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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.

1

(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, Z̄

and n̄ grow at rates of γz and γn such that

(

dZ̄/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.

2

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, τwt 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 + τ

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.

3

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

+ ϕ 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.
4
Macroeconomic Theory
1 Chapter 18: Economic E�ciency
• Economic E�ciency, roughly speaking, is an equilibrium where nothing can be improved
without something else being made worse o�.
→ The concept of e�ciency has nothing to with fairness.
• The Social Planner: a �ctitious, `all-knowing', benevolent entity that is perfectly able to
control and allocate all resources of the economy.
→ Theoretical construct used to obtain e�cient outcomes, which can be used as a benchmark
against sub-optimal outcomes.
1.1 E�ciency in the In�nite-Period General Equilibrium Framework
→ For simplicity, we will assume that labor supply is exogenously �xed at n̄ so that we do not
have to optimize over it (which also means leisure is �xed at l̄)
→ If we had endogenous labor, we would have to optimize over leisure/labor as usual
1.1.1 The Social Planner
→ The Social Planner uses the households' preferences and the �rms' production technology to
allocate the economy's resources to obtain an e�cient equilibrium outcome.
For t = 1, 2, ...
Household's Preferences: V = u(ct, l̄) + βu(ct+1, l̄) + ...
Firm's Production and Investment Technology: f(kt, n̄); invt = kt+1 − (1 −δ)kt
Resource Constraint: f(kt, n̄) = ct + invt
⇒ The Social Planners Optimization Problem:
max
{ct+s,kt+1+s}∞s=0
V =
∞∑
s=0
βsu(ct+s, l̄)
subject to:
f(kt, n̄) − ct − (kt+1 − (1 −δ)kt) = 0 for all t
→ Note that we are only optimizing over the intertemporal dimension. Thus we only need to
take �rst-order condition with respect to ct,ct+1, and kt+1
1
→ One can obtain the e�ciency condition by setting up a sequential Lagrangian and taking the
relevant �rst-order conditions:
L =
∞∑
s=0
{βsu(ct+s, l̄) + λt+s (f(kt+s, n̄) − ct+s − (kt+1+s − (1 − δ)kt+s))}
Writing this out for s = 0 and s = 1:
L = u(ct, l̄) + λt (f(kt, n̄) − ct − (kt+1 − (1 − δ)kt)
+ βu(ct+1, l̄) + λt+1 (f(kt+1, n̄) − ct+1 − (kt+2 − (1 −δ)kt+1)) + ...
FOCs:
∂L
∂ct
= 0 −→
∂u
∂ct
−λt = 0 −→
∂u
∂ct
= λt (1)
∂L
∂ct+1
= 0 −→ β
∂u
∂ct+1
−λt+1 = 0 −→ β
∂u
∂ct+1
= λt+1 (2)
∂L
∂kt+1
= 0 −→−λt + λt+1
(
∂f
∂kt+1
+ (1 − δ)
)
= 0 −→ λt+1
(
∂f
∂kt+1
+ (1 −δ)
)
= λt (3)
Using Equation (1) and (2) into Equation (3):
λt+1
(
∂f
∂kt+1
+ (1 −δ)
)
= λt
β
∂u
∂ct+1
(
∂f
∂kt+1
+ (1 −δ)
)
=
∂u
∂ct
⇒
∂u/∂ct
β∂u/∂ct+1
=
∂f
∂kt+1
+ (1 −δ) (4)
⇒ Equation (4) is the E�ciency condition from the Social Planner's problem.
→ Economic intuition: It is e�cient when the households' marginal rate of substitution of ct
for ct+1 is equal to the �rm's marginal product of capital in period t+1 plus the undepreciated
portion of that marginal unit of capital.
⇒ Note that this is equivalent to the �nancial market equilibrium condition from the `de-
centralized' in�nite-period framework where households and �rms had their own optimization
problems, and equilibrium obtained from market forces.
• First Welfare Theorem of Economics: The equilibrium behavior of households and �rms
in a decentralized framework is economically e�cient.
→ This fails if:
1. distortionary taxes are present (e.g. wage income, interest income taxes)
2. `externalities' are present
3. �rms or households have `market power' to in�uence prices
2
Chapter 18: Economic Efficiency
Efficiency in the Infinite-Period General Equilibrium Framework
Macroeconomic Theory
Week 6
1 Chapter 5: Dynamic Consumption-Labor Framework
We build upon the dynamic consumption-savings framework by adding a labor/leisure decision
for each period. In other words, we are making the labor income endogenous in our dynamic
framework, allowing us to study consumption/savings decisions and labor/leisure decisions si-
multaneously.
1.1 Preferences
Preferences will be represented by the utility function:
V (c1, l1,c2, l2) = u(c1, l1) + βu(c2, l2) (1)
where V (·) is associated with well-behaved indi�erence curves, u(·) is a `sub-utility' function,
and ct and lt are consumption and leisure for t = 1,2.
→ We typically assume that u(·) takes the same function form each period.
→ Note that since V (·) is de�ned over 4 dimensions, we cannot graph it. Nonetheless, you can
think of the associated indi�erence curves as simultaneously de�ned over the (c, l) dimensions
for t = 1,2, the (c1,c2) dimensions we considered in the previous chapter, and an additional
(l1, l2) that we have not yet considered. The indi�erence curves in each of those 4 dimensions
give us 4 MRS's that are relevant for optimal choice.
1.2 Budget Constraints
Recall the budget constraint for the dynamic consumption-savings model in nominal terms:
Ptct + At = Yt + At−1(1 + i) (2)
for t = 1,2. The budget constraint here makes the additional speci�cation income Yt depends
on endogenous labor supply so that:
Yt = (1− τt)Wt(1− lt) (3)
where as in the static consumption-labor model, τt is a tax on wage income, Wt is the nominal
wage rate, and labor supply is given by time not spent on leisure so that nt = 1− lt.
