Ans_to_Final_Fall_07

Ans_to_Final_Fall_07 - Fall 2007 Truman Bewley Economics...

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Unformatted text preview: Fall 2007 Truman Bewley Economics 1§§a Answers to Final Exam Thursday, December 20, 2007 2:00 - 5:30 WLH 208 gm: 1) (10 minutes) Define a competitive equilibrium with transfer payments for an exchange economy, that is, an economy with consumers but no firms. M: A competitive equilibrium with transfer payments consists of ( x, y , p), where 1) x is a feasible allocation 2) peRNandp>0 3)'c=('c,'c,....,'r) eR' I 4) for ail i, Z solves the problem max u‘( x) )(EFlNI + s.t. p.x s pe—‘q I | l I 5) for all n, pnzo, if fixin< Ee in' i = 1 I: 1 Question: 2) (10 minutes) Define local nonsatiation of a utility function. Answer: A utility function u : Rf —'Fl is said to be locally non-garages: if for every x in R“ and every positive number a, there exists an x’ in R” such that llx' - xll < 8 and u(x’) > u(x). Q_u_es_tiqa: 3) (10 minutes) Give an example to show why it is necessary to assume that utility functions are locally nonsatiated in 1the first welfare theorem. Answer: An example is the following Edgeworth box economy. e = (2,0) uA(x1, x2) : x1x 2 A e3: (0, 2) uB(x1, x2) = 0. An equilibrium is (XA’ x5! =((1! 1):(1! 1)! (1r 1)), yet the equilibrium allocation is Pareto dominated by the allocation (XA: x3) = 2)! (oi Quesjm: 4) (10 minutes) Give an example to show why competitive equilibrium may not exist if the economy does not satisfy Walras' law. Answer: An example is the following Edgeworth box economy. eA= (1, 0) uA(x1, x2) = —x1—x2 eB=(0, 1) uB(x1, x2) = -x, —x2. The demand of both consumers at any price vector is (O, 0), so that there is excess supply oi both goods at any price vector. This is impossibie in equilibrium, since by the definition of equilibrium prices are zero if there is excess supply, and in equilibrium some price has to be positive. Question: 5) (15 minutes) Consider the foliowing Edgeworth box economy. e A ll ll (1, 0). uA{x1, x2), |n(x1) + [n(x2). e B ll (0,1). uB(x1, x2) = ln(x1) + 4ln(x2). a) Compute for this economy a competitive equilibrium in which the sum of the prices is 1. b) Compute welfare weights WA and WE for the two consumers such that at the equilibrium allocation the marginal value, as measured by the welfare function wu(x ,x)+wu(x ,x ), AA A1 A2 BB B1 BE of additionai endowments of commodities 1 and 2 equals the equilibrium prices p1 and pa, respectively. Answer: 5a) Because the utility function is Cobb-Douglas, we know that X = X :1 1 4. A1 ? 32 E From ieasibiiity, it follows that wwwmawwwvwwix; Wm val/Ame. WWW we.th X = X = 1 1_ B1? A2? Let p be the price of the first commodity, so that the price of the second commodity is 1 —— p. From the budget equation for consumer A, we have that lp+l(1—p)=p. 2 5 so that 5p + 2(1— p)=10p and hence p = 2/7 and 1 — p = 5/7. Hence, the equilibrium is "‘ — 1 1 1 4 2 5 x r x v : w! — 3 —! — l *l _ ' (A Sp) [12 5112 5117 7]) 33%;: 5b) We need to compute the consumers’ marginai utilities of unit of account, RA and 7L3. These may be computed as follows. 