Distortionary Taxes and Public Investment when Government Promises are not Enforceable

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1 Distortionary Taxes and Public Investment when Government Promises are not Enforceable Marina Azzimonti University of Texas at Austin Pierre-Daniel Sarte Federal Reserve Bank of Richmond Jore Soares University of Delaware March, 2008 Abstract We characterize Markov-perfect equilibria in a settin where the absence of overnment commitment affects the financin of productive public capital. We then use our framework to assess the value of commitment, which we define as the welfare loss incurred when overnments cannot commit to the sequence of actions that produce second-best allocations. Because this calculation relies on numerical approximations, we contrast alternative approaches in the literature. We find that both perturbation and linear quadratic methods deliver accurate steady states, but that the latter can yield misleadin policy implications durin transitions. Ultimately, our analysis suests that the move from a reime with commitment to one with discretion by itself implies only a small welfare loss. This findin stems in part from the reater emphasis that Markov overnments place on short-run ains. In particular, althouh steady state Markov-perfect consumption falls short of its Ramsey counterpart, the tax policy chosen under discretion implies hiher consumption in the short run relative to the Ramsey equilibrium. JEL Classification: E61, E62, H11 Keywords: Public Investment, Commitment, Time consistency, Discretion, Ramsey, Markov- Perfect We thank an anonymous referee, Robert Kin, and Per Krusell for their many helpful comments, and have benefited from interactions with Kartik Athreya, Juan Carlos Hatchondo, Andreas Hornstein, B. Ravikumar, and Alex Wolman. We thank seminar participants at the Federal Reserve Bank of Philadelphia, the University of South Carolina, Claremont Graduate University, the 2003 SED meetins, the 2006 Midwest Macro Meetins, and the 2006 Midwest Theory Meetins. Finally, we thank Kevin Bryan for excellent research assistance. The views expressed in this paper are the authors and do not necessarily represent those of the Federal Reserve Bank of Richmond or the Federal Reserve System. A previous version of this paper circulated under the title Optimal public investment with and without commitment. All errors are our own. Correspondence: Pierre-Daniel Sarte, Research Department, Federal Reserve Bank of Richmond, P.O. Box 27622, Richmond, VA 23261, pierre.sarte@rich.frb.or 1

2 1 Introduction The notion that overnments cannot always commit to a sequence of actions is a subject of increasin interest for economists in eneral and policymakers in particular. To this point, the literature on time consistent fiscal policy has confined itself to simple environments where taxes are used to finance a flow of public oods or services that are rapidly exhausted. In contrast, the benefits of overnment spendin have been mainly documented for durable public oods that can be accumulated over time. 1 This fact is inored in recent studies because introducin public capital (an additional state variable) sinificantly complicates the characterization of the optimal discretionary policy. This paper, therefore, tackles the problem of understandin how the absence of overnment commitment affects the provision of public infrastructure, as well as the implied welfare effects over an economy s transition to its lon-run equilibrium. We solve for Markov-perfect equilibria and provide a quantitative assessment of the value of commitment, which we define as the welfare loss incurred when overnments cannot commit to the sequence of actions that produce secondbest allocations. In doin so, we evaluate the performance of different numerical methods used in approximatin time-consistent policy. Previous work on optimal public investment, includin Glomm and Ravikumar (1994, 1997), or Turnovsky (1997), characterize optimal policy under full commitment only. More recently, several papers have analyzed optimal fiscal policy absent commitment, but in environments where public oods cannot be accumulated. These include, amon others, Klein, Krusell, Rios-Rull (20087), who analyze the trade off between providin a consumable public ood and its financin, Hassler, Storesletten, and Zilibotti (2005), who study time-consistent redistribution under repeated votin, and Azzimonti, De Francisco and Krusell (2006), who explore the distortionary effects of income taxes on the evolution of wealth inequality. In contrast to these papers, our analysis focuses on the provision of a durable public ood that expands the production frontier and which we interpret as infrastructure. Thus, we contribute to the literature on public investment and discretionary policy in mainly three ways. First, at a theoretical level, we show that overnments followin a Markov-perfect policy choose a tax rate such that they trade off marinal inefficiencies arisin in private savins with those arisin in the provision of public infrastructure over two consecutive periods only. The derivation of the overnment Euler equation (GEE) in this case is substantially more involved than those developed in previous work but remains analytically tractable. More importantly, we show that this derivation allows for the application of numerical methods that efficiently and accurately describe transition dynamics. Second, in computin the Markov-perfect policy problem, we compare numerical solutions obtained usin GEE-based perturbation methods recently suested in Krusell, Kuruscu, and Smith (2002), with those that emere under a lobal method that does not require derivation of the GEE. 1 See Fernald (1999), or Hauhwout (2002), for example. 2

