The Effects of Public Spending Externalities Valerio Ercolani∗

Jo˜ao Valle e Azevedo†

Bank of Portugal

Bank of Portugal Nova School of Business and Economics

Final Version: December 2013; First Version: April 2010

Abstract We conduct a positive analysis on the effects of ‘externalities’ produced by government spending. To this effect, we estimate, using U.S. data, an RBC model with two salient features. First, we allow government consumption to directly affect the marginal utility of consumption. Second, we allow public capital to shift the productivity of private factors. We provide an identification analysis that supports the strategy adopted for estimating the parameters governing these two channels. On the one hand, private and government consumption are robustly estimated to be substitute goods. Because of substitutability, labor supply reacts little to a government consumption shock, so the estimated output multiplier is much lower than in models with separabilities. On the other hand, our results point towards public investment being ‘unproductive’.

JEL classification: E32, E62 Keywords: Public Spending Externalities, Fiscal Multipliers, Government Consumption, Government Investment, DSGE Models, Bayesian Estimation, Local Identification ∗

E-mail: [email protected]. We thank the Editor, Paul Klein, and two anonymous Referees, whose suggestions greatly improved the paper. We also thank Emanuela Cardia, Fabrizio Cecchini, Isabel Correia, Carlo Favero, Ricardo F´elix, Nikolay Iskrev, Leonardo Melosi, Caterina Mendicino, Tommaso Monacelli, Nicola Pavoni, Roberto Perotti and Pedro Teles for useful comments. Earlier drafts of the paper have also benefited from discussions with Javier Perez, Gianni Amisano, and Frank Smets. Remaining errors are solely ours. † Corresponding author: E-mail: [email protected].

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1

Introduction

Assessing the mechanisms through which government spending affects the private sector has occupied a large portion of the macroeconomic literature. The current debate on the effects of fiscal stimuli and consolidations across industrialized countries has renewed the interest on such mechanisms.1 This paper contributes to the debate by conducting a positive analysis on the effects of ‘externalities’ produced by government spending. By externality we mean, e.g., that government consumption can affect households’ marginal utility of consumption and, therefore, the marginal rate of substitution between consumption and labor. This occurs if some items of government consumption act as substitutes or complements for private consumption. For example, public health care can reduce the need for private health services, or, public education services can reduce the need for private tutors and schools but, on the other hand, increase the demand for textbooks or personal computers. These potential relations between private consumption and different items of public spending make government consumption, on the aggregate, to be either substitute or complement for private consumption. Thus, assuming a priori separability in preferences between private and government consumption can produce biased estimates of the response of private consumption, labor supply, and, hence, of output to a government consumption shock. In addition, government investment can create externalities for the private sector. More precisely, public capital can act as a shifter of the productivity of private factors, such that a shock to public investment has the potential to create both substitution and wealth effects, affecting the dynamics of several variables such as private investment, consumption as well as output. For example, an efficient system of public highways built in place of an old route can enhance the productivity of private firms operating in that area (e.g. by fostering within-country trade). Again, omitting a priori this role for government investment can bias estimates of its effects. Although the potential effects of government consumption on private consumption through preferences have long been considered, see e.g. Bailey (1971) and Barro (1981), the standard hypothesis of the bulk of RBC (e.g. Baxter and King, 1993) and new Keynesian (e.g. Smets and Wouters, 2007) models is that private and government consumption are separable in preferences or that government consumption is a pure waste of resources. Moreover, public capital is often omitted since it is assumed to be ‘unproductive’ (exceptions are, among others, Baxter and King, 1993, Leeper et al., 2010, or Coenen et al., 2013). Within these models, the so called negative wealth effect is the main driver of spending shocks. If government spending increases, 1

Several papers estimate the effects of the American Recovery and Reinvestment Act in the United States and of the European Economic Recovery Plan in the European Union. Among others, Cogan et al. (2010), Drautzburg and Uhlig (2011) and Coenen et al. (2012) focus on the first Act, whilst Conen et al. (2013) focus on the second Plan. Alesina et al. (2012) study the output effects of fiscal consolidations for a sample of OECD countries.

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then the present discounted value of taxes to be paid by households also increases and so permanent income is lower. The well-known consequence of this effect is the negative correlation between public spending and private consumption conditional on government spending shocks.2 The negative wealth effect impacts positively on labor supply which, in turn, generates an increase in output and a decrease in real wages.3 Finally, private investment usually falls in response to a (temporary) government spending shock. Against this background, it is clear that the externalities we explore have the potential to flip the usual sign of the reactions to government spending shocks or, even if the sign is correct, to assess the likely bias of these responses. For example, evidence of substitutability (complementarity) between private and government consumption leads, ceteris paribus, to a negative (positive) response of private consumption to a government consumption shock. Further, substitutability (complementarity) dampens (exacerbates) the positive reaction of labor supply to a government consumption shock. Also, allowing for the possibility that public capital affects the productivity of private factors may induce government investment shocks to create, on top of substitution effects, positive wealth effects that translate into positive reactions for private consumption, investment, and real wages. That said, if one is interested on the output effects of fiscal stimulus, it is obvious that the responses obtained in Uhlig (2010) - focusing on distortionary taxes - or in Christiano et al. (2011) - focusing on the zero lower bound - or in Monacelli et al. (2010) - focusing on the labor market - can be affected by the government spending externalities. Our objective is thus answering three main questions: is it reasonable to assume separability between private and government consumption? Is there evidence that public capital shifts the productivity of private factors? What are the effects produced by these externalities? To answer these questions we add the two ‘externalities channels’ into an otherwise standard RBC model. First, we allow government consumption to directly affect the marginal utility of consumption. Second, we allow public capital to shift the productivity of private factors. We estimate the model using standard Bayesian techniques, using U.S. data from 1969 through to 2008. Importantly, prior to estimation we provide evidence on the identifiability of the model’s parameters, paying special attention to the parameters governing the ‘externalities’ channels. This analysis offers interesting insights while supporting the adopted estimation strategy. On the one hand, estimation of various versions of the model indicates that government and private 2

Notable exceptions can be found in the literature. For example, Gal´ı et al. (2007) introduce a market imperfection in a new Keynesian model, namely, a share of the population cannot borrow or lend. Because of this, aggregate consumption can increase after a government spending shock. Following a different route, Linnemann (2006) builds a neoclassical model in which leisure and consumption are not separable in preferences. This type of non-separability can allow consumption to react positively to government spending shocks. 3 Real wages surely react negatively within an RBC model since the labor demand schedule remains unchanged. Instead, in versions of sticky prices models, real wages can happen to increase in response to a government spending shock (see Linnemann and Schabert, 2003).

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consumption are robustly estimated to be substitute goods. Substitutability makes labor supply to react little to a government consumption shock. Hence, the estimated output multiplier is much lower compared to the one obtained in models with separabilities. For instance, in our benchmark specification it peaks - on impact - at 0.33, which is approximately one third of the one found in the model with separable government consumption. On the other hand, our results point towards public investment being ‘unproductive’, and very clearly so in the case of defense investment. In the few specifications where non-defense public investment enhances (mildly) the productivity of private factors, a non-defense investment shock generates non-standard effects in the medium-run, such as a positive response of consumption. The remainder of the paper is organized as follows: Section 2 gives an overview of the empirical evidence and relates it to our work, clarifying our contribution to the literature. Section 3 outlines the model while Section 4 describes the identification analysis, the estimation exercise and results. In Section 5, concluding remarks are presented.

2

Review of the Empirical Evidence

Most estimates of the degree of substitutability between private and government consumption and of the effects of public investment on private productivities are obtained within partial equilibrium models. In particular, the empirical evidence for ‘non-separabilities’ of government consumption is most often obtained through estimation of Euler equations and is not conclusive. Aschauer (1985) finds a significant degree of substitutability between the two variables in the case of the U.S. whereas Amano and Wirjanto (1998) find weak complementarity. Focusing on the UK, Ahmed (1986) finds substitutability while Karras (1994), examining the relationship between private and public consumption across thirty countries, finds that the two types of goods are best described as complements (but often unrelated). Fiorito and Kollitznas (2004) split government consumption into two groups: ‘public goods’ and ‘merit goods’. The first includes spending in defense, security forces and judicial system; the second contains health, education and other services that can be provided privately. They use dynamic panel methods motivated by Euler equations and show that, for twelve European countries, public goods slightly substitute while merit goods complement private consumption. The other class of papers related to our work focuses on the importance of public capital in boosting output growth. Aschauer (1989) estimates an aggregate production for the U.S. economy, with inputs being labor, private capital but also public capital, finding that the output elasticity of government capital is 0.39. Following a similar approach, Finn (1993) estimates much lower output elasticities of various items of

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government capital (the largest is 0.16 for highways) and surrounded by great uncertainty. The implication of these two papers is that public capital is an important explanatory factor for changes in the productivity of the economy. Other authors, like Tatom (1991), find, instead, that the best estimate for the mentioned elasticity is zero. Belo and Yu (2013) report movements in stock returns compatible with a specification, very similar to ours, where public investment is directly productive. Other papers perform simulation exercises within general equilibrium models. For example, Leeper et al. (2010) analyze, within an estimated general equilibrium model, scenarios with different values for the output elasticity of public capital. Conditional on choosing the value of 0.1 for this elasticity, they find the posterior mean of the output multiplier ranging from 0.90 to 1.14 within the first three years. Baxter and King (1993), within a fully calibrated framework, find a long-run output multiplier equal to 4.12, conditional on choosing 0.1 for the output elasticity of public capital. Straub and Tchakarov (2007) conduct a calibration exercise for the Euro area within a general equilibrium framework, finding that under reasonable parameter values both permanent and temporary public investment shocks generate a much larger multiplier than the one obtained upon exogenous increases in government consumption.4 Unlike all the papers above, we estimate the parameters of interest within a general equilibrium framework. This allows us to study the effects of government spending shifts on several variables simultaneously while limiting the omitted variables bias associated with partial equilibrium models (e.g., due to lack of consideration of the negative wealth effects generated by government financing needs). Along this line of research, Bouakez and Rebei (2007), henceforth BR (07), were the first to exploit general equilibrium estimation. They find private and government consumption to be complement goods within a small scale RBC model, using Maximum Likelihood estimation and U.S. data. Their result lends support to a crowdingin effect of government consumption on private consumption. Our analysis differs from theirs in various aspects, other than in the results. First, unlike us, BR (07) do not use public spending data to estimate their model; we believe that using this data is essential to identify and measure the elasticity of substitution between private and public consumption. We show in Section 4 that not using public spending data can make the estimation cumbersome while affecting the results. Second, unlike us, they fix some parameters that are essential to determine whether the channels under scrutiny are relevant or not. For example, they fix the weight of private consumption in a constant elasticity of substitution (CES) aggregator for private and government consumption, thus forcing government consumption to affect the agents’ welfare. We show 4

Regarding the analysis within VARs, Perotti (2004) uses a structural VAR identified with institutional information and finds that a government investment shock creates an output multiplier which peaks on impact at 1.68, for the U.S.. Interestingly, Pappa (2009), using a sign-restrictions identified VAR and U.S. state data finds that government investment increases both employment and real wages.

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in Section 4 that fixing the weights in the aggregator affects the estimation results related to the externalities channels. Third, unlike us, they use detrended data; this may generate a loose mapping between the model’s variables and the data, potentially affecting the estimation results. Another interesting and recent paper is Coenen et al. (2013). Using an extended version of the ECB’s New Area-Wide Model (NAWM, see Christoffel et al., 2008), the authors aim at quantifying the impact of the European Economy Recovery Plan on euro area GDP. In addition to standard data, they use 8 time series for fiscal revenues and expenditures for (Bayesian) estimation. Their results point to the existence of significant complementarities between private and government consumption and, in a lower degree, between private and public capital. However, similarly to BR (07), they use detrended data and calibrate the weights in the private-government consumption CES aggregator. Further, they use a CES aggregator for public and private capital but calibrate both the share of total capital and the weights of the two types of capital in the aggregator. This forces public capital to be ‘productive’. Unlike them, we do not impose such restrictions. Finally, it is worth noting that the identification analysis we provide shows that the parameters in the aggregators can be jointly estimated.

3

The Model

We now describe our model economy, making clear the problems solved by households and firms. We also describe the behavior of the government, or fiscal authority. In a nutshell, we will be looking at an otherwise standard RBC model which crucially includes the two aforementioned ingredients aimed at assessing the role of government consumption and investment on private decisions. Further, we borrow from the literature ingredients that have proven useful to fit the data: external habit formation in consumption, monopolistic competition in factor markets, investment adjustment costs, costs of adjusting capacity utilization and (possibly) distortionary taxation. Uncertainty arises from six orthogonal shocks: a preference shock, total factor productivity, investment adjustment, wage markup (wedge) as well as public consumption and public investment shocks. Most often we break public investment into defense and non-defense items which results in the addition of another shock. Given our objectives, including the government spending shocks seems a natural choice. The remaining shocks (implying variation in wedges) have been shown to be important in accounting for business cycle fluctuations (see e.g., Smets and Wouters, 2007, Justiniano et al., 2010, and Albuquerque et al., 2013 for recent examples). In any case, we show in Appendix A that our results are robust to the inclusion of a smaller number of shocks.5 5

There are usually two polar positions when discussing the implications of general equilibrium models. One points to the hopeless misspecification of the model, to the unreasonableness of the specified ‘structural’ shocks or features. The other passes lightly over these issues, interprets every shock and formulation as truly structural and argues forcefully on the ‘speaking of the data’ aspect of the exercises. We combine the two positions. We certainly acknowledge the difficulty of interpreting the

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We choose to use an RBC model since it embeds the necessary features for studying comprehensively our channels of interest. RBC models have been widely used in the literature to study the sources of business cycle fluctuations but also the effects of fiscal policy. Recent examples of calibrated versions of RBC models are, among others, Baxter and King (1993), Cole and Ohanian (2002), McGrattan et al. (2007), McGrattan and Ohanian (2010). Recent examples of RBC models estimated with Maximum Likelihood or Bayesian methods include Leeper et al. (2010) and Schmitt-Groh´e and Uribe (2011).