→ Using equation (3) into equation (2) we have our period 1 and 2 budget constraints:
1
Period 1: P1c1 + A1 = (1− τ1)W1(1− l1) + (1 + i)A0
Period 2: P2c2 + A2 = (1− τ2)W2(1− l2) + (1 + i)A1
From the above period budget constraints, we can derive the lifetime budget constraint by
isolating A1 in the period-1 budget constraint, and using that into the period-2 budget constraint.
Assuming the initial and terminal conditions A0 = A2 = 0:
P2c2 = (1− τ2)W2(1− l2) + (1 + i) ((1− τ1)W1(1− l1)−P1c1)
P2c2
1 + i
=
(1− τ2)W2(1− l2)
1 + i
+ (1− τ1)W1(1− l1)−P1c1
⇒ P1c1 +
P2c2
1 + i
= (1− τ1)W1(1− l1) +
(1− τ2)W2(1− l2)
1 + i
(4)
→ This lifetime budget constraint has the same interpretation as all of the others we have
seen thus far. It equates the present discounted value of lifetime expenditures to the present
discounted value of lifetime resources.
→ While this lifetime budget constraint is in nominal terms, we can transform it into real
terms either by applying the Fisher equation directly to equation (4) or applying the Fisher
equation to each of the period-t budget constraints.
→ There the relative slopes (and intercepts) of the lifetime budget constraint in the (c, l)
dimensions for t = 1,2, the (c1,c2) dimension, and an additional (l1, l2) dimension. Each of
these slopes give the e�ective price ratio that is relevant for optimal choice.
1.3 Optimal Choice
The household chooses the optimal (c1∗,c2∗, l1∗, l2∗) bundle which maximizes their utility sub-
ject to the lifetime budget constraint.
→ c1, c2, l1, and l2 are the endogenous variables
→ P1, P2, W1, W2, τ1, τ2 and i are the exogenous variables
As with the previous models, the optimal choice will be such that the Marginal Rate of Substitu-
tion between each choice variable is equal to the relative prices of those choice variables.
⇒There will be two Intratemporal Optimality Conditions that characterize the consumption-
leisure choice for each period t = 1,2 and one Intertemporal Optimality condition that
characterizes the consumption-savings choice across periods.
⇒ The three optimality conditions together with the LBC fully characterize the solution to
the model, (c1∗,c2∗, l1∗, l2∗).
2
1.3.1 Lifetime Lagrangian Formulation
max
c1,c2,l1,l2
u(c1, l1) + βu(c2, l2)
subject to: P1c1 +
P2c2
1 + i
− (1− τ1)W1(1− l1)−
(1− τ2)W2(1− l2)
1 + i
= 0
⇒L = u(c1, l1) + βu(c2, l2) + λ
(
P1c1 +
P2c2
1 + i
− (1− τ1)W1(1− l1)−
(1− τ2)W2(1− l2)
1 + i
)
Taking FOCs:
∂L
∂c1
= 0 −→
∂u
∂c1
+ λP1 = 0 −→
∂u
∂c1
1
P1
= −λ (5)
∂L
∂c2
= 0 −→
∂u
∂c2
β + λ
(
P2
1 + i
)
= 0 −→
∂u
∂c2
β(1 + i)
P2
= −λ (6)
∂L
∂l1
= 0 −→
∂u
∂l1
+ λ(1− τ1)W1 = 0 −→
∂u
∂l1
1
(1− τ1)W1
= −λ (7)
∂L
∂l2
= 0 −→
∂u
∂c2
β + λ
(
(1− τ2)W2
1 + i
)
= 0 −→
∂u
∂l2
β(1 + i)
(1− τ2)W2
= −λ (8)
The next step is to combine the FOCs to derive the optimality conditions, but given that there
are now four FOCs, you might be wondering how to combine them. The key is to keep in mind
what dimensions of optimization are relevant for each optimality condition.
→ We want (i) one intertemporal optimality condition for consumption-savings across peri-
ods, and (ii) two intratemporal optimality conditions for consumption-leisure in each period.
→ Starting with (i), we want to use Equations (5) and (6):
∂u
∂c1
1
P1
=
∂u
∂c2
β(1 + i)
P2
∂u
∂c1
/
∂u
∂c2
=
β(1 + i)P1
P2
(9)
⇒ Equation (9) is the intertemporal optimality condition in nominal terms. Note that if you
apply the Fisher Equation to this, you would obtain it in real terms.
3
→ Next for (ii), we want to use equation (5) and (7) for t = 1, and equations (6) and (8) for
t = 2. Starting with t = 1:
∂u
∂c1
1
P1
=
∂u
∂l1
1
(1− τ1)W1
∂u
∂l1
/
∂u
∂c1
=
(1− τ1)W1
P1
(10)
⇒ Equation (10) is the intratemporal optimality condition for consumption-leisure in nomi-
nal terms for period one.
→ And for (ii) in t = 2:
∂u
∂c2
1
P2
=
∂u
∂l2
1
(1− τ2)W2
∂u
∂l2
/
∂u
∂c2
=
(1− τ2)W2
P2
(11)
⇒ Equation (11) is the intratemporal optimality condition for consumption-leisure in nomi-
nal terms for period two.
→ Note that we have not talked about an intertemporal optimality condition for leisure across
periods. You should convince yourself that if the two intratemporal optimality conditions for
consumption leisure hold, there is implicitly a condition that holds relating the MRS for leisure
across periods to the relative price of leisure across periods.
4
Chapter 5: Dynamic Consumption-Labor Framework
Preferences
Budget Constraints
Optimal Choice
Macroeconomic Theory
Week 6
1 Chapter 6: Firms
Households represented the demand side of the goods market and the supply side labor and �-
nancial markets. This behavior is represented by the consumption demand function, the labor
supply function and the private savings function respectively. Firms represent the opposite side
of the market than households: the supply side of the goods market, and the demand side of the
labor and �nancial market. Here we look at the �rms' decision-making process that determines
they optimal behavior.
1.1 Some Preliminaries on Firm Technology
⇒ Firms utilize labor hired from households, nt, along with their current stock of capital goods,
kt, to produce output, qt, via a production function, f(kt,nt).
→ Unless otherwise speci�ed, f(kt,nt) will have the following properties:
1. f(·) is twice continuously di�erentiable in each argument
2.
∂f
∂kt
,
∂f
∂nt
> 0→ more inputs generate more output ⇒ positive marginal product (MP)

