Fiscal Policy with Limited-Time Commitment

Transcription

1 Fiscal Policy with Limited-Time Commitment Alex Clymo Andrea Lanteri June 2018 Abstract We propose a theory of optimal fiscal policy consistent with the observation that governments typically inherit their predecessors plans and formulate their policies over a finite future horizon. We use this framework, which we call Limited-Time Commitment (LTC), to make two contributions. First, in a general stochastic model, we show that, under an intuitive sufficient condition, a chain of successive governments with LTC sustains the Full Commitment (FC) Ramsey equilibrium. This equivalence result clarifies in which cases existing recursive methods to solve for FC plans have a positive interpretation based on partial commitment. We apply this result in two benchmark models of optimal fiscal policy (with and without capital), in which the FC policy is time-inconsistent. Second, we investigate the implications of LTC in a model of capital taxation in which the FC outcome cannot be achieved with finite commitment. We show that the internalisation of nearfuture tax distortions represents a key distinction with respect to the literature on No Commitment (NC). In a calibrated version of the model, we find that a single year of fiscal commitment leads to large welfare gains (approximately a third of the difference between FC and NC). Finally, we use LTC to analyse commitment to different fiscal instruments and to study the importance of state-contingency in fiscal plans. Department of Economics, University of Essex. Wivenhoe Park, Colchester CO4 3SQ, UK. a.clymo@essex.ac.uk. Department of Economics, Duke University. 213 Social Sciences, Durham, NC 27708, United States. andrea.lanteri@duke.edu. We thank the Editor Morten Ravn and two anonymous referees for comments that greatly improved the paper. We are very grateful to Albert Marcet for insightful conversations at all stages of this project. We also thank Marco Bassetto, Davide Debortoli, Wouter den Haan, Zhen Huo, Paul Klein, Sarolta Laczó, Ricardo Reis, Victor Ríos-Rull, and Pierre Yared, as well as seminar participants at the Bank of England, Duke TDM workshop, University of Amsterdam, IAE-CSIC/UAB (Barcelona) and EEA Conference Alessandro Villa provided excellent research assistance. All errors are our own. 1

2 1 Introduction Governments in advanced economies tend to formulate their macroeconomic policies as plans for a finite future horizon. In particular, fiscal policy is typically decided upon and announced before or at the beginning of the fiscal year and remains fixed for the duration of the year. 1 Reforms such as tax rate changes and fiscal consolidation plans are also announced before their implementation and typically contain details of short-to-medium run policy plans. 2 Moreover, the political process in representative democracies makes it hard to change contemporaneous policies, with the result that policy changes are typically implemented with a delay. Hence, while clearly there are examples of sudden changes following large economic shocks, the way fiscal policy is conducted in normal circumstances features significant lags between decisions and execution. In contrast, a large part of the literature on optimal fiscal policy assumes either that a single government at the beginning of time has Full Commitment (FC) into the infinite future, or that in each period there is a government with No Commitment (NC) at all, only able to choose contemporaneous policies. Both of these extreme assumptions appear hard to reconcile with the fact that policymakers are in power for a limited amount of time, inherit their predecessors plans, and communicate as if they possessed a degree of commitment over a finite future horizon. Motivated by this apparent distance between observation and theory, in this paper we study fiscal policy when successive benevolent governments inherit the plans of their predecessors and formulate plans for a finite future horizon. In this formulation, which we call Limited-Time Commitment (LTC), governments cannot commit into the infinite future, but instead only possess the ability to commit for a finite horizon. Specifically, we define governments as having L 1 periods of commitment if the time-t government cannot change policies dated time t to t + L 1, and chooses policies dated time t + L. We introduce this novel equilibrium notion of optimal policy in the context of a general stochastic framework, in which the presence of future variables in the competitiveequilibrium constraints faced by the planner induces time-inconsistency under FC. We then apply our LTC equilibrium concept to make two key contributions. Our first contribution is to provide a sufficient condition for equivalence between the outcomes that arise under LTC and FC in a general stochastic framework with time inconsistency. We then find conditions under which this equivalence holds in two seminal optimal policy models, and show that a finite degree of commitment, sometimes a single period, can often sustain FC outcomes in these models. Time-inconsistency of FC plans arises because future allocations appear in the current constraint set. This leads a government under FC to make promises about future policies in order to influence private-sector expectations. Ex-post, the government would like to renege on these (past) promises, if a reoptimisation was allowed. With LTC, a government is allowed to commit to a finite sequence of future policies, which are taken 1 For example, governments in the Euro area announce their fiscal budget plans - the Annual Draft Budgetary Plans - for the following year. In the UK, the budget is announced on Budget Day in March, before the start of the fiscal year, and typically passed shortly afterwards. 2 Examples of significant pre-announced VAT reforms include Japan (announced in 1996, implemented in 1997) and Germany (announced in 2005, implemented in 2007). Alesina et al. (2015) document many examples of multi-year fiscal consolidation plans across different countries. 2

