transition paths between these equilibria.
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crease the value of this parameter to levels suggested by existing empirical studies (as in Bom and Ligthart, 2014a), we find that the welfare-maximizing share of infrastructure investment increases in lock-step. This suggests that Barro’s (1990) finding appears consistent with the findings from an incomplete markets environment, specifically when welfare comparisons are made across steady states (e.g., when ignoring transition dynamics). The next step in the paper is to incorporate the short-run dynamics and transition paths between the stationary equilibria into the welfare calculations, as the share of public investment is adjusted by the government. Once the short-run dynamics are internalized, our results indicate that the long-run welfare-maximizing share of infrastructure investment is actually lower than its corresponding average in the sample period, and also less sensitive to variations in the output elasticity of public infrastructure. It must be noted that infrastructure spending increases significantly in the short run, as the economy transitions from the baseline to the new equilibria. However, the higher public debt that is incurred in the process eventually dictates that this share be lower in the long-run, in order to satisfy the government’s budget constraint. Therefore, welfare-maximization entails an increase in the share of infrastructure spending in the short-run (financed by issuing new public debt), but a convergence to a lower share in the long-run steady state, in order to satisfy the government’s budget constraint. As a consequence, the model predicts a long-run inverse relationship between public debt and infrastructure, consistent with the logic outlined in Aiygari and McGrattan (1998) and Chatterjee et al. (2017). Therefore, the welfare-maximizing path of infrastructure investment implied by our model is qualitatively consistent with that experienced in the United States during the post-war time period. In a related contribution, Cassou and Lansing (1998) find the optimal path for the ratio of public infrastructure to private capital in the U.S. They use a representative agent growth model where the government solves the Ramsey problem and chooses a tax rate and invests in infrastructure optimally. They find an optimal dynamic path where infrastructure investment should be initially high but later decrease and level off as the model approaches long term equilibrium. They compare their optimal path to the US data from 1925 to 1992 and find them to be similar. Our paper differs from Cassou and Lansing (1998) in that our model has heterogenous agents, so it allows us to analyze how the welfare of different households are affected by changing infrastructure policy. Another difference with Cassou and Lansing (1998) is that the government in our model can use debt which is an important way of financing infrastructure construction in the US. In addition, government bonds are assets that affect individual saving and labor supply decisions.
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The paper proceeds as follows. Section 2 presents the model; section 3 describes the calibration and welfare measures. Section 4 presents and discusses the main results. Section 5 investigates sensitivity of our results to the inclusion government consumption in the households’ utility function, and section 5 concludes.
Analytical Framework
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The model economy consists of three types agents: a continuum of infinitely-lived households, a representative firm, and the government. In the next few subsections, we outline the allocation problems faced by these agents.
Households
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Following Aiyagari and McGrattan (1998), households face idiosyncratic shocks to labor productivity, , and lack access to a complete set of state-contingent securities. As is common in the literature, we assume that the labor productivity shock follows a Markov process with associated transition matrix, π(0 |). Given the idiosyncratic shock and the assumption of incomplete markets, households are ex-ante identical, but ex-post heterogenous. This ex-post heterogeneity arises from the precautionary savings motive, as highly productive households increase savings in order to partially insure against the risk of becoming less productive in the future. In this economy, households have two sources of savings, private capital, k, and government bonds, b. Therefore, a household’s total assets, a, is given by a = k + b. Households derive utility from consumption, c, and leisure, l, based on the following period utility function: 1−σ [cη l1−η ] (1) U (c, l) = 1−σ
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They choose consumption, c, leisure, l, labor, n, and next period assets, a0 , in order to maximize (1) subject to the following constraints: c + a0 ≤ [1 + (1 − τ )r]a + (1 − τ )wn + T R
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a0 ≥ a
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n+l ≤1
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where equation (2) is household’s budget constraint, and equation (3) is the household’s borrowing constraint. Inspection of (2) indicates that households earn the market clearing 5
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V (a, ) = max0 U (c, l) + β c,l,n,a
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wage, w, for each efficiency unit of labor they supply, and the market clearing interest rate, r for each unit of savings. Furthermore, regardless of the income source, households are subject to a tax, τ , administered by the government. Lastly, each household receives a lump-sum transfer, T R, from the government. The household’s borrowing constraint, (3), simply states that households are required to maintain a minimum asset balance greater than or equal to a. Given the structure outlined above, the household’s maximization problem can be written as the following dynamic program: #
π(0 |)V (a0 , 0 )
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subject to equations (2)-(4).
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Output in this economy is produced by the representative firm using the following neoclassical technology: Y = KGφ K α N 1−α , φ ∈ (0, 1) and α ∈ (0, 1) (6)
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where K and N are aggregate capital and labor respectively, which are determined by aggregating the individual households’ savings (in private capital) and labor supply decisions. KG denotes the stock of publicly funded infrastructure, and provides spill-overs for the private factors of production. The representative firm solves: max KGφ K α N 1−α − wN − (r + δK )K
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Solving this problem yields the following optimality conditions: w = (1 − r=
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Government
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The government provides the following three public goods: a government consumption good, Gc , public infrastructure, KG , and lump-sum transfers, T R. To raise revenue for these public goods, the government levies a tax, τ , on household income and may sell (or purchase) instantaneous one-period bonds, B, to (from) households. Therefore, the government’s budget constraint is given by: Gc + T R + KG0 − (1 − δG )KG + rB = τ (wN + rA) + B 0 − B
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where KG0 − (1 − δG )KG denotes the resources the government devotes to infrastructure investment, and rB denotes the government’s interest payment on its existing debt.4
Calibration and Welfare Measure
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The calibration of our baseline stationary equilibrium is consistent with that presented in Chatterjee, et al. (2017). However, the current research question, and the nature of the counterfactual experiments considered differs considerably. The next several subsections outline our baseline calibration, choice of shock process, and our preferred measures of aggregate welfare.5
Baseline Calibration
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The model is calibrated to an annual frequency, with specific parameter values chosen so our baseline stationary equilibrium replicates key features of the U.S. economy over the sample period 1990-2015. We set the rate of time preference, β, to 0.95 to match a steady-state interest rate of approximately 4 percent per year. Parameters σ and η in the agents’ utility function are set to target the elasticity of substitution (0.67) and fraction of time allocated to work (0.33) respectively. The depreciation rate of private capital, δ, is set to 10 percent per year, which is consistent with the literature, while the depreciation rate of public capital, δG , is set to 0.06 in order to target the public capital-GDP ratio of 0.67 observed in the data (IMF, 2017). 4
Formal equilibrium definitions (both stationary and transitional) for this economy are available upon request. 5 Details regarding computational methods are available upon request.
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For our production function parameters, we follow the literature and set capital’s share of income, α, to 0.3.6 There is less agreement in the literature on the empirical output elasticity of public infrastructure, φ. Estimates from Bom and Ligthart’s (2014a) metaanalysis suggest a large range for this parameter, depending on the status of a country’s development, and also the time horizon under consideration (short-run versus long-run). We therefore consider three values of φ, specifically 0.05, 0.10, and 0.15, with φ = 0.10 serving as our benchmark case. The parameter restrictions described above pin down all parameters directly faced by private decision makers. However, the government must determine the value of five policy variables, Gc , T R, KG , τ , and B, but so far we have only specified one restriction, i.e., the government’s budget constraint (equation 10). Therefore, we need four additional restrictions to fully pin down government behavior. Following Chatterjee, et al. (2017), we assume that in the stationary equilibrium, each category of public expenditure is proportional to output:7 KG0 − (1 − δG )KG = gI Y ; Gc = gc Y ; T R = gT R Y
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and we set the following shares equal to their average value in the data over the time period we consider: gI = 0.04 following the IMF (2017) data; gc = 0.15, and gT R = 0.09 following the BEA (2017) estimates. Finally, we set τ so the government’s tax revenue as a share of GDP, T S, is approximately 31 percent of output, matching the share observed in the data. TS =
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Given the four policy restrictions described above, the level of public debt in the baseline stationary equilibrium is simply backed out from the government’s budget constraint. See Table 1 for a complete list of parameter values and empirical targets. It is important to examine the model’s fit to aggregate U.S. data for the period 19902015. Table 2 presents this comparison. The private capital-output ratio generated by the model is 2.15, which is close to its corresponding value of 2.20 in the data. Similarly, the debt-GDP ratio generated by the model is 0.72 which is not too different from 0.70, the 6
One could argue that α = 0.3 is on the lower end of recent estimates for capital’s share of income. However, we have re-solved our baseline model under α = 0.36 and our results turn out to be robust to this parameter change (available upon request). 7 The assumption that government spending on various categories of items is proportional to output is consistent with the stability observed in corresponding ratios over the past several decades in the United States.
