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Unformatted text preview: Falt 2008 Truman Bewley Economics 155a Answers to Homework #5
(Due Thursday, October 16) Problem: t3) Compute an equilibrium for the following economy with labor, 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. e
A (2,0,0),u(l,x,x)=l+x +X,e =1,6 =0,
A 12 1 2 A1 e
B y1=2,JT1_.y2=2J—L:. where y1 and y2 are the outputs of produced goods 1 and 2, respectivety, 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. If (2,0,0),u(!,x,x) =lx2x3,e =0,8 =1,
B 1 2 12 B1 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. So we may tentatively assume that all prices equal 1. in this case, the profit
maximization problem for either firm is of the form max[2alf —L]. L20 The solution of this problem is L = 1, so that output is y = 2 and profit is it = y — L = 1. The
wealth of consumer A is wA=p.eA+rt=(1,1,1).(2,0,0) +1 =3 Similarty the wealth of consumer B is w=3. Because of the CobbDouglas form of consumer B’s utility function, B’s consumption of leisure
is ‘ wmawmituwmwmmmramwhwmmmmwwwmww ww
t wrawmwmfwéw «mumem m. WWW mammarmmmwmm Similarly consumer B’s consumption of commodities 1 and 2 are, respectively, x =—2—B=1andx £3~3=—§—. Bi 6 32 z 6 2
Feasibility implies that the consumption of leisure by consumer A is B=e +e —f —L —L =2+2—_1_—1—1=£.
A A0 BO B 1 2 2 2 Similarly and 1 ‘l B
u
<
N
03
ID
to
M 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 ((EAIXA1IXA2)s(EBI X31! X )! (—L1! V1)1(—L2: 3,2): BE = ((3/2, ‘1, 1/2), (1/2, 1, 3/2), (—1, 2), (—1, 2), (1, i, 1)). Problem: 14) Compute an equilibrium for the following economy. 3r? 4/?
U X X =X X =U X X .
A( 1’ 2) t 2 B( 1I 2) e=e :0.
A B Y1={(y11 yz) I 3/12 0, 0 S y2 S 3 2y
Y={(y.y)y20.05ys4 1
2 1 2 1 2 5 9 =1,6 =O,9 =O,9 =1. A1 A2 B1 52 }. Hint: Compute the total output possibiiity set. Use the fact that both consumers have the same
CobbDouglas utility function. i
l
g Answer: The total inputoutput possibility set, Y1 + Y2, is as in the foilowing diagram. The total output possibility set is shaded with dots and its frontier is the heavy tine. The arrows perpendicular to the two straight line segments of the frontier are possible equilibrium price
vectors. income expansion line at
price vector (4, 10) \ \ (39/7, 52/7) ' __..........¢._.__._._._.____.__.._........ 39/7 15/2 10 13 y1 Let use suppose that the equilibrium price vector is (t, 1). Then the total wealth of the
two consumers together is 13. That is, 13 is the maximum value of (i, 1).y, for y in Y1 + Y2. Because the two consumers have the same CobbDouglas utiiity function with coefficients 3/7
and 4/7, the total consumption of commodity 1 is 3 39
x =— 13 =——.
1 7() 7 Simiiarly the total consumption of commodity 2 is 52 7 4
x=—13
2 7() The point (y1, y2) = (39/7, 52/7) is not feasible, as is made clear in the above diagram. This point lies outside the production possibility frontier along a dashed line extending one of the
straight line segments of the frontier. Let us suppose that the equiiibrium price vector is (4, 10). Then the total weatth of the
two consumers is 70, and the totai demand for commodity 1 satisfies the equation 3
4x =—70 =30,
1 7( )
sothat
15
x=——.
‘ 2 Similariy the total demand for commodity 2 satisfies the equation 10x2 = $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 (15/2, 4), then firm 1
maximizes profit by producing at the point y1 = (0, 3), and firm 2's inputoutput possibility vector is y2 = (15/2, 1), a vector that maximizes firm 2’s profits at the price vector (4, 10).
The maximum profits for firm 1 are 751 = (4, 10) .(o, 3) = 30. Since consumer A owns at! of firm t and none of firm 2, A’s wealth is WA 2 711 = 30. Consumer
A’s demand for commodity 1 satisfies the equation 3 90
4x =—3o =—,
A1 7( ) 7
sothat
45
XA1=ﬁ. Consumer A's consumption of commodity 2 satisfies the equation 4
10x =——30,
7,() so that x=£
A2 7' From feasibiEity, we know that consumer B’s demand for commodity 1 is X
H
I
ll 3‘ T 14 7" Similarly 12 16 x =4———=———.
32 7 7
In summary,
45 12 30 $6 15 g
x:x1 1 I : —:—'s “:wr 013: —:1 a '
(ABv1vzp) [[147H7 7]( )[2 ]( )] ...
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