Problem - Consider a simple two-period overlapping generations model. The size of t e population is constant and equal to N for all period. Each person is endowed when young with w units of the single consumption good when young and units when old. Agents have preferences defined over their lifetime consumption:
U(c1t; c2t) = ln c1t + β ln c2t.
At the initial date t = 1, there are N old people, endowed with y > 0 units of the consumption good at date 1. These are also endowed with L units of what is referred to in macro as a 'Lucas Tree'. This tree bears fruit d > 0 each period (think of this as d units of the consumption good at date t). Old people consume d (along with y) and sell their land to the young in exchange for the consumption good at date 1. The price of a tree in terms of the consumption good at date t is Pt. This sort of exchange occurs each period, as long has Lucas Trees have value. You may assume trees bear the same amount of fruit (d) each period.
A. Obtain the saving function for the agent in this economy.
B. Write down the market-clearing condition for trees for each period.
C. In this economy, all the saving of the young will be in the form of a Lucas Tree, if it has value at date t. What assumption(s) do we need to make (on the primitives, β, w, y, L, N, d) to ensure there is an equilibrium with Pt = P* > 0 for all dates t. (this question may be a little messy - if you can't get it, at least answer E.)
D. Can this economy support equilibria where Lucas Trees are exchanged across generations and that display a price sequence for trees with Pt ≠ P* at any date t? Why or why not?
E. Find P* in the case where β = .8, w = 14, y = 2, L = 120, N = 100.