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Ans_to_Final_Fall_08

Ans_to_Final_Fall_08 - Fall 2008 Truman Bewley Economics...

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Unformatted text preview: Fall 2008 Truman Bewley Economics 155a Answers to Final Exam 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. Question; 1) (10 minutes) a) State Walras' law. b) Under what condition or conditions is Walras’ iaw is satisfied. Answer; a) Let p be any price vector. Let EA p) be the set of utility maximizing consumption bundles in consumer i's budget set, for i = 1, ..... , |. Let M p) be the set of utility maximizing l input-output vectors in firm j's set of profit maximizing input-output vectors, for j = t, , J. Then p. Edi—e; — ij_]=0. |=1 i 1 1 ii xi 6 tip), for all i, and yE e nj(p), for aii j. b) Walras’ iaw applies if every consumer’s utility function is locally non-satiated. Question: 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 input-output possibility set should be ciosed and convex and should contain the zero vector, each endowment vector should be strictly positive, and each utility function should be increasing and quasi-concave, that is, the set of points at ieast as desired as one point is convex. _ Answer : 2) Here is one possible example. There are two commodities, one consumer and one firm. The consumer's endowment is e (O, 2). The firm’s input-output possibility set is Y = {( y1, y2) | y1 s 0, y2 s 2 1i—y1 }. The consumer's utility function is defined by the equation x1+ x2, ifx1+ x2< 3,and u(x x) = i’ 2 ifo+X2_>.3. This Robinson. Crusoe example is portrayed in the diagram below. iimwvznw "with:muWigwam?Wmvwwwmmflm‘mxfim- « p=(1!1) O 1 2 3 x1 Suppose there is an equilibrium. Since the utility function is locally non-satiated, the equilibrium allocation is Pareto optimal. Since there is only one consumer, a Pareto optimal attocation must be optimai. It follows that the equilibrium consumption must be the point x = (1, 2), and the equilibrium input-output vector must be y = (—‘i, 2). For this input- output vector to be profit maximizing, the equiiibrium prices of both commodities must be equal, so that p = (1, 1) may be taken to be the equilibrium price vector. The consumer's budget set, therefore, must be {(x1, x2) 6 Fifi x1 + x2 53}. The utility maximizing point in this set is (x1, x2) = (0, 3), not the point (x1, x2) (1, 2). This contradiction proves that no equilibrium exists. wamaimwmrwawrrv lthwmm-«Mwuvmwmm Question: 3) (25 minutes) Consider the Edgeworth box economy with utility functions I! uA(x1, x2) ln(x1) + 2|n( x) 2 and Il u(x,x) B1 2 3|n( x!) + In( x2) and endowment vectors e and e , where e >> 0 and e >> 0. A B A 8 a) Find the allocation (xA, x5) that solves the problem mix [uA(xA) + u3( xa)} S.t.X +X =9 +9. A s A B The optimal values of x , x , x , and x are functions of e , e , e , ande . A1 A2 Bl 32 A1 A2 31 82 b) Find an equilibrium with transfer payments, (xA, x8, p, I), such that p = DW(eA+ as), where W is the vaiue function W(e) = max [uA(xA) + uB(xB)} X.X AB S.t.X +X =6 A B and DW(e) is the derivative of W at e. Ail components of the equilibrium are functions of the components of e and e . A B Answer: a) The maximization probiem may be written as max [ln(x ) + 2in(xA2) + 3|n(eA1+ 931TXA1) + in(eA2+e&—XA2)], x,x A1A2 A1 The first order conditions for this problem are 1 3 = ——————— and _ , Xm 8m + em — XA1 E 2 __ 1 E XAZ 6A2 + 932 "‘ XA2 Solving the first equation, we find that em ‘1" em xm = 4 Sotving the second equation, we find that 2(eA2 + e32) xA2 = ———§————. Therefore 3 e +e ‘ x81: Lif—EL and e +e A2 82 x =—— 32 3 b) The prices are those in an equilibrium with transfer payments in which the marginal utittty of unit of each consumer is 1. Therefore au x , x p1:M:j_:.