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average in the data for the sample period. Finally, the consumption-GDP ratio of the model of 0.59 is somewhat below the that found in the data (0.66). Overall, the model matches the data reasonably well, though not perfectly.
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Shock Process
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We set the borrowing constraint so that households can borrow at most 30% of output a = 0.3Y .8 In addition, we use an income shock process that was estimated by Rohrs and Winter (2017). They used data from the 2007 Survey of Consumer Finances to estimate wealth (financial assets minus liabilities). Given the large disruptions to wealth once the financial crisis started in 2008, it is better to use the 2007 survey to study the wealth distribution and transitions during the preceding period. We use Rohrs and Winters’s (2017) estimates as they are carefully done and allow our model to obtain a reasonable fit to the data. Hence, the labor productivity shock is specified as follows:
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= [0.055, 0.551, 1.195, 7.351] 0.940 0.040 0.020 0.000 0.034 0.816 0.150 0.000 π= 0.001 0.080 0.908 0.012 0.100 0.015 0.060 0.825
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Using the above shock process and parameters described in the previous section, the model generates wealth and income distributions that can be compared to the data. Table 3 presents these comparisons. We present the share of wealth and income held by various quintiles in the data and in each of our three model specifications (e.g., different values of φ). The share of wealth held by the richest quintile (Q5) in our baseline model specification (φ = 0.10) is 92.51%, very close to the 91.57% in the data. For the poorest quintile (Q1), the model generates a wealth share of -2.28% versus -1.60% in the data. In general the model fits the data fairly well, with a tighter fit in the upper quintiles.
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Welfare
Given that our paper focuses on the welfare effects of adjusting public investment and debt policies, we must specify a measure of aggregate welfare. To this end, we adopt the standard 8 This borrowing restriction has been used by Rohrs and Winter (2017) and Chatterjee, et al. (2017) to generate bottom quintile wealth holdings that match well with the U.S. data.
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utilitarian social welfare function: Γ=
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Using equation (12), we derive two alternative welfare measures:
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(14) (15)
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Results
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Equations (14) and (15) both measure welfare effects as the percent of steady state consumption an agent must pay (positive value) or be given (negative value) in order to be indifferent between remaining in the baseline stationary equilibrium versus moving to a new equilibrium under different public investment and debt policies. The difference between the two measures is that while equation (14) compares two stationary equilibria, equation (15) compares the endogenous transition paths between the stationary equilibria. Therefore, the measure presented in equation (14) provides a pure long-run welfare measure ignoring short-run dynamics, while equation (15) provides a long-run welfare measure after taking short-dynamics into account.
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We solve for the baseline stationary equilibrium following the calibration procedure described in the previous section. We also compute stationary equilibria under a variety of alternative infrastructure investment rates. For these alternative equilibria, we adjust infrastructure investment as a share of output, gI , and allow public debt, B, to adjust endogenously to satisfy the government’s budget constraint, holding all other policy variables constant. Given these stationary equilibria, our objective is to determine which of these alternative equilibria dominate in terms of aggregate welfare, and if accounting for the short-run dynamics that occur over the transition path between steady states impacts our results.
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4.1
Welfare Effects: Ignoring Short-run Dynamics
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First, we consider the pure long-run welfare impact of moving from our initial stationary equilibrium to each of the counterfactual equilibria, ignoring the short-run dynamics that may occur along the transition path to the new steady state. This pure long-run welfare impact is found by computing the welfare measure in equation (14) for all pairs of stationary equilibria. Figure 2 presents the welfare profiles under this scenario for each of the three values of φ considered (0.05, 0.10, and 0.15). Inspection of Figure 2 indicates that when φ = 0.05, small welfare gains are realized by increasing infrastructure investment from its baseline level of 4 percent of GDP. In fact, the optimal infrastructure investment share is found to be approximately 5 percent of GDP. This is reminiscent of the welfare-maximization result obtained by Barro (1990) in the context of a representative agent, complete-markets model. Further inspection of Figure 2 indicates that the welfare gains from increasing infrastructure investment grow larger as φ is increased. Specifically, the profile derived when φ = 0.1 suggests a much larger increase in aggregate welfare when infrastructure investment is increased from its benchmark value. While the welfare maximizing infrastructure investment rate lies beyond the upper limit consider in Figure 2, the curvature of the welfare profile, as well as the results under φ = 0.05, suggest that an optimum would likely be reached around gI = 0.10.9 The results under φ = 0.15 echo these findings, suggesting an even stronger welfare impact of increasing infrastructure investment when the stock of infrastructure is more productive. The underlying intuition is that when the output elasticity of infrastructure is set to be above the baseline calibrated spending on this good (φ > gI ), the marginal benefits of increasing infrastructure investment exceed the resource costs. As such, increasing gI towards φ then increases welfare in the long-run equilibrium. It is also worthwhile to note that reducing infrastructure investment, gI , leads to relatively larger welfare losses. For example, consider the baseline case with φ = 0.10. Raising gI by 2% of GDP (from 0.04 to 0.06) results in a welfare gain of approximately 5%. Conversely, reducing gI by 2% of GDP (from 0.04 to 0.02) results in a welfare loss of 10%. Hence, welfare losses can be significant if policy makers reduce infrastructure investment. In summary, our findings suggest that a government which seeks to maximize long-run social welfare should increase infrastructure investment from the baseline level of 4 percent of GDP. While our findings show that the welfare maximizing infrastructure investment rate 9
We choose 0.08 as our upper limit as it is highly unlikely that any developed country could sustain long-run infrastructure investment above 8 percent of GDP.
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is sensitive to the value of φ that is chosen, the qualitative result is consistent across all three specifications. Therefore, even if there is doubt regarding the exact value for φ, infrastructure investment appears currently below its implied long-run optimal level. However, internalizing short-run dynamic adjustments can affect this conclusion significantly, as we describe below.
Welfare Effects: Accounting For Short-run Dynamics
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While our findings from the previous section suggest that long-run infrastructure investment should be increased, it is important to remember that these results were derived without considering the impact of short-run dynamics. When considering the transition dynamics between stationary equilibria, we note that the relationship between public investment spending and public debt varies over time. In the short-run, an increase in debt frees up additional resources for infrastructure spending, leading to a positive association between the two policy variables. However, in the long-run, the government must service its outstanding debt in order to satisfy its budget constraint. This leads to a long-run decline in the share and stock of public infrastructure in GDP, thereby implying a negative long-run relationship.10 Given this differential intertemporal relationship between public investment and debt, it is important for us to reconsider the long-run welfare effects of the alternative infrastructure investment policies after accounting for the short-run dynamics that occur as the economy transitions to its new stationary equilibrium. To this end, we compute the transition dynamics between our baseline stationary equilibrium and each of the alternative equilibria. When computing transition dynamics we can back out from the government budget constraint the size of the long-run debt adjustment and the terminal public debt-GDP ratio that is needed to reach each of the alternative long-run infrastructure investment policies. As such, we treat the debt-GDP ratio as the exogenous variable that is adjusted during the transition path and back out infrastructure investment endogenously. Such a policy experiment is empirically reasonable as government officials typically call for spending adjustments, or the creation of new debt, and then allocate these resources accordingly. The speed with which debt is accumulated (or reduced) may impact which long-run infrastructure investment policy is found to be optimal. Therefore, we compute the transition dynamics under the following two exogenous paths for the economy’s debt-GDP ratio: (i) the public 10
While a long-run inverse relationship between infrastructure investment and debt may seem counterintuitive, it is consistent with the notion that higher debt crowds out private (and public) capital (see Aiyagari and McGrattan,1998 and Chatterjee, et al., 2017).
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debt-GDP ratio equals the baseline value at time 0 and increases (decreases) to its terminal value in one period (year), and (ii) the public debt-GDP ratio equals the baseline value at time 0 and increases (decreases) linearly, reaching its terminal level in ten periods (years).11 The first column of Figure 3 presents the welfare profiles after internalizing short-run dynamics under each value of φ considered, assuming the government completely adjusts the debt-GDP ratio in one period. The welfare profiles reported in these plots are hump-shaped, and they all suggest that the welfare maximizing infrastructure investment rate is less than the benchmark value of approximately 4 percent of GDP. Specifically, when φ = 0.10, the welfare maximizing infrastructure investment rate is approximately 3.25 percent of GDP.12 This is in stark contrast to our results when the short-run dynamics were ignored, which suggested that infrastructure investment as a share of GDP should be increased significantly from its benchmark value, in lock-step fashion with its output elasticity, φ. The welfare profiles for the case when the debt-GDP ratio adjusts linearly over the span of 10 periods (years) are presented in the second column of Figure 3. The qualitative results are consistent with the case where the debt-GDP ratio adjusts instantly. Specifically, the optimal long-run infrastructure investment shares are all lower than the baseline value of 4 percent of GDP. In fact, the values under the 10-period adjustment are even lower than that found when the adjustment takes place within a single time period. Therefore, we find that once short-run dynamics are accounted for, the long-run welfare maximizing infrastructure investment share is lower than the benchmark value or observed value in our sample, and is also not sensitive to variations in the output elasticity of public infrastructure. As such, once transition paths are internalized into the welfare calculations, the welfare-maximization result of Barro (1990) is no longer robust.
