Tax Smoothing, Learning and Debt Volatility

Transcription

1 Tax Smoothing, Learning and Debt Volatility Francesco Caprioli JOB MARKET PAPER 31 October 2008 Abstract In this paper I investigate the optimal fiscal policy when markets are complete and private agents are learning the tax schedule implemented by the government. The main result is that the government should use fiscal variables to manipulate agents expectations, setting low taxes when agents are pessimistic about future consumption and high taxes when they are optimistic about it. Moreover taxes and debt are very persistent, a salient feature in the data that the complete markets optimal taxation framework with rational expectations cannot fit. JEL Classification: E62, H63, D83. Keywords: Optimal Taxation, Learning, Debt Sustainability. I am heavily indebted to Albert Marcet for his excellent supervision and encouragement. I would like to thank Richard Anton Braun, Omar Licandro, Eva-Carceles Poveda, Jordi Galí, Mark Giannoni, Bruce Preston, Michael Reiter and Thijs Van Rens for their very useful comments and suggestions. I am also grateful to Sofia Bauducco, Filippo Ferroni, Stefano Gnocchi, Antonio Mele and Andrea Tesei for detailed feedbacks, which greatly improved this work. francesco.caprioli@upf.edu. Office: (+34)

2 1 Introduction... there is no reason and no occasion for any American to allow his fears to be aroused or his energy and enterprise to be paralyzed by doubt or uncertainty... It is true that the national debt increased sixteen billion dollars... you will be told that the Government spending program of the past five years did not cause the increase in our national income... But that Government spending acted as a trigger, a trigger to set off private activity.. (Franklin Delano Roosevelt, On the Recession (1938)) Just this week, we learned that retails sales have fallen off a cliff, and so industrial production. All signs point to an economic slump that will be nasty, brutish-and long. How nasty?... the unemployment rate will go above 7 percent, and quite possibly above 8 percent... And how long? It could be very long indeed... there is a lot the federal government can do for the economy... Now it is not the time to worry about the deficit.. (Paul Krugman, (2008)) An important issue in public finance theory is how to collect revenues to pay for government expenditures. When lump-sum taxes are not available, fiscal authorities must resort to taxes which distort people s decisions and move the economy away from the first-best. This is why how to set taxes is of crucial importance. The optimal taxation literature (e.g. Lucas and Stokey (1983), Chari et al. (1994), Chari and Kehoe (1999)answers this question looking for the tax profile which minimises the distortionary costs associated with it. The key insight of this literature is that under complete markets the pay-off of the portfolio of state-contingent bonds works as an insurance device. As a consequence, the tax rate should be smooth and respond very little to shocks. This conclusion has been derived in a framework in which agents fully understand the problem faced by the government. As they know the problem, they know the solution too, so that their expectations about future tax rates are model-consistent. In a sense the assumption of rational expectations is very strong because it endows agents with a huge amount of knowledge about market determined outcomes. In reality this knowledge may not be so deep. 1 Then two questions arise: are the fiscal policy recommendations suggested 1 There is a growing evidence against the assumption of fully rational agents. For example, Poterba (1988) and Parker (1999) find that private consumption does not react in anticipation of future tax changes; Feldman and Katuscak (2008) reject the hypothesis of taxpayers being fully rational and fully informed about the tax system with which they interact. Allen and Carroll (2001) find that it takes a very long time 2

3 by the existing literature still valid when agents do not know the problem faced by the policymakers? If not, what is the optimal fiscal policy when agents are learning it? In this paper I give an answer to these two questions. I consider a closed production economy with no capital and infinitely lived agents, where the only source of uncertainty is given by an exogenous government expenditure shock. The problem of the household is to maximise her lifetime expected utility subject to her flow budget constraint. The only difference between this framework and the standard optimal fiscal policy one is that agents do not know the tax schedule the government is going to implement. Hence, their expectations about their own next period s state-contingent marginal utility of consumption are not model-consistent. Agents act like econometricians and use a weighted average of past values of the contingent marginal utility; given the realisation of the shock, each period they update their belief about the marginal utility of consumption contingent on that specific realisation. The government is benevolent and chooses distortionary taxes on labour income and state-contingent debt to maximise households welfare subject to the feasibility constraint, households optimality conditions and the way in which they update their beliefs. Three are the main results. First, the fiscal policy which is optimal under rational expectations generates a very volatile stream of leisure and consumption when it is implemented in a set-up in which agents are bounded rational. The intuition for this result is that the optimal debt management under rational expectations prescribes volatile bond positions across states of nature. Agents perceive this type of debt management simply as a huge wealth effect and therefore they are willing to adjust private consumption. Under standard assumptions on the agents utility function, this higher volatility reduces welfare. Second, in a bounded rationality setting the government should set fiscal variables to manipulate private agents beliefs. To give an intuition for this result, assume that the government expenditure is constant and that the government has zero initial wealth. Under rational expectations, the optimal fiscal rule prescribes balanced budget: the government sets the tax rate to collect enough revenues to finance expenditure. When agents do not have rational expectations, this fiscal rule is still feasible, but it will imply a much longer time for agents to learn the tax rate than if the government followed an expectation-dependent for each household to understand its own consumption function (or a first-order Taylor approximation of it), even considering some form of social learning. Finally, Adam and Padula (2003) and Forsells and Kenny (2003) conclude for not fully rational inflation expectations, and Robert (1997) analyses the consequences of this result in terms of deflation costs. 3

