2.2. EQUILIBRIUM WITH INCOMPLETE MARKETS Since firms have no decision to make, equilibrium is achieved if consumers maximize their utility subject to the budget constraint max ∑By"+q·0≤∑6 and markets for shares and securities clear. ∑=(1, Now change a=0 to aj, change U to j=Uj+q a,, and change ai to di=ai->i Bi, ai. Checking the optimality of the consumers problem and the attainability conditions we see that the economy is still in equilibrium. Definition 5 An equilibrium with incomplete markets consists of an attain able allocation a=(a)ier×(a)ye∈ A and a price vector(q,t)∈R×R such that, for every j, a, E A, maximizes the value of the firm 1=0-∑=m{:(m+∑9)}∑ and, for every i, a; E Ai maximizes ui(i) subject to the budget constraint ∑+∑≤∑"V wnere =+∑+∑(+∑动 Theorem 6 Let(a,,vEAxRXR be an equilibrium with incomplete markets and let(a)jeJ be an arbitrary allocation of portfolios for firms. Then there exists an equilibrium with incomplete markets(a, 9, a) such that (aier x(a)ieJ2.2. EQUILIBRIUM WITH INCOMPLETE MARKETS 5 Since firms have no decision to make, equilibrium is achieved if consumers maximize their utility subject to the budget constraint: max ui(xi) s.t. P j βijvj + q · αi ≤ P j θijvj ; and markets for shares and securities clear: X i αi = 0 and X i βi = (1, ..., 1). Now change αj = 0 to αˆj , change vj to vˆj = vj + q · αj , and change αi to αˆi = αi − P j βijαˆj . Checking the optimality of the consumers problem and the attainability conditions we see that the economy is still in equilibrium. Definition 5 An equilibrium with incomplete markets consists of an attain￾able allocation a = (ai)i∈I ×(aj )j∈J ∈ A and a price vector (q, v) ∈ RH × RJ such that, for every j, aj ∈ Aj maximizes the value of the firm Vj = vj −X h qhαjh = max i ( µi · à yj +X h αjhzh !) −X h qhαjh and, for every i, ai ∈ Ai maximizes ui(xi) subject to the budget constraint X h αihqh +X j βijvj ≤ X j θijVj , where xi = ei +X h αihzh +X j βij à yj +X h αjhzh ! . Theorem 6 Let (a, q, v) ∈ A × RH × RJ be an equilibrium with incomplete markets and let (α0 j )j∈J be an arbitrary allocation of portfolios for firms. Then there exists an equilibrium with incomplete markets (a0 , q, v0 ) 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