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Unformatted text preview: Fall 2008 Truman Bewley K
Economics 1§5a Answers to Homework 4 3 (Due Thursday, October 9) 3: Problem: 1) Solve max (’X + ’X )2
x120,x220 1 2 s.t.px+px=w.
11 22 mr'vwrmnmmwmm mm.” 5‘ ‘ Answer: The solution is not changed if we replace the objective function with the following
monotone transformation of it 2R+2ij, so that the problem becomes max [24—4241—“1
X>0,2K>0 1 st. pjx1+ p2x2=w.
The Lagrangian for this problem is £(X1,X2,?\,) =2E+2E—Mp1x1+p2x2}. From this, we see immediately that the first order conditions are
1 = lp and 1 = .
4x— 1 ix— 2 1 2
Dividing the first of these equations by the second, we obtain W
"O
M and hence x
_2 = 31..
2
x
1 p2
which is the same as E
E
p2
X = _1x E:
2 2 1 E
E
P2 E
Substituting this equation into the budget equation, we obtain
P
p x + _;x = w, E
1 1 p 1 E
E
E
so that E
w E
P p + P
1 2 1 :
By symmetry,
wp E
x = _...1.....
1 p p + I02 Problem: 6) a) Consider the Edgeworth box economy with utitity functions E uA(xE, x2) = 3x1+ x2 and IE
2
E
E
E z
2,
E
r:
i’
E
,5,
3
3 uB(x1, x2) x1+ 3x2 and endowment vectors eA = (2, 0) and eB = (0, 2). Find all equilibria with the price of the first good equal to 1. to) Consider the Edgeworth box economy with utility functions uA(x1, x2) = x1+ 3x2 and II uB(xE, x2) 3x1 + x2 and endowment vectors eA = (2, 0) and eB = (0, 2). Find all equilibria with the price of the first good equal to 1. Answer: a) An easy way to solve this problem is to draw the corresponding Edgeworth box, as
beiow. Three indifference curves for each of consumer A and B are drawn. Since the indifference
curves of consumer A are steeper than those of consumer B, we see by maximizing the utility of
one consumer along the indifference curve of the other that the Pareto cptimai allocations are as
indicated by the thick lines. The only one of these aliocattons that can have a budget line through
it and the initial endowment point is the endowment itself. Therefore it is the only equilibrium
aliocation. The equilibrium prices must give a budget line through the endowment point that
ties on or between the indifference curves of the two consumers through that point. That is, we
must have 9
_2
p1 S S3. J.
3 Since p1: 1, 1/3 5p2 53. Therefore the only equilibrium with the price of the first commodity equal to 1 is {((x.xB).p)=((2,0),(0.2),(1.p2))l1/3Sp2s3}. A Answer: b) The Edgeworth box for this economy is exactly tike that of the previous part of the
problem, except that the iabels of the indifference curves are reversed, as in the diagram below. p=(1:1) If we maximize the utility of one consumer over the indifference curve of the other, we see that
the Pareto optimal allocations are the teft and top edges of the box, as indicated by the thick
lines. The only allocation among these that is an equilibrium is the upper ieft corner, indicated
as E, and the price vector corresponding to this allocation is (1, 1). if we experimented with
some other price vector, such as the one yieiding the budget line xe for consumer B, then B
would choose point x on that line and A would choose point y. Since these points do not coincide,
this cannot be an equilibrium price vector. Therefore, the sole equilibrium with the price of
the first commodity equai to ‘i is ((xA'xB)’p) =((0,2),(2,0),(1,1)) Problem: 7) Consider the Edgeworth box economy where e = (12, 0), e8: (0, 12) and A uA(x1, x2) = uB(x1, x2) = xj’3x2’3. a) Calculate and draw accurateiy the offer curves for each consumer. b) Find all competitive equilibria with the price of the first good equal to 1. l? :5;
$2
a
it '3
a
35s
a %
t
i
2
E
5
g
S
.a.
g
E
F: Answer: a) Because the utility functions are CobbDouglas and each consumer is endowed with
only one commodity, we can calculate immediately that if xA and xB are demand vectors, then xA1 = 1 2/3 = 4 and x32 = (213)12 = 8. Therefore the offer curves are as in the diagram below. Answer b: Since xm =4, feasibility implies that xB1 = 8. Since xB2 = 8, it follows that xA2 = 4. The price of commodity 2, p2, is found by solving the budget equation for one of the consumers, say consumer A. That budget equation is xm + PZXM = (1, p2) eA, which is 4 +4p2= 12. The solution of this equation is p2 = 2. In summary the unique equilibrium is (XAIXBIP) =((4,4),(8,8), (1:2)) Problem: 9) Consider an Edgeworth box economy with utility functions
uA(x1, x2) = min(3x1, x2) and uB(x1, x2) = min(x1, 3x2). a) Find the unique equilibrium when eA= (4, 0) and as = (O, 4) and the price of
good t equal to 1. Compute the utility of person A at the equilibrium. b) Find the unique equilibrium when eA= (6, 0) and as: (0,4) and the price of
good 1equai to ‘l. c) Compute the utility of person A at the equilibrium of part (b) and compare it with his
utiiity in the equilibrium of part (a). Is there anything paradoxical about your finding?
Can you explain intuitively why person A’s utiEity level changes in the way that it does
from part (a) to part (b)? Answer: a) This problem may be solved aigebraically as follows. Because of the nature of the
utility functions, it =3x andx 23x.
A2 A1 at 32 Feasibility implies that x +x =4andx +x =4.
A1 81 A2 BE The solution to these linear equations is In order to find the price of commodity 2, use the budget equation of person A PXA = PeA.
which is 1 + 392 = 4,
so that p :1. In summary, the equilibrium is (xAthip) 2((1I3)I(3I1)I(1r1)) ~ 1‘ mummrmmmmmmr mattwmwmww, mmwmww The utility of person A in this equiiibrium is 3.
Answer: to) The relevant linear equations now become x =3x ,x =3x ,x +x =4,andx +x =6.
A2 A1 B1 B2 A1 81 A2 82 Proceeding as in part a, we find that the equilibrium is (xA,xB, p) =((3/4,9/4), (21/4,?14),(1,7/3)). wmmmwmwmtsrwwmm‘vwmw umwmm. Answer: 0) The utiiity of person A in the equilibrium of part b is 9/4, which is less than 3, the
utility of person A in the equilibrium of part a. This is somewhat surprising, as person A’s
endowment is larger in part b than in part a. The explanation can be understood visually by
considering the superposition of two Edgeworth boxes in the diagram above. The origins of person B in the models of parts a and b are labeled as 03a and 0%, respectively. The endowment points of the two models are indicated by e ande and the equilibrium allocations are indicated !
a b by E and Eb. The dashed tines are indifference curves. The heavy slanted tines are the locus of corners of the indifference curves of persons A and B and are the effective part of the offer
curves of the two people. In the model of part b, the righthand edge of the box is displaced to
the right, and this movement brings the intersection of the offer curves downward along the
offer curve of person A and therefore makes him or her worse off. This movement also
increases the price of good 2, which is the commodity that person A must purchase. It is this
increase in price that makes person A worse off. The increase in the suppty of the commodity
that person A provides turns the terms of trade against him or her. The change in terms of trade
may be seen from the change in the slope of the budget lines in the two equilibria. 3%
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