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MBFM III - Week 11 Tutorial Questions

The objective of this question is to examine our baseline model with augmented with a government budget constraint under contasting government policies.

In this two period OLG model, young individuals are endowed with e1  units of the endowment good and there are no old endowment goods. They can use it to consume when young or to save for old age. Endowments that are not consumed in youth can be convert into units of productive capital one-for-one or sold in return for money in a perfectly competitive goods market. Denote by kt  the amount of capital created by a young individual in period t. Then in period t + 1, this capital produces f(kt ) units of the consumption good for the old individual. Assume that the production function f(k) is strictly increasing in capital so that its marginal product of capital is everywhere positive, f\ (k) > 0 and it exhibits diminishing marginal product of capital, f\\ (k) < 0. For simplicity, capital fully depreciates after use in production so there is no capital left for consumption after use in production. Let the population be constant. Each period, N = 1 young people are born.

There is also a government. The government has to finance spending of g units of the consumption good each period, g < Ne1 . The government can either raise tax revenues through a proportional tax, Tt , on output production by the old or it can print money to finance expenditures.

The total supply of the consumption good is equal to the endowment of the young plus the production of the old. Aggregate demand for the consumption good is equal to the consumption of young and old, investment by the young and government expenditure.

With these assumptions we can write the individual’s problem as

max  {u(c1,t)+ βu(c2,t+1)}

subject to the constraints

Pt c1,t     = Pt e1 − Pt kt − M1,t

Pt+1c2,t+1     = (1 − Tt )Pt+1f(kt )+ M1,t .

1. Consider the steady state of the economy if the stock of money is constant over time, Mt  = M. Solve for the equations characterizing the steady state values of (π, T,k,m1 ). Starting at a steady state, if government expenditures were to be permanently increased, does the tax rate increase in the new steady state (with higher g)? How is the inflation rate a↵ected?

2. Consider the steady state of the economy if the tax rate is zero for all periods, Tt  = 0. Solve for the equations characterizing the steady state values of (π,k,m1 ). Starting at a steady state, if government expenditures were to be permanently increased, do real money balances rise and how does this rise in g a↵ect the inflation rate?