3.1

Households

The economy is populated by a continuum of households. We assume that the representative household derives utility from effective consumption, C˜t , and disutility from working, Lt , in each quarter t. Effective consumption is assumed to be an Armington aggregator of private consumption, Ct , and government consumption, Gt :

[ ] υ υ−1 υ−1 υ−1 υ ˜ υ Ct = ϕ (Ct ) + (1 − ϕ) Gt ,

(1)

where ϕ is the weight of private consumption in the effective consumption aggregator, and υ ∈ (0; ∞) is the elasticity of substitution between Ct and Gt .6 Note that if ϕ = 1 then C˜t = Ct and the standard hypothesis of separability emerges. The lifetime expected utility is given by:  ( )1−σc  A ˜ ˜  Ct − hCt−1 1 1+σL  k εbt  β e  − χ 1+σ (Lt ) E0  , 1−σc L     t=0   ∞ ∑

(2)

where σc denotes the degree of relative risk aversion, σL is the inverse of the Frisch elasticity of labor supply. The parameter h ∈ (0; 1) measures the degree of habit formation in effective consumption whereas A β ∈ (0, 1) is the subjective discount factor and χ is a positive number. C˜t−1 is the aggregate level of

effective consumption at time t − 1 which creates external habit formation in consumption. εbt represents a preference shock, assumed to follow a first-order autoregressive process with an i.i.d.-normal error term: εbt = ρb εbt−1 + ηtb . We impose separability between consumption and labor for tractability while ensuring that hours is constant along the steady-state growth path. To guarantee the existence of a steady-state preference shock, total factor productivity, investment adjustment and wage markup shocks as truly structural shocks, invariant to monetary or fiscal policies. For instance, we accept that movements about the wage markup can be just shifts of relevant margins (wedges, in the analysis of Chari et al., 2009) in the decisions of agents. Still, we retain the formulation of the labor market as the set of equilibrium conditions (resulting in a wedge between the marginal rate of substitution and the wage rate) could be derived from other models. 6 When υ = 0, we have a ‘Leontief’ aggregator, i.e. Ct and Gt become perfect complements. When υ = 1, we have a ‘Cobb˜t = ϕCt + (1 − ϕ) Gt ˜t = Ctϕ G(1−ϕ) . As υ → ∞, we have a linear aggregator of the form C Douglas’ aggregator of the form C t and the two goods are perfect substitutes.

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growth path we are required to use log utility in consumption (σc = 1). This follows from the assumption of separability and the fact that we consider growth of technology (or of the economy) and a neoclassical production function.7 Households maximize their lifetime expected utility by choosing consumption, Ct , labor supply, Lt , next period’s physical capital stock, Kt+1 , the level of investment, It , and the intensity with which the installed capital stock is utilized, ut . Further, in order to justify the existence of a representative agent we complete h the markets by making agents able to trade a full set of one period state-contingent claims, paying Zt+1 at [ ] h t + 1 if state h realizes, at the cost Et Mt,t+1 Zt+1 , where Mt,t+1 = 1/(1 + rt,t+1 ) is a stochastic discount

factor. Here, we present the version of the model with distortionary taxation on labor, consumption and capital, with marginal rates given, respectively, by τ w , τ c and τ k . The agents thus face the following budget constraint (expressed in real terms): [ ] h (1 + τ c )Ct + It + Et Mt,t+1 Zt+1 = [ ] Zth + (1 − τ w )Wt Lt + (1 − τ k ) rtk ut − a (ut ) Kt + Dt − Tt ,

(3)

where rtk is the net return on capital, Wt Lt is labor income, a (ut ) = γ1 (ut − 1) + γ22 (ut − 1)2 represents the cost of using capital at intensity ut (see, e.g., Schmitt-Groh´e and Uribe, 2006), Dt are the dividends paid by household-owned firms while Tt are lump-sum taxes/transfers to/from the government. Since Ricardian equivalence holds in the model, we abstract from government debt and assume that the government balances its budget. The capital stock evolves according to the following equation: [ ( )] I It Kt+1 = (1 − δk ) Kt + It 1 − S eεt , It−1

(4)

where δk is the depreciation rate and the function S(.) introduces investment adjustment costs `a la Christiano ( I )2 It et al. (2005). Specifically, S(.) = κ2 eεt It−1 − eγ , where εIt is a shock to the investment cost function assumed to follow a first-order autoregressive process with an i.i.d.-normal error term (εIt = ρI εIt−1 + ηtI ), and γ is the steady-state growth rate of productivity (see the next Section for details). 7

KPR preferences, see King, Plosser and Rebelo (1988), could achieve the same objective while not imposing separability between consumption and labor.

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3.2

Firms

We assume there is a continuum of monopolistically competitive firms indexed by j ∈ [0, 1], each of which produces a single variety of final goods, Yj,t . They sell Yj,t at price Pj,t to the final goods competitive firms, which combine the differentiated final goods Yj,t in the same way households would choose, using a standard Dixit-Stiglitz aggregator:

[∫

1

Yt =

(Yj,t )

1 1+λp,ss

]1+λp,ss dj

,

(5)

0

where

1+λp,ss λp,ss

represents the elasticity of substitution across goods varieties. The competitive final goods

firm takes Pj,t as given and supplies goods Yt at price Pt to the households and government, for which it ∫1 has to pay a total cost equal to 0 Pj,t Yj,t dj. The profit maximization conditions in the final goods sector generate the following demand schedule for the varieties of final goods: ( Yj,t =

Pj,t Pt

)− 1+λp,ss λp,ss

Yt ,

(6)

while zero profit makes the price of the final good (which we normalize to 1) equal to: [∫

1

Pt =

(Pj,t )

−λ

1 p,ss

]−λp,ss dj

= 1.

(7)

0

Each of the monopolistically competitive firms produces a single variety of final goods, Yj,t , using as inputs capital services, Kj,t , and labor services, Nj,t , from competitive suppliers. Moreover, we augment the standard production function with KtG , representing the ‘productivity’ of public capital. The production function is given by: Yj,t = max((ut Kj,t )α (At Lj,t )1−α (KtG )θg − At Φ, 0),

(8)

where At is a productivity shock.8 The process for ln(At ) has a unit-root and evolves according to: ln(At ) = γ + ln(At−1 ) + εat ,

(9)

where γ is the steady-state growth rate of productivity (and hence of the economy) and εat = ρa εat−1 + ηta , where ηta is an i.i.d.-normal sequence. The parameter Φ represents a fixed cost of production while θg ∈ (0; ∞) is the output elasticity with respect to public capital productivity. 8

The chosen specification implies substitutability between private capital and the productivity of public capital as typical in the literature, see e.g. Aschauer (1989), Finn (1993), Baxter and King (1993) or Leeper et al. (2010). We will later discuss results obtained with a more general specification.

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The productivity of public capital is assumed to evolve according to: G Kt+1 = (1 − δKg )KtG + ξtig ,

(10)

where δKg is the depreciation rate and ξtig is the public investment rate (in our case, public investment, Itg , over total output, i.e., ξtig = Itg /Yt ). We will later specify how ξtig evolves. For now, we refer that ξtig follows a stationary process (which seems consistent with the data), implying that KtG is stationary. This is convenient for technical reasons (see Belo and Yu, 2013 and references therein for a similar specification and reasoning) and avoids keeping track of - poorly measured - public capital. We should further add that Itg comprises only non-defense items, i.e., we assume that public investment in defense, Itg,def , is unproductive.9 The existence of an economy-wide competitive factor market implies that all firms producing final goods varieties pay the same rental rate, rtk , and the same real wage, Wt , while taking into account the demand for their product. Cost minimization subject to the production technology (8), assuming output is positive, yields first order conditions for inputs which can be expressed as relative factor demands: Kt α Wt , = Lt 1 − α rtk ut

(11)

where we omit the index j for firms. Finally, these firms take aggregate output Yt and the price level Pt as given while setting Pj,t so as to maximize the present value of the flow of profits. This results as usual in Pj,t as a markup, equal to λp,ss , over marginal cost.

3.3

Labor Market

There is a continuum of monopolistically competitive households, indexed by i ∈ [0; 1], which set their wage rate Wi,t and supply labor hours Li,t . They sell labor services to a competitive labor aggregator sector which combines differentiated labor hours in the same way final goods varieties’ firms would choose, in a Dixit-Stiglitz form:

[∫ Lt =

1

(Li,t )

1 1+λw,t

]1+λw,t di ,

(12)

0 9

We have estimated versions of the model with potentially productive investment in defense. The production function for α final goods varieties producers becomes Yj,t = Kj,t (At Lj,t )1−α (KtG )θg (KtG,def )θg,def − At Φ, where the productivity of defense G,def G,def capital, Kt , is assumed to evolve according to Kt+1 = (1 − δKg )KtG,def + ξtig,def and ξtig,def is defense investment over ig,def g,def output, i.e. ξt = It /Yt . Estimates of θg,def were always exactly zero. Hence, for reasons of parsimony we will no longer discuss this more general specification.

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where

1+λw,t λw,t

represents the elasticity of substitution across labor varieties. The stochastic parameter λw,t

evolves as λw,t = (1−ρw )λw,ss +ρw λw,t−1 +ηtw , where ηtw is an i.i.d.-normal sequence. The competitive labor aggregator takes Wi,t as given and supplies labor services Lt at wage rate Wt to the final goods varieties’ ∫1 firms, for which it has to pay a total cost equal to 0 Wi,t Li,t di. This generates the following demand schedule by the labor aggregator:

( Li,t =

Wi,t Wt

) 1+λw,t λw,t

Lt .

(13)

Households exploit the demand for Li,t in order to set Wi,t so as to maximize (2), taking aggregate labor demand, Lt , and aggregate nominal wage, Wt , as given. This results in Wi,t as a markup, equal to λw,t , over the marginal rate of substitution between consumption and leisure. Finally, zero profits in the aggregator sector guarantee that in equilibrium the aggregate wage rate Wt is given by: [∫ Wt =

1

−λ1

(Wi,t )

w,t

]−λw,t di .

(14)

0

3.4

Government

First, we specify the evolution of public consumption, Gt , and public investment, Itg . These can always be expressed as a varying fraction of output: Gt = ξtg Yt ; Itg = ξtig Yt . We further specify ξtg and ξtig as follows: ξtg = exp(εgt + ssg )/(1 + exp(εgt + ssg )) ;

ig ig ig ξtig = exp(εig t + ss )/(1 + exp(εt + ss )).

This formulation (basically a reparametrization) ensures ξtg and ξtig are always between 0 and 1. The exogenous shocks εgt and εig t to government consumption and investment, respectively, follow autoregressive processes: εgt = ρg εgt−1 + ηtg ig ig εig t = ρig εit−1 + ηt ,

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(15) (16)

where ηtg and ηtig are normal i.i.d. and mutually independent with mean zero. Note that ssg and ssig fix the steady-state levels of ξtg and ξtig (denoted, respectively, by ξ g,ss and ξ ig,ss ). Therefore: ssg = log(ξ g,ss /(1 − ξ g,ss )) ; ssig = log(ξ ig,ss /(1 − ξ ig,ss )).

Public investment in defense is, we recall, unproductive, but we include it in the model as it is a non-negligible component of public spending. The model is thus augmented with the following equations: Itg,def = ξtig,def Yt ξtg,def = exp(εg,def + ssg,def )/(1 + exp(εg,def + ssg,def )) t t ig,def εig,def = ρig,def εiig,def , t t−1 + ηt

where the superscript def refers to defense items, Itg,def denotes defense investment, ssg,def = log(ξ g,def,ss /(1− ξ g,def,ss )) and ηtig,def is a normal i.i.d. defense investment shock with mean zero. Given this formulation, the paths of ξtg , ξtig and ξtig,def are assumed to be fully exogenous while the paths for Gt , Itg or Itg,def are not. For example, if there is a fall in total factor productivity and output falls government consumption and investment will on average fall. Since the government balances its budget, the following constraint holds: [ ] τ c Ct + τ w Wt Lt + τ k rtk ut − a (ut ) Kt + Tt = Gt + Itg + Itg,def .

3.5

(17)

Market Clearing

∫1 The labor market is in equilibrium when labor demanded by firms, 0 Lj,t dj, equals the differentiated labor ∫1 services supplied by households, 0 Li,t di = Lt , at the aggregate wage rate Wt . The market for capital is in equilibrium when the demand for capital services by the firms equals the capital supplied by households at the market rental rate rtk . The market for one period state-contingent claims is in equilibrium when net supply is zero at market prices. Finally, the final goods market is in equilibrium when the supply by firms equals the demand by households and government: Yt = Ct + It + a (ut ) Kt + Gt + Itg + Itg,def .

12

(18)

3.6

The ‘Externalities’ Mechanisms

Here, we further clarify the functioning of the two non-standard ‘externalities’ channels under analysis: the substitutability/complementarity mechanism and the one allowing public investment to be productive. To analyze the first channel, we look at the derivative of the instant marginal utility of consumption with respect to government consumption. Given a steady-state level of consumption, this is given, in log-linearized form, by:

) ( ) υ−1 ( 1 σc υ Ucg = (1 − ϕ) Gss /C˜ss − , υ (1 − h)

(19)

where Gss and C˜ss are the steady-state levels of government consumption and effective consumption, respectively. The parameters determining the sign to Ucg are the elasticity of substitution between private and government consumption, υ, the coefficient of relative risk aversion, σc , and the level of habit persistence, h. When Ucg is greater than 0, private and government consumption are defined to be complements; when Ucg is less than 0, private and government consumption are defined to be substitutes; when Ucg is equal to 0, private and government consumption are not related in preferences. Obviously, if we set ϕ equal to 1 government consumption does not enter the utility function and Ucg collapses to zero. For values of ϕ less than one, Ucg can be either positive or negative depending on the other parameters in Ucg . In particular, if we use log utility (or σc = 1), Ucg is strictly positive if υ < 1 − h and negative otherwise. Since 0 ≤ h < 1, υ > 1 guarantees that Ucg is negative.10 The potential role for the productivity of public capital is clearly visible in the production function Yj,t = (ut Kj,t )α (At Lj,t )1−α (KtG )θg − At Φ, with θg ∈ (0; ∞). Conditional on θg > 0, KtG has a direct effect on firm’s output and, as a consequence of this, a positive influence on the productivity of the private factors. The higher θg is, the more effective is KtG in boosting firm’s output and private productivities. Notice that if θg = 0 the standard production function obtains and KtG does not produce any externality effects.

3.7

Solution

We start by deriving the first order conditions associated with the households’ and firms’ problems, combining them with market clearing conditions and exogenous processes while recognizing that all firms and households are ex-ante identical. We then stationarize the variables Yt , Ct , It , Kt , Wt , Tt , Gt , Itg and Itg,def dividing them by the level of technology, At . The same treatment is required for the Lagrange multipliers associated with the budget constraint and the capital accumulation equation, respectively Λt and Qt (Tobin’s q). We then rewrite all the equilibrium conditions in terms of the standardized variables. For example, the Euler 10

The described implications hold even if we consider the non-linearized version of Ucg (available upon request).

13

equation induced by a risk free asset, paying gross return (1 + rt ), would be given by: [ βEt [ which becomes βEt

Λt+1 /At+1 At+1 At (1 Λt /At

] + rt ) = 1. Given the evolution of At , this becomes: [

βEt

] Λt+1 (1 + rt ) = 1, Λt

] λt+1 a exp(−(γ + εt+1 ))(1 + rt ) = 1, λt

where λt is the standardized Lagrange multiplier. All the equilibrium conditions can be found in Appendix B. Before proceeding to the estimation, we log-linearize the model equations around the deterministic steady-state. The exception to log-linearization occurs with the variables ξtg , ξtig and ξtig,def , which are fractions of total output.