Understanding the Transition Dynamics
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The results presented in the previous subsection seem to suggest that infrastructure investment should actually be reduced slightly in the United States. However, it is important to
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11 Case (i) may be politically infeasible, but provides a nice comparison to the case where transition dynamics are ignored. When transition dynamics are ignored, we are implicitly assuming that government policy, as well as all aggregate variables in the economy, can instantly adjust to their new long-run level. Under case (i), we still assume that policy variables adjust instantly, but we now allow the economy’s aggregate variables to evolve gradually over time. Case (ii) serves as a more politically feasible case that also allows for a gradual adjustment in the government’s debt policy. 12 While there are small deviations in the welfare maximizing infrastructure investment rate as φ is varied, the results are relatively stable. As such, we will focus attention on the difference between these profiles and that recovered when short-run dynamics are ignored.
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remember these results focus on the long-run infrastructure investment share. So, while it is true that the optimal long-run infrastructure investment share is less than the benchmark observed share of 4 percent of GDP, one must also consider how infrastructure investment evolves over the transition path before making policy prescriptions. Figures 4 and 5 illustrate the intertemporal trade-offs that occur in the economy’s aggregate variables as we move in the direction of the welfare-maximizing stationary equilibrium.13 Specifically, the long run infrastructure investment rate is lowered from 4% of GDP, as suggested by the data, to 3.9% of GDP. This long-run adjustment in the rate of infrastructure investment is associated with a 3 percentage point increase the economy’s debt-GDP ratio (from 0.72 to 0.75). In the short run, an increase in debt allows the government to increase infrastructure spending, leading to an expansionary adjustment for the economy. A such, the higher stock of debt is associated with an increase in the stock of infrastructure as well as its share of spending in GDP above the baseline share of 4 percent of GDP. Consequently, the productive spill-overs for the private sector from this increase lead to an increase in output and labor supply. As the productivity benefits from the higher stock of infrastructure are realized, both private capital and consumption increase along the transition path. These dynamics are qualitatively robust to a 10-period adjustment in debt (column 2 in Figures 4 and 5).14 The higher debt, however, will eventually crowd-out private capital and consumption. Over time, as the economy transitions to its long-run equilibrium, the government must allocate more resources to servicing its now larger debt balance. With all other variables remaining constant, this is achieved by reducing the share of infrastructure spending in GDP over time, leading to a lower share of spending and stock relative to GDP in the long-run stationary equilibrium. Therefore, welfare maximization leads to an intertemporal trade-off in the path of infrastructure spending: an increase above the baseline level of 4 percent in the short-run, and a decrease below this level in the long-run. We find that the short run benefits associated with debt creation and infrastructure investment dominate the long run losses and, as a consequence, the welfare maximizing policy actually leads to a long-run reduction in infrastructure. This result, to some extent, helps rationalize the infrastructure investment patterns observed in the United States over the post-war period. 13
Figure 4 presents the transition paths for variables related to public debt and infrastructure, while Figure 5 presents transition paths for key macroeconomic aggregates including output and consumption. 14 To save space, we focus on the case where φ = 0.10. Results for φ = 0.05 and φ = 0.15 are available upon request.
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Government Consumption in the Utility Function
U (c, l, Gc ) =
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The results in the previous section assumed that government consumption provides no utility benefits to private agents. In this section, we relax this assumption to introduce government consumption as a utility-enhancing public good. In doing so, we also examine the consequences of adjusting government consumption expenditures, instead of debt, in order to satisfy the government’s budget constraint. First, we will recompute the welfare profiles for our original policy experiment (with φ = 0.10 and 1-period policy adjustment) under the following utility function: 1 [cη l1−η ]1−σ + ηG G1−σ 1−σ 1−σ c
(16)
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where, following Christiano et al. (2011), ηG is set to 0.015. This allows us to determine how allowing for utility-enhancing government consumption would impact our primary results. Next, we consider an alternative experiment where changes in the rate of infrastructure investment are offset by adjustments in the government consumption ratio, gc , rather than debt. These results allow us to assess the various trade-offs that exist under different financing schemes. Figure 6 presents the welfare profile from our original policy experiment (with φ = 0.10 and 1-period policy adjustment) under our initial utility function (1) and the modified utility function (16), which includes government consumption.15 Inspection of Figure 6 indicates that these welfare profiles are highly comparable, and that adjustments in the rate of infrastructure investment away from its initial value result in welfare changes that are of a similar magnitude. However, upon closer inspection, it is clear that the welfare maximizing infrastructure investment share is somewhat larger when Gc enters the utility function (3.5% vs 3.25%). Increases in infrastructure investment result in higher output, and since the ratio of government consumption to GDP is fixed across policy experiments, its level increases as we move to the right in Figure 6. Therefore, since higher infrastructure investment is associated more government consumption, allowing Gc to enter the agents’ utility function provides another channel through which infrastructure investment may improve welfare. We have seen from Figure 6 that if we retain the structure of our original policy experiment and simply introduce Gc into our utility function, then our results will only change modestly. Now, lets consider the alternative policy experiment where we offset changes in the rate of infrastructure investment by adjusting government consumption as a share of 15
Note that the solid welfare profile in Figure 6 was also presented in middle left-hand panel of Figure 3.
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Conclusions
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output, gc , rather than debt. The welfare profile from this alternative experiment, presented in Figure 7, is different than those presented earlier (see either Figure 3 or Figure 6). We no longer find that the welfare profile displays a hump. Instead, we find that welfare is increasing over the range of infrastructure investment shares considered (0.02 to 0.0575), though at a decreasing rate. What explains this finding? While our original experiments resulted in a sizable welfare tradeoff between additional infrastructure investment and debt, the alternative experiment considered here does not yield such a large tradeoff. While it is the case that the share of output going to government consumption, gc , must fall as the infrastructure investment share rises in order to maintain the government’s budget constraint, it is important to remember that the level of output being produced is rising. Therefore, the level of government consumption, Gc , which actually enters the agents’ utility function, is not changing much over the range of values considered. As a consequence, the large welfare gains of higher infrastructure investment are only met with modest welfare losses due to reduced government consumption.16 This finding illustrates the different outcomes that can be achieved when various financing schemes are considered, and it also stresses the fact that the welfare impact of infrastructure investment cannot be determined independent of its source of financing.
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In this paper, we have examined the consequences of varying the rate of infrastructure investment for economic welfare in the context of a heterogeneous agent, incomplete-markets economy. Specifically, while the relationship between these two variables have been studied extensively in the representative agent framework, not much is known in the context of incomplete markets and heterogenous agents. We introduce public spending on a stock of infrastructure into the production function in a model similar to that developed by Aiyagari and McGrattan (1998), and calibrate it to match the key aggregate and distributional moments of the U.S. economy over the sample period 1990-2015. We then compute the welfare changes from variations in the rate of infrastructure spending under two scenarios: (i) comparing stationary steady-state equilibria, but ignoring transition dynamics, and (ii) 16
The welfare profile presented in Figure 7 already shows signs of curvature, indicating that a welfare maximizing infrastructure investment share could be found. However, we restrict attention to the range [0.02, 0.0575] as it well covers the range considered by our original experiments. Furthermore, increasing the infrastructure investment share increases output at a decreasing rate and reduces gc at a more or less constant rate, so one will almost surely find a welfare maximizing share.
