Public infrastructure investment, output dynamics, and balanced budget fiscal rules

Journal of Economic Dynamics & Control ] (]]]]) ]]]–]]]

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Public infrastructure investment, output dynamics, and balanced budget fiscal rules$ Pedro R.D. Bom a,n, Jenny E. Ligthart b,c,1 a b c

Department of Economics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria Tilburg University, Netherlands University of Groningen, Netherlands

a r t i c l e i n f o

abstract

Article history: Received 15 September 2011 Received in revised form 19 December 2013 Accepted 14 January 2014

We study the dynamic macroeconomic effects of public infrastructure investment under a balanced budget fiscal rule, using an overlapping generations model of a small open economy. The government finances public investment by employing distortionary labor taxes. The balanced budget rule implies a negative short-run output multiplier that exceeds (in absolute terms) the positive long-run output multiplier. Larger public capital spillovers sharpen the intertemporal output tradeoff. In contrast to conventional results regarding public investment shocks, we obtain dampened cyclical responses for plausible parameter values. The cyclical dynamics arise from the interaction between the labor tax rate, the tax base, and the intergenerational spillover effects. We show that financing scenarios involving public debt creation can substantially reduce the short-run output contraction and the transitional macroeconomic fluctuations induced by public investment. & 2014 Elsevier B.V. All rights reserved.

JEL classification: E62 F41 H54 Keywords: Public infrastructure investment Distortionary taxation Balanced budget fiscal rules

1. Introduction The fraction of GDP devoted to public infrastructure investment has steadily fallen since the late 1970s in most OECD countries, raising concerns of public capital underprovision across researchers and policymakers alike. The OECD (2006, 2012) , for instance, speaks of an ‘infrastructure gap’: at current levels, public infrastructure investment is unable to meet long-run infrastructure needs. The challenge to these countries is how to close the infrastructure gap in a time of fiscal restraint. In fact, severe fiscal imbalances in the aftermath of the recent global economic crisis have led many industrialized countries to adopt (or discuss the adoption of) some type of fiscal rule (Schaechter et al., 2013). In the European Union, for instance, the recently signed ‘Fiscal Compact’ requires member states to adopt national legislation prescribing structural budgets near balance (or in surplus).2 In practice, a balanced budget fiscal rule implies distortionary tax financing of

☆ Part of this paper was written while the first author was employed at Tilburg University and the second author was a Visiting Research Professor at Georgia State University. n Corresponding author. Tel.: þ43 1 4277 37477; fax: þ43 1 4277 37498. E-mail address: [email protected] (P.R.D. Bom). 1 Deceased. 2 The Fiscal Compact—formally entitled the Treaty on Stability, Coordination and Governance in the Economic and Monetary Union—was signed on March 2, 2012 by all European Union countries except the UK and the Czech Republic, and entered into force on January 1, 2013. It requires member states to adopt a structural deficit limit of 0.5 per cent of GDP in national legislation.

0165-1889/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jedc.2014.01.018

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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government spending. The questions arise as to how the balanced budget rule affects the dynamic macroeconomic responses to a permanent public investment impulse, and whether there is a case for exempting public investment from the balanced budget constraint. The present paper addresses these questions. Following Aschauer (1989), who documented sizable output effects of public infrastructure capital in the United States, a few theoretical contributions emerged on the dynamic macroeconomic effects of public infrastructure. A first group of papers studies the real effects of permanent public investment shocks financed by lump-sum taxes. Prominent examples include Baxter and King (1993) and Turnovsky and Fisher (1995), who employ the standard infinitely lived representative agent framework of a closed economy; and Heijdra and Meijdam (2002), who study the intragenerational effects of public investment in an overlapping generations model of an open economy. A more recent strand of literature investigates the effectiveness of public investment as a countercyclical fiscal policy tool, allowing for temporary shocks and distortionary tax financing. In this vein, Leeper et al. (2010) study the implementation delays of public investment in an estimated neoclassical model, whereas Coenen et al. (2012, 2013) use New-Keynesian models to quantify the macroeconomic effects of fiscal stimulus packages. This paper studies the dynamic macroeconomic effects of public infrastructure investment in the context of a balanced budget fiscal rule. We assess the real effects of a permanent public investment shock while focusing on the dynamic tax distortions implied by the balanced budget rule. We develop a dynamic general equilibrium model of a small open economy along the lines of Heijdra and Meijdam (2002), featuring private firms, a government, and Blanchard (1985)–Yaari (1965) overlapping generations of finitely lived households. In contrast to Heijdra and Meijdam (2002), we model endogenous labor supply and allow for nonseparable preferences over private consumption and leisure. The stock of public infrastructure capital provides nonexcludable and nonrival production services to private firms. The government invests in public infrastructure and, given the relative inefficiency of capital income taxes, satisfies the balanced budget requirement by taxing labor income.3 As in Leeper et al. (2010), a key feature of the model is the tradeoff between the production spillovers of public infrastructure capital and the labor market distortions of labor income taxes. We first derive a number of analytical results concerning the long-run effects of a permanent increase in public infrastructure investment financed by distortionary labor taxes. We show that the long-run output, private consumption, and aggregate welfare gains of public investment critically depend on the extent to which the output elasticity of public capital exceeds the public investment-to-GDP ratio. This difference may be currently large in OECD countries. While public investment ratios average at about three per cent of GDP (OECD, 2013), the meta-analysis by Bom and Ligthart (in press) suggests an output elasticity of public capital of 0.08. The long-run gains of a permanent public investment impulse are thus potentially sizable. Even for a conservative public investment ratio of five per cent, we find a long-run output multiplier of 2.25 and a long-run private consumption multiplier of 0.71. We then solve for the dynamic macroeconomic responses to the public investment impulse, assuming that public capital is initially undersupplied. We show that the balanced budget fiscal rule gives rise to an intertemporal output tradeoff of public capital accumulation, whereby output expands in the long run but at the cost of a larger short-run contraction. Because public capital builds up only gradually over time, the tax base effects of public investment are initially small, inducing large short-run increases in the labor tax rate. Labor market distortions initially outweigh production spillovers, causing public investment to depress short-run employment, output, and private investment. We find an impact multiplier of output of  3.68 for benchmark parameter values. A larger output elasticity of public capital sharpens the intertemporal output tradeoff by increasing (in absolute value) both the impact and the long-run output multipliers. Another key finding of the present paper is that balanced-budget permanent shocks to public investment induce substantial macroeconomic fluctuations during transition between steady states. In particular, we find long dampened cyclical responses of output and other key macroeconomic variables in the benchmark quantitative model. The cycles arise from the dynamic interaction between the (time-varying) labor tax rate, the tax base, and the intergenerational spillover effects, which feedback into the labor market through cohort-specific wealth effects on labor supply. Key for this result is a sufficiently large intertemporal elasticity of labor supply, such as the benchmark value of two. We examine how different values of the model's key parameters affect the impulse responses and conduct an extensive numerical search of the parameter regions yielding cyclical dynamics. Fiscal rules that include public investment spending under the balanced budget constraint are commonly criticized on the grounds of economic efficiency and intergenerational equity.4 On the efficiency side, the balanced budget rule generates dynamic distortions from tax rate changes. Because productive public spending expands the tax base, public debt financing could arguably be used to smooth the tax rate over time. On the intergenerational equity side, the balanced budget constraint imposes the entire financing burden of public infrastructure on current generations, despite the crossgenerational nature of its benefits. Indeed, Heijdra and Meijdam (2002) show that future generations disproportionately benefit from public investment in a model of exogenous labor and lump-sum taxes. In a similar setup, Bassetto and Sargent (2006) demonstrate that current generations do not support the efficient provision of durable public goods, unless the balanced budget rule exempts public investment spending. In our model, labor market distortions and intergenerational

3 Source-based capital income taxes are more distortionary than labor income taxes in small open economies facing a perfectly elastic supply of capital (see Gordon and Hines, 2002; Sørensen, 2007). 4 See Blanchard and Giavazzi (2004), Mintz and Smart (2006), and Servén (2007).

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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effects magnify each other, causing large distortionary swings in the labor tax rate and subjecting current generations to especially high tax rates. Under a balanced budget rule, the government could thus fail to secure enough public support among existing generations for a permanently higher level of public investment. We thus depart from the balanced budget rule and study the effects of public investment assuming that the government can borrow for investment purposes. We first consider a tax smoothing scenario in which the labor tax rate is permanently adjusted to a level consistent with long-run government solvency. Because the tax rate does not react to changes in the tax base, the cyclical dynamics vanish and the contraction of employment and output on impact reduces by about two thirds. The intragenerational distribution of net benefits is still biased toward future generations, however, because after-tax wages and the market value of private capital initially fall. Thus, we also consider a variation of the tax smoothing scenario in which the tax rate adjustment is delayed for a number of years. This delay virtually eliminates the impact contraction and improves the output profile until the tax adjustment date, after which the output profile is lower than without delay. Moreover, because generations living at the time of the shock are initially spared from the financing burden of public investment, the time profile of private consumption flattens out. The paper proceeds as follows. Section 2 presents the dynamic macroeconomic framework for a small open economy. Section 3 derives the analytical long-run macroeconomic and welfare effects of a balanced budget public investment impulse. Section 4 solves for the comparative dynamics. Section 5 analyzes numerically the transitional dynamics and longrun effects of a public investment shock. Section 6 concludes the paper.