4

Estimation

We estimate the model described in the previous Section using standard Bayesian techniques, please refer to Appendix C for computational details. Here, we discuss the following issues: first, we present the data along with the mapping to the model (i.e., the measurement equations). Second, we discuss the priors employed in our estimation and the calibrated parameters. Third, we discuss the identifiability of the parameters, paying particular attention to those governing our channels of interest. Fourth, we present estimation results associated with several robustness exercises. Finally, we show the reaction of our model economy to government spending shocks.

4.1

Data and Measurement Equations

We take as observables (denoted with a superscript obs) log differences of quarterly real per-capita output (GDP), consumption, investment and hourly wages as well as a particular transformation of public consumption and public investment (the latter split into defense and non-defense items). Regarding government spending data, we follow the definitions in the National Income Product Accounts (NIPA). Thus, government consumption is the sum of employees compensation, consumption of fixed capital (CFC) and purchases of intermediate goods and services, both at the federal and state/local level. Non-defense public investment consists of investment by both the general government and government enterprisers in structures (such as highways and schools), in non-military equipment and in software. It includes own-account investment by the government, both at the federal and state/local level. Defense public investment includes only 14

federal spending in military equipment.11 A full description of the raw data together with the observables is given in Appendix D. Some descriptive statistics are also presented there. It should be noted that our data covers the period 1969Q1-2008Q3; we believe that using data until 2013 could open (further) issues of misspecification in our empirical model (e.g. driven by the absence of financial frictions and the lack of consideration of the zero lower bound on nominal interest rates).12 Regarding output, consumption, investment and hourly wages, the mapping of data to variables in the model is made through measurement equations that take into account the fact that the solution of the model is in (log) deviations of the stationarized variables from the steady-state. For example, in the case of real output, we have that log(Yt ) − log(Yt−1 ) = log(At yt ) − log(At−1 yt−1 ) = log(At /At−1 ) + (log(yt ) − log(yss )) − (log(yt−1 ) − log(yss )) = γ + εat + ybt − ybt−1

where hats denote log deviations from the steady-state. Thus,

the corresponding measurement equations are the following: obs log(Ytobs ) − log(Yt−1 ) = ybt − ybt−1 + γ + εat

(20)

obs log(Ctobs ) − log(Ct−1 )=b ct − b ct−1 + γ + εat

(21)

obs log(Itobs ) − log(It−1 ) = bit − bit−1 + γ + εat

(22)

obs log(Wtobs ) − log(Wt−1 )=w bt − w bt−1 + γ + εat .

(23)

As for public consumption and investment as a fraction of output, i.e., ξtg,obs and ξtig,obs respectively, the following measurement equations are employed: log(ξtg,obs /(1 − ξtg,obs )) = εgt + log(ξ g,ss /(1 − ξ g,ss )) ig,ss log(ξtig,obs /(1 − ξtig,obs )) = εig /(1 − ξ ig,ss )), t + log(ξ

(24) (25)

which follow from the specifications of ξtg and ξtig in Section 3.4. Similarly, and since public investment is split into defense and non-defense items, the following measurement equation is added: log(ξtig,def,obs /(1 − ξtig,def,obs )) = εig,def + log(ξ ig,def,ss /(1 − ξ ig,def,ss )). t 11

(26)

As Perotti (2004) pointed out, following the 1993 System of National Accounts most countries record purchases of weapons and weapon delivery systems like warships, submarines, military aircraft, tanks, and missile carriers and launchers as government consumption. By contrast, in the U.S. these are recorded as government investment. 12 Interestingly, Fernandez-Villaverde (2010) analyzes the effects of fiscal policy within a DSGE model with financial frictions.

15

4.2

Calibration and Prior Distributions

First, and as common in the literature, we fix (calibrate) several parameters. We set β to 0.995. This value, together with the average growth rate of productivity, γ, which is around 0.004 (or 0.4% per quarter), implies an annual steady-state real interest rate on a risk-free asset around 4%. The depreciation rate of private capital, δk , and the depreciation of the productivity of public capital, δkg , are set at 0.025, implying a common annual depreciation rate of 10% (see Christiano et al. 2005). In versions of the model with distortionary taxation we follow Leeper et al. (2009) in fixing τl = 0.223, τk = 0.184 and τc = 0.028. There are two other distortions in this economy, the wage and price markups, whose steady-state values are set to λp,ss = 0.20 and λw,ss = 0.05, following Christiano et al. (2005). Finally, we will keep the steady-state level of hours, i.e., Lss , fixed at 0.31. This implies writing the parameter χ as a function of other parameters. Thus, χ varies throughout the estimation but guarantees that steady-state hours is always 0.31. We proceed similarly with γ1 which is set equal to the real return on capital as this must occur in equilibrium. We gather these remarks in Table 1. Table 1- Calibrated Parameters Parameter

Value

Justification

β

0.995

Real interest rate (yearly) ≈ 4%

δk

0.025

Depreciation rate (yearly) = 10%

δkg

0.025



τw

0.223

Leeper et al. (2009)

τk

0.184

Leeper et al. (2009)

τc

0.028

Leeper et al. (2009)

λp

0.20

Christiano et al. (2005)

λw

0.05

Christiano et al. (2005)

χ

Varying

s.t. Lss = 0.31

γ1

Varying

k rss , eq’m. relation

Concerning the choice of the priors, we assume they are independent and we keep them mostly uninformative, very much so in the case of the parameters related to the government spending externalities. The utility parameter ϕ follows a uniform distribution with support in [0, 1]. Concerning the parameter υ ∈ (0; ∞), we decide to reparametrize it such that υ := exp(υ b), where now υ b ∈ (−∞; ∞). Then, in assigning the prior to υ b we want to be as agnostic as possible, so we decide again for a uniform distribution with support in [−5, 20] (meaning that υ is in the range [0.018, almost perfect substitutes] , say), which

16

covers a wide range of possibilities in the complementarity/separability space. Regarding the choice of prior for θg we hold to a uniform distribution with support in [0, 4]. Regarding the preferences parameters, for σL (inverse of Frisch elasticity) we specify a Normal distribution with mean 2 and standard deviation of 0.5, thus covering a wide range of admissible values in the literature, whereas for h (habit formation) we specify a Beta distribution with mean 0.7, around what is employed in the literature, and standard deviation 0.2. For the parameters ξ g,ss , ξ ig,ss and ξ ig,def,ss , we specify a Normal distribution with mean equal to the sample average found in the data (1969Q1-2008Q3 ) and a reasonably high standard deviation. The prior for 100γ is a Normal with mean 0.4 and standard deviation of 0.02. All the standard deviations of the shocks follow an Inverse Gamma distribution with mean equal to 0.1 and a standard deviation of 2 while the autoregressive parameters follow a Beta distribution with mean 0.5 and standard deviation of 0.2. The priors for the parameters Φ (fixed cost), κ (adjustment cost parameter), γ2 (capacity utilization adjustment) and α (capital share), have loose priors centered around values close to (or derived from) those in Schmitt-Groh´e and Uribe (2006). Tables 4 and 5, containing also estimation results, summarize these remarks.

4.3

Are the ‘Externalities’ Parameters Identified?

Prior to estimation we check for the identifiability of the parameters to be estimated, with a particular focus on the ‘externalities’ parameters. To this effect, we apply the local identification approach proposed by Iskrev (2010a and 2010b). Specifically, let ms (θ) the mapping from the vector of the deep parameters of the ∂ms (θ) ∂θ′

model, θ, to the unconditional theoretical moments of the observables in the model and let J(θ) :=

be

the Jacobian matrix of this mapping. ms (θ) consists of means, covariances and autocovariances up to order s ≤ T , where T is the sample size. The parameters are locally identified at θ0 if J(θ) has full column rank when evaluated at θ0 . The verification of this rank condition means that the parameters under scrutiny are neither irrelevant, i.e. they affect the model’s implied moments, nor redundant, i.e. their effects cannot be replicated by changing other parameters. Table 2 - Identifiability for selected pairs of parameters θi and θj (ϱi,j is reported in square brackets) Utility Function

Production Function

fixing υ b at:

υ b ϕ θg

fixing ϕ at:

fixing θg at:

−2 (compl)

log(0.3) (no relation in pref)

5 (subst)

≃1 (G affects welfare:NO)

0.75 (G affects welfare:YES)

0

0.1

2

σL [0.93] γ2 [0.70] σa [0.12]

σL [0.65] σL [0.78] σa [0.15]

ϕ [0.97] υ b [0.97] σa [0.15]

κ [0.31] σa [0.15]

σL [0.75] σL [0.76] σa [0.15]

σL [0.65] σL [0.79] α [0.01]

σL [0.65] σL [0.79] α [0.02]

σL [0.65] σL [0.79] σa [0.15]

We perform the identification analysis on the version of the model that allows for non-separable gov-

17

ernment consumption and productive non-defense government investment. First, we evaluate the rank of J(θ) for all parameters at their prior mean and we find full rank.13 Second, we calculated a correlation ( ) 1 (θ0 ) ∂m1 (θ0 ) coefficient, namely 0 ≤ ϱi,j := corr ∂m∂θ , ≤ 1, for all possible pairs of parameters θi and θj .14 ∂θj i If ϱi,j = 1, then the effects on m1 of changing the parameter θi can be offset by changing θj , i.e. θi and θj are not identified at θ0 . We find that for each pair of parameters, θi and θj , ϱi,j is less than one in several representative regions of the parameter space (i.e., not only at the prior mean). Table 2 reports these correlations for different values of our parameters of interest, υ b, ϕ and θg (letting the other parameters set at their prior mean).15 More precisely, each cell contains the most highly correlated parameter with the corresponding one in the first column of the Table (the value of the correlation is reported in brackets). For example, looking at the second column we can observe that, when υ b is set such that C and G are complements, the most collinear parameter with ϕ is γ2 and the associated ϱ is 0.7. The main conclusion is that, given our set of observables, the ‘externalities’ parameters are identified in different regions of the parameter space. Now, BR (07) and Coenen et al. (2013), using a CES aggregator similar to the one in equation (1), fix ϕ and estimate the elasticity of substitution υ. The first paper supports the view that ϕ is poorly identified, without formally proving it, and fixes it to 0.8. The second one sets ϕ to 0.75 in order to roughly equate the marginal utility of private and government consumption, given the data. But our results show that, though ϱϕ,υ

b

is high in some regions of the parameter space, it is always

below unity. Hence, we jointly estimate υ b and ϕ. This avoids imposing either arbitrary numbers for ϕ or quite strong assumptions for its calibration which, of course, can affect the estimation results. We show in Section 4.5.1 that fixing ϕ does indeed affect the estimated value for υ b.16

4.4

Estimation Results

This Section presents the estimation results. We analyze various versions of the model, focusing on the following variations: - Use of distortionary taxation and lump-sum taxation or lump-sum taxation only - Full-sample, i.e. from 1969Q1 until 2008Q3 or only the so-called ‘Great Moderation’, i.e. from 1984Q1 until 2008Q3 13 The theoretical moments used in the analysis are the mean, the covariances and the autocovariances up to the first order for the 7 observables used in the estimation. This gives us 84 moments for 26 free parameters. 14 s (θ0 ) s (θ0 ) The index ϱi,j is the cosine of the angle between the vectors ∂m∂θ and ∂m∂θ . i j 15 We refer two things. First, if not differently specified, υ b is set to log(0.3) such that U cg = 0. Second, we perform the analysis for many points in each of the representative regions. Due to space limitations we report here only some of the results obtained. Full results are available upon request. 16 We also study identification for several variations of our model. Specifically, we check the identifiability of those parameters within the CES aggregator of private and public capital in the firms’ production function, see Section 4.5.2.

18

- Restricted models: without public spending externality channels (ϕ = 1, θg = 0), with the channels in the utility function only (θg = 0) and with the channels in the production function only (ϕ = 1). We start by discussing the results obtained with the distortionary taxation model. Table 3 reports virtually all the estimated specifications, the associated values for the marginal data density, and a summary of the posterior (mean and mode) of the externalities’ parameters, both for the full sample and for the post ‘84 one. The model with the highest marginal data density is, irrespective of the sample, the one with θg restricted to 0. This specification reveals that government consumption affects the marginal utility of consumption, since ϕ is estimated to be less than 1, and that government and private consumption are substitute goods. The estimated value for υ b implies a very high elasticity of substitution, υ. Focusing now on those specifications where θg is estimated, we underline the following facts: first, the estimates for ϕ and υ b are very close to the ones obtained in the version where θg is restricted to 0. Second, the mode of θg is always nil and the posterior mean of θg is above 0 only in some specifications, reaching 0.05 and 0.09.17 Given these findings, we present in more detail results for only two specifications among the ones of Table 3, labeled as the Preferred version, i.e. the one with θg restricted to 0, and the Productive Investment version, i.e. the one without restrictions, both estimated using the full sample. Tables 4 and 5 present the prior distributions along with the estimated mode, mean, and the 5% and 95% percentiles of the posterior distribution of the parameters of the Preferred and Productive Investment specifications. Regarding the parameters of greatest interest for us, i.e., those related to the public spending externalities, we note that ϕ has, in both variations, a mode of 0.65 and mean of 0.63, indicating that government spending does affect the welfare of agents. More interestingly, the mode of υ b is 7.9 in the Preferred version (7.56 in the Productive Investment one) while the mean is 10.16 (respectively 10.69), which indicates strong substitutability between private and government consumption. Conditional on the estimated value for υ b (both mode and mean), Ucg in equation (19) is unambiguously negative. Notice that the large estimated value for υ b (and thus for υ) makes the aggregator in (1) almost linear, i.e. C˜t ≃ ϕ (Ct ) + (1 − ϕ) Gt . This linear specification was indeed suggested by Barro (1981) and estimated by Aschauer (1985). Regarding θg in the Productive Investment specification, the mode is zero but the mean is positive, equal to 0.09, with an associated 90% posterior interval ranging from 0.00 to 0.20. As for the remaining parameters, we note that those related to the constant terms in the measurement equations 17 We recall that we have also estimated models with potentially productive investment in defense. It turned out that models with θg,def (the parameter measuring the productivity of defense capital) restricted to 0 were also very clearly preferred to models with θg,def left unrestricted. Additionally, we have also estimated versions of the model with public investment not split into defense and non-defense items. In this case, estimates of the posterior mode and mean of the parameter measuring the productivity of public investment, θg , were almost always exactly 0. This indicates perhaps the increased difficulty of identifying public investment as a shifter of productivity once defense and non-defense items are considered jointly.

19

Table 3 - Distortionary Taxation, Posterior Mean and Mode of externalities parameters POSTERIOR Post 1969

Parameter υ b := log(υ) ϕ θg Laplace Log D Dens. Log D Dens.

ϕ=1 Mode Mean 0.0

0.0

θg = 0 Mode Mean 7.9 10.2 0.65 0.63 -

2522.3 2534.9

2544.8 2555.1

Unrestr. Mode Mean 7.6 10.7 0.65 0.63 0.0

0.09

2539.6 2551.6

No Channels 2541.4 2548.3

POSTERIOR Post 1984

Parameter υ b ϕ θg Laplace Log D Dens. Log D Dens.

