AP/ECON 3200 Industrial Organization Monopolistic competition: Practice problem set Problem 1. In a monopolistically competitive market with the market demand function given by p = 25 32 Q there are 10 ...rms. Each ...rm has total cost function T C = 12 q 2 + 2q + 8 (i) Find the short run equilibrium in this market. (ii) How will the number of ...rm in this market change in the long run? (iii) Find the long-run equilibrium price, output per ...rm and the number of ...rms. Illustrate it on a diagram. (iv) Suppose the marker demand increased to p = 125 23 Q: How does it aect the long-run equilibrium? (v) Suppose instead the ...xed costs of production increased to 32, while the market demand is still p = 25 32 Q. How does the new long-run equilibrium compare to that in parts (iii) and (iv)? Problem 2. Consider a Cournot duopoly with two ...rms. Demand function for ...rm i's output is pi = ai bi qi rqj where j is another ...rms and r < bi , r < bj (i) Write down the inverse demand function for ...rm i (ii) Calculate demand elasticity for ...rm i with respect to its own price and to the price of its competitor. Show that ...rm i's demand is more sensitive to own price changes than to price changes of the other ...rm (i.e. that two ...rms produce dierentiated products) 1 Answers: Problem 1. (i) Pro...t function of ...rm i: 3X qj 2 j=1 10 i = 25 ! qi 1 2 q + 2qi + 8 2 i Taking the derivative with respect to qi and rearranging we obtain the pro...tmaximizing output level (the best response function): 3X qj 8 j6=i qi = 5:75 Using the symmetry condition and its implication for ...rm-level outputs (qi = qj for any pair of ...rms i and j), in equilibrium each ...rm will produce approximately 1.3 units of output. Total industry output is then Q = 10 1:3 = 13 and price is p = 25 23 13 = 5:5 (ii) With the equilibrium price and ...rm-level output from part (i), the ...rm-level revenue is 7.15 and total cost of production is 11.5. Since ...rms are making losses in the short-run, the number of ...rms will decrease in the long-run. (iii) Taking the derivative of the pro...t function, we obtain a pro...tmaximizing condition which must hold for every ...rm i operating in the market: 3 qi qi 2 = 0 p 2 In the long-run, free entry will ensure that ...rm-level pro...ts are equal to zero. Setting ...rm-level pro...t to zero, we obtain the second equilibrium condition (zero-pro...t condition): pqi 1 2 q + 2qi + 8 2 i =0 Solving this system of equation for p and qi , we obtain the long-run equilibrium price and output per ...rm: p = 7 qi = 2 for all ...rms 2 From, the market demand we obtain total demand (and total supply) Q = 12, and the number of ...rms is N = Qq = 6 (iv) With the new market demand both equilibrium condition remain unaected so p = 7 and qi = 2 continue to be the equilibrium values. Total industry output is Q = 78:6. The number of ...rms will increase to N = Qq = 39: (v) With higher ...xed costs, q = 4; p = 12; N = 3:25: With higher ...xed costs some ...rms will exit and the remaining will operate at larger scale. Problem 2. (i) qi = bj (ai pi ) r (aj b1 b2 r 2 pj ) (ii) @qi pi bj p i = <0 @pi qi bj (ai pi ) r (aj pj @qi rpj>0 = @pj qi bj (ai pi ) r (aj pj ) Increase in own price will decrease ...rm i's demand while increase in the competitor's price will increase it. Given that r < bj , the second elasticity is always smaller than the ...rst one in absolute terms for the any pi = pj , thus variation in the price for the competitor's good has less impact on demand for ...rm i's product. 3