2. The model This section develops a dynamic general equilibrium macroeconomic model for a typical industrialized small open economy. Subsequently, it discusses the behavior of individual households, aggregate households, firms, and the government. 2.1. Individual households The economy is inhabited by finitely-lived households, who face a constant probability of death equal to their rate of birth, denoted by β. Population is constant and normalized to unity. There are infinitely many disconnected generations, reflecting the absence of bequests. Expected lifetime utility at time t of a household born at time v r t is given by the additively time-separable utility function Z 1 Λðv; tÞ  ln Uðv; τÞe  ðα þ βÞðτ  tÞ dτ; α 40; β Z 0; ð1Þ t

where α is the pure rate of time preference. Note that the infinitely lived representative agent framework obtains for β ¼ 0. The sub-utility index Uðv; tÞ is of the constant elasticity of substitution (CES) type: Uðv; tÞ  ½ɛ C Cðv; tÞðsC  1Þ=sC þ ð1  ɛC Þ½1  Lðv; tÞðsC  1Þ=sC sC =ðsC  1Þ ;

0 o ɛC o 1; sC Z 0;

ð2Þ

where Cðv; tÞ denotes private consumption, Lðv; tÞ is hours of labor supplied, ɛC is the consumption weight in utility, and sC is the elasticity of substitution between private consumption and leisure. We normalize the household's time endowment to unity, so that 1  Lðv; tÞ denotes leisure. By choosing a CES sub-utility specification, we model nonseparability between consumption and labor and embed the Cobb–Douglas specification for sC ¼ 1. We define ‘full’ consumption as the market value of private consumption and leisure Xðv; tÞ  PðtÞUðv; tÞ ¼ Cðv; tÞ þ wðtÞ½1  Lðv; tÞ;

ð3Þ

where P(t) is the utility-based consumer price index (see (10) below), wðtÞ  wðtÞ½1  t L ðtÞ is the after-tax real wage rate, w(t) represents the before-tax real wage rate, and tL(t) is a proportional tax on labor income. We use private consumption as numeraire commodity, whose price has been normalized to unity. The household's flow budget constraint is _ tÞ ¼ ðr þ βÞAðv; tÞ þ wðtÞ Xðv; tÞ; Aðv;

ð4Þ

_ tÞ  dAðv; tÞ=dt, with Aðv; tÞ denoting real financial wealth, and r is the exogenously given world rate of interest. In where Aðv; keeping with Blanchard (1985), households contract actuarially fair ‘reverse’ life insurance. While alive, households receive an effective rate of return r þβ on their financial wealth. In the event of death, the insurance company appropriates all the financial wealth of the household. The representative household of cohort v, who is endowed with perfect foresight, maximizes lifetime utility (1) and (2) subject to its budget identity (4) and a no-Ponzi game solvency condition. We solve the household's problem by two-stage budgeting. In the first stage, the household decides on its consumption over time, yielding the individual Euler equation X_ ðv; tÞ U_ ðv; tÞ P_ ðtÞ ¼ þ ¼ r α: Xðv; tÞ Uðv; tÞ PðtÞ

ð5Þ

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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We study the case of a patient nation (i.e., r 4 α), which generates rising individual consumption profiles. By integrating (4), we obtain full consumption as a constant proportion of the household's wealth portfolio: Xðv; tÞ ¼ ðα þβÞ½Aðv; tÞ þ Hðv; tÞ; ð6Þ R1 where Hðv; tÞ  t wðτÞe  ðr þ βÞðτ  tÞ dτ denotes lifetime human wealth of vintage v at time t (i.e., the present discounted value of current and future after-tax returns to labor). In the second stage, the household allocates Cðv; tÞ and 1  Lðv; tÞ so as to maximize (2) subject to (3). Combining the firstorder conditions of this problem gives  sC Cðv; tÞ ɛC ¼ wðtÞsC ; ð7Þ 1  Lðv; tÞ 1  ɛC which we substitute into (3) to find the following demand functions for private consumption and leisure: Cðv; tÞ ¼ ½1 ωN ðtÞXðv; tÞ;

ð8Þ

wðtÞ½1  Lðv; tÞ ¼ ωN ðtÞXðv; tÞ; sC

ð9Þ

1  sC

where ωN ðtÞ  ð1  ɛC Þ ðwðtÞ=PðtÞÞ is the share of leisure in full consumption, which lies in the ð0; 1Þ interval and is constant for sC ¼ 1. Note, in view of (6), that changes in financial or human wealth affect consumption and labor supply decisions via changes in full consumption. The utility-based consumer price index, P(t), is obtained by substituting (8) and (9) into (2): 8 sC sC 1  sC 1=ð1  sC Þ >  for sC a 1 > < ½ɛC þð1  ɛC Þ wðtÞ  ɛC  1  ɛ C ð10Þ P ðt Þ  : 1 wðtÞ > for sC ¼ 1 > : ɛC 1  ɛC

2.2. Aggregate households The size of cohort v at time t is a fraction βeβðv  tÞ of the total population.5 Therefore, the relationship between aggregate full consumption and individual full consumption is Z t Xðv; tÞβeβðv  tÞ dv: ð11Þ XðtÞ ¼ 1

Aggregating (5) over existing generations gives the modified Keynes–Ramsey (MKR) rule AðtÞ XðtÞ  Xðt; tÞ X_ ðtÞ X_ ðv; tÞ ¼ ; ¼ r  α βðαþ βÞ β  XðtÞ XðtÞ XðtÞ Xðv; tÞ

ð12Þ

which says that aggregate full consumption growth equals individual full consumption growth (the first term) minus the ‘generational turnover effect’ (the second term), that is, the wealth redistribution caused by the passing away of generations. Intuitively, old generations have accumulated wealth over the course of their life, whereas new generations are born without financial wealth (i.e., Aðt; tÞ ¼ 0). Consequently, the full consumption level of new generations Xðt; tÞ falls short of the average full consumption level X(t). 2.3. Firms The representative firm hires L(t) hours of labor and rents K(t) units of capital services to produce homogeneous output Y(t) according to a Cobb–Douglas technology: YðtÞ ¼ KðtÞɛY LðtÞ1  ɛY K G ðtÞη ;

0 oɛ Y o 1; η Z 0;

ð13Þ

where ɛ Y is the output elasticity of private capital, η is the output elasticity of public capital, and KG(t) is the stock of public capital, which we model as a pure public good.6 Hence, the stock of public capital gives rise to a positive production externality, measured by η, which augments the private factors of production in a Hicks-neutral fashion. The restriction 0 oη þ ɛY o1 ensures diminishing returns with respect to private and public capital taken together, thus excluding endogenous growth.7 5

We assume large cohorts, so that frequencies and probabilities coincide by the law of large numbers. We assume that the government cannot charge a fee on the use of public capital (i.e., it is nonexcludable) and abstract from public capital congestion (i.e., it is nonrival). Fisher and Turnovsky (1998) and Rioja (1999) explicitly focus on the effects of public capital congestion. 7 One strand of literature considers public capital as a source of endogenous growth. Prominent examples include Barro (1990), Glomm and Ravikumar (1994, 1997), and Chatterjee and Turnovsky (2012). 6

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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To allow non-trivial capital dynamics, we model adjustment costs in private investment. Net capital formation is linked to gross investment I(t) according to     IðtÞ  δ K ðt Þ; Φð0Þ ¼ 0; Φ0 ðÞ 40; Φ″ðÞ o0; ð14Þ K_ ðt Þ ¼ Φ KðtÞ where δ is the rate of depreciation of private capital and ΦðÞ is the installation cost function of private capital. The degree of physical capital mobility of private capital is given by I Φ″ðÞ 5 1; K Φ0 ðÞ

0 o ρA  

where a small ρA characterizes a high degree of physical capital mobility. The firm maximizes the net present value of its cash flow Z 1 VðtÞ  ½YðτÞ wðτÞLðτÞ  IðτÞe  rðτ  tÞ dτ;

ð15Þ

t

subject to the capital accumulation constraint (14) and the stock of public capital. Note that we have normalized the prices of final output and investment goods to unity. Solving the firm's optimization problem yields the following first-order conditions: wðt Þ ¼ ð1  ɛY Þ

1 ¼ qðt ÞΦ0

YðtÞ ; LðtÞ

 IðtÞ ; KðtÞ

ð16Þ



     _ qðtÞ YðtÞ IðtÞ IðtÞ 0 IðtÞ þɛ Y ¼ r þδ  Φ  Φ ; qðtÞ KðtÞ KðtÞ KðtÞ KðtÞ

ð17Þ

ð18Þ

where q(t) denotes Tobin's q, which is defined as the market value of the private capital stock relative to its replacement costs. Eq. (16) represents a downward sloping labor demand relationship in the (w,L) space. Eq. (17) determines the investment–capital ratio as a function of Tobin's q. Finally, Eq. (18) describes the time evolution of Tobin's q; the return on private capital investment (left-hand side)—consisting of the shadow capital gain/loss and the marginal product of private capital—should be equal to the user cost of private capital (right-hand side).8 2.4. Government The government invests IG(t) in infrastructure capital and consumes CG(t) goods. Just like firms, the government faces convex adjustment costs in gross investment. Hence, public capital accumulates according to     I G ðtÞ K_ G ðt Þ ¼ ΦG  δG K G ðt Þ; ΦG ð0Þ ¼ 0; Φ0G ðÞ 4 0; Φ″G ðÞ o 0; ð19Þ K G ðtÞ where ΦG ðÞ is the installation cost function and δG is the depreciation rate of public capital. The government has a single tax instrument available, namely a proportional tax on labor income, tL(t). Any government spending in excess of taxes is financed by public debt. The government budget identity is _ ¼ rBðtÞ þ I G ðtÞ þC G ðtÞ t L ðtÞwðtÞLðtÞ; BðtÞ

ð20Þ

where B(t) denotes the stock of public debt outstanding at time t. To remain solvent, the government must obey the noPonzi game condition limτ-1 BðτÞe  rðτ  tÞ ¼ 0. By combining this solvency condition with (20), we obtain the intertemporal government budget constraint Z 1 BðtÞ ¼ ½t L ðtÞwðtÞLðtÞ  I G ðtÞ C G ðtÞe  rðτ  tÞ dτ; ð21Þ t

which requires the time profile of the labor tax rate to be such that the present discounted value of the stream of future primary surpluses (right-hand side) equal the current stock of public debt (left-hand side). As a benchmark, we assume that the government operates under a balanced budget fiscal rule—i.e., the labor tax rate is such that t L ðtÞwðtÞLðtÞ ¼ I G ðtÞ þ C G ðtÞ holds for all tZ0. Alternatively, we consider a tax smoothing scenario in which the government uses public debt so as to keep the labor tax rate constant at a level consistent with (21). Section 4.1 discusses the financing scenarios in more detail. 8 Without adjustment costs, we have ΦðÞ ¼ IðtÞ=KðtÞ and Φ0 ðÞ ¼ 1. Eq. (17) then reduces to q¼1. In this case, K(t) adjusts instantaneously to its steadystate level. Consequently, Eq. (18) reduces to ɛY ðYðtÞ=KðtÞÞ ¼ r þ δ; which is the familiar expression for the rental rate derived in a static framework.

