The Optimum Quantity of Capital and Debt

What are the optimal long-run levels of capital and government debt, and should capital be taxed at all, in a heterogeneous-agent, incomplete-markets economy of the kind studied by Aiyagari (1995)? Most of the prior literature answers a narrower question – which steady state maximizes welfare – but this paper instead solves the full dynamic Ramsey taxation problem, in which a planner commits at date zero to an entire path of linear labor and capital taxes and government debt to maximize the discounted present value of households’ lifetime utility, and derives three main theoretical results. First, exactly as under complete markets, the long-run pre-tax return to capital equals the rate of time preference – the capital stock satisfies the modified golden rule – even though, unlike the representative-agent case, genuine distributional concerns are present throughout. Second, and in sharp contrast to representative-agent Ramsey economies (where the steady state depends on the initial government debt level, since the planner smooths tax distortions relative to whatever fiscal burden it inherits), the long-run steady-state levels of capital, debt, and both tax rates in this incomplete-markets economy are independent of initial conditions – the same long-run policy is reached no matter where the economy starts. Third, building on this independence result, the authors develop a new Lagrangian computational method – solving for the known terminal steady state analytically first, then finding the transition path of taxes and debt that connects it to the calibrated initial economy via a system of first-order conditions – avoiding the essentially unverifiable global numerical search that would otherwise be required over hundreds or thousands of variables. Quantitatively, calibrating the model to U.S. income inequality with a unit Frisch elasticity of labor supply, the optimal long-run policy features a government debt level of about 1.1 times GDP, a capital income tax around 21 percent (positive, but low relative to most developed economies), and a labor income tax around 50 percent – a pattern of high debt, low capital taxation, and high labor taxation that the paper finds is robust across a wide range of alternative calibrations of labor supply and income-risk parameters. The paper also offers a reinterpretation of Aiyagari’s (1995) original finding of a positive long-run capital tax: rather than existing to correct households’ precautionary over-accumulation of capital, the tax is positive because the planner instead uses government debt to satisfy households’ demand for extra liquidity, which brings the capital stock itself back down to its efficient (modified-golden-rule) level; the capital tax’s remaining role is only to make households willing to hold exactly the optimal quantities of both capital and government debt simultaneously. Moving from a U.S.-calibrated initial steady state to the optimal transition path yields an average lifetime welfare gain of about 2.6 percent of consumption.

Yikai Wang | Macro Paper Warehouse

The Optimum Quantity of Capital and Debt