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comparing stationary steady-state equilibria, while incorporating the short-run dynamics along the transition path between these equilibria. Our results indicate that the implications for the welfare-maximizing share of pubic investment in GDP depend critically on whether one internalizes the transition path between stationary equilibria. When these paths are ignored, and welfare changes are evaluated only by comparing the long-run stationary equilibria, the model implies that the government should increase infrastructure investment above its observed average share of 4 percent of GDP in the data. Further, the welfare-maximizing share of public investment coincides with its output elasticity in the production function, a result familiar from Barro (1990). This result seems to suggest that the current share of public investment in the U.S. is lower than its welfare-maximizing level. However, this insight changes significantly when one internalizes the short-run and transitional adjustment of the economy between stationary equilibria. In this case, welfare-maximization results in an intertemporal trade-off in the path of infrastructure spending. Specifically, to maximize intertemporal welfare, the government must raise the share of infrastructure spending significantly above its baseline level in the short run, while gradually reducing it in the long run as public resources must be reallocated to servicing the larger stock of debt that was accumulated over time to pay for the shortrun increase. This policy is expansionary in the short-run, and the welfare gains along the transition path exceed the long-run losses from higher debt-servicing payments. Further, once the short-run dynamics are accounted for, the link between the rate of public investment spending and the output elasticity of infrastructure is no longer evident. In summary, this paper has attempted to revisit an important public policy issue-the relationship between infrastructure spending and economic welfare-in the context of a quantitative model with incomplete markets and heterogeneous agents. The key insight from our paper is that ignoring the transition path between steady states can lead to a misleading policy prescription, while internalizing these dynamics requires policy-makers to face an intertemporal trade-off in order to maximize welfare. There are several aspects of this question we abstract from, including political economy and open economy issues, as well as those related to labor and capital market frictions. Another relevant question relates to the transformation of public capital into (possibly more productive) private capital, and the related consequences for the welfare-maximizing level of public investment. We hope to pursue these extensions in future work.
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Table 1: Structural Parameters and Empirical Targets Parameter
Value
Targeta
Description Preference Parameters
0.95 1.50 0.36
Rate of Time Preference Coefficient of Relative Risk Aversion Relative Share of Leisure in Utility Production Parameters
0.30 0.05 0.10 0.15 0.10 0.06
Output Elasticity of Private Capital Output Elasticity of Public Capital Depreciation Rate of Private Capital Depreciation Rate of Public Capital
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α φ δK δG
r = 0.04 Literature N = 0.33
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β σ η
Literature Robustness Literature KG = 0.67 Y
Policy Parameters
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Government Consumption (Share of GDP) Government Investment (Share of GDP) Government Transfers (Share of GDP) Income Tax Rate
Gc Y IG Y TR Y
= 0.15 = 0.04 = 0.09 T S = 0.31
Sources for empirical targets come from both the existing literature and aggregate U.S. data for the period 1990-2015. Specifically, data from the World Bank Development Indicators (WDI), the International Monetary Fund’s Investment and Capital Stock database, and the U.S. BEA were used.
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0.15 0.04 0.09 0.38
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gc gI gT R τ
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Table 2: Model Fit to Aggregate Ratios Dataa Model
Description
K/Y B/Y C/Y
Private Capital-GDP Ratio Public Debt-GDP Ratio Consumption-GDP Ratio
a
2.20 0.70 0.66
2.15 0.72 0.59
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Variable
The data represents averages for the U.S. for the sample period 1990-2015. Sources: U.S. BEA and IMF.
Table 3: Distributional Fit
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Wealth Shares
Dataa φ = 0.05 φ = 0.10 φ = 0.15 -2.04 -0.16 2.81 7.44 91.95
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Q1 Q2 Q3 Q4 Q5
-0.40 3.19 12.49 23.33 61.39
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Q1 Q2 Q3 Q4 Q5 a
-2.28 -0.31 2.62 7.46 92.51
-2.10 -0.16 2.57 7.61 92.07
Income Shares
0.44 6.60 14.11 21.66 57.19
0.43 6.55 13.68 22.54 56.80
0.46 6.56 13.56 22.33 57.09
Data Source: U.S. Survey of Consumer Finances, 2007.
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Figure 1: U.S. Infrastructure Investment
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Figure 2: Welfare Profile: Ignoring Transition Dynamics
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Figure 3: Welfare Profile: Including Transition Dynamics
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Figure 4: Transitions Paths: Debt and Infrastructure
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Figure 5: Transitions Paths: Other Macro Aggregates
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Figure 6: Welfare Profile: U (c, l) vs U (c, l, Gc )
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Figure 7: Welfare Profile: Alternative Adjust Gc Experiment
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References Aiyagari, R. & McGrattan, E., 1998, “The Optimum Quantity of Debt,” Journal of Monetary Economics, 42: 447-469.
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American Society of Civil Engineers (ASCE), 2017, “Infrastructure Report Card,” 2017. Aschauer D., 1989, “Does Public Capital Crowd Out Private Capital?” Journal of Monetary Economics, 24(2): 171-188.
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Barro R., 1990, “Government Spending in a Simple Model of Endogenous Growth,” Journal of Political Economy, 98(5): S103-26. Bom P. R. & Ligthart J. E., 2014a, “What Have we Learned from Three Decades of Research on the Productivity of Public Capital?” Journal of Economic Surveys, 28(5): 889-916.
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Bom, P. R., & Ligthart, J. E., 2014b, “Public infrastructure investment, output dynamics, and balanced budget fiscal rules.” Journal of Economic Dynamics and Control, 40, 334-354.
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Calderon C. & Serven L., 2014. “Infrastructure, Growth, and Inequality: An Overview,” World Bank Policy Research Working Paper No. 7034. The World Bank.
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Cassou, S. P., & Lansing, K. J., 1998. ”Optimal fiscal policy, public capital, and the productivity slowdown.” Journal of Economic Dynamics and Control, 22(6), 911-935.
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Chatterjee, S. & Turnovsky, S., 2012, “Infrastructure and Inequality,” European Economic Review, 56(8): 1730-1745. Chatterjee, S., Gibson, J., & Rioja, F., 2017, “Optimal Public Debt Redux,” Journal of Economic Dynamics and Control, 83(C): 162-174. Christiano, L., Eichenbaum, M., & Rebelo, S., 2011, “When is the Government Spending Multiplier Large?,” Journal of Political Economy, 119(1): 78-121. 27
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Desbonnet, A., & Weitzenblum, T., 2012, “Why Do Governments End Up with Debt? Short?Run Effects Matter.” Economic Inquiry, 50(4), 905-919. Gibson J., & Rioja, F., 2017a, “Public Infrastructure Maintenance and the Distribution of Wealth,” Economic Inquiry, 55(1): 175-186.
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Gibson J., & Rioja, F., 2017b, “The Welfare Effects of Infrastructure Investment in a Heterogeneous Agents Economy,” Working Paper.
Glomm, G., & Ravikumar, B., 1994, “Public Investment in Infrastructure in a Simple Growth Model,” Journal of Economic Dynamics and Control, 18(6): 1173-1187.
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Gramlich, E.M., 1994, “Infrastructure Investment: A Review Essay,” Journal of Economic Literature, 32: 1176-1196.
International Monetary Fund (IMF), 2017, ”Investment and Capital Stock Database.”
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Rohrs, S., & Winter, C., 2017, “Reducing Government Debt in the Presence of Inequality,” Journal of Economic Dynamics and Control, 82, 1-20.
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Romp, W., and De Haan J., 2007, “Public Capital and Economic Growth: A Critical Survey,” Perspektiven der Wirtschaftspolitik, 8: 6-52.
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Turnovsky, S.J., 1997, “Fiscal Policy in a Growing Economy with Public Capital,” Macroeconomic Dynamics, 1: 615-639.
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Public Investment, Debt, and Welfare: A Quantitative Analysis Santanu Chatterjee, John Gibson, Felix Rioja PII: DOI: Reference:
S0164-0704(17)30452-4 10.1016/j.jmacro.2018.01.007 JMACRO 3007
To appear in:
Journal of Macroeconomics
Received date: Revised date: Accepted date:
1 November 2017 28 January 2018 29 January 2018
Please cite this article as: Santanu Chatterjee, John Gibson, Felix Rioja, Public Investment, Debt, and Welfare: A Quantitative Analysis, Journal of Macroeconomics (2018), doi: 10.1016/j.jmacro.2018.01.007
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Public Investment, Debt, and Welfare: A Quantitative Analysis ∗ John Gibson‡ Georgia State University
Felix Rioja§ Georgia State University February 13, 2018
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Abstract
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Santanu Chatterjee† University of Georgia
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In this paper, we examine the relationship between infrastructure investment and economic welfare in the context of a heterogeneous agent, incomplete-markets economy. Using a quantitative model to match the key aggregate and distributional features of the U.S. economy over the period 1990-2015, we show that the welfare-maximizing share of public investment in GDP depends critically on whether one internalizes the transition path between stationary equilibria or not. When welfare changes are evaluated by only comparing long-run stationary equilibria, the model implies that the government should increase infrastructure investment above its average share of 4 percent of GDP in the data. However, once the transition path and short-run dynamics are internalized, welfare-maximization generates an intertemporal trade-off in the path of infrastructure spending: a short-run increase significantly above its observed share in the data, but a long-run decline below this share to satisfy the government’s budget constraint.
