R M F P : R C T Giorgio Motta Raffaele Rossi October 10, 2014 Abstract We study Ramsey monetary and fiscal policy in a New-Keynesian model with endogenous government spending, state-noncontingent public debt and distortive taxation. The fiscal authority has access to consumption taxation, in addition to labour income taxation. The contribution of this paper is twofold. First, we characterize analytically the steady-state optimal provision of public goods. We show that this depends on the agents risk aversion, the steady state level of public debt and on the tax instrument adopted. We proof that the optimal size of the public spending is, ceteris paribus, greater under consumption taxation than under labour income tax. Furthermore, if the agents risk aversion is suffi ciently high and the policy-maker can tax consumption, the optimal public spending to GDP ratio is increasing in the long run level of public debt. Second, in the stochastic equilibrium, movements in consumption taxation have non-trivial effects on the agents intertemporal allocation. In turn, the dynamic properties of consumption taxation enable the policy-maker to affect the stochastic discount factor via modifications of the marginal utility of consumption. This extra wedge impacts on the pricing decisions of firms, and hence on inflation stabilization, and greatly improves welfare. Keywords: Ramsey policy, Optimal public spending, Consumption tax, Distortionary taxation, Public debt. JEL: E30, E61, E62. We are grateful to Emanuele Bracco, Stefano Gnocchi, Sarolta Laczò, Dmitry Matveev, Monika Mertz, and Maurizio Zanardi for their comments on a early version of the paper. Furthermore, we benefit from discussions with Michela Cella, Andrea Colciago, Davide Debortoli and Nick Snowden, as well as seminar participants at the Bank of England, University of Milano-Bicocca, University of Durham, University of Vienna, University of Liverpool, University of Lancaster, the 2014 T2M conference in Lausanne, the 2013 CEF conference in Vancouver and the 2014 EEA meeting in Toulouse. All errors are our own. Department of Economics, Lancaster University Management School, LA1 4XY, United Kingdom. E-mail: g.motta@lancaster.ac.uk Department of Economics, Lancaster University Management School, LA1 4XY, United Kingdom. E-mail: r.rossi@lancaster.ac.uk 1
1 Introduction Most of the literature on Ramsey monetary and fiscal policy, e.g. Benigno and Woodford (2003), Schmitt-Grohé and Uribe (2004, 2007), Adam (2011) and Leith and Wren-Lewis (2014), rules out consumption taxation, a policy instrument which is widely used in most industrialised economies. In 2013, for example, the value-added tax on standard items ranged from 15 to 27 percent in all European Union countries. Exceptions include Correia, Nicolini and Teles (2008), who find that in a New-Keynesian model with labour income and consumption taxes, the equilibrium allocations are independent of the degree of price stickiness. More recently, Correia, Farhi, Nicolini and Teles (2013) and Farhi, Gopinath and Itskhoki (2014), stress the role of consumption taxation as a tool to relax a constraint on monetary policy on the nominal interest rate, either as a result of the zero lower bound or when a country is locked in a monetary union. Our paper is closely related and contributes to this literature by highlighting two features of consumption taxation in the context of optimal monetary and fiscal policy. First, we study the role of consumption taxation alongside or as alternative to labor income taxation in shaping the optimal provision of public goods. Second, we analyze the interactions between monetary policy, inflation stabilization, public debt and consumption taxation. In our theoretical model we add a fiscal policy block to a standard closed-economy, stickyprice, cashless DSGE framework, similar to the workhorse model in e.g. Clarida, Galí and Gertler (1999) or Woodford (2003). Public consumption generates utility as in, inter alia, Galí and Monacelli (2008), Adam (2011) and Debortoli and Nunes (2011). All these contributions exclude consumption taxes. As in Correia, Nicolini and Teles (2008) and Correia, Farhi, Nicolini and Teles (2013), the fiscal authority has access to labour income taxes, consumption taxes and state-noncontingent public debt. However, unlike in our paper, they treat government spending as exogenous. More over, like most of the New-Keynesian literature, we abstract from capital accumulation. This modeling approach is particularly convenient as it allows us to better compare our results to previous analyzes and, perhaps more importantly, to obtain a set of original analytical results. The economic environment considered in this paper features three ineffi ciencies. First, firms have market power in the good markets so that they can charge a mark-up over marginal costs. This causes output to be ineffi ciently low. Second, sticky prices in the good market prevent prices from fully adjusting after a shock shocks. As a consequence, the aggregate supply is positively sloped (at least in the short run) and demand-side policies have real allocation effects. Third, fiscal policy uses distortionary taxation to finance public spending and interest payments on outstanding government debt. Hence fiscal policy has additional 2
negative consequences on the economic activity. In turn, as we will discuss in details later, this model generates several channels for which there exist interesting interactions between monetary and fiscal policy instruments. Our analysis is comprised of two parts. In the first one, we study optimal policy in the steady state. In the second one, we analyze optimal stabilization policies during an unexpected economic recession. Regarding the analysis of the optimal steady state, we can summarize our results as follows. First, we re-state the traditional result of optimal zero steady state inflation, e.g. Benigno and Woodford (2003). Then, we study analytically the optimal provision of public goods. More precisely, we analyze the optimal size of government consumption relative to total output. By setting a certain ratio of government spending to GDP, the Ramsey Planner can influence private sector behavior and therefore can reduce ineffi ciency and increase welfare. We show that the incentive for the Ramsey Planner to increase the government