4 fiscal plan. When agents are pessimistic, (i.e. they expect the one-step-ahead tax rate to be higher than they would expect it to be if they were fully rational), government optimality conditions require current expenditure to be financed mainly through debt: in this way the low current tax rate induces agents to revise downwards their expectations about the next period tax rate. 2 In the long-run the tax rate is higher than in a rational expectations framework because the government has to finance the interest paid on a positive amount of debt. 3 In this sense the initial agents beliefs has an effect on the long-run mean value of the tax rate and debt: the more pessimistic (optimistic) the agents are, the higher is the government debt (wealth) in the long-run. One implication of this result is that the model can help explain the wide dispersion across countries in the level of government debt and tax rate. Third, non-rational expectations influence the amount of tax smoothing the government can achieve. The case-study of a perfectly anticipated war is a clear example of the tension between the two conflicting goals the government wants to achieve, tax smoothing on one side and manipulation of beliefs on the other. Under rational expectations it is optimal for the government to accumulate assets before the war and sell them during the war-time. In this way the tax rate is constant in all periods before and after the war. Differently, in a learning framework pessimistic agents do not trust the promises made by the government, which is forced to lower the tax rate to manipulate agents expectations. As a consequence, the government accumulates less assets (than in a RE framework) before the war and finances the war issuing debt. The tax rate after the big shock is much higher than before. 4 Learning introduces a unit-root behaviour in the tax rate, a salient feature in the data which the standard optimal fiscal policy framework with complete markets cannot explain. 5 This persistence affects the power of some widely used tests to check for market completeness and debt sustainability. In line with Marcet and Scott (2007) we find that looking at the behaviour of debt is a much more reliable way to test the bond market structure than looking at the behaviour of tax rates. Similarly, the standard unit-root test in the debt/gdp 2 One implication of this result is that restricting how much a government can become indebted can delay the learning process. 3 The analysis is symmetric for the case of optimistic agents. 4 Manipulating expectations can explain why a benevolent government should run a deficit during peacetime periods, an implication that the Lucas and Stokey (1983) does not have and for which has been criticized. 5 Aiyagari et al. (2002), Battaglini and Coate (2008), Karantounias et al. (2007) and Svec (2008) are all attempts to achieve this result removing alternative assumptions from the baseline model and introducing either market incompleteness, or a political-economic bargaining equilibrium or fear of misspecification by agents. 4

5 ratio used to discriminate between responsible and non-responsible governments can be misleading, since it may cause a fiscal policy plan to be declared unsustainable when instead it is sustainable by construction. Augmenting this test to include the primary surplus in the regressors is a sharper way to distinguish the optimal and sustainable fiscal policy from an unsustainable policy. Finally, we solve the model when the government expenditure is endogenous. We find that, when agents are pessimistic, the fiscal authority would increase the government expenditure compared to the rational expectations value, financing it through debt. This result is in line with some proposals to deal with the recent distress. Many authors have studied the impact of learning on monetary policy design, either when the central bank follows some ad hoc policy rules (see inter alia Orphanides and Williams (2006), Preston (2005a,b, 2006), Preston and Eusepi (2007b,a)) or when it implements the optimal monetary policy (see inter alia Evans and Honkapohja (2003, 2006), Molnar and Santoro (n.d.)). Perhaps surprisingly, fiscal policy has received much less attention. Evans et al. (2007) study the interest rate dynamic learning path in a non-stochastic economy in which the fiscal authority credibly announces a future change in government purchases. Karantounias et al. (2007) and Svec (2008) study the optimal fiscal policy when agents do not trust the transition probabilities of the public expenditure suggested by their approximating model. Up to my knowledge, this is the first paper studying the influence of learning on fiscal policy design. The paper proceeds as follows. Section 2 studies the consequences of implementing the bond policy function under rational expectations when agents are learning. Section 3 solves for the optimal fiscal policy under learning. In Section 4 we present some simulation results. Section 5 gives some policy implications. Section 6 deals with the problem of discriminating between a complete markets model with learning and an incomplete markets model with rational expectations. Section 7 focuses on test sustainability and debt limits. In Section 8 we report some stylized facts about fiscal variables and agents sentiment and we use US data in order to test the model. Section 9 concludes. 5

6 2 The Model 2.1 Case 1: Rational expectations by both the government and agents We consider first the case in which both the government and households have rational expectations. We present the same model as in Lucas and Stokey (1983) in which a benevolent government levies a proportional tax rate τ t on labour income and has access to a complete set of one-period state-contingent bonds to cover an exogenous stream of government expenditure shocks. The resource constraint is given by where c t, l t and g t denote respectively private consumption, leisure, and public consumption. c t + g t = 1 l t (1) The representative agent s problem is to maximise his lifetime discounted expected utility E 0 β t u(c t, l t ) (2) t=0 subject to the period-by-period budget constraint b t 1 (g t ) + (1 τ t )(1 l t ) = c t + g t+1 g t b t (g t+1 )p b t(g t+1 ) (3) where β is the discount factor, τ t is the state-contingent labour tax rate and b t (g t+1 ) denotes the amount of bonds issued at time t contingent on period t + 1 government shock at the price p b t (g t+1). vb t g t+1 g b t t (g t+1 )p b t (g t+1) is defined as the value of government debt. The government pursues an optimal taxation approach: given an initial amount of inherited debt, it chooses the sequence of tax rates and state-contingent bonds to maximise consumer welfare. It can be shown that the Ramsey equilibrium satisfies τ t = G(g t, b g 1) t > 0 (4) 6

