Ans_Midt_06 - Fall 2006 Truman Bewley Economics 155a...

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Unformatted text preview: Fall 2006 - Truman Bewley Economics 155a Answers to Midterm Examination Tuesday, October 24, 2006 Grade Distribution Range 3 0 - 4 0 - 3 9 4 9 50- 60- 70— 80- 90- 100— 110- 120- 130 59 69 79 89 99 109 119 129 140 Question: 1) Consider the Edgeworth box economy with u A(x1, x2) = uB(x x2) = J; + 41;; (=1,0),andeB=(01). a) Find the set of Pareto optimal ailocations and draw it accuratety in' an Edgeworth box diagram. b) Find the utility possibility frontier and draw it accurately. 'Answer: a) The consumers each have the same homothetio utility function, so that the set of Pareto optimal allocations is the diagonal in the box diagram below. 1 0 0 B 1 b) In order to calculate the utility possibility frontier, notice that “224;. Therefore, VAT .x = —. A1 [2 Then, Since The utiiity possibility. frontier is shown below. It is a quarter circle or radius 2. V B -2 Frontier Question: 2) Consider the Edgeworth box economy with uA(x1, x2) = x1 + 2x2, eA = (6, 0), ._ 1I3 2/3 _ uB(x1, x2) — x1 x2 ,ancteB— (0, S). a) Show the set of Pareto optimal allocations in an appropriate Edgeworth box diagram. b) Find and draw accurateiy the utiiity- possibility set, 61L and the utility possibility frontier, GM}. 0) Find the Pareto optimal allocation ( x :8) with xfi1 = 3. A! d) Find positive numbers aA and aB such that the allocation (;(_ :3) is the feasible A! allocation that maxrmlzes the welfare function aAuA(xA) + aBuB( x3) . Answer: a) Along the locus of Pareto optimal allocations, the two consumers’ marginal rates of substitution are equal, so that so that al’-‘B( th’ X32) auA( XA‘I’ XAZ) 6x1 _ ax1 aui3( XB1’ X52) — auA(xA1’ XAZ) ’ 6x2 6x2 which is the same as Hence, and so _1_ 2 xB1 2 x 1 B1 B2 ' A1 A2' and”: mewwauu- u..V-w-.-.\“-:-..,..,,,._Mmw.-_-»-v‘-m,,\-w,,;“MUM“,w K» w-t - m» mummy; The locus of Pareto optima is the diagonat shown in the box diagram below. 0 6 A Pareto Optima b) in order to calculate the utility posstbility frontier, notice that 3x. VA 2 A1 and VB = Xi‘atsnga = X31:1 _XA1: "'g—A' Therefore, ~35— + vB = 1. The utility possibility set is everything to the Southwest of the utility possibility frontier. The utility possibility frontier 6M5)“ and the utility possibility set “IL are shown in the diagram beiow. c) The allocation requested is (xA, x3) = ((3, 3), (3, 3)). d) Any positive vector a perpendicular to the utility possibiiity frontier will do. One possibility is a = (a, a3) = (1, 3). Any positive multiple of this vector will also do. Question: 3) Consider the following economy with two consumers, A and B, and two firms, 1 and 2, and three commodities, 1, 2, and 3. The first commodity may be thought of as labor-leisure time. e =eB=(1,O,O). A uA(x1, x2, x3) : xg’c‘xg’zxg’s. uB(x1, x2, x3) = x1’3xg’3xg’3. Y1={(y1r yzv 0) I V15 0! y2 S —6_y1}' Y2={(y1l01y3) I y1SOFy3_S—8y1}' 6A1:1,6A2=O,6m=0,and932=i. Find an equilibrium for this economy such that p1 = 1. Answer: Because the production possibility sets are cones, the maximum profit earned from each is zero in equilibrium. Because the utility functions are Cobb—Dougias, we know that positive amounts of both commodities 2 and 3 are consumed in equilibrium, so that the profit maximizing vector in Y1 is of the form (yH,—-6y11,0), where y11 < 0. Similarly, the profit maximizing vector in Y2 is of the form (V21: 0, —8y21), where y21 < 0. Because these vectors earn zero profits, it follows that p1y11‘ {3263/11 : 0 and I013,21 _ pasyzt = 0' Since p1 = 1, by assumption, it follows that p2 = 1/6 and p3 = 1/8. Hence, the equilibrium price vector is p = (1, 1/6, 1/8). Because maximum profits are zero, the weaith of each consumer is the vaiue of her or his endowment, which is 1. Because the utility functions are Cobb—Douglas, we have that 1 1 1 1 1 x =——, —-—x =—,and—x =——, f“ 3 6 12 2 8 A3 6 so that 1 4 xA1 = 3—, XAZ = 3, and an = —3—. Similarly, 1 1 1 1 1 x31: E, E—xBQ = E, and —8—xBa = 3—, so that 1 8 x31: 3, xI32 = 2, ande3 = E Therefore, the output of good 2 is y12=xA2+sz=3+2=5, and the output of good 3 is 4 8 y23=XA3+X83='§—+:= It follows that the labor component of the input-output vectors for firms 2 and 3 are, respectively, 5 1 y11=_—andy21:_3' 6 As a check, notice that .5. _1 1_2_ _2 1 xm+XB1_§+3—?”6A1+eer+yA1+yB1_ —6 —2- —6— 6 II I In summary, the equilibrium is , 1 4 1 8 5 1 i 1 x Ix! ! I : “131—! _!2:_-: ——l5r0: _—!0r4! 