spending to GDP ratio increases with the households risk aversion. Furthermore we explore how the optimal size of government spending depends on whether consumption or labor income is taxed and on the level of outstanding public debt. We find that under consumption taxation the optimal share of government spending is, ceteris paribus, always larger then under labor income taxation. Then, we proof that when the agents risk aversion is suffi ciently high and the policy-maker can tax consumption, the optimal public spending to GDP ratio is increasing in the long run level of public debt. Furthermore, we find that taxing consumption allows the policy-maker to achieve higher levels of welfare. This is obtained despite the fact that in our model, due to the absence of capital accumulation, the equilibrium consumption tax base is in general smaller than labour income tax base. We also show that the superiority of consumption taxation is only marginally affected by different degrees of labour supply elasticity and it is robust to the introduction of full profit taxation. This results extend, in a New-Keynesian environment with endogenous government spending, the literature on the superiority of consumption taxation. For example, in a standard RBC model with capital accumulation, Colemann (2000) finds that replacing income taxes with consumption taxes would lead to large welfare gains compared to the existing tax system in the United States. Correia (2010) extends this result to a heterogeneous agents framework. Laczo and Rossi (2014) show that taxing consumption leads to large welfare gains even when the policy-maker is unable to commit to future policies. In the second part of the paper, we analyze the dynamic behavior of the economy under technology shocks. With a steady-state level of public debt (assets), interest rate movements will affect the return on government bonds and hence have budgetary consequences. In other words, unlike in the standard New-Keynesian model without fiscal policy, there are costs 3
associated with movements in the monetary policy instrument so that the Euler equation becomes a relevant constraint for the optimal policy. In turn, movements of consumption tax will affect, not only the leisure-consumption margin, but also the stochastic discount factor (via modification of the marginal utility of consumption in the Euler equation). This effect directly impacts on how agents discount future profits, and hence on firms price setting decisions. Hence the policy-maker can effectively stabilize inflation with an appropriate use of consumption taxation. We find that this extra wedge helps the stabilization of the economy and greatly increases welfare in the stochastic equilibrium when compared to the standard scenario where only labor income taxes are available. Ceteris paribus, the welfare gains of taxing consumption are larger, the stronger the interaction between monetary and fiscal policy, i.e. the higher the steady state level of public debt (assets). This second part of the paper contributes to the literature that stresses the role of consumption taxation as an unconventional instrument for monetary policy. With a transmission mechanism similar to our, Correia, Farhi, Nicolini and Teles (2013) show that when the nominal interest rate is zero, the policy-maker can use consumption taxation as demand managment tool, and hence unwind the negative effects of the zero lower bound. Similarly, Farhi, Gopinath and Itskhoki (2014) show that consumption taxation can be use as monetary policy instrument when a small open economy joins a monetary union. The goal of the paper is twofold. First, it wishes to contribute to the policy debate about the best (or rather the less distortive) way to finance, both in the long-run and along the business cycle, the increase in government spending and in the burden of public debt that the recent economic crisis brought about in most OECD countries. Second, it derives a set of analytical conditions to characterize the optimal provisions of public goods. The paper is organized as follows. Section 2 sets up the model. Section 3 presents the optimal policy exercises. Section 4 concludes. 2 The Model We add a fiscal policy block to the standard sticky-price cashless DSGE framework, similar to the workhorse model in e.g. Clarida et al. (1999) or Woodford (2003). Public consumption has intrinsic value for the agents as in Bohn (1992), Galí and Monacelli (2008), Leith, Moldovan and Rossi (2009) and Adam (2011). Government spending must be financed with linear labor income and/or consumption taxes. Lump-sum taxes or transfer are ruled out. We restrict public nominal debt to be of one-period maturity and to be state-noncontingent as in Schmitt-Grohé and Uribe (2004) and Correia, Nicolini and Teles (2008). 4
Besides presenting the model ingredients, this section derives the implementability constraints characterizing optimal private sector behavior, i.e., it derives the optimality conditions determining households consumption and labor supply decisions, firms price setting decisions, as well as the government s budget constraint. 2.1 Private Sector 2.1.1 Households We consider a continuum over [0, 1] of infinitely-lived households with preferences E 0 t=0 β t u (c t, h t, g t ), (1) where β is the discount factor, c t represents individual consumption, h t denotes hours worked and g t is government spending. E t identifies the rational expectations operator. We impose that utility is separable in its three arguments and u c > 0, u cc < 0, u g > 0, u gg < 0, u h < 0 and u hh 0, where u x defines the derivative of the utility function with respect to the generic variable x. Furthermore, we assume homothetic preferences over public and private consumption, so that gugg u g = cucc u c. There is a continuum of goods, indexed by i. Each i good enters with the same weight in the Dixit-Stiglitz aggregator. This can be written as [ 1 c t = 0 (c i,t ) η 1 η ] η η 1 di ; i [0, 1], (2) where η (1, ) is the price elasticity for differentiated goods, while the aggregate consumption price index is Hence, the demand for good i follows [ 1 P t = 0 ] 1 1 η P 1 η i,t di. (3) ( ) η Pi,t c i,t = c t. (4) P t In each period households maximize (1) subject to the following budget constraint P t c t (1 + τ c t) + R 1 t B t+1 + E t Λ t,t+1 Q t+1 = B t + Q t + P t d t + P t w t h t ( 1 τ h t ). (5) 5