3.

∂2f

∂k2t

,

∂2f

∂n2t

< 0 → MP increases a decreasing rate ⇒ diminishing MP
Graphically, a production function that satis�es the properties of continuous, positive, and
diminishing marginal production looks like the following in two dimensions, in (q,{k,n}) space:
⇒ Capital is accumulated across time through a �ow of real net investment1:
invnett = kt+1 − (1−δ)kt (1)
where δ is the rate of economic depreciation on physical capital.
→ Capital has a `time-to-build' property, where resources spend on investment today do not
become productive until the future period.
1The textbook incorrectly calls net investment in equation (1) 'gross investment'.
1
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1.2 Two-Period Firm Model
Figure 1: Timeline of Events for Model
© Sanjay K. Chugh 77 Spring 2008
Period 2Period 1
Begins with
initial capital k1
Chooses quantity of labor to
hire n1 and level of capital for
next period k2, and then
produces output f(n1, k1)
Chooses quantity of labor to
hire n2 and level of capital for
next period k3, and then
produces output f(n2, k2)
End of the
economy
Figure 25.Timing of events for the firm in the two-period model.
Before we study the firm’s investment decision, let’s briefly consider its decision about
how many workers to hire.In fact, this is something with which you are already familiar
from basic microeconomics.The firm’s demand for labor is a derived demand.To
briefly review:in period 1, the capital 1kis fixed, so that hiring additional workers
increases total output at an ever-decreasing rate according to the law of diminishing
marginal product.The price 1Pis determined by the market (think perfect competition
here), so that marginal revenue product (defined as price of the output good times
marginal product) declines the more labor the firm hires.The derived demand curve for
labor is simply the marginal revenue product curve, as shown in Figure 26.
N
om
in
al
af
te
r-
ta
x
w
ag
e
hours of labor
Labor demand
1.2.1 Objective and Constraints
⇒ The Firm's objective is to choose inputs, kt and nt, to product output, f(kt,nt), at pro�t
maximizing levels.
→ They take as given the nominal wage rate, Wt, the nominal interest rate, it, and the goods
price Pt.
• Lifetime (or Intertemporal) Pro�t Function: expresses the present discounted value
of pro�ts. In nominal terms:
Profit = P1f(k1,n1)−P1invnet1 −W1n1 +
P2f(k2,n2)
1 + i
−
P2inv
net
2
1 + i
−
W2n2
1 + i
(2)
→ Unlike the household's optimization problem, we are setting up the �rm's problem such that the
objective function and budget constraint are in one equation. One way to think about this is that
the �rm wants to produce the pro�t maximizing amount of output over their two-period planning
horizon subject to a constraint that relates present discounted value of output (available resources)
to present discounted value of expenses on investment and labor (possible expenditures), where
investment is de�ned in Equation 1.
→ The pro�t function can be expressed in real terms by application of the Fisher Equation.
1.2.2 Optimal Choice
The �rm chooses the optimal (k2∗,n1∗,n2∗) bundle that maximizes their lifetime pro�t function.
→ k2,n1, and n2 are the endogenous variables
→ P1, P2, W1, W2, and i are the exogenous variables
→ Initial and Terminal conditions: In order to get the full solution to the model, we have to
make assumptions about k1 and k3 since Profit
∗ and inv∗t will depend on their values. However,
we can still gain economic intuition by characterizing the solution with optimality conditions and
assuming k1 and k3 are exogenous.
2
→ For the �rm's problem, we will not be using the method of Lagrange multiplier (although one
could). Instead we will be using the substitution method whereby we substitute the investment
�ow constraint from Equation (1) into the pro�t function in Equation (2).
max
n1,n2,k2
P1f(k1,n1)−P1invnet1 −W1n1 +
P2f(k2,n2)
1 + i
−
P2inv
net
2
1 + i
−
W2n2
1 + i
subject to: invnett = kt+1 − (1−δ)kt for t = 1,2
or alternatively:
max
n1,n2,k2
P1f(k1,n1)−P1 (k2 − (1−δ)k1)−W1n1+
P2f(k2,n2)
1 + i
−
P2 (k3 − (1−δ)k2)
1 + i
−
W2n2
1 + i
(3)
Taking FOCs for n1 and n2:
n1 : P1
∂f
∂n1
−W1 = 0 ⇒
∂f
∂n1
=
W1
P1
(4)
n2 : P2
∂f
∂n2
−W2 = 0 ⇒
∂f
∂n2
=
W2
P2
(5)
⇒ Equations (4) and (5) are the Firm's Labor Optimality Condition.
→ This optimality condition has an important economic interpretation: The �rm hires labor
up until the point where the marginal product of labor is equal to the real wage (i.e. the marginal
cost of the last unit of labor hired). Notice that there is no intertemporal aspect of this condition,
and that it holds for each period t.
Taking the FOC for k2:
k2 : −P1 +
P2
1 + i
∂f
∂k2
+
P2(1−δ)
1 + i
= 0
∂f
∂k2
+ (1−δ) =
P1
P2
(1 + i)
∂f
∂k2
+ (1−δ) =
1 + i
1 + π2
∂f
∂k2
+ (1−δ) = 1 + r2
⇒
∂f
∂k2
= r2 + δ (6)
⇒ Equations (7) is the Firm's Capital Optimality Condition.
→ This optimality condition has an important economic interpretation: The �rm invests into
future capital up until the point where the marginal product of capital is equal to the real interest
