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Unformatted text preview: Fall 2006 Truman Bewley
Economics 155a = . Answersﬂto Final Examination Question: 1) (20 minutes) Define a competitive equilibrium with transfer payments for an
economy a r i=3 j i=1 i] i=1 1:1 ((u_.e,)1 ,(Yw ,(0.)* H ), _. where, for alt i, u: R” —> R and e e R” and, for all j, VCR”.  _+ l + Answer: A competitive or Walrasian eguitibrium for an economy <9’=((U.ae.).1»("'.).J :(BlI ") I l =1 ] =1 ii i=1, j=1 consists of (i, —y_, p), where 1) (I, 7) is a feasibie atlocation,
2) p is a price vector (i.e., p is an Nvectcr such that p > 0), 3) for an i, Y e nip).
1 J _ r~ , ; :...».2;r.§:.5. V}, . = 1:
a bl.)iii§JLeiii\d\_J'L_§z,..1..;.:.42.. 4) for alt i, 7. e at p), I... I J—
5)fora!ln,p=0,it£x <):e'+):y'.
H In i=1 in i 1 i" i=1 , ,  ' is: an ,
Question: 2) (20 minutes) State the second welfare theorem. Be sure to Include ail
assumptions. Answer: Assume that the economy 8’: ( ( ui, e)lI1, (Y): 1, (8”)II1J ) is such that
= = = Ij=1 1) for all i, u_ is continuous, locally non—satiated, and quasi—concave and
I 2) for all j, Y is convex.
F If ( I, 7) is a Pareto optimal allocation such that Y '7 Elli???“ 9‘ it”??? ’” vector, p, and an lvector of transfer payments, 1:, such that (Y, —y—, p, '5) is an equilibrium
with transfer payments. l'>> O, for all i, then there exists a price Question: 3) (15 minutes) Give an example of an Edgeworth box economy with concave utility
functions that has infinitely many equilibrium_§all'ocations. Answer: Many examples would do. Here is one.
uA(x1, x2) = uB(x1, x2) = min(x1,x2). eA=(1,0). eB=(0,1). Every allocation on the diagonal of».the;:Eidgeyyor_th box is an equilibrium allocation, as is
illustrated in the diagram below. ‘ ‘ I f I
I 0
i l B
I I I
l l ‘__ __
__ _____I ____________ __ i _
I' F—
l .I
..... +_..___~—_.._ 4~_.— ~
I i I
I [ I
l r I
I ' q f. I .
.. ;_ .......  r I
l I I
l I f I p
l : g i
l I l l
l I l I
i .E. In, g .' l
l I". " I‘ ' I
l I‘ I
I I I I
_ . . _ _ _ _ __.:.___r._.+__ 4..
l ' F
I I I
I l l e
I l l
0 I I I.
A w {I Question: 4) (35 minutes) Consider the};followingflidgeworth box economy. 2 i _ =
uA(x1, x2) mn(x1, x2), uB(x1, x2)“ ‘ x1x2, eA= (2, O),ancleB =(0,1). a) Show the set of Pareto optimal allocations in a box diagram. b) Compute_a general equilibrium such.that'thelsu'mfof 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. 6) Compute nonnegative numbers a and a8 such that the equiiibrium allocation maximizes the
A welfare function au(x
A A X +aux
A .A2) Ba( .x) 1 B1 32 over all feasible allocations ((x , x ), (x , x )).
A1 A2 Bi 32 Answer: 4a) The set of Pareto optima are the heavy line in the diagram below. p = (1/4, 3/4) 4b) The equilibrium is ( XA, XE, p) = ((1/2'L71/2)’ (3,2, 1,2), (11,4, 3/4)) and is shown
in the above diagram. ‘ 40) Since VA(w) = w, it toliows thatRA = 1. Since _ _ 3 Question: 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. = 1 3
uA(xa, xb) Iln( xa) + 7;_ln( xb), ll u(x, x) _3_in(x) + lin(x),
B a b 4 a 4 r. ande =(1,1) =e.