3 as state variables by future governments. We show that if this finite sequence of future policy instruments is sufficient to uniquely pin down all the future variables that enter the current constraint set, then the time-inconsistency problem can be resolved with LTC, because the future variables become, effectively, current choices for the government. The next government will then inherit this policy plan as a state variable, ensuring that equilibrium allocations are not changed relative to the promised plan, and creating a chain of commitment that sustains the FC equilibrium path. We apply this equivalence result to two key models of optimal fiscal policy that have been studied in the literature (e.g., versions of the models used by Lucas and Stokey, 1983, Judd, 1985, Chamley, 1986). We find the minimum necessary degree of future commitment in each model, and show that the FC solution can be often exactly supported by a finite degree of commitment. We then discuss which specific fiscal instruments or rules help to build commitment, and relate the required degree of commitment to economic features of the model environment. In a model without capital, FC outcomes can be supported with commitment equal to the maximum government debt maturity. In models with capital, we uncover a key role for balanced-budget rules as well as income and consumption taxes in sustaining FC outcomes with LTC. In the literature, these models are often solved under FC using recursive methods, for instance by adding state variables to the planner s problem, such as promised allocations or marginal utilities (Kydland and Prescott, 1980) or multipliers on forward-looking constraints (e.g., Marcet and Marimon, 2017). However, in reality governments choose policy instruments, and cannot commit to equilibrium allocations or multipliers. The existing solution methods are silent as to whether a finite degree of commitment could sustain the FC allocation in equilibrium. Our results clarify when the existing solution algorithms have a positive interpretation as theories of fiscal policy in presence of limited commitment. In so doing, we provide guidance for future researchers on the cases in which assuming FC, even if seemingly unrealistic, leads to the same outcomes as a more empirically plausible setup. Our second contribution is to propose LTC as a positive model of fiscal policy that incorporates realistic commitment assumptions, even in models where exact equivalence with FC does not hold. We explore in depth the properties of our general Markov- Equilibrium concept, which includes pre-existing policy plans as state variables, in the context of a benchmark model of public good provision and capital taxation (Klein et al., 2008), using both analytical and numerical tools. We let governments choose taxes one period in advance and characterize the government s optimality conditions by deriving a Generalised Euler Equation (GEE) for LTC. We then compare this GEE to the optimality conditions with FC and NC. This analysis provides intuition on a key source of welfare gains from LTC (relative to NC): the partial internalization of future tax distortions. In a calibrated version of the model which we solve numerically, we find that this mechanism is sufficient to recover around one third of the welfare difference between FC and NC in steady-state. Next, we simulate the effects of a constitutional reform, imposing one year of commitment to taxes and spending, starting from the steady-state of an economy with NC. The transitional dynamics following this reform involve a non-monotone path for the size of the government (retrenchment followed by government expansion as the whole economy expands) and lead to significant welfare gains (approximately 1.4% of permanent consumption). 3

4 We then introduce shocks, modelled as stochastic shifts in the valuation of government spending, calibrated to match postwar US data. This exercise allows us to assess the importance of state contingency in fiscal commitments in a model that features a trade-off between commitment and state-contingency. We solve for the contingent LTC equilibrium, and discuss how it differs from the non-contingent LTC equilibrium. We find that the high persistence of the calibrated shocks leads to similar outcomes in these two cases. However, we also find that in a counterfactual scenarios with i.i.d. shocks, there would be large gains for state-contingency in LTC plans. Finally, we consider a version of the model with endogenous labour supply, assuming that the government can commit only to taxes, or only to government spending. We show that commitment to taxes alone is more welfare enhancing than commitment to government spending alone, consistent with the fact that in the real world changing tax rates is typically costlier than changing the level of government expenditures. Related Literature. The time inconsistency of optimal policy has been a central issue in the macroeconomic literature since the 1970s. Kydland and Prescott (1977) highlighted that optimal policy in a dynamic model involves ex-ante promises that appear suboptimal ex-post if the government is allowed to reoptimise at a later date. For this reason, the presence of future (expected) variables in the constraint set of Ramsey-optimal plans rules out the use of standard optimal control techniques. Despite its importance both in the academic and in the policy debate, this key insight has not led to a uniform reaction in the literature. A part of the optimal policy research agenda has worked on modifying standard recursive methods to deal with this class of problems under the assumption that the government is endowed with a Full Commitment technology into the infinite future, while another strand of the literature has considered time inconsistency a central, unavoidable problem and has focused on equilibria with No Commitment instead. As leading examples of the first approach, consider the Recursive Contracts method formalised in Marcet and Marimon (2017) which adds the Lagrange multipliers on forward-looking constraints as state variables, allowing the reformulation of optimal policy problems as recursive saddle-point problems, or the Promised Utility approach proposed by Abreu et al. (1990), based on the result that past histories can be summarised by promised utilities. 3 While these approaches can be used to solve recursively many models of optimal policy under FC, our results clarify in which cases these solutions have the interpretation of equilibrium outcomes arising from policy commitments over a future horizon. In such cases, our theory provides a new rationale for studying FC policies. The second part of the literature has focused instead on formulating equilibrium concepts without a commitment technology, starting with the seminal paper on Sustainable Plans by Chari and Kehoe (1990), and the application of Markov perfect NC equilibria to optimal policy games in, for instance, Klein and Ríos-Rull (2003), Krusell et al. (2004) and Klein et al. (2008). In these papers, there is a succession of governments that 3 Relatedly, Kydland and Prescott (1980) and Chang (1998) proposed similar methods based on the addition of future marginal utilities as state variables in the optimal policy problem. In this paper we propose an alternative, intuitive, recursive method to solve for optimal policy with FC, based on using policies as state variables. This method works whenever our equivalence result holds. 4