7 b g t(g t+1 = ḡ) = D(ḡ, b g 1) t > 0 (5) vb g t = V (g t, b g 1) t > 0 (6) The allocation is a time invariant function of the only state variable in this model, g t. The initial holding of government bonds matters for the allocation because it determines the value of the Lagrange multiplier attached to the implementability condition, which can be thought of as the intertemporal consumer budget constraint with both prices and taxes substituted out by the agents optimality conditions. The state-contingent bond holding is a time invariant function and does not depend on the current state of the economy, and the market value of debt is influenced by the current shock only through variations in the state-contingent interest rates. 2.2 Case 2: Rational expectations by the government and learning by the agents We assume now that the government has rational expectations while agents are learning fiscal policy: they have perfect knowledge about their own decision problem, in the sense that they correctly understand their own objective function and constraints, but they do not know the problem that the other agents in the economy, government included, have to solve for. As a consequence, they need to forecast market determined variables, in particular interest rates and tax rates, which are outside of their control although they are relevant to their decision problem. The representative agent s problem is to maximise the lifetime utility Ẽ 0 β t u(c t, l t ) t=0 subject to the flow budget constraint equation (3). Ẽ 0 denotes the agent s subjective expectations. The optimality condition with respect to the state-contingent bond is: p b t(g t+1 ) = β u c,t+1(g t+1 ) u c,t π(g t+1 g t ) (7) 7

8 . where π(g t+1 g t ) is the conditional probability of the government expenditure shock. For simplicity, agents know the distribution of g t.. Combining the optimality conditions with respect to consumption and leisure we get 1 τ t = u l,t u c,t (8) Equation (8) gives the combination of consumption and leisure given the tax rate. It is still open how much the agent consumes and saves: according to the Euler Equation Approach, the optimal consumption choice depends on the forecast of one-step-ahead marginal utility of consumption. Combining the flow budget constraint with equation (7) we get b t 1 u c,t = u c,t (c t (1 l t )(1 τ t )) + β g t+1 g t ũ c,t+1 (g t+1 )b t (g t+1 )π(g t+1 g t ) (9) Given arbitrary beliefs, the current tax rate and the holding of state-contingent bonds issued by the government in the previous period, agents adjust their consumption in order to satisfy equation (9), which gives the optimal consumption rule. We assume that the government implements the policy sequences {b t+j (g t+j+1 )} j=0 coming out of (5). The time-line of the events is the following. At the beginning of period t, agents observe the realisation of the shock g t and the tax rate set by the government at t: using all the information up to t 1, they form their expectations about one-period-ahead state-contingent marginal utility of consumption and decide how much to consume. To close the model it remains to specify the manner in which households form their beliefs. They estimate a theoretical regression using as regressors a constant and the government expenditure shock, which is the only variable appearing in the Minimum State Variable solution of the model under rational expectations. The law of motion of marginal utility of consumption perceived by the agents is given by u c,t = [1; g t 1 ] d t 1 + ǫ t (10) where d t 1 is the coefficient vector of dimension (2 1) and ǫ t is the error term. Since by assumption agents know the probability distribution of g t and observe the realisation of the shock before forming their expectations, forecasts of the endogenous variables 8

9 are given by E t u c,t+j = [1; E t g t+j ] d t 1 = d 0,t 1 + d 1,t 1 ρ j g g t j 1 (11) Equation (11) highlights the fact that, as standard in models of boundedly rational learning, agents are estimating a misspecified model: they believe that the model is stationary, while instead it has time-varying coefficients, whose law of motion is given by d t = d t t V 1 t g t 1 (u c,t x t 1 d t 1) (12) V t = V t t (x t 1x t 1 V t 1) (13) where x t 1 = [1; g t 1 ] and V t 1 is the variance-covariance matrix of the regressors. 6 The question is whether, given a sufficient amount of data, this equilibrium converges to the rational expectations equilibrium. The criterion adopted to judge convergence under a recursive least-squares algorithm is the expectational stability of REE, called E-stability by Evans and Honkapohja (2001). The following proposition holds: Proposition Suppose that the government does not make the fiscal plan contingent on agents beliefs and follows the bond fiscal rule which is optimal under rational expectations. Assume that agents utility is separable into consumption and leisure, logarithmic in consumption and linear in leisure. Assume moreover that the initial bond holding is zero. Then the REE is learnable. Proof. We relegate the proof to the appendix. Even if the fiscal authority does not take into account that agents do not have RE, the allocation would eventually converge to it. If the E-learnability condition would not be satisfied, two things would happen. First, agents would be prone to change their regression scheme; second, it would not make sense to compute log-deviation of the system from a steady state around which there would never be convergence. Even when the rational expectation equilibrium is learnable, along the transition path, the fluctuations in the allocation can be quite large with respect to those under rational expectations. To measure the increase in volatility induced by learning we use numerical methods to simulate the model. The utility function is 6 Alternatively to this scheme, agents can use the constant gain scheme, in which case the term 1 t and (13) is replaced with a constant 0 < γ < 1. In the simulation we consider both cases. in (12) 9

