Optimal Dynamic Fiscal Policy with Endogenous Debt Limits

Since the financial crisis of 2008 and increased government debt levels worldwide, fiscal austerity has been a focal point in public debates. Central to these debates is the natural debt limit, i.e. the level of public debt that's sustainable in the long run, and the design of fiscal policy that is consistent with that limit. In much of the earlier work on dynamic fiscal policy, governments are not allowed to lend, and the upper limit on debt is determined in an ad-hoc manner Aiyagari et. al (2002)'s (AMSS) seminal paper on fiscal policy in incomplete markets relaxed the lending assumption and revisited earlier work of Barro (1979) and Lucas and Stokey (1983) to study the implications on tax policy. Their results implied that taxes should roughly follow a random walk. They also presented examples where the long-run tax rate is zero, and any spending is financed out of its asset income (i.e., government holds debt of the people). However, their approach had some weaknesses. First, it imposed an artificial limit on government debt and therefore did not address the question of a natural debt limit. Second, it assumed, as much of the literature prior to it did, government spending to be exogenous. We relax the assumptions on debt and spending, and we use computational methods that do not rely only on local optimality. While we focus on the models examined in AMSS, we present a framework that can address fiscal policy issues in a self-consistent manner. In particular, we derive the endogenous limits on debt and allow for endogenous government spending. Our approach involves recasting the policy problem as an infinite horizon dynamic programming problem. The government's value function may not be concave and it can also very high curvature, particularly as debt approaches its endogenous limit. In dynamic taxation problems, the government's problem is a mathematical program with complementary constraints (MPCC). We explicitly use the MPCC formulation, which is essential in order to do the necessary global optimization analysis of the government's problem. Our MPCC approach uses the computational algorithms that were developed only in the past twenty years, and allows us to solve the problem reliably and accurately. Using our combination of computational tools and more general economic assumptions, we re-address questions regarding optimal taxation and debt management in a more realistic setup. These tools allow us to determine debt limits implied by assumptions on the primitives of the economic environment and to assess how the level of debt affects both tax policy and general economic performance, and the time series properties of tax rates and debt levels. Our results show that under the more general framework of endogenous government debt limits and spending has substantially different implications than earlier analyses have suggested. First, the behavior of optimal policy is, over long horizons (e.g., 1000 years), is much more complex than simpler models imply. In particular, the long-run distribution of debt is multimodal, and the long-run level of debt is history-dependent. If initial debt is low enough and government spending is not hit with large shocks, then the government will accumulate a "war chest" which allows long-run tax rates to be zero. However, if, in the same model, initial debt is high and/or the government gets hit with a long series of bad spending shocks, then debt will rise to a high level and will not fall even if the government is not hit with bad spending shocks. In the second case, governments with large debt levels will avoid default by reducing spending and use taxes to finance a persistently high debt. We examine the case of fixed government spending and find that the results are dramatically affected. In particular, we illustrate a case where if spending shocks are of moderate size (less than US historical experience) no positive level of debt is feasible. That is, if a government begins with positive debt then there is a sequence of spending shocks such that there is no feasible tax and borrowing policy to finance those expenditures. In such cases, exogenous spending assumptions imply that governments must have their endowed war chests in the beginning and cannot with probability one build up its war chest. These examples illustrate clearly that any analysis of fiscal policy that wants examine historical fiscal policy must consider making spending flexible. The application of our methodology is not limited to optimal tax problems. Optimal macroeconomic policy problems, as well as social insurance design typically involve solving high-dimensional dynamic programming problems. Solving such problems is a complicated, but very important task, as the policy recommendations depend crucially on the accuracy of the numerical results. In much of the optimal macroeconomic policy and social insurance literature, accuracy of the numerical solutions is unclear. Additionally, most solution approaches ignore feasibility issues and impose ad-hoc limits on state variables such as government debt. An accurate approach to solving dynamic policy models requires the ability to handle the high-dimensional nature of the problems as well as the unknown, feasible state space. The methodology offered in this paper can be used for computing high-dimensional dynamic policy problems with unknown state spaces.