In each time period t, households can purchase any desired state-contingent nominal payment Q t+1 in period t+1 at the dollar cost E t Λ t,t+1 Q t+1. The variable Λ t,t+1 denotes the stochastic discount factor between period t and t + 1. Here the only role of state-contingent securities is to define state-contingent prices. We assume that state-contingent claims are in zero net supply. Real dividends on firms profits are denoted by d t, while B t is the quantity of risk-less nominal bonds purchased in period t at price Rt 1 and paying one unit of the consumption numeraire at period t + 1. Taxes on consumption and labor income are, respectively, τ c t and τ h t, and w t is the real wage. The solution for the optimizing household problem is standard and can be written as: u c,t = λ t (1 + τ c t), (6) where λ t stands for the Lagrangian multiplier associated with this program. Labor supply is determined by while the Euler equation is u c,t 1 + τ c t u ( ) h,t 1 τ h = w t t u c,t (1 + τ c t), (7) = βe t [ u c,t+1 ( 1 + τ c t+1 ) R t π t+1 ], (8) where π t = Pt P t 1 is the gross inflation rate. The stochastic discount factor is defined as u E t Λ t,t+1 = βe c,t+1 t P t+1, and absence of arbitrage profits in the asset markets implies that E t Λ t,t+1 = R 1 t. P t (1+τ c t ) u c,t (1+τ c t+1) 2.1.2 Firms A generic good i is produced in a monopolistically competitive market with technology y i,t = a t h i,t, (9) where a t is a common exogenous technology process. Firm i s real marginal costs are: mc t = w t a t. (10) 6
We assume that firms, when resetting their prices, incur in quadratic adjustment costs a lá Rotemberg (1983), i.e., where ϕ represents the degree of price stickiness. problem can be expressed as ( ) 2 ϕ 2 P Pi,t+s t 1, (11) P i,t+s 1 The profit maximizing generic firm i s + ( ) [ ( ) max E t β s uc,t+s (1 + τ c t) Pit+s y it+s y {P i,t } u s=0 c,t (1 + τ c it+s w t+s ϕ ( ) ] 2 Pi,t+s 1 t+s) P t+s a t+s 2 P i,t+s 1 ( ) η Pi,t+s s.t. y i,t+s = y t+s. P t+s We focus on the symmetric equilibrium in which P it = P t holds. Therefore, substituting for mc t, the condition for optimal pricing decision is {[ ] } u c,t+1 (1 + τ c [(1 η) a t + ηw t ] (h t ) ϕ (π t 1) (π t ) + ϕβe t) t ( ) (π u c,t 1 + τ c t+1 1) (π t+1 ) = 0. t+1 (12) 2.2 Government Sector Macroeconomic policies are implemented by two authorities. There is a Central Bank which sets the nominal interest rates on short-term nominal bonds. Furthermore, there is a government choosing the level of public expenditures, labor income tax, consumption tax and public debt. The government finances current expenditure by raising linear labor income and consumption taxes and by issuing new one-period state-noncontingent debt, i.e., b t π t + g t = b t+1 R t + w t h t τ h t + c t τ c t. (13) The fiscal authority credibly commits to repaying its debt. In what follows we assume that government debt and tax policies are such that the no-ponzi constraint lim s + E t [( t+s 1 i=0 ) ] 1 B t+s = 0, (14) R i 7
and the transversality condition are both satisfied. [ ( lim β t+s uc,t+s s + 1 + τ c t+s ) Bt+s P t+s ] = 0, (15) 2.3 Aggregation In the symmetric equilibrium h it = h t. Hence, the economy-wide production function is y t = a t h t. (16) Furthermore, using the household s budget constraint, we can obtain the expression for aggregate profits as d t = a t h t w t h t ϕ 2 (π t 1) 2. (17) Combining the government budget constraint, the definition of profits and the households budget constraint, one can obtain the aggregate resource constraint as y t = a t h t = c t + g t + ϕ 2 (π t 1) 2. (18) With this set of structural equations at hand, we are now ready to define the model s rational expectations equilibrium. Definition 1 (Rational Expectations Equilibrium) A Rational Expectations Equilibrium (REE) is defined by a sequence of private sector decisions {c t, y, h t, w t, π t } t=0 and economic policies { } τ h t 1, τ c t 1, R t 1, g t, b t+1 that, given the initial level of public debt b 0 and the evolution of technology, solves equations (7), (8), (12), (13), (14), (15), t=0 (16), and (18). 3 Ramsey Policy Our policy exercise starts by defining the effi cient allocation within the model. Definition 2 (Social Planner s Program) The Social Planner s Program defines the first best allocation and consists in choosing {c t, h t, gt } t=0 taking as given the technology process {a t } t=0, in order to maximize the utility function of households as in (1) subject to the constraints imposed by the production technology. 8
Proposition 1 The Social Planner s allocation is (u c,t ) = (u h,t) = (u g,t ) (19) a t Proof. See Appendix. In the first best equilibrium the marginal utilities of private and public consumption must equate the marginal disutility of labor, where the latter is scaled by total productivity. This simple allocation rule is optimal because it is equally costly to produce public and private consumption goods. Definition 3 (Ramsey Problem): The Ramsey Problem is to maximize (1) over Rational Expectation Equilibria. A Ramsey outcome is a Rational Expectation Equilibrium that attains the maximum of (1). In this policy problem, the Lagrangian multipliers associated with forward looking variables, i.e. E t π t+1, E t u c,t+1, are additional state variables. Assuming these states initial values equal to zero generally implies transitory non-stationary components in the solution to the Ramsey problem, even in a non-stochastic environment. 1 In other words, the policy problem in the initial period is different than that of any future period. The reason why this occurs is that in the initial period the policy-maker may have the temptation to temporarily increase taxes or generate inflation so as to erode the real value of any outstanding government debt. 2 We do not study these non-stationary deterministic components and focus instead on the time-invariant deterministic long-run outcome. We define this allocation as the Ramsey Steady State (RSS henceforth). This means that we are implicitly imposing an initial commitment on the Ramsey planner not to generate surprise movements in taxes, prices, government spending, or nominal interest rates in period zero. This is standard practice in the optimal taxation literature, see e.g. Schmitt-Grohé and Uribe (2004). 