Factor-biased public capital and private capital crowding out

Accepted Manuscript

Factor-Biased Public Capital and Private Capital Crowding Out Pedro R.D. Bom PII: DOI: Reference:

S0164-0704(17)30109-X 10.1016/j.jmacro.2017.03.002 JMACRO 2932

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Journal of Macroeconomics

Received date: Revised date: Accepted date:

20 July 2016 14 March 2017 18 March 2017

Please cite this article as: Pedro R.D. Bom, Factor-Biased Public Capital and Private Capital Crowding Out, Journal of Macroeconomics (2017), doi: 10.1016/j.jmacro.2017.03.002

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Highlights • Factor-biased public capital is embedded in a small open economy model. • Transitional crowding in for elastic labor supply.

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• Long-run crowding out if public capital augments private capital.

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• The elasticity of substitution between capital and labor plays a critical role.

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Factor-Biased Public Capital and Private Capital Crowding Out Pedro R. D. Bom∗

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Deusto Business School

Abstract

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This paper studies the dynamic effects of public investment on private capital accumulation in a general equilibrium macroeconomic model of a small open economy with factorbiased public capital. I show that public investment induces rather complex private capital dynamics—falling in the short and in the long run, but potentially increasing along transition—if public capital augments private capital and private inputs are gross complements in production. Whether private investment is crowded in or out during transition critically depends on parameters that empirically hard to measure, such as the labor supply elasticity and the elasticity of substitution between private inputs—a small increase in the latter from 0.5 to 0.6, for instance, turns a totally negative transitional effect into a predominantly positive one. These results help rationalize the lack of empirical consensus on the relationship between public and private investment.

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Keywords: Public investment, Public capital, Infrastructure capital, Private capital, Crowding out, Factor bias JEL: E22, E62, F41, H54

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Deusto Business School, University of Deusto. Address: Hermanos Aguirre Kalea 2, 48014 Bilbao, Spain. Tel.: +34 944 139 290. Email address: [email protected] (Pedro R. D. Bom)

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1. Introduction The recent global economic and financial crisis—to which governments of many advanced countries reacted with fiscal stimulus measures strongly based on public capital

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expenditures—sparked a renewed interest in the macroeconomic effects of public investment.1 A central issue for policymakers is how public capital spending affects private capital formation. The empirical evidence on the public-private investment relationship is rather mixed, however.2 This paper studies the dynamic effects of public investment on

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private capital formation in a general equilibrium model of small open economy. Whereas the crowding-in/out effects of public investment have typically been rationalized in terms of the degree of substitution between public and private capital in production, this paper assigns a central role to the dynamic reaction of labor employment and its complementar-

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ity with private capital. In particular, I show how the dynamic interplay between private capital and labor can generate transitional crowding-in effects even when public investment

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crowds out private capital formation both in the short and in long run. From a supply-side perspective, the relationship between public capital spending and

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private capital formation depends on the specification of the production technology. First, public capital can enter the production function in a factor-neutral way (in which case

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the marginal productivity of private capital relative to that of labor is unaffected) or in a factor-biased way (in which case the marginal productivity of one factor changes relative

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to the other). Second, private capital and labor can be specified as gross complements (i.e., 1

Some examples include Leeper, Walker, and Yang (2010); Leduc and Wilson (2012); Bom and Ligthart (2014a); Abiad, Furceri, and Topalova (2015); Clancy, Jacquinot, and Lozej (2016); and Bouakez, Guillard, and Rolleau-Pasdeloup (2017). 2 Voss (2002) finds evidence of crowding out of private investment by public investment shocks in Canada and the United States. Perotti (2004) finds similar results also for Australia, Germany, and the United Kingdom. Afonso and St. Aubyn (2009) document both crowding-in effects of public investment in eight out of 17 developed economies, and crowding-out effects in the remaining nine. The evidence is also mixed for developing economies. Cavallo and Daude (2011) find mostly crowding-out effects in a sample of 116 countries, whereas Eden and Kraay (2014) report large positive effects of public investment on private investment in a panel of 39 countries.

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with an elasticity of substitution below one) or gross substitutes (i.e., with an elasticity of substitution above one) in production. Most studies employ a Cobb-Douglas production specification, which constrains the elasticity of substitution between private capital and

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labor to unity and, consequently, forces public capital to be factor-neutral. According to Chirinko (2008), however, the bulk of empirical evidence suggests an elasticity somewhere in the range 0.4–0.6. To depart from unitary elasticities, I adopt a constant elasticity of substitution (CES) production function and focus on values within the range 0.4–0.6. I

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consider both capital- and labor-augmenting public capital technologies, but focus on the former as a baseline scenario.

I embed the CES production function into a dynamic general equilibrium model of small open economy facing a perfectly elastic supply of capital from abroad at the exogenouslygiven rate of interest. Perfectly competitive firms choose private investment and labor

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taking the stock of public capital as given. To prevent instantaneous adjustment of phys-

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ical capital stocks, I impose adjustments costs on private capital formation. This feature gives rise to a responsive Tobin’s q, which temporarily absorbs shocks to the marginal pro-

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ductivity of private capital. I assume the economy borrows/lends in world capital markets under full commitment and abstract from repayment default and risk premium issues. To

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keep the public sector as simple as possible, I assume the government raises tax revenues by means of a lump-sum tax. This assumption rules out potential distortions from other tax

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instruments and helps focus the analysis directly on the productive role of public capital. In the baseline fiscal scenario, the government adjusts lump-sum taxes so as to keep its budget continuously balanced. I then introduce public debt and allow for the possibility of delaying any required tax adjustments. The household side of the economy consists of infinitely-many overlapping and disconnected generations of finitely-lived agents with a leisure-labor choice. The production and household sectors of the model are therefore connected through endogenous labor supply. I 2

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employ the overlapping-generations structure as a convenient technical device to obtain an endogenously-determined steady state, thus avoiding the unit-root property of small open economy models with fixed interest rates (see Schmitt-Groh´e and Ur´ıbe, 2003). I adopt

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Blanchard’s (1985) perpetual-youth formulation, in which households face a constant (ageindependent) probability of death.3 Although this assumption oversimplifies mortality risk and rules out life-cycle behavior, it captures in a relatively simple way the intergenerational disconnectedness—i.e., absence of perfect intergenerational altruism—observed in

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the data (see, e.g., Barczyk, 2016). Moreover, it encompasses the common infinitely-lived representative agent framework as a special case.

I first study analytically the long-run effects of a permanent public investment increase on private capital. I show that a (private capital-augmenting) increase in public capital crowds out private capital in the long run if (and only if) the elasticity of substitution is

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sufficiently small and the output elasticity of public capital is sufficiently large. Long-run

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private capital is crowded out in this case because public capital decreases its long-run marginal productivity. The labor supply elasticity plays a minor role in the long run,

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affecting the size of the private capital effect through the labor employment effect, but without changing the qualitative crowding-out result.

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I then turn to the impact and transitional dynamics of a public investment shock. I log-linearize the model around the steady state and solve analytically for the impulse re-

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sponse functions. I calibrate the model to the average economy in the euro area and focus on the quantitative and qualitative features of the numerical simulations. The most important result of this exercise is that a capital-augmenting increase in public investment causes rather complex dynamic effects on private capital formation. Private capital falls in the short and long run, but may increase along transition between steady states if labor 3

Heijdra and Meijdam (2002) employ a similar household structure to study the intergenerational welfare effects of public investment, but assume exogenous labor supply.

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supply is relatively elastic or public investment is debt-financed. Moreover, the transitional response of private capital is very sensitive, both quantitatively and qualitatively, to small changes in the output elasticity of public capital and in the elasticity of substitution be-

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tween private capital and labor. A small increase in the elasticity of substitution from 0.5 to 0.6, for instance, suffices to turn the private capital response from entirely negative to predominantly positive.

This paper relates to several studies on the real macroeconomic effects of public invest-

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ment. Fisher and Turnovsky (1998) also study explicitly the effects of public investment on private capital accumulation, but in a model of exogenous labor. In their model, crowding out arises as the result of congestion effects, especially for a low elasticity of substitution between private and public capital. Baxter and King (1993) and Turnovsky and Fisher (1995) study the macroeconomic effects of temporary and permanent shocks to govern-

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ment spending, including its investment component, using a similar neoclassical model of

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a closed economy. Leeper, Walker, and Yang (2010) study the role of public investment as a countercyclical fiscal policy tool, focusing on implementation delays. Bom and Ligthart

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(2014a) investigate how distortionary labor income taxes constrain the effects of public investment under balanced-budget rules. None of these papers, however, consider factor-

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biased public capital and the role of the elasticity of substitution between private capital and labor.

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The rest of the paper is structured as follows. Section 2 describes the main elements of the model. Section 3 derives analytically the steady-state effects of public investment, specifying the conditions under which public capital crowds out private capital in the long run. Section 4 studies the transitional dynamics of a public investment impulse, discussing in turn the solution of the log-linearized model, the calibration strategy, and the simulation results. Section 5 concludes the paper.

