23. DEFAULT Then i(a ) pays one unit if w=19) and nothing otherwise, that is, it i an Arrow security for the state w=Q2. a portfolio consisting of one unit of i(a a-)and minus two units of i (a,a )will yield one unit in state w=Q-1 and nothing otherwise, that is, it is an Arrow security for the state w=Q2 -1. Continuing in this way we can generate Arrow securities for each state. This is a case where capital structure is irrelevant for the individual firm, but not for the equilibrium 2.3.2 Default with incomplete markets To define an equilibrium, we assume that consumers can hold the firms debt but cannot issue debt or sell short the firms equity.(This isn't necessary but simplifies the story Definition 7 An equilibrium with incomplete markets and default consists of an attainable allocation a=(a)ie×(a)∈ A and a price vector(q,)∈ R XR such that a, E A, maximizes the value of the firm V,=U;-qa=max u;(y;(a;)+aii(ai))-qa and, for every i, a: E Ai maximizes ui(ai) subject to the budget constraint a2q+B2U≤6V wnere r=e1+a2(ay)+1((ay)+a(a3) In this case, we have to deal with the valuation problem explicitly: be- cause markets are incomplete, individuals may disagree in their valuation of urity. Only those who value it most highly will hold a positive quantity of a security or equity in equilibrium 2.3.3 Related issues With complete markets, all shareholders agree that value maximization is the right objective function for the firm. With incomplete markets, this may not be the case. The firms choice of yi and a; has two effects, on Vi and on the risk sharing that can be achieved by2.3. DEFAULT 7 Then yˆ ³ α|Ω| j ´ pays one unit if ω = |Ω| and nothing otherwise, that is, it is an Arrow security for the state ω = |Ω|. A portfolio consisting of one unit of yˆ ³ α|Ω|−1 j ´ and minus two units of yˆ ³ α|Ω| j ´ will yield one unit in state ω = |Ω| − 1 and nothing otherwise, that is, it is an Arrow security for the state ω = |Ω| − 1. Continuing in this way we can generate Arrow securities for each state. This is a case where capital structure is irrelevant for the individual firm, but not for the equilibrium. 2.3.2 Default with incomplete markets To define an equilibrium, we assume that consumers can hold the firm’s debt but cannot issue debt or sell short the firm’s equity. (This isn’t necessary, but simplifies the story). Definition 7 An equilibrium with incomplete markets and default consists of an attainable allocation a = (ai)i∈I × (aj ) ∈ A and a price vector (q, v) ∈ RH × R such that aj ∈ Aj maximizes the value of the firm Vj = vj − qαj = max i {µi · (yj (αj ) + αjzˆ(αj ))} − qαj and, for every i, ai ∈ Ai maximizes ui(xi) subject to the budget constraint αiq + βiv ≤ θiV, where xi = ei + αizˆ(αj ) + βi (yj (αj ) + αjzˆ(αj )). In this case, we have to deal with the valuation problem explicitly: be￾cause markets are incomplete, individuals may disagree in their valuation of a security. Only those who value it most highly will hold a positive quantity of a security or equity in equilibrium. 2.3.3 Related issues With complete markets, all shareholders agree that value maximization is the right objective function for the firm. With incomplete markets, this may not be the case. The firm’s choice of yj and αj has two effects, on the value of the firm Vj and on the risk sharing that can be achieved by