Answers_HW_6 - Fall 2009 I Truman Bewley W Answers to...

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Unformatted text preview: Fall 2009 I Truman Bewley W Answers to Homework #6 (Due Thursday, October 22) Mafi-‘X’MWW w» Eroblem: 1) Consider the Edgeworth box economy with reA = (1, 0), eB = (0, 1) . UA(X1: X2) = = min(4x1,‘x2), and uB(x1, x2) = x1 + 2x2. Find the equilibrium with transfer payments, (x,p,'c),thathasu(x ,x ) =1andp =1. 8 BI 32 1 a: i 3 Am: The relevant figure appears below. The heavy line is the set of Pareto optimal atlocations. Two indifference curves for person A are shown as right angles with apexes on the heavy line. Two indifference curves for person B are shown as negatively sloped solid lines. A Pareto optimal allocation to person B is of the form x = (3/4 + t, 4t), where 0 s t g 1/4. B Such an allocation gives person B utility uB(xB) = 3/4 + 9t. if this utility is 1, than t = 1/36. Therefore, the desired allocation is (XA, X8) = ((2/9, 8/9), 7/9, 1/9)). The budget line in the corresponding equilibrium with transfer payments must be parailei to B’s indifference curve, so that the equilibrium price vector is p = (I, 2). The transfer payment of person A is TA: p.(eA—XA) = (t, 2) .[(1, 0) —- (2/9, 8/9)] = —1. it follows that ”CB = 1. Hence the full equilibrium with transfer payments is ((XA,XB), p,(TA,’CB)) =((2/9,8/9), (7/9, 1/9), (1,2),(—1, 1)). _Pr_ot;le_m: 2) Consider the following economy with three commodities, 1, 2, and 3, two firms, firms 2 and 3, and two consumers, A and B. uA(x1, x2 x3) = ln(x2) + ln(x3) = uB(x,, x2, x3). ea: (2, 0, 0). eB=(0, 0,0). Firm 2 produces good 2 from good 1 with the production function y2 = 2(—y2t), where y21 s 0 andy is firm 2's input of good 1. Firm 3 produces goods from good 1 with the production 21 , function ya = —y , where y s 0 and y is firm 3’s input of good 1. (Notice that there is 3! 31 31 constant returns to scale in production, so that there are no profits in equilibrium and hence no need to assign ownership shares to consumers.) a) Compute an allocation that maximizes the sum of the utilities of the two consumers. b) Find an equilibrium with transfer payments the allocation of which is the one calculated in part a. Let the price of good 1 be 1. “VF-m mwmmaimwmmwmfime WWWMWWJZWWW ,% ,3, Answer: a) The welfare maximization problem is max [in(xA2) + ln(an) + |n(x32) + in(st)] x 20.x 20, A2 A3 s.t. x + x s —y A2 32 21 x + x S—y A3 Ba 31 —y —y 52. 21 at The probiem is symmetric with respect to the consumptions of consumers A and B, so that we may assume that x i x = x andx = x x . Since the utility functions are increasing, we A2 82 2 A3 B3 3 may also assume that the constraints hold with equality. After making these assumptions and dividing the objective function by 2, the problem becomes max {In(x) + ln(x)] x20,x20,y so 2 3 2 a 21 s.t.x :— 2 y21 2+y X =__21. 3 2 Men: 5) Consider an economy with two consumer, two goods, and no firms. The endowment vectors of consumers A and B are e and e ,respectively, where e >> O and 6B >> 0. Assume A B A that the utility of each consumer depends on the consumption of the other consumer as well as on his or her own consumption. That is, each consumer cares about what the other consumes, out of sympathy, envy, or because the other’s consumption interferes with or helps his or her own life. For instance, each neighbor might want the other to paint his or her house, but dislike smoke from his or her barbecue. Such effects are known as consumption externaiities. More formally, if XA and x3 are the consumption bundles of consumers A and B, respectively, then their utilities are u(x , x ) and u (x ,x ), respectively. Assume that u and u are A A B B B A A B continuous and that for i = A and B, u is strictly increasing and strictly concave with respect to l x. (That is, u(x,x A A . ) is both strictly increasing and strictiy concave with respect to xA, but , _ B not necessarily so with respect to x , and the symmetric statement applies to us.) In an B equilibrium, consumer A chooses x so as to solve the problem ' A max u(x,x) A B 2 xeFl+ s.t. p.x s p.eA, That is, consumer A holds xB fixed when considering how to choose XA. Similarly, consumer B chooses