10 2 u(c t, l t ) = c1 σ t 1 σ + log(l t) (14) The government expenditure shock g t is assumed to follow a truncated AR(1) process ḡ, if(1 ρ g )g + ρ g g t 1 + ε g t > ḡ g t = g, if(1 ρ g )g + ρ g g t 1 + ε g t < g (1 ρ g )g + ρ g g t 1 + ε g t, otherwise where ε g t N(0, σ ε g ), g is the steady state value, fixed at 0.1 and ḡ, g equal to g + ( ) t ( ) σ 2 ε g t 1 ρ 2 g and g 2 σ 2 ε g t 1 ρ 2 g respectively. In the baseline parametrisation of the model we set σ ε g = 0.01, σ = 1 and β = t To measure the inefficiency of the fiscal policy plan under RE implemented in a context in which agents expectations are formed using a recursive least-squared algorithm rather than being rational, we consider the ratios of consumption and leisure variance when agents learn and the government implements the optimal bond policy rule under rational expectations over their corresponding values when instead both the agents and the government have rational expectations. Table 1: Relative variance ratio ρ g σ 2 c L σ 2 c RE σ 2 l L σ 2 l RE Table 1 shows these ratios change for different values of the autocorrelation coefficient of the government shock. The values are average across 1000 simulations. Two observations are worth noting. First, learning increases the variance of the allocation at least by around 30 per cent with respect to the rational expectations framework. Second, the higher ρ g, the higher is this relative increase. The intuition for these results hinges on the implication of tax smoothing in terms of bond portfolio management; smooth taxes imply volatile bond positions across states of nature. Since optimal consumption under learning depends on the state-contingent bonds issued in the previous period, agents perceive this kind of debt 10

11 management simply as a huge wealth effect and therefore they are willing to adjust private consumption as a consequence. The higher the autocorrelation coefficient for the government shock, the higher is the variance of the bond positions, which translates into a higher variance for consumption. The main implication of this result is that trying to teach the lesson is not a good policy recipe. Implementing a fiscal policy plan without taking into account the way in which not fully rational agents form their expectations induces a stream of consumption and leisure that is much more volatile than in the case of agents assumed to form modelconsistent expectations. Making the policy plan contingent on agents beliefs is much better, in terms of welfare, than obstinately performing the optimal fiscal policy under the rational expectations paradigm. The analysis also raises some doubts about recent developments in the debt management literature, according to which a government should hold extreme debt positions. The first reason to hold such positions relates to the capability to recover the complete markets outcome; almost every equilibrium allocation under complete markets can be replicated through an appropriate position in non-contingent bonds at different maturities. 7 Since the welfare is higher under complete markets than under incomplete markets, the optimal debt management is the one that allows the complete markets outcome to be reached. Due to the low variability of one-period-ahead bond prices across realisations of shock today, Angeletos (2002) and Buera and Nicolini (2004) show that the government can implement the complete markets allocation holding very extreme positions in bonds with different maturities. The second reason for debt volatility is to make the full commitment solution time consistent. Persson et al. (1987) consider a model with nominal frictions where the government in charge has an incentive to engage in a surprise inflation to erode the nominal inherited debt. The unique maturity structure of debt implementing the full commitment solution requires the government in t to leave to its successor a positive amount of nominal bond holdings. 7 Under incomplete markets with only one period bonds, it is not possible to achieve the complete market solution because the implementability condition is replaced with an infinite sequence of period-by-period budget constraints. Aiyagari et al. (2002) for example show that in case of i.i.d. shock, implementing the complete market solution, which generates i.i.d. primary surplus, would imply an explosive path for debt. 11

12 3 Optimal Fiscal Policy When Agents Are Learning The previous section shows that the optimal fiscal policy under rational expectations can lead to very bad outcomes under learning. This suggests that learning should be incorporated in the design of optimal fiscal policy: in this section we study the problem of a government which internalises the fact that agents do not have rational expectations. The pricing equation (7) expresses the state contingent interest rate as a function of households one-step-ahead expectations about state-contingent marginal utility. Understanding their own maximization problem agents realise that consumption at t + 1 depends on the tax rate the government will set at t + 1. It follows that perfect foresight over statecontingent consumption requires perfect foresight over the state-contingent tax rate, which by itself requires a large amount of information about the market. Actually, to forecast the tax rate each household has to solve for the government problem, which by itself implies the following assumptions: Assumption 1 Every agent knows that other agents have the same preferences and constraints and form expectations the way that he does. Assumption 2 Every agent knows that the no-rational-bubble condition holds for all t and all realisations. lim E tβ j u c,t+j b t+j = 0 (15) j Assumption 3 Every agent knows that the resource constraint holds. Assumption 4 Every agent knows that the government has a technology commitment and behaves as a benevolent social planner. The knowledge beyond these assumptions pertains to the external environment of each agent, and it is not required for his maximisation problem. The first three assumptions are necessary in order to compute the relevant constraints of the government, the implementability condition (Assumptions 1 and 2) and the feasibility condition (Assumption 3), while the last one is necessary to understand the objective function of the government. Therefore, the hypothesis implicit in the rational expectations framework of agents able to form modelconsistent expectations endows them with the knowledge of the function (4), which maps 12