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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2.5. Foreign sector and market equilibrium Foreign financial capital F(t) is perfectly mobile across borders. The change in net foreign assets is determined by the balance on the current account of the balance of payments F_ ðtÞ ¼ rFðtÞ þ ZðtÞ;

ð22Þ

where rF(t) denotes the return on net foreign assets and Z(t) are net exports. The goods market clears at each instant of time, yielding YðtÞ ¼ CðtÞ þC G ðtÞ þIðtÞ þI G ðtÞ þZðtÞ. Similarly, the labor market equilibrates instantly via a fully flexible real gross wage. Assets in the household's portfolio are assumed to be perfect substitutes, implying that AðtÞ ¼ VðtÞ þ BðtÞ þ FðtÞ;

ð23Þ

where VðtÞ ¼ qðtÞKðtÞ denotes the firm's stock market value. 3. Analytical long-run effects of public investment This section studies the long-run allocation and welfare effects of a balanced-budget permanent increase in public investment (i.e., dI G 40). The economy is at steady state at time t¼0, when the shock occurs, and reaches a new steady state at t-1.9 3.1. Factor markets and full consumption The public capital accumulation function (19) implies that public investment is proportional to the stock of public capital in the long run: dK G ð1Þ KG ¼ 40: dI G IG

ð24Þ

Similarly, Eqs. (14), (17), and (18) imply that the I=K ratio, Tobin's q, and the marginal product of private capital are fixed in the long run. Hence, it follows that dKð1Þ η K dLð1Þ ¼ þ ; dI G L dI G ð1 ɛ Y ÞyωIG

ð25Þ

where y  Y=K is the average product of private capital and ωIG  I G =Y is the public investment-to-GDP ratio. Eq. (25) decomposes the long-run private capital multiplier into a direct public capital effect (the first term) and an indirect employment effect (the second term). The direct effect arises from the productivity-enhancing nature of public capital, governed by η. The indirect employment effect describes the impact of public capital on private capital via changes in longrun employment, owing to the production complementarity between private capital and labor. Differentiating (16), while using (24) and (25), gives the long-run gross wage multiplier dwð1Þ η ¼ Z0; dI G LωIG

ð26Þ

which is strictly positive for η 40 and nil for η ¼ 0. Intuitively, the gross wage rate is pinned down by the marginal product of labor, which only increases if public investment is productive. As derived in Appendix A, the employment multiplier is   η  ωIG dLð1Þ sL ; ð27Þ ¼ I dI G ωG w 1  t L ð1 þs L Þ where s L  ωLL ðsC 1Þð1 ωN Þ denotes the long-run (uncompensated) wage elasticity of labor supply and ωLL  ð1 LÞ=L is the leisure–labor ratio. A well-documented empirical regularity has it that the long-run elasticity of labor supply is close to zero (Kimball and Shapiro, 2008)—i.e., the substitution and income effects of a permanent wage increase approximately cancel out—so that the long-run employment multiplier is also about zero. Note that, for a positive labor-leisure ratio, a zero long-run elasticity of labor supply implies a unitary elasticity of substitution between consumption and leisure (i.e., sC ¼ 1). If the uncompensated wage elasticity is nonzero, then the sign of the employment multiplier critically depends on η ωIG . To understand the role played by this difference, it is useful to first derive the long-run net wage and full consumption multipliers. The former is obtained by first differentiating the government budget constraint (20)—assuming no public debt _ ¼ BðtÞ ¼ 0) and constant government consumption—to find the long-run effect on the labor tax rate, and financing (i.e., BðtÞ then using (26) and (27) to arrive at   η  ωIG dw 1  tL : ð28Þ ¼ I dIG LωG 1  t L ð1 þ s L Þ 9

For a detailed derivation of the analytical results in this section, see Bom and Ligthart (2013).

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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Fig. 1. Labor market and savings system. The top panel depicts the labor market equilibrium. Aggregating Eq. (9) yields the labor supply curve Ls, whereas Eq. (16) gives the labor demand curve Ld. The bottom panel displays the savings system. The MKR locus denotes the modified Keynes–Ramsey rule (12) and the household budget identity (HBI) is given by the aggregate version of (4).

The expression of the full consumption multiplier follows from plugging (26) and (27) into the differentiated steady-state household budget constraint (4) in aggregate form, while also using the MKR rule (12) in steady state. We find   η  ωIG dXð1Þ ωX ; ð29Þ ¼ dI G ð1 ɛ Y ÞωIG 1 t L ð1 þ s L Þ where ωX  X=Y denotes the output share of full consumption. Assuming that the elasticity of substitution between consumption and leisure is strictly smaller than the upper bound sUC  1 þ1=½ωLL ð1  ωN ÞθL  4 1—which ensures that t L ð1 þ s L Þ o1 in the denominator of the bracketed terms of (27)–(29)—the net wage and full consumption multipliers are strictly positive if and only if η 4 ωIG .10 In this case, the long-run employment 10

Note that (12) implies that dAð1Þ=dI G ¼ ðA=XÞ dXð1Þ=dI G , which is also strictly positive for η 4 ωIG .

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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multiplier is negative for sC o1, zero for sC ¼ 1, and positive for sC 4 1. Intuitively, if the returns to public capital (measured I by η) exceed the resource cost of public investment (given by ωG), the permanent public investment impulse induces net resource augmentation, boosting after-tax wages in the long run.11 Households respond to this positive wealth shock by increasing goods consumption. If consumption and leisure are gross complements (i.e., sC o1), households cut on labor supply and enjoy more leisure; if they are gross substitutes (i.e., sC 41), households cut on leisure and work harder. Fig. 1 illustrates the long-run effects of the public investment impulse on gross wages, employment, full consumption, and financial wealth. Panel (a) depicts the initial labor market equilibrium at E0, the intersection of a downward-sloping labor demand curve (given by (16) and labeled Ld0;1 ) and an upward-sloping labor supply curve (given by (9) in aggregate form, for fixed X and tL, and labeled Ls0). Panel (b) displays full consumption and financial wealth in the initial steady state equilibrium E0, the crossing point of the MKR locus (representing the modified Keynes–Ramsey rule (12) in steady state) and the HBI0 curve (corresponding to the steady-state household budget identity (4) in aggregate form). The steady-state curves are represented by solid lines. A permanent increase in productive public investment (i.e., η 40) raises the marginal product of labor and shifts the long-run labor demand curve to Ld1 , raising gross real wages. To foot the bill of the rise in public spending, the labor tax rate rises in the long run, causing a leftward shift in the labor supply curve that further raises gross wages. After-tax wages, however, increase in the long run only if public investment augments net resources (i.e., η 4 ωIG ). In this case, the HBI locus shifts up in Panel (b), boosting full consumption and financial wealth; this wealth effect, in turn, adds to the leftward shift of the labor supply curve. If, on the contrary, public investment induces net resource withdrawal (i.e., η o ωIG ), net wages fall, the HBI locus shifts down, and the resulting drop in full consumption partly offsets the leftward shift of the labor supply curve. In any case, the long-run labor supply eventually settles at Ls1 , reaching a long-run equilibrium at E1 . If s L ¼ 0 (the case depicted in Fig. 1), the net long-run employment effect is zero. 3.2. Output, net foreign assets, and welfare The long-run output multiplier follows from differentiating (13), and using (24), (25), and (27):   s L ðη  ωIG Þ dYð1Þ 1 : ¼ η þ dI G 1  t L ð1 þ s L Þ ð1  ɛ Y ÞωIG

