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Macroeconomic Theory
Exam 2

Instructions: Write your answers to each question on separate paper. You must show your
work for all derivations and computations for full credit. The point value of each question is
indicated below. Partial credit is assigned based upon the completeness of the entire question
answered not on the proportion of sub-parts answered.

1. (36 points) General Equilibrium: Consider the representative household, who chooses
a path of consumption and leisure over an infinite horizon, {ct+s, lt+s}∞s=0, to maximize
the following objective function:

V =
∞∑
s=0

βsu(ct+s, lt+s)

where u(ct, lt) is a well-behaved utility function, and β is a discount factor. The household
faces the following real budget constraint each period:

at = (1 + rt)at−1 + wtnt − ct − Tt

where at is real wealth, rt is the real interest rate, wt is the real wage rate, nt is labor
supply, and Tt is a lump-sum tax. The household also faces a unitary time endowment
which holds each period:

1 = lt + nt

Also consider the representative firm, who chooses a path of capital and labor input over
an infinite horizon, {kt+1+s, nt+s}∞s=0 to maximize the following real profit function:

Prof =
∞∑
s=0

(
1

1 + rt+s

)s (
f(kt+s, nt+s) − invt+s − wt+snt+s

)
where f(kt, nt) is a well-behaved production function, rt is the real interest rate, wt is the
real wage rate, and k0 is given. For any period t, net investment is defined as:

invt = kt+1 − (1 − δ)kt

where δ is the rate of capital depreciation.

Finally, each period the government purchases an amount of real goods and services equal
to real wage income tax revenue:

gt = Tt

so that government savings is always zero.

(a) Derive the household’s intertemporal and intratemporal optimality conditions in
terms of the general utility function u(ct, lt).

(b) Derive the firm’s intertemporal and intratemporal optimality conditions in terms of
the general production function f(kt, nt).

(c) Using the optimality conditions obtained from parts (a) and (b), derive the equilib-
rium conditions for the financial market, labor market, and goods market.

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(d) Set up the Social Planner’s optimization problem, and use the sequential Lagrangian
to derive the economy’s intertemporal and intratemporal optimality conditions.

(e) Explain whether or not the First Welfare Theorem holds in this scenario, and what
the result implies for the efficiency of the decentralized scenario.

2. (20 points) Neoclassical Growth Model: Consider the two main equations for the
Neoclassical Growth Model with exogenous labor:

∂u/∂ct
β∂u/∂ct+1

=
∂f

∂kt+1
+ (1 − δ)

f(kt, Ztn̄) = ct + (kt+1 − (1 − δ)kt)

where Zt is exogenous labor-augmenting technological progress. In the steady state, ZÌ„
and n̄ grow at rates of γz and γn such that

(
dZÌ„/dt

)
/Z̄ = γz and (dn̄/dt)/n̄ = γn.

Assume that both the production and utility functions take the CES form:

f(kt, Ztn̄) =

(
θkt

γ + (1 − θ)(Ztn̄)γ
)1/γ

u(ct, l̄) =

(
αct

ρ + (1 − α)l̄ρ
)1/ρ

where 0 < θ < 0 is capital’s share of output and γ > 0 determines the elasticity of
substitution between capital and labor, and where 0 < α < 0 is consumption’s share
of utility and ρ > 0 determines the elasticity of substitution between consumption and
leisure. Finally, households are assumed to have a unitary time endowment.

(a) Derive the steady state expressions for capital and output in terms of only exoge-
nous variables.

For parts (b)-(d): If you are unable to obtain an answer to part (a), you may
assume that the steady state expressions for capital and output take the form of
k̄ = ϕkZ̄n̄ and f(k̄, Z̄n̄) = ϕfZ̄n̄, where ϕk and ϕf are exogenous parameters.

(b) Use your solutions from part (a) to mathematically show that this model economy
exhibits the following long-run properties:

i. The output-labor ratio grows at a rate equal to growth in technical progress.

ii. The capital-output ratio is constant.

(c) Compute the steady state expression for consumption.

(d) Suppose that α = 1 so leisure is not valued by households. Compute the long-run
rate of growth in utility from consumption.

