6.1. ADVERSE SELECTION fraction held by the entrepreneur is a. The market value of the firm is defined p(a)=∑r(a,t) Pooling equilibrium Suppose that all types hold a fraction ap of the firm. The equilibrium price p(an)=∑以t Each type t chooses a to maximize u(a, p(a), t), which requires that +(1-a)p(a) t)U(1-a)p(a)≤ tU(apV+(1-app(ap))+(1-tU((1-app(ap)) for all a and t. The easiest way to support this equilibrium is to assume that for every a≠a H(,)=0t≠tm where tmin is the smallest value of t. Then the equilibrium condition reduces tU(av+(1-a)tminV)+(1-t)U((1-a)tminV)< for every a and t, wheret=>v(t)t Separating equilibrium Let a be an increasing function of t and define the beliefs u by putting if a=a(t) (a,t) if t=tmin, a fa-(T) 0 otherwise Define p( in the usual way. Then the strategy a is optimal if tu(av+(1-ap(a))+(1-tU((I-apla< tU(a(t)y+(1-a(t)t)+(1-t)U((1-a(t)tV) or every a and t6.1. ADVERSE SELECTION 3 fraction held by the entrepreneur is a. The market value of the firm is defined by p(a) = X t µ(a, t)tV. Pooling equilibrium Suppose that all types hold a fraction ap of the firm. The equilibrium price is p(ap) = X t ν(t)tV. Each type t chooses a to maximize u(a, p(a), t), which requires that tU(aV + (1 − a)p(a)) + (1 − t)U((1 − a)p(a)) ≤ tU(apV + (1 − ap)p(ap)) + (1 − t)U((1 − ap)p(ap)) for all a and t. The easiest way to support this equilibrium is to assume that for every a 6= ap µ(a, t) = ½ 0 t 6= tmin 1 t = tmin where tmin is the smallest value of t. Then the equilibrium condition reduces to tU(aV + (1 − a)tminV ) + (1 − t)U((1 − a)tminV ) ≤ tU(apV + (1 − ap)tV¯ ) + (1 − t)U((1 − ap)tV¯ ) for every a and t, where t ¯= P t ν(t)t. Separating equilibrium Let α be an increasing function of t and define the beliefs µ by putting µ(a, t) =    1 if a = α(t) 1 if t = tmin,a /∈ α−1(T) 0 otherwise. Define p(·) in the usual way. Then the strategy α is optimal if tU(aV + (1 − a)p(a)) + (1 − t)U((1 − a)p(a)) ≤ tU(α(t)V + (1 − α(t))tV ) + (1 − t)U((1 − α(t))tV ) for every a and t