3 We further aue the more common linear quadratic (LQ) approximation approach developed in Svensson and Woodford (2004) aainst this lobal method. We know of no other papers in the literature that compare these numerical methods for a sinle problem. While both the perturbation and LQ approaches deliver accurate steady state allocations, we find that the approximation errors associated with the latter can yield misleadin policy recommendations in response to chanes in the state variables. In contrast, an application of the perturbation method is able to enerate decision and policy rules that differ minimally from those delivered by the lobal method. Finally, our analysis indicates that while Markov-perfect and Ramsey policies can lead to considerably different allocations in the lon run, movin from an economy with overnment commitment to one with discretion implies only a small welfare loss. This findin stems in part from the reater emphasis that Markov overnments place on short-run ains relative to a Ramsey planner. In particular, althouh the economy with commitment achieves hiher lon-run consumption relative to the reime with discretion, the tax policy chosen under discretion implies hiher consumption in the early staes of the transition relative to the Ramsey equilibrium. This effect, therefore, serves to partially offset welfare losses incurred in the lon run. Ultimately, the absence of overnment commitment results in lower tax rates and, therefore, less public infrastructure bein developed. Because a lower level of infrastructure reduces the marinal product of private capital, the economy operatin under discretion ives rise to lower private investment and lower consumption in the lon run despite its lower taxes. This paper is oranized as follows. Section 2 describes the basic economic environment. In section 3, we define the competitive equilibrium iven a stationary policy rule. Section 4 characterizes the Markov-perfect equilibrium that yields the optimal policy. Section 5 contrasts numerical solution methods, and we calculate the value of overnment commitment in Section 6. Section 7 offers some concludin remarks. 2 Environment Consider an economy populated by infinitely many households whose preferences are iven by X U = β t u(c t ), t=0 where 0 < β < 1 is a subjective discount rate, and households period utility, u(c t ),satisfies u c > 0, u cc < 0, and the usual Inada conditions. The size of the population is normalized to one. A sinle consumption ood is produced usin the technoloy y t = F (k t,l t,k t ), where k t and k t denote the date t stocks of capital in the private and public sector respectively. We interpret k either as public capital in the form of physical infrastructure or as a more eneral 3

4 public ood, such as education, that enhances the productivity of other factor inputs. Labor input is denoted by l t, and we assume that F exhibits constant returns to scale with respect to k private capital and labor. We denote the public capital elasticity of output, F k y,byθ (0, 1), and assume that F kk > 0. These assumptions follow alon the lines of earlier work, notably by Glomm and Ravikumar (1994,1997). Since leisure is non-valued, we assume that aents supply labor inelastically and set l t =1 t. 2 To simplify notation, we define f(k t,k t ) F (k t, 1,k t ). Both types of capital can be accumulated over time and evolve accordin to k t+1 = i kt +(1 δ k )k t, (1) and k t+1 = i k t +(1 δ k )k t, (2) where i k and i k denote private and public investment respectively. Private capital depreciates at rate δ k, and public capital at rate δ k,whereδ i (0, 1). The overnment taxes households to finance public infrastructure as well as conspicuous expenditures that do not provide direct utility to households. Followin Cassou and Lansin (1998), we assume that nonproductive expenditures are proportional to output, t /y t = φ with φ (0, 1), and that the ratio φ is exoenous. As such, this specification allows us to introduce expenditures that are quantitatively important in calibration exercises but whose endoeneity we do not explicitly model. Our analysis in this paper will focus exclusively on productive capital expenditures. 2.1 The First-Best Solution It is useful to first characterize efficient allocations in this environment. These allocations will then be used as a benchmark with which to compare the decentralized solutions that emere under two key policy frictions: distortionary taxation and the inability to make credible promises reardin future tax policy. Pareto-optimal allocations are found by solvin the problem of a benevolent planner who chooses sequences of consumption, private capital, and public capital so as to maximize households lifetime utility subject to max {c t,k t+1.k t+1 } X β t u(c t ), (P ) c t + k t+1 + k t+1 + t = f(k t,k t )+(1 δ k )k t +(1 δ k )k t. t=0 2 This assumption is made for simplicity, and helps keep the derivation of the GEE below somewhat concise despite the additional state variable. See Klein et al. (2008) for an extention with endoenous labor in a settin without public capital. 4

5 The first-order necessary conditions imply that 3 u ct = β[f kt+1 (1 φ)+1 δ k ]u ct+1 (3) and that u ct = β[f kt+1 (1 φ)+1 δ k ]u ct+1. (4) Therefore, in the absence of frictions, the standard result obtains such that when δ k = δ k,itis optimal to invest in k and k up to the point where their marinal products are equalized at each date, f kt+1 = f kt+1. We refer to the optimal levels of private and public capital as the unconstrained optimum. It is straihtforward to show how to obtain these allocations in a decentralized competitive equilibrium when the overnment has access to lump sum taxation. In the absence of such an instrument, however, a distortion emeres that creates a wede, or ap, in conditions (3) and (4). In addition, to the deree that policies other than lump sum taxes are used, such policies will enerally be time inconsistent 4. In eneral, we define a wede in the efficient private investment decision by kt = u ct + βu ct+1 (f kt+1 (1 φ)+1 δ k ), (5) where kt =0 t under first-best allocations. Similarly, we define a wede in public investment by kt = u ct + βu ct+1 fkt+1 (1 φ)+1 δ k. (6) 3 The Decentralized Economy iven Policy Because lump sum taxes are almost never observed, we focus on a decentralized economy where the overnment uses distortionary income taxes to finance its public capital. In that settin, a new overnment comin into office typically has an incentive to disreard promises made by its predecessors. Hence, settin taxes once and for all at time zero results in policy announcements that are not credible. In other words, absent a commitment technoloy, the Ramsey policy is time inconsistent. Our analysis centers on the determination of optimal time consistent tax rates, and contrasts the findins with those that obtain under full commitment. In particular, we study Markov-perfect equilibria whereby sequential overnments choose policy optimally based on the state they inherit when takin office. 5 3 We use u ct to indicate u c(c t), f kt+1 to indicate f k (k t+1,k t+1), etc... to simplify notation when the context is clear. 4 See Kydland and Prescott (1977) for an early formal treatment of time inconsistent policy. 5 An alternative approach finds the set of all possible sustainable equilibria, and characterizes the problem usin reputational mechanisms that rely on trier strateies involvin reversions to the worst possible equilibrium (Chari and Kehoe [1993]). Such mechanisms, however, are enerally not reneotiation proof. 5