rate plus the depreciation rate (i.e. the opportunity cost of the last unit of capital purchased).
Notice that since the �rm is choosing future capital, it is dated t + 1.
3
1.2.3 Labor and Capital Demand
⇒ The Labor Demand Function gives the optimal quantity of labor, nt∗ chosen by the �rm
at every possible real wage rate, wt.
→ It is characterized by the �rm's labor optimality condition for any period t:
wt =
∂f
∂nt
→ Since we're looking for the qualitative relationship between wt and nt at the �rm's optimal
choice, we could di�erentiate the optimality condition:
∂wt
∂nt
=
∂2f
∂n2t
< 0
which says that the �rm's demand curve is downward sloping in (nt,wt) space:
⇒ The Capital Demand Function gives the optimal quantity of future capital, kt+1∗ chosen
by the �rm at every possible real interest rate, rt+1
→ It is characterized by the �rm's capital optimality condition for any period t + 1:
rt+1 =
∂f
∂kt+1
+ δ
→ Since we're looking for the qualitative relationship between rt+1 and kt+1 at the �rm's
optimal choice, we could di�erentiate the optimality condition:
∂rt+1
∂kt+1
=
∂2f
∂k2t+1
< 0
which says that the �rm's demand curve is downward sloping in (kt+1,rt+1) space:
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⇒ The �rm's demand functions for capital and labor are considered derived demand func-
tions, since the �rm's demand for these inputs arises from the demand for their output.
→ Given a functional form for the production function q = f(kt,nt), one could derive the
derived demand functions for n∗t and k
∗
t in terms of q and other exogenous parameters. This is
done by substituting the speci�c functional form for the production function into each respective
optimality condition, and re-arranging for the relevant endogenous variable.
5
Chapter 6: Firms
Some Preliminaries on Firm Technology
Two-Period Firm Model
Macroeconomic Theory
1 Chapter 8: Infinite-Period Framework
All of the dynamic models we have build thus far consist of only two periods. Here we expand
those models to an infinite amount of periods. The main takeaway is that the economic intuition
gained from the simplified two-period framework holds when the number of periods is expanded.
• Infinite-period Framework: an infinite sequence of overlapping two-period frameworks.
→ The representative household or firm has an “Infinite Planning Horizon”
→We use indices t,t + 1, t + 2, ... instead of 1,2,3, ...
→ Nothing before period t can be changed
1.1 Households
Households choose consumption/savings and labor/leisure each period over an infinite horizon
to maximize utility subject to a budget constraint.
1.1.1 Preferences
Preferences are represented by:
V (ct, lt,ct+1, lt+1,ct+2, lt+2...) = u(ct, lt) + βu(ct+1, lt+1) + β
2u(ct+2, lt+2) + ...
or more compactly:
V =
∞∑
s=0
βsu(ct+s, lt+s) (1)
where each sub-utility function u(·) has the usual properties.
1.1.2 Budget Constraint
In the absence of taxes, households have the following period-t real budget constraint:
ct + at = (1 + rt)at−1 + wtnt (2)
with the unitary time endowment:
1 = lt + nt (3)
→ One could use this period-t budget constraint to derive a lifetime budget constraint over infinite
periods. We will not be doing that in course.1
1If you are curious, the infinite period LBC can be derived by expressing the budget constraint as a difference
equation and solving forward to get:
∑∞
s=0
ct+s/(1 + rt+1+s)
s =
∑∞
s=0
(wt+snt+s)/(1 + rt+1+s)
s with the usual
terminal and initial conditions imposed.
1
1.1.3 Optimal Choice
Since the model as a recursive structure where the events occur each period, we can derive a single
intertemporal and a single intratemporal optimality condition that will hold for each period using
a sequential Lagrangian.
max
{ct+s,at+s,lt+s}∞s=0
V =
∞∑
s=0
βsu(ct+s, lt+s)
subject to:
ct + at − (1 + rt)at−1 −wt(1− lt) = 0 for all t = 1,2,3, ...
The sequential Langrangian can be written in the same form we learned for two periods, but for
an infinite amount of periods as follows:
L =
∞∑
s=0
{βsu(ct+s, lt+s) + λt+s (ct+s + at+s − (1 + rt+s)at−1+s −wt+s(1− lt+s))} (4)
Equation (4) might look intimidating, and you might be thinking ‘How do do I take for the first
order conditions for this?’ It could be helpful to first write out the summation for a few periods:
L = u(ct, lt) + λt (ct + at − (1 + rt)at−1 −wt(1− lt))
+ βu(ct+1, lt+1) + λt+1 (ct+1 + at+1 − (1 + rt+1)at −wt+1(1− lt+1))
+ β2u(ct+2, lt+2) + λt+2 (ct+2 + at+2 − (1 + rt+2)at+1 −wt+2(1− lt+2)) + ...
The summation would go on forever in this pattern as s approaches ∞, but writing it out until
s = 2 as above is sufficient for our purposes. So now what? The key thing to keep in mind is
that since the model has a recursive structure, the optimality conditions hold for any t period.
This means that we could take the FOCs as usual for any t period:
→ For intertemporal optimality condition we need to optimize over ct and ct+1.
→ For intratemporal optimality condition we need to optimize over ct and lt.
→ Note that we can take these FOCs while ignoring the expanded Lagrangian for s > 1.

