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Unformatted text preview: Fall 2006 Truman Bewley
Economics 155a Final Exam ination
Tuesday, December 19, 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. ‘ '5 ’ " ' 1) (20 minutes) Define a competitive equilibrium with transfer payments for an economy ((u.e)1 .(trjlj=1.(6._)‘ £1). i i i=1 I] i=1 j: where, for all i, u_: Ft” a R and e e R” and, for all f, YICFiN. I + + j + is tri‘ni; equal ‘ ti“: 2) (20 minutes) State the second welfare theorem. Be sure to include all assumptions. 3) (15 minutes) Give an example of an Edgeworth box economy with concave utility functions
that has infinitely many equilibrium allocations: _ 4) (35 minutes) Consider the following Edgeworth box economy. uA(x1, x2) = mln(x1, x2), ué(xi.;>:'ixé)p;§gi:‘Ix1x2 I,
ex: (2, 0), andeB =(O, 1). a) Show the set of Pareto optimal allocations in a box diagram. b) Compute_a general equilibrium such that the sum of the prices is one. Show the equilibrium
allocation, x, in the box diagram. 0) Compute the marginal utilities of unit ofiacoount'of the two consumers in the equilibrium. d) Compute nonnegative numbers a and aB such that the equilibrium allocation maximizes the
A weifare function au(x ,x )+au(x ,x)
AA A1 A2 BB 81 B2 over all feasible allocations ((x , x ), (x , x )).
A1 A2 B: B2 5) (10 minutes) Consider the following insurance model with two states, a and b, and two
consumers, A and B, and one commodity ginie‘aCEt state: u(x, x) =__1_ln(x) +Eln(x),
A b 4 a 4 b a u(x,x) =£ln(x) +_1_ln(x),
B a b 4 a 4 b 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. 6) (30 minutes) Consider the followingiinsurzance consumers, A and B, and two states, a ahdlia’f _ u(x,x,x,x)=l’xx +i/xx, A a1 a1 b1 b2 2 a! 82 2 b1 b2 u(x',x,x,x) l(x +x)+_1_x,
8 a1 a1 b1 b2 2 a1 a2 2 b1 model with two commodities, 1 and 2, two II e = (e ,e ,e ,e ) = (0;?7'1EQO,'1)~,._éhd A A3? A32 Ab1 ADE e=(e ,e e ,e )=(1,0,i,0). B Bat Baz' Bb1 Bb2 Compute an ArrowDebreu equilibrium (This should be a hard probiem.) 7) (10 minutes) Let u(x,x,x)=3x +2x +x2’3,
A12 3 1 2 3 C
A
“X
M"
X
co
5.;
H xi’2+2x +3x,
1 2 3 E
O
rx
A X
I‘D—X
X
CA3
V
II in( x1) + 2ln(x2) + 3ln(x3). l;
it Let V(e,e,e)= max [u(x)+u(x)+u(x)]
1 2 a XERa,XER3.XEH3 :B' c C
A +5 + C G ' I t H" s';,~ J
s.t.x+x+xs(e e,e).’
C 2 3 i
A B 1 Compute a subgradient for V at the point (91, e2, e3) = (5, 5, 5) . 8) (20 minutes) Consider the Samuelson'ioveﬂapping generations model with one commodity
and with u(x , x) =~exo—_1e‘2x1 and (e,e) .= (3/2, 3/2).
0 1 2 0 1 a) Draw a diagram showing the set of feasible stationary altocations and on the diagram indicate
which of these are Pareto optimal. inﬂadditionygive 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, e1) = (3/2, 3/2), is the equilibrium allocation. Be sure to find the equilibrium interest rate. 9) (40 minutes) Consider the Diamond model with f(K, L) = min(2K, L) and uo(x) = u1(x) = 24;.
.. the), CV; ~~ \t3'.;":, t, i
a) Compute a stationary spot price equitibrium‘for this model 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, C(r). (Hint: Remember that equilibrium profits are zero when returns to
scale in production are constant.) in) Calculate the equilibrium ailocation when r = —0.5.
c) Demonstrate that the equilibrium aiiocation is not Pareto optimal if r = —0.5. d) Write down a welfare maximization problem that is solved by the equilibrium allocation if
r > 0. ' 10) (20 minutes) Let 3 : ((ul, e)ll 1) be a pure trade economy (i.e., one without production) such that, for all i, u}: RxFi” e Fl and u; has the form u(xo, x, , xN) = x + vl(x1, , xN), 1 0 where x e R and v: R“ ~ R is continuous, strictly concave, and strictly increasing. Assume 0 i + E; that, for all i, ei0 = 0 anolei > O, for n = 1, 2, , N. a) Show that if ((71, w{iiHS)”‘i's‘tfa’r't7equiiibrium for 8, then the equilibrium allocation ( x , ...., x) solves the. problem b) Show that all equilibria have the same allocation. 2,
i?
i:
if wthizmmw Maciuousz sizietlylca ‘ . _. ...
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