13 realisations of government expenditure into the tax rate. 8 Since we want to analyse situations outside the rational expectations equilibrium, in what follows we assume that Assumptions 1-4 do not belong to the agents information set. As a consequence, the optimality conditions for each agent are summarised by u l,t u c,t = 1 τ t (16) p b t(g t+1 ) = βũc,t+1(g t+1 ) u c,t π(g t+1 g t ) (17) Equation (16) coincides with the intratemporal condition under rational expectations, equation (8), while ũ c,t+1 indicates non-rational expectations over the one-step-ahead statecontingent marginal utility of consumption. For simplicity we assume that the government expenditure shock can take only two realisations, g H and g L. Since in the rational expectations equilibrium the marginal utility of consumption is constant given the realisation of the shock, agents expectations are equal to and their law of motion is give by: Ẽ t u c,t+1 (g t+1 = g H ) = γ H t Ẽ t u c,t+1 (g t+1 = g L ) = γ L t γ i t = { γ i t 1 + α t (u c,t (g t = g i ) γ i t 1), if g t = g i γ i t 1, if g t = g j (18) with i = H, L and where α t follows an exogenous law of motion. Definition 1. A competitive equilibrium with non rational agents is an allocation {c t, l t, g t } t=0, state-contingent beliefs about one-step-ahead marginal utility of consumption {γ i t} t=0 for i = H, L, a price system {p b t } t=0 and a government policy {g t, τ t, b t } t=0 such that (a) given the price system, the beliefs and the government policy the allocation solves the household s problem; (b) given the allocation and the price system the government policy satisfies the sequence of government budget constraint (3); and (c) the goods and the bonds markets clear. 8 The reader might argue that if the government announces the state contingent tax plan, and it is fully credible, then the agents have perfect foresight about state-contingent marginal utility, and we are back to the rational expectations equilibrium. This is true if and only if the agents know the resource constraint: only in this case in fact, given the tax rate in the next period, agents can pin down future consumption from the ratio between marginal utility of leisure and marginal utility of consumption. 13

14 Let x t = [γ H t I(g t+1 = g H ) + γ L t I(g t+1 = g L )] (19) where I is the indicator function and define t A t k=0 x k 1 u c,k (20) The log of A t can be interpreted as the sum of past forecast errors in predicting the marginal utility that agents have made from period 0 to period t. Under rational expectations, this variable is constant and equal to 1, while under learning this is not the case any more, unless the initial beliefs about state contingent marginal utilities are exactly equal to the actual marginal utility. Proposition Assume that for any competitive equilibrium β t A t u c,t 0 a.s. 9 Given b 1, γ 1 H, γl 1, a feasible allocation {c t, l t, g t } t=0 is a competitive equilibrium if and only if the following constraint is satisfied 10 E 0 β t A t (u c,t c t u l,t (1 l t )) = A 0 u c,0 b 1 (21) t=0 with initial condition A 1 = 1 Proof. We relegate the proof to the appendix. With respect to the rational expectations case, there are two new state variables. The first one is the vector of last period beliefs about today s contingent marginal utility. The second one, A t 1, is the product of the ratios until period t 1 between expected marginal utility and actual marginal utility. The recursive formulation for A t is A t = A t 1 γ H t 1I(g t = g H ) + γ L t 1I(g t = g L ) u c,t (22) Definition 2. The government problem under learning is subject to (21), (22), (18) and (1). max {c t,l t,γ H t,γl t,at} t=0 E 0 β t u(c t, l t ) (23) 9 Using the results of Proposition we show that this is actually the case. 10 Because of learning, future endogenous control variables do not appear in the budget constraint at time t, and therefore it is not necessary to summarize all the constraints into the implementability condition. Nevertheless we use the intertemporal budget constraint and not the period-by-period one since in this way do not need to calculate the Lagrange multiplier associated with it. 14 t=0

15 Let g t = (g 0,..., g t ) denote the history of government expenditure. Attach the multipliers, β t π t (g t )λ 1,t (g t ), β t π t (g t )λ 2,t (g t ), β t π t (g t )λ 3,t (g t ) and β t π t (g t )λ 4,t (g t ) to constraints (21), (18) for i = H, L, (1) and to (22). The Lagrangian is L =E 0 β t {u(c t, l t ) + (A t (u c,t c t u l,t (1 l t ))) t=0 + λ 1,t ((γ H t γ H t 1 )I(g t = g L ) + (γ H t (1 α t )γ H t 1 α tu c,t )I(g t = g H )) + λ 2,t ((γ L t γ L t 1 )I(g t = g H ) + (γ L t (1 α t )γ L t 1 α tu c,t )I(g t = g L )) γt 1 H + λ 3,t (1 l t c t g t )} + λ 4,t (A t A I(g t = g H ) + γt 1 L I(g t = g L ) t 1 ) A 0 u c,0 b 1 u c,t Assuming b 1 = 0, the first-order necessary 11 conditions t 0 are: c t : u c,t + A t (u cc,t c t + u c,t ) λ 1,t α t u cc,t I(g t = g H ) λ 2,t α t u cc,t I(g t = g L ) + u cc,t λ 4,t A t 1 γ H t 1 I(g t = g H ) + γ L t 1 I(g t = g L ) u 2 c,t = λ 3,t (24) l t : u l,t + A t (u l,t u ll,t (1 l t )) = λ 3,t (25) γ H t : λ 1,t βe t {λ 1,t+1 I(g t+1 = g L ) + (1 α t+1 )λ 1,t+1 I(g t+1 = g H )+ + λ 4,t+1A t u c,t+1 I(g t+1 = g H )} = 0 (26) 11 As standard in the optimal fiscal policy literature, it is not easy to establish that the feasible set of the Ramsey problem is convex. To overcome this problem we check that the solution to the first-order necessary conditions of the Lagrangian is unique. 15