2

Taking FOCs:

∂L

∂ct

= 0 −→

∂u

∂ct

+ λt = 0 −→

∂u

∂ct

= −λt (5)

∂L

∂ct+1

= 0 −→ β

∂u

∂ct+1

+ λt+1 = 0 −→ β

∂u

∂ct+1

= −λt+1 (6)

∂L

∂lt

= 0 −→

∂u

∂lt

+ λtwt = 0 −→

∂u

∂lt

1

wt

= −λt (7)

∂L

∂at

= 0 −→ λt − (1 + rt+1)λt+1 = 0 −→ λt = (1 + rt+1)λt+1 (8)

Remember that with the sequential Lagrangian we need to optimize over at to get an expres-

sion that links together the Lagrange multipliers across periods.

Combining the FOCs:

→ Intertemporal Optimality Condition: Using Equations (5) and (6) into (8):

−λt = −(1 + rt+1)λt+1

∂u

∂ct

= β

∂u

∂ct+1

(1 + rt+1)

⇒

∂u/∂ct

β∂u/∂ct+1

= (1 + rt+1) (9)

→ Intratemporal Optimality Condition: Using Equations (5) and (7):

∂u

∂ct

=

∂u

∂lt

1

wt

⇒

∂u/∂lt

∂u/∂ct

= wt (10)

→ The solution to the model, {ct+s∗, lt+s∗}∞s=0, is fully characterized with an intertemporal

and intratemporal optimality condition for each period and a LBC with initial and terminal

conditions on wealth.

→ We will not be solving the infinite period models in this class. With the exception of a few

special cases, computational methods are typically needed to obtain the solution.

3

2 Firms

The representative firm chooses labor and capital inputs to produce output at profit maximizing

levels over an infinite horizon.

→ The firm’s real profit function over an infinite horizon:

Profit =

∞∑

s=0

(1 + rt+s)

−s{f(kt+s,nt+s)− (kt+1+s − (1− δ)kt+s)−wt+snt+s} (11)

As with the households’ optimization problem, it will help to write out the summation for a

few periods. We’ll again expand to s = 2.

Profit =f(kt,nt)− (kt+1 − (1−δ)kt)−wtnt

+

(

1

1 + rt+1

)

(f(kt+1,nt+1)− (kt+2 − (1−δ)kt+1)−wt+1nt+1)

+

(

1

1 + rt+2

)2

(f(kt+2,nt+2)− (kt+3 − (1−δ)kt+2)−wt+2nt+2) + …

→ For intertemporal optimality condition we need to optimize over kt+1.

→ For intratemporal optimality condition we need to optimize over nt.

→ Note that we can take these FOCs while ignoring the expanded profit function for s > 1.

Taking FOCs:

nt :

∂f

∂nt

−wt = 0 ⇒

∂f

∂nt

= wt (12)

kt+1 : −1 +

(

1

1 + rt+1

)(

∂f

∂kt+1

+ (1−δ)

)

= 0 ⇒

∂f

∂kt+1

= rt+1 + δ (13)

where Equations (12) and (13) are the firms intratemporal and intertemporal optimality condi-

tions, which hold for all t.

→ The solution to the model, {nt+s∗,kt+1+s∗}∞s=0, is fully characterized by the optimality

conditions for each period, and initial and terminal conditions of the capital stock given demand

for the firms’ output.

⇒ If the economic intuition from two-period models is the same as that from infinite period

models, why bother with more complex infinite period models? When using macroeconomic mod-

els to make quantitative statements about the dynamics of aggregates — such as consumption,

savings, capital, labor, and output — the infinite period framework is more useful because it allows

for multiple periods that reasonably correspond to a monthly, quarterly, or annual frequency.

4

Chapter 8: Infinite-Period Framework

Households

Firms

Macroeconomic Theory

1 Neoclassical Growth

The Neoclassical Growth Model is a variant of our in�nite-period framework that is a applied

to study the determinants of long-run growth in economic output. The main take-away is the

neoclassical growth model asserts that long-run economic growth is driven by increases in `tech-

nological capabilities’ and population growth.

1.1 The �Long-Run”

• Steady State: a situation in the in�nite-period framework where all real variables stop

�uctuating (or grow at a constant rate) over time absent external disturbances.

→ For any variable xt, in a steady state xt = xt+1 = xt+2 = … = x̄

⇒ Why does the model converge to a steady state?

→ Consider the economy’s intertemporal optimality condition (i.e. �nancial market equilib-

rium condition):

∂u/∂ct

∂u/∂ct+1

= β

(

∂f

∂kt+1

+ 1 −δ

)

(1)

• Case 1: Growing Economy → ct < ct+1 < ... and kt+1 < kt+2 < ...
→ This implies that the left-hand side (LHS) of Equation 1 is greater than 1, which itself
implies that the right-hand side (RHS) is also greater than 1.
→ if k is growing, then ∂f/∂k is shrinking over time −→ RHS decreases towards 1, which
implies that LHS is also decreasing towards 1.
• Case 2: Shrinking Economy → ct > ct+1 > … and kt+1 > kt+2 > …

→ This implies that LHS of Equation 1 is less than 1, which itself implies that RHS is also

less than 1.

→ if k is shrinking, then ∂f/∂k is growing over time −→ RHS increases towards 1, which

implies that LHS is also increasing towards 1.

⇒ Putting together the implications from Cases 1 and 2, the positive but diminishing

marginal product of capital implies that the economy settles at a steady state.

1.2 Technological Progress in Neoclassical Growth Model

• Technological Progress: increases in the productivity of the factors of production

→ more output for a given amount of inputs to the production function.