A B Compute an ArrowDebreu equiiibrium such that the price of a contingent claim on one unit of
the commodity in state a is 1. Answer: Because the economy is symmetric with respect to states and consumers and the utility
functions are CobbDouglas, you should be able to compute the equilibrium in your head. The
symmetry implies that the two prices are equai, so that they may be set equat to one. That is, p
= (1, 1). Then the income of each consumer is 2. Since consumer A spends a quarter of his or
her income on contingent ctaims for state a, an=1/2. B f 'b'l't, y east lly ‘X )1
xBa=3/2. By the symmetry of the economy, 4 ‘. ' Ln'iiia‘it S‘tti’C’h iliatii’iié 1r t‘si' ' bi = 112 and xAb = 3/2. in summary, the equilibrium is (xA, x8, p) =((1/2,3/2), (3/2, 1/2), (1, 1)). Question: 6) (30 minutes) Consider the‘foi'loWing" insurance model with two commodities, 1 and
2, two consumers, A and B, and two states, a and b. u(x ,x,x ,x) —l’xx.+_1_lx x,
Aa1 a1 b1 b2 2 a132 2 b1b2 _1_(x +x ) +lx ,
B a1 a1 b1 b2 2 a1 a C
15
X
X
X
X
v
ll e =(e ,e ,e ,e )=(0,1,0,1),and A Aa1 A82 Ab1 Ab2 e ,e )=(1,0,1,0). s Bat Ba2’ Bb1 Bb2
Compute an ArrowDebreu equilibrium. (This shouid be a hard problem.) Answer: Because person B does not have any desire for commodity 2 in state b, Because of the linearity of person Bis utility Tiunctiionu,” we can guess that
pat = pa2 = ps1 :1/2‘ Because of the symmetry and strict concavity of person A’s utility function and since pan = pee,
we may conclude that = = r ’ A
xa1 xa‘2 z. ,,,1.._I,.U)y_i,,’,. Therefore, person A’s utility maximization problem in the equilibrium becomes Setting the derivative of the Lagrangian rwith'reispeotgto 2 equal to zero, we see that 2t=1. A Setting the derivative of the Lagrangian with respect to xb1 equait to zero, we see that ‘ Therefore xb1=1. The budget condition than implies that x = X = 1 1 ' :stsztgt:qistiilWIli'i [ESPECt tU A. t K':K,%L,i€,{"t w v w ' ~ v :=/' 7;
Ki. “mewtam wwwmwcmwmww«Mamime The symmetry of person A’s utility "functienww‘ithn respect to xb1 and xb2 and the fact that
xm = xAb2 imply that pb1 = pbz. In conclusion, the equilibrium prices are pat =pa2: pm : pb2:1’
and the equilibrium is (XA: x3! = 0! 1: r .i I 1!, ' Question: 7) (10 minutes) Let u(x,x,x)=3x+2x+x2’3, A1 2 3 1 2 3
u(x,x,x)=x"2+2x+3x, B 1 2 3 1 2 3
u(x,x,x)=n(x)+2n(xr)r4l3n(w>€).‘ C1 2 3 1 2 3 Let
Ve,e,e = max u x +u x +u x
(12 a) XERSXERBHRSIJA) 8(5) 0(0)]
A é'B +ICC s.t.x+x+xs(e,e,e).
A :3 c _12 3., Compute a subgradient for V at the point (e1, e2, e3) = (5, 5, 5) . Answer: We can guess from the linear terms in the utility functions of persons A and B that
DV(5, 5, 5) = (3, 2, 3). In order to see that this answer is valid, we have to verify that at the
optimum, x > O, that either x > 0 or x > O, and that xi33 > 0. A1 A2 « 52
In order to verify that xm > O,’ we set the partial derivatives of uB and uC with respect to
x1 equal to 3 in order to calculate xBi and x61.
aUB( X3) = 1 = 3,
6X 2 x
1 B1
30 3!“; (KW; _, UV}: {,1 x = J_
31 6 7 in: the ,....v%.~.‘._..._..ﬂ.~...“wu‘nune. and”. we and hence
x = 1—
B1 3 6 u rrrrr h 5?
Similarly, “4%) = 1_ z 3, 6X X 1 C1 so that 0‘ 3 Therefore x =5—x —x =51_—1_>0. A1 B1 C1 H I x
in order to verify that either x > 0 or x > 0, set the partial derivative of uC with
A2 32
respect to x2 equal to 2.