6U 1/2,1/5 6U 1/2,4/5 ._i‘(___)_=?up and L;=?tp, ax A1 B1 1 ‘l sothat ;=7t(2/7) and ;=M2/7), 1/2 A 1/2 B so that AA: 7L3 = 7 and hencewA= 11M :11? =1/7l.B= wB. Question: 6) (25 minutes) Consider the following Edgeworth box economy. e A ll (10,0). uA(x1, x2) = min(2x1, x2). e B (0,5). uB(x1, x2) = min(x1, 3x2). a) Show accurately the set of Pareto optimal allocations for this economy in a box diagram. b) Show accurately the utility possibility frontier for this economy in a diagram. c) Compute a competitive equilibrium for this economy in which the sum of the prices is 1. 3 d) Show the competitive equilibrium allocation of part c as a point E in the diagram of part a. e) Show the pair of utility levels of the equilibrium allocation of part c as a point E in the diagram of part b. t) For what pairs of welfare weights (wA, wB) does the equilibrium allocation maximize the welfare function wu(x A X +WUX X A 'A2) BB( ' ) A! B! 32 among alt feasible altocations? Meg: 6a) The Pareto optimal allocations are the shaded areas in the diagram betow. The point the two shaded areas have in common may be calculated using the following equations. x =2x x =3x x +x :10 x +x :5 A2 A1 B1 :32 At 31 A2 32 From these, we see that 10—): =x =3x =3(5—x)=3(5—2x )=15—6x. A A2 A1 A! 1 BI 82 so that 5x :5 A1 and hence X =1 A1 and so Answer: 6b) The utility possibiiity frontier is shown as the heavy line in the next diagram. (1, 2) V B 10 E 9 (3.1) 0 2 5 VA 343%: 60) The equilibrium allocation is clearly the point the two shaded areas have in common in the diagram of part a. This allocation is (XA!XB) = 2)! (9! The price vector may be found from the budget equation for consumer A as follows. p+2(1—p)=10p, where p is the price of commodity 1 and 1 — p is the price of commodity 2. From this equation, amemwmrmJacarmuw it follows immediately that p = 2/11. Hence the equiiibrtum is ((1,2), (9,3),(2/11, 9111)). A >< x E it Answer: 6d and 6e) The points E are shown in the above diagrams. M; St) The arrows in the diagram of part b indicate the range of directions for the weight vectors. The set of weight vectors is uw,w)eamw ¢m1_s_as3r A B + A 2 w Question: 7) (15 minutes) Consider a firm that produces one output (y) from capitat (K) and labor (L) according to the production function y = 3(KL)“3. The price of output is 1 and the wage of one unit of labor is 100 and the firm has one million (1,000,000) units of capital. The firm hires iabor so as to maximize its profit. How much would be the firm willing to pay per unit for 8 units of extra capital, where a is arbitrarily small and positive? M: The quantity we are to calculate is the Kuhn-Tucker coefficient 7L for the maximization problem max [3(KL)“3 —100L] 1.20 s.t. K S 1,000,000. The Lagrangian for this problem is 3(KL)“a —100L -—?LK. The Lagrangian is maximized at K = 1,000,000 and at the levei of L that solves the constrained maximization problem. Hence we may find L and l by setting the two partial derivatives of the Lagrangian equal to zero at K = 1,000,000. Setting the derivative with respect to L equaE to zero, we find that 100 = 100 L2f3 and hence L = 1. Setting the partial derivative with respect to K equal to zero, we see that it: _§_SK“3 =(106)-2'3=1o-4. 