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2. The Model The model consists of a production sector, a household sector, a public sector, and a foreign sector.4 The economy is small and open, so that the interest rate is exogenously

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determined in world capital markets. Financial capital is perfectly mobile across borders. The model is specified in real terms. Time is continuous. I first discuss the production technology and the behavior of firms in Section 2.1. Section 2.2 turns to the behavior of individual households and the aggregate household sector. Section 2.3 describes the public

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sector. Section 2.4 closes the model with a description of the foreign sector and market equilibrium. 2.1. Firms

The production sector of the economy consists of infinitely-many identical firms pro-

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ducing a homogeneous good under perfect competition in the output and factor markets.

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Firms have access to a production technology that transforms private capital, K(t), and labor, L(t), into output, Y (t). Firms choose K(t) and L(t) taking as given the existing stock of public capital, KG (t), which is provided by the government. I allow for flexi-

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ble substitutability between private capital and labor by assuming a constant elasticity of

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substitution (CES) production specification: σY n σY −1 o σ −1 σY −1 Y σY σY + [EL (t)L(t)] , Y (t) = [EK (t)K(t)]

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(1)

where EK (t) and EL (t) are capital- and labor-augmenting technical change terms (discussed below). Denoting the marginal productivities of private capital and labor by YK (t) and

YL (t), the elasticity of substitution between the two factors is σY ≡ 4

d ln[L(t)/K(t)] d ln[YK (t)/YL (t)]

> 0.5

Detailed mathematical derivations of the model are provided in the Technical Appendix to this paper (Section 2). 5 The CES production function embeds the Leontief technology for σY = 0, and the linear technology for σY → ∞. The paper does not consider these limiting cases.

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Private capital and labor are said to be gross complements if σY < 1, and gross substitutes if σY > 1. The case σY = 1 corresponds to the Cobb-Douglas production function. Chirinko (2008) notes that the bulk of the empirical evidence suggests a value for σY in the range

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0.4–0.6. As a baseline, therefore, I assume gross complementarity between private capital and labor.

Apart from private capital and labor, private sector output also depends on public capital services, which are assumed to be proportional to the stock of public capital. In

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this paper, public capital is defined as core infrastructure that is directly productive to private firms—e.g., roads and highways, water systems, energy utilities, etc. Public capital is modeled as a pure public good, provided by the government free of charge and not subject to congestion.6 Public capital, KG (t), enters the production function through the

i = {K, L},

(2)

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Ei (t) ≡ ρi KG (t)ηi ,

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technical change terms EK (t) and EL (t) in a factor-augmenting fashion:

where ηi ≥ 0 represents the elasticity of the factor-augmentation term with respect to public

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capital, and ρi > 0 is a scaling factor. If ηK = ηL , public capital is factor-neutral—i.e., it

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augments private inputs in the same proportion. In this paper, however, I allow for factorbiased public capital, which requires ηK 6= ηL . Focusing only on pure factor-augmentation

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cases, private capital-augmenting public capital is captured by ηK > ηL = 0, whereas labor-augmenting public capital is described by ηL > ηK = 0. In light of its interpretation as core infrastructure, it seems natural to assume public capital to be private capitalaugmenting. The labor-augmenting specification may be more appropriate for other, less directly productive, forms of public spending—such as education or research grants. Hence, the baseline specification of public capital assumes ηK > ηL = 0. 6

Fisher and Turnovsky (1998) and Dioikitopoulos and Kalyvitis (2008) explicitly focus on the effects of public capital congestion.

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To understand the importance of the elasticity of substitution between private capital and labor in connection with the type of factor-augmentation induced by public capital,

YK (t) = YL (t)



ρK ρL

 σYσ −1 Y

KG (t)

(ηK −ηL )(σY −1) σY



K(t) L(t)

− σ1

Y

,

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consider the ratio of marginal productivities of private capital and labor:

(3)

If public capital is purely capital-augmenting (i.e., ηK > ηL = 0) and private inputs are gross substitutes (i.e., σY > 1), an increase in KG (t) raises the marginal productivity of

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capital relative to that of labor. Hence, public capital is biased towards private capital. In the empirically-plausible case of gross complementarity (i.e., σY < 1), however, an increasing in capital-augmenting KG (t) lowers the relative marginal productivity of private capital. Public capital is then biased towards labor.

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The production specification (1) features constant returns to scale across private inputs. Denoting the output elasticity of private factor j = {K, L} by θj (t) ≡ Yj (t)j(t)/Y (t),

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it must then hold that θK (t) + θL (t) = 1. Moreover, because public capital generates a production externality, the production technology exhibits increasing returns to scale across

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all inputs—i.e., θK (t)+θL (t)+θG (t) ≥ 1, where θG (t) ≡

∂Y (t) KG (t) ∂KG (t) Y (t)

≥ 0 denotes the output

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elasticity of public capital. Note that this elasticity can be written as θG (t) = θK (t)ηK + θL (t)ηL , a convex combination of ηK and ηL . Because I focus on pure factor-augmentation

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cases, θG (t) = θK (t)ηK if public capital augments private capital, and θG (t) = θL (t)ηL if public capital augments labor. I rule out endogenous growth by requiring diminishing returns to reproducible factors (i.e., private and public capital), which amounts to imposing ηK + θK (t) < 1; this condition is met for plausible parameter values (see Section 4.2). I postulate convex adjustment costs to private capital. Capital adjustment costs are not only empirically relevant but also technically convenient in generating nontrivial transitional dynamics in private capital in the case of small open economy facing an exogenously7

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given interest rate.7 In particular, I assume that net capital formation relates to gross investment, I(t), according to     I(t) ˙ − δ K(t), K(t) = Φ K(t)

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(4)

where δ is the depreciation rate of private capital and Φ(·) is a strictly concave capital installation function (i.e., Φ0 (·) > 0 and Φ00 (·) < 0) featuring zero net capital formation and adjustment costs at the origin (i.e., Φ(0) = 0 and Φ0 (0) = 1).

V (t) ≡

Z

t

∞

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The representative firm maximizes the net present value of its cash flow:

[Y (τ ) − w(τ )L(τ ) − I(τ )] exp{r (t − τ )}dτ,

(5)

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given the production technology (1) and the existing stock of public capital, KG (t), and subject to the capital accumulation constraint (4). In (5), w(t) and r denote the gross

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wage rate and the exogenously-given interest rate. Note, moreover, that the price of output and investment goods are both normalized to unity. The co-state variable of the

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firm’s optimization problem, denoted by q(t), corresponds to Tobin’s q—i.e., the market value of installed capital relative to replacement cost—and is governed by

(6)

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      I(t) I(t) 0 I(t) − Φ − (r + δ) − YK (t), q(t) ˙ = −q(t) Φ K(t) K(t) K(t)

where q(t) ˙ ≡ dq(t)/dt (a notational convention I use for all time derivatives in this paper). Because capital adjustment costs prevent the capital stock from adjusting instantaneously, shocks to its marginal productivity (denoted by the the last term) are temporarily absorbed 7

Cooper and Haltiwanger (2006) find that, although a combination of convex and non-convex adjustment costs is necessary to fit the data at the plant level, convex adjustment costs fit the aggregate reasonably well.

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by Tobin’s q. The static first-order conditions are:

1 = q(t)Φ0



I(t) K(t)



(7) .

(8)

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w(t) = YL (t),

Equation (7) is a standard labor demand function setting the wage rate to the marginal productivity of labor, whereas (8) pins down the optimal investment level conditional on the existing stock of capital and Tobin’s q. Note that (8) boils down to q(t) = 1 in a model

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without adjustment costs, in which case (6) reduces to the standard condition YK (t) = r+δ. 2.2. Households

Following Heijdra and Meijdam (2002) and Bom and Ligthart (2014a), I assume over-

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lapping generations of finitely-lived households. Households face a constant instantaneous probability of death measured by β, the same rate at which new households are born. The

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population size is thus constant and can be normalized to one. Because households do not have a bequest motive, generations are disconnected. Households insure against mortality

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risk by contracting actuarially-fair ‘reverse’ life insurance, which adds an extra return on financial wealth equal to the probability of death, β (cf. Blanchard, 1985). Individual

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households optimally decide on private consumption spending, C(v, t), and on the split of one unit of time between labor, L(v, t), and leisure, 1 − L(v, t). At time t, a household

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born at v ≤ t maximizes Λ(v, t) ≡

Z

t

∞

{εC ln C(v, t) + (1 − εC ) ln[1 − L(v, t)]} exp{(α + β)(t − τ )}dτ,

(9)

subject to the flow budget constraint ˙ t) = (r + β)A(v, t) + w(t) − T (t) − X(v, t). A(v, 9

(10)

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In the objective function (9), α and εC denote the pure rate of time preference and the consumption weight in instantaneous utility. In the constraint (10), A(v, t) denotes the real stock of financial wealth, w(t) is the (age-independent) real wage, T (t) are lump-