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Keywords: Infrastructure, public investment, heterogeneous agents, public debt, welfare, transitional dynamics. JEL Classification: E2, E6, H3, H4, H6
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We would like to thank the Editor, William Lastrapes, an anonymous referee, and B. Ravikumar for helpful comments that helped improve this paper significantly. † Department of Economics, University of Georgia, Athens, GA 30602, USA. Telephone: 706-542-1709. Email: [email protected] ‡ Department of Economics, Georgia State University, Atlanta, GA 30303, USA. Telephone: 404-413-0202. Email: [email protected] § Department of Economics, Georgia State University, Atlanta, GA 30303, USA. Telephone: 404-413-0164. Email: [email protected]
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Introduction
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Adequate provision of public infrastructure is a critical component of supporting productive economic activity within a country. Firms need reliable road networks, airports, port facilities, and electrical and water networks to produce goods and deliver them efficiently to markets. However, the rate at which countries invest in infrastructure is not constant over time. For example, while the United States experienced a period of rising infrastructure spending as a share of GDP between between 1947-1960, this share has steadily declined since then (see Figure 1). The declining share of infrastructure investment in the United States since the 1970s has raised several concerns on its consequences for private productivity and economic growth. Moreover, the American Society for Civil Engineers (ASCE) have assigned a failing grade of D+ to the stock of U.S. infrastructure for the past several years (ASCE, 2017). These concerns have, subsequently, led recent administrations to request the U.S. Congress for legislation on an “infrastructure bill” that would significantly expand funding for new projects as well as maintenance and repair of the existing stock. Therefore, it is important to know, from society’s point of view, what level of infrastructure investment yields the highest welfare gains? This is the key question we try to answer in the paper. The literature on the link between public infrastructure spending and economic growth is voluminous. On the empirical side, the findings of a positive association go back to the work of Aschauer (1989). The theoretical foundations for analyzing public investment in dynamic macro models go back to Barro (1990) and Glomm and Ravikumar (1994), among others.1 Using an “AK” endogenous growth model, Barro (1990) demonstrated that economic welfare is maximized when the share of infrastructure investment in GDP is set equal to its output elasticity in the aggregate production function. Subsequent papers, however, have refined this result in other contexts, such as the presence of transition dynamics, installation costs for infrastructure and aggregate uncertainty, as in Cassou and Lansing (1998), Turnovsky (1997) and, more recently, Chatterjee and Turnovsky (2012). The majority of the theoretical literature on the link between public infrastructure and welfare has relied on the representative agent complete-markets modeling framework. However, infrastructure investment can play a key role in determining economic activity in a heterogeneous agent incomplete markets framework. Specifically, when agents are ex-post heterogenous, government investment in infrastructure can affect not only aggregate productivity, but also the precautionary savings motives for individual households. This, in 1
See Gramlich (1994), Romp and den Haan (2007), and Bom and Ligthart (2014a) for surveys of the empirical literature. Calderon and Serven (2014) provide a comprehensive review of the theoretical literature.
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turn, generates a differential impact on household labor supply and savings decisions and, ultimately, economic welfare. Moreover, these effects also change the economy’s wealth and income distributions over time. A representative agent framework is not equipped to analyze these transmission mechanisms. While Chatterjee et al. (2017) and Gibson and Rioja (2017a, b) introduce public infrastructure into the incomplete-markets framework in other contexts, the present paper is the first to characterize the welfare-maximizing rate of infrastructure investment, both in the short-run as well as the long run. 2 We extend the incomplete-markets framework of Aiyagari and McGrattan (1998) by introducing a stock of government-provided infrastructure into the aggregate production function and solve for a baseline stationary equilibrium that is calibrated to match key features of the U.S. economy, both at the aggregate and distributional levels, over the sample period 1990-2015. We then compute the welfare implications of moving from this equilibrium to various counterfactual stationary equilibria with alternative infrastructure investment rates. These welfare results are then compared to those where the full transition path between the stationary equilibria has been incorporated into the welfare calculations.3 Since new investment in infrastructure is typically financed by issuing public debt, we consider scenarios where changes in the economy’s infrastructure investment rates are financed by an appropriate adjustment of the share of public debt in GDP. In conducting our numerical experiments, we focus on how the implied welfare-maximizing share of public infrastructure spending obtained from our model compares with its average share in the data during the sample period of 1990-2015. Our results suggest that when we compute welfare effects by comparing stationary equilibria (thereby ignoring the transition paths between these equilibria), the long-run welfaremaximizing share of infrastructure investment in GDP is (i) larger than its corresponding share of 4 percent observed in the data during our sample period, and (ii) depends on the assumed output elasticity of infrastructure in the model. Specifically, when we set this parameter to a low value of 0.05, we find that it is optimal for the government to allocate approximately 5 percent of GDP to infrastructure investment in the long run. As we in-
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Bom and Ligthart (2014b) also try to characterize the welfare-maximizing level of public infrastructure in a representative agent framework. They use a small open economy overlapping generations Blanchard-Yaari model and find that OECD countries should invest about 6% of GDP in infrastructure. The governments source of funding in their model is labor income taxes. Our paper differs in that we utilize a heterogenous agents framework and that we allow the government to use debt financing. 3 Several recent papers, including Desbonnet and Weitzenblum (2012), Rohrs and Winter (2017), and Chatterjee et al. (2017) have demonstrated that welfare implications derived from comparing stationary equilibria can be very different from those derived from internalizing the transition paths between these equilibria.
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crease the value of this parameter to levels suggested by existing empirical studies (as in Bom and Ligthart, 2014a), we find that the welfare-maximizing share of infrastructure investment increases in lock-step. This suggests that Barro’s (1990) finding appears consistent with the findings from an incomplete markets environment, specifically when welfare comparisons are made across steady states (e.g., when ignoring transition dynamics). The next step in the paper is to incorporate the short-run dynamics and transition paths between the stationary equilibria into the welfare calculations, as the share of public investment is adjusted by the government. Once the short-run dynamics are internalized, our results indicate that the long-run welfare-maximizing share of infrastructure investment is actually lower than its corresponding average in the sample period, and also less sensitive to variations in the output elasticity of public infrastructure. It must be noted that infrastructure spending increases significantly in the short run, as the economy transitions from the baseline to the new equilibria. However, the higher public debt that is incurred in the process eventually dictates that this share be lower in the long-run, in order to satisfy the government’s budget constraint. Therefore, welfare-maximization entails an increase in the share of infrastructure spending in the short-run (financed by issuing new public debt), but a convergence to a lower share in the long-run steady state, in order to satisfy the government’s budget constraint. As a consequence, the model predicts a long-run inverse relationship between public debt and infrastructure, consistent with the logic outlined in Aiygari and McGrattan (1998) and Chatterjee et al. (2017). Therefore, the welfare-maximizing path of infrastructure investment implied by our model is qualitatively consistent with that experienced in the United States during the post-war time period. In a related contribution, Cassou and Lansing (1998) find the optimal path for the ratio of public infrastructure to private capital in the U.S. They use a representative agent growth model where the government solves the Ramsey problem and chooses a tax rate and invests in infrastructure optimally. They find an optimal dynamic path where infrastructure investment should be initially high but later decrease and level off as the model approaches long term equilibrium. They compare their optimal path to the US data from 1925 to 1992 and find them to be similar. Our paper differs from Cassou and Lansing (1998) in that our model has heterogenous agents, so it allows us to analyze how the welfare of different households are affected by changing infrastructure policy. Another difference with Cassou and Lansing (1998) is that the government in our model can use debt which is an important way of financing infrastructure construction in the US. In addition, government bonds are assets that affect individual saving and labor supply decisions.
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The paper proceeds as follows. Section 2 presents the model; section 3 describes the calibration and welfare measures. Section 4 presents and discusses the main results. Section 5 investigates sensitivity of our results to the inclusion government consumption in the households’ utility function, and section 5 concludes.
Analytical Framework
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The model economy consists of three types agents: a continuum of infinitely-lived households, a representative firm, and the government. In the next few subsections, we outline the allocation problems faced by these agents.
Households
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Following Aiyagari and McGrattan (1998), households face idiosyncratic shocks to labor productivity, , and lack access to a complete set of state-contingent securities. As is common in the literature, we assume that the labor productivity shock follows a Markov process with associated transition matrix, π(0 |). Given the idiosyncratic shock and the assumption of incomplete markets, households are ex-ante identical, but ex-post heterogenous. This ex-post heterogeneity arises from the precautionary savings motive, as highly productive households increase savings in order to partially insure against the risk of becoming less productive in the future. In this economy, households have two sources of savings, private capital, k, and government bonds, b. Therefore, a household’s total assets, a, is given by a = k + b. Households derive utility from consumption, c, and leisure, l, based on the following period utility function: 1−σ [cη l1−η ] (1) U (c, l) = 1−σ
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(2)
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(3)
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(4)
where equation (2) is household’s budget constraint, and equation (3) is the household’s borrowing constraint. Inspection of (2) indicates that households earn the market clearing 5
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V (a, ) = max0 U (c, l) + β c,l,n,a
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wage, w, for each efficiency unit of labor they supply, and the market clearing interest rate, r for each unit of savings. Furthermore, regardless of the income source, households are subject to a tax, τ , administered by the government. Lastly, each household receives a lump-sum transfer, T R, from the government. The household’s borrowing constraint, (3), simply states that households are required to maintain a minimum asset balance greater than or equal to a. Given the structure outlined above, the household’s maximization problem can be written as the following dynamic program: #
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subject to equations (2)-(4).
