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Unformatted text preview: Fall 2008 Truman Bewley Economics 155a Answers to Homework #1 (Due Thursciay, September 18) Problem: 1) Consider the Edgeworth box economy where the endowment of consumer A is (1, 0) and the endowment of consumer B is (0, 1). For each of the following three cases, find and sketch the set of Pareto optimal allocations and the utitity possibility set and find the allocations that maximize the sum of the utilities ot the two consumers; By “sketch the set of Pareto optimal allocations,” I mean find the coordinates of a few points on the utility possibility frontier and fill in the remaining part of the curve. In maximizing the sum of the utilities in part b, use the symmetry of the problem. In order to see where the sum of the utilities is maximized, it is important to have a fairly accurate sketch of the utility possibility set. = ix x = x . a) uA(x1, x2) 1 2 and uB(xi, x2) x1 2 E
b) u(x,x) :x“6x1’3andu(x,x) =x1’3x1’6. A 1 2 1 2 B 1 2 1 2 c)u(x,x)=xx2andu(x,x)=x2x. A 1 2 1 2 B 1 2 1 2 Answer: a) Because u and u are homothetic, the set of Pareto optimal allocations is the
A B diagonal of the Edgeworth box, as shown below. x A2 Set of
Pareto Optima!
Allocations Because uA and u are linearly homogeneous, the utility possibility frontier is a straight line as
B sketched below. u n s . . u .  u  , I I I I I I I I I I I I . Utility Possibility Frontier l . V . . . . _ . _ . . . _ . . . _ . 1 . . u  .  n . . . . . . . . . . . . . . . u . u . . .  . . . .
. . . . . . . . . . . . . . . . . . . . . . u . . .
 . . . t . . u . u . . . . . . . ; . . . . . . . . The allocations that maximize the sum of utility is the entire set of Pareto optimal allocations, namely,{(x,x)x =x ,x =x =‘t—x}
A B A? A2 B1 32 A1 Answer: b) In order to find the set of Pareto optimal allocations, we must solve the equation au(x ,x )lax au(x ,x )lax A A1 A2 A1:MRS =MRS = B B1 32 Bt_
au(x ,x)/ax A B au(x ,x)/ax A A1 A2 A2 B 31 32 32 Since the total endowment of each commodity is one, this equation becomes au(x ,x)/ax au(1—x ,1—x)/ax
A A1 A2 1 B A A2 1 1
au(x ,x)/ax au(1—x ,1—x)/ax
A A1 A2 2 B A1 A2 2 Y which in turn becomes 11‘3 __ 116
x 1 (1 XAZ) 1_ A2 _ —
6 XSIS 3 (.1 ___X )2l3
A1 = A1 '
1 X1116 1 (1 man)“3
3 x213 6 (1 __x )5I6
A2 A2 which impiies that and hence that x —xx =4x —4xx.
A2 A1A2 A‘E A1A2 Solving this equation for xAz, we see that , 4x X : A1 _
A2 1 + 3x
A1 The set of Pareto optimal allocations looks approximately as in the drawing below. Set of
Pareto
OptimaE
Allocations The point (0.727, 727) on the frontier may be found by letting xm1 = 1/3. Then xA2 = 2/3, so that x = 2/3 and x = 1/3. At these values for the allocation, v = v = 21,3 5 0.727.
B1 B2 A B 3112 The concavity of the utility possibility frontier is a consequence of the concavity of the utility
functions. The symmetry of the utitity poesibifity set with respect to the two axes follows from
the symmetry of the problem. if the allocation ((XIXM)I(XB,X)):((X :X)I(XIX)) A1 1 32 A1 A2 B1 BZ is feasible and achieves utility ievels (VA, VB) = (VA, 73), then the allocation ((XA1'XA2)’ (X81, X82)) = xez’ X31)’ ( XAz’ an is feasible and achieves the utility tevels (v , VB) = (V , 7A). From the symmetry and
. A B
convexity of the utility possibility set, it follows that the point on the frontier where the sum of 4 the utilities IS maxrmlzed Is the pomt where VA: V = 2“ . The corresponding allocation [S
B 112
3 ((x ,x ),(x ,x )) =((1/3,2/3),(2/3,1/3)). A1 A2 31 32 Answer: 0) The utiiity functions of part b are the sixth power of the utility functions of part a.
Since the sixth power is an increasing function, the utiiity functions of part c are monotone
transformations of the corresponding utility functions of part b. Therefore the indifference
curves and hence the set of Pareto optima are the same in the two parts. The utility possibility
frontier, however, differs, because it depends on the shape of the utility functions, not just on
the indifference curves. In attempting to plot the utility possibility frontier, notice that the
points (VA, VB) = (t, O) and (VA, VB) = (0, 1) belong to the frontier. If we calculate the utility
levels at the allocation ((xm, xAé), (xB , x )) = ((1/3, 2/3), (2/3, 1/3)), we find that
1 BE
VA 2 VB 2 4/9. Therefore the utility possibiiity frontier is not concave and the utility possibility set is not convex. if we calcuiate more points, we see that the utility possibility
frontier is convex. This is so because the utility functions are convex. The utility possibility
set looks approximately as follows. I Z i Z I Z I I 2 Utility Possibility Frontier u . .  .  < < ~ ~  v v     . u u u . . .
. . . . . . . . . . . . . a . u . . . . . . . .
. . . . . . . . . . . . . x . . . . r . 1 . u . The allocations that maximize the sum of the utilities are the ones corresponding to the points
(1, O) and (0, 1) on the utility possibility frontier. That is, they are the allocations ((X :X)I(X , X
A1 A2 B1 B2 and ((xm,xA2).(xB.xB2)) =((0.0).(1,1)) 1 Problem: 2) Find the optimum allocation and draw the feasibie set for each of the foliowing
three Robinson Crusoe economies, where L is the input of labor, I? is the consumption of leisure, x is the consumption of food, and y is the production of food. Indicate Crusoe’s
indifference curves and the optimum on the drawing. In a commodity vector, the first
component is laborleisure time and the second is food. a) y= f(L) =2L, e: (1, O), u(l?, x) =32’3X2’3.
b) y = f(L) = L, e = (1, O), u(f, x) = min(f, 2x).
0) y=f(L) =3 L,e=(t,0), u(£’, x) =E+2x. Answer: a) The first order condition for an optimum is 6u(x, I)
m _dﬂU
au(x, l) — dL ’
6X which implies that so that i=2.
3 Feasibility implies that ﬂ=x=2L=ﬂ1—ﬂ=2~ﬂ and hence and so The corresponding diagram is as follows. The feasibility set is indicated by the letter 5)“. 0 1 L, Answer: b) The diagram looks as follows. The optimum is at the point (1/3, 2/3), as is evident
from the diagram. The indifference curves are the each made up of two half lines meeting at
right angles at their endpoints. Answer: c) If we apply the equation au(x, If)
a? w(L)
au(x, f) C“—
ax we find that ___= £i_i_
2 I 1...... which is infeasible. if L = 1, so that E: O and x = y = 3, then df( L) =
dL = au(0, 3)/af > 3_ ~__________
2 au(o,3)/ax ii
2 so that we have a corner solution to the maximization problem. The optimum occurs at this
point. The diagram is as follows. The parallel straight lines are indifference curves. Egyféﬁg @Eigai ...
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