5 can only choose contemporaneous policy instruments. Hence, any ability to formulate credible promises about future allocations is ruled out by assumption. In our paper, we explore an intermediate assumption on the commitment technology, namely that a succession of governments can announce policies for a finite future horizon. We allow for some commitment to future outcomes as in the first strand, but we apply the Markov perfect equilibrium concept typical of the NC literature. Closest to our work, in terms of motivation, are the Quasi Commitment approach of Schaumburg and Tambalotti (2007), and the Loose Commitment approach formulated by Debortoli and Nunes (2010, 2013), which build on an early contribution by Roberds (1987). These papers assume that a government can formulate a plan into the infinite future, but with some probability in every future period a new government is elected and allowed to change the plan. Like ours, this game represents an intermediate point in the FC vs. NC debate. Differently from these papers, however, LTC gives the government full commitment within a limited, deterministic time horizon. While this may seem a technical difference, it turns out to be important. We show that in some models where LTC with one period of commitment leads to allocations equivalent to FC, Loose Commitment with on average one period of commitment leads to allocations closer to NC. In this sense a result of this paper is that once you limit the commitment technology of the government, the details of how you do so matter for the results. 4 An interesting example of limited commitment is provided by Klein and Ríos-Rull (2003), who assume that the government can set capital taxes one period ahead, but only set current labour taxes. They show that capital taxes are on average high, compared to the FC equilibrium which has average capital taxes near zero. We solve a deterministic version of the same model and show that if instead the time-t government chooses both the time-t + 1 capital and labour taxes then LTC sustains the FC policy and allocation. We thus show that even small changes to timing assumptions on policy choices can have large effects on the size of the inefficiencies caused by limited commitment. 5 Furthermore, the paper is related to the political economy literature on optimal rules in presence of time inconsistency (Amador et al., 2006, Halac and Yared, 2014). Unlike in our paper, the source of time inconsistency in this literature is present bias, and this leads to a microfoundation of optimal saving rules to limit governments incentives to overspend. Instead, our source of time inconsistency is the presence of forward-looking agents. We show that the combination of a limited degree of commitment to taxes and 4 In related work, Brendon and Ellison (2018) propose to substitute the assumption of plans formulated from the perspective of time-0 with a Pareto criterion, according to which a plan is selected if in every period there is no plan that can make all current and future governments better off. An alternative approach is given in the reputational equilibria literature, starting with the seminal contribution of Barro and Gordon (1983). Bassetto (2016) distinguishes the communication and implementation of future policies, and shows that there is no separate role for communication if the government has the same information as the private sector. Other papers explore the extent to which FC outcomes can be supported by adding extra policy instruments, e.g. Alvarez et al. (2004), Conesa and Dominguez (2012) and Laczó and Rossi (2015). 5 Domeij and Klein (2005) study a tax reform in the presence of implementation lags where they solve the Full Commitment problem under the assumption that policies can only be changed starting from some period T onwards. They assume Full Commitment from time T onwards, while we are interested in implementation lags as a source of commitment. Martin (2015) studies the effects of within-period commitment to fiscal policy. In contrast, we analyse the effects of commitment over a finite future horizon. 5

6 imposed fiscal rules (such as balanced budgets) can be used as institutional tools to mitigate the adverse consequences of limited commitment. 6 Finally, our work takes a first step to connect the literature on optimal fiscal policy with the literature on the economic effects of anticipated policy changes. Mertens and Ravn (2012) emphasise the importance of anticipation effects in their empirical analysis of tax changes in U.S. post-war data. House and Shapiro (2006), Mertens and Ravn (2011) and Leeper et al. (2012) study fiscal anticipation effects in the context of DSGE models of the economy. We see LTC as a natural framework to study how governments may strategically exploit the link between promised policies in the near future and current equilibrium outcomes. The rest of the paper is organised as follows. Section 2 informally introduces the concept of LTC in two simple examples. Section 3 formalizes our equilibrium concept. Section 4 provides the main equivalence result. Section 5 applies the result to two benchmark models of optimal fiscal policy. Section 6 studies a model where our equivalence result fails and provides numerical results. Section 7 discusses our contribution relative to two existing approaches. Section 8 concludes. 2 LTC in two examples In this section we provide an intuitive introduction to the notion of fiscal policy with LTC using two examples. In the first example, we present a model where the FC solution can be exactly supported with LTC. In the second, we present a model where there is no finite degree of commitment that can exactly support the FC solution, and discuss the effects of LTC. 2.1 Example 1: A model where FC can be sustained with a single period of commitment Our first example is a simple deterministic version of the optimal fiscal policy model of Lucas and Stokey (1983), in which the government only has access to one-period bonds. We informally introduce the notion of optimal policy with LTC and argue that a single period of commitment is sufficient to exactly sustain the FC outcome in this model. To avoid notational clashes with the general framework presented in Section 4, here and wherever we refer to a specific model we use upright text to denote variables. Preferences of a representative household over consumption, c t, and labour, l t, are represented by the following utility function β t u(c t, l t ) (1) t=0 with standard assumptions u c > 0, u cc < 0, u l < 0, u ll < 0. The budget constraint is given by c t + q t b t+1 = w t l t ( 1 τ l t ) + bt (2) 6 Relatedly, Athey et al. (2005) study the optimal design of monetary institutions in the context of time inconsistency stemming from the role of expectations in inflation and output determination. 6

