Consider an overlapping generations model in which agents live two periods. Each agent inelastically supplies one unit of labour services when young. For an agent born in period t, utility is given by:
Ut = c1t1-θ1-θ + (1/(1+ρ)) c2t+1 1-θ
where c1t is consumption when young, c2t+1 is consumption when old, 0<ρ<1 is the rate of time preference, and θ>0 is a parameter. The agent’s budget constraint in each period of life are given by:
st + c1t = wt (1 – τt),
c2t+1 = (1 + rt+1) st
where st is savings when young, wt is the wage rate, τt is the tax rate on wages, and rt+1. The production function is:
Yt = A Ktα Lt1-α
where A is a fixed parameter representing technology, Kt is the aggregate capital stock, and Lt is the number of workers, equivalently the population of the young generation. To simplify, assume that population is fixed over time. The government spends all of its tax revenue each period such that:
Gt = Lt wt τt
Derive the Euler equation relating consumption when old to consumption when young as a function of the interest rate. Interpret θ – 1.
B.
Now, set θ = 1, and solve for the consumption in both periods and for savings as a function of the wage rate and the interest rate. Does saving depend on the interest rate? Explain in terms of income and substitution effects.
C.
Assume perfectly competitive markets, write expression for the real wage rate and the interest rate. Use these results to solve for the reduced form for the steady-state stock of capital.
D.
How does increase in government spending affect the capital stock? Compare the effects in the OLG model with those of representative agent Ramsey model. Explain.
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