6 Throuhout the paper, we assume that the overnment balances its budet every period. While the presence or absence of overnment debt can matter importantly for the types of policies that emere as optimal, we maintain the balanced budet assumption in this paper for mainly three reasons. First, we wish to contrast our findins reardin the lack of overnment commitment with previous studies of optimal public investment carried out under full commitment. We shall show that under full commitment, our framework nests findins in Glomm and Ravikumar (1994) amon others. Second, Azzimonti, Sarte and Soares (2006) show that overnment debt can be a powerful instrument that helps mitiate the time consistency problem in questions of optimal taxation. Therefore, an environment where domestic overnment debt is in limited use, as in developin economies, is also one where the time consistency problem is likely to have the most impact. Finally, under a specific parameterization of the model, the framework we write down is simple enouh to admit an intuitive closed form solution. Suppose that the tax rate households face at any date stems from a stationary policy rule that depends only on the states of the economy at that date, Ψ(k, k ). Since the overnment maintains a balanced budet, public investment satisfies i k = Ψ(k, k )y t. We can then express new outlays of public capital, k,as 0 k 0 =[Ψ(k, k ) φ] y +(1 δ k )k. (7) Throuhout the analysis, we denote next period s value of a iven variable x by x The Recursive Competitive Equilibrium In order to address how overnments miht choose discretionary policy optimally in our environment, we first need to describe how households and firms make decisions iven that a tax policy, Ψ(k, k ), is in place. Firms There exists a lare number of homoeneous small size firms that act competitively, where k acts as a common externality with respect to each firm s production. We denote by r and w the rental price of private capital and the wae. Takin these prices as iven, each firm maximizes profits and solves Π = f(k, k ) rk wl. max k, l The correspondin optimality conditions yield r = f k (k, k ) and w = f(k, k ) f k (k, k )k. Households At each date, households decide how much to consume and save, as well as how much capital to rent to firms. Takin the policy rule Ψ(k, k ) as iven, households maximize their lifetime utility subject to their budet constraint and the law of motion describin the accumulation of private capital. Because the policy rule Ψ is stationary (i.e. it does not depend on time as a separate 6

7 arument), we can write the household problem recursively as follows, V (k, k )=max u(c)+βv (k 0,k) 0 ª (P C ) c,k 0 subject to where c + i =(1 Ψ(k, k )) [wl + rk], (8) k 0 = i +(1 δ k )k. (9) The solution to the representative household s dynamic optimization problem yields the familiar Euler equation (henceforth EE), u c = βu 0 c (1 Ψ(k 0,k))r δ k, (EE) Havin described the behavior of households and firms, we can define the recursive competitive equilibrium iven taxes: Definition 1: Given the policy rule Ψ(k, k ), a recursive competitive equilibrium is a set of functions, V (k, k ), w(k, k ), r(k, k ),l(k, k ), H(k, k ) and K G (k, k ), such that 1. H(k, k ) solves equation (EE), l(k, k )=1,andV (k, k ) solves (P C ), 2. prices reflect competitive factor markets, r(k, k )=f k,andw(k, k )=f(k, k ) f k (k, k )k, 3. the overnment Budet Constraint holds, K G (k, k )=(Ψ(k, k ) φ)[w(k, k )l + r(k, k )k]+(1 δ k )k. In (EE), taxes distort private incentives to consume and save, but also induce hiher future returns to private investment throuh the development of public infrastructure. Specifically, the return to private investment, r 0, depends on k 0 throuh the marinal product of private capital. Aquestionthenimmediatelyarisesastowheretoset the tax rate, or equivalently, the share of public investment in output. 4 Description of the Markov-perfect problem We define a stationary Markov equilibrium followin Klein et al. (2008) where, in our setup, the tax rate depends on the stocks of public and private capital. Unlike Klein et al. (20087), however, and more enerally models with a sinle state variable, the derivation of the overnment Euler equation which characterizes the solution is substantially more involved with two states. We show that such a derivation remains analytically tractable. More importantly, it also turns out to be 7

8 helpful for the application of numerical methods that efficiently and accurately describe transition dynamics. 6 A Markov-perfect equilibrium can be described as a sequence of successive overnments, each choosin a sinle tax rate based on the state it inherits when takin office. In makin this choice, each overnment correctly anticipates the optimal decision rule adopted by its successors. In equilibrium, future policymakers choices are time consistent if and only if they coincide with the rule that the current policymaker anticipated them to choose optimally. This implies that the current planner s policy choice of τ must also follow Ψ(k, k ). In order for Ψ(k, k ) to be subame perfect, no overnment must ever have an incentive to deviate from this rule. Joint deviations are not feasible since, by assumption, a new overnment chooses policy every period and, consequently, cannot enter bindin contracts with future overnments. It is sufficient, therefore, to analyze the problem of a overnment that is allowed to deviate in the current period by settin a tax rate τ 6= Ψ(k, k ), under the assumption that Ψ(k, k ) is forever followed in the future. Thus, we now describe how an arbitrary deviation perturbs equilibrium allocations, and we allow the overnment to choose the best possible deviation τ. InaMarkov- perfect equilibrium, it must be the case that the optimal deviation τ coincides with the policy rule Ψ(k, k ). From the overnment budet constraint, we can define the evolution of public capital as a function of current states and an arbitrary tax rate, G(k, k, τ), such that G(k, k, τ) =(τ φ)f(k, k )+(1 δ k )k. (10) The representative household s budet constraint is iven by c =(1 τ)[w(k, k )l + r(k, k )k]+(1 δ k ) k k 0, where prices are written explicitly as a function of current states. Because of the representative household assumption, this constraint can also be expressed as an economy-wide resource constraint that implicitly defines a consumption function C, C(k, k, τ,k 0 ) (1 τ)f(k, k )+(1 δ k ) k k 0. (11) Let us define the evolution of private capital under the one-period deviation as k 0 = H(k, k, τ), where H solves households optimal consumption-savins decision when the current tax rate is iven by τ, and all future tax rates obey Ψ(k, k ). Since households take policy as iven, equation (EE) then becomes u c C(k, k, τ,k 0 ) = βu c [C(k 0,k, 0 Ψ(k 0,k),k 0 00 )] f k 0 (1 Ψ(k0,k)) 0 ª + 1 δ k, (12) where k 0 = G(k, k, τ) and k 00 = H(k 0,k). 0 6 In related work, Azzimonti, De Francisco, and Krusell (2006) study a politico-economic equilibrium also with mulitple state variables but where taxes are determined by majority votin. 8