16 γ L t : λ 2,t βe t {λ 2,t+1 I(g t+1 = g H ) + (1 α t+1 )λ 2,t+1 I(g t+1 = g L )+ + λ 4,t+1A t u c,t+1 I(g t+1 = g L )} = 0 (27) A t : (u c,t c t u l,t (1 l t )) + λ 4,t βe t λ 4,t+1 γ H t I(g t+1 = g H ) + γ L t I(g t+1 = g L ) u c,t+1 (28) From equation (28) λ 4,t = (u c,t c t u l,t (1 l t )) + βe t λ 4,t+1 γ H t I(g t+1 = g H ) + γ L t I(g t+1 = g L ) u c,t+1 (29) Multiplying both sides by A t we get λ 4,t A t = A t (u c,t c t u l,t (1 l t )) + βe t λ 4,t+1 A t γ H t I(g t+1 = g H ) + γ L t I(g t+1 = g L ) u c,t+1 = = A t (u c,t c t u l,t (1 l t )) + βe t λ 4,t+1 A t+1 (30) where the last equality follows from equation (22). Iterating forward we obtain λ 4,t A t = E t β j A t+j (u c,t+j c t+j u l,t+j (1 l t+j )) (31) Inserting (31) into (24) we get j=0 u c,t + A t (u cc,t c t + u c,t ) λ 1,t α t u cc,t I(g t = g H ) λ 2,t α t u cc,t I(g t = g L ) u cc,t E t β j (32) A t+j (u c,t+j c t+j u l,t+j (1 l t+j )) = λ 3,t u c,t j=0 The first term on the left side of equation (32) represents the benefit for the government by increasing consumption by one unit. The second one measures the impact of the implementability constraint on the allocation, weighted by the distortion A t represented by non rational expectations. The third and fourth terms reflect the fact that the government takes 16

17 into account how agents update their expectations on the basis of the current consumption. The last term on the left represents the derivative of all future expected discounted primary surpluses with respect to current consumption. This is because from equation (21) each primary surplus (in terms of marginal utility) at t+j, j 0 is pre-multiplied by the product of past ratios between expected and actual marginal utility. In choosing optimal consumption today, the government is implicitly choosing the factor at which all future primary surpluses are discounted. The term on the right is the shadow value of output. A similar interpretation holds for the optimality condition with respect to leisure. Several comments are necessary. First, the optimal allocation is history-dependent through the presence of A t 1 : differently from Lucas and Stokey (1983), the allocation is not any more a time-invariant function of the current realisation of the government shock only, but depends on what happened in the past. Given the state variables g t, A t 1, the vector of past beliefs γt 1 i, i = H, L and α t 1, the problem becomes recursive and standard solution techniques can be applied. Second, it is simple to show that the optimality condition in a complete markets and rational expectations framework is a very special case of (32). Under rational expectations the following equalities hold γ H t 1 γ L t 1 = u c,t (g t = g H ) t = u c,t (g t = g L ) t which implies that A t = A t 1 = 1 t. The Lagrangian collapses to L =E 0 β t [u(c t, l t ) + (u c,t c t u l,t (1 l t )) + λ 1,t (γt H t=0 γ H t 1) + λ 2,t (γ L t γ L t 1) + λ 3,t (1 l t c t g t )] u c,0 b 1 The first-order conditions with respect to γ H t and γ L t are γ H t : γ L t : λ 1,t = βe t λ 1,t+1 (33) λ 2,t = βe t λ 2,t+1 (34) 17

18 which imply that the only solution is λ 1,t = λ 2,t = 0. Combining the first-order condition with respect to consumption and leisure we get u c,t + (u cc,t c t + u c,t ) = u l,t + (u l,t u ll,t (1 l t )) (35) which is exactly the optimality condition found in a rational expectations framework (see Lucas and Stokey (1983)) in which expectations do not depend on the current consumption level and in which there is no distortion into agents beliefs that the government has to manipulate optimally. Third, using the recursive formulation of A t, the intertemporal budget constraint at t b t 1 A t u c,t = E t β j A t+j (u c,t+j c t+j u l,t+j (1 l t+j )) j=0 and combining (32) and (25) the optimal allocation satisfies the following equations: u c,t + A t (u cc,t (c t b t 1 ) + u c,t ) λ 1,t α t u cc,t I(g t = g H ) λ 2,t α t u cc,t I(g t = g L ) = u l,t + A t (u l,t u ll,t (1 l t )) (36) Equation (36) looks very similar to the first-order condition with respect to consumption in the incomplete market model of Aiyagari et al. (2002). In the two frameworks in fact the excess burden of taxation is not constant, although for different reasons. In the absence of a full set of state-contingent bonds, as in Aiyagari et al. (2002), the excess burden of taxation is time-varying because of the incomplete insurance offered by the financial bonds: since the interest payment on last period debt is fixed across realisations of the current government shock, the government in each period has to adjust the stream of all future taxes to ensure solvency. 12 In a complete markets model with learning, what makes the excess burden of taxation time-varying is the cost of issuing state-contingent debt. Although market completeness implies that in each period the government can fully insure against expenditure shocks, the state contingent interest rates change as time goes by because agents expectations change. When agents stop updating their beliefs because the forecast error is zero, A t+j = A t 1 j 0, and the excess burden of taxation becomes constant again. Equation 32 expresses the actual marginal utility of consumption as a function of agents beliefs about it. Figure 1 shows the T-mapping for a log-log utility function and a given value of A t The left panel displays the actual marginal utility of consumption contingent on 12 In a complete market framework with rational expectations the excess burden of taxation is constant because the variable which adjusts to ensure solvency is the pay-off of the portfolio of contingent bonds. 13 The shape of the mapping is robust to different values of this variable. 18

