macro question

Description

Question 2. (110 points) Consider the following closed economy.

Households. The economy is populated by a representative household

with  identical individuals. Each individual is endowed with one unit of

time. The household maximizes lifetime utility

U(0) =

Z

1

0

et[log c (t) +  log (1  l (t))] dt, ,  > 0

2

where  is the individual discount rate,  is preference for leisure and c is

consumption per capita. The household faces the flow budget constraint

˙a = ra + (1  ) wl + G/  c,

where a is assets holding, r is the rate of return on assets (the interest rate),

w is the wage rate, l is the fraction of time that each individual allocates

to work,  is the tax rate on labor income, and G/ is a lump sum transfer

from the government.

Firms. The economy is also populated by N production firms that sell a

homogenous good whose price is normalized to one. The good can be either

consumed or invested. Firms have access to the production technology

Yi = (Ki) (TiLi)1 , 0 <  < 1, i = 1, …,N

where Ki is capital, Li is labor and Ti is labor-augmenting technology described

by the relation

Ti = Zi + 

XN

j6=i

Zj , 0 <  < 1

where Zj is knowledge accumulated by firm j and  is a parameter governing

spillovers across firms. Capital accumulates according to

˙K

i = Ii.

For simplicity we assume that capital does not depreciate and that knowledge

accumulates as a by-product of investment,

˙Z

i = ˙Ki,

so that after suitably normalizing terms we can write Zi = Ki. Finally, we

assume that N is su¢ciently large that each firm in equilibrium commands

a negligible market share so that we can focus on a competitive equilibrium

where increasing returns are treated as external to the firm (i.e., firms take

Ti as given).

Aggregation. Sincewehave N firms, we need to specify the following

aggregation rules:

Y =

XN

i=1

Yi; K =

XN

i=1

Ki; L =

XN

i=1

Li; I =

XN

i=1

Ii; V =

XN

i=1

Vi,

3

that define aggregate output, capital, employment, investment and stock

market value (think of the household as holding shares of a fully diversified

equity fund whose price per share is V ).

Government. Thegovernmentcannotborrowandsatisfies thebudget

constraint

wL = G.

In other words, we assume that the government sets tax rates and rebates

in a lump-sum fashion the revenues to the household.

Answer the following questions.

1. Write down the Hamiltonian for the household’s problem and derive

the Euler equation. Interpret the Euler equation as an equation characterizing

the reservation after-tax rate of return on assets demanded

by savers. What are the e§ects of tax rates on this rate of return?

2. Write down the Hamiltonian for the firm and derive the equations

characterizing the behavior of the firm. Interpret carefully the equation

characterizing the after-tax rate of return on capital generated by

firms. What are the e§ects of tax rates on this rate of return?

3. Define the variable x  C/K, where C = c is aggregate consumption.

Show that the equilibrium of the labor market yields aggregate

employment L as a downward sloping function of x. Explain why the

relation is downward sloping. Next, show that the Euler equation for

saving and the resource constraint of the economy yield the di§erential

equation

x˙

x

= x  (1  )



1 +  (N  1)

N

L(x)

1

 .

Use this equation to discuss the equilibrium dynamics of this economy.

Is the equilibrium trajectory unique? What is the equilibrium value

of aggregate employment?

4. Show that the equilibrium just discussed yields

log c (t) +  log (1  l (t)) = u0 + gt,

where u0 and g are expressions that depend only on the fundamentals.

Next, show that

U(0) =

1





u0 +

g





.

4

This is a handy expression for evaluating the welfare e§ects of policies.

Interpret it carefully. In particular, observe that

dU (0)

du0

=

1



and

dU (0)

dg

=

1

2 ,

so that

dU (0)

dg

=

1



dU (0)

du0

.

Notice that we typically think of  = .02 so that in practice the welfare

gain from a given dg is 200 times (!) the welfare gain from an equal

du0.

5. Assume an unanticipated, immediate, permanent increase in . What

happens to welfare in this economy? Note that according to the characterization

above, you need to show what happens to both u0 and

g.