Question

3. Consider the Diamond model with logarithmic utility (In(Ct)) and Cobb-Douglas production. Describe in words and using equations how the following affects kt+1 as a function of kt: (a) A fall in n. (b) A downward shift in the production function (that is, f(k) takes the form Bka, and B falls). (c) A rise in a

Answer 1

Diamond model

It is a present a canonical overlapping generation (OLG) model, like the one originally proposed by Diamond (1965), building on Samuelson (1958).

Features

• Two generations are alive at any point in time, the young (age1) and old (age2).
• This size of the young generation in period t is given by Nt=N0=t.
• Household work only in the first period of life, earning income Y1,t. They earn no income in the second period of life (Y2,t+1 = 0).
• They consume part of their first period income and save the rest to finance their consumption when old.
• Labour and capital markets are perfectly competitive and the aggregate production technology is CRS,Y = F(K,L).

Let's normalize everything by the period t young population Nt,writing normalised variable in lower case. Thus the per young capita aggregate production function becomes

f(kt) = F(Kt,Nt)/Nt = F(Kt/Nt,1).

The perfect competition assumption implies that wages and net interest rates are equal to the marginal products of labour and capital, respectively;

Wt = f(kt)-ktf`(kt),

rt = f`(kt).

To make further progress,we need to make specific assumptions about the utility function and the aggregate production function. Assume that utility is CRRA,u(•) = •^1-p/(1-p) and assume a Cobb Douglas aggregate production function F(K,L)= K^sum L^1-sum , f(k)=k^sum.

CRRA function

The domain of the CRRA function: From an economic point of view it is desirable that the domain of our utility functions include c=0. Starvation is a real life possibility.