• Labor-Augmenting Technological Progress: Technological progress is modeled as in-

creases in `e�ective’ labor input. (required for existence of steady state)

1

→ Let Zt be the level of technology embodied by the labor input, nt. Then the `e�ective’

labor input is Ztnt, and total output is given by f(kt,Ztnt).

→ Assume that Zt grows by a rate of γz in the steady state such that

dZ̄/dt

Z̄

= γz

1.

1.3 Neoclassical Growth Model

→ For simplicity, we will assume that labor supply is exogenous, n̄, but grows at a rate of γn in

steady state such that

dn̄/dt

n̄

= γn. (Loosely thought of as population growth)

→ Exogenous labor means that leisure is also exogenous, l̄.

Households:

max

{ct+s,at+s}∞s=0

V =

∞∑

s=0

βsu(ct+s, l̄)

subject to:

ct + at − (1 + rt)at−1 −wt(1 − l̄) = 0 for all t = 1, 2, 3, …

Firms:

max

{kt+1+s}∞s=0

Profit =

∞∑

s=0

(1 + rt+s)

−s{f(kt+s,Zt+sn̄) − (kt+1+s − (1 − δ)kt+s) −wt+sn̄}

⇒ Since the First Welfare Theorem will hold, the economy’s optimality conditions can be

obtained either by solving for general equilibrium in a decentralized fashion using the house-

holds’ and �rms’ problems one at a time, or by using the Social Planner’s framework.

→ With exogenous labor, we need two equilibrium conditions for the economy:

Intertemporal Optimality Condition (Financial Market Equilibrium Condition):

∂u/∂ct

β∂u/∂ct+1

=

(

∂f(kt+1,Zt+1n̄)

∂kt+1

+ 1 −δ

)

(2)

Aggregate Resource Constraint (Goods Market Equilibrium Condition):

f(kt,Ztn̄) = ct + (kt+1 − (1 −δ)kt) (3)

→With endogenous labor we would also need the economy’s intratemporal optimality condition

(i.e. labor market equilibrium condition)

1

Since for any variable, x,

dx̄/dt

x̄

is the instantaneous rate of growth, this tells us that steady capital grows

continuously at the rate of technological progress plus the rate of labor supply growth.

2

→ With a functional form for the production function, we can solve directly for the steady

state capital stock, output, and consumption.

EXAMPLE: Suppose f(kt,Ztn̄) = k

α

t (Ztn̄)

1−α. Solve for steady state capital and output.

→ Impose steady state on intertemporal optimality condition:

∂u/∂ct

β∂u/∂ct+1

=

(

αkα−1t (Ztn̄)

1−α + 1 −δ

)

∂u/∂c̄

β∂u/∂c̄

=

(

αk̄α−1(Z̄n̄)1−α + 1 −δ

)

1

β

=

(

αk̄α−1(Z̄n̄)1−α + 1 −δ

)

→ Solve for k̄:

1

β

− 1 + δ = αk̄α−1(Z̄n̄)1−α

1

β

− 1 + δ

α

=

(

k̄

Z̄n̄

)α−1

1

β

− 1 + δ

α

1/(α−1)

=

k̄

Z̄n̄

⇒ k̄∗ = Z̄n̄

1

β

− 1 + δ

α

1/(α−1)

(4)

→ Use expression for k̄ into the production function:

f(k̄∗, Z̄n̄) =

Z̄n̄

1

β

− 1 + δ

α

1/(α−1)

α

(Z̄n̄)1−α

f(k̄∗, Z̄n̄) = (Z̄n̄)α(Z̄n̄)1−α

1

β

− 1 + δ

α

α/(α−1)

⇒ f(k̄∗, Z̄n̄) = Z̄n̄

1

β

− 1 + δ

α

α/(α−1)

(5)

3

→ Equations (4) and (5) are the expressions for steady state capital and output. Using these

expressions into the resource constraint evaluated at the steady state, one can also obtain an

expression for steady state consumption:

f(k̄, Z̄n̄) = c̄ + (k̄ − (1 −δ)k̄)

f(k̄, Z̄n̄) = c̄ + δk̄

⇒ c̄∗ = f(k̄∗, Z̄n̄) −δk̄∗

1.3.1 Technological Progress and Economic Growth

What does technological progress (and population growth) imply for steady state growth of capital

stock and output?

→ Want steady state growth rate of capital in terms of steady state growth rates of technology

and labor.

→ Consider the expression for the steady state capital stock:

k̄ = Z̄n̄

1

β

− 1 + δ

α

1/(α−1)

→ Taking the natural log of both side yields:

log k̄ = log Z̄ + log n̄ + ((1/(α− 1)) log

1

β

− 1 + δ

α

→ Next, taking the total derivative of the above equation with respect to time:

dk̄/dt

k̄

=

dZ̄/dt

Z̄

+

dn̄/dt

n̄

dk̄/dt

k̄

= γz + γn (6)

⇒ Steady state capital grows at the rate of technological progress plus population growth.

→ Intuition: The marginal product of capital increases as e�ective labor input exogenously

grows, which increases the desired future capital stock by the �rm thereby lowering the marginal

product of capital. In a steady state these two e�ects on the marginal product of capital cancel

each other out.

4

→ Repeating this process for our expression for steady state output:

f(k̄, Z̄n̄) = Z̄n̄

1

β

− 1 + δ

α

α/(α−1)

log f(k̄, Z̄n̄) = log Z̄ + log n̄ + ((α/(α− 1)) log

1

β

− 1 + δ

α

df̄/dt

f̄

=

dZ̄/dt

Z̄

+

dn̄/dt

n̄

df̄/dt

f̄

= γz + γn

⇒ Steady state growth in output is determined by growth rates in technology and population.

