CHAPTER 5. DYNAMIC CONTRACTING for t=l, .,T-1, where w is a known constant. The first-order condition for efficient risk sharing takes the form. Ae-Az (u)= Be-By(u) which implies that a(a) is an affine function of y(w). Then the fact that (w)+y(w)= w implies that both a(w) and y(w) are affine functions of w Suppose that y(w)= Aw +u, where< A< 1. Then in order to implement the first-best, risk-sharing scheme, the debt contract adopted at date T-l when wT-1 is observed must satisfy minWT-1+a, dT-1l-mT-1= A(wT-1+a)+u minwT-1+b, dT-1)-mr and since 0 <A< l this req +b-m7-1=X(r-1+b)+ (1-入)(r-1+b)+ So there are unique values of debt and transfers at date T-l that implement the first best. Obviously, the adding-up condition implies that the same values of dr-1 and mT-1 will give the firm (o) At dates t <T-1, the problem is more complicated, because we have to choose dt and mt to give the venture capitalist the equilibrium status quo utility level rather than to give it a particular income level. As a result the calculations are more complicated. However, the critical problem have seen is to ensure that the spanning conditions are satisfied. Recall that (d,t, mt)can be chosen so that Elexpf-B(minwr, dt-mthwt=el-exp-B(Ar +phwt which is equivalent to El-expf-Bminfwr, dt ))wr=e-B(+m )EI-exp -B(Awr)Hwt8 CHAPTER 5. DYNAMIC CONTRACTING for t = 1, ..., T − 1, where w1 is a known constant. The first-order condition for efficient risk sharing takes the form: Ae−Ax(w) = Be−By(w) which implies that x(w) is an affine function of y(w). Then the fact that x(w) + y(w) ≡ w implies that both x(w) and y(w) are affine functions of w. Suppose that y(w) = λw + µ, where 0 <λ< 1. Then in order to implement the first-best, risk-sharing scheme, the debt contract adopted at date T − 1 when wT −1 is observed must satisfy: min{wT −1 + a, dT −1} − mT −1 = λ(wT −1 + a) + µ min{wT −1 + b, dT −1} − mT −1 = λ(wT −1 + b) + µ and since 0 <λ< 1 this requires dT −1 − mT −1 = λ(wT −1 + a) + µ wT −1 + b − mT −1 = λ(wT −1 + b) + µ or dT −1 = wT−1 + λa + (1 − λ)b mT −1 = (1 − λ)(wT −1 + b) + µ. So there are unique values of debt and transfers at date T −1 that implement the first best. Obviously, the adding-up condition implies that the same values of dT −1 and mT −1 will give the firm x(w). At dates t<T − 1, the problem is more complicated, because we have to choose dt and mt to give the venture capitalist the equilibrium status quo utility level rather than to give it a particular income level. As a result, the calculations are more complicated. However, the critical problem as we have seen is to ensure that the spanning conditions are satisfied. Recall that (dt, mt) can be chosen so that E[− exp{−B(min{wT , dt} − mt)}|wt] = E[− exp{−B(λwT + µ)}|wt], which is equivalent to E[− exp{−B min{wT , dt}}|wt] = e−B(µ+mt) E[− exp{−B(λwT )}|wt]