Name: Final Exam EconS 527 (December 12 th, 2016)

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1 Name: Final Exam EconS 527 (December 12 th, 2016) Question #1 [20 Points]. Consider the car industry in which there are only two firms operating in the market, Trotro (T) and Fido (F). The marginal production cost is c>0 and the inverse demand structure for the two products is: p T = a βq T γq F and p F = a γq T βq F where β >0 and β 2 > γ 2 a. Discuss how a value of β>0 and γ=0 affects the demand structure. Provide some intuition. b. Calculate and draw the best response functions for quantity competition in differentiated products. c. Solve the Cournot game with differentiated product. Make sure you solve for the output level of each firm, market price and profits. Discuss how profits are affected by an increase on γ. d. Compare your results (profits and output levels) with the case in which firms sequentially decide their output level (Stackelberg game). Assume that leader firm is Trotro. Solution a. When β>0 and γ=0 these two products are highly differentiated, so a change in the price of Fido will have a negligible effect for Trotro. b. Trotro solves the following maximization problem: Hence the first order condition with respect to q H is: Max qt π T = (a βq T γq F ) q T cq T Solving for q H, we obtain Trotro s best-response function: By symmetry, Fido s best-response function is: π T q T = a bq T γq F c = 0 R T (q F ) = q T = a γq F c (A) R F (q T ) = q F = a γq T c (B) q T a-c/γ R F(q T) a-c/ R T(q F) a-c/ a-c/γ q F c. Substituting (B) into (A),

2 a c q T = γ a γq H c / 4β 2 4β 2 q T = (a c) γa + γ 2 q T + γc/ γ 2 q T 4β 2 q T γ 2 q T = (a c) γ(a c) Solving for qh, q T = ( γ)(a c) 4β 2 γ 2 And by symmetry, q F = ( γ)(a c) 4β 2 γ 2 Substituting the above quantities into the demand functions, Profits, p T = p F = a β π T = π F = (a β ( γ)(a c) ( γ)(a c) 4β 2 γ 2 γ 4β 2 γ 2 = a ( γ)(a c) 4β 2 γ 2 [β γ] ( γ)(a c) ( γ)(a c) ( γ)(a c) 4β 2 γ 2 γ 4β 2 γ 2 ) 4β 2 γ 2 Hence when γ increases firms profits decreases d. Operating by backward induction, Fido maximizes the following profit function Max π F = (a γq T βq F ) q F cq F Hence the first order condition with respect to q V is: Solving for q V, we obtain Valley s best response function: π F q F = a q F γq T c = 0 R F (q T ) = q F = a c γq H In the first stage, Trotro maximizes its profits considering Fido s BRF (I) Max π T = (a βq T γ ( a c γq H )) q T cq T Hence the first order condition with respect to q T is: Solving for q T, π T a c = a q q T γ T + γ2 q T β = 0 4β 2 q T 2γ 2 q T = 2aβ γ(a c)

3 q T = 2aβ γ(a c) (II) 2( 2 γ 2 ) Substituting II into I we obtain, q F = 2aβ γ(a c) a c γ[ 2( 2 γ 2 ) ] First, note that q T > q F in a sequential move game, second, comparing the results in part (d) with those in part (c) we obtain that Trotro will produce a higher output under the Stackelberg competition than Cournot. In the case of Fido, the firm will produce a higher output level under Cournot than Stackelberg competition. Question #2 [15 Points]. Consider the following Cobb-Douglas utility function u(x, y) = x 2 y Identify the Walrasian and Hicksian demands for goods x and y, that is, x(p, w) and h(p, u), respectively. In addition, using the Roy's Identity demonstrate that your result x(p, w) is correct. Solution In order to find the Walrasian demand, let us solve the UMP max x 2 y applying the Lagrangian we obtain, taking F.O.C s.t. x + p yy= w Γ = x 2 y λ( x + p y y w) x = 2xy λ = 0 (1) y = x2 λp y = 0 (2) λ = x + p y y w = 0 (3) Hence, from (1) and (2) we have that, x = 2 p y y, hence substituting it into (3) we obtain 2 p y y + p y y w = 0 y = w 3p y Therefore, the walrasian demands are x x (p, w) = 2w 3 and x y (p, w) = w 3p y In order to obtain the Hicksian demand, we need to solve the EMP min x + p y y s.t. x 2 y u applying the lagrangian we obtain Γ = x + p y y μ(x 2 y u)