9 Althouh each overnment only chooses taxes for the period in which it is in office, this choice also affects individual and areate behavior in subsequent periods. As savins and investment in public capital adjust followin a chane in the current tax rate, so do future taxes as a byproduct. Put another way, since Ψ is followed tomorrow onwards, variations in τ that cause k 0 and k 0 to chane will also cause the rate τ 0 = Ψ(k 0,k) 0 to chane, thus affectin future savins and investment (k 00 and k). 00 Hence, by affectin future states with current policy decisions, the current policymaker possesses some leverae over future overnments throuh Ψ. Formally, the Markov problem describin optimal discretionary policy at any date is where V (k, k ) is determined by max u(c(k, k, τ,h(k, k, τ)) + βv (H(k, k, τ),g(k, k, τ)) (P M ) τ V (k, k )=u(c)+βv (H(k, k ), K G (k, k )), (13) H(k, k, τ) satisfies condition (12), and C (1 Ψ(k, k ))f(k, k )+(1 δ k ) k H(k, k ). A Markov-perfect equilibrium is found when, for any pair {k, k }, τ = Ψ(k, k ). Formally, Ψ(k, k ) armax u(c(k, k, τ,h(k, k, τ)) + βv (H(k, k, τ),g(k, k, τ)). τ When the overnment optimally chooses not to deviate from the rule Ψ in settin τ, theresultin policy functions must be consistent with the recursive competitive equilibrium described in Definition 1. Hence, H(k, k, Ψ(k, k )) H(k, k ) and G(k, k, Ψ(k, k )) K G (k, k ). 4.1 The Government Euler Equation Assumin that the policy functions are differentiable, the first-order condition associated with problem (P M )is h i u c C τ + β Vk 0 H τ + Vk 0 G τ =0. (14) The expressions for V k and V k (where V is iven in equation 13) involve H k and H k. Since households Euler equation (EE) depends on the policy rule Ψ(k, k ),thecalculationofh k and H k usin the implicit function theorem in turn involves derivatives of the policy function, Ψ k and Ψ k. This feature of the problem imposes an additional burden on solvin for the Markovperfect equilibrium since, to arrive at the unknown policy Ψ (or even just the tax rate in the steady state), one already needs to take into account how this policy chanes with the states of the economy. We saw in section 2.1 that in the presence of lump sum taxes, the overnment would set all distortions to zero so that first-best allocations could be attained. In contrast, in our settin, distortionary taxes induce wedes in the intertemporal conditions describin the efficient provision of private and public capital. The followin proposition states that the optimal discretionary policy is such that it sets a linear weihted sum of these distortions to zero. 9

10 Proposition 1. Let k = u c + βu 0 c(fk 0 (1 φ)+1 δ k) and k = u c + βu 0 c(fk 0 (1 φ)+ 1 δ k ). Then, the overnment s first order condition (14) may be re-written as an Euler equation as follows H τ k + G τ k + β{ H τ 0 0 k + G e0 τ 0 k } =0, (15) where H τ 0 = ξbk 00 and G e0 τ = ξbk 00, with B H i = G i G i τ H τ Proof. See Appendix A. i = k, k and ξ = H τ B 0 k Hτ +B0 k Gτ Bk 00H0 τ +B. k 00 G0 τ The derivation of the overnment Euler equation (GEE) in terms of a weihted sum of deviations from efficient intertemporal decisions is somewhat involved, but perhaps most intuitive in that form. Furthermore, it is worth notin that althouh the economy is dynamic, so that there are potentially an infinite number of distortions, only those in the current and subsequent period matter directly. The recursive nature of the problem toether with the envelope theorem ensure that other wedes arehandledoptimally. Whyisitthattwoinefficiency wedes in the current and subsequent period matter in proposition 1? In Klein et al. (2008), or Azzimonti et al. (2006), the overnment attempts to manipulate one intertemporal distortion usin one policy instrument, income taxes. As a result, the GEE in Klein et al. (20087), for instance, involves tradin off one intertemporal wede, the one that distorts savins, with one intratemporal wede, the one between the marinal utilities of private and public consumption. In our settin, however, the overnment attempts to manipulate two intertemporal marins, those that determine private capital, k, and public capital, k, but still havin access to only one instrument, income taxes. Thus, relative to the earlier frameworks with a sinle state variable, the optimal policy with public investment dictates tradin off two intertemporal wedes over time. 7 The first term in proposition 1 depicts the increase in the inefficiency of private savins induced by a marinal increase in distortionary taxes. This inefficiency is captured by the intertemporal savins distortion that arises with distortionary taxes, k > 0, scaled by the reduction in household savins that takes place when the tax rate increases, H τ < 0. 8 Similarly, the second term G τ k in proposition 1 summarizes how chanes in current taxes affect the inefficiency of public capital provision. In particular, this effect is characterized by the wede k, present whenever τ > 0, scaled by the rise in public investment implied by a marinal increase in τ, G τ > 0. 9 The third and fourth terms in the weihted sum of distortions in proposition 1 may be interpreted in an analoous way and account for how current policy affects next period s wedes throuh its effect on future levels of private and public capital. In sum, therefore, overnments followin a Markovperfect policy choose a tax rate such that they trade off marinal inefficiencies that arise in private 7 In eneral, what affects the number of intertemporal trade offs in the GEE involves both the number of state variables and the number of independent financin instruments. 8 To ³ see this, simply substitute households first-order condition (EE) into the definition of k to obtain k = τ 0 βu 0 c fk 0 (1 φ)+1 δ.thus, k > 0 when τ 0 > 0, which implies a level savins that is lower than optimal. 9 Recall ³ that in the first best economy, the optimal level of optimal public capital is determined by the condition u c +βu 0 c fk 0 (1 φ)+1 δ =0. With distortionary taxes, this equality no loner necessarily holds and k 6=0. 10