3 3.1 Analytical Results As it is well known from Lucas and Stokey (1983), there exists a continuum of RSS, each of which associated with an outstanding level of public debt. In other words, without specifying the steady state level of public debt, the model displays an indeterminacy of degree one. 1 In section 3.6 we present a special case where the Ramsey policy is time consistent. This means that the Lagrangian multipliers associated with E tπ t+1, E tu c,t+1 are zero at all time. 2 However contrary to a flexible price economy, the Ramsey Planner does not find optimal to set P 0 =. 3 Furthermore, given the presence of sticky prices and state-noncontingent government bond, our primal form of the Ramsey problem can no longer be reduced to a unique intertemporal budget constraint in period 0 and a feasibility constraint holding in every period. However we could still write our policy problem in terms of a sequence of intertemporal implementability constraints. For a detailed discussion, see Aiyagari et. al. (2002) and Schmitt-Grohé and Uribe (2004). 9
Proposition 2 (Optimal Inflation): Despite this indeterminacy, all the RSS are characterized by π = 1 (20) and R = 1 β (21) Proof. See Appendix. It is therefore suboptimal to use steady-state inflation to reduce the real value of any outstanding level of public debt or to erode the real value of profits and in turn RSS allocations are independent of the degree of price stickiness. This is a common result in the literature that considers this class of models, e.g. Benigno and Woodford (2003) and Schmitt-Grohé and Uribe (2007). In order to understand the indeterminacy problem, it suffi ces to recall the first order condition (FOC) of the Ramsey problem (see Appendix A) with respect to b t+1, i.e. γ 4,t R t βe t γ 4,t+1 π t+1 = 0, (22) where γ 4 is the Lagrangian multiplier on the government budget constraint. At steady-state, the Euler equation implies R = 1. Therefore, at steady-state, (22) is satisfied for any values β of γ 4. Hence, in order to pin down the RSS one has to fix the outstanding level of public debt such that (22) becomes redundant, i.e. public debt becomes exogenous at steady-state. This means that in order to find the RSS, it is possible to recast the policy problem as one featuring constant real public liabilities/assets, i.e. we can rewrite the (13) as x + g t = w t h t τ h t + c t τ c t, (23) where x = (1 β) b 0 represents the steady-state level of real government liabilities. As in Adam (2011), when optimal monetary policy is in place, this modified version of the Ramsey problem leads to the same real allocation as the general problem in Definition (3) and is used to obtain a number of analytical results. 4 Proposition 3 (First Best Decentralized Equilibrium): the Ramsey Steady State and the Social Planner s allocation at steady state coincide under the following necessary condi- 4 Detailed discussion of this can be found in Appendix B, Subsection 6.5. 10
tions where x opt = b opt 0 (1 β). π = 1 (24) 1 τ h η = 1 + τ c η 1 with τ h (, 1) and τ c ( 1, ) (25) x opt = c h h (1 + τ c ) + η 1 τ h 1 with x opt < 0 η (26) Proof. See Appendix. Corollary 4 The Ramsey Planner cannot replicate the Social Planner Allocation for a generic level of outstanding public debt with τ h (, 1) and τ c ( 1, ). A few things are worth stressing. First, condition (25) indicates that the first best allocation requires at least one fiscal instrument between τ h and τ c to be a subsidy, i.e. to be negative. 5 Second, the first best allocation requires negative public debt. Put it differently, the Ramsey Planner cannot obtain the first best allocation with a generic level of debt, as this would imply either τ c < 1 or τ h > 1. 6 The reason for this result is the following. Consider, for sake of simplicity, a model with no public spending, no public debt ( x = 0) and perfect competition (η and w = 1). The monetary policy implements a zero inflation policy, i.e. π = 1. In this scenario, the market clearing condition is h = c and consumption taxes and labor income taxes have the same equilibrium tax base. Then, the decentralization of the first best implies τ c = τ h. This tax policy would raise zero revenues in equilibrium, i.e. τ c c = τ h h. However, in the presence of monopolistic competition, i.e. η <, the η first best policy implies τ h = 1 + τ c as the policy-maker needs to compensate households for the fact that the real wage fells short of ( equating the) marginal rate of substitution η 1 η 1 1 between consumption and leisure. Given that + τ c η h > τ c c, this policy implies η 1 η 1 negative revenues in equilibrium. Moreover, using numerical solutions which we describe in details below, we find that the policy-maker would always find a marginal welfare advantage in increasing consumption taxation and subsiding labor when the level of outstanding public debt is greater than x opt. This means that a potentially revenue-neutral combination of an increase in the consumption 5 In particular, if τ h > 0, then first best allocation requires τ c < 0. However it is possible to find negative values of τ h for which the first best requires τ c < 0. At the same time, if τ c > 0, the effi cient allocation requires τ h < 0. 6 One may wonder if this result relies on the absence of profit taxation. As it is well known, the optimal profit taxation in this class of models is generally equal to 100%, i.e. the Ramsey planner finds it optimal to confiscate all the profits in the economy, see for instance, Correia, Nicolini and Teles (2008) and Correia, Farhi, Nicolini and Teles (2013). We proof in the Appendix that even with the introduction of full profit taxation, i.e. τ d = 1, the effi cient level of public debt is negative. 11
