Make_Up_Final_B,155a,_06

Make_Up_Final_B,155a,_06 - Fall 2006 Truman Bewley...

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Unformatted text preview: Fall 2006 Truman Bewley Economics 155a Make-up 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 quasi-concave utility functions and with a feastbte allocation, (3? , .x-B), 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 andaB-suchxthat 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 Arrow-Debreu 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‘ns-iu‘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 Arrow-Debreu 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 r-s 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: tit-ateggpvtrfldfhent pts'worit ha i ...
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Make_Up_Final_B,155a,_06 - Fall 2006 Truman Bewley...

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