8K K2106 gm: 8) (30 minutes) Consider the foliowing economy with two consumers, A and B, two firms, 2 and 3, and three commodities, 1, 2, and 3. The first commodity is iabor and the second and third are produced from labor. The endowments and utiiity functions of the consumers are as foliows. X2X3 . eA (40, 0, 0). uA(x1, x2, x3) ll eB (40, 0, 0). uB(x1, x2, x3) = xg’4xg’4. Notice that there is no utility for leisure. Firm 2 produces commodity 2 from labor L according to the production function y =3L, 2 2 where L is the amount of iabor used in the production of commodity 2. Firm 3 produces 2 commodity 3 from labor L according to the production function where L is the amount of iabor used in the production of commodity 3. 3 Find an equilibrium with transfer payments in which the price of iabor is 1 and consumer A consumes 6 units of commodity 3. Answer: By assumption, p1 = 1 . Because the production functions are homogeneous of degree 1, profits are zero. Therefore the nature of the production functions implies that p2 = 1/3 and p3 = 2. Let wA be the wealth of consumer A. Because consumer A’s utility function is Cobb- Douglas, we see that paxA3 = (1/2) WA, where an' is consumer A's consumption of good 3. Then 2an = (1/2)wA, so that wA = 4xA3. Since Xm = 6, by assumption, it follows that wA = 24. Therefore, using obvious notation —13—xA2 = gwA = 12 and so xA2 = 36. Similarly ~19:sz = fiwa = %(80 — WA) = 21(56) = 14, so that x!32 = 14. Also 2xBa = —:~wB = %(56) = 42, so that x = 21. B3 Therefore the output of good 2 is y2=XA2+x52=36+42=78, and Similarly y3=XA3+xBS=6+2t =27, and L3=2y3=54._ As a check on thelicalculations';we'see‘that‘ L2+ L3=26 +54 =80 =eA1 +931. Consumer As transfer payment, “CA, obeys the equation, wA : eA1 —’EA, so that TA = 9m —wA = 40 — 24 = 16. Therefore consumer B’s transfer payment is 1:3 2 4A = ——16. In summary, (xAI x3! (_L2! ygl 0): (—1—:3! 0! V3)! pr T) = ((0, 36, 6), (O, 42, 2?), (—26, '78, 0), (—54, 0, 27), (1, 1/3, 2), (16, ~16”. Question: 9) (35 minutes) There are two farmers, A and B, who transform iabor into food. Each is endowed with one unit of labor and nothing eise. There are two states of the world, 1 and 2,,and each occurs with probability 1/2., Eoogcproduction takesone period, so that iabor input. occursbefore the state of the werld is known? 'In‘state‘i, farmer A’s prodUction function is : |_ , yAI A where yA1 is farmer A’s output in state {and LA is farmer A’s input of labor. in state 2, farmer A's production function is "-352 = 2erneiar pavmehf'f'r obeys ’the‘eqtsafron; W? o ' »- 1' . F“: '1’ ”' ” with the obvious notation. In state 1, farmer B's production function is y =2L, 31' B and in state 2, farmer B’s production qutiorLjs , , .v": V} ,‘m"; \,' U UV “a: '1'» ,y :L, B2 B again with the obvious notation. The utiiity function ofeach farmer is ‘ - 1- r'irr-I-si it: eff-{i1 "ii ’7‘? " u(i?,x1, x2) :1? + 1311M + auteur Einstein 1 and L'Qis’rarmér A'é‘iihmstioi int-52% ' * where Bis the consumption of teisure and'xfigéand xL-ware the farmer’s consumption of food in states 1 and 2, respectively. a) Compute the input Qt'iabgr.otgeachjarmeh "h o'iifiéhgjioperateskinfisoiation with no market for insurance. b) Compute an Arrow-Debreu equilibriumwith the price .of labor equal to 1. c) Is the input of labor in part b greater-or lessthanthatin part a? Answer: a) The maximization problem for an isolatedfarmer is maxi‘i — - 2L " L20 2 The first order condition of the maximum is- ' 1 4/E_1 + _. 