11 savins, H τ k, with those that arise in the provision of public infrastructure, G τ k,acrosstwo consecutive periods. The next section centers on a specific parametric example that captures an intuitive deenerate case. 4.2 A Special Case: Constant Markov-Perfect Policy As a first step towards the analysis of Markov-perfect equilibria, it is important to reconize that, dependin on the nature of preferences and technoloy, the commitment problem on the part of the overnment does not always bind. In particular, consider the case where households period utility is loarithmic, u(c) =loc, technoloy is Cobb-Doulas, y = kk θ α l 1 α, capital depreciates fully within the period, δ k = δ k =1, and there are no unproductive expenditures, φ =0.Itiswell knownthatforthesespecifications of preferences and technoloy, the household savins function in the decentralized equilibrium depends only on the current level of taxes and not the entire policy stream. Specifically, future policy chanes lead to income and substitution effects on current savins that exactly offset each other. Under these assumptions, the household Euler equation reduces to c 0 = αβcfk 0 [1 Ψ(k0,k)]. 0 Given policy, we uess that the recursive competitive equilibrium is such that households save a constant proportion, s (0, 1), of after tax income, H(k, k )=s(1 Ψ(k, k ))kk θ α. It is then straihtforward to show that equation (EE) holds if and only if s = αβ. It follows that H(k, k )=αβ(1 Ψ(k, k ))y, (16) and that consumption is also proportional to after tax income c =(1 αβ)(1 Ψ(k, k ))y. (17) Given equations (10) and (16), it is straihtforward to obtain expressions for H τ, G τ, H e τ 0, G e0 τ that can can be substituted into the GEE in Proposition 1. Moreover, expressions for k and k are iven by equations (5) and (6) respectively. Basic alebraic manipulations then immediately show that the constant tax policy τ = βθ solves the GEE, as summarized in proposition Proposition 2. When household preferences are loarithmic, u(c) =lo(c), technoloy is Cobb Doulas, y = k α kl θ 1 α, capital depreciates fully within the period, δ k = δ k =1,andthere are no unproductive expenditures φ =0, optimal discretionary income tax rates are independent of the states, Ψ(k, k )=βθ (k, k ), and thus constant throuh time. It is precisely because savins at any date depend only on contemporaneous taxes in equation (16), and hence cannot be directly influenced by future policy, that optimal discretionary tax rates are constant in this case. The fact that household behavior is independent of past or future taxes 10 See Azzimonti, Soares, Sarte (2007) for details of the calculations. 11

12 also implies that the commitment problem does not bind. Put another way, there is no value, in this example, in havin future policy differ from current policy so that the Markov problem becomes deenerate. Under more eneral parameterizations, chanes in current taxes, τ, affect private savins, k 0, and public investment, k,whichthenaffect 0 future policy throuh the rule, τ 0 = Ψ(k 0,k). 0 A chane in τ 0 then feeds back into households current decisions by way of a substitution effect, that alters the net return to today s savins, and an income effect, that chanes disposable income tomorrow. This feedback creates a political intertemporal link between successive overnments (i.e. between choices of τ and τ 0 ). In contrast, the key to proposition 2 is that tomorrow s policy has no effect on today s choices by households so that this intertemporal link between overnments breaks down. The political decision problem, therefore, essentially collapses to a static one with a constant policy as its solution 11 5 Numerical Solutions for a Calibrated Economy In this section, we carry out numerical simulations of our economy with public investment. The parameters we use are standard and selected alon the lines of other studies in quantitative eneral equilibrium theory. Followin Klein et al. (2008), we assume that utility is loarithmic, u(c) =loc, and that households discount the future at rate β =0.96, where a time period represents a year. Turnovsky (2004) estimates depreciation rates for private and public capital such that δ k =0.05 and δ k =0.035 respectively. The share of private capital in output in the U.S. is approximately 36 percent which implies α = The one non-standard parameter relates to the elasticity of public capital with respect to output. Estimates of θ vary sinificantly across studies. Glomm and Ravikumar (1997) cite values ranin from as low as 0.03 (Eberts, 1986) to as hih as 0.39 (Aschauer, 1989). We set θ to 0.25 to approximate a consensus view, but alternative values ranin between 0.15 and 0.35 do not materially alter the findins below. Government spendin on oods and services, net of public investment, averaed approximately 17 percent of output over the period so that we set φ = Overview of Alternative Numerical Methods Numerical work on optimal time consistent policy has enerally adopted one of three alternative methods: (i) a Global Approximation Method (henceforth GM), (ii) a Linear Quadratic Approximation Method (LQ), and (iii) a Perturbation Method. Global approximation methods that solve for equilibria on a discretized state space are simplest, in the sense that they do not necessarily require derivation of the GEE, and accurate. A sinificant 11 Under lo preferences, absent the political intertemporal dimension, one can show that the policymaker s decision problem is one where the effects of policy do not interact with the state variables. 12

