Equilibrium Defaultable Corporate Debt and Investment

TL;DR

This paper addresses the challenge of proving existence and uniqueness in dynamic capital structure models with defaultable debt by introducing a dual approach and change of measure, enabling analysis of equilibrium strategies.

Contribution

It provides a novel method using duality and measure change to establish existence and uniqueness in models with investor break-even conditions.

Findings

Existence and uniqueness can be proved using duality and measure change.

Firm decisions reflect state-dependent capital and debt targets.

The approach is applicable to other dynamic firm models with similar conditions.

Abstract

In dynamic capital structure models with an investor break-even condition, the firm's Bellman equation may not generate a contraction mapping, so the standard existence and uniqueness conditions do not apply. First, we provide an example showing the problem in a classical trade-off model. The firm can issue one-period defaultable debt, invest in capital and pay a dividend. If the firm cannot meet the required debt payment, it is liquidated. Second, we show how to use a dual to the original problem and a change of measure, such that existence and uniqueness can be proved. In the unique Markov-perfect equilibrium, firm decisions reflect state-dependent capital and debt targets. Our approach may be useful for other dynamic firm models that have an investor break-even condition.