At a cost-minimizing input mix, the MRTS (ratio of marginal products) must equal the ratio of factor prices, or. f r

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1 ECON 311 NAME: KEY Fall Quarter, 2011 Prof. Hamilton Final Exam 200 points 1. (30 points). A firm in Los Angeles produes rubber gaskets using labor, L, and apital, K, aording to a prodution funtion Q = f(l,k). The wage rate for labor is w and the rental rate for apital is r. (a) Write out (or desribe in words) the ondition whih must hold at a ost-minimizing input mix. Briefly provide intuition on why this must be so. Show this outome on a diagram. K K US K C Q = Q 0 L US L C L At a ost-minimizing input mix, the MRTS (ratio of marginal produts) must equal the ratio of fator pries, or fl w f r K fl fk This implies that the marginal produt per dollar is equated aross eah fator of prodution,. w r fl fk If this were not the ase (say if ) then the firm ould lower its osts by alloating less w r dollars to the less produtive fator (apital) and more dollars to the more produtive fator (labor). (b) Suppose the firm an produe rubber gaskets in Mexio. The rental rate on apital is the same in Mexio as it is in Los Angeles, but the wage rate in Mexio is 10 times lower than in Los Angeles (i.e., r M = r LA and w M < w LA ). Show on your diagram how the ost-minimizing input mix differs for Mexio and Los Angeles for a given level of output.

2 2. (30 points). A onsumer derives utility uxy (, ) from onsuming two goods (X and Y) whose pries are P X and P Y. She has a fixed inome of $I. Draw a 2-panel diagram that shows demands for eah good, X* and Y*, at an initial level of pries and inome in the upper panel and a point on the demand funtion for X* at the initial prie level in the lower panel. Suppose the prie of good X dereases from P 0 X to P 1 X < P 0 X. Carefully label the inome effet and substitution effet of the prie hange for good X in the upper panel of your diagram. In the lower panel of your diagram, draw the ordinary demand urve and the ompensated demand urve assoiated with eah indifferene urve. Carefully label the regions in the lower panel of your diagram that represent ompensating variation, equivalent variation, and the hange in onsumer surplus. Y I = P x 0 X + P y Y I = P x 1 X + P y Y U 1 U 0 SE IE X 0 * = X 0 * X $/X P x 0 P x 1 X * (Px, Py*, I*) X (Px, Py*, U 0 *) X 0 * = X 0 * X Compensating variation (CV) is the yellow-shaded region, the hange in onsumer surplus is the sum of the yellow-shaded region and the blue-shaded region.

3 3. (20 points). TRUE, FALSE, EXPLAIN. For eah of the following, state whether you think the answer is true or false, then bak up your answer with a brief 1-2 sentene disussion. (a) (10 points) Andy has the utility funtion U 4X 2 Y. Andy likes the bundle (X,Y) = (2,4) twie as muh as the bundle (1,1) FALSE. Utility funtions are ordinal, not ardinal, and this means the atual utility value assoiated with any bundle is meaningless. Andy prefers (2,4) to (1,1), beause U(2,4) = 12 is greater than U(1,1) = 6, but we annot say he prefers the bundle twie as muh. (b) (10 points) The indifferene urves for an individual with rational preferenes annot interset. Support your answer with a diagram. TRUE. Interseting indifferene urves are not onsistent with rational preferenes. To see this, onsider the following figure. In the figure, there is more of both X and Y at point C than at point B. Therefore C is preferred to B by non-satiation. However, by the definition of indifferene urve U 1, B ~ A, and by the definition of indifferene urve U 2, C ~ A. Therefore, by transitivity, B ~ C whih ontradits rational preferenes. Y A B C U 2 U 1 X* Quantity of X

4 4. (50 points) A firm produes in an industry with the prodution funtion Q K L where input pries for apital and labor are given by r = $1 and w = $2, respetively. (a) (20 points) Find the supply funtion, Q(p), in the ase where both inputs are variable inputs. Firm s Problem (FP): min. C = wl + rk s.t. Q K L = 2L + K + [Q - K L 0.5 ] FOC: (1) d /dl = 2 - L -0.5 = 0 (2) d /dk = 1 - K -0.5 = 0 (3) d /d = Q - K L 0.5 = 0 From (1), (2) (K/L) 0.5 = 2 K = 4L whih means the firm should employ 4 times as muh Capital as Labor inputs at the ost-minimizing input mix. Substituting K =4L into the prodution funtion, we find Q = (4L) L 0.5 = 2L L 0.5 Q = 3L 0.5 => L*(y) = (Q/3) 2 K* = 4L* K*(y) = 4(Q/3) 2 The ost funtion is: (Q) = 2L*(y) + K*(y) = 2(Q/3) 2 + 4(Q/3) 2 = 6(Q/3) 2 => (Q) = (2/3)Q 2 Supply solves the profit maximization problem for the ompetitive firm: = pq (2/3)Q 2 FOC: =p 4/3Q = 0 => Q(p) = 3p/4

