14.123 Microeconomic Theory III Problem Set 2 The due date for this assignmen t is Thursda y February 25 (You can return it on Tuesda y March 1 without penalty.) 1. There are two ur ns, A and B, each consisting of 100 balls, some are black and some are red. In urn A there are 3 0 red b alls, but the number of r ed balls in urn B is not known. W e draw a ball from urn A with color α and a ball from urn B with color β. Consider the following acts: ½ ½ 100 if α = red 0 if α = red fA,r =0 if α = blac k fA,b = 100 if α = black ½ ½ 110 if β = red 0 if β = red fB,r =0 if β = black fB,b = 110 if β = blac k Let c be the choice function induced by º.Find the sets c ({fA,r,fA,b,fB,r,fB,b}) that are consisten t with 110 Â 100 Â 0 and Sa vage’s postulates. 2. Exercise 6.C.19 in Mass-C o lell, Whinston, and Green (Assume that the asset returns are independen t.) 3. Exercise 6.D.3 in Mass-C olell, Wh inston , and Green 4. Consider a mon opolist w ho faces a stochastic deman d. If he produces q units, he incurs ¯a zero marginal cost and sells the good at price P (θ, q) where θ ∈ £ θ, θ¤ is an unknown demand shock where P and C twice differentiable. Assum e that the p rofit function is strictly concave in q for each given θ,and P (θ, q)+ qPq (θ, q) is increasing in θ, where Pq is the d erivativ e of P with respect to q. The monopolist is expected profit maximizer. (a) Show that there exists a unique optimal production level q∗. (b) Show that if the distribution of θ changes from G to F where F first-order sto-c hastically dominates G, then the optimal productio n level q∗ weakly increases. (c) Tak e P (θ, q)= φ (θ) − γ (q). Suppose that there are t wo identical monopolists as abo ve in two independent but identical markets. Find conditions under which the monopolists have a strict incen tive to m erg e and share the profitfrom each market equally. 1MIT OpenCourseWarehttp://ocw.mit.edu 14.123 Microeconomic Theory III Spring 2010 For information about citing these materials or our Terms of Use, visit: