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Unformatted text preview: Fall 2007 Truman Bewley Economics i5§a
Midterm, Tuesday, October 23 You have the full 75 minute period to do this examination. A time is recommended for
each question and the maximum grade points for each is proportional to this time. The times add
up to 90 minutes, so that you are certainly not expected to do the whoie exam. Your grade witi
be 100xS/60, where S is your point score. Thus if you get all 90 points, your grade woutd be
150. Your full grade will count in your final average, even if it exceeds 100. 1) (15 minutes) Consider the Edgeworth box economy ll 2x+x UX X
A(1’2) 1 2 uB(x1, x2) m;n( 2xi, x2) eA=eB=(2,1). a) Find the set of Pareto optimal allocations and draw them accurately in the appropriate
Edgeworth box diagram. b) Find and draw accurately the utility possibility set and the utility possibiiity
frontier. c) Find a competitive equilibrium such that the sum of the prices is 1. 2) (10 minutes) Draw and explain an Edgeworth box economy that demonstrates that the
conclusion of the second welfare theorem may not be valid if some utitity functions are locally
satiated. ' 3) (15 minutes) Consider the following economy with one consumer, two firms, and three
’ commodities. u(x,x,x)=tnx +2inx
12 3 1 3 e = (3, 0, O) Y={ly,y)ly 50,3; S—2y}
1 1 2 1 2 1 Y2 = {(312, y3)r v2 3 0, y 5—3312} 3
The single consumer owns both firms. Find a competitive equilibrium with the price of commodity 1 equal to 1. MENwWw'vNWl m m: warm 4) A group of I consumers distributes a bundte of commodities among its members so as to
maximize the sum of their utilities. Let u_: Ft” a R be the utility function of consumer i, for
I + i: 1, ,I. wnmu»;«wm.:.xtm u a) (10 minutes) Show that if each u is continuous, then the problem max iu(x_) xeFl.N,....,xeRNi=1i I
1 + I + has a solution, where x e RN.
‘3‘ Let V(x) be the maximum vaiue in the problem justdetined, so that the V(x) is the total utility of ali I consumers when the bundle x is distributed among them so as to maximize the
sum of their utilities. ' b) (5 minutes) Show that if each of the u_ is strictly increasing, then V is strictly increasing. Mamitim‘wm’w»tmmn.m.mwmhﬁ aaimmune;tmwmmmmmmma.»mmmmmwwmmwswam:wanesamWrmmwwmwmwwwmww c) ( 10 minutes) Show that if each of the u_ is concave, then V is concave.
I 5) Let ((u_, e_) :1) be a pure trade economy, that is, one with no production. Assume that there are N commodities and that each of the utility functions u. is continuous, strictly increasing, and
I concave. The consumers torm disioint groups, 61, ,G , such that the union of all the Gk is the set of aii consumers. Each group buys and sells as a single entity and trades and distributes
consumption among members so as to maximize the sum of their utilities. a) (10 minutes) Define a notion of competitive equiiibrium for this economy with 2
groups and that refiects the assumption that the groups make all the decisions. t b) (15 minutes) Suppose that the economy with groups achieves an equilibrium in
which each group consumes a positive amount of at least one commodity. The groups then
disband, each consumer trades on her or his own, and the economy reaches a new
equiiibrium. Could the aiiocatton of the new individualistic equiiibrium Pareto dominate
the aliocation of the originai equilibrium with trading by groups, where Pareto
domination is defined in terms of the utilities of the individual consumers? You may use
the results of problem 4, even if you could not prove they are true. (Hint: This problem
requires you to put together several results proved in class. The argument is pretty
short.) ...
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