(5)
Firms
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Output in this economy is produced by the representative firm using the following neoclassical technology: Y = KGφ K α N 1−α , φ ∈ (0, 1) and α ∈ (0, 1) (6)
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where K and N are aggregate capital and labor respectively, which are determined by aggregating the individual households’ savings (in private capital) and labor supply decisions. KG denotes the stock of publicly funded infrastructure, and provides spill-overs for the private factors of production. The representative firm solves: max KGφ K α N 1−α − wN − (r + δK )K
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Solving this problem yields the following optimality conditions: w = (1 − r=
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Government
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The government provides the following three public goods: a government consumption good, Gc , public infrastructure, KG , and lump-sum transfers, T R. To raise revenue for these public goods, the government levies a tax, τ , on household income and may sell (or purchase) instantaneous one-period bonds, B, to (from) households. Therefore, the government’s budget constraint is given by: Gc + T R + KG0 − (1 − δG )KG + rB = τ (wN + rA) + B 0 − B
(10)
where KG0 − (1 − δG )KG denotes the resources the government devotes to infrastructure investment, and rB denotes the government’s interest payment on its existing debt.4
Calibration and Welfare Measure
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The calibration of our baseline stationary equilibrium is consistent with that presented in Chatterjee, et al. (2017). However, the current research question, and the nature of the counterfactual experiments considered differs considerably. The next several subsections outline our baseline calibration, choice of shock process, and our preferred measures of aggregate welfare.5
Baseline Calibration
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The model is calibrated to an annual frequency, with specific parameter values chosen so our baseline stationary equilibrium replicates key features of the U.S. economy over the sample period 1990-2015. We set the rate of time preference, β, to 0.95 to match a steady-state interest rate of approximately 4 percent per year. Parameters σ and η in the agents’ utility function are set to target the elasticity of substitution (0.67) and fraction of time allocated to work (0.33) respectively. The depreciation rate of private capital, δ, is set to 10 percent per year, which is consistent with the literature, while the depreciation rate of public capital, δG , is set to 0.06 in order to target the public capital-GDP ratio of 0.67 observed in the data (IMF, 2017). 4
Formal equilibrium definitions (both stationary and transitional) for this economy are available upon request. 5 Details regarding computational methods are available upon request.
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For our production function parameters, we follow the literature and set capital’s share of income, α, to 0.3.6 There is less agreement in the literature on the empirical output elasticity of public infrastructure, φ. Estimates from Bom and Ligthart’s (2014a) metaanalysis suggest a large range for this parameter, depending on the status of a country’s development, and also the time horizon under consideration (short-run versus long-run). We therefore consider three values of φ, specifically 0.05, 0.10, and 0.15, with φ = 0.10 serving as our benchmark case. The parameter restrictions described above pin down all parameters directly faced by private decision makers. However, the government must determine the value of five policy variables, Gc , T R, KG , τ , and B, but so far we have only specified one restriction, i.e., the government’s budget constraint (equation 10). Therefore, we need four additional restrictions to fully pin down government behavior. Following Chatterjee, et al. (2017), we assume that in the stationary equilibrium, each category of public expenditure is proportional to output:7 KG0 − (1 − δG )KG = gI Y ; Gc = gc Y ; T R = gT R Y
(11)
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and we set the following shares equal to their average value in the data over the time period we consider: gI = 0.04 following the IMF (2017) data; gc = 0.15, and gT R = 0.09 following the BEA (2017) estimates. Finally, we set τ so the government’s tax revenue as a share of GDP, T S, is approximately 31 percent of output, matching the share observed in the data. TS =
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Given the four policy restrictions described above, the level of public debt in the baseline stationary equilibrium is simply backed out from the government’s budget constraint. See Table 1 for a complete list of parameter values and empirical targets. It is important to examine the model’s fit to aggregate U.S. data for the period 19902015. Table 2 presents this comparison. The private capital-output ratio generated by the model is 2.15, which is close to its corresponding value of 2.20 in the data. Similarly, the debt-GDP ratio generated by the model is 0.72 which is not too different from 0.70, the 6
One could argue that α = 0.3 is on the lower end of recent estimates for capital’s share of income. However, we have re-solved our baseline model under α = 0.36 and our results turn out to be robust to this parameter change (available upon request). 7 The assumption that government spending on various categories of items is proportional to output is consistent with the stability observed in corresponding ratios over the past several decades in the United States.
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average in the data for the sample period. Finally, the consumption-GDP ratio of the model of 0.59 is somewhat below the that found in the data (0.66). Overall, the model matches the data reasonably well, though not perfectly.
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Shock Process
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We set the borrowing constraint so that households can borrow at most 30% of output a = 0.3Y .8 In addition, we use an income shock process that was estimated by Rohrs and Winter (2017). They used data from the 2007 Survey of Consumer Finances to estimate wealth (financial assets minus liabilities). Given the large disruptions to wealth once the financial crisis started in 2008, it is better to use the 2007 survey to study the wealth distribution and transitions during the preceding period. We use Rohrs and Winters’s (2017) estimates as they are carefully done and allow our model to obtain a reasonable fit to the data. Hence, the labor productivity shock is specified as follows:
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= [0.055, 0.551, 1.195, 7.351] 0.940 0.040 0.020 0.000 0.034 0.816 0.150 0.000 π= 0.001 0.080 0.908 0.012 0.100 0.015 0.060 0.825
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Using the above shock process and parameters described in the previous section, the model generates wealth and income distributions that can be compared to the data. Table 3 presents these comparisons. We present the share of wealth and income held by various quintiles in the data and in each of our three model specifications (e.g., different values of φ). The share of wealth held by the richest quintile (Q5) in our baseline model specification (φ = 0.10) is 92.51%, very close to the 91.57% in the data. For the poorest quintile (Q1), the model generates a wealth share of -2.28% versus -1.60% in the data. In general the model fits the data fairly well, with a tighter fit in the upper quintiles.
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Welfare
Given that our paper focuses on the welfare effects of adjusting public investment and debt policies, we must specify a measure of aggregate welfare. To this end, we adopt the standard 8 This borrowing restriction has been used by Rohrs and Winter (2017) and Chatterjee, et al. (2017) to generate bottom quintile wealth holdings that match well with the U.S. data.
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utilitarian social welfare function: Γ=
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1 η(1−σ) P R a V0 (a, )f0 (a, )da ∆Γ = 1 − P R a V1 (a, )f1 (a, )da 1 " P R PT # η(1−σ) t β U (c (a, ), l (a, ))f (a, )da 0 0 0 ∆Γ = 1 − P R aPTt=0 t a t=0 β U (ct (a, ), lt (a, ))ft−1 (a, )da
(14) (15)
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Equations (14) and (15) both measure welfare effects as the percent of steady state consumption an agent must pay (positive value) or be given (negative value) in order to be indifferent between remaining in the baseline stationary equilibrium versus moving to a new equilibrium under different public investment and debt policies. The difference between the two measures is that while equation (14) compares two stationary equilibria, equation (15) compares the endogenous transition paths between the stationary equilibria. Therefore, the measure presented in equation (14) provides a pure long-run welfare measure ignoring short-run dynamics, while equation (15) provides a long-run welfare measure after taking short-dynamics into account.
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We solve for the baseline stationary equilibrium following the calibration procedure described in the previous section. We also compute stationary equilibria under a variety of alternative infrastructure investment rates. For these alternative equilibria, we adjust infrastructure investment as a share of output, gI , and allow public debt, B, to adjust endogenously to satisfy the government’s budget constraint, holding all other policy variables constant. Given these stationary equilibria, our objective is to determine which of these alternative equilibria dominate in terms of aggregate welfare, and if accounting for the short-run dynamics that occur over the transition path between steady states impacts our results.