4 L 4E, _ which implies that " and hence “ 'fi‘ tr-.|_ V'pa‘rt'ia'? 1 6 1 6 ' Answer: b) Because the production functions are homogeneous of degree 1, profits are zero in equilibrium and therefore ' [I —A. p1+ 2p2 2p1 + p2 = 1, where p1 and p2 are the prices of food in statest and2, respectively. It fotlows that . max » ‘ -' 7 _ r, ,7 ' 'I20,x120.x220 72 1 2 s.t.i’+_1_x +lx :1. 31 3 2 From the symmetry of this problem, it is clear that X1: x2. Therefore the maximization problem may be written as «we ' max in? 3r 120.x20 s.t,f+_2__x=1. 3 1;: ‘rE 9- i {g a: g The Lagrangian for this problem is .BJr r '4‘? '+ ., 3 so that the first order conditions are 7L = 1 and r ‘ 1 2 24:3 and hence 5:2 4 and so ' x=i,£.=1—_2_x 16 3 n _L I [m lco ii _I. i colon " il 01 m 3 o. i- || "A i l'fi Ii In summary, the equilibrium is )1 (—2! 8 As a check, observe that food consumption: in2 e‘ither‘state'equais 9/8, Which is the same as food production in either state. ' 7 Answer: 0) The labor input of isolated tarmers 3+2/J; l 16’ is less than labor input of farmers in an Arrow-De’breu equilibrium, which is SIB, because 3+2 2 <6,since[§<3/2. Quefiflgm: 10) (35 ‘minutes)'Consider the "Dia‘nion'dfiiinoadel' With production function y = 2 JKL and utility function u(x)+u(x)=’x +’x. 0 0 1 1 0 1 > - ---:: agesei‘ve that food consumptionineithgémstateequals sis, WithliiS ijau -« l=iecaIl that in the Diamond modei, each consumer is endowed with one unit of labor in youth and none in old age. a) Calculate a stationary spot price equilibrium, (IOU), :1“), EU), Eh), P(r), W(r), G(r), T(r)), with P(r) = 1 and for each interest rate r exceeding —1. t2) @999???"E‘fiioi’hefiifi‘itevil-illufiiéigfifii'iiiixis?fifigflfijafétifligfiiiiimn‘a’iiiir[flififis’isafir an?! thatfljliyiunded socr'ai security Is introduce le' eeping the regular tax'at thebieiie'l‘i‘fil)‘. What will be the direction of the effect on the stationary spot price equiiibrium interest rate? 0) Suppose that we start with a stationaryzspot price equilibrium with interest rate r = 1 and that an arbitrarily small positive amount:ofifp‘a'y-asi‘y‘ouigd’social security is introduced while keeping the regular tax at the level T(‘l). rWhat Will be the direction of the effect on the stationary spot price equilibrium interest rate? ' Answer: a) The equation af(K, L) :1 + r aK becomes { —1-—=1 + r, sothat K=—_‘_. (1+ r)2 Outputis y=f(K,1) = 2 . 1+ r Output net of capital is y_K=fi_2r____1_.:1+2r (1+r)2 (1+r)2 (1+r)2' Theequation du1(x1) _ duo(x0) (1+ r) dx dx becomes 1+ r 2 1 2.]; zfl’ sothat (1+ nJZ = J; andhence x1 = (1 + r)2xo. The equation 13’ x +x1=y-K impiies that 1 + 2r (1+(1+r)2)xo=—————————-;. (1 + r) Therefore X _ 1 + 2r °_ (1+ r)2(2+2r + :2) and hence 1+2r X=—“—”—z- 2+2r+r 1 The equation (1+ r)(G+ K): x1 implies that x G: 1 1 2 1+r (1+r)(1+2r+r2) (1+r) WM: 1+3r+2r2—2—2r-—r2 (1+r)2(2+2r+r2) (1+r)2(2+2r+r2) Ii —1+r+ r2 (1 + r)2(2+2r + r2) Hence T~ rG— “r + r2+ r3 _ —r + r2+ r3 _ __.