ϕ=1 Mode Mean 0.0

0.05

1713.9 1725.3

θg = 0 Mode Mean 8.9 14.3 0.66 0.47 -

1722.3 1734.3

Unrestr. Mode Mean 7.7 11.4 0.66 0.51 0.0

0.0

1710.4 1721.7

No Channels 1715.7 1727.9

have a mode very close to the mean of the specified prior. Further, the posterior for these parameters is concentrated around the mode. Most structural parameters not related to public spending have coverage from the intervals found in the literature and a posterior distribution generally concentrated around the mode, cf. the 5% − 95% percentile for h, α or κ and γ2 . The parameter surrounded by greater uncertainty is the inverse of the Frisch elasticity of labor supply, σL , whose mode is away from its mean in the two specifications and has a looser posterior distribution.18 As for the shocks parameters, we note the strong persistence (high autoregressive parameter) of most shocks, the mild persistence of the preference shock and the low persistence of the productivity shock (see Table 5).19 Appendix A contains additional results and comparisons with what obtains in the post ‘84 sample for the model with distortionary taxation. We notice an ample stability of most parameters across the two samples but, in general, a lower standard deviation of most shocks in the post ‘84 sample. Now we discuss briefly estimation results obtained in the model with (only) lump-sum taxation. Table 6, reveals that, under lump-sum taxation, the qualitative findings obtained in the model with distortionary taxation are confirmed. That is, models with θg restricted to 0 are usually the preferred ones. Further, the 18

This lack of robustness is further confirmed when looking at results for the post ’84 sample only (see Appendix A), where the distribution of σL is concentrated around low values that seem to be compensated by low values of ρw , the auto-regressive parameter of the wage wedge (or markup) shock, and high values of its standard deviation, σw . 19 Allow us to refer that we have checked the rank condition for identification at the posterior mean, concluding that all the parameters are identified.

20

Table 4 - Priors and Posteriors of estimated parameters 1969Q1-2008Q3 Distortionary taxation. Preferred specification (θg = 0) vs. Productive Investment PRIOR Parameter

Distr.

POSTERIOR

Mean

St. Dev.

Mode

Preferred Mean 5%

95%

Productive Investment Mode Mean 5% 95%

A. Utility function h σL υ b := log(υ) ϕ

Beta N ormal U nif orm U nif orm

0.7 2

0.1 0.5 [−5, 20] [0, 1]

0.77 0.32 7.90 0.65

0.81 0.98 10.16 0.63

0.73 0.49 0.65 0.52

0.89 1.43 17.98 0.70

0.77 0.31 7.56 0.65

0.80 0.86 10.75 0.63

0.72 0.21 2.87 0.52

0.87 1.32 19.99 0.74

0.05 0.3

0.02 0.02

0.023 0.38 -

0.023 0.39 -

0.00 0.38 -

0.043 0.40 -

0.020 0.38 0.0

0.019 0.39 0.09

0.00 0.38 0.00

0.038 0.41 0.20

B. Production function Φ α θg

N ormal N ormal U nif orm

[0, 4]

C. Investment Adj. costs κ/100 γ2

N ormal N ormal

4 0.0685

0.5 0.002

4.74 0.063

4.73 0.064

4.08 0.061

5.36 0.066

4.74 0.063

4.79 0.063

4.12 0.060

5.45 0.065

N ormal N ormal N ormal N ormal

0.4 0.16 0.025 0.008

0.02 0.01 0.001 0.001

0.40 0.169 0.026 0.009

0.40 0.169 0.026 0.009

0.37 0.160 0.024 0.007

0.43 0.178 0.028 0.010

0.40 0.169 0.027 0.009

0.40 0.169 0.027 0.009

0.37 0.161 0.025 0.007

0.43 0.178 0.028 0.010

D. Constant terms γ ∗ 100 ξ g,ss ξ ig,ss ξ g,def,ss

Table 5 - Priors and Posteriors of Shocks parameters 1969Q1-2008Q3 Distortionary taxation. Preferred specification (θg = 0) vs. Productive Investment PRIOR Parameter

Distr.

POSTERIOR Preferred Mean 5%

95%

Productive Investment Mode Mean 5% 95%

Mean

St. Dev.

Mode

0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.73 0.08 0.95 0.96 0.97 0.94 0.95

0.72 0.11 0.95 0.95 0.97 0.95 0.95

0.67 0.02 0.92 0.92 0.96 0.91 0.93

0.76 0.18 0.98 0.98 0.99 0.98 0.98

0.73 0.08 0.95 0.96 0.97 0.94 0.95

0.73 0.10 0.95 0.95 0.98 0.94 0.95

0.68 0.02 0.92 0.92 0.96 0.91 0.93

0.77 0.17 0.98 0.99 0.99 0.98 0.98

0.1 0.1 0.1 0.1 0.1 0.1 0.1

2.0 2.0 2.0 2.0 2.0 2.0 2.0

3.64 0.024 0.039 0.023 0.015 0.030 0.086

3.57 0.024 0.040 0.034 0.015 0.030 0.087

3.13 0.022 0.030 0.021 0.014 0.028 0.079

3.98 0.027 0.052 0.047 0.017 0.033 0.095

3.63 0.024 0.039 0.023 0.015 0.030 0.086

3.61 0.024 0.040 0.032 0.015 0.030 0.087

3.16 0.022 0.030 0.020 0.014 0.028 0.080

4.03 0.026 0.050 0.044 0.017 0.033 0.096

A. Autoregressive Parameters ρb ρa ρI ρw ρg ρig ρig,def

Beta Beta Beta Beta Beta Beta Beta

B. Standard deviation of shocks σb σa σI σw σg σig σig,def

Inv Inv Inv Inv Inv Inv Inv

Gamma Gamma Gamma Gamma Gamma Gamma Gamma

21

mean and the mode of ϕ are always estimated to be less than 1, and usually higher vis-`a-vis the distortionary taxation case. The mean and mode of υ b fall always in the substitutability region, ranging from 4.1 to 12.7. Finally, the mode of θg is always nil, although in some instances its posterior mean is away from zero, ranging from 0.01 to 0.26.20 Table 6 - Lump-Sum Taxation Posterior Mean and Mode of externalities parameters POSTERIOR Post 1969

Parameter υ b := log(υ) ϕ θg Laplace Log D Dens. Log D Dens.

ϕ=1 Mode Mean 0.0

0.26

θg = 0 Mode Mean 4.5 12.7 0.73 0.76 -

2458.9 2467.8

2511.6 2483.0

Unrestr. Mode Mean 4.1 9.7 0.72 0.74 0.0

0.0

2482.4 2492.8

No Channels 2420.4 2443.5

POSTERIOR Post 1984

Parameter υ b ϕ θg Laplace Log D Dens. Log D Dens.

ϕ=1 Mode Mean 0.0

0.01

1629.3 1638.4

θg = 0 Mode Mean 8.0 10.2 0.75 0.78 -

1678.6 1674.8

Unrestr. Mode Mean 8.9 10.0 0.77 0.76 0.0

0.0

1657.2 1665.8

No Channels 1638.9 1643.8

All in all, the results suggest clear evidence of strong substitutability between public consumption and private consumption and a very weak evidence on the positive effects of non-defense public investment on private sector productivity. The next sub-section analyzes further the robustness of these results.

4.5

Robustness Checks

This Section aims at checking if our results are robust along several dimensions. First, we assess the robustness of the estimated relation of substitutability between private consumption (C) and government consumption (G), explaining the divergence between our results and those in BR (07). Second, we present several variations of the model aimed at evaluating the role of public investment in shifting private productivities. Finally, we check for the sensitivity of our results to a few additional assumptions. Unless otherwise stated, all the specifications are estimated using our full sample, i.e., 1969Q1-2008Q3.21 20 21

Full results concerning all the variations considered are available upon request. Due to space limitations we only report the most important results. All the details are available upon request.

22

4.5.1

The Substitutability between C and G

We start by considering different specifications for the prior of the elasticity of substitution between private and government consumption within our Preferred Model. In the first exercise, we consider a quite loose Normal prior for υ b, centered at -1.0 (implying complementarity between private and government consumption). In the second one, we estimate directly υ (recall, υ b := log(υ)) and consider, for this parameter, a Gamma prior with mean 0.1 and variance 9.0, implying that the mass of the prior distribution is concentrated at values close to zero (or strong complementarity). Table 7 presents the results using the full sample. Previous results are reported here for comparison. Using the Normal prior for υ b we still find strong substitutability between private and public consumption. The 5th percentile of the posterior distribution for υ b is relatively low but it still implies substitutability. As for the Gamma prior specification for υ, the posterior mean and mode imply significant substitutability and the 90% credibility interval is always in the substitutability region. We should refer that a graphical analysis of the posterior reveals some flatness for high values of υ (υ b), where the effective consumption aggregator is quasi-linear, but there is discrimination between low and high values of υ (υ b). This probably justifies the relatively large width of the 90% credibility interval, although we should stress that substitutability is never questioned. This is also consistent with the results of our identification analysis, where the highest ϱi,j ’s were found in the (strong) substitutability region.

Table 7 - Priors and Posteriors of estimated parameters 1969Q1-2008Q3 Variations in the Utility Parameters PRIOR Parameter

Distr.

Mean

St. Dev.

Mode

POSTERIOR θg = 0 Mean 5%

95%

Preferred Model - Uniform Prior for υ b := log(υ) υ b ϕ

U nif orm U nif orm

[−5, 20] [0, 1]

7.90 0.65

10.16 0.63

0.65 0.52

17.98 0.70

10

3.78 0.53

4.73 0.55

0.37 0.42

16.27 0.68

3

2.14 0.67

4.81 0.62

1.11 0.47

9.10 0.77

Specification A - Normal Prior for υ b υ b ϕ

N ormal U nif orm

−1.0 [0, 1]

Specification B - Gamma Prior for υ υ ϕ

Gamma U nif orm

0.1 [0, 1]

As referred before, our result of substitutability contrasts with the one of complementarity found by BR (07).22 We now check whether following closely BR (07)’s estimation strategy reduces the divergence between 22

Recall that Coenen et al. (2013) find, using euro area data, the same results as BR (07). We focus here on the comparison between our results and BR (07)’s ones since they also use U.S. data.

23

our results and theirs. To this effect, we estimate several variations of our Preferred Model: (i) versions with ϕ fixed, (ii) versions not using government spending data, as in the estimated general equilibrium model of BR (07), (iii) versions using BR (07)’s sample, i.e., 1948Q1-2005Q4; finally, (iv) versions using HP detrended data. The first conclusion is that fixing ϕ affects significantly the estimate of the elasticity of substitution. That is, higher values of ϕ result in a lower estimate of this elasticity, as also noticed by BR (07). Specifically, fixing ϕ to 0.8, BR (07)’s value, we obtain an estimate for the mode of υ b equal to 0.63 which is much lower than what we obtained (7.9) conditional on an estimated ϕ of 0.65 (see Table 4). Notice that the specifications with ϕ estimated tend to be preferred to the ones where ϕ is fixed.23 Secondly, excluding government consumption from the list of observables lowers significantly, again, the estimate for υ b, although we should point that estimating this version is rather cumbersome computationally.24 Third, and very importantly, we estimate the model with ϕ fixed to BR (07)’s value and G not used as an observable; we obtain an estimated mode for υ b equal to -2.10, implying a υ equal to 0.12. This value, conditional on the other parameters, makes Ucg unambiguously positive, meaning that C and G are estimated to be complement goods. Fourth, we should refer that using either detrended data or BR (07)’s sample does not change substantially our original estimated value of υ b.25 Finally, we try to get as close as possible to the BR (07)’s estimation setup by estimating a version of our Preferred Model that includes all the above-mentioned features, from (i) to (iv). The results clearly speak in favor of complementarity. Again, the estimation was rather cumbersome. To sum up, we are able to estimate variations of our Preferred Model that generate results similar to BR (07)’s. However, it seems clear that our setup is preferable to that used by BR (07). First, we estimate ϕ (avoiding imposing arbitrary restrictions for it) and these models tend to be preferred to models with ϕ fixed at values around those employed by BR (07). Further, we use G as observable and this seems more appropriate if we are interested in measuring the elasticity of substitution between C and G. Importantly, using G also improved the reliability of the results. In fact, the versions without G in the list of observables 23

More precisely, the Laplace Log Data Density associated to our original estimation is 2544.8 (see Table 3). The ones associated to the model that fixes ϕ to 0.8 or 0.9 are 2544.2 and 2542.2, respectively. 24 In this version, and in order to keep the number of observables equal to the number of shocks, we have removed the wage mark-up shock (since it seems to be the least important shock in the model, in the sense that removing it affects little the results, see also Appendix A). Then, in order to circumvent numerical problems arising during the estimation, we follow BR (07)’s strategy and fix some parameters. We fix ξ g,ss , h, and α to 0.169, 0.77, and 0.35, respectively, which are as close as possible to the estimated modes (see Table 4). Under this specification, the estimated modes for ϕ and υ b equal 0.91 and -1.09, respectively. 25 Regarding the use of detrended data, we estimate a version of the model where we eliminate growth, i.e. γ¯ = 0, and change the measurement equations for output, consumption, investment and wages such that, as usual, log deviations of these variables from their respective steady-states (in the model) are mapped to HP filter detrended log data. For example, regarding output, we use the following measurement equation, HP log(Ytobs ) = ybt , where HP log(Ytobs ) denotes the HP cycle of log(Ytobs ). The measurement equations associated with public spending items do not change since they are measured as fractions of output and are already stationary.

24

were very difficult to estimate (e.g., with routines not reaching convergence). In this sense, our experience is similar to BR (07)’s, who admitted the difficulties in estimating their specification.

4.5.2

The Role of Public Investment

We start by considering two different priors for θg . Within our Productive Investment specification, we try a Normal and an Inverse Gamma prior. In the first case, we try priors with different means and variances and in all cases we obtain a mode for θg equal to zero. We get a positive mean in a few instances.26 Regarding the Inverse Gamma prior, we notice that the posterior mode and mean obtained are positive but very close to the mode of the respective prior. Further, the posterior distributions had a shape close to that of the prior. These are clear signs that the shape of the priors is driving the estimation results associated to θg .27 Using both Normal and Inverse Gamma priors we obtain a Log Data Density lower than the one generated by the model where θg is restricted to zero, meaning that the data again favored the specification with ‘unproductive’ public capital. Second, we try a different specification for the production function, under the model with non-separable government consumption, namely: e tα (At Lt )1−α − At Φ, Yt = K where

[

e t = ϕk (Kt ) K

υk −1 υk

+ (1 − ϕk ) (At KtG )

(27)

υk −1 υk

]

υk υk −1

,

(28)

similarly to Coenen et al. (2013). First, we find evidence that the parameters ϕk and υk are identified in different regions of the parameter space. As in the specification of the priors for the utility function parameters, we specify a uniform distribution for ϕk with support in [0, 1] and try several priors for υk , finding that most often the posterior mode and mean of ϕk are tightly concentrated around 1. In the sole trial where we find a value below one, the posterior mode and mean are 0.88 and 0.85, respectively, while the values of υk suggested substitutability between private and public capital. In this case, however, the Log Data Density clearly favors a specification with ϕk restricted to 1. Third, we check for the plausibility of time-to-build effects on public capital. That is, we consider a 26

For example, considering a Normal prior with mean equal to 0.2 and standard deviation equal to 0.5, the estimated posterior mean (obtained with an MH using 200000 draws) is 0.247 with [0, 0.602] as the associated 90% posterior interval. In general, it should be noticed that, consistently with the parameter space of θg , all the Normal priors have been truncated to zero. 27 For example, the specification that uses the prior with mean and standard deviation equal to 0.1 and 2.0 (and mode equal to 0.05) delivers a posterior mode and mean equal to 0.05 and 0.07, respectively. Further, the specification that uses the prior with mean and standard deviation equal to 0.3 and 2.0 (and mode equal to 0.14) delivers a posterior mode and mean equal to 0.13 and 0.16, respectively.