7 where q t is the price of a one-period discount bond, b t+1, issued at t that repays one unit of consumption at t + 1. w t is the wage, and τ l t the proportional tax on labour income. 7 The resource constraint reads c t + g = l t (3) where g is a constant level of government expenditure that needs to be financed with the proportional labour income tax. Output is produced using a linear technology in labour: y t = l t, hence firms profit maximisation implies a unit wage: w t = 1. The government s budget is implied by the agent s budget constraint and the resource constraint. The agent s first order conditions with respect to consumption, labour effort and bonds, together with the resource constraint, can be summarised by an intratemporal optimality condition and an Euler equation for each t = 0, 1,... u l(c t, c t + g) u c (c t, c t + g) = 1 τ l t (4) q t u c (c t, c t + g) = βu c (c t+1, c t+1 + g) (5) We refer to Lucas and Stokey (1983) for a detailed treatment of the FC optimal policy in this setup. At time 0, the FC government announces a plan for the whole path of taxes, {τ l t} t=0 in order to maximise (1) subject to the sequence of competitive equilibrium constraints (2), (3), (4), and (5). In this model, the source of time inconsistency of the FC policy is the presence of the future choice variable c t+1 in (5), the Euler equation for bonds. In particular consider a government who starts with an initial stock of debt, b 0 > 0. In this case, the optimal FC path for taxes features a lower tax rate at time zero, followed by a constant higher tax in all future periods: τ l 0 < τ l 1 = τ l 2 =... The difference between the time-0 and t > 0 policies reflects the time inconsistency: the FC government has an incentive to use the initial tax rate to twist the interest rate and decrease the value of outstanding initial debt. In particular, the FC government lowers τ l 0 to increase initial consumption, c 0, relative to future consumption and receive a more favourable interest rate (higher q 0 ) when rolling over its initial debt. For any t > 0, this would come at the cost of raising the interest rate the period before, but at time 0, there is no time -1 Euler equation which will be affected by the choice of c 0. However, if we consider allowing the government to reoptimise at any time t > 0, it is clear that the government would follow exactly the same logic: in order to decrease the value of outstanding debt, b t, the government would like to offer another tax cut at time t, although it promised not to do this initially. Thus, the FC policy is time inconsistent, because the governments at time t and t + 1 disagree about how to value the choice of c t+1. These are precisely the incentives faced by governments in the No Commitment game, which is discussed further in Section 5. This twisting of the interest rate reflects the very tight link that competitive equilibrium places between taxes and consumption in this model. In particular, notice that the static labour optimality condition, combined with goods-market clearing, imply that 7 Here and throughout the paper we ignore the possibility that the government can issue lump-sum rebates. This is without loss of generality, and all results go through if the rebate is treated as a policy instrument, symmetrically with the other taxes. 7

8 consumption is determined solely by the level of the contemporaneous labour tax in this model, irrespective of the path of future taxes. That is, (4) defines a function c t = c(τt). l 8 Assume there is a positive, albeit finite, degree of commitment. In particular, consider a game where a new government is in power in each period. Due to institutional features that make it too costly to modify current policies and implementation lags, governments cannot affect the current tax rate, and can only choose next period s tax rate. Thus, each government takes the contemporaneous tax rate, τt, l as a state variable and chooses the policy for the following period, τt+1. l As detailed in Section 3, we denote this game the Limited-Time Commitment game with one period of commitment. In a Markov-perfect equilibrium of this game, the government at time t chooses next period s policy as a function of the natural state variables (b t, τt), l giving a policy function τt+1 l = τ l (b t, τt). l As in the NC game, we solve for a symmetric equilibrium, assuming that each government plays against its successors, understanding how the state variables it leaves affect future policies. With a single period of commitment to taxes, the choice of c t+1 is taken completely out of the time-t + 1 government s hands, and instead c t+1 is chosen by the time-t government through their choice of τt+1. l Thus, while without any commitment c t+1 was a future choice variable in the time-t constraint set, with just a single period of commitment it becomes simply a current choice variable in the time-t constraint set. This removes the following government s ability to twist the interest rate and affect the value of outstanding debt. It is then possible to show that starting from an initial condition consistent with the FC plan, a chain of successive governments with LTC will sustain the whole FC plan as the unique equilibrium. This model is an example of a situation where a finite amount of commitment can exactly support the FC solution. We formally provide the conditions under which this result attains in Section 4, but the model above provides an intuitive example: if a finite sequence of policy instruments (here: τt+1) l can uniquely pin down all future choice variables appearing in time-t constraints (here: c t+1 ) then time inconsistency can be overcome with a finite amount of commitment. This equivalence result and its applications to benchmark models of optimal fiscal policy constitute the first key contribution of the paper. 2.2 Example 2: A model where FC cannot be sustained with any finite amount of commitment Our framework allows us to think about fiscal policy in presence of LTC, regardless of whether this leads to the same outcomes as FC. Thus, our next example is a deterministic model of optimal public good provision based on Klein et al. (2008), in which no finite amount of commitment can sustain the FC allocation. We use this model to argue that a small degree of commitment may still lead to important differences relative to the NC equilibrium. Preferences of the representative household are now represented by the utility function β t [u (c t ) + w (g t )] (6) t=0 8 Under standard conditions on the utility function u, this mapping is unique. 8

