Instructor: Marc-André Letendre

Deadline: September 29, 1998 (in class)

Question 1

Let the generation size be N(t) = 2 for all t ³ 0. Let the total endowment be Y(t) = 2 for all t ³ 1. Let preferences be represented by the utility function u(c_t^h(t), c_t^h(t+1)) = [c_t^h(t)]^0.5+[c_t^h(t+1)]^0.5.

(a) Show that the following allocation is feasible.

Generation 0

c₀¹(1) = 0.5

c₀²(1) = 0.5

Generation 1

c₁¹(1) = 0.25        c₁¹(2) = 0.75

c₁²(1) = 0.75        c₁²(2) = 0.25

Generations t, for t ³ 2

c_t¹(t) = 0.5        c_t¹(t+1) = 0.5

c_t²(t) = 0.5        c_t²(t+1) = 0.5

(b) Is this allocation efficient?

(c) Find another allocation Pareto superior to this allocation.

Question 2

Consider an endowment economy where generation size is N(t) = 100 for all t ³ 0, where member h of generation t receives endowment [w_t^h(t),w_t^h(t+1)] and has the utility function
u_t^h = ln(c_t^h(t))+bln(c_t^h(t+1)) where b = 0.95. We want to solve for the time t competitive equilibrium interest rate and the time t consumption allocation.

(a) Write down the budget constraint member h of generation t faces when young, the budget constraint he faces when old and derive his lifetime budget constraint.

(b) Maximize member h of generation t utility subject to her budget constraint. Solve for the consumption function when young and the savings function.

(c) Write down the time t good market clearing condition and use the old budget constraint to derive the competitive equilibrium condition S_t(r(t)) = 0.

In the economy we are working with, there are three types of people. The three types differ with respect to their endowments.

(d) Given these endowments, compute the aggregate savings function S_t(r(t)).

(e) Impose the competitive equilibrium condition to solve for the time t equilibrium interest rate. Then compute the time t consumption allocation.

Note: The notation is borrowed from McCandless with Wallace and is identical to what we have used in class. If you are confused with notation, please visit the course web page where a table of notation is available.

Econ 320A web page

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