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sum taxes, and X(v, t) ≡ w(t) [1 − L(v, t)] + C(v, t) denotes ‘full’ consumption—i.e., the combined market value of consumption and leisure. I assume taxes cannot exhaust labor income (i.e., T (t) < w(t)L(t)), so that consumption and saving can both be positive for an individual with no financial assets (as is the case at birth). Note the presence of β as

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an extra component in the discount factor (reflecting the mortality risk), and also as an extra return on financial wealth (representing the insurance premium on that risk). The standard infinitely-lived representative agent framework obtains for β = 0. Solving the household’s problem by two-stage budgeting gives, in the first stage, the

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Euler equation for individual full consumption

(11)

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X˙ (v, t) = r − α, X (v, t)

which governs the intertemporal allocation of full consumption. The second stage gives

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first-order conditions for consumption and leisure demand:

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C(v, t) = εC X(v, t),

(13)

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w(t) [1 − L(v, t)] = (1 − εC )X(v, t),

(12)

which determine the intratemporal allocation of full consumption across private consumption and leisure. To aggregate individual-level variables, note that the size of each living cohort v is

a fraction β exp{β(v − t)} of total population. Hence, aggregating a generic individual Rt variable x(t, v) amounts to finding x(t) = −∞ x(v, t)β exp{β(v − t)}dv. For financial

wealth, aggregating A(v, t) and taking its time derivative delivers the aggregate version of 10

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the flow budget constraint (10): ˙ = rA(t) + w(t) − T (t) − X(t). A(t)

(14)

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Notice how the extra return component β washes out in the aggregate, as it merely represents transfers of financial wealth between generations. Following the same procedure for

˙ β(α + β)A(t) X(t) =r−α− , X(t) X(t)

(15)

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full consumption gives the aggregate Euler equation

which augments (11) with a ‘generational turnover’ effect (represented by the last term). Thus, an economy with positive aggregate financial wealth (i.e., A(t) > 0) in steady state ˙ (i.e., with X(t) = 0) implies r > α, which in turn yields rising consumption profiles

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at the individual level. Equations (12) and (13) keep the same form in the aggregate:

2.3. Public Sector

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C(t) = εC X(t) and w(t) [1 − L(t)] = (1 − εC )X(t).

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The government spends on public capital investment, IG (t), and public consumption, CG (t). To finance its spending, the government levies a lump-sum tax on households, T (t),

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˙ and issues government bonds, B(t). The government’s flow budget constraint is ˙ B(t) = rB(t) + IG (t) + CG (t) − T (t),

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(16)

where B(t) denotes the stock of public debt outstanding at time t. Note that, in order to focus on the productivity spillovers of public capital, I abstract from spillover effects of government spending on the consumption side and from market distortions that would potentially arise under alternative tax instruments.8 Imposing the no-Ponzi game condition 8

Bom and Ligthart (2014a) consider the case of proportional labor income taxes.

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limτ →∞ B(τ ) exp{−r(τ − t)} = 0 in (16) gives the government’s intertemporal budget constraint

t

∞

[T (τ ) − IG (τ ) − CG (τ )] exp{−r(τ − t)}dτ,

(17)

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B(t) =

Z

which requires the path of lump-sum taxes to be such that the present discounted value of the infinite stream of primary balances covers the current stock of public debt.

I assume the government satisfies the intertemporal constraint (17) by adjusting lump-

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sum taxes in a discrete fashion, given the current stock of public debt and the chosen paths of the spending components. Moreover, the government can choose the implementation timing of any required tax adjustment. In general, lump-sum taxes are adjusted k ≥ 0 periods after a spending shock. As a baseline financing scenario, I assume instantaneous tax

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adjustments (i.e., k = 0), in which case the government’s budget is continuously balanced. As an alternative scenario, the government resorts to debt-financing in order to delay the

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tax adjustment by k > 0 periods.

Similar to private capital, government capital accumulates according to

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    IG (t) ˙ KG (t) = ΦG − δG KG (t), KG (t)

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(18)

where ΦG (·) is a strictly concave installation function of public capital—satisfying the same

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properties as Φ(·) in (4)—and δG is the rate of depreciation of public capital.

2.4. Foreign Sector and Market Equilibrium Financial capital moves unrestrictedly across borders. Denoting net exports by Z(t), net

foreign assets, F (t), evolves according to F˙ (t) = rF (t) + Z(t). I assume that the available assets—i.e., shares of domestic firms, government bonds, and international bonds—are perfect substitutes in the household’s portfolio. Hence, financial equilibrium amounts to 12

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A(t) = B(t) + V (t) + F (t), where V (t) ≡ q(t)K(t) denotes the stock market value of firms. I abstract from nominal and real rigidities in the labor and goods markets. The goods

3. Long-Run Effects of Public Investment

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market thus clears at all instants of time: Y (t) = C(t) + CG (t) + I(t) + IG (t) + Z(t).

As argued in Section 4.1, the model is characterized by a unique and locally-stable steady state. Before discussing the dynamics of the model, however, this section studies

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analytically the marginal long-run level effects of a public investment increase (dIG > 0) financed by lump-sum taxes. Because the analysis amounts to comparative statics, I drop time indices and use the subscripts 0 and 1 to denote variables in the initial and new steady states, respectively. For intuition, I complement the analytical results with a graphical

3.1. Full Consumption and Labor

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illustration of the long-run forces at work (see Figure 1).

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Let us start by the long-run effect of public investment on full consumption. Consider the consumption-saving subsystem defined by equations (14) and (15) in steady state—

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˙ ˙ i.e., with A(t) = 0 and X(t) = 0 imposed. Dropping time indices and solving for X, the

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resulting equations are:

(19)

X =

(20)

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X = rA + w − T,

rωX β(α + β) A= A, r−α ωA

where ωX ≡ X0 /Y0 and ωA ≡ rA0 /Y0 are steady-state shares. Steady-state conditions (19) and (20) can be depicted as straight, positively-sloped lines in the X–A space, conditional on w and T (see Panel (a) of Figure 1). Denote these lines by A˙ = 0 and X˙ = 0, respectively. The X˙ = 0 line is determined solely by fixed parameters and is thus unaffected by public investment. The position of the A˙ = 0 line, however, depends on w and T , which are 13

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affected by public investment; an increase in w shifts A˙ = 0 up, while an increase in T shifts it down. For w = w0 and T = T0 , its initial position is at A˙ = 0|0 . Note that w is determined in the labor market (discussed below), which is depicted in Panel (b); the two

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panels are therefore connected. Solving the steady-state conditions A˙ = 0 and X˙ = 0 leads to an expression for full consumption as linear function of w − T . Differentiating this function with respect to IG gives one equation of a linear system of long-run multipliers (see Appendix A). By solving

  I ωX θG (1 + ωLL ) − ωG dX = I , dIG ωG θL (1 + ωLL ) − ωT

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this system for dX/dIG one arrives at the reduced-form multiplier

(21)

where ωLL ≡ L0 /(1 − L0 ) is the leisure-labor ratio (and also the Frisch elasticity of labor

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supply), θL ≡ w0 L0 /Y0 is the labor share of aggregate income, θG ≡ YG0 KG0 /Y0 is the I output elasticity of public capital, ωG ≡ IG0 /Y0 is the public investment-to-GDP ratio,

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and ωT ≡ T0 /Y0 is the tax revenues-to-GDP ratio. Noting that (1 + ωLL )θL − ωT = ωX − ωA > 0, public investment raises full consumption in the long run if and only if

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I dw/dIG = θG (1 + ωLL )/ωG > 1—i.e., if and only if the long-run increase in wages more

than compensates the long-run tax raise needed to finance the public investment increment.

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This case is represented in Panel (a) of Figure 1 by an upward shift of the A˙ = 0|0 line.

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Let us now turn to the labor market, which is described by labor demand and labor supply:

  σ1 Y Y , w = EL L X w = (1 − εC ) , 1−L σY −1 σY

(22) (23)

where the former corresponds to (7) and the latter obtains from aggregating (13). Panel (b) of Figure 1 depicts the two curves in the w–L space as Ld and Ls , respectively. The 14

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Ld curve slopes downward and stands initially at Ld0 , whereas the Ls curve slopes upward and is initially at Ls0 . An increase in public investment—and thus in the stock of public capital— affects the labor market directly through EL , and indirectly through X and K

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(which is embedded in Y ). The effect of X on L reflects the wealth effect in labor supply and connects the labor market with the consumption-saving subsystem depicted in Panel (a). The dependency of L on K follows from factor complementarity in production and links the labor market with the capital market represented in Panel (c) (discussed below).

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Equations (22) and (23) implicitly define the equilibrium level of labor employment as L = L(K, KG , X). By fully differentiating this function with respect to IG and solving for dL/dIG (see Appendix A) one finds

  I dL ωLL − θG ωT θL ωG = , I dIG θL (1 + ωLL ) − ωT yK θL ωG

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(24)

where yK ≡ Y0 /K0 is the average productivity of capital in the initial steady state. Note

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that the long-run labor multiplier depends neither on the elasticity of substitution nor on the particular type of factor augmentation, only on the overall output elasticity of public

PT

capital. Proposition 1 summarizes the signs of the long-run effects of public investment on

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full consumption and labor.