9 where c t continues to denote consumption and g t is a public good, and we assume that u, w > 0, and u, w < 0. We assume that labour is not valued and inelastically supplied at l t = 1. 9 The budget constraint of the household is c t + k t+1 = w t l t + k t [ 1 + rt ( 1 τ k t )]. (7) Output is produced with production function y t = k α t l 1 α t. Using the fact that labour is inelastically supplied at l t = 1, firms profit maximisation implies that the pre-tax return on assets is given by r t = αk α 1 t δ and the wage by w t = (1 α)k α t. In order to simplify notation, let f (k) k α + (1 δ) k. The resource constraint of the economy reads c t + k t+1 + g t = f (k t ), (8) and the government balanced-budget constraint is 10 τ k t (f (k t ) 1) k t = g t. (9) The competitive equilibrium is fully characterised by (8) and the following Euler equation for capital, where we substitute taxes from (9). ( u (c t ) = βu (c t+1 ) f (k t+1 ) g ) t+1 (10) k t+1 This Euler equation highlights the time inconsistency issues in this model. There are two future choice variables in the constraint: c t+1 and g t+1. Starting with the latter, the FC government understands that promising high government spending tomorrow (high g t+1 ) implies higher capital taxes tomorrow, which will discourage investment today. This counteracts the welfare benefits of providing more government services, leading the FC government to set a relatively lower level of steady-state government spending. If we consider a government without commitment, at time t they observe an already installed level of capital and ex-post decide the level of capital taxation. Thus, they do not take into account the effect of capital taxes on investment when choosing the level of taxes, and will hence set higher taxes and higher government spending. Ultimately however, the private sector anticipates the higher taxes, leading to lower investment and welfare. Now consider the case of Limited-Time Commitment, where governments again have a single period of commitment. The government at time t is not allowed to change τt k or g t, and thus has as natural state variables (k t, g t ). 11 The government chooses g t+1 (or equivalently τt+1) k subject to the Euler equation (10) and resource constraint (8). That the time-t government chooses g t+1 in the LTC game removes one source of time inconsistency from the model, since g t+1 goes from being a future choice variable in the current constraint (10), to a current choice variable. Intuitively, since the government at 9 We show in Section 5 that if labour is elastically supplied the FC solution can be supported with a single period of commitment. 10 Differently from Section 5, here we allow for a deduction for depreciation before capital taxes are charged. 11 For a given k t, any value of τt k implies a unique value of g t via (10) and vice versa, meaning that only one needs to be held as a state variable. 9

10 t cannot change g t, this removes the ability of the government to act on its temptation to increase g t, and hence τ k t, once capital is already installed. However, this is not enough to completely eliminate time inconsistency from the model and sustain the FC equilibrium, because c t+1 also appears in (10) and it is not uniquely pinned down by the policy variables at t + 1. To see this, consider the resource constraint at t + 1: c t+1 + k t+2 + g t+1 = f (k t+1 ). (11) Notice that fixing g t+1 leaves open the choice of how to allocate output net of government spending to either consumption c t+1 or new capital k t+2, which in turn depends on taxes at t+2. Moreover, letting the government at t commit for two periods, that is, choose g t+2 (and τ k t+2), is also not sufficient to resolve the time inconsistency, because consumption at t + 2 is jointly determined with capital k t+3, which in turn depends on fiscal policy at t + 3. Using this logic, it is clear that no finite amount of commitment can sustain the FC equilibrium. Nonetheless, in Section 6 we show that even a small degree of commitment can be quite powerful in sustaining outcomes that are preferable to the NC equilibrium in terms of welfare. The reason for this is that, different from what would happen in absence of any commitment, the government with LTC internalizes the distortionary effect of taxes in the near future on current investment decisions. 3 Equilibrium with LTC In this section we describe a generic model economy, and define two notions of optimal policy: Full-Commitment Ramsey equilibrium and Limited-Time Commitment equilibrium. We argue that LTC nests both the NC equilibrium, as a special case without any degree of commitment, and the FC equilibrium, in the limit with infinite commitment, under the right initial conditions. We then provide conditions under which the LTC game can support the FC solution with a finite amount of commitment. 3.1 Environment and competitive equilibrium Time is discrete and indexed by t = 0, 1, 2,... The economy is populated by a continuum of households, a continuum of firms, and a government or a sequence of governments. A vector of exogenous variables z t Z evolves stochastically with the Markov property that z t is sufficient information to calculate the probability distribution over z t+1. We restrict ourselves to settings where the shocks are drawn from either finite or countably infinite distributions. 12 That is, the set of possible values for z t can be expressed as a, possibly infinite, list Z = { z 1, z 2,...}. Denote by P z the associated Markov transition matrix. We call b t B the endogenous state variables (b 0 being an initial condition), c t C the remaining variables constituting allocations (for instance consumption and hours 12 This is to avoid technical discussions which we do not think add to the substance of the paper. We conjecture that all the results should go through, suitably modified, in a model with uncountably infinite distributions. 10

