$9\mathrm{.}14.$ The Modigliani$-$Miller theorem. $($Modigliani and Miller, $1958.)$ Consider the analysis of the effects of uncertainty about discount factors in Section $9\mathrm{.}7.$ Suppose, however, that the firm finances its investment using a mix of equity and risk$-$free debt. Specifically, consider the financing of the marginal unit of capital. The firm issues quantity $b$ of bonds; each bond pays $1$ unit of output with certainty at time $t+$ for all $0.$ Equity holders are the residual claimant; thus they receive $(K(t+))-b$ at $t+$ for all $0.$

$($a$)$ Let $P(t)$ denote the value of a unit of debt at $t,$ and $V(t)$ the value of the equity in the marginal unit of capital. Find expressions analogous to $(9\mathrm{.}35)$ for $P(t)$ and $V(t).$

$($b$)$ How, if at all, does the division of financing between bonds and equity affect the market value of the claims on the unit of capital, $P(t)b+V(t)?$ Explain intuitively.

$($c$)$ More generally, suppose the firm finances the investment by issuing $n$ financial instruments. Let $d_i(t+)$ denote the payoff to instrument $i$ at time $t+$; the payoffs satisfy $d_1(t+)+$cdots$+d_n(t+)=(K(t+)),$ but are otherwise unrestricted. How, if at all, does the total value of the $n$ assets depend on how the total payoff is divided among the assets?

$($d$)$ Return to the case of debt and equity finance. Suppose, however, that the firm's profits are taxed at rate $,$ and that interest payments are taxdeductible. Thus the payoff to bond holders is the same as before, but the payoff to equity holders at time $t+$ is $(1-)[(K(t+))-b].$ Does the result in part $($b$)$ still hold? Explain.

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