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Unformatted text preview: Fall 2009 Truman Bewley
Economics 350a Answers to Homework #5
(Due Thursday, October 15) Problem: 13) Compute an equilibrium for the following economy with fabor, two produced
goods, two consumers, A and B, and two firms, 1 and 2. In commodity vectors, the first component is laborleisure time, the second component is the first produced good, and the third
is the second produced good. 9
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B y1=2JT1.y2=2,[L:, where y1 and ya are the outputs of produced goods 1 and 2, respectively, and L1 and L2 are the inputs of labor into the production of goods 1 and 2, respectively. Let the price of labor be one. (2,0,0),u(l,x,‘x) 21x2x3,6 :0,B =1,
B 1 2 12 B1 B2 Hint: Use the fact that person A’s utility function is linear to guess the equilibrium price
vector. Answer: if consumer A consumes a positive amount of every commodity, then the'price of all
commodities must be the same in equilibrium because of the nature of consumer A’s utility function. 30 we may tentatively assume that all prices equai 1. In this case, the profit
maximization problem for either firm is of the form max [2E —L]. LED The solution of this problem is L = 1, so that output is y = 2 and profit is 1: = y — L = 1. The
wealth of consumer A is wA=p.eA+'Jt:(1,1,1).(2.0,0) +1=3 Similarly the weaith of consumer B is w=3. Because of the CobbDouglas form of consumer B’s utility function, B's consumption of leisure
is w Similarty consumer B’s consumption of commodities 1 and 2 are, respectiveiy, x =——2~3=1andx —3—3=§—. at 6 32 = 6 2
Feasibility implies that the consumption of leisure by consumer A is H=e +e —l.’ —L —L =2+2~_1_—1—1=§_.
B 1 2 A A0 BO Simitarly and 3 t N)
N Since consumer A consumes a positive amount'of every commodity, the tentative hypothesis
made earlier about A’s consumption is correct. in summary, the equilibrium is ((Eix )!(EB!X31!X )r(—L1iy1)l(_L2!y2)Ip) ,X
A MM 32 = ((3/2, 1, 1/2), (1/2, 1, 3/2), (—1, 2), (—1, 2), (1, 1, 1)). Problem: 14) Compute an equilibrium for the following economy. 3/7 417
X U X,X =X
A(12) 12 =u x,x.
8(12)
e=e=0. A B Y1={(y1,y2)ly120.0sy2s3y1}. 2y1 Y2={(y1. ya) IV 20, Osy s4 } 1 2 5 ,
e =1,9 =o,e =o,e =1. A1 A2 B1 82 Hint: Compute the total output possibility set. Use the fact that both consumers have the same
CobbDouglas utility function. Answer: The total inputoutput possibility set, Y1 + Y2, is as in the following diagram. The total output possibility set is shaded with dots and its frontier is the heavy line. The arrows
perpendicular to the two straight line segments of the frontier are possibte equilibrium price
vectors. Income expansion line at
price vector (4, 10) ‘$(39r7, 52/7) \ 39/7 15/2 10 13 y Let use suppose that the equitibrium price vector is (1, 1). Then the total wealth of the
two consumers together is 13. That is, 13 is the maximum value of (1, 1).y, for y in Y1 + Y2. Because the two consumers have the same CobbDouglas utility function with coefficients 3/7
and 4/7, the total consumption of commodity 1 is ' 3 39
x =—13 =—.
1 7‘) 7 Simitarly the totat consumption of commodity 2 is 4 52
x=—13 =———.
2 71 ) 7 The point ('y1, y2) = (3 9/7, 5 2/7) is not feasible, as is made ctear in the above diagram. This point lies outside the production possibility frontier atong a dashed line extending one of the straight line segments of the frontier. a:
i‘é
r Let us suppose that the equilibrium price vector is (4, 10). Then the total wealth of the
two consumers is 70, and the total demand for commodity 1 satisfies the equation 4x = 3—(70) =30, 1
so that X1: g
2 .
Similarty the total demand for commodity 2 satisfies the equation 10x2 = 171—(70) = 40, so that x=4. The point (y1, y2) = (15/2, 4) lies on the production possibility frontier. It is at the intersection of the frontier with the income expansion line for price vector (4, 10), as is
shown in the diagram. If the price vector is (4, 10) and the total production vector is (1512, 4), then firm 1
maximizes profit by producing at the point y1 = (0, 3), and firm 2’s inputoutput possibility vector is y2 = (1 5/2, 1), a vector that maximizes firm 2's profits at the price vector (4, 10).
The maximum profits for firm 1 are 71:1 = (4, t0) .(0, 3) = 30. Since consumer A owns all of firm 1 and none 0? firm 2, A’s wealth is WA = 1121 = 30. Consumer
A's demand for commodity 1 satisfies the equation 3 90
4XA1= = 7—,
so that
X : 5—5—
A‘ 14' Consumer A’s consumption of commodity 2 satisfies the equation WMWWWW" “er and,” MN ,w, 10x (30) . 
xi“: so that x:£
A2 7‘ From feasibility, we know that consumer B’s demand for commodity 1 is 15 45 30 B1 7 14 7
Similarly 12 16
x =4——=—.
'32 7 7 In summary, ﬁsﬂfﬂi‘dﬂ‘ mwwwzrmmemwmikmwmc muwmwmmmmmwaacm‘nmﬁa»  wcwrmrwwm wwwa—mmmﬁ' :mwmmmmmmw4wmw Wsz ...
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