  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.
