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Unformatted text preview: Fali 2005 or _ Truman Bewley
I Economics 155a
Final Examination
Tuesday, December 13, 2005, 2:00  5:00, WLH 113
You have the full 180 minute examination period to finish this test. The grade points for each question are proportional to the time recommended for it. The times for all the questions add up to 150 minutes, so that theoretically you have 30 extra minutes to took over your
answers. 7_ .. : 1) (10 minutes) State the Minkowski separation theorem.
2) (15 minutes) State the second weifare theorem. 3) (25 minutes) Consider the following Edgeworth box economy. u.i\(X1’ x2) = ( 4;: + 2E)2 = “5(x1'x2)’; eA=(4,1). eB=(1,4). a) Find the set of Pareto optimal allocations and draw it in a box diagram. Indicate the
endowment point. 'b) Find the utility possibility frontier and draw it.
c) Find a competitive equilibriumiwith{theiprice of good 1 equal to 1. d) Find positive numbers aArandaéusuch that the competitive equilibrium aliocation maximizes the welfare function aAuA(xM, xm) + aaua(xB1, x32) over the set of feasible
allocations (xA, xB) . ,Iiiisilr titC'~.JiLiEii. Arrr‘ n_ (my! ,t 'OieuithiiCi Liiau \,€1Li(}‘ H . . .. “7.3..” . :ttutvt..." ...V 4) (25 minutes) Consider the followingirecohomyéwith three commodities, 1, 2, and 3, two
firms, firms 2 and 3, and two consumer's,”A and’B". uA(x1, x2 x3) = ln(x2) + ln(x3) = uB(xt, x2, x3).
eA= (2, 0, 0). eB=(0,O,O). Firm 2 produces good 2 from good 1 with th_e.production function y2 = 2(—y1) , where y1 s 0 and
y1 is firm 2’s input of good 1. Firm 3 produces :good 73 from good 1 with the production function
y3 = —y1, where 3/1 S 0 and y1 is firm 3's input of good 1. (Notice that there is constant returns to scale in production, so that there are no profits in equilibrium and hence no need to assign
ownership shares to consumers.) a) Compute an allocation that maximizes the sum of the utilities of the two consumers. b) Find an equitibrium with transfer payments the allocation of which is the one
caicuiated in part a. Let the price of good1 be 1, 5) (20 minutes) Consider the following insurance problem. There are two consumers, A and B,
two events, a and b, and one period. The endowment of consumer A is eAz (9%, em) = (1, O).
The endowment of consumer B is eB=(e =(O,1). Ba’ 83b) The utility function of consumer A is 1 at 2 'x
uA(xa, xb) = —3—e a—Ee ". The utiiity function of consumer B is u(x x)=lx+£x
Ba’b a 3b' 3 Compute an ArrowDebreu equitibrium in which the sum of the prices is one.
' ‘ r mu;:stagétrﬁgtgttciﬁprgbtc":1, lé‘isfstz: ‘ » f ‘ ’ .2 mag.g»...«,...m,r_m...d........m... m.» . . V 6) (30 minutes) Consider the foltowing one: commodity Samuelson model. e = (1, t). u(x0, x1) = min(3xo + x1; 2xo + 5x1). a) Show the set of feasible stationary allocations in a diagram. Show indifference curves
in the diagram. b) Show in a separate copy otithei'lsaméﬁ'diagram which of the feasible stationary
allocations is Pareto optimal. ' " c) Find a stationary spot price equilibrium in which the price of the one commodity is one, there is a constant interest rate, and where the endowment is the equilibrium allocation. d) Find a stationary spot price equilibrium such that the stationary allocation (x0, x1) = (0, 2) is the equilibrium allocation. 9) What welfare function is maximized by the endowment allocation? 7) (25 minutes) Consider the totlowing model with two consumers, A and B, and with two commodities, 1 and 2. 7 uA(x1, x2) = x1+ n(x2). ‘ uB(x1:, x2) = x1+ 2n( x2). eA:(eA1’eA2)' es=(es1'ei32)' Assume that eA1 > 10, e81 >10, 9A2 > 0, eEe > 0, ancieﬂ2 + e82 $1. a) Compute an equilibrium wheresthepricszgi 9.0041 is,.,1 The equiiibrium pricept good.
2 and the equitibrium altocattons will be formulas in terms of the endowments. Hint:
The endowments of good 1 are so large that each consumer consumes a positive amount of
good 1 in equilibrium. None of. the tricks I gave you in class for computing equitibria
work in this case. You have to find another trick. b) Show that the price of good 2 and the consumer’s consumption of good 2 depends only
on 9A2 + e82. ...
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