25

modified version of our Productive Investment Model with the following production function: G Yt = max(Ktα (At Lt )1−α (Kt−N )θg − At Φ, 0),

(29)

where N is an integer. We estimate this version of the model trying two different values for N : 4 and 12. Using N equal to 4 (12) allows public capital to start affecting output only one (three) year(s) after the realization of the public non-defense investment shock. In both specifications we have tried different priors for θg such as a uniform prior and a normal one. Results have not changed with respect to the original ones, i.e., we find a mode of θg equal to zero in both specifications. We also checked for time-to-build effects using a production function of the type of (27) and (28) but we do not find any evidence supporting a θg greater than zero. To sum up, within our framework, arguing that public capital plays a role in shifting the production frontier is difficult. On the one hand, we are aware that there is little variation in the public investment to output ratio, which is one of our observables (see Table D3 in Appendix D). This certainly makes identification of the effects of public investment difficult. On the other hand, we believe it is possible to reconcile the existence of public investment with the fact that the estimated θg is zero. For example, suppose ¯ G , more ports and highways add ports and highways are productive but that, above a certain threshold K ¯ G )θg , (K G )θg ). This rationalizes K G and can explain nothing to production, or Yt = Ktα (At Lt )1−α min((K t t ¯ G ) does not shift private productivities. why variation in KtG (above K Finally, we also check for an alternative channel through which public capital can affect the economy, namely, through the preferences of households. We estimate a version of our Preferred Model, where we insert the public capital in the aggregator (1).28 With such specification we do find the capital stock affecting households’ preferences. Specifically, public capital seems to be a substitute for private consumption. For example, new public hospitals allow individuals to use the newly created public structures instead of private ones. This generates a fall in the demand for private health services, inducing a fall in total private consumption, ceteris paribus. While we acknowledge that this alternative ‘preference channel for public capital’ is worth studying carefully, this investigation goes beyond the scope of this paper. 28

[ ] υ υ−1 υ−1 ˜t = ϕ (Ct ) υ + (1 − ϕ) (G∗t ) υ−1 υ Specifically, the aggregator becomes C , where G∗t = Gt + λKtG,Stock and λ is a

parameter controlling the impact of non-defense public capital on household’s utility. Notice that KtG,Stock represents the stock of non-defense public capital (and not our original KtG ). As a consequence, we have to add an equation governing the process G,Stock of the public capital stock, namely, Kt+1 = (1 − δKg )KtG,Stock + Itg , to our model.

26

4.5.3

Other Robustness

We have also estimated a variation of our Productive Investment Model with a different specification for the ( I )2 adjustment costs, namely: Kt+1 = (1 − δ) Kt − κ2 eεt KItt − δ Kt . We consider a Uniform prior for κ with support in [0, 200]. With θg unrestricted, we still get the mode of θg equal to zero while the mode of κ is around 94. The remaining parameters are very much in line with what obtains in the original specification for the investment adjustment costs. Importantly, we get a mode of ϕ equal to 0.64 while that for υ b is 15.78. Further, when we set θg = 0 the main results are almost unchanged, i.e., the mode of κ, ϕ and υ b are 93.95, 0.65 and 15.96, respectively. We acknowledge that we use a closed economy model while the data is taken from an open economy.29 As a rough robustness exercise, we estimate a version of our Productive Investment Model in which we take out Net Exports from the observable output, i.e., Ytobs := (GDP−Net Exports). Private and government consumption are, again, estimated to be substitute goods. Specifically, the modes for υ b and ϕ are estimated to be equal to 14.3 and 0.65, respectively, while the mode of θg is again 0. Finally, in Appendix A, we analyze the importance of the seven shocks of our benchmark model. The main conclusion is that our main results survive to reasonable perturbations in the number and types of shocks used in the estimation. Given that the main findings related to the substitutability results seem robust, next we turn to the analysis of the responses of our Preferred economy, i.e., the one presented in Table 4 and 5, to government consumption shocks, comparing these responses to what obtains in the case of separable government consumption.30 Given that in some instances we get positive estimates for θg , we also present the dynamics generated by a non-defense public investment shock within the Productive Investment version of the model.

4.6

Impulse-Response Analysis

In this Section we analyze the dynamic effects of government consumption and non-defense investment shocks on selected model variables. Figures 1 and 2 embed the estimated model variables’ reactions to a government consumption shock (within the Preferred economy) and to a government investment shock (within the Productive Investment economy), respectively. The size of the shocks is set to the estimated posterior mean of the corresponding standard deviation and the impulse response functions are expressed as deviations from the steady-state (in percentage points). Within the Figures, each plot presents two lines, 29

We should anyway refer that most of the papers that estimate closed-economy DSGE models share the same problem, e.g., see Smets and Wouters (2007) and Schmitt-Groh´e and Uribe (2011). 30 Results for the analysis in the next Sections under the new specifications studied here are available upon request. Allow us to refer that the main conclusions are robust to these specifications.

27

a black and a dark grey one. The black one is the posterior mean of the estimated responses obtained in the Preferred Model (Figure 1) and in the Productive Investment Model (Figure 2), and is named Posterior Mean. The dark grey line is the reaction obtained by fixing, in the two models, all the parameters at the respective posterior mean, with the difference that the ‘externalities’ parameters, ϕ and θg , are set to 1 and 0, respectively. The latter summarizes the reactions when the externality channels are shut down, and is labeled No Channels.31 The shaded area within each plot draws the 80% Bayesian posterior credibility interval of the estimated impulse response functions. The behavior of the variables in Figure 1 can be explained as follows: because of substitutability, the increase in government consumption lowers the marginal utility of consumption, leading households to substitute part of their private consumption with the newly available government consumption. As a consequence, private consumption (black line) decreases more than private consumption in the No Channels specification (dark grey line). That is, in the Preferred Model, both the negative wealth effect and the substitutability effect sum up. Additionally, households work less in the Preferred framework since - for given negative wealth effect - the marginal utility of consumption is lower than in the No Channels Model. As a result, wages decrease less in the Preferred economy. Importantly, the impact on output is much smaller in the Preferred Model because of the lower increase in labor supply. To accommodate the new path for private consumption, the return to capital increases by less on impact, so that investment is crowded-out less than in the No Channels specification.32 Our estimated reaction for private consumption conflicts with part of the recent empirical evidence based on structural autoregressions (SVAR) represented by, among others, Fatas and Mihov (2001), Blanchard and Perotti (2002), Perotti (2004) and Perotti (2007). These papers find that a government spending shock generates a positive response of private consumption. However, on the other hand, those papers based on the narrative approach, which aims at isolating exogenous shifts in government (military) spending, find opposite results. Indeed, Ramey and Shapiro (1998), Burnside et al. (2004) and Ramey (2011) find that private consumption reacts negatively to government spending shifts, though in some cases the response is weak. It is worth noting that our estimated channel of substitutability favors a negative reaction of private consumption to government spending shocks, but we certainly acknowledge that other channels, not considered in the present analysis, might overturn this crowding-out effect.33 31 Hence, this restricted model is not estimated in order to guarantee that the only parameters changing are those related to public spending externalities. However, we should note that the estimated posterior mean of the impulse response functions obtained with the imposed restrictions are very similar to the reported ones. 32 We note that in the case of a government consumption shock, the reported responses obtained within the Preferred Model are very similar to what obtains in the Productive Investment Model. 33 For example, allowing for the presence of liquidity constrained agents in a framework of sticky prices (e.g., see Gal´ı et al., 2007) could generate a positive reaction of private consumption to government spending shocks, despite C and G being

28

0,3

0,25

oni ta iv e D tea ts -y da et S st ino p eg tan ec re P

no tia iv ed tea ts -y da et S st ino p eg tan ec re P

0,2

Posterior Mean

0,15

0,1

0,05

0,25

Posterior Mean 0,15 No Channels 0,1

0,05

1E-16 4 8 12

4 8 12

oni ta iv ed tea ts -y da teS st ni op eg at ne cr Pe

24

48

72

100

24

48

72

-0,05

I 72

100

-0,005

-0,01

Posterior Mean No Channels

-0,015

-0,02

-0,025

n iot a vie d tea ts -y da et S st ni op eg at ne cr eP

-0,04

4 8 12

24

48

72

-0,1

-0,2 Posterior Mean

-0,3

No Channels

-0,4

4 8 12

48

0

0,45 24

48

72

-0,02

24

C

0,1

100

0

0 4 8 12

n o tia iv ed tea ts -y d ae tS st in o p eg tan ec re P

0,2

0

0,005

0,2

Y

G/Y

W

-0,06

-0,08

-0,1

-0,12 Posterior Mean -0,14 No Channels -0,16

100

L

0,4 n o tia iv0,35 ed tea 0,3 ts -y 0,25 ad teS tsn 0,2 io p0,15 eg tan ec 0,1 re P0,05

Posterior Mean

No Channels

0 4 8 12

-0,18

24

48

72

0,3

rk

oni ta 0,25 iv ed tea 0,2 ts -y da et 0,15 S st ni op eg 0,1 at ne cr 0,05 Pe

Posterior Mean

No Channels

0 4 8 12

24

48

72

100

Figure 1 Impulse Response Functions (IRFs) of the Preferred Model: effects on output (Y), consumption (C), investment (I), wages (W), hours (L) and return on capital (rk ) of a one standard deviation government consumption shock. The black line is the posterior mean of the IRF and the grey area is the associated 80% Bayesian credibility interval. The dark grey line is the IRF obtained with the “externalities” channels closed (ϕ = 1, θg = 0, θg,def = 0) and the other parameters set to the posterior mean obtained from the Preferred Model (denoted by No Channels)

29

100

0,1

n o tia iv e D tea ts -y ad et S st in o p eg at n ec re P

1,2

IG/Y

0,09

n o tia iv e d tea ts -y ade tS st in o p eg ta enc re P

0,08 0,07 Posterior Mean

0,06 0,05 0,04 0,03 0,02 0,01

0,8 Posterior Mean

0,6

0,4

0,2

0,2

24

48

72

100

4 8 12 0,25

C

0,15

0,1

0,05

0 4 8 12

24

48

72

100

-0,05 Posterior Mean -0,1 No Channels -0,15

n iot ai ve d et at sy da et S ts in o p gea t ne cr eP

24

48

72

100

No Channels

0,15

0,1

0,05

0 24

48

72

100

No Channels

0,15

0,1

0,05

0 4 8 12

24

48

72

100

-0,05

4 8 12

0,2

0,16

n oi 0,14 ta vie d0,12 tea ts -y 0,1 da et 0,08 S st ni 0,06 o p eg at 0,04 ne cr 0,02 Pe

n io ta 0,15 vie d tea ts -y 0,1 d ae tS st n i o 0,05 p eg at n cer 0 e P

Posterior Mean

No Channels

24

48

72

24

48

72

-0,02

-0,1

100

Posterior Mean

No Channels

4 8 12

24

48

72

-0,05

0,15

q

0,1

n o tia iv0,05 e d tea 0 ts -y d ae-0,05 tS st -0,1 in o p-0,15 eg at n ec -0,2 re P

4 8 12

24

48

72

100

Posterior Mean No Channels

-0,25

-0,3

Figure 2 Impulse Response Functions (IRFs) of the Productive Investment Model: effects on output (Y), consumption (C), investment (I), wages (W), hours (L), return on capital (rk ), Tobin’s q (q), Private Capital (Private K) and Public Capital Productivity (Public K) of a one standard deviation government investment shock. The black line is the posterior mean of the IRF and the grey area is the associated 80% Bayesian credibility interval. The dark grey line is the IRF obtained with the “externalities” channels closed (ϕ = 1, θg = 0, θg,def = 0) and the other parameters set to the posterior mean obtained from the Productive Investment Model (denoted by No Channels)

30

100

rk

0 4 8 12

100

No Channels

L

Posterior Mean

72

Posterior Mean

Private K

n 0,14 o tia iv e 0,12 d tea ts 0,1 -y 0,08 ad teS st 0,06 in o p 0,04 eg at 0,02 en rce P 0 -0,02

0,18

W

48

-0,05

0,16

0,2

4 8 12

24

No Channels

0,18

Posterior Mean

I

-0,05

0,2

n o tia iv e d tea ts -y d ae tS st n i o p eg at n ec re P

4 8 12

0 4 8 12

Posterior Mean

Y

n 0,2 o tia iv e d tea 0,15 ts -y ad et 0,1 S st in o p eg 0,05 at n ec re P 0

1

0

n io ati ve d et at sy d ae St ts in o p gea t n ec erP

0,25

Public K

100

Turning the focus to Figure 2, we recall that, conditional on a positive value for θg , a non-defense government investment shock increases the productivity of private factors. On the one hand, this type of shock can be seen as a persistent technology shock where both positive wealth effects and substitution effects manifest themselves. First, the shock translates into an increase in households’ permanent income which positively impacts on leisure, consumption, Tobin’s q and, consequently, on investment. Secondly, the rise in productivity entails an increase in the real wage that leads agents to substitute leisure for labor. Thirdly, the immediate increase in the marginal product of capital makes investing more attractive than consuming. Private capital responds gradually to the accumulation of investment. On the other hand, the newly created public investment has to be financed through taxes. The associated negative wealth effect depresses leisure, real wages, and consumption. Thus, the net effects of a public investment shock are potentially ambiguous. In our case, we observe that in the Productive Investment economy both consumption, Tobin’s q (labeled q), and real wages decrease on impact, suggesting that, initially, the negative wealth effect dominates the positive one. Indeed, the posterior mean of the IRFs mimics the behavior of the ones of the No Channels economy. This is also true for labor, return to capital and output, which indeed increase on impact. Gradually, the increase in private productivities starts dominating the negative wealth effect, leading to an increase in Tobin’s q and so in investment. This contributes to the build up of private capital, which, together with the build-up of public capital productivity, directly sustains output growth. Consistent with this fact, wages and consumption go up after the initial decrease. In the Productive Investment economy all the variables show a much higher degree of persistence vis-`a-vis the No Channels world; this occurs because private and public capital take time to accumulate and to release their effects on the economy. This process is fueled by the high estimated persistence of the public investment process.34 Importantly, and in accordance with our estimation results, it is worth noticing the ample width of the posterior intervals. The next Section completes the quantitative analysis, resorting to the analysis of dynamic multipliers induced by the public spending externalities.