11 worked), p t P R Np the prices and τ t T the policy instruments chosen by the government(s). Households preferences are represented by the following utility function: E 0 t=0 β t r(c t, b t, z t, τ t ) (12) where r : C B Z T R is the instantaneous return function, and β 0 is the discount factor. E t is the expectations operator conditional on information up to time t. Let z t = (z 0,..., z t ) denote the history of shocks up to time t. Any sequence of government policies and equilibrium prices can be written as functions of the histories of shocks: {τ t (z t )} t=0 and {p t(z t )} t=0. Households and firms take these functions as given in their optimisations. Following the general formulation in Marcet and Marimon (2017), we can summarise these equilibrium conditions with three sequences of constraints: a transition equation for the endogenous states, a set of constraints involving only contemporaneous allocations and a set of constraints involving future variables. For any time t = 0, 1,... these constraints are b t+1 = l(b t, z t, c t, p t, τ t ) (13) N E t n=0 k(b t, z t, c t, p t, τ t ) 0 (14) h n (b t+n, z t+n, c t+n, p t+n, τ t+n ) = 0. (15) N 1 is a parameter governing how many periods ahead future variables enter into the time-t constraints via (15). 13 As is well known in the literature, the presence of these future variables in the constraint set defining competitive equilibria is the reason for the time inconsistency of FC policies. We wrote (15) such that it potentially contains all future allocations, prices, policies and so on up to N periods ahead. However, in practice, only some subset of these variables will actually appear in (15) in a given model. Thus, in Definition 1 we explicitly label the future variables which enter into constraint (15) as problematic : Definition 1. Let i = 1,..., N c index the individual elements of c t, such that c t (c 1 t,..., c Nc t ). We call any element i problematic if any c i t+s (with s > 0) appears in constraint (15). The same definition applies to elements of p t+s and τ t+s, and to b t+s (with s > 1). Intuitively, any variable (consumption, capital tax,...) which has a future value appearing in a forward-looking constraint is labeled as problematic. For the endogenous state b t, it is allowed for an element of b t+1 to appear in the constraints without it being labelled problematic. For example, in the Lucas-Stokey example of Section 2.1 we have c t = (c t, l t ). Consumption, c t, is problematic because it appears as c t+1 in the 13 We write (15) as a sum of separate h n functions, but all results go through for h functions which are not time separable. 11

12 Euler equation, while labour, l t, is not. 14 Since we assume the functions h n describing the forward-looking constraints are not time varying, the definition of a variable as problematic does not depend on time: it is consumption which is problematic, and not consumption at time t. Importantly, it is often possible to rewrite the competitive equilibrium conditions of a model in different, equivalent ways, featuring different versions of (15) with different values of N. 15 This does not cause any fundamental issues for our analysis. However, if one wants to identify the minimum amount of commitment required for LTC to exactly support FC in a given model, one must first identify the representation of the model with the smallest N. We define competitive equilibrium in the standard way: Definition 2. Given an initial condition (b 0, z 0 ) B Z and a policy sequence {τ t (z t )} t=0, a competitive equilibrium is a sequence of functions {c t (z t ), p t (z t ), b t (z t )} t=0 which satisfy (13), (14) and (15) for all t = 0, 1,... and histories z t Z t. We summarise the variables of the economy in a vector y t (z t ) (c t (z t ), p t (z t ), b t (z t ), τ t (z t )), and use the notation y {y t (z t )} t=0 to denote plans. Let Π (b 0, z 0 ) denote the set of plans which are competitive equilibria. That is, y Π (b 0, z 0 ) if and only if it is a feasible competitive equilibrium from initial state (b 0, z 0 ). Define the truncation of any plan from time t as y t {y s (z s )} s=t. Once time t is reached, a plan is feasible if and only if it satisfies the competitive equilibrium constraints at times t, t+1,... Importantly, constraints dated t 1 and before can be ignored, because these are now in the past. Due to the recursive nature of the definition of competitive equilibrium, it must be that any truncated plan y t is a valid competitive equilibrium from t onwards if and only if y t Π (b t, z t ). Wherever possible, we will suppress the functional notation (c t (z t ) and similar) unless it will lead to confusion. We restrict our analysis to the subset of initial conditions for which a competitive equilibrium exists for at least one policy sequence, and denote this set B B Z. We maintain the following regularity assumption throughout the text: Assumption 1. The environment is such that r is bounded over all competitive equilibrium paths. The discount rate is strictly less than one: 0 β < 1 This assumption serves two purposes. Firstly, it ensures that discounted utility is well defined in the limit of an infinite horizon, in the sense that the limit is sure to converge. More importantly, it rules out potential extraneous equilibria of the LTC game achieving negative infinite utility. 16 The assumption does not rule out unbounded utility functions, 14 Labour is not problematic because we substituted out l t+1 from the Euler equation using l t+1 = c t+1 +g. If we had not done so it would also be considered problematic. Such cases where it is ambiguous whether a variable is problematic or not are not a problem for our analysis, as long as one is careful to include all constraints in the definition of competitive equilibrium. 15 For example, the basic Lucas and Stokey (1983) model s implementability condition can be written as either u c,tb t = u c,tc t + u l,t l t + βu c,t+1 b t+1 for t = 0, 1,... or (by iterating forward on this equation) as u c,tb t = [ ] s=t βs t u c,sc s + u l,s l s, also for t = 0, 1,... The first gives N = 1 and the second gives N =. 16 If the government today thinks that future governments will play a policy leading to negative infinite utility, whatever policy the current government follows, its value is negative infinity. Hence it is weakly optimal for the government to pursue the same policy, leading to this policy being a Markov Perfect equilibrium. 12

