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Time-Inconsistent Optimal Quantity of Debt

A key feature of the infinite-horizon heterogeneous-agents incomplete-markets (Inf-HAIM) framework is that the equilibrium interest rate of public debt lies below the time discount rate (regardless of capital). This happens because of a positive liquidity premium on asset returns due to imperfect risk sharing. This fundamental property of standard Inf-HAIM models, however, implies that the Ramsey planner's fiscal policy may be time-inconsistent---because the planner has a dominate incentive to issue plenty of debt such that all households are fully self-insured against idiosyncratic risk whenever the interest rate of government borrowing is lower than the household time discount rate. But such a full self-insurance allocation may be infeasible---because to achieve it the optimal quantity of debt may approach infinity or the optimal labor tax rate may approach 100%. This is puzzling from an intuitive perspective because near the point of full self-insurance the marginal gains of increasing debt should be less than the marginal costs of financing the debt under distortionary taxes. We show that this puzzling behavior originates from the assumption that the planner must commit to future plans at time zero. Under such a full commitment, the Ramsey planner opts to exploit the low interest cost of borrowing to front load consumption by sacrificing future consumption in the long run---because future utilities are heavily discounted compared to the inverse of the interest rate on government bonds. We demonstrate our points analytically using a tractable Inf-HAIM model featuring non-linear preferences and a well-defined distribution of household wealth.

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