tax and a decrease in the labor income tax, the latter restricted to being non negative, always increases effi ciency. However, while this policy will never replicate the first best allocation, it would create extreme tax and subsidy positions. 7 Hence, this policy would be extremely diffi cult to implement due, for example, to the high costs associated in verifying hours worked. Similarly, a very high consumption tax rate would perhaps lead to a significant amount of unreported barter. Therefore, we allow the policy-maker to use both tax instruments when x x opt, while under a generic level of steady-state debt(asset)-to-gdp, we study the cases where the Ramsey Planner is constrained in using only one tax instrument, i.e. either τ h = 0 or τ c = 0. 8 Proposition 5 Under labor income taxation, the Ramsey Planner sets u g u h (27) and ( η 1 u g u c g + x ). (28) η h }{{} (0,1) Proof. See Appendix. This proposition shows that it is optimal to set public consumption below the level suggested by the Social Planner. 9 This is because there exists a wedge between the marginal utility of leisure and the marginal utility of consumption, which is composed of two components. First, the monopolistic power that firms hold. Second, the distortive nature of fiscal policy used to finance public spending and public debt. By reducing government spending, the Ramsey Planner can lower the tax rate and hence shrinks the wedge between consumption and leisure. Moreover, while (28) does not give a precise analytical mapping between u g and u c, we can nevertheless provide some intuition about this relationship. Consider, for instance, any increase in ineffi ciency, due, for example, to more market power or higher outstanding debt. This would dampen economic activity. The Ramsey Planner by adjusting government size relative to GDP and therefore the level of taxation, can affect households labor supply and private consumption and potentially reduce ineffi ciency. For example, by imposing u g > u c, i.e. a government to consumption ratio lower than the first best, the policy-maker can sustain a given level of outstanding debt with lower taxation. Lower taxa- 7 For example Coleman (2000) finds that in a perfectly competitive economy with capital accumulation, the Ramsey planner would set τ c = 692% and τ h = 692%. Monopolistic distortion amplifies even further these fiscal positions, i.e. within standard parametrizations consumption taxation is higher than 1000%. 8 This is common practice in the literature when optimal policy implies extreme tax positions, e.g. Coleman (2000), Correia (2010) and Martin (2010). 9 This is true as long as private and public goods are not inferior goods, as it is assumed in this paper. 12
tion would increase labor supply. 10 The sign of c determines whether a higher labor supply h implies a higher or lower consumption level. In the Appendix we show that ( ) c = h sign c ( ) + cu cc c x 1h u c η The sign of (29) depends, inter alia, on the relative measure of risk aversion cucc u c, the degree of monopolistic competition η and the level of outstanding debt x. When (29) is positive, i.e. sign (29) when risk aversion is low, shrinking the government size (relative to GDP) below the first best, allows the Ramsey Planner to increase the labor supply and private consumption, thus reducing ineffi ciency. The opposite is true as (29) turns negative, i.e. when risk aversion is high. In this case the policy-maker has a strong incentive to set the government spending-to-gdp ratio greater than the first best allocation, i.e. u g < u c. With such a policy the Ramsey Planner can induce an increase in consumption and therefore reduce the wedge between the marginal utility of private consumption and the marginal utility of leisure. The desire for such a policy, i.e. u g < u c, is decreasing both in the degree of monopolistic competition and in the size of public debt. In the particular case of perfect competition and no public debt, i.e. η and x = 0, the RHS of (29) collapses to 1. In this case, if the utility is logarithmic in private and public consumption, the Ramsey Planner finds it optimal to set the share of government spending over total output as in the first best. 11 Next we analyze the scenario where the Ramsey Planner has access only to consumption taxation, i.e. τ h = 0. Proposition 6 Under consumption taxation the Ramsey Planner sets u g > u h. (30) Furthermore, if cu cc u c > 1 = u c > u g = 1 = u c = u g (31) < 1 = u c < u g u g is decreasing (increasing) in x if cucc u c = 1 for any outstanding level of public debt. and uc u c u g > (<) 1. In the special case where cucc u c = 1, 10 We are implicitly assuming that the substitution effect on labour supply always prevails, which is consistent with a Laffer curve in government revenues. 11 In the case of log utility and positive public debt and/or monopolistic competition, the Ramsey planner finds it c optimal to set the share of public spending-to-gdp lower than the first best, i.e. > 0. h 13
Proof. See Appendix. As under labor income taxation, the Ramsey Planner finds it optimal to set the level of government spending below the Social Planner level. This is due to the monopolistic features of the goods markets and the distortive nature of fiscal policy. Furthermore, (31) clarifies whether the optimal allocation requires the share of government over total output to be above or below the first best level. Interestingly, this is now only function of consumers risk aversion. If this is higher (lower) than 1, the optimal government spending-to-gdp ratio is set above (below) the Social Planner level. By allocating a high share of government spending, the Ramsey Planner is imposing, for a given level of outstanding debt, a higher tax rate. Assuming that consumption is a normal good, higher taxation implies lower consumption. In order to understand the implications of lower consumption on the labor supply schedule we need to study the sign of ( ) h = c sign ( ) [ uh ( cucc u c 1 u h+ x )] c c [ ( ) ] 1 h u hh c u h + 1 u h u c u hh u c x sign. (32) In the Appendix we show that the above expression is negative when cucc u c > 1 and positive otherwise. This means that when cucc u c > 1, the lower consumption generated by higher taxation implies an increase in labor supply, which in turn pushes the economy closer to the first best. On the contrary, when cucc u c < 1, the Ramsey Planner sets u c < u g. The resulting lower taxation pushes consumption, and therefore labor supply, upward, i.e. h > 0. c For the same reason, the optimal policy calls for an increase in the government spendingto-gdp ratio when public debt increases. Moreover, for the particular case in which risk aversion is 1, the Ramsey Planner, independently of the outstanding level of public debt, sets the marginal utility of private consumption equal to the marginal utility of public spending, as prescribed by the Social Planner. A simple comparative static exercise can show that for a given degree of risk aversion and a positive (or at least not too negative) outstanding debt, the optimal share of government spending-to-gdp is greater under consumption taxation than under labor income tax. 12 3.2 Computational Issues As discussed above, we assume that in period 0 the economy is in the RSS. When an analytical expression is missing, we rely on non-linear numerical solution algorithms. 13 Firstly, 12 Take for instance the case of logaritmic preferences, i.e. cucc u c = 1. Under labour income taxation u g > u c, while under consumption taxation u g = u c. 13 The different Ramsey problems are described in detail in the Appendix. 14
