ECN121A (INDUSTRIAL ORGANIZATION) PROBLEM SET 1

ECN121A (INDUSTRIAL ORGANIZATION)PROBLEM SET 1DUE BEGINNING OF CLASS: JANUARY 16, 2015Question 1: [30 points] Consider the production of some product A. (Assume if q A = 0, thenC (q A ) = 0).(a) [10 points] Does the cost function C (q A ) = 1 + q A exhibit economies of scale? Explainwhy/why not.(b) [10 points] Does the cost function C (q A ) = q2 exhibit economies of scale? Explain why/whyAnot.(c) [10 points] Does the cost function C (q A ) = (1 + q A )0.5 + 100 exhibit economies of scale?Explain why/why not.Question 2: [35 points] A monopolist faces demand of q( p) = 230 p and has cost functionC (q) = 30q.(a) [5 points] What is the monopolists maximization problem?(b) [5 points] What is the rst order condition corresponding to this problem?(c) [5 points] Draw a diagram showing demand, and the relevant curves corresponding to themonopolists problem. Indicate the (price, quantity) chosen by the monopolist.(d) [5 points] What are the monopolists prots, consumer surplus, and deadweight loss ofmonopoly?(e) [5 points] How much would consumers be willing to pay to have the monopolist set priceequal to its marginal cost?(f) [5 points] If the monopolist set price equal to marginal cost, but had a membership feeequal to what you found in part e, how would total surplus change compared to part d?Question 3: [35 points] There are two groups of consumers who have inverse demands p1 =120 q1 and p2 = 60 0.5q2 . The monopolist has cost function C (q) = 20q.(a) [5 points] If the monopolist can choose a separate p1 and p2 for each group, what will theybe?(b) [10 points] If the monopolist can only charge a single price, what will it be? Do bothgroups purchase? If so, how much does each group buy? [Hint: Add up demands, notinverse demands.](c) [5 points] How do the prices p1 and p2 compare to the uniform price you found in part b?(d) [5 points] Does the monopolist prefer to set a separate p1 and p2 , or does the monopolistprefer to set a single price?1(e) [10 points] How much would group 1 consumers pay to have a ban on price discrimination? How much would group 2 pay? (Might be negative)Question 4: [20 points] Consider the example in class (price discrimination slides, pg. 6). Astudents demand for beer is ps (qs ) = 10 0.5qs , a faculty members demand is p f (q f ) = 20 q f ,and beer is sold by a monopolist with cost function c(q) = 2q. A monopolist can verify a buyer asa student by checking his/her student ID.(a) [10 points] Suppose the monopolist knows there are 9 faculty members and only one student, but cannot tell them apart. Calculate the prices the monopolist would charge if itsallowed to price discriminate, and the price the monopolist would charge if its bannedfrom price discriminating. Who would support a ban on price discrimination?(b) [10 points] Suppose the monopolist knows there are 5 faculty members and 5 students, butcannot tell them apart. Calculate the prices the monopolist would charge if its allowed toprice discriminate, and the price the monopolist would charge if its banned from pricediscriminating. Who would support a ban on price discrimination?Question 5: [45 points] Consider the example in class (price discrimination, p.17). The individualinverse demand of a group 1 individual is p1 = 20 0.5q1 and the individual inverse demand ofa group 2 individual is p2 = 40 q2 . The monopolists cost function is C (q) = 2q. There are 100consumers and a fraction of them are in group 1 and a fraction 1 are in group 2. A monopolistmust choose ( p, T ), a price and a membership fee. A consumer who pays the membership fee T isthen permitted to purchase items at a price of p per unit.(a) [5 points] Suppose ( p, T ) = (2, 0) so that the membership fee is zero. What is the consumer surplus (CS1 ) of a group 1 individual and the consumer surplus (CS2 ) of a group 2individual?(b) [5 points] Suppose ( p, T ) = (2, 0). If the monopolist then increases the membership fee sothat it is equal to CS2 , will group 1 individuals still purchase?(c) [5 points] Suppose ( p, T ) = (2, 323) so that the membership fee has increased to 323. Hasthe monopolist maximized his prots or can he do better, even without adjusting his priceper unit?(d) [5 points] Suppose ( p, T ) = ( p, 0). What is CS1 as a function of p?(e) [5 points] Suppose ( p, T ) = ( p, CS1 ( p) ) where is some positive number. Can this pairof price and membership fee ever maximize the monopolists prots?(f) [5 points] Assume that both groups of consumers still purchase. In class we found that for= 0.5 the maximization problem of the monopolist was50[ T + ( p 2)q1 ] + 50[ T + ( p 2)q2 ]s.t.T = CS1 ( p)where you found CS1 ( p) in part d. Write the maximization problem out for any arbitrary[0, 1].2(g) [5 points] Solve for price as a function of . When = 0.5 you should get the same pricewe found in class.(h) [5 points] How does price change as the fraction of group 1 individuals increases?(i) [5 points] As gets very small, what happens to price? Does this make sense? Should themonopolist still bother with group 1 consumers or should it re-consider serving both typesand focus on the more lucrative group 2 consumers?