Economics 302: Intermediate Microeconomics II Assignment 3 (Due in my office 3:00, Thursday March 10) 1. Consider two firms, 1 and 2, each producing an identical good simultaneously. This good has market demand given by the (inverse) demand function p = 10 - Y, where p is price, and Y = yi + 2 is market quantity. y, represents the amount produced by firm i. These firms have cost functions as follows: C, = cy, where c = c2 = 1. (60 pts) a) Solve algebraically for these firm's reaction functions, expressing each firm's optimal output level given some level of its competitor's output. b) Graph these reaction functions and show the equilibrium point. Include isoprofit contours through the equilibrium point for both firms. c) Solve algebraically for the equilibrium: Determine the equilibrium market price, as well as each firm's equilibrium quantity and profit. d) Is your answer to part c) the only equilibrium possible? Explain. 2. Take the same industry outlined in question 1 and imagine that firms choose prices rather than quantities. Consumers split themselves evenly across the firms if the firms set the same prices, otherwise all consumers shop at the lower-priced firm. Define a Nash equilibrium in prices pi and p2. Solve for the equilibrium and explain your work. (20 pts). Consider the following duopoly industry. Demand is given by p = a - 191', where p is price, and Y = y1 + y: is market quantity. y; represents the amount produced by rm i. The two firms have the same marginal cost equal to c. Assume a is large enough and there is no xed cost. (50 points) (1) In a Cournot model (two rms decide their output levels simultaneously and independently), solve each rm's Output quantity and prot, market price, and market quantity. (2) Now suppose two rms collude to decide their output levels together and split the market demand equally. Solve each rm's output quantity and prot, market price, and market quantity. (3) Now suppose two rms move sequentially and firm 1 is the leader (Stackelberg model). Solve each firm's output quantity and prot, market price, and market quantity. (4) Put your results 'om the previous three sub-questions in the following table. -_-__-\"\" Com ------ Collusion Stackelberg Model (5) Compare your results and explain. (Cournot vs. Collusion and Cournot vs. Stackelberg) 2. Sixty nine percent of adults think that musicians should be allowed to sing potentially offensive 20 songs. You randomly select twenty adults and ask them whether they think musicians should be allowed to sing potentially offensive songs. The random variable represents the number of adults who think musicians should be allowed to sing potentially offensive songs. Find (a) the mean, variance, standard deviation and interpret the results, (b) probability that exactly 15 people think musicians shold be allowed to sing potentially offensive songs, (c) probability that more than 10 people think musicians should be allowed to sing potentially offensive songs, B 1/ 4