19 the government expenditure shock being low (average with respect to the expected marginal utility of consumption contingent on the government expenditure shock being high), as a function of the previous period belief, γt 1. L The right panel display the same for the government expenditure shock being high. Since the slope is negative, the E-stability condition is satisfied. Figures 2 and 3 show the tax rate and the state-contingent bond policy functions which guarantee that the convergence between actual and expected marginal utility holds. The tax rate is a decreasing function of the previous period expected marginal utility; symmetrically, the state-contingent bond is an increasing function of it. 3.1 Some examples In order to characterise the optimal fiscal policy in the framework we are studying, in what follows we consider some examples restricting the government expenditure shock to a specific form. Although they are not meant to offer a realistic description of the reality, these examples highlight the large differences in the optimal fiscal plan with and without rational expectations Constant government expenditure shock. Consider the case in which the government expenditure is known to be constant. The policy recipe under rational expectations would be to fix the tax rate at the level to balance the budget every period: since there are no intertemporal distortions to smooth there is no role for debt. With learning, the government problem is L = β t [u(c t, l t ) + A t (u c,t c t u l,t (1 l t )) t=0 + λ 1,t (γ t (1 α t )γ t 1 α t u c,t ) + λ 2,t (1 l t c t g)] A 0 u c,0 b 1 (37) where the notation is the same as before and The optimality conditions t 0 are: A t = A t 1 γ t 1 u c,t (38) u c,t + A t (u cc,t c t + u c,t ) λ 1,t α t u cc,t u cc,t β j A t+j (u c,t+j c t+j u l,t+j (1 l t+j )) = u l,t + A t (u l,t u ll,t (1 l t )) u c,t j=0 (39) 19

20 λ 1,t β(1 α t )λ 1,t+1 + β b t A t = 0 Equation (39) gives the mapping T between agents beliefs about (marginal utility of) consumption and actual (marginal utility of) consumption. In the next proposition we characterise the properties of this mapping. Proposition Let the utility function be u(c t, l t ) = logc t + l t (40) and that the term α t is small enough to be ignored. Then, in the set γ t 1 > 0 the mapping T: R + R + has the following properties: T is increasing and concave. T has one fixed point. The least squares learning converges to it. Proof. We relegate the proof to the appendix. The solid lines in figures 4, 5 and 6 show the optimal fiscal plan under rational expectations. Whereas the dashed lines show the optimal fiscal plan when agents adopt a constant gain algorithm to update their belief while supposing that in the initial period the expected consumption is lower than the actual consumption prevailing at t = In Proposition we have shown that the expected marginal utility converges to actual one. The next proposition characterises this value. Proposition Given an initial value for the government bond holding b 1 the allocation under learning does not converge to the allocation under rational expectations implied by the same initial bond holding, unless agents have the correct initial belief about marginal utility. However, for any initial belief held by agents, γ 1, there exists a b 1 such that lim t c L t (γ 1) = c RE t (b 1 ) (41) Proof. We relegate the proof to the appendix. 14 Since for optimistic agents the evolution of the system is symmetric, we do not report it. 20

21 For any initial belief about the marginal utility of consumption, there is always an initial level of government wealth such that the allocation under learning converges to the one under rational expectations starting with that initial government wealth. Figure 10 shows this relation assuming that in equation (37) b 1 = 0. Given the parameters values used, the solution of the Ramsey problem under rational expectations implies that the marginal utility of consumption is constant and equal to 2.5. For all values of initial belief higher (lower) than this reference value, the learning allocation coincides with the solution of a Ramsey problem in an economy populated by rational agents and endowed with a positive (negative) initial government debt One big shock Consider the case in which expenditure is known to be constant in all periods apart from T, when g T > g t. Under rational expectations the government runs a positive primary surplus from period 0 to T 1, using it to buy bonds. At T the government finances the high public consumption level by selling the accumulated assets and possibly by levying a tax rate on labour income. From period T + 1 onwards the tax rate is just sufficient to cover the expenditure and to service the interest on the bonds issued at T. On the other hand, in an economy populated by pessimistic agents, the government can accumulate less assets because it has to stimulate the economy to manipulate expectations. The big shock at T is financed by increasing debt much more than under rational expectations. Figures 7-9 illustrate the optimal plan under rational expectations (the solid lines) and under learning (the dashed lines), assuming T = 10, g t = 0.1 and g T = Cyclical shocks Suppose that g t = g H t = j H T g t = g L otherwise with j = 1, 2,..., T. H is the length of time over two subsequent bad shocks and T is the H last period in which a bad shock can occur. The rational expectations policy recipe is the same as before: the tax rate is constant in all periods when g t = g L and increases very little when the bad shock hits the economy, due to the assets the government accumulates during the good shock periods. Under learning with pessimistic agents, before the first realisation of the bad shock the tax rate is lower than under RE and increases between any two subsequent 21