13 such as the commonly used Constant Relative Risk Aversion (CRRA) form, as long as the environment is such that the possible values of utility in equilibrium are themselves bounded. 3.2 Full-Commitment Ramsey equilibrium In a FC equilibrium, a single benevolent infinitely-lived government endowed with the ability to credibly commit into the infinite future announces a contingent plan at t = 0 and then implements it. Denote a policy plan by τ {τ t (z t )} t=0. In order for the government to be able to pin down a unique competitive equilibrium, the following assumption on the mapping from policy plans to allocations is required. Assumption 2. Given an initial condition (b 0, z 0 ) B and a policy plan τ, there exists a unique competitive equilibrium, y. That is, there is a unique sequence of functions {c t (z t ), p t (z t ), b t (z t )} t=0 which satisfy (13), (14) and (15) for all t = 0, 1,... and histories z t G t. This assumption allows us to map an infinite sequence of policies to a single competitive equilibrium, ensuring that the government knows exactly which equilibrium is selected for a given policy choice. 17 Under this assumption we can equivalently define the government s problem as choosing a conditional path for policies, τ, or simply choosing the associated competitive equilibrium, y. This allows us to state the FC government s problem, for any (b 0, z 0 ) B, as V F C (b 0, z 0 ) = max E 0 {τ t,c t,b t+1,p t} t=0 t=0 β t r(c t, b t, z t, τ t ) (FC) subject to (13), (14) and (15), and initial conditions (b 0, z 0 ). This is solved by a policy sequence τ F C (b 0, z 0 ) { τt F C (z t ) }. For expositional simplicity, we maintain the t=0 assumption that the optimal policy is unique for any initial conditions, but all of the results of the paper go through if there are multiple optimal policies which all achieve the same value. Definition 3. Fix an initial condition (b 0, z 0 ) B. Let τ F C (b 0, z 0 ) be the policy sequence that solves (FC). A Full-Commitment (FC) Ramsey equilibrium is the competitive equilibrium, y F C (b 0, z 0 ), associated with τ F C (b 0, z 0 ). 3.3 Limited-Time Commitment equilibrium The Limited-Time Commitment setup can be described as a game, where successive one-period-lived governments indexed by t choose only a finite sequence of policy instruments, taking as given the strategies of the following governments. Each government is benevolent and maximises (12) subject to the competitive equilibrium conditions (13), 17 Bassetto (2005) argues that a full specification of out-of-equilibrium government strategies is necessary to guarantee that the desired equilibrium is obtained. In this paper we implicitly assume that there is a more general formulation of the government strategy, involving binding promises about outof-equilibrium paths, that guarantees uniqueness of the desired Ramsey equilibrium. 13

14 (14) and (15). In other words, as is standard in the literature, governments act as Stackelberg leaders with respect to the private sector, internalizing the effects of policy choices on competitive equilibrium outcomes. In our exposition, we follow the literature (for instance Klein et al., 2008) in restricting attention to symmetric Markov perfect equilibria, where each government chooses a common best-response function mapping a small set of natural state variables into the chosen sequence of policy instruments Contingent LTC In a stochastic environment, we must take a stand about whether the government is able to make plans contingent on future shocks or not. A state-contingent version of LTC is consistent with the view that governments are able to commit to near-term fiscal plans, but harder to reconcile with the notion of implementation lags, which would make state-contingency almost impossible. On the other hand, non-contingent LTC plans are consistent both with our motivation based on fiscal announcements and with the presence of implementation lags in fiscal policy. In this paper, we explore both alternatives, both in our theoretical analysis and by comparing them numerically in Section 6. We start by specifying the game where governments can make fully state-contingent plans, which we denote contingent LTC. Let L = 0, 1, 2,.. index the duration of commitment. The key assumption of the LTC game is that the government at time t is not able to choose policies from time t to time t + L 1, and is only able to choose policy at time t + L. This choice is made given a state variable s t S, which will feature past policies as states and which we hence describe later. In the contingent LTC game, the government is able to make fully state-contingent plans. Thus, the government at time t makes a plan for τ t+l contingent on the history of shocks between t + 1 and t + L. Denoting this partial history of shocks by zt+1 t+l (z t+1,..., z t+l ), the government chooses a policy function τ : S Z L T such that τ t+l = τ(s t, zt+1 t+l ). In the case of L = 0, which corresponds to the No Commitment solution, the government simply chooses a value for current policy: τ t = τ(s t ). Define t+l as a vector listing all the values of τ t+l (zt+1 τ t+l ) across the possible shock realisations between t+1 and t+l. This is the vector of contingent values for τ t+l which the time t government must choose. Similarly, define t ) as the vector-valued policy function summarising τ(s τ(s t, zt+1 t+l ). Thus, we equivalently can write the policy function as τ t+l = τ(s t, zt+1 t+l ) or = t ). τt+l Given this structure for choices, τ(s the government s state variables need to be augmented with the pre-committed state-contingent paths for policy between t and t+l 1, chosen by the governments from time t L to t 1. These are summarised in the state variable τt L. Intuitively, this state variable should contain the pre-committed value of τ t and the pre-committed contingent values of τ t+1 to τ t+l 1 depending on the realised shocks z t+1 to z t+l 1. That is, τt L is given by: { τt L (τ t, τ t+1 (z t+1 ), τ t+2 (zt+1 t+2 ),..., τ t+l 1(zt+1 t+l 1 )) L > 0 (16) L = 0 for some value τ t and functions (τ t+1,..., τ t+l 1 ) which the time t government takes as given, but have intuitively been chosen by previous governments. When L = 0 there are no promises to take as states, and τt L is given by the empty set. 14

