debtequity

The increase in the bankruptcy level induces an additional marginal loss in output KL ′ (â)dâ.
Replacing dâ by its value in (26) shows that the last term in (25) is the additional loss of output
due to the increase in the bankruptcy level induced by the increase in debt.
Although there is still bankruptcy in the second best equilibrium, Property (iii) shows that
the probability of bankruptcy is lower and the equity-debt ratio is higher that in the first-best
equilibrium. Since R B sb < RD∗ = E(ã), D sb = D(R B sb ) < D(RB∗ ) = D ∗ : the Central Bank lends less
to banks than in the first best. The constraint that the return on equity is at least E(ã) implies
that the supply of equity is at least that of the first best: e(Rsb B ) ≥ e(E(ã). It follows that the
debt-equity ratio 1−α
α
is lower that in the first best, or equivalently α sb > α ∗ : the share of equity
in the financing of investment is increased relative to the first best. We noted in Section 3 that the
return on equity can ∫ be decomposed into the product of the leverage multiplier 1 α
and the return
∞
on capital r(â) = (a − â)f(a)da, where the return on capital is a strictly decreasing function of
â
â. Since the leverage multiplier is lower in the second best than in the first-best while the return
on equity is as large, it must be that the return on capital is higher, which is possible only if the
bankruptcy level is lower than in the first-best: â sb < â ∗ . This in turn implies that the probability
of bankruptcy is lower in the second-best than in the first-best equilibrium. Finally, the return on
equity Φ(α, R D ) viewed as a function of the equity ratio and the rate of interest charged to banks is
decreasing in each variable. Since α sb > α ∗ , and Φ(α, R D ) ≥ E(ã) = Φ(α ∗ , R D∗ ) does not decrease
in the second best equilibrium, it must be that R D decreases to reestablish a sufficient return on
equity. Thus R D sb < RD∗ : banks are charged a lower rate for there loans in the second-best than in
the first-best equilibrium.
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