→ Intuition: Steady state output grows at same rate as each of the factors of production.

5

Macroeconomic Theory

1 Chapter 14: Real Business Cycles

We utilize the in�nite-period general equilibrium framework to study a theory of business cycles.

• Real Business Cycle (RBC) Theory asserts that short-run �uctuations to economic ac-

tivity are �e�cient� responses to temporary technological shocks.

1.1 Technological Shocks in Real Business Cycle Model

• Technological Shock: unexpected, temporary changes to `total factor productivity’

• Total Factor Productivity (TFP): Portion of output not determined by capital or labor.

→ Let Θt be the level of TFP, then output is modeled as Θtf(kt,nt).

→ Graphically, TFP shocks look as follows:

1.2 Real Business Cycle Model

Households:

max

{ct+s,at+s,lt+s}∞s=0

V =

∞∑

s=0

βsu(ct+s, lt+s)

subject to:

ct + at − (1 + rt)at−1 −wt(1 − lt+s) = 0 for all t = 1, 2, 3, …

Firms:

max

{kt+1+s,nt+s}∞s=0

Profit =

∞∑

s=0

(1 + rt+s)

−s{Θt+sf(kt+s,nt+s)−(kt+1+s−(1−δ)kt+s)−wt+snt+s}

⇒ Since the First Welfare Theorem will hold, the economy’s optimality conditions can be

obtained either by solving for general equilibrium in a decentralized fashion using the house-

holds’ and �rms’ problems one at a time, or by using the Social Planner’s framework.

1

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→ We need three equilibrium conditions for the economy:

Intratemporal Optimality Condition (Labor Market Equilibrium Condition):

∂u/∂lt

∂u/∂ct

=

∂Θtf(kt,nt)

∂nt

(1)

Intertemporal Optimality Condition (Financial Market Equilibrium Condition):

∂u/∂ct

β∂u/∂ct+1

=

(

∂Θt+1f(kt+1,nt+1)

∂kt+1

+ 1 −δ

)

(2)

Aggregate Resource Constraint (Goods Market Equilibrium Condition):

Θtf(kt,nt) = ct + (kt+1 − (1 −δ)kt) (3)

→ Unlike the steady state of Neoclassical Growth Model, we cannot solve analytically for the

equilibrium values of economic aggregates in the RBC model. However, we can use the economy’s

optimality conditions to characterize what happens following a TFP shock.

⇒ How does a temporary increase in Θt and Θt+1 a�ect equilibrium in the model?

→ First, recall that �rms hire labor and invest into future capital until each respective marginal

product equals its marginal cost:

∂Θtf(kt,nt)

∂nt

= wt

∂Θt+1f(kt+1,nt+1)

∂kt+1

= rt+1 + δ

where Θ directly a�ects each marginal product. From the above �rm’s optimality conditions:

→ Θt ↑ → MPn ↑ −→ �rm’s demand for labor increases at wage rate

→ Θt+1 ↑ → MPk ↑ −→ �rm’s demand for investment increases at real interest rate

→ Second, an increase in the marginal products of labor and capital will increase the RHS of

the labor market and �nancial market equilibrium conditions.

→ MPN ↑→ return to labor increases −→ quantity of n

supply

t ↑ along given supply function,

which causes MRSl,c ↑ so that Equation (1) holds in equilibrium

→ MPK ↑ → return to capital increases −→ quantity of s

supply

t ↑ along given supply func-

tion, which causes MRS1,2 ↑ so that Equation (2) holds in equilibrium

→ Third, an increase in nt,t+1,kt+1, and Θt,t+1 increases AS in t = 1, 2

→ AD↑ (from increase in investment demand) so that Equation (3) holds in equilibrium.

2

→ Graphically this change looks as follows in the labor, �nancial, and goods markets:

⇒ This is considered to be an e�cient response to the technological shock because the social

planner would choose the change to resource allocation that would be obtained from �rms and

households interacting in decentralized markets.

3

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Neoclassical Growth

The “Long-Run”Technological Progress in Neoclassical Growth ModelNeoclassical Growth Model

Macroeconomic Theory

1 Chapter 7: Intertemporal Fiscal Policy

We introduce a central government that collects taxes from households (and/or �rms) and makes

consumption expenditures. In doing so, we discuss the government’s intertemporal budget con-

straint which involves saving or borrowing across periods when taxes are unequal to expenditures.

• Primary Budget De�cit (Surplus): exists in any period where tax revenue is smaller

(larger) than government expenditures.

• Secondary Budget De�cit (Surplus): exists in any period where tax revenue plus net

interest income is smaller (larger) than government expenditures.

→ We will generally refer to a secondary ‘budget de�cit (surplus)

⇒ Let sgovt denote the (secondary) budget surplus, tt denote tax revenue, gt denote govern-

ment expenditures and bt denote net government wealth (all in real terms). Then:

s

gov

t ≡ tt + rtbt−1 −gt (1)