ð30Þ

The first term in brackets corresponds to the private capital accumulation effect. The second term describes the employment effect and only plays a role for nonzero long-run elasticities of labor supply. For s L 4 0, public investment stimulates both private capital accumulation and employment in the long run, so that the output multiplier increases with the public capital externality, η. If  1 o s L o 0, larger values of η increase the private capital accumulation effect but decrease the employment effect. (The negative employment effect dominates the positive private capital effect for s L r  1, but such long-run labor supply elasticities are implausibly small.) As argued in the previous section, an empirically plausible case is that of Cobb– Douglas preferences (i.e., sC ¼ 1, so that s L ¼ 0). The employment effect then vanishes, reducing the output multiplier to dYð1Þ=dIG ¼ η=½ð1 ɛ Y ÞωIG . In this case, net resource augmentation (i.e., η 4 ωIG ) implies a long-run output multiplier larger than one.12 The output multiplier (30) emphasizes the role of private capital accumulation in driving the long-run output effects of public investment. In a small open economy facing an exogenously given rate of interest, the question arises as to how public investment affects the long-run net foreign asset position. By differentiating (23), while using (25) and (29), and noting that Tobin's q returns to its initial value in the long run, we find    η  ωIG s L ðη  ωIG Þ dFð1Þ ωA  ηþ ; ð31Þ ¼ I dI G 1 t L ð1 þ s L Þ rð1  ɛY ÞωG 1  t L ð1 þ s L Þ where we have used that A¼qK in the initial steady state. Eq. (31) identifies in curly brackets the two factors affecting the long-run stock of net foreign assets. The first term represents the long-run change in total financial wealth (i.e., domestic and foreign assets), which depends on whether public investment augments or withdraws net resources. The term in square brackets describes the long-run expansion of the private capital stock (i.e., domestic assets only). The difference therefore measures the long-run change in net foreign assets. For plausible parameter values, the multiplier is negative and rises with η (see Section 5.4). Finally, the long-run instantaneous welfare effect of public investment follows from totally differentiating XðtÞ ¼ PðtÞUðtÞ with respect to IG and using multipliers (26), (27) and (29): PωX ð1 þ ωN Þðη ωIG Þ dUð1Þ : ¼ dI G ð1 ɛ Y ÞωIG ½1 t L ð1 þ s L Þ

ð32Þ

11 Turnovsky and Fisher (1995, p. 760) also discuss the critical role played by the net resource augmentation (or withdrawal) effect in the long run. In their model, however, the production function features the flow of public investment (rather than the stock of public capital), so that this effect reduces to F g  1, where Fg is the marginal product of public investment. 12 This implication holds more generally for  1 r s L r 1=t L  1; see Bom and Ligthart (2013).

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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Hence, provided that sC o sUC , public investment improves long-run aggregate welfare if and only if it augments net resources (i.e., η 4ωIG ). For lower (but positive) output elasticities of public capital, output and the stock of private capital may still increase, but after-tax wages and private consumption fall, thereby decreasing welfare.13 4. Solving for the comparative dynamics We now turn to studying the dynamic effects of public investment. We log-linearize the model around an initial steady state in which the stocks of public debt and net foreign assets are zero (i.e., Bð0Þ ¼ Fð0Þ ¼ 0). Appendix B reports the loglinearized equations. A tilde denotes a relative change; that is, X~ ðtÞ  dXðtÞ=X, where X is the steady-state value of X(t). Variables with a tilde and a dot represent the time rate of change relative to the initial steady state, that is, _ ~  rdAðtÞ=Y and X~ ðtÞ  dX_ ðtÞ=X ¼ X_ ðtÞ=X. For financial assets and human capital, we use a slightly different notation: AðtÞ _ ~ _ ~ A ðtÞ  rdAðtÞ=Y. Finally, for the labor tax rate we employ: t L ðtÞ  dt L ðtÞ=ð1 t L Þ. 4.1. Public investment shock and financing scenarios We study the effects of a permanent and unanticipated increase in public investment occurring at time t¼0, implying that I~ G ðtÞ ¼ I~ G for all t Z0. The policy change is unanticipated in the sense that it is simultaneously announced and implemented. Public consumption is assumed not to change: C~ G ðtÞ ¼ 0 for all t Z0. Using the log-linearized public capital accumulation function (see (B.5) in Appendix B), it follows that the public investment shock expands the stock of public capital over time according to K~ G ðtÞ ¼ ð1 e  χ G t ÞI~G ;

ð33Þ

0 o χ G  ðI G Φ0G ðÞÞ=K G 51

is the elasticity of the public capital installation cost function. where We consider two financing scenarios, both of which assuming an initial steady state with no public debt (i.e., Bð0Þ ¼ 0). In the benchmark scenario, the government commits to a balanced budget fiscal rule. Hence, the labor income tax rate varies in response to the public investment shock so as to keep the budget balanced at each instant of time t~ L ðt Þ ¼

i ωIG I~ G tL h ~ ~ ðt Þ ; L ðt Þ þ w  ð1  ɛY Þð1  t L Þ 1  t L

ð34Þ

_ ~ ¼ 0. which follows from the log-linearized budget constraint (B.6) with B~ ðtÞ ¼ BðtÞ We subsequently assume a tax smoothing scenario in which the government minimizes the dynamic distortions of labor income taxation by carrying out a one-off tax rate adjustment of size t~ L . Because public investment entails transitional effects on the tax base, this scenario requires public debt financing. We allow the tax adjustment to occur at time t ¼ k Z0, and thus to be either immediate (for k ¼0) or delayed until time t ¼ k 4 0: ( 0 if 0 r t o k t~ L ðtÞ ¼ uðt kÞt~ L ; where uðt  kÞ  : ð35Þ 1 if t Z k We assume that k is revealed when the public investment shock occurs at time t¼0. In the following, we use the binary variable dD to indicate the financing scenario; in particular, dD ¼1 in the balanced budget case and dD ¼0 in the tax smoothing scenario. 4.2. The reduced-form dynamic system The dynamic equations of the model can be reduced to a system in two predetermined variables (i.e., the private capital stock and financial assets) and two non-predetermined variables (i.e., Tobin's q and full consumption)14 2 _ 3 2 32 3 2 3 rωI 0 0 0 K~ ðtÞ 0 K~ ðtÞ ρA ωA 6 _ 7 6 rɛ  7 6 7 6 7 Y 6 q~ ðtÞ 7 6 Y 1 ξyk ~ 7 6 γ q ðtÞ 7 0 7 r  rɛ qðtÞ 6 7 6 ωA 76 ωA ξyx 7þ6 7; ð36Þ 6 ~_ 7¼6 76 6 7 6 r  α ~ 6 X ðtÞ 7 6 X ðtÞ 5 4 0 7 0 0 r α  ωA 7 5 4 5 4 54 ~ _ γ A ðtÞ AðtÞ rωw ξwk 0 rðωw ξwx  ωX Þ r A~ ðtÞ where ωI  I=Y is the private investment-to-GDP ratio, ωA  rA=Y is the output share of income from financial wealth, and ωw  w=Y denotes the output share of after-tax wages. The ξij terms are given by ξyk  ɛY ð1 þ ω LL ÞΘ;

ξyx  ðɛ Y  1ÞωLL Θ; ξwk  ɛY ð1 þ dD θ L ÞΘ;

ξwx  ωLL ½ɛY þ dD θ L ðɛ Y  1ÞΘ;

13 It follows from (8) in aggregate form that goods consumption, C(t), is a fraction 1 ωN ðtÞ of full consumption. Hence, abstracting from changes in ωN ðtÞ—which are negligible for sC near unity—the sign of the long-run goods consumption multiplier follows from that of full consumption. 14 Strictly speaking, the variable A~ is not completely predetermined. The non-predetermined part of it, however, is already determined by the investment system.

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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1

where Θ  f1 þω LL ½ɛ Y ð1 þ dD θ L Þ  dD θ L g and θ L  t L =ð1 t L Þ. The parameter ω LL  ωLL þ s L denotes the Frisch (marginal utility of wealth-constant) intertemporal elasticity of labor supply. The shock term γ q ðtÞ follows from using (33)–(34) in (B.2): i rɛY h  ξyg 1  e  χ G t I~G þ dD ξyd I~G þ ð1 dD Þξys uðt kÞt~ L ; γ q ðt Þ   ð37Þ ωA where ξyg  ηð1 þω LL ÞΘ, ξyd  ω LL ωIG ð1 þ θ L ÞΘ, and ξys   ð1  ɛY Þω LL Θ. The first term in brackets, which only materializes for t40, corresponds to the expansion of the public capital stock. The second and third terms regard the tax rate increase in the balanced budget and tax smoothing scenarios; the latter only materializes for t Z k. Similarly, γ A ðtÞ  rωw ½ξwg ð1 e  χ G t ÞI~ G þ dD ξwd I~G þ ð1  dD Þξws uðt kÞt~ L ;

ð38Þ ð1 þ θ L ÞωIG

where ξwg  ηð1 þ dD θ L ÞΘ, ξws  fω LL ½ɛY þdD θ L ðɛY 1Þ  1gΘ, and ξwd  1  ɛY ξws : Let us first focus on two special cases giving rise to real characteristic roots. First, the trivial case of exogenous labor ~ supply (i.e., ωLL ¼ 0) yields a recursive dynamic system that can be decomposed into an investment subsystem [qðtÞ; K~ ðtÞ] ~ and a savings subsystem [X~ ðtÞ; AðtÞ]. Both subsystems are saddle-path stable, each featuring two real characteristic roots of opposite sign. If households have infinite life spans (i.e., β ¼ 0), the generational turnover effect drops from (12). For a steady state to exist, therefore, the knife-edge condition r ¼ α must be imposed, implying one zero root in full consumption, one negative real root, and two positive real roots. Hence, the model features a hysteretic steady state. For the general case of endogenous labor supply, the dynamic system is non-recursive. We conduct an extensive numerical analysis of the characteristic roots in Section 5.2. Saddle-path stability holds for a wide range of plausible parameter values, although the dynamic properties of the system crucially depend on ωLL , sC , and β. In the benchmark parameterization (see below), for instance, the characteristic polynomial corresponding to (36) yields complex-valued roots. We detail the analytical solution of the log-linearized model in a companion mathematical appendix (see Bom and Ligthart, 2013). In the remainder of the paper, we focus on the numerical properties of the model and on the quantitative impulse responses.

5. Quantitative dynamic effects of public investment To quantify and visualize the dynamic macroeconomic effects of the unanticipated and permanent public investment impulse, we perform a simulation analysis based on the parameter setting of Section 5.1. Section 5.2 investigates the numerical properties of the dynamic system. Section 5.3 shows the transitional dynamics. Finally, Section 5.4 provides numerical results on the short-run and long-run public investment multipliers.