—4_’and ax? xAi eA1 + eBi p = M = .2_ = __:3»_ 2 . 6x2 )(A2 eA2 + eE52 The transfer payment of consumer A is TA = p.(eA — xA) :[ 4 73 ){e _eA1+eEI1e _2(eA2+e|32)] ’ ' A! ’ A2 eA1 + eB1 6A2 + 932 4 3 :[ 4 3 )[3em_eat era—2852] 9A1 + e31 9A2 + e22 4 3 = 39A: _ ea: + eA2 _ 2932 9M + ei51 eA2 + e82 mewwmmmwwm , M W WWW-NMWW ' mmwir'rtwmmwwmmm _ em _ 36m 2832 _ eA2 1:3 2 4A _ —————-— + ———————. 6A1 + em 9A2 + ea: Question: 4) (30 minutes) Farmers A and B live on an island where they are isolated from the rest of the world. Farmer A has 200 acres of farm iand 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 long 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 iand she or he wants in short or long kernel seed. Only long kernel grain will sprout on farmer B’s land, where it produces nothing in dry weather and 8 busheis per acre in wet weather. The utility function of each farmer is u(x,x ) =lln(x) +lln(x ), D W 2 D 2 W where xDandx are grain consumption in dry and wet weather respectively. The two farmers W plan their economic life for each season by means of an Arrow-Debreu equilibrium. Compute such an equilibrium in which the price of a contingent claim on 1 bushei of grain in wet weather is 1. You should find the price of a contingent claim on 1 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 kernei seed, respectively. Answer: The fine from 400 on the vertical axis to 1600 on the horizontal axis in the diagram below is the output possibility frontier for farmer A, where yW andyD are output in wet and dry weather, respectiveiy. 0 400 800 1600 yw We may guess from this diagram that if the price of a bushel of grain contingent on wet weather is one, then the price of a bushel of grain contingent on dry weather is 4. With these prices, the weaith of farmer A in unit of account is 1600. Because of the Cobb-Douglas form of the utility function, we know that farmer A must consume 800 bushels of grain in wet weather and 200 in dry weather. This is the consumption point xA in the above diagram. Farmer B produces 800 bushets of grain in wet weather and none in dry weather. This farmer's 5 umrwmmww “tweenmurawwmwwmmomwwmwwnw E i 3 production point is the point yB in the diagram above. Therefore her or his wealth is 800, so that he or she consumes 400 bushels of grain in wet weather and 100 in dry weather. This is the consumption point xB in the diagram above. Farmer B produces 400 more bushels in wet weather than she or he consumers, so that consumer A must produce 400 less than the 800 bushels she or he consumers. Hence farmer A produces 400 bushels in wet weather and so produces 300 in dry weather. This 300 equals the 200 she or he consumes plus the 100 consumed by consumer B in dry weather. Farmer A devotes one quarter of her or his 200 acres, Le, 50 acres to long kernel seed. The other 150 acres are planted in short kernel seed. In summary, the equilibrium allocation is x=(x A = (800, 200) , AW’ XAD) yA = (yAw. yAD) = (400, 300) . xB = (wa, xBD) = (400, 100) , ya = (yaw. ya.) = (800, 0). The price vector is p=(I0W,|0D =(1,4). Farmer A devotes 50 acres to long kernei seed and 150 acres to short kernel seed. Farmer B devotes 100 acres to long kernet seed and no acres to short kernel seed. Question: 5) (30 minutes) Consider the Samuelson overlapping generations model with the endowment of each generation equat to e: (e , e) = (2, O) and with the utility function of each 0 . generation equal to u(x, x) = min(x +3x,3x + x). O 1 0 1 0 1 a) Draw in a single diagram the set of feasibie stationary allocations and the endowment and some indifference curves for a single generation. b) Compute all stationary spot price equilibria with the price of the singie good equal to 1 and with interest rate r, where r> ~1. The ailocation and tax or allocations and taxes depend on r. c) For what values of r is the stationary equilibrium allocatt0n(s) Pareto optimal? d) For which values of r does the stationary equitibrium atlocation(s) maximize a soctai welfare function and what is this function? vktamehmmww stratum:wwwwfimmmnuwwmwmm t‘mmwnwwmwmmwmwwwhtmr mmrwwmmw , -; i t t E i t t t . a Answer: a) 1 (3,1)or(1,1/(1+ 2)) Feasibility Jin’e 2 (1, 3) or (1, 1m — 2/3)) Indiif'e’rence curve 1 O 1 2 x o b) Case 1: —1 < r< —2/3. Then I = (2, O) = e, and T =0. This equilibrium is illustrated in the following diagram, where the budget line is thick. ’ (3, 1) or (1, 1/(1 + 2) ) (ll/’11“ + r)) (1, 3) or (1, 1/(1 — 213)) We“ wax-wei- ww w mmm MWmWwWinmwmmnwwanmWW “W a“ E weave-twee “ mum" w Mm Case 2: r = —2/3. in this case, ;=(_x-o, Z—YO), where 1 s 3:052. T = 2 — (Y0 + 3(2 — 1—0)) = —-4 + 2:0. This case is illustrated in the next diagram, where the budget line is again thick. X 1 (3, 1) or (1, 1/(1 + 2)) 2 (1, 3) or (1, mt — 2/3)) (1, mt -— 213)) 1 2 —~x 0 0 1 i 2 x O ., 0 Case 3: ~23 < r< 2. In this case, :r— = (1, 1) and T = 2 — (1 + t/(1+r)) = r/(1 + r). This case is illustrated in the next diagram, where two possible budget lines are shown as thick lines. (3, 1) 0r (1, 1/(1 + 2) ) ‘WTWWWWMM—wfirfiwmwfimVimtffihvk‘h‘rfilfi5‘7m7m‘lfi’zw mwmmww :é IE in? is § g t: gt é, ti r Case 4: r = 2. In this case, ; = (:0, Z—YD), where 0 5:031, and T = 2 - (xD+(2— x0)/3) = 4/3—(2/3) x again the budget line is thick. 0, This case is iliustrated in the next diagram, where wwawmm‘m m'wxmm WNW W' wwww W333; immwwwwwwwmtwmlwmtm Wmcwz-w (3, 1) or (1, 1/(1+ 2)) r I (1, 1/3)’,r’ (1, 3) or (1, 1/(1 — 2/3)) 0 2 X 0 Case 5: r> 2. In this case, 3: = (O, 2) and T = 2 — 2/(1 + r) = 2r/(1 + r). This case is iilustrated in the diagram below, where once again the budget line is thick. c) The stationary equilibrium allocations are Pareto optimal for r >_—2/§_._ At r = —2/3, oniy one equilibrium allocation is Pareto optimal, namely, the allocation ( x0, x1) = (1, 1). d) The stationary equilibrium allocations maximize a welfare function for r > 0. The weifare function is (1 + r)u(x_1'o, x—1,t) + 1230(1 + i’)_1U(Xw, X“), where it“ 0 is given and equals the value of :0 in the stationary equilibrium atiocation ( x 0, x1}. -1 Question: 6) (40 minutes) Consider a Diamond overlapping generations model with production function f( K, L) : 3K113L2r3 and where each generation has utility function u(xo) +u1(x1) =2Jx—D+2‘J;(:. 