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Unformatted text preview: Fall 2008 Truman Bewley Economics 155a
Answers to Homework #2
(Due Thursday, September 25) Probtem: 4) For the Edgewcrth box economies listed below: i) find the set of feasible
allocations and the set ofﬁaretogptiﬂal allocations and show them on a box diagram, ii) find a Pareto optimal allocation x = ( x A, xB) that gives the consumers equal utltity, that is, is such that uA( YA) = ua( :5), iii) indicate the endowment allocation e and the allocation 7 on the box diagram, iv) find and draw the utility possibility set and indicate the utility vectors
corresponding to the endowment allocation and to the allocation x , and v) find all vectors of the form a = (aA, as), such that a > O and aAuA( xA) + aBuB( x B) zaAuA(xA) + aBuB(xB), for all
feasible allocations x = (xA, x3). I a) eA=(1,O),eB=(0, 1), uA(x1, x2) = ’x1x2 =uB(x1, x2).
b) eA=(O,1),eB=(1,O),u(x x2) =2x1+x2,u3(x1, x2) = x1+2x2.
c) eA= (2,0),ea=(0, 1), uA(x1, x2) =x1 + x2=uB(x1, x2).
d) e =(1,0),e =(0,1),u(x,x) =min(2x,x),u(x,x) = min(x,2x).
A B A 1 2 1 2 B 1 2 1 2
e) eA= (1, 0),e8= (0, 2), uA(x1, x2) = mln(x1, x2) = uB(x1,x2).
Answer:a)
1 0B
9
0 1 .sgJMaLaWM. a.» Noam“; i) The set of feasible allocations is the box above. The set of Pareto optimal allocations is
the diagonal, becatﬁe both consumers have the same homothetic utilit_y function. ii) The Pareto optimal allocation x that gives both consumers the same utility is ( x A, x8) = ((1/2, 1/2), (1/2, 1/2)). iii) The allocation I is at the center of the box, and the endowment allocation, e, is at the lower righthand corner of the box. iv) The utility possibility set “it is all the points on the line segment going from (1, 0) to (0, 1) in the diagram below together with all the
points southwest of that line segment. The utility vector of the endowment allocationis at the origin and that of the allocation x is at the midpoint of the line segment. mm lav/mt my».:twmwwqg.awwwmtwwwmmwmmwmm‘ «sis,memww “’mmsmwwrma 'r W)». massew." We em a“ w,_\,_\;$.M;ms._a,_;éwewxa;~“we. .. M, _ .,._ ”HM H q» a. w...a.vs.:..,...s_v::,uMyaag_.,./,,,,_wwwﬂ. aamwom...“ , mm N Answer: b) o T:
A i) The set of feasible allocations is the box shown above. The Pareto optimal allocations are the
thick lines at the bottom and righthand edges of the box. indifference curves of person A and B are indicated as IA and IB, respectively. ii) The altocation x is at the lower righthand corner of the box. iii) The endowment allocation is at the upper lefthand corner of the box. iv) The
utility possibility frontier is the tine going from the point (0, 3) to (2, 2) to (3, 0) in the
diagram below. The utliity possibility set elf, is everything on this frontier and southwest of it. The utility vector of x is at (2, 2), and the utility vector of the endowment, e, is at (1, 1). a v) {(a,a)(a,a) >>Oahd s
A B A B .532}.
a
B j.
2 c) i) The set of feasible allocations is the whole box in the diagram below, as is the set of
Pareto optimal allocations. (Because the consumers have the same iinear utility functions,
their indifference curves are tangent at every point in the box.) Indifference curves are
s_traight lines of slope minusgne. ii) An allocation gying the same utility to each consumer is x, where xA = (1, 1/2) = x . iii) The allocation x is at the center of the box in the diagram.
B The endowment allocation e is at the lower righthand corner of the box. 1/2 §
F
E mmwﬁmwwzewmmmrm»«Wm mmuwztmww mmmmtnwm; WWm,=—.umy.mw,m® aw»;"gzmnm‘mwmnwwirwN» iv) The utility possibility frontier is fine going from (0,, 3) to (3, 0) in the diagram below.
The utility possibility set GUI is everything on this frontier and southwest of it. The utility vectors corresponding to eand to x are as indicated; u(e) = (2, 1); and u( x ) = (3/2, 3/2). é
'é Answer: d) i) The set of feasible allocations is the box in the above diagram. The set of Pareto optimal
allocations is shaded with dots. Sampte indifference curves for persons A and B are labeted as IA and IB, respectively. The Pareto optimal allocation giving each consumer the same utility is x = (:A, x3) = ((1/3, 2/3), (2/3, 1/3)). it is shown in the box, as is the endowment allocation e. iv) The utility possibility set 5% is as shown in the diagram below. Answer: e) i) The set of teasibie allocations is the box in the above diagram. The set of Pareto optimal
allocations is the region in the box shaded with dots. Sample indifference_curves for person A and B and labeled as IA and IB, respectively. ii) The allocation x is the allocation xA= :B=(1/2,1). iii) 3: and e are as indicated in the box diagram. iv) The utility possibility set all: is as shown in the diagram below. E
E
§
§
3‘,
i E
é
.y 93.95%.»a.5£2:,_,“5.255uﬁaﬁﬁigﬁgwtit?i; i515. 10 ...
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