2. 1. ARROW-DEBREU MODEL WITH ASSETS Equilibrium requires that P·ah,vh∈H. Otherwise firms could increase profits without bound. But under this condi- tion, any portfolio is optimal. Thus equilibrium with securities requires only that attainability be satisfied We can do the same thing with traded equity. If equity is fairly priced, there is no reason for anyone to trade it 2.1.4 Irrelevance of capital structure a1=(x2,a2,B1)∈A≡X1×RBxR a=(9,0)∈A≡Y×R (a)ier×(a3)jeJ Definition 1 An allocation a=(oilier X(ai)iej is attainable if j∈J i∈I Definition 2 An attainable allocation a=(ai)erx(ai)ieJ is weakly efficient if there does not exist an attainable allocation a=(aier x(a')ie sUC that ui(ai)<ui(ai for all i. An attainable allocation a=(aiier x(ailieJ is(strongly) efficient if there does not exist an attainable allocation a (a')iel x(a')ieJ such that ui(i)<ui(ai) for all i and ui(ai)<ui(ai) for Definition 3 A Walrasian equilibrium consists of an attainable allocation a=(ai)iel X(ai)ieJ and a price vector(p, )E XR such that, for every j, a; E A; maximizes the value of the firm2.1. ARROW-DEBREU MODEL WITH ASSETS 3 Equilibrium requires that qh = p · zh, ∀h ∈ H. Otherwise firms could increase profits without bound. But under this condi￾tion, any portfolio is optimal. Thus equilibrium with securities requires only that attainability be satisfied: X i αi +X j αj = 0. We can do the same thing with traded equity. If equity is fairly priced, there is no reason for anyone to trade it. 2.1.4 Irrelevance of capital structure ai = (xi, αi, βi) ∈ Ai ≡ Xi × RH × RJ aj = (yj , αj ) ∈ Aj ≡ Yj × RH a = (ai)i∈I × (aj )j∈J Definition 1 An allocation a = (ai)i∈I × (aj )j∈J is attainable if X i∈I xi = X j∈J yj X i∈I αi +X j∈J αj = 0 X i∈I αi = 1. Definition 2 An attainable allocation a = (ai)i∈I×(aj )j∈J is weakly efficient if there does not exist an attainable allocation a0 = (a0 i)i∈I × (a0 j )j∈J such that ui(xi) < ui(xi) for all i. An attainable allocation a = (ai)i∈I × (aj )j∈J is (strongly) efficient if there does not exist an attainable allocation a0 = (a0 i)i∈I × (a0 j )j∈J such that ui(xi) ≤ ui(xi) for all i and ui(xi) < ui(xi) for some i. Definition 3 A Walrasian equilibrium consists of an attainable allocation a = (ai)i∈I × (aj )j∈J and a price vector (p, q) ∈ X × RH such that, for every j, aj ∈ Aj maximizes the value of the firm Vj = vj −X h qhαjh = p · Ã yj +X h αjhzh ! −X h qhαjh