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3. (22 points) Fiscal Policy: Consider the infinite-period general equilibrium framework
with a government. The representative household chooses a path of consumption and
leisure over an infinite horizon, {ct+s, lt+s}∞s=0, to maximize the objective function:

V =
∞∑
s=0

βsu(ct+s, lt+s)

subject to the following real period-t budget constraint:

ct + at = (1 + rt)at−1 + wtnt(1 − τwt ) − tt

where at is real wealth, rt is the real interest rate, wt is the real wage rate, nt = 1 − lt is
labor supply, Ï„wt is a proportional tax rate on wage income, and tt is a lump-sum tax.

The representative firm chooses capital and labor input to maximize profits. (You may
assume that the firm here is identical to the firm in question 1.)

The government faces the following real period-t budget constraint:

gt + bt = bt−1(1 + rt) + Tt

where Tt = tt + Ï„
w
t wtnt is the government’s total tax revenue from households.

(a) Write down expressions for real private savings, government savings, and national
savings.

For parts (b)-(d): Suppose that the government makes two tax temporary changes
in period t: (i) the tax rate on wage income is decreased (τwt ↓); and (ii) the lump-sum
tax is increased (tt ↑) as needed so that total tax revenue remains constant in period t.

Furthermore, assume that the substitution effect dominates the income effect for
household labor supply decisions.

(b) Explain how this policy change will affect each of the three definitions of savings
from part (a) (if at all), holding constant wt and rt. Make reference to both household
optimality conditions in your answer.

(c) Explain whether or not Ricardian Equivalence holds for this policy change.

(d) Use supply-and-demand diagram of the Labor Market and the Financial Market to
show how the policy would affect equilibrium prices and quantities in each market.

(e) Use a supply-and-demand diagram for the Goods Market to show a possible equilib-
rium outcome on GDPt and Pt as a result of this policy change.

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4. (22 points) MIU Model: Consider the representative household in the Money-in-the-
Utility model, who chooses a path of consumption, leisure, and nominal money balances
over an infinite horizon, {ct+s, lt+s, Mdt+s}∞s=0, to maximize the following objective func-
tion:

V =
∞∑
s=0

βs


cαt+s

(
Mdt+s
Pt+s

)1−α
+ Ï• ln(lt+s)




where 1 > α > 0 and ϕ > 0 are exogenous preference parameters, and β is a discount
factor. The household faces the following nominal budget constraint each period:

Ptct + At + M
D
t = (1 + it)At−1 + M

D
t−1 + Wtnt

where Pt is the aggregate price level, At is nominal wealth, it is the nominal interest rate,
Wt is the nominal wage rate, and nt is labor supply. The household also faces a unitary
time endowment which holds each period so that 1 = lt + nt.

The representative firm chooses labor and capital to produce output. In nominal terms:

max
{nt+s,kt+s+1}∞s=0

Profit =
∞∑
s=0

(1 + it+s)
−s{Pt+s

(
kθt+sn

1−θ
t+s

)
− Pt+sinvt+s − Wt+snt+s}

where 0 < θ < 1 is capital’s share of output and invt = kt+1 − (1 − δ)kt.

The central bank conducts monetary policy by (exogenously) targeting a desired nominal
interest rate through changes to the money supply, MSt .

(a) Derive the household’s nominal intertemporal, intratemporal, and consumption-
money optimality conditions in terms of the specific utility function given above.

(b) Derive the firms’ nominal intratemporal and intertemporal optimality conditions in
terms of the specific production function given above.

(c) Use your answers from part (a) and (b) to derive the equilibrium conditions for the
labor market, the financial market, the money market, and the goods market.

(d) Assume that the aggregate supply is function upward sloping because some goods
prices quickly adjust in response to changing economic conditions. (To avoid compli-
cating the firm’s optimization problem beyond what we discussed in class, maintain
the assumption that the firm does not choose Pt.)

i. The United States is currently experiencing high levels of inflation. Explain how
the central bank can use monetary policy to reduce inflation. Make reference to
the nominal interest rate and consumption in your answer.

ii. Use a Money Market diagram and a Goods Market diagram to illustrate your
answer above.

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