CHAPTER 2. THE MODIGLIANI-MILLER THEOREM and,for every i, a; E A: maximizes i(zi)subject to the budget constraint h ∑ P Cith+ 9+ aih2 Theorem4Let(a,p,q)∈X×RH∈AxX× r be a walrasian equilibriun and let(a)ieJ be an arbitrary allocation of portfolios for firms. Then there exists a Walrasian equilibrium(a, p, q) such that a=(,a3),v Note also that, by the previous argument, V;=V for every j There are two aspects to the Modigliani-Miller theorem: one says that the firms choice of financial strategy a; has no effect on the value of the firm(or shareholder's welfare); the other says that the choice of a, has no essential impact on equilibrium. Here we are making the second(stronger 2.2 Equilibrium with incomplete markets To simplify, and avoid some thorny issues about the objective function of the firm. we assume that production sets are singletons Y={},vj∈ We start by assurning that firms do not trade in securities a,=0. There are ies, so that consumption bundles can only be achieved by trading securities ∑+∑a+∑4 CHAPTER 2. THE MODIGLIANI-MILLER THEOREM and, for every i, ai ∈ Ai maximizes ui(xi) subject to the budget constraint p · xi +X h αihqh +X j βijvj ≤ p · ei +X j θijVj +p · ÃX h αizh +X j βij à yj +X h αjhzh !!. Theorem 4 Let (a, p, q) ∈ X×RH ∈ A×X×RH be a Walrasian equilibrium and let (α0 j )j∈J be an arbitrary allocation of portfolios for firms. Then there exists a Walrasian equilibrium (a0 , p, q) such that a0 = (a0 i)i∈I × (a0 j )j∈J a0 i = (xi, α0 i, β0 i), ∀i a0 j = (yj , α0 j ), ∀j. Note also that, by the previous argument, Vj = V 0 j for every j. There are two aspects to the Modigliani-Miller theorem: one says that the firm’s choice of financial strategy αj has no effect on the value of the firm (or shareholder’s welfare); the other says that the choice of αj has no essential impact on equilibrium. Here we are making the second (stronger) claim. 2.2 Equilibrium with incomplete markets To simplify, and avoid some thorny issues about the objective function of the firm, we assume that production sets are singletons: Yj = {y¯j}, ∀j ∈ J. We start by assuming that firms do not trade in securities αj = 0. There are no Arrow securities, so that consumption bundles can only be achieved by trading securities. xi = ei +X j θijyj +X h αijzh +X j βijyj