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Unformatted text preview: Fall 2009 Truman Bewley
Economics 350a
Answers to Homework #9
(Due Thursday, November 12) 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(x x1) = log x0+ (0.5)log x1. 01 a) Compute formulas for a stationary spot price equilibrium, (x x1, P, r, G,T),
with nonnegative interest rate r and with P = ‘l. 0! b) Show on a diagram ail feasible stationary allocations and those allocations that
are Pareto optimal. Show in similar diagrams the allocations and budget sets for the .equilibria with interest rates r = 0 and r = 1. indicate the tax payments on the diagrams
for each of these equilibria. Answer: a) To find the stationary equilibrium allocation, we must solve the equations au(x,x) au(x,x)
0 1 0 1 (1+r) 1 0 The first equation is the first order condition for utility 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 1+r
2 _1_ ‘_
x x The sotution of these equations is x: 2 andx=1+r.
3+r 13+r Letting the price be P = t, the tax T satisfies the budget equation E
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it awmtneawwmwmmmmmo amtt:er 3+r 3+r 3+r It foilows that the government debt is 6:1: 1 .
r 3+r in summary, the equilibrium is ((x.?),P,r,G,T):[[2 .t+r}1,r. 1. r .
01 3+r3+r 3+r3+r Answer: b) The feasible stationary allocations are the points on the line segment connecting the points at distance 1 from the origin on the coordinate axes. The set of Pareto optimal atlocations
are the heavy part of this line. x 1 1 Pareto Optima
1/3 Feasibie Stationary
Aliocations
0 2/3 t ' x
o
2 warev:heme:wetwat“m'MwywasramwWAWﬁﬁwkﬂ’mkammaﬁWuwmmm’imamwwiﬁvﬁ‘ﬁwiWewtwﬁﬁmwﬁﬁimﬂﬁmﬁw‘1‘ W“ cm7timxmwwwwwimnw'wmwwwwmwm a 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. %
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D The next diagram portrays the stationary equilibrium with interest rate 1. The budget set is shaded and the equitibrium 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. a MWwmwﬁmﬂwmmﬁﬁﬁﬂ f/2 1/2 3/4 1 x Probtem: 2) Consider a Samuelson model with one commodity in each period and where the
endowment of every consumer is e = (1, 0). Let the utility function of every consumer be u(x,x)=x+2x.
01 D 1 a) Draw a diagram that shows the set of feasible stationary allocations, the endowment,
and sampte indifference curves. b) indicate on a second copy of the diagram which of the feasible stationary altccations
are Pareto optimat. c) Compute a stationary spot price equilibrium, (:0, :1, P, r, G,T), for this economy with positive interest rate r and with P = 1. (3) State a social welfare maximization probtem that is solved by the allocation of the
stationary spot price equilibrium that you just found. Answer: a) t Feasible Atiocations :2 E
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E Pareto Optimai Allocation Feasibie Allocations Warm w wﬂmwrre; 0 Answer: o) The allocation is the only Pareto optimai allocation, namely, (:0, x1) = (0, 1). The government debt, G, satisﬁes the equation amxmmmmwawwwwwwwmt‘WWMmw (1 + r)G= Px1. SinceP=1andx1=1, The tax is T=rG= r.
1+r in summary, the equilibrium is ((x , x ),P, r, G,T) =((0,1),t, r, 1/(1 + r), r/(1+ r)) O 1 Answer: d) (t + r)2x11 + )3“ + r)“[xm+ 2(1 + r)"xn]. Problem: 7) Consider a Samuelson 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 u(x,x)= min(2x +x,x +2x).
0 1 0 1 D 1 Compute all the stationary spot price equilibria, (Z), :1, P, r, G,T), in which P = r = 1. Answer: !t we set 2x0 + x1 equal to x x0. It follows that the
0 diagram for this example is as beiow. The allocations of stationary spot price equilibria with interest rate 1 are shown as a heavy tine. If (x0, x1) = (x , 1 — x ) is such an allocation, then
0 O the corresponding government debt is G(xo) = x /( 1 + r) = (1 — x0) l2. Since the interest rate
1 + 2x1, we obtain the equation x1 1/2 O 1/2 1 x is 1, the tax is the same as the government debt. in summary, the set of stationary spot price
equilibria is PM
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