5.1. Parameter values We choose the parameter values so as to match the characteristics of a typical small open economy in the OECD area (see Table 1). The time unit is one year. We assume a probability of death β of 1.82 per cent to reflect an average expected life span of 55 working years. The world rate of interest is fixed at 4 per cent. We assume that both private and public capital C depreciate at the rate of 10 per cent. Following Baxter and King (1993), the ratio of public consumption to GDP (ωG) is set to I 20 per cent. In addition, the ratio of public investment to GDP (ωG) takes on a value of 5 per cent, which is somewhat above the average for industrialized countries, but more closely in line with data for southern European member states. Our quantitative results depend crucially on the size of the output elasticity of public capital, η. Based on Bom and Ligthart's (in press) meta-analysis of estimated values of η, we employ η ¼ 0:08. We perform a sensitivity analysis on this parameter later on. Because our model features labor market distortions of public investment, its quantitative implications greatly depend on the leisure–labor ratio (ωLL ) and on the intertemporal (Frisch) elasticity of labor supply (ω LL ¼ ωLL þ s L ). Kimball and Shapiro (2008, p. 1) claim that ‘[modest long-run elasticities of labor supply are] one of the best-documented regularities in economics.’ In our model, a zero long-run (uncompensated) elasticity of labor supply implies ωLL ¼ ω LL , which in turn requires a unitary elasticity of substitution between consumption and leisure. As a benchmark, therefore, we set sC ¼ 1, so that s L ¼ 0; later, we consider values of sC slightly different than one. A more controversial issue concerns the choice of appropriate values of ωLL and ω LL . Microeconometric studies typically find intertemporal elasticities of labor supply below unity (e.g., Chetty et al., 2011). Macroeconomic models, on the other hand, require much larger elasticities to explain employment fluctuations over the business cycle—Prescott (2006), for instance, considers values of at least two. Some recent contributions rationalize this discrepancy by arguing that adjustments along the extensive margin generate larger elasticities at the aggregate level than at the micro level (see Prescott, 2006 and Hall, 2009a,b). In this vein, Rogerson and Wallenius (2009) find that micro elasticities as small as 0.05 are consistent with macro elasticities larger than two. In line with these arguments, we assume ωLL ¼ ω LL ¼ 2 in the benchmark parameterization. We later investigate the sensitivity of our results to different elasticities. Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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Table 1 Chosen and implied parameter values in the benchmark model. Description Panel (a): chosen values Birth rate Rate of interest Depreciation rate of private capital Depreciation rate of public capital Output elasticity of public capital Parameter of the installation function for private capital Parameter of the installation function for public capital Public consumption-to-GDP ratio Public investment-to-GDP ratio Private consumption-to-GDP ratio Leisure–labor ratio Elasticity of substitution between consumption and leisure Panel (b): selected implied values Private investment-private capital ratio Public investment-public capital ratio Output-private capital ratio Output-public capital ratio Tobin's q Balanced budget labor income tax rate Output elasticity of private capital Elasticity of the private capital accumulation function Elasticity of the public capital accumulation function Preference weight of private consumption in utility function Leisure-full consumption ratio Frisch elasticity of labor supply Pure rate of time preference

Parameter/share

Value

β r δ δG η z zG ωCG ωIG ωC ωLL sC

0.018 0.040 0.100 0.100 0.080 0.532 0.532 0.200 0.050 0.550 2.000 1.000

I=K I G =K G Y=K Y=K G q tL ɛY ρA χG ɛC ωN ω LL α

0.110 0.110 0.550 2.200 1.210 0.350 0.288 0.171 0.091 0.370 0.630 2.000 0.039

Notes: Panel (a) shows the parameters and shares of the benchmark analysis. Panel (b) presents implied values of selected economic variables and shares.

We assume the following installation cost functions for private and public investment: I   þz I  z ln K ; Φ K z

 ΦG

IG KG

  z G ln

IG þ zG KG ; zG

ð39Þ

where z and z G are constants. From (39) and the definitions of ρA and χ G , we derive ρA ¼ ðI=KÞ=ðI=K þzÞ and χ G ¼ ðI G =K G Þz G =ðI G =K G þ z G Þ. Setting z ¼ 0:532 and using δ ¼ 0:10 yields I=K ¼ 0:11 in steady state. The latter together with z implies steady-state adjustment costs of about 0.2 per cent of GDP. Similarly, choosing z G ¼ 0:532 and using δG ¼ 0:10 gives rise to I G =K G ¼ 0:11. In this way, we arrive at adjustment costs of similar size for public capital. These parameters imply ρA ¼ 0:171 and χ G ¼ 0:091. The values of the remaining parameters and macroeconomic ratios are implied by the steady state conditions. The rate of time preference is chosen so that A¼ qK holds in steady state, which yields α ¼ 0:039 o r. By setting the output share of private consumption to 0.55, we find a ratio of investment to output of 0.20. The implied output elasticity of private capital is 0.29, implying that the no-endogenous growth condition η o1 ɛ Y ¼ 0:71 is easily met. The implied ratio of output to private capital is 0.55, which is slightly lower than the value found by Cooley and Prescott (1995). For the public capital stock, we derive Y=K G ¼ 2:20. This ratio is roughly in line with Kamps (2006), who finds a value of about two. In keeping with the average for OECD countries, the labor income tax rate is 0.35. The implied preference parameters are ωN ¼ 0:63 and ɛC ¼ 0:37. 5.2. Roots and stability Panel (a) of Fig. 2 analyzes model stability in the balanced budget case for various values of ωLL and sC . The negatively sloped solid curve represents the upper bound on the parameter region that yields a stable solution. Provided that ωLL is not too large for a given sC , the model has a unique and locally saddle-point stable steady state. We find two negative and two positive roots, which are potentially complex valued. In the stable complex case (in which the roots feature two negative and two positive real parts), the analytical solution for the transition paths of the variables includes cosine and sine terms, giving rise to endogenous dampened oscillations in key variables (Bom and Ligthart, 2013). The dotted line demarcates the upper bound of the stable, non-cyclical region; it approaches the sC -axis as sC -1 and intersects the vertical axis at the benchmark value of ωLL . The benchmark calibration indicated by Point C (sC ¼ 1; ωLL ¼ 2) lies within the stable, cyclical region. Panel (b) of Fig. 2 shows the stability regions for different values of ωLL and β. In the case of infinitely lived households (i.e., β ¼ 0), the roots are either unstable (for large ωLL ) or stable but non-cyclical (for small ωLL ). If we allow finite horizons Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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Fig. 2. Stability regions for various values of ωLL ; sC ; and β. The dotted line represents the upper bound of the stable, non-cyclical region and the solid line demarcates the lower bound of the unstable region. The area in between the solid line and the dotted line represents parameter combinations for which the model yields stable, cyclical dynamics. Point C denotes the benchmark calibration. Panel (a): ωLL and sC , and Panel (b): ωLL and β.

(i.e., β 40), then a region of cyclical stability arises for intermediate values of ωLL . Notice that the dynamic properties of the model are sensitive to ωLL ; although the calibration point C (β ¼ 0:018; ωLL ¼ 2) lies in the cyclical region, labor supply elasticities closer to one generate non-cyclical dynamics. Nevertheless, as shown by the dotted line, higher values of β decrease the minimum value of ωLL that is necessary to keep the economy in the cyclical region. The dynamic properties of the model crucially depend on the balanced budget assumption. In fact, a similar analysis for the tax smoothing scenario yields non-cyclical stability over the entire parameter ranges considered in Fig. 2. We thus conclude that the macroeconomic responses to a permanent public investment impulse exhibit dampened cycles if a balanced budget fiscal rule is combined with finite household planning horizons, a large (but not too large) intertemporal elasticity of labor supply, and an elasticity of substitution between consumption and leisure not too far from unity. Note, finally, that the size of the public capital externality, η, does not affect the characteristic roots.15

15 This result stems from the assumption of a Cobb–Douglas production technology and does not hold for technologies allowing for non-unitary elasticities of substitution between private capital and labor. See Bom et al. (2010) for an exposition of this case under lump-sum taxation.

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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5.3. Impulse responses We use the analytical transition paths derived in Bom and Ligthart (2013) to generate the impulse responses to the public investment shock. To allow for different adjustment speeds of variables, we plot impulse response functions for 200 time periods. The public investment impulse amounts to I~G ¼ 0:1 and occurs at time t ¼0. When discussing the impulse responses, ‘short run’ refers to the first five or so years after the shock, whereas ‘long run’ indicates steady state adjustments. For intuition, we analyze the impulse responses in light of Fig. 1, where the transitional dynamics are represented by dotted dynamic paths. Sections 5.3.1 and 5.3.2 study the allocation and welfare effects in the balanced budget scenario for benchmark parameters. Section 5.3.3 considers alternative parameter values. Section 5.3.4 studies the tax smoothing scenarios.