15—1— ' (iiy‘yip) [[3 3H3 3H6 H2 HGBD Question: 4) Give an example of an economy with no production that has continuous and strictly concave utility functions and strictly positive endowment vectors and yet has no competitive equilibrium. Answer: Consider the Edgeworth box economy eA=(1,1) =eB, _ 2 2 __ uA(x1, x2) — —x1—-x2 — uB(x1, x2). The utility function is strictiy concave, and the endowment vectors are strictiy positive. No matter what the price vector is, both consumers consume nothing, so that there is an excess supply of both commodities. In equilibrium, a commodity in excess supply has price zero. Since both prices cannot be zero, there can be no equilibrium. Question: 5) Give an example of a Robinson Crusoe economy which has at least one feasibie allocation but no optimal aliocation, and the utility function is increasing in the sense that u(x) > u(y) ifx >> y. Answer: i give three examples. Example 1: Y={(y1. y2)| V150. y2< V1}-e=(1:0)- u(x1, x2) = xtxz. The feasible set does not contain its frontier, because Y is not closed. It follows that there is no vector maximizing utility. See the diagram below. Example 2: 1 Y={(y1, 3/2) I 3/150. 3123 T171 —1}.e=(1,0). u(x1, x2) : x1+ x2. Because the feasible set is unbounded, there is no maximum. See the diagram below. Example 3: Y={(y1. ya) I V130, yzs y’1}-e=(1,0). x1, if x2 > 0, and u(x,x) = ‘ 2 0,ifx2=0. The utility’inoreases toward 1 as you move along the production possibility frontier to the right until you reach'the point (1, 0), and then it dropsvto zero. Therefore, there is no maximum. See the figure beiow. The utiiity function is discontinuous at the horizontat axis. Question: 6) Consider an economy with I consumers and N commodities, where I 2 2, N 2 2, and the utility of each consumer depends on the consumption of all consumers. That is, the utility function of each consumer i is u(x , , x1), where xk 6 Hi“, for k = 1, , I. Assume i 1 that for each i, ei >> 0 and ui is continuous. Show that there exists a Pareto optimal aliocation. . t Answer: The set of feasible aliocations, (if = {(x1, , xl) [ x. 6 Ft“, for all i, and Z I + i=1 xs '2 e_}, ' l is compact. Let U: 8} eR be defined as U(x1, , XI) = Z ui(x1, , XI). Since each utitity i=1 function u. is continuous, U is continuous. Since 55' is compact, U achieves a maximum on (it, say E at x = ( x1, , XI). The atlocation x isParetc optimal, because if the allocation x Pareto dominates it, then U(x) > U( I) , which contradicts the fact that U achieves a maximum at x . Question: 7') Consider an economy with 2 consumers and N commodities, where N 2 2 and the utiiity of each consumer depends not onty on her or his own consumption but on the utility levels of the other consumer. That is, the utility of consumer t is 10 mtwat-tum.“mm,,:.:.:,:.;,:.u.w;,.r: a w wvv" u1(x1, v2), where U1: Fifoi a H, xi 5 RT, and v2 is the utility level of consumer 2, and u2(x2, v1), where u2: fo R e R, x2 6 Rf, and v1 is the utility level of consumer 1. Assume that U1 and u2 are continuous and bounded. Show that given fixed values for x 1 and x2, there exist numbers vi and v2 such that .v=u(x 1 1, v2) and v:2 = u2(x2, v1). 1! Answer: Since u1 and u2 are bounded, there exist numbers b1 and b2 such that and b1S‘u2(x2r v) Sb, for all x1, x2, and v. Approach 1: 1 2 For all x, x, and v. Let u: {131, b2}x[b1, b2] —» [b1, b2]x[b1, b2] be defined by the equation u(v1, v2) = (u1(x1, v2), u2(x2, v1)). Since u1 and u2 are continuous, u is continuous. Since [b1, b2] ><[b1r ha] is compact and convex, the Brouwer fixed point theorem implies that u has a fixed point, (v v2). From the 1! definition of u, we see that v1: u1(x1, v2) and v2: u2(x2, v1). ' Approach 2: Thefunctionu(x, u(x, .)): [b, b2] » [b 2 2 , b] is continuous. Since the intervaE [b1, b2] 1 1 1 2 1 is compact and convex, the Brouwer fixed point theorem implies that there is v2 5 [b1, b2] such that v2 =-u2(x2, u1(x1, v2)). Let v1: u1(x1, v2). Then v1 and v2 have the desired properties. 11 ...
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