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Unformatted text preview: Fall 2006 Truman Bewley
Economics 155a Makeup Finai Examination
Wednesday, January 24, 2006 You have three hours or 180 minutes to’do this examination. There are times suggested '
for each of the ten questions, and these times equal the points for each of them. The total of the
times is 220. A total of 140 points will count as a grade of 100, so that your grade on the exam
will be the total number of points awarded times (100/140). Hence your grade on this exam
could be as high as 157.14. I will not be very generous with partial credit, so try to answer
correctly the questions you do answer. i use no curve in grading, so that a 100 is a very good grade. 1) (20 minutes) Consider an economy ((U; 8)I ,(Y)J ). I i E1 jj=1 Where. for all i. U; R” a R and e e B'T‘andﬁfogialij;.1¥}CFtN.r Define for this economy a) a feasible allocation,
b) a Pareto optimal allocation, and c) a welfare function. nu Hgt_bu vary gone.sus n ' rim 3 LA pqw 2) (15 minutes) State formally a theorem about an economy, ((ui, ei)i1=1, (Yi)i:1), that says that every Pareto optimal aliocation maximizes a weighted sum of the consumers’ utility
functions. Be sure to state all the assumptions. 3) (15 minutes) Give an example 'of‘lan Edgeworth box economy, ((uA, eA), (uB, eB)), with continuous and quasiconcave utility functions and with a feastbte allocation, (3? , .xB), such A that x A >> 0 and :8 >> 0, where the economy and allocation do not satisfy the second welfare theorem. A good drawing is enough. 4) (35 minutes) Consider the following Edgeworth box economy. uA(x1, x2) = min(x1, x2), ua(x1, x2) = 2x1 + x2, eAz (2, 0),andeB= (0,3). a) Show the set of Pareto optimal allocations In a b0); diagram. b) Compute_a genera! equilibrium such that thesum of the prices is one. Show the equilibrium
allocation, x , in the box diagram. 0) Compute the marginal utilities of unit of account of the two consumers in the equilibrium. d) Compute non—negative numbers aA andaBsuchxthat the equitibrium allocation maximizes the welfare function all X X +aU X X
AA( A1’ A2) a B( 51’ 52) over alt feasible aiiocations ((x , x ), (x , x )).
A1 A2 B1 B2 5) (20 minutes) Consider the following insurance model with two states, a and b, and two
consumers, A and B, and one commodity in each state. 1 2
uA(Xa' Xb) : + ER = uB(Xa’ xb)’
andeA= (16, 2). GB: (8, 4): where x and xb are the quantities of the one commodity consumed in states a and b, respectivety. a Compute an ArrowDebreu equilibrium such that‘the price of a contingent claim on one unit of
the commodity in state a is 1. it An MWMWWWWMWW 6) (25 minutes) Consider the following insurance model with three commodities, labor and
goods 1 and 2, with two consumers, A and B, with two firms, 1 and 2, with two states, a and b, and with two periods. Each consumerﬂisendowedfwlth 1 unit of labor (L) or leisure (3) time in period 1 and nothing else. Leisure is c'c‘J‘nsiu‘med"'in‘pieriod 1 and goods 1 and 2 are consumed in
period 2. Labor is used in period 1 and output of good 1 or 2 appears in period 2. UA(B’Xa1'x32’Xb1'Xb2) :9 + 2’lixa1’ usw’ Xat’ xaz’ Xbi' sz) :E + 24Xb2' Y1={(—L,ya1,0,yb1,0) L2 0,y;1= L}, and Y2 : {(_Ll Oi yaz, O, yb2) I L 2 O, ya.2 = ybz = Compute an ArrowDebreu equilibrium with the price of tabor—Ieisure time in period 1 equal
to 1. 7) (20 minutes) Consider the foiiowing economy with three consumers, A, B, and C, and three
commodities, 1, 2, and 3. C
A
X
X
X l —x“3x2’3,e =(3,0,0),
A 1 2 3 1 2 A ‘I (ll uB(x1, x2, x3) = xyzxg", eB =(0, 3,0), 1 1 =
1 2 3 Zin(i<1)+Eln(x3),ec (0,0,6). C
rs
X
X
x
v
I] This economy has a unique equilibrium allocation and that allocation maximizes a welfare
function that is a weighted sum of the utilities ofthe consumers. if the weights are normalized
so that the weight on the utility of consumer A is it, what is the derivative of the maximized
value of weifare with respect to eca at the equilibrium allocation? .egcontﬁrnywiti'. iixém; 8) (25 mjnmes) conSider the Samuegro‘ﬂi,Q‘leilappilflg'generations model with two commodities '
and _ if r. , :1... u(x x 01, 02, x x )2 n(xo1) + in(x62') + in(x11) + 2in(x12) and 11’ 11 e=(e 01'602'911’312):(1’1'0'0)’ where, for n = 1 and 2, xon is the consumption of good n in youth, x1n is the consumption of good n in old age and so on. Calculate a stationary"spotiprice equilibrium with interest rate r, where
r 2 0 and the price of good 1 is 1. Be sure to calculate the price of good 2 and the tax as a
function of r. (This should be a somewhat hard problem.) 9) (25 minutes) Consider the Diamond model with
f(K, i.) = 2 KL and
u0(x) = u1(x) = x. Compute a stationary spot price equilibrium for this modei with interest rate r such that r > —1 and with the price of output equal to one. Be sure to compute the tax, T(r), and government
debt, G(r). Hint: The formulas vary depending on the vaiue of r. n (13‘ 10) (20 minutes) Corn farmers in the state of iowa can buy actuarialiy fair insurance against
hail. That is, the insurance premium‘each farmerpays equals the expected value of the
insurance compensation for iosses. All the corn farmers in the state take fuil advantage of this
insurance. It is proposed that the State government prevent hail storms altogether by seeding
thunderstorms with dry ice. An argument against this proposal is that the seeding program is
not worth a cent, because the farmers are already fully insured against hail storms. Is this
argument valid? Defend your view with the most rigorous argument you can make. forthis'  tan: titateggpvtrﬂdfhent pts'worit ha i ...
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 DonaldBrown
 Pareto optimal allocation, Truman Bewley, edgeworth box economy, unique equilibrium allocation

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