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Unformatted text preview: Fail 2009 Truman Bewley Economics 359a Answers to Homework #19
(Due Thursday, November 19) 9) We know that the aliocation of a stationary spot price equiiibrium in the Samuelson modei is
Pareto optimal if the sum of its sequence of equilibrium prices is summable when discounted to
period 0 at market interest rates. This fact might lead you to suspect that stationary
equilibrium aliocations wouid be Pareto optimal if some asset were present that yielded a fixed
positive return indefinitely, for then the price of the asset would be finite and equal to the
discounted present value of its future returns. The foliowing problem tests this idea. Consider a Samuelson modei with one commodity in every period and where each
consumer has endowment(eo, e1) 5 Rf and utility function u: Rf e R. Suppose that there is an asset that yields i units of the singlecommodity in every period, where i > 0. This problem concerns stationary equitibria, ( x0, x1, P, r, G, T), for this modei, where P = 1 and G = T = O. The feasibility condition for the model is —x_+:se+e+i. (9.112)
D 1 0 1 Assume that P = i and T = 0 , so that there are no taxes and the budget constraint is x + 1 se+ 1 . (9.113) Assume that the utiiity function is strictly increasing and that e = 0. Because e1 = 0, the
1 young cannot have negative savings. That is, they cannot borrow against future income. They
can only invest, lend, or buy government debt. Recail that in a spot price equilibrium the
interest rate r is that paid by government to holders of its debt. This rate cannot be less than or
equal to t, for if it were, buyers of government debt would be paid nothing or a negative
quantity in the next period and so they would never buy the debt, that is, lend to the government.
The rate r cannot be infinite either, because the government coutd not pay such a rate.
Therefore, it may be assumed that —1 < r < oo. a) Show that if a stationary equilibrium with no taxes exists, then its interest rate is
positive, so that its aiiocation is Pareto optimal. b) Calculate a stationary spot price equilibrium with no taxes when u(x,x) = n(x) + n(x)ande =1.
0 ‘E O 1 0 The equilibrium wili depend on the asset yield i. 0) Make a drawing showing the set of feasible stationary altocations and the budget set of a consumer in a stationary equilibrium with no taxes and when eo = i = 1. d) Prove that if so = 0, then no stationary spot price equilibrium without taxes exists. Answer: a) Since the utittty function is increasing, the budget constraint holds with equality.
Since s1 = 0, it foltows that Hence x1=(t+r)(eo—xo). (A) Because —1 < r < co, (1 + r)"1 S 0, so that 20. CD

X
ll 1+r Because the P t > 0, the feasibility inequality holds with equality, so that x+x=e+i.
0 10 If we substitute equation (A) into this equation and cancel like terms, we obtain the equation r(eo— X0) = i. Since i > 0 and e0— :0 2 0, it follows that t‘ > 0. Since the utitity function is strictly increasing, it is locally nonsatiated with respect to
x0 and x1 separately. Hence corollary 9.15 (in section 9.3) implies that the stationary equilibrium atlocation is Pareto optimat if r > 0.
Answer: b) The equations that determine the equilibrium allocation are x+x=e+e+i
O 1 0 t (1 + r) au(xo, x1) = au(x0, x1) 6x 6x
1 0 On substituting the actuat vatues for the endowment and utility function, these equations become 2 mama '7" A tiny mmnAwwxmmwmtwwm»yxwwttwmwﬁmwwwwmﬂﬂww . ”Human“ nutsWt:wmmmwrwumwmmtmm acrhstm‘mmwmmm My:,mTNWMbmmw‘w‘amwlt‘wﬁtnh)’NwéNSHC‘l w’ Hence and so x=1+iandx=ﬂ.i__rl_(_1_ii_).
° 2+r ‘ 2+r On substituting these equations into the budget equation x
x+ 1 =1,
0 1+r
weobtain
1+i+t+i=1’ 2+r 2+r so that r = 2i. Hence In summary, the equilibrium is ((x , X ).P, r,G.T) = ([1. 1+2i].1,2i,0,0].
° 1 2 2 Answer: c) If i = 1, then :0 = 1/2, ~x—1 = 3/2, and r = 2. The appropriate diagram is below. The heavy line is the set of feasible stationary allocations. The shaded triangle is the budget set. The point x = (1/2, 3/2) is the equilibrium allocation. The endowment is inside the
feasibiiity line because of the earnings of i = 1 from the asset. 1mm immmwm Wmmmwvwmm ', ammwrwwstsmmmawsmewww . ., i
3} Since x 20, x120, and (t + r)‘1 > 0, it foliows that x O = x = 0. Since i units of the
1 0
commodity are avaitable in every period and i > 0, there is excess supply, and so the price P
must be 0. Since the price of the singie commodity cannot be 0 in equilibrium, there exists no
equilibrium. mittamew;m7mcﬂi:WWTWVWMWKWWWWWWWWQWWQWWWWWWMZWWM‘ﬁKMiﬁm‘ﬂﬂwﬁ mv,u‘vMW'vi~"3x?i"aimE13a1‘damimeNW‘WWWVkaﬁWz‘ﬁMWWW wwwwmwmw _ ‘ mmcwmmmmwwmmwmwmw,.«mvw,w ...
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