we compute the exact non-linear RSS by using the OLS projection approach proposed by Schmitt-Grohé and Uribe (2004, 2007, 2012). This consists in exploiting the insight that the Ramsey equilibrium conditions are linear in the vector of Lagrange multipliers, γ i. Then, using the perturbation method proposed by Schmitt-Grohé and Uribe (2004), we compute the accurate second-order approximation of the Ramsey s FOCs around the non-stochastic steady state of these conditions. We then use this solution to simulate the Ramsey equilibrium in the face of a technology shock. 14 The shock realizations and all the other structural parameters used for the simulations are kept constant through the different fiscal scenarios. This means that any difference between fiscal arrangements are attributable entirely to the properties of the economic policies. Moreover, we measure welfare in terms of percentage of consumption units, i.e. ϖ required by a generic policy X to reach the same level of utility as under policy Y, i.e. E 0 t=0 β t [ u ( c X t (1 + ϖ), h X t, g X t )] = E0 β [ t u ( c Y t, h Y t, gt Y t=0 )]. (33) 3.3 Parametrization We specify preferences to satisfy the conditions stated in Section 2, i.e. u (c t, h t, g t ) = c1 σ t 1 σ ω h h 1+φ t g 1 σ t 1 + φ + ω g 1 σ with σ, φ > 0. Each period represents a quarter with the discount factor, β, set to 0.9913. The elasticity of demand is chosen in order have a steady state gross markup of 1.2 (η = 6), which is in line with the macro literature. Given the importance of the CRRA parameter for our results, i.e. σ = cucc u c, we solve the model with a set of values of risk aversion that are generally found in the literature, i.e. σ (0.8, 2). However, we set σ = 1, i.e. log utility, as a benchmark value. The utility parameter ω h is chosen so that households supply between one fifth and one third of their time to work in the decentralized equilibrium when the steady-state level of public debt is zero, i.e. ω h = 19.792. We further fix ω g in order to have in the decentralised allocation with labour income tax and no public debt, a ratio of government spending over total output of 20%, i.e. ω g = 0.2641. We set the inverse Frisch elasticity of labor supply φ to 1, a value generally used as a benchmark in the macroeconomic literature, e.g. Adam and Billi (2008). The price stickiness parameter is selected such that the log-linearized version 14 In the case of effi cient steady-state, i.e. when the government is allowed to accumulate a large asset position, the second order approximation approach gives the same welfare ranking of the correspondent linear-quadratic (LQ) representation of the problem, see Woodford (2003). For this reason, part of our results are readily comparable with the literature that adopts the LQ approach. 15
of the Phillips curve (12) is consistent with the estimates of Sbordone (2002), (ϕ = 17.4). The quarterly standard deviation of the technology shocks is 0.6% and it has a quarterly persistence equal to 0.7. Furthermore, we show the implications of varying the long run level of public debt. 15 Table 1 collects the parametrization adopted. 3.4 Steady State Results This section explores the quantitative implications of different fiscal scenarios for the RSS allocations (with particular focus on the Ramsey public spending) and welfare. In particular, here we show how the various fiscal arrangements interact with government spending and long-run level public debt in determining the Ramsey allocations. Figure (1) presents the RSS allocations when public debt is allowed to vary from -100% to 200% of GDP. Increasing the level of outstanding public debt implies an increase in the (distortive) tax rate and hence a loss of effi ciency. Higher taxes imply a lower after tax real wage. However, despite the consumption tax being higher than the labor income tax, it has a lower impact on the real wage. This is because consumption is generally more inelastic than leisure. Hence households respond less, ceteris paribus, to a variation in consumption tax than to a variation in labor income tax. As a consequence, under consumption taxation the increase in public debt implies a lower response of the gap variables. In order to better understand the reason why consumption taxation is, ceteris paribus, less distortive than labor income tax, let us consider a simplified version of the model, i.e. a one-period, perfectly competitive model where agents have preferences given by log (c) h and face a budget constrained given by (1 + τ c ) c = wh ( 1 τ h) + q. Wages are constant at 1 and q is a transfer equal to the tax revenues. In this simple case hours worked are equal to the tax wedge, i.e. h = ζ = 1 τ h 1+τ c and private consumption follows from c = wh. This means that consumption taxes and labor income taxes have the same tax base. Now, let us consider the case where one of the two tax rates is set to zero. Given our policy analysis, this exercise is particularly instructive. In this case, one achieves the same equilibrium labor supply with τ h = 1 ζ and τ c = 0 or with τ c = 1 ζ 1 and τ h = 0. In the first case, i.e. with labor income taxes, the tax revenues are given by ζ (1 ζ) and have a peak at ζ = h = 0.5. The tax revenues are equal to 1 ζ in the second case of consumption taxation, and are increasing to one as the tax wedge ζ, labor supply and therefore available resources approach to zero. Transfers q approach one, but they are treated as income before consumption taxes: when the household attempts to consume this transfer income, she has to pay taxes approaching 100%, so that she is indeed left only with the resources originally produced. In other words, for any level of tax revenues, under consumption taxation the system produces more resources then 15 As a benchmark value, we fix the debt-to-gdp ratio at 90%, a value consistent with Figure 1 for the year 2012. 16