5 (b) (10 points) Does this prodution funtion exhibit inreasing, dereasing, or onstant returns to sale? Suppose we started with K=1 and L=1 and then quadrupled the use of eah fator. Would output quadruple? With K=1, L=1: Q = = 2. With K = L = 4: Q = = 4 Output doubles with a fourfold inrease in both fators, so the prodution funtion is haraterized by dereasing returns to sale. () (10 points) Suppose that the market prie is p = $20. Find the optimal output level, Q*. Calulate the firm s average ost at Q* and the equilibrium profit level in a ompetitive market. AC = (Q)/Q = 2Q/3 = 2(15)/3 = $10 If p = $20, Q* = 3(20)/4 => Q* = 15 units Profit, in equilibrium, is: * = 20(15) (2/3)(15) 2 = 15[20 2(15)/3] => * = $150 (d) (10 points) Provide a diagram that shows marginal ost, average ost, the market prie, and profit. Shade the region that represents profit. $/Q P* =$20 MC profit AC AC* = 10 Q* =15 Quantity (Q)

6 5. (40 points). Suppose Mark Jaobs and Prada ompete in a Cournot duopoly market for handbags. Consumers onsider Mark Jaobs and Prada handbags to be perfet substitutes and inverse demand for handbags is P = 1,200 2Q, where Q = q M + q P is the total quantity of handbags produed by both firms. Suppose Mark Jaobs has the ost funtion (q M ) = 100q M and Prada has the ost funtion (q P ) = 200q P. (a) (15 points) Calulate the reation funtions, q M R and q P R. Mark Jaob s Profit: M = [1200 2q M 2q P ]q M 100q M FOC: d M /dq M = q M 2q P 100 = 0 Reation funtion: q R M = q P Prada's Profit: P = [1200 2q M 2q P ]q P 200q P FOC: d P /dq P = q M 4q P 200 = 0 Reation funtion: q R P = q M (b) (10 points) Show the equilibrium quantity of handbags produed by eah firm as the intersetion of two reation funtions on a graph. q M 500 q P R 275 q M * =200 Cournot equilibrium q M R q P * = q P

7 () (15 points) Find the equilibrium quantity of handbags produed in a Cournot duopoly. What is the market prie of handbags in a Cournot duopoly? Cournot equilibrium is the intersetion of the reation funtions. q M = [ q M ] = q M => (3/4)q M = 150 => q M * = 200 Plugging this bak into Prada's reation funtion: q P = q M = (200) => q P * = 150 Market prie: P* = Q* = (q M * + q P *) = (350) => P* = $500 Mark Jaob s Profit: M *= $500(200) 100(200) => M *= $80,000 Prada's Profit: P *= $500(150) 200(150) => M *= $45, (30 points) Pete Rose is seen to plae an all-or-nothing $100,000 bet on the Raiders to win the AFC West in 2011; that is, if the Raiders win the AFC West, Pete Rose wins $100,000 and if the Raiders do not win the AFC West, Pete will lose $100,000). If Pete has a logarithmi utility-of-wealth funtion, um ( ) ln( m), and his urrent wealth is $1,000,000, what must he believe is the minimum probability that the Raider win the AFC West? (Round all answers to 2 deimal plaes.) Pete must reeive greater expeted utility from the bet than his utility level from not plaing any bet at all. If he does not bet, his utility is: um ( ) ln(1, 000, 000) If Pete bets, he will have either $1,100,000 or $900,000 in wealth, depending on the outome of the bet. His expeted utility is: Eum ( ( )) p ln(1,100, 000) (1 p) ln(900, 000) p (1 p) p Beause Pete is seen to make the bet, Pete must believe that p > 13.82, or p>0.55. THE END. HAVE A GOOD WINTER BREAK!