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4.1
Welfare Effects: Ignoring Short-run Dynamics
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First, we consider the pure long-run welfare impact of moving from our initial stationary equilibrium to each of the counterfactual equilibria, ignoring the short-run dynamics that may occur along the transition path to the new steady state. This pure long-run welfare impact is found by computing the welfare measure in equation (14) for all pairs of stationary equilibria. Figure 2 presents the welfare profiles under this scenario for each of the three values of φ considered (0.05, 0.10, and 0.15). Inspection of Figure 2 indicates that when φ = 0.05, small welfare gains are realized by increasing infrastructure investment from its baseline level of 4 percent of GDP. In fact, the optimal infrastructure investment share is found to be approximately 5 percent of GDP. This is reminiscent of the welfare-maximization result obtained by Barro (1990) in the context of a representative agent, complete-markets model. Further inspection of Figure 2 indicates that the welfare gains from increasing infrastructure investment grow larger as φ is increased. Specifically, the profile derived when φ = 0.1 suggests a much larger increase in aggregate welfare when infrastructure investment is increased from its benchmark value. While the welfare maximizing infrastructure investment rate lies beyond the upper limit consider in Figure 2, the curvature of the welfare profile, as well as the results under φ = 0.05, suggest that an optimum would likely be reached around gI = 0.10.9 The results under φ = 0.15 echo these findings, suggesting an even stronger welfare impact of increasing infrastructure investment when the stock of infrastructure is more productive. The underlying intuition is that when the output elasticity of infrastructure is set to be above the baseline calibrated spending on this good (φ > gI ), the marginal benefits of increasing infrastructure investment exceed the resource costs. As such, increasing gI towards φ then increases welfare in the long-run equilibrium. It is also worthwhile to note that reducing infrastructure investment, gI , leads to relatively larger welfare losses. For example, consider the baseline case with φ = 0.10. Raising gI by 2% of GDP (from 0.04 to 0.06) results in a welfare gain of approximately 5%. Conversely, reducing gI by 2% of GDP (from 0.04 to 0.02) results in a welfare loss of 10%. Hence, welfare losses can be significant if policy makers reduce infrastructure investment. In summary, our findings suggest that a government which seeks to maximize long-run social welfare should increase infrastructure investment from the baseline level of 4 percent of GDP. While our findings show that the welfare maximizing infrastructure investment rate 9
We choose 0.08 as our upper limit as it is highly unlikely that any developed country could sustain long-run infrastructure investment above 8 percent of GDP.
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is sensitive to the value of φ that is chosen, the qualitative result is consistent across all three specifications. Therefore, even if there is doubt regarding the exact value for φ, infrastructure investment appears currently below its implied long-run optimal level. However, internalizing short-run dynamic adjustments can affect this conclusion significantly, as we describe below.
Welfare Effects: Accounting For Short-run Dynamics
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While our findings from the previous section suggest that long-run infrastructure investment should be increased, it is important to remember that these results were derived without considering the impact of short-run dynamics. When considering the transition dynamics between stationary equilibria, we note that the relationship between public investment spending and public debt varies over time. In the short-run, an increase in debt frees up additional resources for infrastructure spending, leading to a positive association between the two policy variables. However, in the long-run, the government must service its outstanding debt in order to satisfy its budget constraint. This leads to a long-run decline in the share and stock of public infrastructure in GDP, thereby implying a negative long-run relationship.10 Given this differential intertemporal relationship between public investment and debt, it is important for us to reconsider the long-run welfare effects of the alternative infrastructure investment policies after accounting for the short-run dynamics that occur as the economy transitions to its new stationary equilibrium. To this end, we compute the transition dynamics between our baseline stationary equilibrium and each of the alternative equilibria. When computing transition dynamics we can back out from the government budget constraint the size of the long-run debt adjustment and the terminal public debt-GDP ratio that is needed to reach each of the alternative long-run infrastructure investment policies. As such, we treat the debt-GDP ratio as the exogenous variable that is adjusted during the transition path and back out infrastructure investment endogenously. Such a policy experiment is empirically reasonable as government officials typically call for spending adjustments, or the creation of new debt, and then allocate these resources accordingly. The speed with which debt is accumulated (or reduced) may impact which long-run infrastructure investment policy is found to be optimal. Therefore, we compute the transition dynamics under the following two exogenous paths for the economy’s debt-GDP ratio: (i) the public 10
While a long-run inverse relationship between infrastructure investment and debt may seem counterintuitive, it is consistent with the notion that higher debt crowds out private (and public) capital (see Aiyagari and McGrattan,1998 and Chatterjee, et al., 2017).
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debt-GDP ratio equals the baseline value at time 0 and increases (decreases) to its terminal value in one period (year), and (ii) the public debt-GDP ratio equals the baseline value at time 0 and increases (decreases) linearly, reaching its terminal level in ten periods (years).11 The first column of Figure 3 presents the welfare profiles after internalizing short-run dynamics under each value of φ considered, assuming the government completely adjusts the debt-GDP ratio in one period. The welfare profiles reported in these plots are hump-shaped, and they all suggest that the welfare maximizing infrastructure investment rate is less than the benchmark value of approximately 4 percent of GDP. Specifically, when φ = 0.10, the welfare maximizing infrastructure investment rate is approximately 3.25 percent of GDP.12 This is in stark contrast to our results when the short-run dynamics were ignored, which suggested that infrastructure investment as a share of GDP should be increased significantly from its benchmark value, in lock-step fashion with its output elasticity, φ. The welfare profiles for the case when the debt-GDP ratio adjusts linearly over the span of 10 periods (years) are presented in the second column of Figure 3. The qualitative results are consistent with the case where the debt-GDP ratio adjusts instantly. Specifically, the optimal long-run infrastructure investment shares are all lower than the baseline value of 4 percent of GDP. In fact, the values under the 10-period adjustment are even lower than that found when the adjustment takes place within a single time period. Therefore, we find that once short-run dynamics are accounted for, the long-run welfare maximizing infrastructure investment share is lower than the benchmark value or observed value in our sample, and is also not sensitive to variations in the output elasticity of public infrastructure. As such, once transition paths are internalized into the welfare calculations, the welfare-maximization result of Barro (1990) is no longer robust.
Understanding the Transition Dynamics
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The results presented in the previous subsection seem to suggest that infrastructure investment should actually be reduced slightly in the United States. However, it is important to
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remember these results focus on the long-run infrastructure investment share. So, while it is true that the optimal long-run infrastructure investment share is less than the benchmark observed share of 4 percent of GDP, one must also consider how infrastructure investment evolves over the transition path before making policy prescriptions. Figures 4 and 5 illustrate the intertemporal trade-offs that occur in the economy’s aggregate variables as we move in the direction of the welfare-maximizing stationary equilibrium.13 Specifically, the long run infrastructure investment rate is lowered from 4% of GDP, as suggested by the data, to 3.9% of GDP. This long-run adjustment in the rate of infrastructure investment is associated with a 3 percentage point increase the economy’s debt-GDP ratio (from 0.72 to 0.75). In the short run, an increase in debt allows the government to increase infrastructure spending, leading to an expansionary adjustment for the economy. A such, the higher stock of debt is associated with an increase in the stock of infrastructure as well as its share of spending in GDP above the baseline share of 4 percent of GDP. Consequently, the productive spill-overs for the private sector from this increase lead to an increase in output and labor supply. As the productivity benefits from the higher stock of infrastructure are realized, both private capital and consumption increase along the transition path. These dynamics are qualitatively robust to a 10-period adjustment in debt (column 2 in Figures 4 and 5).14 The higher debt, however, will eventually crowd-out private capital and consumption. Over time, as the economy transitions to its long-run equilibrium, the government must allocate more resources to servicing its now larger debt balance. With all other variables remaining constant, this is achieved by reducing the share of infrastructure spending in GDP over time, leading to a lower share of spending and stock relative to GDP in the long-run stationary equilibrium. Therefore, welfare maximization leads to an intertemporal trade-off in the path of infrastructure spending: an increase above the baseline level of 4 percent in the short-run, and a decrease below this level in the long-run. We find that the short run benefits associated with debt creation and infrastructure investment dominate the long run losses and, as a consequence, the welfare maximizing policy actually leads to a long-run reduction in infrastructure. This result, to some extent, helps rationalize the infrastructure investment patterns observed in the United States over the post-war period. 13
Figure 4 presents the transition paths for variables related to public debt and infrastructure, while Figure 5 presents transition paths for key macroeconomic aggregates including output and consumption. 14 To save space, we focus on the case where φ = 0.10. Results for φ = 0.05 and φ = 0.15 are available upon request.