x so as to soive the problem B maxu(x ,X) x582 B A . s.t. p.x s p.e . B in addition, the market excess demand for each good is non-positive and the price of any good in excess supply is zero. a) Prove that an equilibrium exists. b) is an equilibrium allocation Pareto optimal? Give a proof or counter example. Arm: The proof is similar to that of theorem 4.24, and l wiii not fili in the details that can be found in that proof. First of alt, we must truncate the commodity space suitably. Let b be a positive number suchtnat eA + eB < b, for n = 1, , N. Let B = {x 6 WW x s b, for all n}. n F! + n For each xB e R” and p 6 AN“, let T = h th t EA(xB, p) {x e B i p.x s p.eAand uA(x, x8) 2 uA(z, x3) , for all z E B suc a p.zsne }.’ . A Symmetricaily, for each xA e R”, tet §;(xA, p) = {x e B E p.x 5 p.eB and uB(x, xA) 2 uB(z, xA), for all z e B such that p.z s pea}. A first task is to show that E and E are continuous functions. I give the proof for g: A B Since is convex and compact and u(x, x ) is continuous and strictiy concave with respeCt to x, 8 it foilows that §7(XB, p) exists and contains a single point. To show that g; is continuous, let A i3 i; t 2 E (xg, p“) be a convergent sequence in BxAN" converging to (XB, p). I must show that Iim Elma, p") = Elna, p) ._ If §:(x:, p”) does not converge to Elna, p), then there exists a [1...» positive number s and a subsequence §T( xnx, pnk) such that l§:(x;t, pnk) —— EB XE, P) I > 8. for A B ali k. By the Bolzano-Weierstrass theorem, we may assume that §:(x;k, pnk) converges to some . ~T .' ”.T“ "s“., |k,'tfl|wthat vector xA Thenle §A(XB,p)|2e Sincepx§A(xBx,pu) pieA for at z 00 s p.x _<_ p.eA. Let x e B be such that p.x s p.eA. I show that uA(xA, KB) 2 u (x, x8). Because A eA >> 0 and p >> 0, it foilows that p.62A > 0. if t is such that 0 < t < 1, then tx e B and p.(tx) = tp.x s tp.eA< p.eA. Therefore, if k is sufficiently large p“k.(tx) < p"x.eA. That is, tx A belongs to the budget set defined by pnk and hence uA(§:(x;k, pnk) , X?) 2 u (tx, xgk) . Since uA is continuous and Iim §T(x“x, pnk) = x and lim x"k= x, it follows that A B A k4.» k__, a B uA(xA, x3) 2 uA(tx, XE). Since t may be made arbitrariiy‘close to 1, the continuity of uA implies that u (x , x ) 2 u (x, x ). Therefore x = §T(x , p), which contradicts the A A B A B A A B inequality IXA— §:(XB, p) I 2 e > 0. This contradiction proves that i; is continuous. I may now proceed with a fixed point argument. Let 2: BxBxNHeRN be defined by the equation z(xA, xB, p) = §A(XB, p) + §B(XA, p) — eA— e8. Because uA(x, x3) and uB(x, x) are increasing with respect to x, the proof of Walras' law applies. That is, p.2(xA, XE, P) = 0, for all p. Letg: BxBxAN"»RN be defined by the equation x s x r = a y l r I F ' ' 9( A B p) (maX(0 p1+z1le x3 9) max(o P2+zzle "B p)» Since p.g(xA, xB, p) 2 pp > 0, it follows that g(xA, xB, p) > 0. Let h: BxBxAMaAN“ be defined by the equation slx, x. P) h(xA, xB, p) = A B g1( XA' XE. P) + 92(XA. XE, p) Let H: Bx BxAN‘B BxBxAN’1 be defined by the equation H( xA' XB, P) = (§:(XB’ P), ESXA’ P) ! h(xAr xB’ P)) - i: 5.9 swam“ Since the function H is continuous and the set BxBxAN" is compact and convex, the Brouwer fixed point theorem implies that H has a fixed point, ( x A, x B, p) . The argument given in the proof of theorem 4.24 implies that z( x A, :3, p) s 0. Because 3.z( x A, x B. P) = 0, if follows that, for all n, p < 0, if 2(xA, 73’ p) < 0. Because 7A=E;(xa, p) and n n X3 = §;(XA, p), it follows that each consumer anticipates correctly the consumption of the other. Because any feasible allocation is in the interior of B, it foilows by an argument given in the proof of theorem 4.24 that = N _ RN xA E §A(xB, p) {x E Ft+ | 9.x s p.eA and uA(x, x8) 2 uA(z, x3), for alt z e + such that p.z s p.e }. A That is, —;A maximizes consumer A’s utility in the untruncated budget set. A similar argument proves that — = N. , 2 , ,r H RN xBe§B(xA,p) {XER+ipX speBanduB(x,xA) ua(z xA) era 26 + such that p.z s peg}. Therefore ( xA, x8, p) is an equilibrium. WKWWWWWWWWWW" ‘ ‘“$3351!meY‘VWWWWWWLWZKWWWVW" "‘W awfl'vfifihfifivflh-LWWW‘I w m. W:wwymuwwymmawwmflwmm “ . ~ «umtummy-mmmmmmvawmwwwemrmm '«mwr—ww'ww ...
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