4.7

Dynamic Multipliers

We analyze here public spending multipliers associated with the estimated effects of government spending shocks on output, consumption and investment. We use the notion of present value multipliers formulated substitutes. 34 Interestingly, the estimated dynamics for both the return to capital, rk , and Tobin’s q, a scale measure of market value, are in line with the results of Belo and Yu (2013) who focus on the asset pricing implications of government investment. Indeed, they find that government investment forecasts both aggregate total factor productivity and the risk premium, which is related to the firms’ return to capital. Further, these effects are found to be stronger the larger is the forecast horizon. They also show that these empirical results are consistent with a Neoclassical model, similar to ours, that allows government capital, or transformations of it, to affect private productivities.

31

in Mountford and Uhlig (2009); the present value multiplier of output, say, t quarters after an increase in government consumption (or investment) is: t ∑

φt =

k=1 t ∑

(1 + rss )−k ∆yk (1 + rss )−k ∆gk

k=1

where ∆yk represents the actual deviation of stationarized output from its steady-state at time k, ∆gk represents the actual deviation of stationarized public consumption (investment) from its steady-state at time k, and rss is the steady-state real interest rate on the risk-free asset. The expression generalizes for the case of consumption and investment.35 Tables 8 and 9 show the present value multipliers for Y , C and I and for various periods in response to a government consumption and a government investment shock, conditional on the Preferred and Productive Investment specifications, respectively. We look at the posterior mean of the multipliers and also at an 80% Bayesian credibility interval. Looking at the multipliers produced by a government consumption shock (Table 8), we see that - under the Preferred specification - the one related to output reaches its maximum at 0.33 on impact, and then slowly decreases. The output multiplier calculated within the No Channels specification turns out to be 0.99 on impact, i.e. three times bigger than the one in our Preferred economy.36 The multiplier value obtained under the Preferred specification is close to the impact output multiplier found by Mountford and Uligh (2009), which equals 0.31. This is obtained within a VAR identified through sign restrictions, where taxes are forced to adjust so as to fully finance the increase in government spending during the first four quarters after the shock. Clearly, our 80% posterior interval does not even contain the impact values found by Blanchard and Perotti (2002), which are 0.90 (under a deterministic detrending of the data) and 0.84 (under a stochastic one).37 As expected, the multipliers for consumption and investment are negative, though the 80% posterior Since the solution of the model is in (log) deviations of the stationarized variables from the steady-state, we use yˆk ∗ yss and gˆk ∗ gss as approximations for ∆yk and ∆gk , respectively. 36 Notice two things. First, if we estimate directly the No Channels specification we still obtain a posterior mean for the impact output multiplier around unity, or 1.02. Second, our No Channels multiplier is close to unity mostly because of the low estimated value for σL . Simulating the No Channels Model for higher values of σL , which implies lower values of the Frisch elasticity, delivers lower values for the multipliers. For example, setting σL = 2 delivers an output multiplier around 0.85 in the No Channels economy. 37 It is again worth noting that many of the analyzes of the effects of government spending focus on military spending, instead of government consumption. Among others, Barro and Redlick (2010) estimate an output multiplier ranging from 0.6 to 0.7 at the median unemployment rate (reaching 1.0 when the unemployment rate is around 12%); also, they find a crowding out effect for investment and net exports. Hall (2009)’s range for the output multiplier is 0.7-1.0. Finally, Ramey (2011) finds output multipliers in the range 0.6-1.2 (at peak GDP). 35

32

Table 8 - Dynamic Multipliers, Preferred vs No Channels Model, Government Consumption Shock Quarters Y Preferred

1

2

4

8

12

24

48

72

100

0.33

0.28

0.25

0.19

0.17

0.16

0.14

0.14

0.13

(0.14,0.50)

(0.12,0.43)

(0.11,0.35)

(0.09,0.29)

(0.08,0.27)

(0.07,0.25)

(0.06,0.24)

(0.05,0.24)

(0.04,0.0.24)

Y No Channels

0.99

0.89

0.77

0.65

0.60

C Preferred

−0.75

−0.79

−0.82

−0.85

−0.86

(−0.87,−0.64)

(−0.89,−0.68)

(−0.92,−0.73)

(−0.96,−0.77)

(−0.98,−0.75)

0.55 −0.87

0.51

0.49

0.48

−0.89

−0.89

−0.90

(−1.19,−0.63)

(−1.24,−0.61)

(−1.04,0.−0.73) (−1.13,−0.67)

C No Channels

−0.24

−0.32

−0.43

−0.52

−0.56

−0.61

−0.63

−0.64

I Preferred

−0.001

−0.001

−0.001

−0.001

−0.001

−0.002

−0.002

−0.003

−0.003

(−0.002,0.000)

(−0.003,0.000)

(−0.004,0.000)

(−0.007,0.000)

(−0.009,0.000)

(−0.014,0.000)

(−0.020,0.000)

(−0.024,0.000)

(−0.026,0.000)

I No Channels

−0.003

−0.004

−0.005

−0.009

−0.012

−0.019

−0.65

−0.029

−0.035

−0.039

48

72

100

80% Bayesian credibility interval in parenthesis

Table 9 - Dynamic Multipliers, Productive Investment vs No Channels Model, Government Investment Shock Quarters Y Productive I.

1

2

4

8

12

0.99

0.89

0.81

0.88

1.03

1.57

2.50

3.12

3.56

(0.86,1.15)

(0.74,1.07)

(0.62,1.05)

(0.54,1.31)

(0.53,1.68)

(0.57,2.85)

(0.65,4.98)

(0.71,6.39)

(0.75,7.32)

Y No Channels

1.01

0.90

0.76

0.63

0.57

C Productive I.

−0.40

−0.46

−0.51

−0.49

(−0.47,−0.29)

(−0.54,−0.33)

(−0.61,−0.37)

(−0.66,−0.28)

C No Channels I Productive I. I No Channels

24

−0.24

−0.32

−0.42

−0.51

0.51

0.47

0.46

0.44

−0.42

−0.15

0.31

0.64

0.90

(−0.66,−0.11)

(−0.61,0.47)

(−0.51,1.39)

(−0.46,2.12)

(−0.42,2.67)

−0.59

−0.61

−0.63

−0.64

−0.55

0.01

0.01

0.02

0.04

0.06

0.14

0.33

0.48

0.59

(0.00,0.01)

(0.00,0.02)

(0.00,0.04)

(0.00,0.09)

(0.00,0.14)

(0.00,0.32)

(0.02,0.74)

(0.03,1.07)

(0.04,1.31)

−0.003

−0.005

−0.007

−0.012

−0.017

−0.029

−0.046

−0.055

−0.059

80% Bayesian credibility interval in parenthesis

interval for investment contains zero. In the case of consumption, the multipliers are clearly below those obtained in the No Channels Model, especially at short horizons, whereas for investment they are above, but still negative. Turning now to Table 9 we observe that, as expected, the multipliers’ posterior intervals are very large and often include zero. Focusing on the values at the estimated mean, government investment multipliers become large only after some quarters. The output multiplier starts slightly below 1, at 0.99, similar to the No Channels multiplier, but increases as public capital productivity builds up, reaching 1.03 after three years and 1.57 after six. The associated 80% posterior intervals contain the multipliers found by Leeper et al. (2010) and Baxter and King (1993) but not the ones of Perotti (2004). In our case, even the investment multiplier builds up over time, reaching, for instance, 0.14 after six years. The consumption multiplier starts at negative values, similar to those in the No Channels case, becoming positive after more than six years. We should stress that, whenever the consumption multiplier is positive, the associated posterior interval contains zero, contrary to what occurs with investment.

5

Conclusions

This paper has posed attention on the potential externalities produced by public expenditures, focusing on how these externalities affect the response of the economy to two government spending shocks: a government consumption shock and a government investment shock. To this effect, we have built an otherwise standard RBC model extended with two important features. First, we have allowed government consumption to affect 33

the welfare of agents, by entering directly households’ utility function. Second, we have allowed public capital to shift the productivity of private factors, by entering in the firms’ production function. We have provided an identification analysis that supports the strategy adopted for estimating the parameters governing these two channels. On the one hand, our results question the standard hypothesis of separability between private and government consumption, as the two goods are robustly estimated to be substitutes. Because of substitutability labor supply reacts little to a government consumption shock, so the estimated output multiplier is much lower than the one measured in models with separable government consumption. On the other hand, we do not find clear evidence supporting public investment as a shifter of the production frontier. In a few specifications we find that non-defense public investment enhances mildly the productivity of private factors while investment in defense appears, robustly, not to have any such impact. Our results show that incorporating the channels we study into general equilibrium models can be important to understand and measure more thoroughly the expected impacts of fiscal stimuli and consolidations, as well as to conduct welfare analysis of fiscal policy. It will be worth investigating how our measures interact with several important features of fiscal and monetary policy such as debt smoothing details or the zero lower bound on nominal interest rates. This is left for future research.

34

References [1] Ahmed, S. (1986), “Temporary and Permanent Government Spending in an Open Economy”, Journal of Monetary Economics, 17 (3), 197-224. [2] Albuquerque, R., M. Eichenbaum and S. Rebelo (2013), “Valuation Risk and Asset Pricing”, Northwestern University, mimeo. [3] Alesina A., C. Favero and F. Giavazzi (2012), “The Output Effect of Fiscal Consolidations”, Bocconi University, mimeo. [4] Amano, A.R. and T.S. Wirjanto (1998), “Government Expenditures and the Permanent-Income Model”, Review of Economic Dynamics, 1 (3), 719-30. [5] An, S. and F. Schorfheide (2007), “Bayesian Analysis of DSGE Models”, Econometric Reviews, 26 (2-4), 113-172. [6] Aschauer, D. A. (1985), “Fiscal policy and aggregate demand”, American Economic Review, 75 (1), 117–127. [7] Aschauer, D. A. (1989), “Is public expenditure productive?”, Journal of Monetary Economics, 23 (2), 177-200. [8] Bailey, M. J. (1971), “National Income and the Price Level: A Study in Macroeconomic Theory”, 2nd Edition, New York: McGraw–Hill. [9] Barro, R. (1981), “Output Effects of Government Purchases”, Journal of Political Economy, 89 (6), 1086-1121. [10] Barro, R. and C. Redlick (2010), “Macroeconomic Effects from Government Purchases and Taxes”, Harvard University, mimeo. [11] Baxter, M. and R. G. King (1993), “Fiscal Policy in General Equilibrium”, American Economic Review, 83 (3), 315-334. [12] Belo, F. and J. Yu (2013), “Government Investment and the Stock Market”, Journal of Monetary Economics, 60 (3), 325-339.

35

[13] Blanchard, O. J. and R. Perotti (2002), “An Empirical Characterization of the Dynamic Effects of Changes in Government Spending and Taxes on Output”, Quarterly Journal of Economics, 117 (4), 1329-1368. [14] Bouakez, H. and R. Rebei (2007), “Why Does Private Consumption Rise after a Government Spending Shock?”, Canadian Journal of Economics, 40 (3), 954-979. [15] Brooks, S. P. and A. Gelman (1998), “General Methods for Monitoring Convergence of Iterative Simulations”, Journal of Computational and Graphical Statistics, 7 (4), 434-455. [16] Burnside, C., M. Eichenbaum and J. Fisher (2004), “Fiscal shocks and their consequences”, Journal of Economic Theory, 115 (1), 89-117. [17] Chari, V. V., P. J. Kehoe and E. R. McGrattan (2009), “New Keynesian Models: Not Yet Useful for Policy Analysis”, American Economic Journal: Macroeconomics, 1 (1), 242-266. [18] Christiano, L., M. Eichenbaum and C. Evans (2005), “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy”, Journal of Political Economy, 113 (1), 1-45. [19] Christiano, L., M. Eichenbaum, and S. Rebelo (2011), “When is the Government Spending Multiplier Large?”, Journal of Political Economy, 119 (1), 78-121. [20] Coenen, G., C.J. Erceg, C. Freedman, D. Furceri, M. Kumhof, R. Lalonde, D. Laxton, J. Linde, A. Mourougane, D. Muir, S. Mursula, C. de Resende, J. Roberts, W. Roeger, S. Snudden, M. Trabandt and J. in’t Veld (2012), “Effects of Fiscal Stimulus in Structural Models”, American Economic Journal: Macroeconomics 4 (1), 22–68. [21] Coenen, G., R. Straub and M. Trabandt (2013), “Gauging the Effects of Fiscal Stimulus Packages in the Euro Area”, Journal of Economic Dynamics and Control, 37 (2), 367–386. [22] Cogan, J.F., T. Cwik, J.B. Taylor and V. Wieland (2010), “New Keynesian versus old Keynesian government spending multipliers”, Journal of Economic Dynamic and Control, 34 (3), 281–295. [23] Cole, H. and L. Ohanian (2002), “The U.S. and U.K. Great Depressions though the Lens of Neoclassical Growth Theory”, American Economic Review, 92 (2), 28-32. [24] Drautzburg, T. and H. Uhlig (2011), “Fiscal Stimulus and Distortionary Taxation”, NBER Working Paper Series 17111.