__.______—____—_____4 (1+ r)2(2+2r+ r2) 2+2r + r2+4r+4r2+2r3+2r2+2r3+ r _ —r+r2+r3 2+6r +7r2+4r3+ r4 FinaiEy 14 W=ai(K,t) 2R: 1 I aL i+r in summary, (de.mh),NU.PU).WU)JXQ,TUH 1+2? ' r _a1:£L;;eaai—-. 1 7(-i+r)2’ 1+r — (1 + r)2(2+2r + r2) ' (2 +‘2r + r2) —1+r+r2 ~r+f+r3 _u~ioa2w2riahiaiagoflege ‘ This equiiibrium exists if r > —1/2. If r s ~1/2, there is no output available for consumption. Answer: b) There will be no change in the interest rate. 7- Answer: 0) it is necessary to calculate the sign of JPN-high" “gig. cw;w clr 6H1) =__d_ —r+ r2+ r3 _=. 20(74)—1(3s) : 44 >0 Cir Cir 2+6r +7r2+4r3_+r4 r5} 400 400 Therefore a small amount of pay-as-you:go social security increases the interest rate. 'i (.3 p z = I r?‘ ‘1‘ r . ~7‘ ‘%'I_J“::'x§.,."_:.4_ . II: _I .. 7 yr :-> I ‘ Question: 11)'(30‘ minutes)ConsiderifSamiielsoiti mo’del Withgone cdmmodity In each period and where each consumer has endowment e = (e , 91) = (4, O) and utiiity function 0 u(x , x) = min (x +2x,2x +E-‘x-)"7'f 3‘” i 0 1 1 0 0 1 a) In a diagram, show the set of 'feasibie stationary allocations, the endowment allocation e, and a few indifferencecuryes;,gtkthe'iutili‘ty junction-ml ' b) Indicate on the diagram which allocations are Pareto'optimal. c) What stationary ailocation or allocations are part of a stationary spot price equilibrium with interest rate 0? - - - . . d) What stationary allocation or allocations are part of a stationary spot price equilibrium with interest rate r, where O < r < 1? = - a: {a} i-ém‘fijék'fi.55%;; in a. goof?- -_:f:::;:;{:. 400 , . . jiia‘; \A L,;A , « Mm;mfiirmwmmmv m w :é g “a i “W wwmmwwwwm «7" MW immwmwwumumm‘wm mainmz’m‘vt‘y e) What stationary allocation or allocati'omn‘smare"partof a stationary spot price equilibrium with interest rate r, where r = 1? " f) What stationaryjalgloc t' _ . . ‘ lloc t". 1“ ' fr- ' a’r’yhepotprice equilibrium with interest rate rt‘Where r V " ' - ' 9) Define a stationary spot price equitibrium with stationary allocation'x = (x0, x1) = (1, 3) and commodity price 1. You must calculate'the- intereet"_rate, the tax, and government debt. h) What stationary allocation or allocations are optimal with respect to the catching up criterion? ' i) What stalwart"attaeattsfiforan; ptitt‘tar‘amonfiE'Ztli‘téétsmle allocations with respect to the welfare function 2u(x o,x_11) + iil]u(x ,x)? '4, i=0 2 to H Answer: a) t - k‘i‘lli'ttté—EFV allam Horror atlocations {are stationary s:in tutti» artntttitiwwu': 1 4 2 Answer: b) The Pareto optimal allocations are indicated in the above diagram by a thick tine. Answer: Answer: Answer: 9) {(x , x) [x0 + x1: 4, 0 sxo $2.}. Answer: f) (x0, x1) = (0,4). Answer: g) r =1. (1 + r)G= x1, sothatZG= x1andhenceG= 3/2. T .= re, so mart;--tar2-:_-::Inassmmaryrgihé' ‘ x, r, P.G,T) = (1,3,1,1,3/2,3/2). (XO’ 1 Answer: h) (x0, x1) = (2, 2). Answer: E) {(x0, x1) « awn-s.“ ' r , .f'\£i:;‘svt:§., (2'1 r-x..'X-:)‘ ). is“; \‘3(¥L--1ir‘ir}érr;(“KiJ53 ...
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Ans_to_Final_Fall_07 - Fall 2007 Truman Bewley Economics...

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