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Unformatted text preview: Fali 2008 ‘ Truman Bewley Economics 155a Answers to Homework #9
(Due Thursday, November 13) Problem: 1) Consider a Samuelson model in which consumers are endowed with one unit of good
in youth and none in old age and where their utility functions are u(x0, x1) = log xtJ + (0.5)log x1. a) Compute formulas for a stationary spot price equilibrium, (:0, x
with non—negative interest rate r and with P = 1. P: rr GsT)s 11 b) Show on a diagram all feasible stationary ailocations and those allocations that
are Pareto optimal. Show in similar diagrams the allocations and budget sets for the
equiiibria with interest rates r = 0 and r = 1. indicate the tax payments on the diagrams
for each of these equiiibria. Answer: a) To find the stationary equilibrium allocation, we must solve the equations (1+r)au(x0,x1) = 6X 6X
1 0 cubic, x1) The first equation is the first order condition for utiiity maximization over the budget set. The second equation is the feasibility condition. . On substituting for the utility function u and the
endowment e, these equations become ‘l+r
2 .L _1_
x x 0 1 The solution of these equations is x: 2 andx=1+r.
° 3+r 13+r Letting the price be P = i, the tax T satisfies the budget equation ii
6; a
a
31’ HWWMMWW wruwew ammwwmmw .é
1
§
i,
?
. g i
i
i
f
E x+ 1:1—T
°1+r
Therefore
.X _
01+r 3+r 3+r 3+r It foilows that the government debt is 6:1: 1 .
r 3+r In summary, the equilibrium is ((;!;)IP!rIGIT):[{ 2 l1+r]l1ir! 1 i r 
0 1 3 + r 3 + r 3 + r 3 + r
Answer: to) The feasible stationary atlocations are the points on the line segment connecting the points at distance 1 from the origén on the coordinate axes, The set of Pareto optimal allocations
are theheavy part of this line. x 1 1 Pareto Optima
1/3 Feasible Stationary
Allocations
0 2/3 1 x
o The next diagram portrays the stationary equilibrium with interest rate zero. The
budget set is shaded and the equilibrium allocation is x = (2/3, 1/3). There is no tax. 1/3 2f3 1 x
0 The next diagram portrays the stationary equilibrium'with interest rate 1. The budget set is ’ shaded and the equilibrium allocation is x = (1/2, 1/2). The tax is 1/4 and can be thought of
as the distance from 3/4 to 1 on the horizontal axis. ' E
s Problem: 3) Consider a Samuelson model with one commodity in each period and where each
consumer is endowed with one unit of the commodity in youth and none in old age. The utility
function of each consumer is u(x,x)=24[x_+24];_.
01 0 1 a) Compute a stationary spot price equitibriurn, ( x :1, P, r, G,T), with r: P = 1. o! b) Compute a stationary spot price equilibrium, ( x
r _= 0. o, 71, P, r, G, T), with P =1 and c) Show that the allocation of the equilibrium of part (a) is Pareto optimal. d) Compute the utility of a typical consumer in each of the equilibria of parts (a) and (b). e) Why is the allocation of the equilibrium of part (a) not Pareto dominated by that of
part (b)? f) Show the equilibrium allocations of parts (a) and (b) in a twodimensional diagram. Answer:  first compute a stationary spot price equilibrium with interest rate r and with P = i.
The equilibrium allocation satisfies the equations aux,x aux,x
(O1) (01) (t+r) 6x 6x
1 o which implies that (1+r)Jx:=J;1_ and hence x =(1+r)2x.
1 0 Substituting this equation into the feasibility equation, we obtain x +(1+r)2x =1,
0 0 so that (1+ r)2 x: 1 ___________.
2+2r+r2 —_andx=
0 2+2r+r2 1 Government debt, G, satisfies the equation _ 1+r G: 1: 71 : .
2+2r+r2 t+r The tax is then r(1+r) T=rG=____.___,_____.
2+2r+r2 In summary, the equilibrium is ((x , x), r, P,G,T) 0 (1+ r)2 = 1 ’ 1+: r(1+r)
2+2r+r2 2+2r+r2 lrl1l~—————————!———
2+2r+r2 2+2r+r2 Answer: a) On substituting r = 1 into the above formula, we obtain ((xo. 7,), P, r, 6,1“) ll ((115,4/5), 1, 1,215,2/5). Answer: b)_On substituting r = 0 into the above formula, we obtain ((x0, 71), P, r, G,T) =((1/2,1/2), 1,0,1/2,0). Am: c) The ailocation of the equilibrium of part a is Pareto optimal, because it maximizes
over the set of feasible allocations a social weifare function that gives positive weight to every
consumer. This social welfare function is E+t§02t(2ﬁ+2ﬁ). Answer: d) if r = 1, the utility level of a consumer is 2P+2ﬁ=e£.
5 5 5 If r = 0, the utility level of a consumer is 2J1+2JI=2JE
2 2 Notice that 25>§f5~,
r 5 since 10 2 >643, because if we square both sides of this inequality; we find that
200 > 36(5) = 180. Answer: e) In order to switch from the stationary allocation (x0, x1) = (1/5, 4/5) to the
stationary allocation (x0, x1) = (1/2, 1 l2), the oEd person at the time of the switch must give up 4/5 — 1/2 = 3/10 units of consumption. Answer: f) In the figure that follows, the feasibility line is heavy and the budget time for the
equilibrium with r = 1 is the steeper tins of normal thickness. The vector (1, 1/2) is perpendicular to the budget line. 4/5 1/2 O 1/5 1/2 1 X 0 Problem: 7) Consider a Samuetson model with one commodity in each period, where each
consumer is endowed with one unit of the commodity in youth and none in old age and has utility function ll u(x,x) min(2x +x,x +2x).
0 1 1 0 t 0
Compute all the stationary spot price equilibria, (:0, Y1, P, r, G,T), in which P = r = 1. Answer: It we set 2x + x equal to x + 2x , we obtain the equation x = x . It foilows that the
1 0 1 0 1 0
diagram for this example is as below. The allocations of stationary spot price equilibria with interest rate 1 are shown as a heavy line. If (x , x ) = (x , 1 —x ) is such an allocation, then
0 1 0 0 the corresponding government debt is G(x ) = x1/(1 + r) = (1 — x0) l2. Since the interest rate
0 mww ‘ :«mwrm jawW  'Waxmwmumm V memeW, 1/2 1/2 1 x is 1, the tax is the same as the government debt. In summary, the set of stationary spot price
equiiibria is {((x , x ),P, r,G,T) =((xo,1——x_o),1,1,(1—;0)l2,(1—;—0)/2) 10s; 31/2}. 0 1 ...
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