13 disadvantae of this method, however, relates to its potentially lenthy computational time which rows exponentially with the number of state variables. As an alternative, LQ approximations allow for rapid computations of steady states and transition dynamics. They are widespread in the literature and easily handle multiple state variables. Examples of this approach are found, for instance, in Krusell and Rios-Rull (1999), and Benino and Woodford (forthcomin), in the context of fiscal policy, but are more prevalent in the monetary policy literature as illustrated in Dotsey and Hornstein (2003), Svensson and Woodford (2004), and Kin and Wolman (2004) amon many others. Developments related to the perturbation method are more recent and, to this point, have been used mostly to approximate steady state equilibria (see Klein et al. 2008, or Azzimonti et al. 2006). In this paper, we use the perturbation method to also approximate transitions to the steady state. To our knowlede, our work offers the first comparisons of relative accuracy between different approaches in a time consistent policy problem. Surprisinly, we find that all methods deliver accurate estimates of the steady state allocations. Transition dynamics obtained usin either an LQ approximation or a perturbation method with linear policy functions, however, can prove sinificantly inaccurate. In particular, transition paths for taxes and public investment computed from an LQ approximation are noticeably different from those suested by the lobal method solution. On a more practical level, the strict parametric form imposed on the savins and income tax rules under the LQ approach can lead to misleadin policy recommendations. The Perturbation Method essentially consists of approximatin decision rules with n-th order polynomials. In our environment, conditional on Ψ, the overnment s budet constraint (7) immediately ives us the rule G. Consequently, we need only approximate the savins rule H and the tax policy Ψ. Given the assumed functional forms for H and Ψ, calculatin the derivatives that appear in the GEE is now straihtforward. In essence, this method recasts the problem in terms of findin unknown coefficients. For comparison with the LQ approach, we use the perturbation method with two different approximations of the policy functions, one that assumes a linear functional form for H and Ψ, referred to as method PL, and one that assumes a quadratic form for H, referred to as method PQ. Under method PL, we specify H as follows, H PL (k, k, τ) =a k 0 + a k 1k + a k 2k + a k 3τ, (18) Next period s public capital satisfies the overnment s budet constraint, G PL (k, k, τ) =(τ φ) k α k θ +(1 δ k )k, (19) and taxes are linear in the state variables, Ψ PL (k, k )=a τ 0 + a τ 1k + a τ 2k. (20) There are 7 unknown a parameters and 2 unknown steady state values (i.e. k M and k M ). The correspondin system of 9 equations needed to solve for these parameters consists of the Euler 13

14 equation, the GEE, the derivatives of the Euler equation with respect to τ, k and k, the derivatives of the GEE with respect to k and k, as well as equations (18) and (19) evaluated at the steady state. See Klein, Krusell and Rios Rull (2008) for additional details. 12 Under the second approximation used with the perturbation approach, method PQ, Ψ and follow equations (19) and (20), but H is quadratic in its aruments. As a result, the number of unknown a parameters rises to 15. The required additional equations in this case are iven by the second derivatives of the Euler Equation. The LQ Method re-states problem (P M ) as one with a quadratic objective subject to a set of linear constraints. The end result is one where all decision rules are linear in the state variables, includin the decision rule for G in contrast to equation (19) under method PL. Thus, the fundamental difference between the LQ and the perturbation approach is that the latter solves the oriinal problem while only approximatin the decision rules but not necessarily the objective function or constraints. The relative accuracy of these methods are aued aainst a Global Method carried out as follows. Given arbitrary initial uesses for equilibrium taxes, Ψ n (k, k ),savins,h n (k, k ),and the value function, V n (k, k ), on a discretized state space, households savins for iven tax rates, H(k, k, τ; Ψ n ), are obtained by solvin the Euler equation in (12) at each rid point. Similarly, the evolution of public capital, G(k, k, τ; Ψ n ), can be directly calculated from the overnment s budet constraint (7). This process immediately yields private consumption from equation (11), C(k, k, τ; Ψ n ), which can then be used to compute households indirect utility for arbitrary tax rates on a rid. One can then directly solve the overnment s problem (P M ) without derivin the GEE. The solution to this problem yields an updated optimal tax function, Ψ n+1 (k, k ),andinturna new savins function, H n+1 (k, k ), and value function, V n+1 (k, k ). By iteratin on this procedure, we search for an equilibrium where the policy and value functions have convered. In particular, Ψ n Ψ n+1 <²suests that a policymaker solvin problem (P M ) will not deviate from the rule expected to be followed in the future. Because H(k, k, τ; Ψ n ) and G(k, k, τ; Ψ n ) may imply values of k 0 and k 0 respectively that are not on the discretized state space, we approximate the policy rule, Ψ n (k, k ), and welfare function, V n (k, k ), at each iteration with quadratic polynomials so that they may be evaluated at arbitrary points. 13 Additional points are added to the rids for k, k,andτ, until solutions no loner differ across different discretizations of the state space. 12 A Technical Appendix with a detailed description of all methods used in this paper is available from the authors. 13 Given that this problem involves three different rids, interpolatin between points proved computationally costly. The results presented here rely on rids containin 100 points for τ, 40 points for k, and40 points for k.quadratic polynomial fits for Ψ n (k, k ) and V n (k, k ) were associated with R 2 statistics in excess at each iteration and for each function. Usin Matlab version 7.1 on a PC with a 2 GHz processor and 2GB of RAM resulted in a computin time of approximatly two days. In contrast, the other three methods can be solved in less than a minute. 14