under labor income tax and hence is less distortive. Figure (2) quantifies, in percentage of permanent steady state consumption units, the loss suffered by households as steady state public debt increases. Under both fiscal scenarios, higher values of steady state public debt imply an increase in ineffi ciency and hence a deterioration of welfare. However there is a surprisingly high gain from using consumption taxation over labor income taxation as a fiscal instrument. This gain is increasing in the debt-to-gdp ratio, passing from 24% when government debt to GDP is 100% (here we put 0) to 42% when public debt is 200% of total income. As public debt increases, the fiscal authority has to devote more and more resources to pay its burden. Therefore, the lower distortive effects of consumption taxation generate a relative gain as the ineffi ciency generated by the burden of public debt increases. Figure (3) shows the difference between the Ramsey government spending to GDP ratio and the Social Planner allocation when public debt is at its benchmark value, i.e. 90% of GDP and we allow σ to vary in the interval [0.8, 2] while all the other parameters are kept at their benchmark values. This figure may be seen as a graphical representation of Proposition 5 and 6. A negative (positive) number identifies a scenario where the Ramsey Planner sets the ratio between public spending and total output below (above) the first best, i.e. g y g < (>). Under both fiscal scenarios, this difference depends critically on the parameter y controlling the risk aversion in the CRRA utility function. In particular, as consumers become more risk adverse, optimal policy calls for reducing private consumption in favor of public spending. By directly affecting the size of government spending and therefore the level of distortive taxation, the policy-maker can influence labor supply and private consumption and potentially reduce ineffi ciency. As discussed above, for a give level of risk aversion, optimal policy under consumption taxation implies a higher share of government spending over GDP than under labor income taxation. 16 Figure (4) presents the optimal public spending rule as public debt increases for different degrees of risk aversion. Under labor income tax, the optimal share of public spending is decreasing in public debt. This is because the strong distortionary effects of higher tax rates prevail over the incentive of the Ramsey Planner to increase g/y as the degree of risk aversion increases. This is consistent with (29), i.e. for a given level of risk aversion a higher level of steady state public debt generates an incentive to reduce the share of public consumption. Differently, under consumption taxation, the optimal spending rule depends crucially on whether risk aversion is below or above unity, as presented in Proposition 6. Therefore when σ = 0.8, i.e. cucc u c < 1, the Ramsey Planner, by cutting the share of government as public debt increases, can boost both consumption and labor supply. On the contrary, when σ = 2, 16 In particular, under labour income tax g = g c for σ = 1.3868 = y y (c x 1 h), i.e. c = 0. h ε 17
i.e. cucc u c > 1, the optimal policy increases the government spending-to-gdp ratio in order to increase hours worked and therefore total output. Finally, as presented in Proposition 6, in the benchmark case of log utility in consumption, i.e. cucc u c = 1, the optimal share of government spending is always at the first best, independently of the level of steady state public debt. We perform a set of robustness checks both on some key structural parameters of the model as well as on the fiscal instruments available to the government. Results from this exercise are reported in Table 2. Decreasing the degree of market power, i.e. increasing η, it reduces ineffi ciency: the overall production increase, and so do private and public consumption. As a result, under both fiscal scenarios, as η becomes bigger, the welfare loss decreases. When consumption taxes are excluded, lower levels of monopolistic power are consistent with higher government spending-to-gpd ratios. This is consistent with Proposition (5) and (29). The reason is that as the ineffi ciency due to market power gets smaller, the Ramsey planner has an incentive to push g towards the effi cient level. Furthermore, in this fiscal scenario, as η y gets bigger, the optimal level of labour income tax decreases. This is due to the increase of the labour-income tax base that a greater degree of market competitiveness brings about. As a result, the fiscal authority can sustain a higher level of government spending, and ceteris paribus, a higher public spending-to total output ratio, with a lower labour income tax rate. Differently, when consumption taxes are available, reducing monopolistic power does not effect the optimal provision of public spending (as share of the GDP). This is a direct consequence of Proposition 6, where the Ramsey ratio of g depends solely on the degree of y risk aversion. Furthermore, in the optimal allocation, the increase in η reduces the ineffi ciency in the system, but it does not affect the consumption tax base. Therefore τ c stays constant at its benchmark value as the system moves towards perfect competition. 3.5 Ramsey Dynamics 3.5.1 First Best Steady State We now turn our attention to the optimal policy in a stochastic setting. To this end we perturb the model with a technology shock as described in 3.4. Studying Ramsey policy in the face of this type of shock is standard practice in the literature (e.g. Schmitt-Grohé and Uribe (2004), Correia et al. (2008), Adam (2011), Leith and Wren Lewis (2013) ) and allows us to better disentangle our contribution. We start our analysis by considering a situation where the policy-maker implements first best at steady-state. This implies that the government is allowed to accumulate large asset positions (i.e. negative debt) and that one (or both) taxes can be negative. Both these assumptions will be dropped later. We refer the 18
