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}âˆžs=0, to maximize

the following objective function:

V =

âˆžâˆ‘

s=0

Î²su(ct+s, lt+s)

where u(ct, lt) is a well-behaved utility function, and Î² is a discount factor. The household

faces the following real budget constraint each period:

at = (1 + rt)atâˆ’1 + wtnt âˆ’ ct âˆ’ 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}âˆžs=0 to maximize the following real profit function:

Prof =

âˆžâˆ‘

s=0

(

1

1 + rt+s

)s (

f(kt+s, nt+s) âˆ’ invt+s âˆ’ 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 âˆ’ (1 âˆ’ Î´)kt

where Î´ 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 householdâ€™s intertemporal and intratemporal optimality conditions in

terms of the general utility function u(ct, lt).

(b) Derive the firmâ€™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 Plannerâ€™s optimization problem, and use the sequential Lagrangian

to derive the economyâ€™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:

âˆ‚u/âˆ‚ct

Î²âˆ‚u/âˆ‚ct+1

=

âˆ‚f

âˆ‚kt+1

+ (1 âˆ’ Î´)

f(kt, ZtnÌ„) = ct + (kt+1 âˆ’ (1 âˆ’ Î´)kt)

where Zt is exogenous labor-augmenting technological progress. In the steady state, ZÌ„

and nÌ„ grow at rates of Î³z and Î³n such that

(

dZÌ„/dt

)

/ZÌ„ = Î³z and (dnÌ„/dt)/nÌ„ = Î³n.

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

f(kt, ZtnÌ„) =

(

Î¸kt

Î³ + (1 âˆ’ Î¸)(ZtnÌ„)Î³

)1/Î³

u(ct, lÌ„) =

(

Î±ct

Ï + (1 âˆ’ Î±)lÌ„Ï

)1/Ï

where 0 < Î¸ < 0 is capitalâ€™s share of output and Î³ > 0 determines the elasticity of

substitution between capital and labor, and where 0 < Î± < 0 is consumptionâ€™s share

of utility and Ï > 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

kÌ„ = Ï•kZÌ„nÌ„ and f(kÌ„, ZÌ„nÌ„) = Ï•fZÌ„nÌ„, where Ï•k and Ï•f 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 Î± = 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}âˆžs=0, to maximize the objective function:

V =

âˆžâˆ‘

s=0

Î²su(ct+s, lt+s)

subject to the following real period-t budget constraint:

ct + at = (1 + rt)atâˆ’1 + wtnt(1 âˆ’ Ï„wt ) âˆ’ tt

where at is real wealth, rt is the real interest rate, wt is the real wage rate, nt = 1 âˆ’ 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 = btâˆ’1(1 + rt) + Tt

where Tt = tt + Ï„

w

t wtnt is the governmentâ€™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 (Ï„wt â†“); and (ii) the lump-sum

tax is increased (tt â†‘) 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}âˆžs=0, to maximize the following objective func-

tion:

V =

âˆžâˆ‘

s=0

Î²s

ï£«

ï£cÎ±t+s

(

Mdt+s

Pt+s

)1âˆ’Î±

+ Ï• ln(lt+s)

ï£¶

ï£¸

where 1 > Î± > 0 and Ï• > 0 are exogenous preference parameters, and Î² is a discount

factor. The household faces the following nominal budget constraint each period:

Ptct + At + M

D

t = (1 + it)Atâˆ’1 + M

D

tâˆ’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}âˆžs=0

Profit =

âˆžâˆ‘

s=0

(1 + it+s)

âˆ’s{Pt+s

(

kÎ¸t+sn

1âˆ’Î¸

t+s

)

âˆ’ Pt+sinvt+s âˆ’ Wt+snt+s}

where 0 < Î¸ < 1 is capitalâ€™s share of output and invt = kt+1 âˆ’ (1 âˆ’ Î´)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 householdâ€™s nominal intertemporal, intratemporal, and consumption-

money optimality conditions in terms of the specific utility function given above.

(b) Derive the firmsâ€™ 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 firmâ€™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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