15 5.2 Comparison Between Methods Table 1 summarizes the steady state tax rates and capital allocations under optimal discretionary policy usin the four methods we have just described. Interestinly, all methods yield steady state solutions that are very close. The perturbation method with quadratic decision rules (PQ) yields solutions that are closest to those obtained with the lobal method (GM). Table 1. Markov-Perfect Steady States: Method Policy a, τ k k Global Method Perturbation Method (Quadratic Decision Rules) Perturbation Method (Linear Decision Rules) Linear Quadratic Method a. The tax rate is expressed in percentae terms. We can further assess the performance of each method by examinin how well it ultimately solves the Euler equation (12), and the overnment Euler equation (15), at points outside the steady state. To this end, let ν EE (k i,k,j ) denote the (demeaned) error associated with households Euler equation evaluated at iven points k i and k,j when usin the solutions for H, K G and Ψ from each of the four methods we have described. Similarly, let ν GEE (k i,k,j ) denote the error that arises from evaluatin the overnment Euler equation usin these solutions. Because these errors vary over two dimensions, k and k,wefirst present results where we interate over each dimension one at a time. Fiure 1, panel A, shows the root mean square error (RMSE) associated with the Euler equation h i for different values of k, averaed over k for each k, ρ EE (k) = 1 Pn k 1/2, n k j=1 ν EE(k, k,j ) 2 where n k denotes the number of rid points describin public capital. Conversely, Fiure 1, panel B, illustrates the RMSE computed from the Euler equation for different values of k, averaed over h k for each k, ρ EE (k )= 1 P i nk 1/2, n k i=1 ν EE(k i,k ) 2 wherenk denotes the number of rid points for private capital. It is immediately apparent from the fiures that the linear quadratic and linear perturbation methods display the larest errors amon the four numerical approaches, with correspondin RMSE statistics that tend to increase rapidly as we move away from the steady state shown in Table 1. In contrast, the perturbation method with quadratic decision rules performs just about as well as the lobal method, and even slihtly better near the steady state in one case. Fiure 1, panel C, depicts the RMSE associated with the overnment Euler equation for different h i values of k, averaed over k, ρ GEE (k) = 1 Pn k 1/2. n k j=1 ν GEE(k, k,j ) 2 Fiure 1, panel D, plots its h counterpart, ρ GEE (k )= 1 P i nk 1/2. n k i=1 ν GEE(k i,k ) 2 As before, methods that rely solely on linear approximations are least accurate, with the linear quadratic approximation yieldin notably larer 15

16 errors that increase steeply outside the steady state. Interestinly, with respect to the GEE, panels C and D suest that the perturbation method with linear decision rules (PL) performs almost as well as when computed with quadratic decision rules (PQ). Here, the lobal method always yields the smallest errors. InTable2,wereportthetotalRMSE,ρ, when both dimensions of the state space are taken into account and across both optimality conditions, n 1 k n X X k ρ = ν EE (k i,k,j ) 2 n k n k i=1 j=1 1/2 n 1 k n X X k + ν GEE (k i,k,j ) 2 n k n k i=1 j=1 1/2. (21) While the lobal method shows the smallest areate RMSE, the perturbation method with quadratic decision rules proves nearly as accurate. In contrast, the LQ approximation and the perturbation method with linear rules are associated with RMSE statistics that are 5 to 10 times larer. Table 2. Accuracy Comparisons Between Methods Method Root Mean Square Error a Global Method 4.06e 004 Perturbation Method (Quadratic Decision Rules) 5.56e 004 Perturbation Method (Linear Decision Rules) 4.20e 003 Linear Quadratic Method 2.24e 003 a. The RMSE is computed over rids whose bounds are set at ±10 percent of steady state values. To sum up, all approaches considered here yield similar steady state tax rates and capital allocations. 14 As discussed in Judd (1996, 1999), and Benino and Woodford (2006), by inorin hiher order effects that may be important in providin a welfare rankin of different policies, the LQ approach produces linear decision rules that may poorly approximate optimal policies away from the steady-state. In our framework, we find that a perturbation approach based on the GEE can perform nearly as well as a lobal method. While the lobal approach is conceptually simpler, derivin the GEE allows for the implementation of a numerical method that takes virtually no computational time. 5.3 Implications for Transition Dynamics Table 3 shows what the different methods we have just reviewed imply for the response of optimal discretionary policy to chanes in the states of the economy. Specifically, we compute Ψ(k,k ) and Ψ(k,k ) k evaluated at the steady state. In eneral, the Markov-perfect policy prescribes a tax rate rule that increases with private capital and decreases with public capital. This suest that Markov 14 Total RMSE statistics at the steady state are on the order of k 16

17 overnments purposefully create incentives for the private sector to invest when private capital falls below its steady state, Ψ(k,k ) k > 0. In contrast, optimal discretionary tax rates rise when public infrastructures fall short of their lon-run equilibrium, Ψ(k,k ) k < 0. In this case, hiher taxes are necessary to fund public investments. Table 3. Policy Responses to Chanes in Economic States a : Method Ψ(k,k ) k Ψ(k,k ) k Global Method Perturbation Method (Quadratic Decision Rules) Perturbation Method (Linear Decision Rules) Linear Quadratic Method a. Chanes in the tax rate are expressed in percentae terms. Interestinly, the response of optimal discretionary tax rates is rouhly of the same manitude with respect to chanes in either type of capital. Moreover, the perturbation methods yield policy responses that are quantitatively similar to those obtained with the lobal method. The linear quadratic approximation, however, considerably underestimates the response of optimal policy to an increase in private capital. More enerally, Fiure 2 illustrates the policy functions describin the evolution of public capital and Markov-perfect tax rates with respect to each state variable. The left-hand panels hold public capital at its steady state and show the policy functions with respect to private capital. Conversely, the riht-hand panels hold private capital fixed at its steady state and illustrates how the policy variables move with public capital. Althouh the policy rules can differ off steady state, especially in the case of the LQ approximation when public capital is held fixed, these functions all intersect at the steady state values for public and private capital. These plots, therefore, are consistent with all numerical methods ivin us accurate estimates of the lon-run Markov-perfect equilibrium in Table 1. The policy functions in Fiure 2 show relatively little difference between the Perturbation method with quadratic decision rules (PQ) and the lobal method. In contrast, the LQ approximation considerably overstates the optimal discretionary tax rate, and hence the evolution of public capital, when private capital falls below steady state (Fiure 2, panels A and C). In addition, Fiure 3 shows that LQ estimates of households decision rules notably understate optimal savins when private capital falls short of its lon run equilibrium. To ain a sense of the quantitative implications from these different policy functions for transition dynamics, Fiure 4 depicts the evolution of both types of capital, tax rates, and consumption when the economy starts off with stocks of public and private capital that are initially 5 percent below their respective steady states. We envision, therefore, a developin economy that is transitionin to its lon-run equilibrium. The most strikin difference across methods concerns the 17