4 taking F.O.C x = 2μxy = 0 (4) y = p y μx 2 = 0 (5) λ = x2 y u = 0 (6) Hence, from (4) and (5) we have that x = 2 p y y, hence substituting it into (6) we obtain Therefore the hicksian demands are Finally, using the Roy s Identity we know that y = u1/3 2/3 4 1/3 ( ) p y h x (p, u) = 2 1/3 u 1/3 ( p y ) 1/3 and h y (p, u) = u1/3 4 1/3 ( 2/3 ) p y Hence, we first need to identify the indirect utility function, v(p,w). Substituting, xx(p,w) and xy(p,w) into u(x,y) we obtain v(p, w) = ( 2w 2 ) ( w ) = 4w3 3 3p y 27p 2 x p y Hence, v(p, w) = 8w3 27p 3 x p y v(p, w) w = 12w2 27p 2 x p y And 8w3 27p 3 x p y 12w 2 = 27p 2 x p y 2w 3 = x x (p, w) Question #3 [20 Points]. Let us examine the Market for Lemons model discussed the last week of class. Consider that the owner of the good used car must sell her car because she is leaving the country. In addition, assume that the market price of used and new cars are exogenously given by p U = UG 4, and p N = N G, respectively. Analyze the problem of the buyers, the problem of the lemon used-car seller and the 4 problem of the good used-car seller and show that nobody buys a used car. Clearly identify your assumptions and discuss the results. Solution Under these prices, we seek to characterize the demand and supply patterns of our agents. New buyers (who do not yet own a car): Recalling our assumption that NL = UL = 0 and N G > U G > 0, if a new buyer buys a new car, then his or her expected utility is Vb = 0.5N G + - p N = 0.25N G. In contrast, if a new buyer 0:5NL buys a used car then Vb = 0.5U G +0.5U L -p U = 0.25U G : Hence, under the assumed prices, new buyers buy new cars.

5 Good-used-car sellers: The question simply assumes that good-used-car owners must leave the country and therefore sell their good-used cars. Lemon-used-car sellers: If a lemon-used-car seller sells his or her lemon and buys a new car, then Vs;L = 0.5N G - p N + p U = 0.25N G U G. In contrast, if she or he buys a used car then Vs;L= 0.5U G - p U + p U = 0.5U G. By assumption N G > U G, and 0.25N G U G > 0.5U G, since 0.25N G >0.25U G. Hence, lemon-used-car sellers buy (and sell) new cars. Altogether, under the assumed prices, nobody buys a used car. Question #4 [15 Points]. Assume that the market for Green Tea is composed by two leading firms: Asian (Brand A) and Lupita (Brand L). The direct demand functions facing each producer are given by q A (p A, p B ) = 100 2p A + p B and q B (p A, p B ) = 60 2p B + p A Note: The demand functions are not symmetric (i.e., they have different intercepts). Assume zero production cost (c A = c B = 0) and solve the following problems: (a) Suppose that the two firms merge. However, they decide to keep selling the two brands separately and charge, possibly, different prices. Compute the prices p A and p B which maximize joint industry profit and the aggregate industry profit. Question #5 [10 Points]. Define and discuss the Independence Axiom (IA). Provide an example in which the IA is NOT violated. Definition: A preference relation satisfies I.A. if, for any three lotteries, L, L and L, and α (0,1) we have L L if and only if α L+(1- α )L α L +(1- α)l. Question #6 [20 Points]. The certainty equivalent of a lottery is the amount of money you would have to be given with certainty to be just as well-off with that lottery. Suppose that your von Neumann-Morgenstern utility function over lotteries that give you an amount x if Event 1 happens and y if Event 1 does not happen is U(x, y, π) = π x + (1 π) y where π is the probability that Event 1 happens and 1 π is the probability that Event 1 does not happen. a. If π = 0.5, calculate the utility of a lottery that gives you $10,000 if Event 1 happens and $100 if Event 1 does not happen. b. If you were sure to receive $4,900, what would your utility be? c. Given this utility function and π =.5, write a general formula for the certainty equivalent of a lottery that gives you $x if Event 1 happens and $y if Event 1 does not happen. d. Calculate the certainty equivalent of receiving $10,000 if Event 1 happens and $100 if Event 1 does not happen.

6 Solution a). 5 10, = = 55 b) If you were sure to receive $4,900, then you receive $4,900 in both events. π 4,900 + (1 π) 4,900 = 70, where π can be any probability. c) (.5x y 1 2) 2 d) Using the equation in part c) (.5x y 1 2) 2 = (.5(10,000) (100) 1 2) 2 = 3,025