36

[25] Fat´as A. and I. Mihov (2001), “The Effects of Fiscal Policy on Consumption and Employment: Theory and Evidence”, CEPR Discussion Papers, 2760. [26] Fernandez-Villaverde, J. and J. F. Rubio-Ramirez (2005), “Estimating Dynamic Equilibrium Economies: Linear versus Nonlinear Likelihood”, Journal of Applied Econometrics, 20 (7), 891–910. [27] Fernandez-Villaverde, J. (2010), “Fiscal Policy in a Model With Financial Frictions”, American Economic Review P&P, 100 (2), 35-40. [28] Finn, M. (1993), “Is All Government Capital Productive?”, mimeo Federal Reserve Bank of Richmond. [29] Fiorito, R. and T. Kollitznas (2004), “Public goods, merit goods, and the relation between private and government consumption”, European Economic Review, 48 (6), 1367-1398. [30] Gal´ı, J., J. D. L´opez-Salido and J. Vall´es (2007), “Understanding the Effects of Government Spending on Consumption,” Journal of the European Economic Association, 5 (1), 227-270. [31] Geweke, J. F. (1999), “Using Simulation Methods for Bayesian Econometric Models: Inference, Developments and Communication”, Econometric Reviews, 18, 1-126. [32] Hall, R. (2009), “By How Much Does GDP Rise if the Government Buys More Output?”, Brookings Papers on Economic Activity. [33] Iskrev, N. (2010a), “Local Identification in DSGE Models”, Journal of Monetary Economics, 57 (2), 189-202. [34] Iskrev, N. (2010b), “Evaluating the Strength of Identification in DSGE Models. An a priori Approach”, Working Papers 32, Banco de Portugal. [35] Justiniano, A., G. Primiceri and A. Tambalotti (2010), “Investment Shocks and Business Cycles”, Journal of Monetary Economics, 57 (2), 132-145. [36] Justiniano, A., G. Primiceri and A. Tambalotti (2011), “Investment Shocks and the Relative Price of Investment”, Review of Economic Dynamics, 14 (1), 101-121. [37] Karras, G. (1994), “Government spending and private consumption: Some international evidence”, Journal of Money, Credit and Banking 26 (1), 9–22. [38] King, R. G., C. Plosser and S. Rebelo (1988), “Production, growth and business cycles : I. The basic neoclassical model” Journal of Monetary Economics, 21 (2-3), 195-232. 37

[39] Leeper, E. M., M. Plante and N. Traum (2009), “Dynamics of Fiscal Financing in the United States”, NBER Working Paper Series 15160. [40] Leeper, E., T. B. Walker and S.S. Yang (2010), “Government investment and fiscal stimulus”, Journal of Monetary Economics, 57(8), 1000-1012. [41] Linnemann, L. (2006), “The Effects of Government Spending on Private Consumption: A Puzzle?”, Journal of Money, Credit, and Banking 38 (1), 1715-1736. [42] Linnemann, L. and A. Schabert (2003), “Fiscal Policy in the New Neoclassical Synthesis”, Journal of Money, Credit and Banking, 35 (6), 911-929. [43] McGrattan, E. R. and L. Ohanian (2010), “Does Neoclassical Theory Account for the Effects of Big Fiscal Shocks? Evidence From World War II”, International Economic Review, 51 (2), 509-532. [44] Monacelli, T., R. Perotti, and A. Trigari (2010), “Unemployment fiscal multipliers”, Journal of Monetary Economics, 57 (5), 531-553. [45] Mountford, A. and H. Uhlig (2009), “What are the Effects of Fiscal Policy Shocks?”, Journal of Applied Econometrics, 24 (6), 960-992. [46] Pappa, E. (2009), “The Effects of Fiscal shocks on Employment and The Real Wage”, International Economic Review, 50 (1), 217-244. [47] Perotti, R. (2004), “Public Investment: Another (Different) Look”, Universita Bocconi, mimeo. [48] Perotti, R. (2007), “In Search of the Transmission Mechanism of Fiscal Policy”, NBER Working Papers 13143. [49] Ramey, V. A. (2011), “Identifying Government Spending Shocks: It’s all in the Timing”, The Quarterly Journal of Economics, 126 (1), 1-50. [50] Ramey, V. A. and M. D. Shapiro (1998), “Costly Capital Reallocation and the Effects of Government Spending”, Carnegie-Rochester Conference Series on Public Policy, 48(1),145-194. [51] Schmitt-Groh´e, S. and M. Uribe (2006), “Optimal Fiscal and Monetary Policy in a Medium-Scale Macroeconomic Model”, NBER Macroeconomics Annual 2005, 20, 383-462. [52] Schmitt-Groh´e, S. and M. Uribe (2011), “Business Cycles With A Common Trend in Neutral and Investment-Specific Productivity”, Review of Economic Dynamics, 14 (1), 122-135. 38

[53] Smets, F. and R. Wouters (2007), “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach”, American Economic Review, 97 (3), 586-606. [54] Straub, R. and I. Tchakarov (2007), “Assesing the Impact of a Change in the Composition of Public Spending: a Dsge Approach”, ECB Working Paper 479. [55] Tatom, J. A. (1991), “Public capital and private sector performance”, Federal Reserve Bank of St. Louis Review, 3-15. [56] Uhlig, H. (2010), “Some fiscal calculus”, American Economic Review, 100 (2), 30–34. [57] Walker, A.M. (1969), “On the Asymptotic Behaviour of Posterior Distributions”, Journal of the Royal Statistical Society, 31 (2), 80-88.

39

Appendix A: Further Results Sub-Samples Comparison: Distortionary Taxation, Unrestricted Versions

Table I - A: Priors and Posteriors of selected parameters 1969Q1-2008Q3 Distortionary taxation, Unrestricted Parameter

Distr.

PRIOR Mean

St. Dev.

POSTERIOR 1969-2008 Mode Mean 5% 95%

POSTERIOR 1984-2008 Mode Mean 5% 95%

A. Utility function h σL υ b := log(υ) ϕ

Beta N ormal U nif orm U nif orm

0.7 2

0.1 0.5 [−5, 20] [0, 1]

0.77 0.31 7.56 0.65

0.80 0.86 10.75 0.63

0.72 0.21 2.87 0.52

0.87 1.32 19.99 0.74

0.85 0.38 7.70 0.66

0.95 0.16 11.41 0.51

0.93 0.01 4.02 0.40

0.97 0.32 19.92 0.63

0.05 0.3

0.02 0.02

0.020 0.38 0.0

0.019 0.39 0.09

0.00 0.38 0.00

0.038 0.41 0.20

0.007 0.34 0.00

0.015 0.33 0.001

0.00 0.31 0.00

0.029 0.35 0.002

B. Production function Φ α θg

N ormal N ormal U nif orm

[0, 4]

C. Investment Adj. costs κ/100 γ2

N ormal N ormal

4 0.0685

0.5 0.002

4.74 0.063

4.79 0.063

4.12 0.060

5.45 0.065

4.50 0.062

4.74 0.062

4.12 0.060

5.32 0.064

N ormal N ormal N ormal N ormal

0.4 0.16 0.025 0.008

0.02 0.01 0.001 0.001

0.40 0.169 0.027 0.009

0.40 0.169 0.027 0.009

0.37 0.161 0.025 0.007

0.43 0.178 0.028 0.010

0.41 0.159 0.025 0.008

0.41 0.158 0.025 0.008

0.41 0.151 0.024 0.007

0.41 0.167 0.026 0.009

D. Constant terms γ ∗ 100 ξ g,ss ξ ig,ss ξ g,def,ss

Table II - A: Priors and Posteriors of Shocks parameters 1969Q1-2008Q3 Distortionary taxation, Unrestricted Parameter

Distr.

PRIOR Mean

St. Dev.

POSTERIOR 1969-2008 Mode Mean 5% 95%

Mode

1984-2008 Mean 5%

95%

A. Autoregressive Parameters ρb ρa ρI ρw ρg ρig ρig,def

Beta Beta Beta Beta Beta Beta Beta

0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.73 0.08 0.95 0.96 0.97 0.94 0.95

0.73 0.10 0.95 0.95 0.98 0.94 0.95

0.68 0.02 0.92 0.92 0.96 0.91 0.93

0.77 0.17 0.98 0.99 0.99 0.98 0.98

0.71 0.10 0.96 0.88 0.97 0.89 0.98

0.70 0.13 0.95 0.29 0.97 0.90 0.97

0.65 0.02 0.92 0.12 0.95 0.83 0.96

0.75 0.23 0.98 0.49 0.99 0.97 0.99

0.1 0.1 0.1 0.1 0.1 0.1 0.1

2.0 2.0 2.0 2.0 2.0 2.0 2.0

3.63 0.024 0.039 0.023 0.015 0.030 0.086

3.61 0.024 0.040 0.032 0.015 0.030 0.087

3.16 0.022 0.030 0.020 0.014 0.028 0.080

4.03 0.026 0.050 0.044 0.017 0.033 0.096

2.75 0.021 0.032 0.033 0.013 0.022 0.054

2.83 0.022 0.032 0.078 0.014 0.022 0.055

2.37 0.020 0.024 0.056 0.012 0.019 0.049

3.22 0.025 0.041 0.10 0.015 0.024 0.061

B. Standard deviation of shocks σb σa σI σw σg σig σig,def

Inv Inv Inv Inv Inv Inv Inv

Gamma Gamma Gamma Gamma Gamma Gamma Gamma

40

Post-1984 Sample: Preferred (θg = 0) vs. Unrestricted specification

Table I - B Priors and Posteriors of selected parameters 1984Q1-2008Q3 Preferred specification (θg = 0) vs. Productive Investment PRIOR Parameter

Distr.

POSTERIOR

Mean

St. Dev.

Mode

Preferred Mean 5%

95%

Productive Investment Mode Mean 5% 95%

A. Utility function h σL υ b := log(υ) ϕ

Beta N ormal U nif orm U nif orm

0.7 2

0.1 0.5 [−5, 20] [0, 1]

0.85 0.35 8.91 0.66

0.94 0.17 14.3 0.47

0.93 0.01 2.44 0.36

0.96 0.37 26.5 0.58

0.85 0.38 7.70 0.66

0.95 0.16 11.41 0.51

0.93 0.01 4.02 0.40

0.97 0.32 19.92 0.63

0.05 0.3

0.02 0.02

0.007 0.34 -

0.014 0.33 -

0.00 0.31 -

0.028 0.35 -

0.007 0.34 0.00

0.015 0.33 0.001

0.00 0.31 0.00

0.029 0.35 0.002

B. Production function Φ α θg

N ormal N ormal U nif orm

[0, 4]

C. Investment Adj. costs κ/100 γ2

N ormal N ormal

4 0.0685

0.5 0.002

4.50 0.062

4.84 0.061

4.16 0.058

5.51 0.064

4.50 0.062

4.74 0.062

4.12 0.060

5.32 0.064

N ormal N ormal N ormal N ormal

0.4 0.16 0.025 0.008

0.02 0.01 0.001 0.001

0.41 0.159 0.025 0.008

0.41 0.158 0.025 0.008

0.37 0.150 0.024 0.007

0.44 0.166 0.026 0.009

0.41 0.159 0.025 0.008

0.41 0.158 0.025 0.008

0.41 0.151 0.024 0.007

0.41 0.167 0.026 0.009

D. Constant terms γ ∗ 100 ξ g,ss ξ ig,ss ξ g,def,ss

Table II - B Priors and Posteriors of Shocks parameters 1984Q1-2008Q3 Preferred specification vs. Unrestricted PRIOR Parameter

Distr.

POSTERIOR θg = 0 Mean 5%

Mean

St. Dev.

Mode

0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.71 0.10 0.96 0.88 0.97 0.89 0.98

0.71 0.13 0.95 0.35 0.97 0.90 0.98

0.1 0.1 0.1 0.1 0.1 0.1 0.1

2.0 2.0 2.0 2.0 2.0 2.0 2.0

2.74 0.021 0.031 0.034 0.013 0.022 0.054

2.77 0.022 0.034 0.070 0.014 0.022 0.055

Unrestricted Mean 5%

95%

Mode

95%

0.65 0.02 0.91 0.10 0.95 0.83 0.96

0.76 0.22 0.99 0.57 0.99 0.97 0.99

0.71 0.10 0.96 0.88 0.97 0.89 0.98

0.70 0.13 0.95 0.29 0.97 0.90 0.97

0.65 0.02 0.92 0.12 0.95 0.83 0.96

0.75 0.23 0.98 0.49 0.99 0.97 0.99

2.36 0.019 0.022 0.055 0.013 0.019 0.048

3.21 0.024 0.047 0.083 0.014 0.024 0.062

2.75 0.021 0.032 0.033 0.013 0.022 0.054

2.83 0.022 0.032 0.078 0.014 0.022 0.055

2.37 0.020 0.024 0.056 0.012 0.019 0.049

3.22 0.025 0.041 0.10 0.015 0.024 0.061

A. Autoregressive Parameters ρb ρa ρI ρw ρg ρig ρig,def

Beta Beta Beta Beta Beta Beta Beta

B. Standard deviation of shocks σb σa σI σw σg σig σig,def

Inv Inv Inv Inv Inv Inv Inv

Gamma Gamma Gamma Gamma Gamma Gamma Gamma

41

Results with different sets of shocks Our benchmark model contains seven orthogonal shocks: public investment in defense, non-defense public investment, government consumption, preference, technology, investment adjustment costs and the wage markup shock. In order to understand the importance of the seven shocks in the benchmark model we estimated several versions of the model with shocks removed, one at a time. Notice that once we remove a shock, we must also eliminate one observable. We eliminated the observable more obviously related to the shock or, whenever estimation proved unfeasible, a close alternative. Notice that the amount of specifications that can be tried is very large. For each shock removed one can consider removing as many variables as those currently in the model (except if stochastic singularities arise). We focus here on the Productive Investment version of the model and estimate the externalities parameters, ϕ, υ b and θg using our full sample, i.e., 1969Q1-2008Q3. Table III below summarizes the results. The first column indicates the shock removed, the second the observable removed. We report the posterior mode of ϕ, υ b and θg . Notice that when we eliminate the government spending variables, the parameters representing the fraction of that item in output are not identified (e.g., the government consumption to output ratio in steady state, ξ g,ss , is patently not identified if G is not observed). In these cases we fix the parameters to what we obtained in our Productive Investment Model. The first main conclusion is that estimation of ϕ and υ b is little affected if we remove one of the following shocks: the shock to public investment in defense (and the corresponding observable), the shock to nondefense public investment (and the corresponding observable), the wage markup (and wages as observable) and the investment adjustment shock (along with investment as observable). The parameter ϕ is still around 0.65 in these cases and, although there is some variation in υ b, the values found are quite high, indicating strong substitutability. Within these variations the mode of θg changes in two instances: when we eliminate the shock to public investment in defense (and the corresponding observable), in which case θg seems not identified with the mode going towards 4.0, the boundary of the prior, and also when the shock to investment adjustment costs and investment (as observable) are eliminated (θg = 0.12), suggesting that public investment is productive. What changes substantially the estimates of ϕ and υ b (with no effect on θg ) is the removal of the government consumption shock (along with government consumption as observable). In this case it is very difficult to estimate the model and ϕ seems to be not identified, with the mode going towards 1.0. Also, removing wages and the technology shock has little effect on the estimation of ϕ and υ b but only if some parameters are fixed; we had to fix h to 0.77 (obtained in the Productive