15 The state vector is then s t (b t, z t, τ L t ). Given a government s policy τ, we are able to calculate how the state is updated between t and t + 1 depending on the realised shock, z t+1, by taking (16) one period forward. In the simplest case of Limited-Time Commitment, with L = 1, each government takes the current policy, τ t, as given and makes a state-contingent plan for next period s policy, τ t+1 = τ(s t, z t+1 ). Next period, the shock z t+1 is revealed and the relevant policy is implemented, leaving s t+1 = (b t+1, z t+1, τ t+1 ) as the state variable for the t + 1 government. More commitment, represented by a larger L, involves taking more periods of policy as given, and hence choosing a new policy farther in the future Definition of recursive LTC game The following section defines the Markov perfect equilibrium of the contingent LTC game. In the section, and throughout the paper, we refrain from using recursive notation and continue to index variables by time in recursive formulations. This is to improve clarity, given the importance of timing assumptions in the LTC game. Consider the government in power in period t and suppose all governments from time t + 1 onwards are expected to play a common policy function τ t+j+l = τ(s t+j, z t+j+l t+j+1 ) for j = 1, 2,... (alternatively: = t+j )). The following defines a function φ τt+j+l which is used to recursively compute competitive τ(s equilibrium. Definition 4. Given a policy function τ played by all future governments, a state s t = (b t, z t, τ L t ), and a current policy choice τ t+l, the competitive equilibrium system given by (13), (14) and (15) for s = t, t + 1,... defines a unique time-invariant solution for the vector (b t+1, c t, p t ) given by the function (b t+1, c t, p t ) = φ (s t, τ t+l ; τ). 18 We further define C(s t ; τ) as the set of values for t+l which are consistent with some competitive equilibrium. That is, if and only if τ t+l C(s t ; τ) then a competitive equilibrium exists and implies the values for (b τ t+1, c t, p t ) given in φ (s t, τ t+l ; τ). As indicated in the notation below, we restrict governments to choose policies for which competitive equilibria exist. Following Klein et al. (2008), any symmetric Markov-perfect equilibrium can be written as functions v and such that, for all s t S, the policy function t+l = t ) solves τ τ τ(s max r(c t, b t, z t, τ t ) + βe t v(s t+1 ) (LTC1) t; τ) τt+l C(s subject to (b t+1, c t, p t ) = φ (s t, τ t+l ; τ) and the value function v(s t ) is computed from v(s t ) = r (c t, b t, z t, τ t ) + βe t v(s t+1 ) (LTC2) where (b t+1, c t, p t ) = φ (s t, τ; τ). Note that the state tomorrow, s t+1 = (b t+1, z t+1, τ L t+1), can be computed given the transition for b t+1, and using (16) taken one period forward to construct τ L t+1 using the policy choice τ t+l and the realised value of z t Without further assumptions on the environment, there is no guarantee that it is possible to define a such a function φ uniquely. That is, even though Assumption 2 pins down a unique competitive equilibrium for an infinite sequence of government policies, the choice of a single policy τ t+l combined with a future policy function τ might not. This could make it impossible for a government to know which equilibrium its policy choice pins down. This issue arises in both the LTC and No Commitment games, and we thus maintain as an assumption that the environment and permissible policy functions uniquely pin down equilibrium in this sequence game. 15

16 3.3.3 Non-Contingent LTC The non-contingent LTC game is identical to the contingent formulation above, with the exception that the government at time t can only choose a non-contingent value for the future policy τ t+l, giving policy function τ t+l = τ(s t ). The government s state is still augmented with the previously chosen policies, so that s t (b t, z t, τt L ). Since these policies are non-contingent, τt L is now simply a list of policies to be implemented between t and t + L Paths generated by LTC game Given an equilibrium policy function of either the contingent or non-contingent LTC game and an initial condition s τ 0, we can iterate the policy function forwards to solve for an implied contingent policy plan τ (s 0 ) = {τ t (z t )} t=0. Importantly, when L > 0 we must specify as part of our initial conditions the pre-committed policies from the point of time 0: τ0 L. Thus, we cannot solve for the paths generated by the LTC game without specifying some rule for these initial conditions. Definition 5. Given initial conditions s 0 = ( ) b 0, z 0, τ0 L, a symmetric Markov perfect Limited-Time-Commitment (LTC) equilibrium is the competitive equilibrium, y (s 0 ), associated with the equilibrium policy plan τ (s 0 ) Consider either non-contingent LTC in a deterministic environment, or contingent LTC in a stochastic environment. Take as initial conditions the first L periods of the optimal Full Commitment policy. That is, take τ0 L = (τ0 F C, τ1 F C (z 1 ),..., τl 1 F C (zl 1 )), where τt F C (z t ) is the time-t element of τ F C (b 0, z 0 ). Then, as L the equilibrium trivially coincides with the FC equilibrium, as in the limit the entire path of taxes coincides with the initial conditions. In this case, LTC nests both NC (L = 0) and FC (L ). 4 Equivalence result In this section we provide a sufficient condition under which the unique equilibrium of the LTC game is the allocation generated under Full Commitment. We prove this result for the case of contingent LTC in a stochastic environment. The crux of the proof relies on assuming that a finite sequence of government policies can uniquely pin down certain allocations through competitive equilibrium restrictions. Hence we are careful to define what it means for a variable to be uniquely determined: Definition 6. For any t, L > 0, and t t t+l, we say that the natural states (b t, z t ) and a partial policy sequence {τ t+s (zt t+s )} L s=0 uniquely determine a variable x t from time t if all implied competitive equilibria from time t onwards feature the same value of x t (zt t ) regardless of the future policy choices {τ t+s (zt t+s )} s=l+1. This allows us to state the main assumption: Assumption 3. There exists an L, with N L <, such that the following holds for all t = 0, 1,... and histories z t. 16