Suppose that we have a two-period economy with a representative consumer

whose utility is given by u(c; l) = log(c) in each period. The consumer is

endowed with capital K in period 1 and sells it to the rm for a price pk in the

rst period. They invest in second period capital K0 in period 1 and likewise sell

it for the price p0

k in the second period. They are endowed with h units of labor

to split between leisure and labor as they see t. The government taxes capital

income in each period and uses it to nance (G;G0) of goods consumption.

Consumers and the government both have access to the market interest rate r.

Suppose that = 1. Then the consumer's rst period budget constraint is given

by:

c + K0 + s = w(h l) + pkK(1 )

The rm has standard Cobb-Douglas production and purchases capital and

labor from the consumer. The rst period resource constraint is:

c + G + K0 = z(Kd)(Nd)1

Please complete the following:

i) Derive the consumer's lifetime budget constraint.

ii) Dene a competitive equilibrium in this environment.

iii) Solve for a series of equations summarizing the CE.

iv) What is the Euler Equation from the CE?

v) What is the social planner's problem? Find the associated Euler Equation

and compare it to (iv). What does this suggest about the optimality of

the CE?

vi) From the planner's perspective, why would a labor income tax be preferred

to a capital income tax in this environment?

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