Consider an economy with N consumers, a representative firm, and a government.

Each consumer has an endowment of one unit of time and receives all the income of the economy Y , but she also has to pay lump-sum taxes T to the government. The consumers maximize their utility u(C) = C when they choose to consume a fraction 1s of their income, and invest the remaining s fraction in new capital. Assume 0 < s < 1.

Since all consumers are the same we can aggregate to a single representative consumer.

Aggregate consumption and investment are given by: C = (1 s) (Y T) I = s (Y T)

Aggregate capital accumulates according to the following equation: K' = (1 ) K + I

where 0 < 1

The firm operates a technology that uses capital, labor and public goods (as roads and other infrastructure) to produce.

Total output is given by: Y = zGKN 1 where 0 < , < 1.

The firm must pay for labor and capital, but its use of public goods (G) is free of charge. To be clear the firm does not choose how much G to demand.

The government must finance investment in public goods using lump sum taxes T. Public goods each period are given by G = gN, where 0 < g < 1 is a constant.

Population grows at a constant rate n > 0. So that N' = (1 + n) N.

(a) What is output per-worker in this economy?

(b) Find the per-worker capital accumulation equation.

(c) Show in a diagram how to determine the steady state level of per-worker capital k .

(d) Prove that there exists a positive steady state level of per-worker capital k . Is the steady state unique?

(e) Show in a diagram or diagrams the effect of an increase in g in this economy. Derive the formula for k g and y g .

(f) How does it relate to a change in z in the model covered in class? What does this tell you about the role of the government in the economy?