Ans_Midt_09 - Fall 2009 Truman Bew!ey Economics 350a...

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Unformatted text preview: Fall 2009 Truman Bew!ey Economics 350a Answers to Midterm Examination Tuesday, October 27 Problem: 1) Suppose there are two commodities and that a consumer has utility function u(x1, x2) = 2 ’x1 + In x2, where Ma 2 0 and x2 2 0. Suppose that the consumer’s endowment is e : (2, 0). Find the consumer's offer curve and draw, it accurately. Answer: The consumer’s maximization probtem is max(2J; + |n(x )) eri2 1 2 s.t.px +px =2p. 11 22 1 The Lagrangian is .L’ = 2 1/; + ln(x2) — ?t(p1x1 + p2x2) . The first order conditions are '1 =lp1andL=hp, ’x X 2 2 1 so that p1 J: = l = p x . Substituting this equation into the budget equation, it 2 2 px +px =2p, ‘i1 2 2 1 weseethat pixt + pt‘fi‘ : 2[01' andhence x + J—x‘=2. 1 t inspection shows that the solution to this equation is x = 1 . This equation defines the offer 1 % 2% e: E g ‘ “ mm’m ‘ ‘wmxwawirwannwmmmnwzmwwm -- armm-ww curve, which appears as 0 in the diagram below. If p1 = 0 or p2 = 0, there is no point in the budget set that maximizes utility. Problem: 2) Consider the Edgeworth box economy 24!: + ln(x2), 2|n(x1) + ln(x ). 2 II eA = (6,0), uA(x1, x2) ll eB = (0,6), uB(x1, x2) a) Find a competitive equilibrium with the price of commodity'1 equal to 1. it) Find weights aA and aB such that aA + aB = 1 and the equilibrium allocation maximizes the welfare function a u (x , x ) + a u (x , x ) over alt feasible A A A1 A2 B B B1 B2 allocations. Answer: 2a) Using the result of problem 1, x + ’x =6, A1 A1 so that XM = 4. Hence feasibility implies that x81 = 6 — 4 = 2. Because person B’s utility function is Cobb-Dougtas, we see immediately that x82 = 6/3 = 2. Hence feasibility implies that xA2 = 6 —- 2 = 4. The budget equation of person A is x +px =6, A1 2A2 that is, 4 + 4p2 = 6, and hence p2 = 112. The equilibrium is (XA’ xB. p) = ((4. 4). (2, 2), (1, 112)). 2b) The marginal utility of wealth of consumer A, AA, satisfies the equation __1.. = A p X A 2 A2 That is, 1_ = 32. 4 2 so that AA = i. The marginal utility of wealth of consumer B, AB, satisfies the equation 2 1— = A p X B 2 82 That is, 1_ = E, 2 2 so that AB = 1. The weights are proportional to (A: 22‘) = (2, 1). Since the weights must add to 1, they are (aA, a3) = (2/3, 113). Problem: 3) State Walras’ Law and state conditions under which it is valid. Answer:- Walras' Law says that for an economy, if x. 6 l;( p), for all i, and y. e n_(p), for all j, I l I l | .J then p. XXX-H31) — 2y] 2 0, for all price vectors p, whether it be an equilibrium price i=1 ' )«1 1 vector or not. This law applies if, for all E, the utility functions u_, are iocally non-satiated. Problem: 4) Give an example of a Robinson Crusoe economy (u,_‘e, Y) with no optimal allocation. Answer: A Robinson Crusoe economy has an optimal allocation if the utility function is continuous and the set of feasibie allocations is non-empty and is closed and bounded. Violation of any of these conditions can give an exampie. For instance, suppose that e = (1, 0), W-mmmm’t‘m‘fi‘flWW—Wm l? i g u(x1, x2) = x2, andY= {(y , y2) E R2} y1 s 0 and y2 < —-y1}. In this example, the input-output 1 possibility set is not closed, so that the set of feasibie allocations is not closed. The consumption allocation (0, 1 — t/n) is feasible, for all positive integers n, and the utiiity of this allocation is 1 — iln. if x is any feasible consumption allocation, then u(x) < 1 and so u(x) < u(0, 1 — 1m) = 1 —— 1/n, for some n. However, the consumption allocation (0, t) is not feasible, and so there is no optimal allocation. The visual picture is as foilows. Problem: 5) An economy ((u_, e)! , (Y)J ) is in equilibrium at (( x , _y-), E), and all the l E l=1 j j=1 u_tility functions are iooally non-satiated. Firm 1 produces nothing in the equilibrium. That is, y1 = 0. A government obliges firm 1 to produce 116 Y , where y at 0. The economy then 1 ‘— reaches a new equilibrium (( x , y), E) . Could the new equilibrium allocation (3:, y ) Pareto dominate the original equilibrium aiiocation ( x , y )? Explain why or why not. Answer: Begausg the utility functions are locally non-satiated, the original equilibrium allocation ( x , y ) is Pareto optimal and therefore cannot be Pareto dominated by the allocation ( x , y ). Question: 6) Two consumer workers A and B can produce goods 1 and 2. The two consumers have access to the same technology, but have different types of labor. Each consumer worker is endowed with one unit of their own labor. With her or his labor, consumer worker A can produce 3 units of good 1 per unit of labor time but oniy 1 unit of good 2 per unit of labor time. That is, if consumer worker A devotes Li units of labor time to the production of good 1 and L2 units of labor time to the production of good 2, then she or he produces 3i.1 units of good 1 and L2 units of good 2. Consumer B can produce 1 unit of good 1 per unit of labor time and 2 units of good 2. Both consumers have utility function u (x1, x ) = In x1 + In x2. Find a competitive 2 equilibrium for the economy consisting of the two consumer workers and in which the price of good 1 is 1. (Hint: Draw the production possibility frontier for goods 1 and 2 and use the fact that both consumers have the same Cobb—Douglas utility function.) Answer: The ‘output possibility sets, YA and YB, for persons A and B are as pictured below. The total input-output possibility set, YA + YB, is as in the next diagram. mmw-«wmumrvmww nasmmnfimwmywm (1, 1/2) The vector (1, 3) is perpendicular to the upper face and the vector (1, 1/2) is perpendicular to the right-hand face. Let x and x be the totai consumptions of goods 1 and 2, respectivety, . 1 2 and let w be the total of the incomes of consumers A and B. Because the two consumers have the same Cobb-Douglas utility function, we know that There are three possibilities, p2 = 3, p2 = 1 ori < p2< 3. If p2 = 3, then 2 2 y1 = x1: 3X2 = 3y2, which means that (y1, y2) lies to the right of the upper face of YA + Y3 and this is inconsistent with equilibrium. ff p = l, theny = x = ix = ly , which means that 2 2 1 1 2 2 2 2 (y1, V2) lies to the left of the right-hand face of YA + Y3 and this is inconsistent with equilibrium. Therefore 1_ < p2< 3, which means that consumer A maximizes profit or 2 revenue by producing (ym, yAz) = (3, 0), and consumer B maximizes profit or revenue by producing (ym, y ) = (0, 2), andso(x1, x2) = (ym + yB1,yA2 + yBE) = (3,2). Hence 82 w‘a w 1. ”‘0 ll x 1.. II M Because consumer A’s production process has constant returns to scale, she or he earns no profits and the wage for consumer A's labor is :E E g}? E E § 3, it Mwwsxiivlfimwwmamw which is the same as consumer A’s weaith, since A has 1 unit of labor. Similarly the wage for consumer B’s labor and B’s wealth are «mmammwmw;rmqquwmmcwMmmnummw m: Wm?» A1 2p 2 w-wmnwwwmuw w ‘ mm Tug-w v wmm‘mwrwmamwmmmw mwcmmrmw W 8:1. p 2 x=l B22 In summary, the equilibrium is (xA, x3, yA, yB, WA, WE, (p1, p2)) = ((3/2, 1), (3/2, 1), (3, 0), (0, 2), 3, 3, (1, 3/2)). , ...
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