Proposition 1 (Signs of Full Consumption and Labor Multipliers). Define θG ≡ ωT

AC

and θG ≡

I θL ωG

I ωG 1+ωLL

≥0

≥ 0. Then, θG < θG , and the long-run multipliers of full consumption and

labor, given by (21) and (24), can be signed as follows:           

dX(∞) dIG

≤ 0 and

dL(∞) dIG

> 0,

dX(∞) dIG

>0

and

dL(∞) dIG

≥ 0, if θG < θG ≤ θG ,

dX(∞) dIG

>0

and

dL(∞) dIG

< 0,

Proof. That

dX(∞) dIG

if θG ≤ θG , if θG > θG . ωI

Q 0 if and only if θG Q 1+ωGLL ≡ θG and 15

dL(∞) dIG

R 0 if and only if

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θG Q

I θL ωG ωT

≡ θG follows directly from the numerators of the bracketed fractions of (21)

and (24), after noting that θL (1 + ωLL ) > ωT . This last inequality can also be rearranged as

I ωG 1+ωLL

<

I θ L ωG , ωT

so that θG < θG .

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For a graphical interpretation of Proposition 1, notice that the public capital externality is split into small, moderate, and large values. For values of θG smaller than θG , Ld0 shifts to LdS , whereas the A˙ = 0|0 schedule moves down; because the wealth effect is negative, the Ls0 curve moves rightward to LsS . The full consumption multiplier is then negative,

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while the labor multiplier is positive. For values of θG in the moderate range (θG , θG ), Ld0 moves to LdM in Panel (b) of Figure 1. In Panel (a), wages increase by more than taxes, so that A˙ = 0|0 line shifts up to A˙ = 0|M , which in turn triggers a wealth effect that shifts Ls0 leftward to LsM . As a result, the long-run full consumption and labor effects are both

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positive. Finally, for large values of θG —argued below as the empirically relevant case (see Section 4.2)—labor demand shifts to Ld1 and the A˙ = 0|0 line jumps to A˙ = 0|1 , causing a

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strong wealth effect on labor supply to Ls1 . The long-run effect on full consumption is then

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positive but the labor multiplier is negative. 3.2. Long-Run Crowding Out

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To find the long-run effect of public investment on private capital, consider the steady˙ state conditions obtained by imposing K(t) = 0 in (4) and q(t) ˙ = 0 in (6):

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1 = q0 , Φ0 (Φ−1 (δ)) " #   σ1 σY −1 Y 1 Y I σ q = EK Y − . r K K q =

(25) (26)

Panel (c) of Figure 1 depicts the two conditions in the q–K space as K˙ = 0 and q˙ = 0, respectively. The former is represented by an horizontal line at the unique steady-state value of q0 , whereas the latter gives rise to a downward-sloping curve. The capital market 16

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is initially at E0 , the crossing point of q˙ = 0|0 and K˙ = 0, with an equilibrium stock of capital of K = K0 . Public investment affects long-run private capital directly through EK and indirectly through L (which is embedded in Y ), both of which affect the q˙ = 0 curve.

1 dK = I dIG yK θL ωG

  θG − ηK (1 − σY ) + ωLL

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As derived in Appendix A, the reduced-form private capital multiplier reads I θL ωG − θG ωT θL (1 + ωLL ) − ωT



.

(27)

The combined first two terms within brackets (i.e., θG − ηK (1 − σY )) capture the direct

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effect of public capital on private capital productivity, whereas the last term captures the indirect productivity effect of public capital through changes in labor employment. While the indirect labor effect has the same sign as the labor multiplier (determined in Proposition 1), the sign of the direct productivity effect critically depends on the technology parameters

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θG and ηK (i.e., size and factor-augmentation type of the public capital spillover), and σY (i.e., elasticity of substitution between private capital and labor). In particular, the private

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capital multiplier can be negative for relatively large values of ηK and small values of σY . Proposition 2 gives the necessary and sufficient conditions under which public investment

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crowds out private capital in the long run.

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Proposition 2 (Long-Run Crowding Out). Define the threshold values

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σY

  I ωLL θL ωG − θG ωT ≡ θL − Q 0, ηK θL (1 + ωLL ) − ωT I ωLL θL ωG ≡ ≥ 0. ωLL θK ωT + θL [θL (1 + ωLL ) − ωT ]

ηK

Then, the long-run multiplier of private capital, given by (27), is strictly negative if and only if σY < σY . A necessary (yet not sufficient) condition for this is that ηK > ηK . Proof. The necessity and sufficiency of σY < σY follows trivially from requiring that the expression in curly brackets of (27) be strictly negative. To show the necessity of ηK > ηK , 17

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note that σY ≥ 0; σY < σY can only hold, therefore, if σY > 0. Solving this inequality for ηK delivers ηK > ηK . Proposition 2 states that public investment crowds out private capital in the long

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run if private capital and labor are strong complements in production (‘small’ σY ) and if public capital sufficiently augments private capital (‘large’ ηK ). In this case, public capital augments private capital but, because of the strong complementarity with labor, decreases its marginal productivity. Intuitively, a larger stock of public capital causes

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an overabundance of private capital relative to labor, to which firms optimally respond by cutting on private investment. This case is depicted with solid lines in Panel (c) of Figure 1. Conditional on the initial level of labor employment (i.e., L = L0 ), public capital decreases the marginal productivity of private capital, shifting the q˙ = 0|0 curve to the

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left. The long-run effect in labor employment may weaken (if positive) or reinforce (if negative) the leftward shift of the q˙ = 0|0 curve. In any case, under the conditions of

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Proposition 2, q˙ = 0|0 moves to, say, q˙ = 0|1 , with a lower long-run stock of private capital. If public capital is instead labor-augmenting or insufficiently productive—even if capital-

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augmenting—then the q˙ = 0|0 curves moves to, say, q˙ = 0|H , giving rise to a positive

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private capital multiplier.

4. Transitional Effects of a Public Investment Shock

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This section studies the transitional dynamics of a permanent public investment shock

in the log-linearized model. Section 4.1 describes the log-linearization procedure and the solution of the model. Section 4.2 lays out the calibration strategy and reports the parameter values. Finally, Section 4.3 discusses the results of the numerical simulations.

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4.1. Log-Linearization and Model Solution I log-linearize the model around its initial steady state. Hereafter, I denote the initial steady-state value of a generic variable x(t) by x. The log-linearized variables are defined,

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in most cases, as x˜(t) ≡ dx(t)/x and x˜˙ (t) ≡ dx(t)/x ˙ = x(t)/x. ˙ Exceptions include the various types of financial assets—i.e., A(t), B(t), and F (t)—whose log-linear versions are defined as z˜(t) ≡ rdz(t)/Y and z˜˙ (t) ≡ rdz(t)/Y ˙ , for z = {A, B, F }; and lump-sum taxes, which are log-linearized as T˜(t) ≡ dT (t)/Y . The complete log-linearized model is presented

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in Appendix B.

I consider a permanent and unanticipated impulse to public investment occurring at time t = 0—i.e., I˜G (t) = I˜G for all t ≥ 0. Public consumption is kept fixed at its initial level, so that C˜G (t) = 0. As discussed in Section 2.3, the government is allowed to delay

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I ˜ T˜(t) = exp{kr}ωG IG ,

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the lump-sum tax raise by k ≥ 0 periods, after which taxes are adjusted according to

I where ωG ≡ I/Y denotes the public investment-to-GDP ratio.9

(28)

Note that the bal-

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˜ G (t) = anced budget case obtains for k = 0. Public capital accumulates according to K

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(1 − exp{−σG t})I˜G , where σG ≡ IG Φ0G (·)/KG > 0 is the elasticity of the public capital installation function. The economy approaches the new steady state as t → ∞.

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To solve the log-linearized model, I first combine equations (B.7), (B.8), and (B.11) (see ˜ Appendix B) in a static system for Y˜ (t), L(t), and w(t), ˜ conditional on the state variables 9

See Section 3.2 of the Technical Appendix for additional details.

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˜ ˜ G (t), and X(t). ˜ K(t), K The quasi-reduced form solution of this system can be written as   ˜  Y (t)         L(t)  ˜  = Υ    w(t) ˜

 ˜ K(t)   , ˜ X(t)   ˜ KG (t)

(29)

where I have defined the coefficient matrix  θK (σY + ωLL ) −θL ωLL σY   θK ωLL −σY ωLL   θK ωLL θK



θG (σY + ωLL ) − (1 − σY )ωLL θL ηL    ωLL [θG − ηL (1 − σY )]   θG − ηL (1 − σY )

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1 Υ≡ σY + ωLL θK



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

Denote the typical element of Υ by υij , for i = {y, l, w} and j = {k, x, g}.