5.3.1. Allocation effects in the benchmark case Fig. 3 shows the benchmark impulse responses for various labor supply elasticities. We first focus on the benchmark value of ωLL ¼ 2 (solid line). On impact, the balanced budget rule requires a substantial increase in the labor tax rate. Afters tax wages fall, inducing households to substitute from work to leisure. In Panel (a) of Fig. 2, the labor supply curve L0 shifts d to the left along the labor demand curve L0;1 . In addition, higher labor taxes cause a negative aggregate wealth effect. As shown in Panel (b) of Fig. 2, the economy jumps down to E1. Households respond to this negative wealth effect by cutting on goods consumption and working harder, which partly offsets the leftward shift of the labor supply curve. On s impact, therefore, the labor supply curve shifts to L1 and the economy jumps to E1, with lower employment and higher pretax wages. Because private capital is predetermined and employment declines, output falls on impact. Also, the private capital–labor ratio rises, deteriorating the market value of installed capital (i.e., Tobin's q) and inhibiting private capital formation. As a consequence of lower private investment, the private capital stock starts falling shortly after the shock. Because labor and private capital cooperate in production, the marginal product of labor deteriorates. In Panel (a) of Fig. 1, the labor d demand curve shifts from Ld0;1 to L2. To balance the budget, the tax rate increases for two more years, further reducing net wages. Full consumption starts recovering immediately after the impact drop, however, as newborn generations gradually replace old generations. Intuitively, public capital spillovers take time to materialize, disproportionately benefiting young and future generations. In Panel (b) of Fig. 1, the economy moves along the dotted line from E1 to E2. The increasing labor tax rate and the positive aggregate wealth effect both contribute to shift the labor supply curve further to the left. Two years s d after the shock, the labor supply curve L2 crosses the labor demand curve L2 at E2, where employment reaches its trough. At this point, hours of work and output are 2.8 per cent and 2 per cent below their pre-shock values, respectively. As the stock of public capital expands, the marginal product of labor gradually improves. In Panel (a) of Fig. 1, the labor demand curve starts a long rightward excursion, raising employment and gross wages along the way. The expanding tax base allows for a decreasing tax rate, which brings the labor supply curve closer to its original position. The resulting increase in after-tax wages raises full consumption and, eventually, financial assets. In Panel (b) of Fig. 1, the economy moves along the dotted dynamic path from E2 toward E3. The positive wealth effect increases aggregate goods consumption and leisure, partly offsetting the rightward shift of the labor supply induced by the decreasing tax rate. Because public capital also improves the marginal product of private capital, Tobin's q recovers, stimulating private capital accumulation. Output monotonically increases during this phase, returning to the pre-shock level 13 years later and expanding for another 20 years. d s Thirty five years after the public investment shock, the labor demand and labor supply curves reach L3 and L3 in Panel (a) of Fig. 2. At E3 the changes in employment and output peak at 3.3 per cent and 4.3 per cent of their pre-shock values, respectively. Full consumption (and thus goods consumption) continues to rise, leading households to reduce labor supply. The shrinking tax base, in turn, forces the labor tax rate to increase again, although by a smaller amount than the initial increase. Both the tax rate increase and the positive wealth effect contribute to shift the labor supply curve to the left, approaching its long-run position at Ls1 . As public capital expands toward its long-run level, the labor demand curve approaches Ld1 . After going through a number of dampening oscillations, the labor market eventually settles in the new steady state, E1 . Panel (b) of Fig. 2 shows that full consumption and financial assets also spiral toward E1 . In the new steady state, output, private capital, financial assets, gross and after-tax wages, and private consumption are larger than in the initial steady state, whereas Tobin's q and employment return to their pre-shock levels. In the benchmark case of ωLL ¼ 2, therefore, the dynamic output response to a permanent public investment impulse shows long dampening cycles. Output drops at impact, remains depressed (below the pre-shock level) for 12 years, further expands for another 23 years, only to drop again, initiating a new (smaller) cycle. The time profile of output closely follows the employment response, itself a mirror image of the tax rate response. When the tax rate is high, households substitute away from labor, shrinking the tax base and forcing a higher increase in the tax rate. This tax base–tax rate interaction, together with the age-specific wealth effects on labor supply, is the source of the cycles. Key for this result is the ‘large’ intertemporal elasticity of labor supply. If ωLL ¼ 1, the employment and output responses are no longer cyclical; the initial contraction and subsequent expansion are, therefore, substantially less pronounced. If ωLL ¼ 0, labor supply does not substitute intertemporally; output increases monotonically, reflecting the accumulation of private and public capital. Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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Fig. 3. Dynamic effects of a permanent public investment impulse for various ωLL . The vertical axis reports the relative change in the respective macroeconomic variable. All parameters except ωLL are set at their benchmark values (Table 1). The size of the public investment impulse amounts to I~ G ¼ 0:1.

5.3.2. Welfare effects in the benchmark case Panel (a) of Fig. 4 depicts the instantaneous welfare effects of a change in public investment for various values of the intertemporal elasticity of labor supply. We compute the welfare responses from the log-linear version of (3) in aggregate form, U~ ðtÞ ¼ X~ ðtÞ  P~ ðtÞ. If ωLL ¼ 0, the welfare profile is monotonically rising, starting from slightly negative on impact to a positive long-run value. Because leisure is constant, the welfare profile simply replicates the private consumption profile (see Fig. 3). For ωLL 4 0, the dynamic welfare effects are non-monotonic, being positive in the short run and long run, but dropping below zero between periods 15 and 60, approximately. The welfare swings are mainly driven by fluctuations in leisure consumption, thereby increasing with ωLL . In fact, the transitional variation in goods consumption is one order of magnitude smaller than the variation in leisure, so that the welfare profile roughly mirrors the employment profile. Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

P.R.D. Bom, J.E. Ligthart / Journal of Economic Dynamics & Control ] (]]]]) ]]]–]]]

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Fig. 4. Dynamic welfare effects of public investment and optimal public investment-to-GDP ratio. Panels (a) and (c) depict the relative change in instantaneous utility—as defined by Eq. (2) in aggregate form—for various values of ωLL and η. Panels (b) and (d) show the optimal public investment-toGDP ratio for various values of ωLL and η. In Panels (b) and (d), the areas below the dots denote the parameter combinations for which the lifetime welfare of an infinitely lived household—given by Eq. (1) with β ¼ 0—is improved after the permanent public investment increase. The other parameters are set at their benchmark values (Table 1). Point C denotes the calibration point. The size of the public investment impulse amounts to I~ G ¼ 0:1. Panel (a) I I instantaneous welfare: various ωLL , Panel (b) optimal ωG: various ωLL , Panel (c) instantaneous welfare: various η, and Panel (d) optimal ωG: various η.

How sensitive is the discounted stream of welfare changes to the intertemporal elasticity of labor supply? As shown in Panel (b) of Fig. 4, very little. The dots represent, for various values of ωLL , the second-best optimal public investment-to-GDP ratio—i. R1 e., the value of ωIG that maximizes ΛðtÞ  t ln UðτÞe  αðτ  tÞ dτ—given the balanced budget fiscal rule. For ωLL ¼ 0, in which case labor taxes do not distort the labor market, the optimal level of public investment is around 7.5 per cent of GDP. For ωLL 4 0, the optimal share of public investment drops to about 5.6 per cent, slightly increasing with ωLL to about 6 per cent for ωLL ¼ 2:2 (above which point the dynamic system is no longer stable). The benchmark calibration point C lies within the area of welfare gains, suggesting that, from a welfare perspective, public investment should be increased to around 6 per cent of GDP. The welfare effects of public investment critically depend on the size of the public capital externality. Panel (c) displays the instantaneous welfare effects for several values of η. The welfare profiles display dampened cycles, irrespective of η. If public capital is unproductive, the welfare effects are always negative. Productive public investment generally increases instantaneous welfare, except in periods 15–60, when instantaneous welfare drops significantly. Larger values of η exacerbate both the positive and the negative welfare effects. Nevertheless, as shown in Panel (d), the second-best optimal ωIG rises linearly with the size of the public capital spillover. Note that the slope of the line is below unity, showing that a given η sustains a smaller second-best optimal GDP share of public investment, reflecting the deadweight loss of labor tax financing. 5.3.3. Other specifications Panels (a) and (b) of Fig. 5 show the responses of output and private consumption in the cases of infinitely lived households (i.e., β ¼ 0) and unproductive public spending (i.e., η ¼ 0), for various values of the intertemporal elasticity of labor supply. Panel (c) allows non-unitary elasticities of substitution between private consumption and leisure. Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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P.R.D. Bom, J.E. Ligthart / Journal of Economic Dynamics & Control ] (]]]]) ]]]–]]]

Fig. 5. Other specifications: infinite horizons, unproductive public spending, and various elasticities of substitution between consumption and leisure. The top panels show the relative change in output and the bottom panels depict the relative change in private consumption. All remaining parameters are set at their benchmark values. The size of the public investment impulse amounts to I~ G ¼ 0:1. In Panel (a), the benchmark value ωLL ¼ 2 gives rise to dynamic instability (see Panel (b) of Fig. 1); thus, we use ωLL ¼ 1:5. Panel (a) infinite-horizon model (β ¼ 0), Panel (b) unproductive public spending (η ¼ 0), and Panel (c) various sC .

Panel (a) shows that the cycles disappear in the infinite-horizon model for any value of ωLL in the stability region (see Panel (b) of Fig. 1). Without intergenerational spillovers, all future costs and benefits of public investment accrue to the infinitely lived representative agent, who adjusts full consumption once and for all at the time of the shock. The wealth effect triggers a negative response of labor supply—which in turn causes a temporary drop in employment and output—but only at impact. During transition, the wealth effect is switched off, thereby eliminating the cyclical responses of employment and output. Panel (b) depicts the case of unproductive public spending (i.e., η ¼ 0). The output response remains cyclical in the benchmark case of ωLL ¼ 2, showing a 40-year long post-shock depression—but much less severe than in the productive public spending case—followed by an equally long and mild expansion, before returning to the pre-shock level in the long run. If ωLL ¼ 1, the output response is non-cyclical, falling in the first ten years and slowly converging back to the pre-shock level. If labor supply is inelastic (i.e., ωLL ¼ 0), output is insensitive to unproductive public spending. In all three cases, private consumption falls over time—with small cyclical fluctuations in the case ωLL ¼ 2—reflecting the higher labor tax rate required to balance the government budget. Finally, Panel (c) considers various elasticities of substitution between leisure and private consumption. The solid lines replicate the benchmark impulse responses for sC ¼ 1. As shown by the dashed line, a slightly larger elasticity (sC ¼ 1:2) exacerbates the cyclicality of the impulse responses. Intuitively, a larger value of sC increases the intertemporal elasticity of substitution (ω LL ) for a given leisure–labor ratio (ωLL ), which amplifies the employment effects of changes in after-tax wages. An elasticity smaller than one (sC ¼ 0:8), on the other hand, gives rise to smoother (yet cyclical) responses, as shown by the dotted line. In addition, public investment increases private consumption even on impact. Because private consumption and leisure are gross complements in this case, the impact drop in hours of work (i.e., increase in leisure time) raises the marginal utility of private consumption.