42

Investment Model), otherwise its mode converges towards 1.0. Next, removing the preference shock is also very problematic, whether we remove consumption or wages from the observables. With consumption removed, and not surprisingly, most parameters of the utility function become very difficult to estimate (in particular the habit parameter, h, the inverse of the Frisch elasticity, σL , as well as ϕ), in such a way that we were unable to stabilize the estimation procedure (results are not reported). When we remove wages instead, the habit parameter, h, is unreasonably high (0.99). In this case, however, we still find a mode of ϕ equal to 0.66 and a mode of υ b equal to 7.27. The results above suggest that our main results can be obtained within a model with fewer shocks visa-vis the original model. In pursuit of a model with fewer shocks, and taking into account these results, ` we decided to estimate a version of the model removing the non-defense public investment shock (and corresponding observable) and the wage markup shock (removing wages from the observables). These seem to be the least important if the focus is on the parameters ϕ, υ b and θg . As expected, the posterior modes of these parameters in this simplified model do not change much compared to our benchmark specification. We obtain, ϕ, υ b and θg equal to 0.65, 1.67 and 0.0, respectively. Only υ b is farther away from the benchmark value although 1.67 still implies a very large elasticity of substitution (recall that υ := exp(υ b) , so υ equals 5.31). In order to further reduce the number of shocks, we tried to estimate the Preferred Model (i.e., with θg = 0 ) removing the non-defense public investment shock (and corresponding observable), the defense investment shock (and corresponding observable) as well as the wage markup shock (removing wages from the observables). The results make clear that for the purpose of estimating the relation in preferences between C and G this specification is sufficient. In fact, the estimated modes of ϕ and υ b (0.65 and 17.88, respectively) still indicate a strong substitutability. In the same vein, we have estimated the Productive Investment Model (i.e., with θg unrestricted) removing the government consumption shock (and corresponding observable), the defense public investment shock (and corresponding observable) as well as the wage markup shock (removing wages from the observables). In this case, we find a mode of θg equal to zero, ϕ equal to 0.12 (rather low, confirming the unreliable results obtained once G is removed from the observables) and υ b equal to 17.88, supporting again strong substitutability. Given these consequences for ϕ and υ b, we decided to estimate yet another version with these parameters fixed, specifically, ϕ = 0.66 and υ b = 8.0. In this case the estimated mode of θg is slightly positive, equal to 0.035. In short, if we had assumed from the onset that θg,def = 0, the main results in the paper could have been obtained with only 5 shocks instead of 7: technology, preference, investment adjustment costs, public investment (non-defense) and consumption. We thus agree the model could survive without the wage markup 43

shock, although there is hindsight in this conclusion. If we reduced the paper to estimating the relation in preferences between public and private consumption we could have further dropped the public investment shock (non-defense). Table III - Mode of selected parameters with shocks and observables removed Sample: 1969Q1-2008Q3

Shock removed

Observable removed

εig,def t εig t εgt λw,t εa t εIt εbt εig,def , λw,t t ig,def εig , λw,t t , εt εgt , εig,def , λw,t t εgt , εig,def , λw,t t

ξtig,def ξtig ξtg,obs Wtobs Wtobs Itobs Wtobs ξtig,def,obs , Wtobs ig,obs ξt , ξtig,def,obs , Wtobs ξtg,obs , ξtig,def,obs , Wtobs ξtg,obs , ξtig,def,obs , Wtobs

Notes Baseline

h = 0.77

ϕ = 0.66 υ b = 8.0

ϕ 0.65 0.65 0.65 ≈ 1.00 0.62 0.58 0.62 0.66 0.65 0.65 0.12 −

Estimates υ b θg 7.56 0.00 9.12 11.49 NA 2.87 7.22 8.90 7.27 1.67 17.88 12.92 −

0.00 4.00 0.00 0.00 0.00 0.12 0.00 0.00 0.00 0.00 0.035

g

Notes: εbt - Preference shock, εat - Technology, εIt - Investment Adjustment, λw,t - Wage markup, εt - Government ig

ig,def

consumption, εt - Government investment (non-defense), εt

- Government investment (defense)

Ytobs - Output, Ctobs - Consumption, Itobs - (non-defense) Investment, Wtobs - Wages, ξtg,obs - Government consumption ig,obs

to Output ratio, ξt

ig,def,obs

- Government investment (non-defense) to Output ratio, ξt

(defense) to Output ratio

44

- Government investment

Appendix B: Equilibrium Conditions with Transformed Variables We restrict attention to the model with distortionary taxation. The version with lump-sum taxation obtains with all marginal tax rates equal to zero. Equilibrium conditions follow from the first order conditions (F.O.C.s) of households’ and firms’ problems while imposing symmetry, fiscal policy equations, market clearing conditions and processes for the exogenous processes. We then stationarize the variables Yt , Ct , It , Kt , Wt , Tt , Gt , Itg and Itg,def dividing them by the level of technology, At . Lower case letters indicates transformed variables. The same treatment is required for the Lagrange multipliers associated with the budget constraint and the capital accumulation equation, respectively λt and qt (Tobin’s q). • Aggregator (consumption):

[ ] υ υ−1 υ−1 υ−1 e ct = ϕ (ct ) υ + (1 − ϕ) (gt ) υ

(30)

• Consumption F.O.C.: [ ( )1 ] ] υ [ e c t (e ct exp(γ + εat ) − γe ct−1 )−1 (1 + τ c )λt exp(−(γ + εat )) = e ϕ ct εbt

• Labor supply F.O.C.:

(31)

b

eεt χLσt n λt = (1 − τ w )wt • Risk free asset F.O.C.:

[ βEt

(32)

] λt+1 a exp(−(γ + εt+1 ))(1 + rt ) = 1 λt

(33)

• Investment F.O.C.: {[ λt = λt qt Et

κ 1− 2

(

εIt it exp(γ

e

+ εat )

it−1

)2 ] −e



γ

it−1

exp(γ +

I εat )κeεt

( )} a εIt it exp(γ + εt ) γ e −e + it−1

(34) ( ( ) )] 2 a a εit+1 it+1 exp(γ + εt+1 ) a εIt+1 it+1 exp(γ + εt+1 ) εIt+1 γ e λt+1 qt+1 exp(−(γ + εt+1 )) e κe −e it it

[ + βEt

it

45

• Next period capital F.O.C.: [ ] k λt qt = Et βλt+1 exp(−(γ + εat+1 ))rt+1 ut+1 − a (ut+1 )](1 − τ k ) + qt+1 (1 − δ) where a (ut ) = γ1 (ut − 1) +

γ2 2

(35)

(ut − 1)2 represents the cost of using capital at intensity ut .

• Capital law of motion: [ kt+1 = (1 − δ) kt exp(−(γ +

εat ))

+ it

κ 1− 2

( )2 ] a γ εIt it exp(γ + εt ) −e e it−1

(36)

• Capacity utilization F.O.C.: rtk = a′ (ut ) = γ1 + γ2 (ut − 1)

(37)

• Marginal rate of substitution consumption/labor: mrst = eεt

χLσt n (1 + τ c )λt

(38)

1 + λw,t =

wt (1 − τ w ) mrst

(39)

b

• Wage markup:

• Production function: yt = exp(−α(γ + εat ))(ut kt )α (Lt )1−α (KtG )θg − Φ

(40)

• Factor demands: (1 − α)

α • Marginal cost:

yt exp(γ + εat ) = ut rtk kt (1 + λp,ss ) (

mct =

yt = wt Lt (1 + λp,ss )

ut rtk



wt1−α

αα (1 − α)

1−α

46



1 (KtG )θg (KtG,def )θg,def

(41)

(42)

(43)

• Price markup: 1 = 1 + λp,ss mct

(44)

• Evolution of the productivity of public capital: G Kt+1 = (1 − δKg )KtG + ξtig

(45)

ig ig ig ig ig,ss /(1 − ξ ig,ss )) where ξtig = igt /yt = exp(εig t + ss )/(1 + exp(εt + ss )) and ss = log(ξ

G,def Kt+1 = (1 − δKg,def )KtG,def + ξtig,def

(46)

where ξtig,def = ig,def /yt = exp(εig,def +ssig,def )/(1+exp(εig,def +ssig,def )) and ssig,def = log(ξ ig,def,ss /(1− t t t ξ ig,def,ss )) • Government consumption: ξtg = gt /yt = exp(εgt + ssg )/(1 + exp(εgt + ssg ))

(47)

where ssg = log(ξ g,ss /(1 − ξ g,ss )) • Balanced government budget: [ ] τ c ct + τ w wt Lt + τ k rtk ut − a (ut ) kt + tt = ξtg yt + ξtig yt + ξtig,def yt

(48)

• Shocks processes: εbt = ρb εbt−1 + ηtb εIt = ρi εIt−1 + ηtI εat = ρa εat−1 + ηta λw,t = (1 − ρw )λw,ss + ρw λw,t−1 + ηtw εgt = ρg εgt−1 + ηtg ig ig εig t = ρig εit−1 + ηt ig,def εig,def = ρig,def εiig,def t t−1 + ηt

47

(49)

Appendix C: Bayesian Estimation and Inference Bayesian estimation entails specifying prior distributions for the parameters that are not fixed. Let P (θ|m) be the prior distribution of the parameter vector θ ∈ Θ for some model m ∈ M and L(XT |θ, m) be the likelihood function for the observed data XT = {xt }Tt=1 , conditional on the parameter vector θ and model m. The likelihood is computed starting from the solution to the log-linear approximation of the model. The solution can be cast in state-space form which makes easy the application of the Kalman filter and thus computation of the likelihood. The posterior distribution of the parameter vector θ for model m, P (θ|XT , m), is then obtained combining the likelihood function for XT with the prior distribution of θ: P (θ|XT , m) ∝ L(XT |θ, m)P (θ|m),

(50)

P (θ|XT , m) can be numerically maximized to obtain the mode of the posterior distribution, which is often seen as a point estimate of the parameter vector θ. Use of the Metropolis-Hastings algorithm allows to obtain numerically the distribution P (θ|XT , m) as well as distributions of functions of the parameter vector θ (e.g., impulse response functions), see An and Schorfheide (2007). As discussed in Geweke (1999), Bayesian inference also provide tools to compare the fit of various models. For a given model m, the marginal likelihood is:

∫ L(XT |m) =

L(XT |m)P (θ|m)dθ,

(51)

θ∈Θ

which gives an indication of the overall likelihood of a model conditional on observed data. The following characterizes briefly our estimation procedure. The model is solved up to a log-linear approximation around the deterministic steady-state. Once the solution is obtained, the model can be cast in state-space form, and the likelihood function can be computed using the Kalman filter. In every variation we start by finding the mode of the posterior distribution, or the parameter vector θM maximizing P (θ|XT , m). Initial values in the numerical algorithms are sometimes important so we repeated the procedure using various initial values in order to guarantee as much as possible that the mode is indeed found. Only then we ran the Metropolis-Hastings algorithm using 3.105 iterations while monitoring its convergence. Specifically, we use the Random-Walk Metropolis algorithm which generates Markov chains with stationary distributions that correspond to the posterior distribution of interest. Practically, we strictly follow the procedure described in Section 4.1 of An and Schorfheide (2007).38 That said, the following information is worth noting. First, the acceptation rate in the Metropolis38

It is worth noticing that other methods can be used to estimate general equilibrium models. For example, to see a comparison between linear and non-linear methods refer to Fernandez-Villaverde and Rubio-Ramirez (2005).

48

Hastings algorithm is around 25%. Second, we monitor the convergence of the Markov Chain generated by the Metropolis Hastings algorithm to the posterior distribution of interest by using the diagnostics proposed by Brooks and Gelman (1998). Intuitively, different Markov Chain sequences obtained with the algorithm should be similar once the effects of starting values vanish. Hence, second moments of the parameters calculated across sequences should not differ much from second moments calculated within a sequence. Using a rule-of-thumb that the two moments are equal if their difference is below 20%, should ensure convergence. If this is not the case, either the effect of the initial replications has not worn off, or the number of draws taken is too low. In our case, the number of draws we use is sufficient to achieve convergence.

Appendix D: Data and Observables Table D1 presents the raw data used to build the observables and the corresponding sources. All the data is taken from two sources: the NIPA, available trough the Bureau of the Economic Analysis (BEA) website, and the Bureau of Labor Statistics (BLS). We use the September 30, 2010 vintage of data. Table D2 clarifies the mapping between the raw data and the observables used in the estimation. We follow the common practice in the literature (e.g., see Smets and Wouters, 2007) in expressing output, consumption, investment and wages in per-capita real terms. Notice that our private consumption variable includes only non-durable goods and services, consistent with the definition of consumption in the fiscal policy literature, e.g., BR (07), Perotti (2007) and Ramey (2011). Also, as typical in the early RBC literature and even now (e.g., Justiniano et al. 2010, 2011), durable consumption is added to private investment. Finally, notice that the observables for the government spending variables are expressed as a share of output. Table D3 reports some descriptive statistics for the observables. The (average) government consumption to output ratio, non-defense investment, and defense investment are, 0.164, 0.026 and 0.008, respectively. All government spending variables exhibit a high degree of persistence. The government investment items show a low degree of variability over time.

49

Table D1 - Data Sources Variables

Designation

Gross Domestic Product (Nominal) Personal Cons. Expenditures (Nominal) Personal Cons.Expenditures - Durables (Nominal) Private Fixed Domestic Investment (Nominal) Federal Cons. Expenditures (Nominal) State & Local Cons. Expenditures (Nominal) Federal Gross Investment, Non-Defense (Nominal) Federal Gross Investment, Defense (Nominal) State & Local Gross Investment (Nominal) Gross Domestic Product Deflator Hourly Compensation, Non Farm Sector (Nominal) Civilian noninstitutional population, 16 years and over

GDP C Durables PFI G Federal G StateLocal IG Federal IG Defense IG StateLocal GDPDEF Wages POPULATION

Source U.S. Dep. of Commerce U.S. Dep. of Commerce U.S. Dep. of Commerce U.S. Dep. of Commerce U.S. Dep. of Commerce U.S. Dep. of Commerce U.S. Dep. of Commerce U.S. Dep. of Commerce U.S. Dep. of Commerce U.S. Dep. of Commerce Bureau of Labor Statistics Bureau of Labor Statistics

All series are seasonally adjusted at annual rates.

Table D2 - Observables for measurement equations Ytobs =(GDP/GDPDEF)/POPULATION Ctobs =((C-Durables)/GDPDEF)/POPULATION Itobs =((PFI+Durables)/GDPDEF)/POPULATION Wtobs =(Wages/GDPDEF) g,obs =(G Federal+G StateLocal)/GDP ξt ξtig,obs =(IG Federal+IG StateLocal)/GDP ξtig,def,obs =IG Defense/GDP

Table D3 - Descriptive statistics for the observables Variables

Mean

SD

AC(1)

Min

Max

∆ log Ytobs ∆ log Ctobs ∆ log Itobs ∆ log Wtobs ξtg,obs ξtig,obs ξtig,def,obs

0.369 0.487 0.317 0.351 0.164 0.026 0.008

0.841 0.493 2.199 0.601 0.011 0.002 0.003

0.234 0.305 0.374 0.127 0.985 0.945 0.965

-2.456 -1.238 -10.135 -1.335 0.143 0.022 0.005

3.338 1.575 6.605 2.875 0.184 0.034 0.015

SD: standard deviation; AC(1) first order autocorrelation. Full sample: 1969Q1-2008Q3.

50

CODE BEA BEA BEA BEA BEA BEA BEA BEA BEA BEA - BLS - BLS

A191RC1 DPCERC1 DDURRC1 A007RC1 A957RC1 A991RC1 A798RC1 A788RC1 A799RC1 GDPDEF PRS85006103 LNU00000000Q

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Within these models, the so called negative wealth effect is the main driver of spending shocks. If government spending increases,. 1Several papers estimate the effects of the American Recovery and Reinvestment Act in the United States and of the European. Economic Recovery Plan in the European Union. Among others ...

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