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Unformatted text preview: Fall 2006 Truman Bewtey
Economics 3 55a Maketgp‘Final 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 equai the points for each of them. The total of the
times is 220. A total of 140 points wilt count as a grade of 100, so that your grade on the exam
will be the totai number of points awarded times (100/140). Hence your grade on this exam
could be as high as 157.t4. i wiil notbe. verygenerous with partial credit, so try to answer
correctly the questions you do. 7 ‘ I " ’ ' ' 1) (20 minutes) Consider an economy ((U 8) (Y).J (9)E ) i l I t l 1:1 ij i=1 i=1 where, for all i, u_: FtN a Ft and e e Ft” and, for ail j, YCR”. Define for this economy + i + j + a) a feasible allocation, ,raiilu ujf EEO “(JO tug} VlWxn‘uildn b) a Pareto optimal allocation, and c) a welfare function. 2) (15 minutes) State formally a theorem that says that every Pareto optimal allocation
maximizes a weighted sum of the consumers' utiiity functions. Be sure to state all the
assumptions. " ~  3) (15 minutes) Give an example of an Edgeworth box economy, ((uA, eA), (LIB, e B,)) with continuous and quasi—concave utility functions and of a feasible allocation, (I— , x B,) such that
A x A >> 0 and x B >> 0, such that the economy “and! allocation do not satisfy the second welfare
i":(il¥§.3,i1i}Iiiii VUJI Lit“... an, theorem A good drawing is enough. 4) (25 minutes) Consider the following,Edgreworthxbox economy. x x =minx x =
UA( 1’ 2) ( 1' 2)' L1:3(x1’M2) 2x1+x2’ eA= (2, 0),andeB= (0, 1). a) Show the set of Pareto optimal alfocations in a box diagram. b) Compute _a general equilibrium Suchihﬁtiifhié1.39"?“ the prices is one. Show 1the 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 and as such that aA = 1 and the equilibrium allocation maximizes the weEfare function au(x ,x)+au(x ,x )'
AA A1A2 BB B1 BZ over all feasible allocations ((x , x ), (x , x )).
A1 A2 B1 82 4. if. 5) (10 minutes) Consider the following insurance model with two states, a and b, and two
consumers, A and B, and one commodity in each state.
u(x x)=1x“5+2x”5
A a’ b ”5' a ‘5' b '
u(x X)=1X1’5+2X1’5
B a’ b 3 a 3 b ’ mt}: stir'5: i! e =(12, 0),andeB=(0,6_). A Compute an ArrowDebreu equilibrium such that the price of a contingent claim on one unit of
the commodity in state a is 1. “(WWWmmomrxnruwwm w~rmmmmmvw¢w "1r mamw ' «w; e m“ i .m A? «mam, we; mwwemewt. 6) (30 minutes) Consider the foilowingtnsurance model with two commodities, t and 2, two
consumers, A and B, and two states, a and b. u(x,x,x,x) ifxx +l/xx,
Aa1a1bib2 2 3182 2 b1b2 l(xw 4 x) '+ ix“,
B a1 a1 b1 b2 2 a1 32 2 hi C
PS
X
x
x
X
It e=(e .e ,e ,e )=(0.1.O.1)tand
A Aai Aa2 Abi Ab2 e ,e )=(1,0,1,0). B 331 332’ Bb1 Bb2 Compute an ArrowDebreu equilibrium." (This'smiold be a hard problem.) 7) (10 minutes) Let u(x,x,x)=3x +2x +x2’3,
A1 2 3 1 2 3 x“2 + 2x + 3x Li X X X
B( 1I 2I 3) 1 2 3’ uc(x1, x2, x3) = In( x1) + 2ln( x2) + 33n( x3).
Let
Wei, e2, e3) = x ER3 1116:: x E§a[uA(xA) + uB(xB) + uC(xG)]
A +' B +' C C s.t.x +x +x s(e,e,e).
A B C 1 2 3 Compute a subgradtent for V at the point (91, 92’ ea) ,7: (5, 5, 5). 8) (20 minutes) Consider the Samuetson overlapping generations model with one commodity
and with u(x , x) = —e"xe—.3_e‘2x1 and (e.,..e;)ﬂ=(3/2 3/2)
0 1 2 O] f7 ‘74 i," '_ a) Draw a diagram showing the set of feasible stationary attocations and on the diagram indicate
which of these are Pareto optimai. In addition, give a precise formula for the set of Pareto
optimal stationary allocations. b) Define a stationary spot price equilibrium such that the endowment allocation,
(x0, x1) = (e0, 91) = (3/2, 3/2), is theequilibrium allocation. Be sure to find the equilibrium interest rate. 9) (40 minutes) Consider the Diamond model with f(K, L) = min(2K, L) waitZion{cuterlagéiitrta:
and u0(x) = u1(x) =25. a) Compute a stationary spot price equilibrium for this modet with interest rate r such that r > —t and with the price of output equal to one. Be sure to compute the tax, T(r), and government debt, G(r). (Hint: Rememberﬁthat]equiiibrium profits are zero when returns to
scale in production are constant.) 5? ' b) Calculate the equilibrium ailocaticn when r = —0.5. 0) Demonstrate that the equilibrium allocation is not Pareto optimal if r = —O.5. d) Write down a welfare maximization probtem that is solved by the equitibrium allocation it
r > O. , teat eétitiiis‘tri’irtf:”r6? {1‘43“ iii Iiu‘l I"{'it\._'t1 Uh”: f ﬁW—WﬁiﬁWﬁﬂw‘ikWﬁWﬁ wmmmm v:wtWNW:mmtwwmwwtmmﬁm ntmnwumsw :mrvamwrwmrmw News" 10) (20 minutes) Let 3 = ((u, ei):1) be a pure trade economy (i.e., one without production)
i = such that, for all i, ul: RthN a R and Lil haethe _tc_rm’_ui(xo, x, , xN) = x + v(x , , xN), __ ‘ 1 o i 1
where x 6 Pi and v: Ft" —» R is continUou's; stridtlyjconcave, and strictly increasing. Assume
G i +
that, for all i, e =Oande >0, forn=1,2, ....,N.
__ _ é
a) Show that if (( x , ...., x ), P) is an equilibrium for 6', then the equilibrium
aliocation ( x , , x ) soives the problem i
1 I 2
1
max )3 u(x)
(x1.....,xi) i=1 ' '
I I
st 2 x = 2 e
i=1 i i=1 ' b) Show that all equilibria have the same ailocation. =4; ...
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