problems ( Numbers in brackets at the beginning of problems below refer to related examples and problems in the text.) 14.1. [Related to text examples 14.1 and 14.2, and problems 14.1 and 14.2.] This problem continues Problem 13.1. Recall that demand for hot dogs on Main Street in Smalltown is: Q = 100 - 20P Assume that there are no fixed costs and, as before, that MC =$1.00. The purpose of this problem is to examine duopoly equilibria in this market (different behavioral assumptions lead to different equilibria) and to compare these equilibria to both the competitive and monopoly equilibria as well as to each other. a. The demand and MC functions are graphed on page 197. Indicate the competitive price and quantity with * and (*. Indicate the monopoly price and quantity with P., and @. [See your answers from the previous chapter.] As you proceed through the problem, you will be adding additional solutions to this diagram. W DELL MayDocument 5 4 3 - price 0 0 20 40 80 100 hot dogs b. The town grants vending rights to Doug's Dogs and Frank's Franks, at no charge. Other vendors are excluded. Assume for now that both Doug and Frank are Cournot Duopolists-they take each other's production as fixed and adjust their own to maximize their profits. Let q, represent Doug's production and q- represent Frank's: 0 = 4p + qp. Write Doug's profits as a function of 4p and qp [Hint: as a first step, invert the demand function and replace Of with go + 9plbe Acrobat Standard DC 2 19 (+ 12:2% DID = (1) Suppose that Doug knows qp If he assumes that this will not change once Frank knows qp. what value of qp should Doug choose to maximize his profits as a function of qp?Sign This is Doug's "reaction function." Draw it on the diagram below. . 50 40 30 Doug's output 20 10 - 0 10 20 30 ( 40 50 Frank's Output Now, find Frank's reaction function and add it to the diagram. OF = (3)3 19 (+ 122% 790... Sign In 9F = (3) Suppose that Doug had been producing at the monopoly solution before Frank arrived on the scene. Indicate his output level on the reaction function graph. What output would Frank initially choose to sell? (4) Explain why this would not be an equilibrium, and describe what would happen. (5)_ Draw the path that output would follow toward the new equilibrium on the reaction function diagram. What would equilibrium P, 9p. 9, and 0 be? 4F = (8)_ - 9p = (7) P = (6)_> Doug's profits are II, = (10) "Their combined O = (9) and Frank's profits are II = (11) _: Plot P and O on the graph in a and label profits are = II (12)Draw the path that output would follow toward the new equilibrium on the reaction function diagram. What would equilibrium P, 9p: 9, and 0 be? P = (6)_> 4D = (7)_ 4p = (8) Q = (9) _ _ Doug's profits are IIn = (10) and Frank's profits are II = (11) "Their combined . profits are = II (12) : Plot P and O on the graph in a and label them Pc and Cc (Cfor Cournot).4 179 Sign In c. Explain why the behavior of both, Frank and Doug in b is naive. (D) Assume that Frank continues to behave naively, but that Doug figures out that Frank is reacting to his decisions. Specifically, Doug knows that Frank's choice of q, is a function of his own choice of qp: as you found in b (Frank's reaction function). Doug takes this into account by substituting Frank's reaction function into his own profit equation, to get: Ilp = (2). Find Doug's new profit-maximizing choice of qp. Frank will choose qp = (4)_ 4D = (3) Total quantity produced is q = (5)_Q 4 12216- ID = (3)_ . Frank will choose qp = (4). Total quantity produced is q = (5)_ The price is P = (6) Doug's profits are II, = (7)_ IIF = (8) and Frank's profits are Their combined profits are II = (9) Plot the new O and P on the graph in a, labelling them Pryand OLF(LF for Leader-Follower). Also, plot the point (qp, qp) on the reaction function diagram in b and label it LF. d. In c it was assumed that Frank continued to be naive. Suppose instead that he figures out Doug's behavior at the same time that Doug figures out his own. That is, they both simultaneously decide to lead, mistakenly thinking that the other will follow. His choice of output level will then be go = (1) [Hint: does his problem differ in any way from Doug's in c?] Total output will . Doug's profits _ and the price will be P = (3). be Q = (2)_ Their and Frank's profits are IIF = (5). are IID = (4)_ . Plot the new O and P on the graph combined profits are II = (6)/ in a and label them @, , and Prz. Also, plot the point (9p, 9p) on the reaction el it /.L. Finally, what do you speculate willment 4 (+ 12216- d. In c it was assumed that Frank continued to be naive. Suppose instead that he figures out Doug's behavior at the same time that Doug figures out his own. That is, they both simultaneously decide to lead, mistakenly thinking that the other will follow. His choice of output level will then be qp = (1). [Hint: does his problem differ in any way from Doug's in c?] Total output will be Q = (2)_ _ and the price will be P = (3). Doug's profits are II, = (4)_ and Frank's profits are IIF = (5). Their combined profits are II = (6)/ . Plot the new Q and P on the graph in a and label them @,, and Prz. Also, plot the point (9p, q;) on the reaction function diagram in b and label it LL. Finally, what do you speculate will happen once they both realize that the other is not following? (7)_Models of Imperfect Competition e. Compare the outcomes in b through d with both the competitive and monopoly outcomes. What do you find? What general conclusions, if any, can be drawn from this example