6%”8 = A = 2,
6x x
2 02
so that
x = i.
02 Sincex +x =5 —x =5~1 >0, itfollowsthateitherx >Oorx >0.
A2 32 C2 A2 32 In order to verify that x > 0. set the partial derivatives of u and u with respect to 82 A 0
x3 equal to 3.
alJA( XA) = 2 = 3,
ax 3x113 I
3 A3 so that so that II
A X
03 Therefore x =5—x —x =5—[ET—1>0. Question: 8) (20 minutes) Consider the Samuelson overlapping generations model with one
commodity and with u(x , x) =—e"‘o—1_e‘2"1 and (e, e) = (3/2, 3/2).
0 1 2 D 1 a) Draw a diagram showing the set of feasible stationary atlocations and on the diagram indicate
which of these are Pareto optimal. 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, e) = (3/2, 3/2), is the equilibrium allocation. Be sure to find the equilibrium interest rate. Answer: 8a) We must solve the problem max [—e'xu — .Le’zxt] x,x 2 0 1 at x + x = 3.  is; 0 1 ' ' A in additiort,zgivti .g E. o. t mm»: earmtcsmwmtw By substitution, this problem becomesfﬁi‘ max [—e'xo — lam"0’1. X 2 0 The first order condition for this problem is e—xo = e—2(3 —xo) which implies that ~x =—6+2x.
0 0 Hence and so x =2andx :1.
0 1 The requested diagram is as follows. Pareto Optima 3/2   0 3/2 2 3 x 10 . w..” .,....._.rk.w...._.,.m. M N . . . .._ A. The set of Pareto optima is 2 _ .
{(xo,x1) ER+ X0+X1—3,Xo $2}. Answer: 8b) In order to catculate the interest rate r, use the equation (1 + r) ___(.__9__..11_ : (—02—,
ax ax
1 0 which becomes
~2x —x (1 + r)e s=eo. Substituting 3/2 for X0 and x1, we obtai'h‘i'ilir. 7 7' , ,. (1+ r)e“a = e'“. Therefore
1 + r = e3"2 and hence ‘
r: e‘”2 ~15 3.4817. _ Because the endowment is the consumption allocation, there are no taxes and there is no
government debt. Question: 9) (40 minutes) Consider the Diamond model with t(K, L) = min(2K, L)
and uo(x) = u1(x) = 24/; a) Compute a stationary spot price equiiibrium 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). (liint: Remember that equilibrium profits are zero when returns to
scaie in production are constant.)
b) Calculate the equitibrium allocation when r = —0.5. 0) Demonstrate that the equilibrium allocation is not Pareto optimai if r = —0.5. wisr'isumﬁtiori 'él‘lricat? ‘ 11 J;
I:
if; E :5
a; d) Write down a welfare maximizationfprohteni:thatis sotved by the equilibrium altooation if
r>0. ' Answer: 9a) Since L = 1, the production function is y = f(K, 1) = min(2K, 1). Therefore in
any stationary equitibrium, capitat = K = 1/2 and output = y = 1. Totai consumption is,
therefore, x0 + X1 = y  K = 1 — 1/2 = 1 /27‘.,.__;.;:«. .. ' Using the equation du(xj) du(o.5 —x1)
dx dx dx (1 + r) du(x0) so that 1+r= 1 E [0.5 —x1 I Hence i“.r’\'vli\ .» .. :;,:
Huh”: x.) , * I_\i\, .. (1+ r) [0.5 —x1 1 Continuing the aigebra, we see that (1 + r)2(0.5 —x1)= x1
and so [(1 + r)2 + 11x = WW? ’2)? and hence x =lMandx=ii—. ; 1 2(t+r)2+1 O 2(1+r)2+t By assumption, the price of output is P”: 1"; Constant returns to scale implies that
Py=W+(1+r)K, where W is the wage. Therefore 12 W:1_1+r=2—1—r=i—r_ Notice that this formula makes sense only. ifrrSZ.,1:.5‘E:‘Therefore, the equiiibrium is defined only if
—1 < rs 1. The tax is T=W—x — X: =2~1i+r'—‘1____L_.___—_L____L:1____.