reader to Proposition 3 in Section 3.2 on how a Ramsey Planner can achieve the first best in this environment. In practise, we fix the steady state value of consumption taxation to 16%. This value corresponds to the average level of consumption taxation in the industrialized countries found by Gordon and Li (2009). Then, the labor income tax rate and the level of the government assets are pinned down by (25) and (26) respectively. Here we run three policy exercises. In the first one the policy-maker can respond to shocks by using both taxes. In the other two, we restrict the fiscal authority to use only one tax instrument in the face of shocks. The three scenarios share the same steady state allocation so that any difference can be attributed entirely to the dynamic properties of the tax instruments adopted. Results are reported in Figure (5). When the policy-maker has access to both taxes, first best is attainable in the stochastic economy under consideration. In other words, the Ramsey Planner can commit to a statecontingent policy such that in face of a technology shock the response in the decentralized economy coincides with the Social Planner s allocation. In this case, Ramsey policy is also time-consistent, i.e. if a policy-maker were given the opportunity to revise this policy in any future period, it would find it optimal not to do so. Absence economic policy, in a sticky-price environment a negative technology shock would imply an increase of inflation and a positive output gap, i.e. price would increase but not as much as required in a flexible price world, thus imply an ineffi ciently high level of output. As it is well know from Galí (2001), a welfare maximizing policy-maker would therefore tighten aggregate demand as to push output towards the effi cient level. This policy would also stabilize inflation. Here the Ramsey Planner looses monetary policy on impact and promises to keep it above the steady state level for some periods after the shock. At the same time, she increases the consumption tax and promises to cut it in the next period. As we discuss in more details later, this contingent tax policy represents one of the main advantages of using consumption taxation as a demand management tool. Indeed, the combined optimal use of monetary and fiscal instruments allows the policy-maker to fully control the stochastic discount factor and hence offset firms desire to changing prices, thus replicating the flexible price equilibrium. Furthermore, in order to mimic the Social Planner solution, the Ramsey Planner has to avoid distorting households consumption-leisure choice. In other words, the policy-maker moves the consumption and the labor income tax rate in the same measure and in opposite direction so that they do not create a wedges between the marginal utility of leisure and that of consumption. Finally the Ramsey equates the marginal utility of private and public consumption as required by the first best equilibrium by varying her assets positions. In other words, if the government is allowed to take large assets position and can use both consumption and labor income taxation, she can offset both the static and the dynamic 19
ineffi ciencies of the model. Table 3 reports the model s campionary moments of this exercise. 17 Given the shape of the utility function (σ = 1), dynamic effi ciency requires output, consumption, government spending and real wages to be perfectly correlated with each other and with the technology process, while hours worked and inflation should remain at their steady state values. These results are closely related to Correia et al. (2008). They show that with consumption and labor income taxation, full profit taxation and exogenous government spending, the Ramsey Planner can mimic the flexible price equilibrium, so that any real allocation is independent from the degree of nominal rigidities. Here we show that the first best allocation can be implemented without profit taxation and with endogenous government spending as long as the government is allowed to take large asset positions. Next, we analyze fiscal scenarios in which the government is constrained in using only one tax instrument. We start with the case where only consumption taxation is available. IRF s are presented in Figure (5). In this case, while the policy-maker can obtain long run effi ciency via asset accumulation, the absence of labor income taxation makes it impossible to reach dynamic effi ciency. In particular, the optimal tax policy required to balanced the government budget constraint distorts the leisure-consumption decision, thus creating an ineffi ciency wedge in the labor supply and therefore in the real wages. At the time of the shock, output, consumption and public spending all drop by roughly the same amount as the case where the Ramsey Planner has access to both taxations, thus resulting in very small movements of gap variables. 18 This policy, coupled with an initial cut in the nominal rate and an increase in consumption taxation, is aimed to generate an increase on impact of inflation. This helps to reduce the cut in the government assets necessary to balanced the government budget constraint. As for the case with two tax instruments, this policy is reverted in the period after the shock. Fiscal policy commits in reducing consumption taxation while monetary policy increases the nominal rate above its steady state value. These combined policies affect the real stochastic discount factor and imply a one-period deflation episode, which in turn pushes the price level near its steady state value. Table 3 reports the implied simulated moments of this policy experiment. Compared to the previous scenario, the ineffi ciency generated by the absence of labor income taxation generates an incentive for the policy-maker to reduce the volatility of output, consumption, 17 Given the quasi-random walk properties of public debt, unconditional moments are not available. Hence, like Schmitt-Grohé and Uribe (2004), Arseneau and Chugh (2008) and Nieman and Pichler (2011), we calculate the campionary moments of the model. In particular we simulate the model 5000 times for 100 periods. For each simulation we calculate the statistics of interest. Then we report the median value. 18 As in Benigno and Woodford (2003) and Adam (2011), consumption, output, government spending, taxes and debt all follow a near ramdom walk pattern. However, given the optimal consumption tax policy, the long run values of these variables are very close to their steady state counterparts. 20