5.3.4. Tax smoothing Fig. 6 shows the impulse responses in the tax smoothing scenarios. We first assume immediate tax rate adjustment (i.e., k ¼0). Subsequently, we consider a delayed tax rate adjustment by five years (i.e., k ¼5) and ten years (i.e., k ¼10). To better visualize the short-run and medium-run implications of the tax smoothing scenarios, we depict the impulse responses for 100 periods only. The cyclical dynamics disappear in all tax smoothing scenarios considered. As discussed above, the cyclical responses emerge from the dynamic interaction between the time-varying labor tax rate and the labor income tax base—a constant tax Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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Fig. 6. Tax smoothing scenarios. The vertical axis reports the relative change in the respective macroeconomic variable. All parameters are set at their benchmark values (Table 1). The size of the public investment impulse amounts to I~G ¼ 0:1.

rate (up to a level shift) neutralizes this interaction. Moreover, the one-off tax rate adjustment occurring at time t¼k is much smaller (for the values of k considered) than the short-run labor tax rate change in the balanced budget scenario, because it accounts for the long-run tax base expansion induced by productive public investment. In the case of no delay (i.e., k ¼0), the labor tax rate adjustment amounts to just a third of the impact tax rate change in the balanced budget scenario. As a result, employment, private capital, and output contract much less in the short run, whereas private consumption jumps up even on impact. If the tax rate adjustment is delayed (i.e., k 4 0), then the public investment shock barely depresses output on impact. Employment slightly falls upon the shock, recovers within a year, and further increases for k years. Additionally, Tobin's q jumps up on impact and remains above its pre-shock value for almost k years, thereby stimulating private investment and avoiding the initial reduction in the private capital stock. At time t¼k, however, the tax rate adjustment induces a discontinuity in the time profiles of employment, output, and Tobin's q, which sharply fall and remain at a lower level thereafter (compared to the case of no delay). Hence, the tax adjustment delay softens the short-run contraction and improves the medium-run output profile, but reduces the output expansion at longer horizons. Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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Postponing the financing burden of public investment evens out the output profile because the productive effects of public capital accumulation materialize only gradually over time. In the case of a ten-year delay, for instance, output has expanded enough by time t ¼10 that it remains above the pre-shock level after the tax adjustment. Also, by sparing the initially old generations from the tax burden of public investment, the tax adjustment delay allows for a more equitable intergenerational distribution of its net benefits. For this reason, the private consumption profile becomes virtually flat for k ¼10. 5.4. Impact and long-run multipliers Table 2 presents the impact multipliers (evaluated at t¼ 0) and long-run multipliers (evaluated at t-1) of the unanticipated and permanent public investment increase. In the benchmark parameterization, the long-run output multiplier amounts to 2.25, reflecting the larger stocks of public and private capital in the long run (the employment multiplier is zero for sC ¼ 1). The long-run private capital multiplier is 4.08, owing to ‘crowding-in’ of public investment by private investment. A large fraction of the additional stock of private capital is foreignly owned, as indicated by the net foreign assets long-run multiplier of 2.08. The long-run output expansion comes at the cost of a much larger short-run contraction, however. The increase in labor taxes generates an impact employment multiplier of  1.72 and an impact output multiplier of  3.68. Moreover, public investment severely crowds-out private spending on impact: each unit increase in public investment decreases private consumption by 0.04 units and private investment by 1.84 units. Because the benchmark case assumes Cobb–Douglas preferences (i.e., sC ¼ 1, so that s L ¼ 0), the long-run multipliers reported in Table 2 are invariant to the size of ωLL (see Section 3). In contrast, the impact multipliers of output and employment are rather sensitive to this parameter, falling to 1.63 and  1.25 for ωLL ¼ 1, and further reducing to zero for ωLL ¼ 0. Even more sensitive to the value of ωLL is private investment, whose negative impact multiplier shrinks fourfold for ωLL ¼ 1 and turns positive for ωLL ¼ 0. Setting the elasticity of substitution between consumption and leisure below unity (sC ¼ 0:8, so that s L ¼  0:15) generates a negative long-run employment multiplier [see (27)], which in turn decreases (in absolute value) all remaining long-run multipliers. In the short run, the lower value of sC leads to less negative multipliers of employment, output, and private investment, and switches the sign of the private consumption multiplier into positive. An elasticity of substitution larger than unity (sC ¼ 1:2, implying s L ¼ 0:15), on the contrary, exacerbates both short-run and long-run multipliers. Under Cobb–Douglas preferences, unproductive public spending (i.e., η ¼ 0) affects neither employment nor the private capital stock in the long run, leaving the steady-state level of output also unchanged. Larger public capital externalities raise the long-run multipliers of private capital and output and lower the negative long-run net foreign assets multiplier, but at the cost of deepening the short-run contraction in employment and output. This intertemporal tradeoff arises from the fact

Table 2 Macroeconomic multipliers of a permanent increase in public investment. Multiplier

Benchmark

Alternative specifications ωLL ¼ 0

dYð0Þ dI G dYð1Þ dI G dCð0Þ dI G dCð1Þ dI G dIð0Þ dI G dIð1Þ dI G dLð0Þ dI G dLð1Þ dI G dKð1Þ dI G dFð1Þ dI G

Tax smoothing

ωLL ¼ 1

sC ¼ 0:8

sC ¼ 1:2

η¼0

η ¼ 0:05

η ¼ 0:10

k ¼0

k ¼5

k ¼10

 3.6756

0.0000

 1.6334

 3.2988

 4.0788

 0.0866

 2.3297

 4.5728

 1.2541

 0.1214

 0.1811

2.2464

2.2464

2.2464

2.0672

2.4571

0.0000

1.4040

2.8080

2.2464

2.2464

2.2464

 0.0473

 0.0167

 0.0515

0.1232

 0.2213

 1.1629

 0.4657

0.2316

0.0635

0.0739

0.1102

0.7139

0.7139

0.7139

0.5778

0.8739

 1.1898

0.0000

1.1898

0.5362

0.3416

0.1243

 1.8411

0.2734

 0.4110

 1.4631

 2.4074

 0.1206

 1.1959

 2.2712

 0.2414

0.2241

0.4098

0.4493

0.4493

0.4493

0.4134

0.4914

0.0000

0.2808

0.5616

0.4493

0.4493

0.4493

 1.7202

0.0000

 1.1466

 1.5438

 1.9089

 0.0405

 1.0903

 2.1401

 0.5869

 0.0568

 0.0848

0.0000

0.0000

0.0000

 0.0597

0.0702

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

4.0839

4.0839

4.0839

3.7581

4.4668

0.0000

2.5524

5.1048

4.0839

4.0839

4.0839

 2.0806

 2.0806

 2.0806

 1.9001

 2.2928

 4.7460

 3.0801

 1.4143

 2.7895

 3.5658

 4.4326

Notes: Unless indicated otherwise, all parameters are set at their benchmark values (Table 1), where ωLL ¼ 2, sC ¼ 1, and η ¼ 0:08. In the benchmark case and in the alternative specifications, the government is assumed to continuously balance its budget. In the tax smoothing cases, the government resorts to public debt financing to keep the labor tax rate fixed (up to the one-off adjustment at time t¼ k).

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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that higher productivity of public capital steepens the time profile of after-tax wages, which induces households to postpone work effort into later stages of their life cycles. Notice that the impact multiplier of private investment also falls with the productivity of public capital. This seemingly counterintuitive result—in view of the productivity-enhancing nature of public capital—stems from the short-run employment decline, which discourages private capital formation via a fall in Tobin's q. In contrast, the short-run private consumption multiplier turns positive for high values of η. Relaxing the balanced budget constraint by allowing public investment to be debt-financed reduces the negative impact multipliers of output, employment, and private investment, and turns into positive the impact multiplier of private consumption. In the non-delayed tax smoothing case (i.e., k ¼0), the employment and output impact multipliers are  0.59 and 1.25, only about a third of their benchmark values. If, moreover, the labor tax rate is adjusted with a ten-year delay (i.e., k¼ 10), the short-run employment and output multipliers further reduce to 0.08 and  0.18, whereas the private investment multiplier also turns positive. Because of the unitary elasticity of substitution between private consumption and leisure, the long-run multipliers of output, private capital, and employment are invariant to the financing scenario. The longrun private consumption effect, on the other hand, is lower under tax smoothing—and decreases with k—reflecting the deterioration of the economy's long-run net foreign asset position.