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 equiiibrium with r = 1 and that there is no social security. The government now introduces an infinitesimal amount of pay-as-you-go social security without changing the reguiar tax found in part a (for r = 1). b) Does the introduction of social security increase or reduce r? 0) Does the introduction of sociai security increase or reduce the stationary equilibrium capital stock? cl) Does the introduction of social security increase or reduce the stationary equilibrium amount of government debt? e) is the equiiibrium with r = 1 and with no social security Pareto optimal? Would a new equilibrium with a small but positive amount of sociai security be Pareto optimal? Answer: a) By the first order conditions for profit maximization, K’m : 1 + r, so that 10 m'u‘i/WWVREvNTmm‘v/X‘r‘tfivgrwhmore immwmwwwfiflmfimfimw “it-awmvwmmimmwwrw ‘X‘i‘iflWXfiWfiWfiMMWQ'WXWVWWWWWMMN statesman, «M‘s-«tibiw‘vixmw war-7 w» ”we-z» “+0” (1+r) 1+r Total output of the produced good is 1 11'2 3 :3 =__..._. y [1+r] 1+r Again by the first order conditions for profit maximization, the wage is W=2KH3=—2-—-—. Total output net of capital input is 3 i = 2+3r 1+r (1+r)1+r (1+r)in+r By the first order conditions for maximization of utifity over the budget set, it”;wwwwmmmmwWWram“;emewwmwwwiwwwmmmmwmmwrmdmmmmwmm W" ' Wm a u1(x1) 6u0(x0) “ t ”T = T' sothat 1+ r = 1 i: ii; andhenoe x =(1+ r)2x0. Substituting this last equation into the feasibility equation x +x1:y—-K, we find that 2+3r (1+r)41+r xo(t+(1+ r)2) = and hence « ywxnwmwwmww Meow-.WNW‘WM i w ,, who.» 9,. 2+3r 0 (1+ r)(2+2r+r2)4i1+ r and so (2+3r)(1+ r)2 = (2+3r)4/1+r' 1 (1+r)(2+2r+r2)q’1+r (2+2r+r2) The budget equation implies that the tax is 3:: 1% [5 E5 ii: 3 $3 :3 '3 3 3% 55 3% T5 a; 5?} e g '3 1:? V it ____—.__________—______.—_—— _L + "C C: + D 73 + N) -| + l" Iii .,.‘ fl TS + N "-1 + ‘39 v _|. + yéwa-W «am: aim—pm; m w" 2(1+r)(2+2r+r2)—- (2+3r) —(2+3r)(1+r) (1+r)(2+2r+r2)q/1+r 2(2+2r+ r2+2r+2r2+r3) —(2+3r)—(2+5r+3r2) (1+ r)(2+2r+r2)rJ1+r mwwwrw =2(2+4r+3r2+r3)-—(2+3r)—(2+5r+3r2) i (1+r)(2+2r+r2)th+r ‘ 4+8r +6r2+2r3—-2—3r —2—5r—3r2 (1+ r)(2+2r+r2)q/1+r 31'2+2r3 (1+ r)3’2(2 +2! + r2) I! l i wmwmm By the definition of government debt, G, ___£r_:_2__r2_____1__ (1+ r)3’2(2+2r+r2) (1+ r)” 3r +2r2—2—2r— r23 (‘i + r)3’2(2 +2r + r2) 12 —2+r+r2 (1 + r)3’2(2+2r+ r2). In summary, x: 2+3r ° (r+r)3’2(2+2r+r2)' x=(2+3r)41+r 1 (2+2r+r2) K=__1__.__ (1+r)3’2' w: 2 1+r’ 3r2+2r3 T=———-T—-———,and (1+r) (2+2r+r2) G —2+r+r2 = (1+ r)3’2(21—2r + {2). b) The direction of the effect of social security on t has the same sign as the derivative or the tax function with respect to r at r = 1. 511(1) d r3r2+2r3 dr Cir (1 + r)3’2(2 +2r + r2) (240 -155) J? (105)12—5(31/2) J? = 200 Therefore social security increases the interest rate r. 0) Because social security increases at, it decreases the capital stock, 01) Since r6 = TO + T1 + T = T , the introduction of social security does not change rG. Since social security increases r, it must decrease government debt, G.‘ e) The new equilibrium with social security would be Pareto optimal, since the interest rate would exceed 1 and therefore be positive and the equilibrium with social security has the same r=1 >0. 200 13 T? g 1% it E . gig it 3% g; f? I; ii a E t g E c if 3»: l allocation as an equilibrium with the same interest rate and no social security but just a single lump-sum tax paid by youth. WWW-mixWWWimmmmmwwwwwlmmkmwwvmwmmwwmwmmwxmmWmfiwmtmfimwwm 14 ...
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