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5
Government Consumption in the Utility Function
U (c, l, Gc ) =
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The results in the previous section assumed that government consumption provides no utility benefits to private agents. In this section, we relax this assumption to introduce government consumption as a utility-enhancing public good. In doing so, we also examine the consequences of adjusting government consumption expenditures, instead of debt, in order to satisfy the government’s budget constraint. First, we will recompute the welfare profiles for our original policy experiment (with φ = 0.10 and 1-period policy adjustment) under the following utility function: 1 [cη l1−η ]1−σ + ηG G1−σ 1−σ 1−σ c
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where, following Christiano et al. (2011), ηG is set to 0.015. This allows us to determine how allowing for utility-enhancing government consumption would impact our primary results. Next, we consider an alternative experiment where changes in the rate of infrastructure investment are offset by adjustments in the government consumption ratio, gc , rather than debt. These results allow us to assess the various trade-offs that exist under different financing schemes. Figure 6 presents the welfare profile from our original policy experiment (with φ = 0.10 and 1-period policy adjustment) under our initial utility function (1) and the modified utility function (16), which includes government consumption.15 Inspection of Figure 6 indicates that these welfare profiles are highly comparable, and that adjustments in the rate of infrastructure investment away from its initial value result in welfare changes that are of a similar magnitude. However, upon closer inspection, it is clear that the welfare maximizing infrastructure investment share is somewhat larger when Gc enters the utility function (3.5% vs 3.25%). Increases in infrastructure investment result in higher output, and since the ratio of government consumption to GDP is fixed across policy experiments, its level increases as we move to the right in Figure 6. Therefore, since higher infrastructure investment is associated more government consumption, allowing Gc to enter the agents’ utility function provides another channel through which infrastructure investment may improve welfare. We have seen from Figure 6 that if we retain the structure of our original policy experiment and simply introduce Gc into our utility function, then our results will only change modestly. Now, lets consider the alternative policy experiment where we offset changes in the rate of infrastructure investment by adjusting government consumption as a share of 15
Note that the solid welfare profile in Figure 6 was also presented in middle left-hand panel of Figure 3.
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Conclusions
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output, gc , rather than debt. The welfare profile from this alternative experiment, presented in Figure 7, is different than those presented earlier (see either Figure 3 or Figure 6). We no longer find that the welfare profile displays a hump. Instead, we find that welfare is increasing over the range of infrastructure investment shares considered (0.02 to 0.0575), though at a decreasing rate. What explains this finding? While our original experiments resulted in a sizable welfare tradeoff between additional infrastructure investment and debt, the alternative experiment considered here does not yield such a large tradeoff. While it is the case that the share of output going to government consumption, gc , must fall as the infrastructure investment share rises in order to maintain the government’s budget constraint, it is important to remember that the level of output being produced is rising. Therefore, the level of government consumption, Gc , which actually enters the agents’ utility function, is not changing much over the range of values considered. As a consequence, the large welfare gains of higher infrastructure investment are only met with modest welfare losses due to reduced government consumption.16 This finding illustrates the different outcomes that can be achieved when various financing schemes are considered, and it also stresses the fact that the welfare impact of infrastructure investment cannot be determined independent of its source of financing.
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In this paper, we have examined the consequences of varying the rate of infrastructure investment for economic welfare in the context of a heterogeneous agent, incomplete-markets economy. Specifically, while the relationship between these two variables have been studied extensively in the representative agent framework, not much is known in the context of incomplete markets and heterogenous agents. We introduce public spending on a stock of infrastructure into the production function in a model similar to that developed by Aiyagari and McGrattan (1998), and calibrate it to match the key aggregate and distributional moments of the U.S. economy over the sample period 1990-2015. We then compute the welfare changes from variations in the rate of infrastructure spending under two scenarios: (i) comparing stationary steady-state equilibria, but ignoring transition dynamics, and (ii) 16
The welfare profile presented in Figure 7 already shows signs of curvature, indicating that a welfare maximizing infrastructure investment share could be found. However, we restrict attention to the range [0.02, 0.0575] as it well covers the range considered by our original experiments. Furthermore, increasing the infrastructure investment share increases output at a decreasing rate and reduces gc at a more or less constant rate, so one will almost surely find a welfare maximizing share.
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comparing stationary steady-state equilibria, while incorporating the short-run dynamics along the transition path between these equilibria. Our results indicate that the implications for the welfare-maximizing share of pubic investment in GDP depend critically on whether one internalizes the transition path between stationary equilibria. When these paths are ignored, and welfare changes are evaluated only by comparing the long-run stationary equilibria, the model implies that the government should increase infrastructure investment above its observed average share of 4 percent of GDP in the data. Further, the welfare-maximizing share of public investment coincides with its output elasticity in the production function, a result familiar from Barro (1990). This result seems to suggest that the current share of public investment in the U.S. is lower than its welfare-maximizing level. However, this insight changes significantly when one internalizes the short-run and transitional adjustment of the economy between stationary equilibria. In this case, welfare-maximization results in an intertemporal trade-off in the path of infrastructure spending. Specifically, to maximize intertemporal welfare, the government must raise the share of infrastructure spending significantly above its baseline level in the short run, while gradually reducing it in the long run as public resources must be reallocated to servicing the larger stock of debt that was accumulated over time to pay for the shortrun increase. This policy is expansionary in the short-run, and the welfare gains along the transition path exceed the long-run losses from higher debt-servicing payments. Further, once the short-run dynamics are accounted for, the link between the rate of public investment spending and the output elasticity of infrastructure is no longer evident. In summary, this paper has attempted to revisit an important public policy issue-the relationship between infrastructure spending and economic welfare-in the context of a quantitative model with incomplete markets and heterogeneous agents. The key insight from our paper is that ignoring the transition path between steady states can lead to a misleading policy prescription, while internalizing these dynamics requires policy-makers to face an intertemporal trade-off in order to maximize welfare. There are several aspects of this question we abstract from, including political economy and open economy issues, as well as those related to labor and capital market frictions. Another relevant question relates to the transformation of public capital into (possibly more productive) private capital, and the related consequences for the welfare-maximizing level of public investment. We hope to pursue these extensions in future work.
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Table 1: Structural Parameters and Empirical Targets Parameter
Value
Targeta
Description Preference Parameters
0.95 1.50 0.36
Rate of Time Preference Coefficient of Relative Risk Aversion Relative Share of Leisure in Utility Production Parameters
0.30 0.05 0.10 0.15 0.10 0.06
Output Elasticity of Private Capital Output Elasticity of Public Capital Depreciation Rate of Private Capital Depreciation Rate of Public Capital
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β σ η
Literature Robustness Literature KG = 0.67 Y
Policy Parameters
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= 0.15 = 0.04 = 0.09 T S = 0.31
Sources for empirical targets come from both the existing literature and aggregate U.S. data for the period 1990-2015. Specifically, data from the World Bank Development Indicators (WDI), the International Monetary Fund’s Investment and Capital Stock database, and the U.S. BEA were used.
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Table 2: Model Fit to Aggregate Ratios Dataa Model
Description
K/Y B/Y C/Y
Private Capital-GDP Ratio Public Debt-GDP Ratio Consumption-GDP Ratio
a
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2.15 0.72 0.59
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The data represents averages for the U.S. for the sample period 1990-2015. Sources: U.S. BEA and IMF.
Table 3: Distributional Fit
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Wealth Shares
Dataa φ = 0.05 φ = 0.10 φ = 0.15 -2.04 -0.16 2.81 7.44 91.95
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Q1 Q2 Q3 Q4 Q5
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-2.10 -0.16 2.57 7.61 92.07
Income Shares
0.44 6.60 14.11 21.66 57.19
0.43 6.55 13.68 22.54 56.80
0.46 6.56 13.56 22.33 57.09
Data Source: U.S. Survey of Consumer Finances, 2007.
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Figure 1: U.S. Infrastructure Investment
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Figure 2: Welfare Profile: Ignoring Transition Dynamics
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Figure 3: Welfare Profile: Including Transition Dynamics
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Figure 4: Transitions Paths: Debt and Infrastructure
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Figure 5: Transitions Paths: Other Macro Aggregates
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Figure 6: Welfare Profile: U (c, l) vs U (c, l, Gc )
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Figure 7: Welfare Profile: Alternative Adjust Gc Experiment
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Barro R., 1990, “Government Spending in a Simple Model of Endogenous Growth,” Journal of Political Economy, 98(5): S103-26. Bom P. R. & Ligthart J. E., 2014a, “What Have we Learned from Three Decades of Research on the Productivity of Public Capital?” Journal of Economic Surveys, 28(5): 889-916.
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Desbonnet, A., & Weitzenblum, T., 2012, “Why Do Governments End Up with Debt? Short?Run Effects Matter.” Economic Inquiry, 50(4), 905-919. Gibson J., & Rioja, F., 2017a, “Public Infrastructure Maintenance and the Distribution of Wealth,” Economic Inquiry, 55(1): 175-186.
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Gramlich, E.M., 1994, “Infrastructure Investment: A Review Essay,” Journal of Economic Literature, 32: 1176-1196.
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Rohrs, S., & Winter, C., 2017, “Reducing Government Debt in the Presence of Inequality,” Journal of Economic Dynamics and Control, 82, 1-20.
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