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Unformatted text preview: Faii 2009 Truman Bewley Economics 359a Homework #1
(Due Thursday, November 19) Do problem 9 on page 447 at the end of chapter 9. This problem should read as folloWs. 9) We know that the allocation of a stationary spot price equilibrium in the Samuelson model 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 allocations would 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 following problem tests this idea. Consider a Samuelson model with one commodity in every period and where each
consumer has endowment (e0, e1) 6 Rf and utility function u: Rf a Ft. Suppose that there is an asset that yields i units of the sigle commodity in every period, where'i > 0. This problem concerns stationary equitibria, ( x 0, x1, P, r, G, T), for this model, where P = 1 and G = T = 0. The feasibility condition for the model is :(‘+_X_Se+e+i. (9.112)
0 1 0 1 Assume that P = i and T = O , so that there are no taxes and the budget constraint is + 1 59+ 1 . (9.113) Assume that the utility 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. Recali 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 could 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 allocation is Pareto optimai. b) Calculate a stationary spot price equilibrium with no taxes when u(x,x) 0 1 in(x0) + |n(x1) ande =1. 0 The equilibrium will depend on the asset yield i. 0) Make a drawing showing the set of feasible stationary allocations and the budget set of a consumer in a stationary equilibrium with no taxes and when e = i = 1.
0 d) Prove that if eo = 0, then no stationary spot price equilibrium without taxes exists. mmwmmw .MWWZ WW. . i “5" Am ...
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