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Exam Economics I (Intermediate Economics) B. Koebel / T. Riechmann Prüfg.Nr. 5024 Winter 2002/2003 February 20, 2003 Name, First name Student number Degree/semester Instructions: • You have 90 minutes to answer all four questions. • Use of non–programmable calculators is allowed. • Please put your name on all sheets. • Please hand in all exam materials. 1 Question 1 An industry faces the demand curve y = 250 − p with y giving the demanded quantity and p giving the price of the good. Each firm has constant marginal costs of 4 and no other costs. a) Consider a competitive supply side of the market. Write down the first order condition for profit maximization of a competitive firm in this market. What is the equilibrium price and the equilibrium quantity in this market? b) Consider a monopolistic supply side of the market. Write down the first order condition for profit maximization of the monopolist in this market. What is the equilibrium price and the equilibrium quantity in this market? c) Consider a duopolistic supply side of the market. Write down the reaction function of firm number 1 in this market, assuming both firms decide at the same time. What is the (Cournot) equilibrium price and the (Cournot) equilibrium quantity in this market? Answer 1 a) π = py−c FOC: dπ ! = p − MC = 0 dy ⇔ p = 4 y? = 246, b) p? = 4 π = p y − c with p = 250 − y FOC: dπ ! = MR − MC = 0 dy ⇔ 246 − 2 y = 0 y? = 123, c) π = p yi − c with reaction function: p? = 127 p = 250 − (yi + y−i ) 1 R1 (y2 ) = 123 − y2 2 ? ? y = 164, p = 86 2 Question 2 The demand function is given by x = A p−γ with x giving the demand, p the price and a and γ as positive parameters. a) Derive the price elasticity of demand, ε. What is the economic meaning of the price elasticity of demand? What is elastic, what is inelastic demand? b) Denote revenues as a function of demand x and price p. How do revenues change as a reaction to an increase of the price, if demand is inelastic? c) Is the good in focus a Giffen good? Explain your answer both verbally and analytically. Answer 2 a) ε = dx p dp x: ε = −γ A p−γ−1 γ = − A p−γ x γ = − x x = −γ p x first important step Price elasticity of demand gives the reaction of demand to a change in price in relative terms. Elastic demand means, in relative terms, the reaction of demand is stronger than the initiating price change. Inelastic demand means, in relative terms, the reaction of demand is weaker than the initiating price change. b) R(x, p) = x(p) · p. If demand is inelastic, an increase of the price will increase revenues. c) No, it is not. For a Giffen good, the demand should increase with an increasing price, i.e. dd xp > 0. This is not the case, as the first derivative shows: dx = −γ A p−γ−1 < 0 . dp Question 3 A monopolist faces two separate markets with the demand curves given as D1 (p1 ) = y1 = 100 − p1 D2 (p2 ) = y2 = 100 − 2 p2 with D1 = y1 and D2 = y2 giving the quantities and p giving the price on the market. Let the monopolist’s costs be given as C(y) = 20 y . 3 a) Write down the equations of the inverse demand curves p1 (y1 ) and p2 (y2 ). b) Assume that the monopolist can price discriminate by charging a different price in each market. aa) Write down the specific revenue functions R1 (y1 ) for the first and R2 (y2 ) for the second market. bb) Write down the specific marginal revenue functions MR1 (y1 ) for the first and MR2 (y2 ) for the second market. cc) What are the profit maximizing quantities and prices on the markets? c) Assume the monopolist is unable to price discriminate. Thus, he faces the aggregate demand y = D(p) of both markets. aa) What is the function of aggregate demand D(p)? (Derive the explicit equation of aggregate demand for the given problem.) bb) For the aggregate demand function D(p) = y = 200 − 3 p, compute the function of inverse aggregate demand. cc) What is the optimal quantity and price for the non–discriminating monopolist? Answer 3 a) inverse demand curves: p1 (y1 ) = 100 − y1 1 p2 (y2 ) = 50 − y2 2 b) aa) revenue functions: R1 (y1 ) = 100 y1 − y21 1 R2 (y2 ) = 50 y2 − y22 2 bb) marginal revenue functions: MR1 (y1 ) = 100 − 2 y1 MR2 (y2 ) = 50 − y2 cc) optima: y?1 = 40; y?2 = 30; p?1 = 60 p?2 = 35 c) aa) aggregate demand: D(p) = y = D1 (p) + D2 (p) = 200 − 3 p 4 bb) inverse aggregate demand: p(y) = cc) y? = 70, p? = 130 3 200 1 − y 3 3 = 43.3 Question 4 Consider a game in simultaneous moves, where the payoffs are given in the following table: Player B b1 b2 b3 a1 2, 0 1, 1 4, 2 Player A a2 3, 4 1, 2 2, 3 a3 1, 3 0, 2 3, 0 a) What are the Nash–equilibria (in pure strategies)? b) Which strategies survive the iterated elimination of dominated strategies? c) Do Nash–equilibria get lost? Answer 4 a) Nash–equilibria at (a1 , b3 ) and (a2 , b1 ). b) (a) a3 is strictly dominated by a1 . → eliminate a3 . (b) Now, b2 is strictly dominated by b3 . → eliminate b2 . (c) The surviving strategies are a1 , a2 , b1 and b3 . c) No Nash–equilibria get lost. 5