18 evolution of tax rates (Fiure 4, panel D). Specifically, while the solution computed usin the lobal method predicts a small decrease in tax rates of less than 1 percent initially, the LQ approximation prescribes a tax rate increase of almost 5 percent (from 26 percent to 31 percent). Moreover, the lobal method suests a smooth radual increase in tax rates throuhout the transition to the steady state while the LQ method shows an abrupt decline in tax rate over the first five periods. As the economy moves towards its lon-run equilibrium, these differences in tax rates imply a noticeably smoother consumption profile under the lobal method than when computed with the LQ approximation (Fiure 4, panel C). Our example suests that the choice of numerical methods in problems of optimal discretionary policy should depend in part on the oal of the investiation. To the deree that one is mostly interested in lon-run equilibrium properties of the Markov-perfect problem, the LQ method is straihtforward, fast, and can accommodate a lare number of state variables without additional complications. If the focus is instead on transition paths, then the perturbation method seems a preferable choice. The perturbation method, however, presents additional challenes in that the number of parameters characterizin the policy functions increases exponentially with the deree of the polynomials that describe those functions and the number of state variables. This introduces a trade-off between the ains from usin a larer polynomial and the error introduced by the alorithm used to solve for its parameters. For the model at hand, with two state variables, method PQ enerates decision and policy rules that are very close to those obtained with the lobal method (Fiures 2 and 3), but at a fraction of the computational time. 6 The Value of Government Commitment Havin described the optimal public investment problem under discretion, we briefly presentthe more conventional Ramsey settin to assess the importance of a commitment technoloy. We show that the special case of an optimal constant policy, τ t = βθ t 0, described in Proposition 2 also emeres as a solution to the full commitment problem under the same functional form assumptions. In this sense, our framework nests previous results in the literature, notably by Glomm and Ravikumar (1994). While this findin implies that the commitment assumption is innocuous in their settin, that is not the case in a more eneral settin. The objective of this section, therefore, is to compare allocations under the two institutional extremes of full commitment and no commitment, and to aue the accuracy of different numerical methods in measurin the value of commitment. 6.1 Maximization Problem Consider a benevolent overnment that, at date zero, is concerned with choosin a sequence of tax rates consistent with the development of public infrastructure that maximizes household welfare. In doin so, this overnment takes into account the decentralized optimal behavior of households 18

19 and firms, as summarized by the Euler equation and prices bein equated to marinal products, and can commit to future policy. The Laranian correspondin to the Ramsey problem can be written as: max {c t,τ t,k t+1,k t+1 } t=0 L = X β t u(c t ) (P R ) X β t μ 1t {βu c (c t+1 )[(1 τ t+1 )F k (k t+1,k t+1 )+1 δ k ] u(c t )} t=0 t=0 X β t ª μ 2t (τ t φ) F (k t,k t )+(1 δ k )k t k t+1 t=0 X β t μ 3t {(1 τ t )F (k t,k t )+(1 δ k )k t k t+1 c t }. t=0 The constraint associated with the multiplier μ 1t implies that our benevolent planner takes households consumption-savins behavior as iven. He can, however, influence the intertemporal allocations they choose by alterin tax policy over time. Because the capital stocks are fixed at date 0, thefirst-order conditions associated with (P R ) will enerally differ at t =0and t>0. As first noted in Kydland and Prescott (1980), this suests an incentive to take advantae of initial conditions in the first period with the promise never to do so in the future. It is exactly in this sense that the optimal policy may not be time consistent since, once date 0 has passed, a planner at date t>0 who re-optimizes would want to start with a tax rate, τ t,thatdiffers from what was chosen for that date at time 0. While Ramsey plans are typically time inconsistent, we introduced a parameterization in section 4.2 that we arued made the time consistency problem non-bindin. To see why, recall that under that parameterization, household savins depended only on the current tax rate throuh disposable income. A Ramsey policymaker would then reconize that the choice of tax rate at any date never has an effect on either past or future savins and, in this sense, period 0 is no different than any other period. Put another way, in that example, the overnment never has an incentive to renee on past promises so that the resultin Ramsey policy is time consistent. We summarize this findin in the followin proposition. Proposition 3. When preferences are loarithmic, u(c) =lo(c), technoloy is Cobb Doulas, y = k α k, θ capital depreciates fully within the period, δ k = δ k =1, and there are no unproductive expenditures φ =0, the optimal sequence of tax rates with commitment is time invariant and reproduces the Markov-perfect policy, τ t = βθ t 0. Proof: See Appendix B. An implication of proposition 3 is that in this specific example, there are no welfare ains from adoptin a technoloy that constrains overnments to fulfill their promises. In other words, whether or not overnments can commit to future policy is immaterial. 19