0 1+r 2 2(1+r)2+3 2(1+r)2+1
Government debt is
6:3: 1_2_1+ r_1_;_*_1'___—"1_L,
r r 2 2(1+r)2+1 2(1+r)2+1 ifr¢0. lfr=0,then G=x :J._(_1__i_'22_.
1 2(1+ r)2+1 Answer: 9b) (x , x1) = (0.4, 0.1). 0 Answer: 90) The stationary allocation (x0, x) = (07.25.720.25) is feasible and Pareto dominates the
'8. .725 'i ‘ _' Z '1 . ‘
stationary allocation (X0, x1) = (0.4, 0.1 ), because the old person of period zero does better under that allocation (0.25, 0.25) and the iifetime utiiity of any other person under the
allocation (0.25, 0.25) is
1 = 2 _ + 2 [.1 2,
4 4
which exceeds the lifetime utiiity,
2 f... + 2 L1— : _6_..
u 1 O 1 0 4’1 0 of the same person under the allocation (0.4, 0.1). In order to check the last statement, notice
that  .213 if and only if M>3, if and only if
10 > 9, which is true. Answer: Sci)
°° —t
(21:: (1 + r)2 x4:1 + a)“ + r) (2 IX“) + 2 Ix”)
s.t. K+x +x smin(2K ,L),fort>0,
t 10 1—1.1 _t1 It K +x +x smin(1, L),and
0 00 —1,1 t 0 SL151,fOJ’tZO. Question: 10) (20 minutes) Let 8 = ((u, ei)II1) be a pure trade economy (i.e., one without production) such that, for all i, sul:Rth”»R and u has the form u(x , x, , x) = x + v(x , , x ), where x e R and v: Ft“ ~ Ft is continuous, strictly
i i 1 N ' 0 1 N 0 0 I I
ccncave and strictly increasing. Assume that, for al! i, ei0 = 0 and e_ > 0, for n = t, 2, , N.
a) Show that if “:1, , :1), P) is an equilibrium for 8, then the equilibrium allocation ( x1, , x ) solves the probiem
I .  I
(x1,....,xl) [=1
I I
st Ex = )je
I=I i =1 b) Show that all equilibria have the same allocation. 14 ' ‘1nmwmﬁswmuwwmwmw:wwmmomt m mmmmwssm WWW. we“ » mmwmmwmm .l M Answer: 10a) We may normalize the equilibriumso that P0 = 1. In an equilibrium, for each i, x] = (x , x1,...., XN) solves the problem
‘0 i i max [x + v.(x1,,xN)} xeﬁszJornzt 0
o n N
s.t.x + [P
0 n N .. r.
x :9 + ZPe*..>"“~~'
":1 :1 i0 n=1n'ln The Lagrangian for this problem is N
.4 = x0 + vl(x1, , xN) —7Li[x0 + "gianni. The first order condition of maximizing this Lagrangian implies that A: 1, so that the Lagrangian is .L’_=x +V(x,....,x)—[xo+ ENIP _ x].
r 0 t 1 N n=1 n n The Lagrangian for the problem max [1 u.(x) (x1,....,xl) i=1 ' E t
st Xx = Xe
=1 i =1i
is
.2 I 1 _N.,P
= x+vx, .,x — xiiip x
i=2?! 10 i( it iN)] §1 i0+ "g1 n H1
Since
I
.2: 2.3,
i=1 ' and an equilibrium allocation maximizesfﬂl, it follows that an equilibrium allocation maxxmlzes
i r .2. Since an equilibrium allocation is also feasible, it follows from the sufficiency part of the
KuhnTucker theorem that an equilibrium allocation solves the problem Answer: 10b) Since the equilibrium allocation solves the problem max i u_( x .) (x......x,) i=1 ' max Zv(x_, .....,x)
(x x ,x .. x)i= ' ‘1 '
11 IN 25 IN
I I
st£x=£e,forn=1,.....,l\l
3:15“ i=1in Since the functions v_ are strictly concave, the solution of this probiem is unique. Therefore the
 components of the allocation corresponding to commodities 1 through N are unique. It is not
true that the components corresponding to commodity zero are unique. 16 ...
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