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The next step consists of setting up a dynamic system for the state variables of the model. Using equations (B.1)–(B.4) in conjunction with (B.9) and (28), this system can

rωI σA ωA

0

0

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  ˙ ˜  K(t)       q˜˙(t)      =     X(t) ˙    ˜    ˙ ˜ A(t)

r

K − σrθ υyx Y ωA

0

0

0

r−α

− r−α ωA

rωw υwk

0

r(ωw υwx − ωX )

r

0

rθK σ Y ωA

(1 − υyk )

  ˜   K(t)   0      q˜(t)   γq (t)     −   0   X(t)  ˜      ˜ A(t) γA (t) 



    ,   

(30)

AC

CE



ED

be written as

where ωI ≡ I/Y is the ratio of private investment to GDP, σA ≡ −(I/K)[Φ00 (·)/Φ0 (·)] is the elasticity of the marginal installation function of private capital, ωA ≡ rA/Y denotes

the ratio of income from financial assets to GDP, ωw ≡ w/Y is the ratio of wages to GDP, and ωX ≡ X/Y is the ratio of full consumption to GDP. The terms υyk and υyx correspond to the elements in the first row, first and second columns of Υ. Similarly, υwk and υwx 20

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correspond to the third row, first and second columns of Υ. Finally, γq (t) and γA (t) denote the shock terms through which the public investment impulse affects the system: rθK [υyg − (1 − σY )ηK ] (1 − exp{−σG t})I˜G , σY ωA   I ˜ γA (t) ≡ −r ωw υwg (1 − exp{−σG t}) − exp{kr}ωG IG ,

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γq (t) ≡

where υyg and υwg correspond to the elements in the last column, first and third rows of Υ.

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The stability properties of the system are governed by the eigenvalues of the 4 × 4 Jacobian matrix on the right hand-side of (30). The trace of this matrix is 3r − α > 0, while its determinant is strictly positive for r > α and w > T .10 Hence, the Jacobian matrix possesses either four positive eigenvalues, or two negative and two positive eigenvalues (four

M

negative eigenvalues are ruled out by the positive trace). Plausible calibrations of the model yield the latter case, with two negative and two positive real eigenvalues (see Section 4.2),

ED

implying a unique and locally saddle-path stable steady state.11 Two special cases of (30) are worth noting. First, the system decouples into two recursive subsystems when labor

PT

supply is exogenous. In this case ωLL = 0, so that υyx = 0; the capital market subsystem ˜˙ ˜˙ K(t)– q˜˙(t) can then be solved independently of the consumption-saving subsystem X(t)–

CE

˜˙ A(t). Second, the system has a zero eigenvalue if r = α (in which case the third row of the Jacobian matrix only consists of zeros), which gives rise to a hysteretic steady state.

AC

Solving the log-linearized model amounts to: (i) finding the impulse response functions

˜ ˜ ˜ in the dynamic system (30) to the public of the state variables K(t), q˜(t), X(t), and A(t)

investment shock; (ii) using these impulse response functions to recover those of the static 3

K θL In particular, the determinant of the Jacobian matrix is (r − α) (ωw − ωT ) σA ω3r (σωYI θ+ω > 0 (see LL θK ) A Section 3.5 of the Technical Appendix). 11 Bom and Ligthart (2014a) show that distortionary taxation can lead to complex eigenvalues, in which case the dynamic responses to the public investment shock are cyclical. This possibility does not arise for plausible parameter values under lump-sum taxes.

10

21

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˜ and I(t)) ˜ system (29); and (iii) obtaining the solution for the remaining variables (e.g., C(t) using the relevant log-linearized expressions in Appendix B. The analytical derivation of the impulse response functions is provided in Section 4 of the Technical Appendix. Here,

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I focus on the numerical simulations of the model. 4.2. Parameter Values

I calibrate the model for an average small open economy in the euro area.12 The calibration proceeds in three steps. First, I choose the values of key aggregate ratios in

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order to match the corresponding euro area averages over the period 1995-2015. These values include the GDP expenditure shares, private and public capital stocks, public debt, net foreign assets—all relative to GDP—and the depreciation rates of private and public capital. Next, I select the values for a few structural parameters—which are not directly

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matchable to data—based on the empirical literature. The remaining parameters are finally pinned down by the steady state conditions. Table 1 reports the main parameter values.

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The GDP expenditure shares of the average economy in the euro area for the period 1995-2015 are ωC = 0.565 for private consumption, ωI = 0.183 for private investment,

PT

C I ωG = 0.201 for government consumption, ωG = 0.031 for government investment, and

ωZ = 0.019 for net exports. The average ratios of private and public capital to GDP

CE

are K/Y = 2.388 and KG /Y = 0.588, whereas the depreciation rates implicit in the consumption of fixed capital are δ = 0.064 and δG = 0.044.13 The public debt-to-GDP

AC

ratio is set at B/Y = 0.675, the euro area average for the period 2000-2008.14 12

The data used are mostly from the European Commission’s AMECO database. The only exceptions are the data for public capital stocks, which come from the IMF Investment and Capital Stock Dataset, 2017 (see IMF, 2017); and the data for net foreign investment positions, which are obtained from the Eurostat. 13 AMECO’s database contains information only on the capital stock of the whole economy. To find the private capital stock, I first subtract the average stock of public capital over the same period using the IMF’s data. 14 Public debt increased considerably in the euro area countries after the 2008 financial crisis. Because such high ratios mostly reflect transitional public debt adjustments, I exclude the recent years from the calculated average.

22

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Next, I select the values of the birth/death rate, the leisure-labor ratio (also the Frisch elasticity of labor supply), the output elasticity of public capital, and the elasticity of substitution between private capital and labor. The birth/death rate is fixed at β = 0.018,

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reflecting an average individual working life of 55 years. Concerning the baseline Frisch elasticity of labor supply, I follow the recommendation by Chetty et al. (2011, 2012) of calibrating macro models with the value ωLL = 0.75. The empirical magnitude of this parameter has been subject to intense dispute, however. Whereas microeconometric

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studies usually find smaller elasticities, macroeconomic models often employ values of two or higher. Hence, I also consider the alternative values of ωLL = 0 and ωLL = 2. For the baseline output elasticity of public capital, I use Bom and Ligthart’s (2014b) finding of θG = 0.131 for ‘core’ public capital. I later check the sensitivity of the results to the alternative values of θG = 0.05 and θG = 0.2. Regarding the elasticity of substitution

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between private capital and labor, Chirinko (2008) argues that the bulk of the empirical

ED

evidence suggests a value in the range 0.4–0.6. I take the midpoint of this interval, σY = 0.5, as the baseline calibration value, and later conduct sensitivity analysis to the alternative

PT

values of σY = 0.4 and σY = 0.6.

The values of the remaining parameters are implied by the steady-state conditions,

CE

after normalizing initial output to one. Following Bom and Ligthart (2014a), I specify the private capital installation function as Φ(x) ≡ z¯ [ln(¯ z + x) − ln(¯ z )]. Because Φ(I/K) = δ

AC

in steady state, the value of z¯ = 0.188 follows from imposing I/K = 0.077 and δ = 0.064. Similarly, the public capital installation function is ΦG (x) ≡ z¯G [ln(¯ zG + x) − ln(¯ zG )]; for

IG /KG = 0.053 and δG = 0.044, it follows that z¯G = 0.116. This implies, via (8), a steadystate Tobin’s q value of q = z¯/(I/K + z¯) = 1.409. The world interest rate is implied by

the steady-state condition for net foreign assets: r = −Z/F = 0.052. The steady-state capital share of GDP is thus θK = (rqK + I)/Y = 0.358, which implies a labor share of θL = 1 − θK = 0.642. 23

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The baseline leisure-labor ratio determines steady-state labor employment at L = 1/(1 + ωLL ) = 0.571, which then implies a gross wage rate of w = θL Y /L = 1.125. From the government budget constraint (16), the tax revenues-to-GDP ratio is ωT =

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C I rB/Y + ωG + ωG = 0.268. The initial stock of financial assets is A = qK + B + F = 3.669,

which requires steady-state full consumption to be X = rA + w − T = 1.046. The pure rate of time preference is then determined by (20) as α = (rX − β 2 A)/(X + βA) = 0.048 < r. The leisure share of full consumption follows from the household’s first-order condition

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(13): 1 − εC = w(1 − L)/X = 0.461. The consumption share of full consumption is then εC = 0.539. The implied elasticities of the private and public capital installation functions are σA = (I/K)/(I/K + z¯) = 0.290 and σG = (¯ zG IG /KG )/(¯ zG + IG /KG ) = 0.037. In the baseline case where public capital is capital-augmenting, ηK = θG /θK = 0.366. But I also

case ηL = θG /θL = 0.204.

