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Unformatted text preview: Fall 2008 Truman Bewley Economics 155a
Final Examination, Friday, December 19, 2:00 — 5:00 pm You have three hours or 180 minutes to do this examination. There are times suggested for each of the 6 questions, and these times equal the points for each of them. The total of the
times is 150, so that you have 30 minutes to read the questions and check your work. 1) (10 minutes) a) State Walras’ law. b) Under what condition or conditions is Walras’ law satisfied. 2) (15 minutes) Give an example to show why an equilibrium may not exist if some consumer’s
utility function is not continuous. In this example, each inputoutput possibility set should be
closed and convex and shoutd contain the zero vector, each endowment vector should be strictly positive, and each utiiity function should be increasing and quasiconcave, that is, the set of
points at least as desired as one point is convex.  3) (25 minutes) Consider the Edgeworth box economy with utility functions ll uA(x1, x2) n(x1) + 2n( x2) and uB(x1, x2) = 3n( x1) + In( x2) and endowment vectors e and e , where e >> 0 and e >> O.
A B A B a) Find the allocation (xA, xB) that solves the problem max [uA(xA) + uB(xB)} s.t.x +X =9 +e.
A B A B The optimal values of x , x , x , and x are functions of e , e , e , ancte .
A1 A2 B1 I32 A1 A2 BI B2 [3) Find an equilibrium with transfer payments, (xA, xB, p, 1:), such that p= DW(eA+eB), where W is the value function W( e) = max [uA(xA) + uB(xB)} x,x
AB S.t.X +X =8
A B and DW(e) is the derivative of W at e. All components of the equilibrium are functions of the components of e and e .
A B 4) (30 minutes) Farmers A and B live on an island where they are isoiated from the rest of the
world. Farmer A has 200 acres of farm land and farmer B has 100 acres. The growing season
is wet with probability one half and dry with probability one half. The farmers grow grain, and
there are two kinds of seed, short and long kernel. On farmer A's land, short kernel seed
produces 2 bushels per acres when the season is dry and nothing when it is wet, and tong kernel
seed produces nothing when the season is dry and 8 bushels per acre when the season is wet.
Farmer A can plant whatever proportion of land she or he wants in short or long kernel seed.
Only long kernel grain wilt sprout on farmer B’s land, where it produces nothing in dry
weather and 8 bushels per acre in wet weather. The utility function of each farmer is u(x,x ) =lln(x) +l’ln(x ),
D W 2 D 2 W where xDand xW are grain consumption in dry and wet weather respectively. The two farmers plan their economic life for each season by means of an ArrowDebreu equilibrium. Compute
such an equilibrium in which the price of a contingent claim on 1 bushel of grain in wet weather
is 1. You should find the price of a contingent claim on t bushel of grain in dry weather, each
farmer's consumption in dry and wet weather, and how much land farmer A devotes to each of
short and long kernel seed, respectively. 5) (30 minutes) Consider the Samuelson overlapping generations model with the endowment of
each generation equal to e = (so, ei) = (2, 0) and with the utility function of each generation equal to u(x, x) = min(x +3x,3x + x).
0 ‘i 0 1 O 1 a) Draw in a single diagram the set of feasible stationary allocations and the endowment and
some inditference curves for a single generation. b) Compute all stationary spot price equilibria with the price of the single good equal to 1 and
with interest rate r, where r > ——1. The aliocation and tax or allocations and taxes depend on r. c) For what values of r is the stationary equilibrium allocation(s) Pareto optimal? d) For which values of r does the stationary equilibrium allocation(s) maximize a social
welfare function and what is this function? SE
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i: 6) (40 minutes) Consider a Diamond overlapping generations model with production function
f( K, = 3K113L213 and where each generation has utility function u(x0) +u1(x1) =2R+2Jxﬁ. a) Compute a stationary spot price equilibrium with interest rate r, where r > 0, and where the
price of the produced good is 1. Ali variables depend on r. ' Suppose the economy is in a stationary equilibrium with r = 1 and that there is no social
security. The government now introduces an infinitesimal amount of payasyougc social
security without changing the regular tax found in part a (for r = 1). b) Does the introduction of sociai security increase or reduce r? 0) Does the introduction of social security increase or reduce the stationary equilibrium capital ' stock? d) Does the introduction of sociai security increase or reduce the stationary equilibrium amount
of government debt? 9) Is the equilibrium with r = t and with no social security Pareto optimal? Would a new
equilibrium with a small but positive amount of social security be Pareto optimal? awrunrawmwmidway» wrww imipbﬁﬁy‘. cwmay:i;mmmmartwaysaw .iaaw.m&‘ahtm“$awiﬁﬁ, wwmwmm ...
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