22 bad shocks, generating resources devoted to reducing debt, which increases whenever the bad shock occurs. After the last bad realisation of the shock, the tax rate falls and then gradually increases over time to ensure intertemporal solvency Bad shock of unknown duration Suppose that the shock can take two realisations, g L and g H, with the following transition probabilities matrix ( 1 0 P = π H,L π H,H where π i,j is the probability that tomorrow the shock is in state j, being today in state i, with g t = g H at t = 0. This example corresponds to an absorbing Markov chain, where the low realisation of the shock is the absorbing state and the high one is the transient state. Under rational expectations, the government finances the bad shocks partly through taxes and partly by issuing debt. Numerical results, not reported here, confirm the role of fiscal policy as stabiliser of expectations: the accumulation of public debt is higher and longer under learning than under rational expectations, the difference being due to the opportunity of inducing the agents to revise their expectations downwards. 4 Simulation Results In this section we apply numerical methods to compute the Ramsey equilibrium described in the Section 3 when the expenditure shock is correlated. We assume a discount factor equal to 0.95 and a gain parameter equal to The government expenditure shock is a two-state Markov process with values equal to g H = 0.1 and g L = 0.05 and transition probabilities equal to π H,L = π L,H = 0.2, π H,H = π L,L = Table 6 summarises some statistics for the allocation and the fiscal variables under rational expectations. Table 7 summarises the same statistics under learning after convergence of beliefs for initially pessimistic and optimistic agents. Reported values are average across 1000 simulations. 15 The choice of the updating parameter is not easy because it requires a trade-off between filtering noises and tracking structural changes. Milani (2007) estimates a New-Keynesian model with one-period-ahead expectations and finds that the best fitting specification has a gain coefficient in a range between and Orphanides and Williams (2004) find that a value for k in the range fits the expectations data from the Survey of Professional Forecasters better than using higher or lower values. Evans et al. (2007) also use the same value for k. 16 For the case of i.i.d shock case the results are very similar to those with serially correlated shock, and therefore are not reported. ) 22

23 Table 2: Tax smoothing and learning γt 1 L γt 1 H E j β γt 1 L γt 1 H E j β Two points are worth noting. First, if γ 1 i > u c RE(g t=0 = g i ) for i = H, L, in the long run the consumption level under learning is lower than under rational expectations. Since at the beginning consumers were pessimistic, the accumulation of debt necessary to induce them to revise their expectations upwards requires higher taxes in the long run than under rational expectations. Because of this, consumption is lower and leisure is higher (than under rational expectations). The average primary surplus is higher as well. In this sense we can say that beliefs are self-fulfilling in the long run: the lower is the initial expected consumption, the lower the actual consumption after convergence. Exactly the opposite is true with optimistic agents, i.e. when γ 1 i < u c RE(g t=0 = g i ) for i = H, L. Second, the amount of tax smoothing the government can achieve in the long run is influenced by the initial beliefs. Suppose we estimate the following equation τ j t = α j + β j g j t + ǫ t (42) where the index j refers to the simulation. The higher β j, the more the tax rate responds to the government shock, and therefore the lower the tax smoothing. The reference value under RE for E j β j is equal to Table 2 reports the mean β across 1000 simulations as a function of the initial expectations. Table 8 shows the same statistics for the system during the first 30 periods of transition, respectively under RE and under learning for pessimistic agents. All the variables are more volatile before convergence than after convergence. This is due to the fact that before convergence agents expectations and the product of past ratios between what agents 23

24 expected in terms of future consumption and actual realised consumption change over time, and this volatility is added to that of the government shock. Notice that the market value of government debt and the labour tax rate are the most volatile endogenous variables, since they have to adjust to influence agents beliefs. Notice also that tax rate volatility while agents are learning is double that after convergence. As concerns autocorrelation, learning creates persistence in the system, which displays more correlation than after convergence. Importantly, the most autocorrelated variables are the fiscal variables, the market value of debt and the tax rate. The top panel in Fig. 14 displays the persistence of the debt/gdp ratio, the primary surplus/gdp ratio and the government expenditure shock under rational expectations: the bottom panel displays the same variables under learning. In both cases we consider the first 50 periods of simulated data. 17 Although in the model debt is the most persistent variable, it is not as persistent as in the data. Another way to compare persistence under learning and under rational expectations is to look at autoregressions of tax rates in the two frameworks when the shock is i.i.d.. Table 9 shows that while under rational expectations the coefficient on the lagged tax rate is close to zero and not statistically significant, under learning it is high and statistically significant. We conclude this section measuring the goodness of the learning equations used by agents to predict one-step-ahead state-contingent marginal utility of consumption. We use the Epsilon-Delta Rationality criterion (EDR), as formalized in Marcet and Nicolini (1998). Define π ǫ,t P( 1 T T [u c,t γ t 1 ] 2 < 1 T t=0 T [u c,t E t 1 u c,t ] 2 + ǫ) which is a function of ǫ, δ and T. E t 1 u c,t denote the expectations of an agent who knows the whole economic structure of the model. The learning mechanism (18) with α t = α satisfies EDR for (ǫ,δ,t) if π ǫ,t 1 δ. Table 5 shows this π ǫ,t for different values of ǫ (across columns) and T (across rows). Reported values are computed out of 1000 simulations. Since the discount factor is equal to 0.95, each period corresponds to one year. 17 To measure the persistence of a variable, say y, we use the k-variance ratio, defined as t=0 P k y = V ar(y t y t k ) kv ar(y t y t 1 ) 24