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consider the alternative technology scenario of labor-augmenting public capital, in which

ED

For the baseline parameter values, the threshold values defined in Proposition 1 are θG = 0.018 and θG = 0.075. Hence, the baseline output elasticity of public capital θG =

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0.131 implies a positive long-run effect of public investment on full consumption and a negative long-run effect on labor employment. In terms of Proposition 2, one finds σY =

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0.678 and ηK = 0.024. The baseline values σY = 0.5 < σY and ηK = 0.366 > ηK thus imply long-run crowding out of private capital. The same result applies to the alternative

AC

values considered. Concerning the stability properties of the model, the eigenvalues of the Jacobian matrix of (30) are −0.014, −0.094, 0.145, and 0.071, so that the system is saddle-path stable. 4.3. Simulation Results This section uses the parameterized model to simulate numerically the dynamic macroeconomic responses—in particular, the response of private capital formation—to a 10% increase in public investment. Section 4.3.1 discusses the baseline results. Section 4.3.2 24

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considers debt-financing. Section 4.3.3 conducts a sensitivity analysis to key parameters. 4.3.1. Baseline Model Figure 2 reports the impulse responses to a permanent public investment shock in

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the baseline parameterization (solid lines), varying only the labor supply elasticity to the alternative values of ωLL = 2 (dashed lines) and ωLL = 0 (dotted lines). For intuition, I complement the numerical results with the graphical illustration in Figure 3.

In the baseline model, Tobin’s q strongly falls on impact but recovers in about ten

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years. After a short-lived rise above its initial level, it falls again and approaches its initial level from below. As a consequence, private investment is crowded out during the whole transitional process. It partially recovers from its initial decline during the first 25 years, but falls again later towards its lower long-run level. The stock of private capital thus

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exhibits a non-monotonic transitional path, falling in the first 10 years after the shock, slightly recovering for about 20 years, but then falling again towards its lower long-run

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equilibrium. Panel (a) of Figure 3 illustrates the transitional dynamics of Tobin’s q and private capital. Starting at E, the economy jumps down to E0 on impact and approaches

PT

E∞ in the long run, after completing the dynamic path represented by the dotted arrow. That q˙ = 0 shifts to the left, crowding out private capital in the long run, follows from the

CE

conditions stated in Proposition 2, which hold for baseline parameter values (see Section 4.2). Understanding the impact effect and complex transitional dynamics towards the new

AC

steady state requires analyzing the response of labor employment, however. To understand the short-run effect, first note that Tobin’s q would fall on impact

even if labor supply were exogenous (see below), in view of the lower long-run marginal productivity of capital. In the case of endogenous labor, however, a negative wealth effect on labor supply exacerbates the impact drop in Tobin’s q and, therefore, the short-run crowding-out effect. The wealth effect arises as forward-looking households anticipate 25

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higher wages from gains in labor productivity. In Panel (b) of Figure 3, this wealth effect moves the economy from E to E0 at the time of the shock. The impact increase in full consumption prompts households to increase goods consumption and cut on labor supply.

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In panel (c), the labor supply curve Ls shifts to Ls (0), reducing labor and increasing wages. Because private capital is predetermined, the capital-labor ratio rises and private capital productivity falls, which lowers Tobin’s q and reduces private capital formation.

Over time, as anticipated, the stock of public capital gradually builds up, increasing

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labor productivity and wages. In Panel (c) of Figure 3, the labor demand curve Ld starts a long, continuous shift to the right towards Ld (∞), inducing along the way both an intertemporal substitution effect and a wealth effect in labor supply. In a first transitional phase, the intertemporal substitution effect dominates, so that labor employment expands and the capital-labor ratio falls, which stimulates Tobin’s q and private capital formation.

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This phase comes to a halt about 50 years after the shock when, with high employment and

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wages, the wealth effect becomes relatively more important. In Panel (c), the leftward shift of the labor supply curve dominates the rightward shift of the labor demand curve, which

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bends the dotted transitional path backwards. From this point onwards labor employment falls towards its new steady-state level, dragging private capital formation with it.

CE

The importance of endogenous labor in the private capital response to public investment becomes apparent when the labor supply elasticity is changed. Increasing the labor supply

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elasticity to ωLL = 2, for instance, magnifies both the wealth effect and the intertemporal substitution effect. In Figure 2 (dashed lines), private investment is then more strongly crowded out on impact, but also recovers much faster, raising the private capital stock substantially above its initial level for a long transitional period. Exogenous labor supply (dotted lines), on the other hand, switches off both wealth and intertemporal labor supply effects, so that the crowding-out effect on private capital formation is weaker on impact, but also more persistent over time. In short, the labor dynamics give rise to the non-monotonic 26

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private capital response, which is exacerbated for larger labor supply elasticities. 4.3.2. Allowing for Public Debt In order to mitigate the initial short-run contraction in private capital and labor employ-

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ment, the government may choose to keep lump-sum taxes temporarily fixed and finance the public investment impulse with public debt. This section studies how this alternative mode of financing affects the transitional effects of public investment in the baseline model. In particular, the public investment increment is financed entirely with public debt

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for k years, after which lump-sum taxes are permanently increased so as to ensure government solvency. Figure 4 shows the impulse response functions for the cases of k = 10 (dashed lines) and k = 20 (dotted lines), and compares them to the baseline case of a continuously-balanced budget (i.e., k = 0; solid lines).

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Delaying the lump-sum tax increase lowers the transitional response of full consumption. Debt-financing the public investment impulse causes a negative wealth effect because it

ED

later implies a larger lump-sum tax raise, which increases with k.15 By lowering the full consumption response, the tax delay increases labor supply along the transitional

PT

path, which reduces the negative response of Tobin’s q and, hence, the crowding-out effect on private capital accumulation. In the case of a ten-year delay (i.e., k = 10), labor

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employment, Tobin’s q, and private capital accumulation still drop on impact but recover quickly, giving rise to a transitional crowding-in period. A longer tax-adjustment delay of

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20 years (i.e., k = 20) totally eliminates the short-run crowding-out effect and significantly

extends the crowding-in transitional phase. Hence, debt-financing the public investment impulse can change both the size and the sign of the transitional effect on private capital formation. In the long run, however, the crowding-out effect on private capital remains. 15

Note that, since taxes are non-distortionary, this wealth effect is the only operative channel whereby public debt matters in the model.

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4.3.3. Alternative Parameters This section examines the sensitivity of the simulated impulse responses to changes in the values of key model parameters. Panel (a) of Figure 5 considers small deviations

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of the baseline elasticity of substitution σY = 0.5 (solid lines) to the alternative values σY = 0.4 (dashed lines) and σY = 0.6 (dotted lines). Although both values are still below σY = 0.678—so that public investment is known beforehand to crowd out private capital in the long run—the transitional dynamics of private investment and private capital are

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qualitatively very different. Not surprisingly, the value σY = 0.4 gives rise to stronger transitional crowding out. A small increase to σY = 0.6, however, suffices to generate a long transitional crowding-in effect on private investment. Despite the crowding-out effect in the short and long run, private capital formation rises above its initial level for a long transitional phase in between. This result highlights the quantitative importance of the

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elasticity of substitution in shaping the transitional effects of public investment on private

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capital information.

Panel (b) of Figure 5 studies how different assumptions about the factor-augmenting

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role of public capital affect the dynamics of private capital formation. In particular, it compares the baseline case of capital-augmentation (i.e., ηK > ηL = 0; solid lines) with

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the alternatives scenarios of factor-neutrality (i.e., ηL = ηK ; dashed lines) and laboraugmentation (i.e., ηL > ηK = 0; dotted lines). The results are markedly different.

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Whereas private capital formation is crowded out during the entire transition between steady states in the baseline case, it is totally crowded in if public capital is instead factorneutral or labor-augmenting. Moreover, these alternatives scenarios generate qualitatively similar results, differing mainly in the magnitude of the responses. Identifying the factoraugmenting role of public capital is thus pivotal to understanding the relationship between public and private investment. Finally, Panel (c) of Figure 5 varies the baseline output elasticity of public capital 28

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θG = 0.131 (solid lines) to the alternative values of θG = 0.05 (dashed lines) and θG = 0.20 (dotted lines). Note that, while the latter alternative value is larger than the upper limit θG = 0.075—like in the baseline case—the former falls within the intermediate range (θG ,

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θG )=(0.018, 0.075), which switches the long-run labor multiplier to positive. Also, the corresponding values of ηK (0.140 and 0.559) both exceed ηK = 0.024, so that the private capital multiplier remains negative. Despite the qualitatively similar long-run effects of public investment on private capital formation, the transitional dynamics are substantially

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different on both quantitative and qualitative grounds. In particular, although the higher elasticity θG = 0.20 exacerbates the crowding-out effect, a slightly smaller elasticity of θG = 0.05 gives rise to moderate crowding-in effects in the first decades of transition. The public-private investment relationship is therefore also sensitive to the magnitude of this

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5. Concluding Remarks

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parameter.