6. Conclusions This paper studies the dynamic macroeconomic effects of a permanent impulse to public infrastructure investment under the constraint of a balanced budget fiscal rule. To this end, we build a dynamic general equilibrium model of a small open economy where the government resorts to labor income taxation to finance public infrastructure investment. On the household side, we assume overlapping and disconnected generations of finitely lived households. The dynamics of the model are governed by the interplay of public capital production spillovers, labor market distortions, and intergenerational spillover effects. We argue that the long-run gains of a permanent increase in public investment, even if constrained by a balanced budget rule, are potentially sizable in OECD countries. Our analytical results show that public investment increases output, private consumption, and aggregate welfare in the long run if the output elasticity of public capital exceeds the public investmentto-GDP ratio. Most OECD countries currently operate public investment ratios below 5 per cent. Pinning down the output elasticity of public capital has stimulated a large body of literature after Aschauer (1989). We have meta-analyzed some of this literature in previous work and found an average value of 0.08 for mostly OECD economies (Bom and Ligthart, 2014). For a public investment ratio of 5 per cent, we find a long-run output multiplier of 2.25 and a long-run consumption multiplier of 0.71. The balanced budget fiscal rule severely constrains the macroeconomic responses to a permanent public investment impulse, however, even when public capital is initially undersupplied. Because public capital builds up only gradually over time, the labor market distortions initially outweigh the production spillovers. In the short run, therefore, the balancedbudget public investment shock contracts employment, output, and private investment. For benchmark parameter values, the impact output multiplier amounts to  3.68. An important political implication of this result is that governments may fail to secure sufficient public support among existing generations for a permanently higher level of public investment. Moreover, balanced budget rules exacerbate the macroeconomic transitional fluctuations induced by public investment shocks. In particular, we find dampened cyclical macroeconomic responses to the permanent increase in public investment. The cycles arise from the dynamic interaction between the time-varying labor tax rate and the tax base, and obtain for finite household planning horizons and large intertemporal elasticities of labor supply. The sensitivity of the impulse responses to the intertemporal elasticity of labor supply emphasizes the need to better grasp the magnitude of this parameter at the macro level. The short-run contraction and the transitional fluctuations can be substantially mitigated if the government is allowed to borrow for purposes of investment spending. In a pure tax smoothing scenario—where the labor tax rate is immediately adjusted to a level consistent with long-run government solvency—the initial output contraction reduces by two thirds and the cyclical responses vanish. If, in addition, the labor tax rate adjustment is delayed by a number of years, the short-run output depression virtually disappears. For a ten-year delay, output falls slightly on impact but remains above the pre-shock level thereafter, whereas private consumption and private investment increase even on impact. By postponing the financing burden to future generations, the tax adjustment delay should help gather public support toward a permanently higher level of public investment. The results in this paper therefore support the view of Bassetto and Sargent (2006), among others, that fiscal rules should exempt public investment from the balanced budget constraint.

Acknowledgments The authors would like to thank Lex Meijdam, Mark Rider, and two anonymous referees for helpful discussions and valuable suggestions. Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

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Appendix A. Deriving the long-run employment multiplier Eq. (9) can be fully differentiated with respect to IG to arrive at a general expression for the labor supply effect: dLð1Þ 1  L dwð1Þ 1  L dt L ð1Þ 1  L dωN ð1Þ 1  L dXð1Þ ¼    : dI G w dIG 1  t L dI G ωN dI G X dI G

ðA:1Þ

To obtain the effect of public investment on the long-run tax rate, we differentiate the balanced-budget government constraint, I G ðtÞ þ C G ðtÞ ¼ t L ðtÞwðtÞLðtÞ, for fixed CG(t): dt L ð1Þ 1 t L dwð1Þ t L dLð1Þ  ¼  : dI G wL w dIG L dIG

ðA:2Þ  sC

To derive dωN ð1Þ=dIG , we fully differentiate ωN ðtÞ  ð1  ɛC Þ wðtÞ=PðtÞ   dωN ð1Þ 1 dwð1Þ 1 dt L ð1Þ 1 dPð1Þ ¼ ωN ð1  sC Þ   : dI G w dI G 1 t L dIG P dI G

1  sC

:

The long-run change in the price index follows from differentiating (10):   dPð1Þ 1 dwð1Þ 1 dt L ð1Þ ¼ ωN P  : dIG w dI G 1  t L dI G

ðA:3Þ

ðA:4Þ

To obtain an expression for dXð1Þ=dI G , we note that the steady-state version of Eq. (12) implies A=X ¼ ðr  αÞ=βðα þ βÞ, so that dAð1Þ A dXð1Þ r  α dXð1Þ ¼ ¼ ; dI G X dI G βðα þ βÞ dI G

ðA:5Þ

which, combined with the differentiated steady-state (aggregate) version of (4), gives dXð1Þ X dwð1Þ X dt L ð1Þ ¼  : dIG w dI G 1  t L dI G

ðA:6Þ

Substituting (A.2)–(A.4) and (A.6) into (A.1) while using (26), and then solving for dLð1Þ=dI G yields the employment multiplier (27). Appendix B. Summary of the log-linearized model (a) Dynamic equations: i rωI h~ _ I ðt Þ  K~ ðt Þ K~ ðt Þ ¼ ωA

ðB:1Þ

i rɛY h ~ q~_ ðt Þ ¼ r q~ ðt Þ  Y ðt Þ  K~ ðt Þ ωA

ðB:2Þ

" # ~ AðtÞ _ X~ ðt Þ ¼ ðr  αÞ X~ ðt Þ  ωA

ðB:3Þ

_ ~ þω w ~ ðtÞ  ωX X~ ðtÞ A~ ðtÞ ¼ r½AðtÞ w

ðB:4Þ

_ K~ G ðtÞ ¼ χ G ½I~G  K~ G ðtÞ

ðB:5Þ

_ ~ þ wðtÞÞ ~ þ ωI I~G þωC C~ G θL ð1 t L Þt~ L ðtÞ  θL t L ðLðtÞ ~ B~ ðtÞ ¼ r½BðtÞ G G

ðB:6Þ

(b) Static equations: ~  K~ ðtÞ ~ ¼ ρA ½IðtÞ qðtÞ

ðB:7Þ

~ ~ wðtÞ ¼ Y~ ðtÞ  LðtÞ

ðB:8Þ

~ þ ηK~ G ðtÞ; Y~ ðtÞ ¼ ɛ Y K~ ðtÞ þð1  ɛY ÞLðtÞ

ðB:9Þ

h i ~ ðtÞ  ω~ N ðtÞ  X~ ðtÞ ~ ¼ ωLL w LðtÞ

ðB:10Þ

ωN ω~ N ðt Þ þ X~ ðt Þ 1  ωN

ðB:11Þ

h i ~ ωA qðtÞ ~ þ K~ ðtÞ F~ ðtÞ ¼ AðtÞ

ðB:12Þ

C~ ðt Þ ¼ 

Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i

P.R.D. Bom, J.E. Ligthart / Journal of Economic Dynamics & Control ] (]]]]) ]]]–]]]

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(c) Definitions: ~ ðtÞ P~ ðtÞ ¼ ωN w

ðB:13Þ

h i ~ ðtÞ  P~ ðtÞ ω~ N ðtÞ ¼ ð1 sC Þ w

ðB:14Þ

~ ðtÞ ¼ wðtÞ ~  t~ L ðtÞ w

ðB:15Þ ωCG

ωIG

 C=Y,  I G =Y, ωw  w=Y, ωLL  ð1 LÞ=L, ωX  X=Y, Notes: the following definitions are used: ωA  rðqK=YÞ, ωI  I=Y, ρA  ðI=KÞðΦ″=Φ0 Þ 4 0, and χ G  I G Φ0G ðÞ=K G 40. A tilde denotes a relative change, for example, C~ ðtÞ  dCðtÞ=C. However, for ~  rdAðtÞ=Y) and for labor taxes we use financial assets we scale by steady-state output and multiply by r (e.g., AðtÞ t~ L ðtÞ  dt L ðtÞ=ð1  t L Þ. References Aschauer, D.A., 1989. Is public expenditure productive? J. Monet. Econ. 23, 177–200. Barro, R.J., 1990. Government spending in a simple model of endogenous growth. J. Polit. Econ. 98, S103–S125. Bassetto, M., Sargent, T.J., 2006. Politics and efficiency of separating capital and ordinary government budgets. Q. J. Econ. 121, 1167–1210. Baxter, M., King, R.G., 1993. Fiscal policy in general equilibrium. Am. Econ. Rev. 83, 315–334. Blanchard, O.J., 1985. Debt, deficits, and finite horizons. J. Polit. Econ. 93, 223–247. Blanchard, O.J., Giavazzi, F., 2004. Improving The SGP Through A Proper Accounting Of Public Investment. Discussion Paper No. 4220, Centre for Economic Policy Research. Bom, P.R.D., Heijdra, B.J., Ligthart, J.E., 2010. Output Dynamics, Technology, And Public Investment. CentER Discussion Paper, No. 10-58, Tilburg University, Tilburg. Bom, P.R.D., Ligthart, J.E. What have we learned from three decades of research on the productivity of public capital? Journal of Economic Surveys. http://dx. doi.org/10.1111/joes.12037, in press. Bom, P.R.D., Ligthart, J.E., 2013. Public Infrastructure Investment, Output Dynamics, and Balanced Budget Fiscal Rules: Mathematical Appendix. Mimeo, University of Vienna. Download from 〈http://www.pedrobom.com/BLMA13.pdf〉. Chatterjee, S., Turnovsky, S., 2012. Infrastructure and inequality. Eur. Econ. Rev. 56, 1730–1745. Chetty, R., Guren, A., Manoli, D., Weber, A., 2011. Are micro and macro labor supply elasticities consistent? A review of evidence on the intensive and extensive margins. Am. Econ. 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Please cite this article as: Bom, P.R.D., Ligthart, J.E., Public infrastructure investment, output dynamics, and balanced budget fiscal rules. Journal of Economic Dynamics and Control (2014), http://dx.doi.org/10.1016/j.jedc.2014.01.018i