This paper investigates the dynamic effects of public investment on private capital for-

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mation in a general equilibrium macroeconomic model of a small open economy. The model allows for factor-biased public capital by combining asymmetric factor-augmentation with

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a constant—but in general different than one—elasticity of substitution between private inputs. I show that a permanent impulse to public investment crowds out private capital

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in the long run when public capital directly augments private capital and the elasticity of substitution is smaller than one. This case arises for plausible calibrations of the model. This paper also shows that the dynamics of private capital formation to a public in-

vestment shock are rather complex, falling in the short and long run, but potentially rising during transition between steady states. This transitional crowding-in effect can occur if public investment is tax-financed but is especially likely in the case of debt-financing. The crucial element for this transitional non-monotonicity lies in the dynamics of labor em29

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ployment, which interplays with private capital via factor complementarity in production. The dynamic effects of public investment on private capital formation are therefore very sensitive to several key parameters for which little empirical consensus exists, namely the

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elasticity of substitution between private capital and labor, the elasticity of labor supply, and the output elasticity of public capital. Small variations within plausible ranges of values give rise to rather different, both quantitatively and qualitatively, dynamics of private capital formation. The complex dynamics of private capital formation, as well as its

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sensitivity to key parameters and financing mode, may help explain the mixed empirical evidence on the private-public investment relationship.

The results in this paper have important policy implications. Public investment is usually regarded among policymakers as a potent fiscal policy tool to boost long-run private investment and growth, especially when directed towards core public capital items that

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directly complement private capital. The present paper shows that this argument misses

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important general equilibrium considerations, however, most notably the dynamic interplay between private capital and labor. Limited by a small degree of technical substitutability

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between factors, firms are more likely to save on capital investment when faced with an increased stock of public capital. To boost long-run private investment, the government

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should instead favor those components of public investment that are more likely to augment labor, such as education, science, technology and innovation.

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Capital-augmenting public investment, on the other hand, may play an important role in addressing wealth inequality issues. By favoring labor productivity and wages at the expense of capital productivity, this type of public capital disproportionately benefits wage earners relative to capital owners. Conversely, decreasing core public investment may promote wealth inequality. This observation suggests a link between the general decline in public investment-to-GDP ratios—observed in most advanced economies over the last few decades—the rise in wealth inequality, and the fall in the labor share of income. I intend 30

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to study this link in future research. Acknowledgement

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This paper greatly benefited from discussions with Ben Heijdra and Jenny Ligthart. I would also like to thank Pedro Gomes, Heiko Rachinger, and an anonymous referee for useful comments and suggestions. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Appendix A. Reduced-Form Long-Run Multipliers

This section derives the reduced-form long-run multipliers of public investment. Fully differentiating (7) with respect to public investment gives an expression for the long-run

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change in the wage rate:

(A.1)

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YL θK dK YL [θG − ηL (1 − σY )] dKG YL θK dL dw = + − . dIG σY K dIG σY KG dIG σY L dIG

Solving the steady-state conditions A˙ = 0 and X˙ = 0 gives steady-state full consump-

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tion as linear function of w − T . Fully differentiating this function with respect to IG gives

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the quasi-reduced form long-run multiplier of full consumption:

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dX ωX = dIG (1 + ωLL )θL − ωT



 dw −1 . dIG

(A.2)

Denote by L = L(K, KG , X) the equilibrium level of labor implicitly defined by the

steady-state labor demand and labor supply functions, Ld and Ls . A quasi-reduced form ex-

pression for the long-run multiplier of labor obtains from fully differentiating L(K, KG , X) with respect to IG : dL ωLL = dIG ωLL θK + σY

  L0 dK L0 dKG L0 dX θK + [θG − ηL (1 − σY )] − σY . (A.3) K0 dIG KG0 dIG X0 dIG 31

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Because q and I/K are fixed in the long run, so is the marginal productivity of private capital. By totally differentiating YK with respect to K, KG , and L, setting the resulting

dK K0 dKG K0 dL = [θG − ηK (1 − σY )] + . dIG θL KG0 dIG L0 dIG

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expression to zero, and then solving for dK/dIG , one arrives at

(A.4)

Noting that the long-run change in the capital stock is a multiple of the public investment change—i.e., dKG /dIG = KG0 /IG0 = 1/Φ−1 (δG )— equations (A.1)–(A.4) form a

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linear system of four equations in four unknowns: dw/dIG , dK/dIG , dL/dIG , and dX/dIG . The solution of this system for dw/dIG is

(A.5)

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θG (1 + ωLL ) dw = . I dIG ωG

The solutions for dX/dIG , dL/dIG , and dK/dIG correspond to (21), (24), and (27) in the

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main text.

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Appendix B. Log-Linearized Model The model is log-linearized around its initial steady state. For most variables, I define

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x˜(t) ≡ dx(t)/x and x˜˙ (t) ≡ dx(t)/x. ˙ The exceptions are A(t), B(t), and F (t), which are defined as z˜(t) ≡ rdz(t)/Y and z˜˙ (t) ≡ rdz(t)/Y ˙ , for z = {A, B, F }; and lump-sum taxes,

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which are defined as T˜(t) ≡ dT (t)/Y . Using these notational conventions, the dynamic equations (4), (6), (10), (15), (16) and

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(18) are log-linearized as follows:

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rθK ˜ ˜ − (1 − σY )ηK K ˜ G (t)], [Y (t) − K(t) q˜˙(t) = r˜ q (t) − σY ωA rωI ˜ ˜ ˜˙ [I(t) − K(t)], K(t) = ωhA i ˜˙ ˜ + ωw w(t) ˜ A(t) = r A(t) ˜ − T˜(t) − ωX X(t) , # " ˜ ˜˙ ˜ − A(t) , X(t) = (r − α) X(t) ωA h i ˜˙ ˜ + ω I I˜G (t) + ω C C˜G (t) − T˜(t) , B(t) = r B(t) G G

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˜˙ G (t) = σG [I˜G − K ˜ G (t)], K

(B.1) (B.2) (B.3) (B.4) (B.5) (B.6)

C I ≡ CG /Y , and ≡ IG /Y , ωG where ωA ≡ rA/Y , ωI ≡ I/Y , ωw ≡ w/Y , ωX ≡ X/Y , ωG

σG ≡ IG Φ0G (·)/KG .

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In turn, the log-linearized expressions of the static equations (1), (7), (8), (12), (13),

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and A(t) = q(t)K(t) + F (t) + B(t) are:

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˜ + θL L(t) ˜ + θG K ˜ G (t), Y˜ (t) = θK K(t) i 1 h˜ ˜ − (1 − σY )ηL K ˜ G (t) , Y (t) − L(t) w(t) ˜ = σY ˜ − K(t)], ˜ q˜(t) = σA [I(t)

(B.7) (B.8) (B.9) (B.10)

˜ ˜ L(t) = ωLL [w(t) ˜ − X(t)],

(B.11)

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˜ ˜ C(t) = X(t),

˜ − ωA [˜ ˜ ˜ F˜ (t) = A(t) q (t) + K(t)] − B(t),

(B.12)

where σA ≡ −(I/K)(Φ00 (·)/Φ0 (·)) and ωLL ≡ (1 − L)/L. References Abiad, A., Furceri, D., Topalova, P., 2015. The Macroeconomic Effects of Public Investment: Evidence from Advanced Economies. IMF Working Paper WP/15/95, International Monetary Fund.

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Afonso, A., St. Aubyn, M., 2009. Macroeconomic Rates of Return of Public and Private Investment: Crowding-In and Crowding-Out Effects. The Manchester School 77, 21–39. Barczyk, D., 2016. Ricardian Equivalence Revisited: Deficits, Gifts and Bequests. Journal of Economic Dynamics and Control 63, 1–24.

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Baxter, M., King, R. G., 1993. Fiscal Policy in General Equilibrium. American Economic Review 83, 315–334.

Blanchard, O. J., 1985. Debt, Deficits, and Finite Horizons. Journal of Political Economy 93, 223–247. Bom, P. R., Ligthart, J. E., 2014a. Public Infrastructure Investment, Output Dynamics, and Balanced Budget Fiscal Rules. Journal of Economic Dynamics and Control 40, 334–354.

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Bom, P. R., Ligthart, J. E., 2014b. What Have We Learned from Three Decades of Research on the Productivity of Public Capital? Journal of Economic Surveys 28, 889–916. Bouakez, H., Guillard, M., Roulleau-Pasdeloup, J., 2017. Public Investment, Time to Buid, and the Zero Lower Bound. Review of Economic Dynamics 23, 60–79.

Cavallo, E., aude, C. D., 2011. Public Investment in Developing Countries: A Blessing or a Curse? Journal

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of Comparative Economics 39, 65–81.

Chetty, R., Guren, A., Manoli, D., Weber, A., 2011. Are Micro and Macro Labor Supply Elasticities

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Consistent? A Review of Evidence on the Intensive and Extensive Margins. American Economic Review: Papers and Proceedings 101, 471–475.

Chetty, R., Guren, A., Manoli, D., Weber, A., 2012. Does Indivisible Labor Explain the Difference between

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Micro and Macro Elasticities? A Meta-Analysis of Extensive Margin Elasticities. NBER Macroeconomics Annual 27, 1–56.

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Chirinko, R. S., 2008. σ: The Long and Short of It. Journal of Macroeconomics 30, 671–686. Clancy, D., Jacquinot, P., Lozej, M., 2016. Government Expenditure Composition and Fiscal Policy

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Spillovers in Small Open Economies Within a Monetary Union. Journal of Macroeconomics 48, 305–326. Cooper, R. W., Haltiwanger, J. C., 2006. On the Nature of Capital Adjustment Costs. Review of Economic Studies 73, 611–633.

Dioikitopoulos, E. V., Kalyvitis, S., 2008. Public Capital Maintenance and Congestion: Long-Run Growth and Fiscal Policies. Journal of Economic Dynamics and Control 32, 3760–3779. Eden, M., Kraay, A., 2014. ‘Crowding in’ and the Returns to Government Investment in Low-Income Countries. Policy Research Working Paper No. 6781, World Bank. Fisher, W. H., Turnovsky, S. J., 1998. Public Investment, Congestion, and Private Capital Accumulation.

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Economic Journal 108, 399–413. Heijdra, B. J., Meijdam, L., 2002. Public Investment and Intergenerational Distribution. Journal of Economic Dynamics and Control 26, 707–735. IMF, 2017. Estimating the Stock of Public Capital in 170 Countries, International Monetary Fund.

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Leduc, S., Wilson, D., 2012. Roads to Prosperity or Bridges to Nowhere? Theory and Evidence on the Impact of Public Infrastructure Investment. NBER Macroeconomics Annual 27, 89–142.

Leeper, E. M., Walker, T. B., Yang, S.-C. S., 2010. Government Investment and Fiscal Stimulus. Journal of Monetary Economics 57, 1000–1012.

Perotti, R., 2004. Public Investment: Another (Different) Look. mimeo, Universit`a Bocconi.

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Schmitt-Groh´e, S., Uribe, M., 2003. Closing Small Open Economy Models. Journal of International Economics 61, 163–185.

Turnovsky, S. J., Fisher, W. H., 1995. The Composition of Government Expenditure and Its Consequences for Macroeconomic Performance. Journal of Economic Dynamics and Control 19, 747–786. Voss, G. M., 2002. Public and Private Investment in the United States and Canada. Economic Modelling

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19, 641–664.

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Figure 1: Long-Run Effects of a Public Investment Impulse Panel (a): Consumption-saving subsystem X X˙ = 0

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A˙ = 0|1

E1

X1

A˙ = 0|M

EM

XM

A˙ = 0|0

E0

X0

A˙ = 0|S

AS

A0

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ES

XS

A1

AM

A

Panel (b): Labor market

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w

E1

Ls1

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w1 wM

LsM

ES

E0

Ls0 LsS

L1 L0 LM

LdM LdS Ld0 L

LS

Panel (c): Capital market q

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CE

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wS w0

Ld1

EM

q0

E1

E0

K˙ = 0

EH

q˙ = 0|H q˙ = 0|0 q˙ = 0|1 K1

KH

K0

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K

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Figure 2: Numerical Responses to a Public Investment Shock: Baseline Model w(t) ˜

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q˜(t) 0.15

2.5 ! =0.75 LL

0.1

! LL =2

2

! LL =0

0.05

! =0.75 LL

1.5

! =2 LL

0

! =0 LL

-0.05

1

-0.1 0.5 -0.15

-0.25

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0 -0.2

-0.5

0

10

30

60

100

150

time

˜ I(t) 0.8

0

10

30

60

100

150

time

˜ X(t)

2

! =0.75 LL

0.6

! LL =2 ! LL =0

0.4

1.8 1.6

! =0.75 LL

1.4

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0.2 0 -0.2 -0.4

! LL =2 ! =0

1.2

LL

1 0.8

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0.6

-0.6 -0.8 -1 0

10

30

60

0.4 0.2 0

100

150

0

10

30

60

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time

˜ K(t)

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0.5

˜ L(t) ! =0.75

! =0.75

LL

! LL =0

30

! LL =2 ! LL =0

0.8 0.6 0.4 0.2 0 -0.2 -0.4

-1

10

LL

1

! LL =2

-0.5

0

150

1.2

AC

0

100

time

-0.6 60

100

150

0

time

10

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100

150

time

Notes: The time unit is a year. For each variable, the vertical axis measures the percent deviation from its steady-state value. The public investment shock amounts to I˜G = 0.1. The baseline parameter values are reported in Table 1.

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Figure 3: Transitional Effects of a Capital-Augmenting Public Investment Impulse Panel (a): Capital market

E∞

q

E

E0

K˙ = 0

q˙ = 0

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q(0)

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q(t)

K(∞)

K

K(t)

Panel (b): Consumption-saving system X(t)

X˙ = 0

ED

E0

X(0)

A˙ = 0

E

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X

E∞

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X(∞)

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A(0)A

A(∞)

A(t)

Panel (c): Labor market

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w(t)

Ld

w(∞)

w(0) w

Ld (∞)

E∞

Ls (∞)

Ls (0) Ls

E0

E

L(0) L(∞) L

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L(t)

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Figure 4: Numerical Responses to a Public Investment Shock: Allowing for Debt-Financing q˜(t)

w(t) ˜

0.15 k=0 k=10 k=20

0.1

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2.5

2

1.5

0

1

-0.05

0.5

-0.1

0

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0.05

-0.15

k=0 k=10 k=20

-0.5

0

10

30

60

100

150

time

˜ I(t) 0.5

0

10

30

60

100

150

time

˜ X(t)

2

k=0 k=10 k=20

M

0

1.5

ED

-0.5

-1 0

10

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60

1

k=0 k=10 k=20

0.5

0

-0.5

100

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0

10

30

60

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time

˜ K(t)

0.4

CE

0.2

0

AC

-0.2

100

150

time

˜ L(t) 1 k=0 k=10 k=20

k=0 k=10 k=20

0.8

0.6

0.4

0.2

-0.4

0

-0.6

-0.2

-0.8

0

10

30

-0.4 60

100

150

0

time

10

30

60

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150

time

Notes: The time unit is a year. For each variable, the vertical axis measures the percent deviation from its steady-state value. The public investment shock amounts to I˜G = 0.1. The parameter k measures the tax-adjustment delay (in years). The parameter values are reported in Table 1.

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Figure 5: Numerical Responses to a Public Investment Shock: Alternative Parameter Values Panel (a): Alternative values of σY ˜ I(t)

˜ K(t)

0.4

0.4 Y

0.2


0

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< =0.5

0.2

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8



-1

-1.2

-1.2

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-1

-1.4

-1.4

0

10

30

60

100

150

time

0

10

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150

time

Panel (b): Different types of factor-augmentation ˜ I(t) 2.5

˜ K(t)

2.5

2

M

2

1.5

1

1.5

1 2K >2L =0

2K >2L =0 2 =2

0.5

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K

0

-0.5

-1 10

30

60

K

0

-0.5

-1 150

0

10

30

60

time

150

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Panel (c): Alternative values of θG ˜ K(t) 0.2 0

AC

0

100

time

˜ I(t)

0.2

L

2L >2K =0

2L >2K =0

100

PT

0

2 =2

0.5

L

-0.2

-0.2

3 G =0.131

3 =0.131

3 G =0.050

G

3 G =0.050

-0.4

-0.4

3 =0.200 G

3 G =0.200

-0.6

-0.6

-0.8

-0.8

-1

-1

-1.2

-1.2

-1.4

-1.4 0

10

30

60

100

150

0

time

10

30

60

100

150

time

Notes: The time unit is a year. For each variable, the vertical axis measures the percent deviation from its steady-state value. When varying one parameter, the remaining parameters are kept at their baseline values (see Table 1).

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Description

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Table 1: Baseline Parameter Values

Parameter/Share Value

ωC ≡ C/Y ωI ≡ I/Y C ωG ≡ CG /Y I ωG ≡ IG /Y ωZ ≡ Z/Y K/Y KG /Y B/Y F/Y δ δG

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Values based on euro area averages Private consumption-to-GDP ratio Private investment-to-GDP ratio Government consumption-to-GDP ratio Public investment-to-GDP ratio Net exports-to-GDP ratio Private capital-to-GDP ratio Public capital-to-GDP ratio Public debt-to-GDP ratio Net foreign assets-to-GDP ratio Private capital depreciation rate Public capital depreciation rate

0.565 0.183 0.201 0.031 0.019 2.388 0.588 0.676 -0.371 0.064 0.044

β ωLL ≡ (1 − L)/L θG σY

0.018 0.750 0.131 0.500

Implied values Parameter of the private capital installation function Parameter of the public capital installation function Rate of interest Output elasticity of labor Pure rate of time preference Preference weight of consumption in utility Elasticity of the private capital installation function Elasticity of public capital installation function Capital-augmentation elasticity

z¯ z¯G r θL α εC σA σG ηK

0.188 0.116 0.052 0.642 0.048 0.539 0.290 0.037 0.366

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Selected values Birth/death rate Leisure-labor ratio Output elasticity of public capital Elasticity of substitution capital/labor

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