→ if sgovt > 0 then government is running a surplus

→ if sgovt < 0 then government is running a de�cit
1.1 Two-Period Partial Equilibrium Fiscal Model
We will start with a dynamic two-period model of the government's intertemporal budget for
simplicity. We later generalize this to the in�nite-period framework.
Figure 1: Timeline of Events
Spring 2014 | © Sanjay K. Chugh 119
Period 1 Period 2
b0 b2
Government activities
during period 1:
government spending
and tax collection
b1
Government activities
during period 2:
government spending
and tax collection
Beginning
of analysis
End of
analysis
�
Figure 39.Timing of events for the government.
Unlike household or �rm, there is assume to be no utility or pro�t maximization here. Tax
and spending policies are assumed to be exogenous.
1
1.1.1 Government Budget Constraint
⇒ Periods t=1,2
→ Resources available to government: real tax revenue, tt; real wealth, bt−1; net real interest
income, bt−1rt
→ Possible expenditures: real expenditures, gt; real wealth for beginning of next period, bt
−→ Thus the period-t budget constraint is:
gt + bt = bt−1(1 + rt) + tt (2)
⇒ Lifetime Budget Constraint
Evaluating Equation (2) for t = 1,2 we have:
g1 + b1 = b0(1 + r1) + t1
g2 + b2 = b1(1 + r2) + t2
→ We can derive the lifetime budget constraint for the government by combining periods-1 and -2
budget constraints and imposing the corresponding initial and terminal conditions b0 = b2 = 0:
g2 + b2 = (t1 + b0(1 + r1)−g1) (1 + r2) + t2
g2 = (t1 −g1) (1 + r2) + t2
g2
1 + r2
= t1 −g1 +
t2
1 + r2
⇒ g1 +
g2
1 + r2
= t1 +
t2
1 + r2
(3)
→ Equation (3) is the Lifetime Budget Constraint (LBC) for the government, which equates
the present discounted value of lifetime expenditures to the present discounted value of lifetime
resources.
⇒ Implication of the LBC for Tax Policy: For a given path of real interest rates and govern-
ment expenditures, any changes to tax revenue now must be o�set by tax changes in the future.
→ Re-arranging Equation (3) for t2:
t2 = g1(1 + r2) + g2 − t1(1 + r2)
Di�erentiating with respect to t1:
dt2
dt1
= −(1 + r2)
⇒ dt2 = −(1 + r2)dt1 (4)
→ Intuition: Since the left-hand side of Equation (3) does not change when t1 changes, t2 must
change in an o�setting fashion so that the LBC holds.
2
1.1.2 Ricardian Equivalence
As special case in our dynamic framework, the Ricardian-Equivalence Proposition asserts
that the timing of taxes do not matter for private agent's economic behavior because they in-
ternalize the government's lifetime budget constraint into their own.
→ Requires all taxes to be lump-sum (or non-distortionary) taxes, where the amount
owed by an economic agent does not depend on their choices.
EXAMPLE: Consider the representative household in the two-period consumption-savings model
with exogenous labor income. Suppose that the government collects lump-sum taxes tt from
them. Their period-t budget constraint would then be:
ct + at = (1 + rt)at−1 + yt − tt (5)
This yields a lifetime budget constraint for the household of:
c1 +
c2
1 + r2
= y1 +
y2
1 + r2
− t1 −
t2
1 + r2
(6)
where the initial and terminal conditions a0 = a2 = 0 have been imposed.
→ Consider the intertemporal dimension for c1 and c2. The relevant price for optimal choice
is given by the slope of the LBC, ∂c2/∂c1. Using Equation (6) to compute that slope:
c2 = y1(1 + r2) + y2 − c1(1 + r2)− t1(1 + r2)− t2
⇒
∂c2
∂c1
= −(1 + r2)
(i) From above, the relevant price for the intertemporal decision does not depend on taxes
(ii) From equation (4), changes to timing of will not change the PDV of lifetime resources
⇒ Because of (i) and (ii), Ricardian Equivalence holds.
EXAMPLE: Suppose that the government collects distortionary taxes on the representative
households' consumption so that tt = τtct for t = 1,2. Using this expression into Equation (6),
and re-arranging for c2 to compute the intertemporal price of consumption:
c1 +
c2
1 + r2
= y1 +
y2
1 + r2
− τ1c2 −
τ2c2
1 + r2
c1(1 + τ1) +
c2(1 + τ2)
1 + r2
= y1 +
y2
1 + r2
c2 =
(
1 + r2
1 + τ2
) (
y1 +
y2
1 + r2
)
− c1
(1 + τ1)
(1 + τ2)
(1 + r2)
⇒
∂c2
∂c1
= −
(1 + τ1)
(1 + τ2)
(1 + r2)
⇒ Since the distortionary tax a�ects the intertemporal price of consumption, tax changes will
a�ect private behavior and Ricardian Equivalence will fail.
3
1.2 National Savings and Generalization to the In�nite-Period Framework
Whether using the two-period model or the in�nite period framework, we de�ne real national
savings as the sum of private savings and government savings for a given period t.
→ Recall the de�nition of real private savings:
s
priv
t ≡ at −at−1
→ Using the general expression for the household's period-t budget constraint with labor
and interest income, in addition to lump-sum taxes, we have:1
s
priv
t = wtnt + rtat−1 − ct − tt (7)
→ Using our de�nition of the real budget surplus for government savings, we have:
s
gov
t ≡ tt + rtbt−1 −gt (8)
⇒ National Savings, snatt , is the sum of private and government savings for a given period t :
snatt = s
priv
t + s
gov
t
snatt = (wtnt + rtat−1 − ct − tt) + (tt + rtbt−1 −gt)
snatt = wtnt + rtat−1 − ct + rtbt−1 −gt (9)
Intuitively: National savings is household income less consumption, plus government income less
expenditures.
1.2.1 Ricardian Equivalence and National Savings
⇒ Under non-distortionary taxation, Ricardian Equivalence will hold:
→ From inspection of Equation (7), a decrease in tt will increase private savings.
→ From inspection of Equation (8), a decrease in tt will decrease government savings
→ From inspection of Equation (9), a decrease in tt will not a�ect national savings because
the increase in private savings and government savings are exactly o�setting.
⇒ Under distortionary taxation, Ricardian Equivalence will fail, and changes to the timing
of taxation will a�ect national savings through the impact on private savings.
1
Note that yt = wtnt in the previous example with exogenous labor income
4
1.2.2 Financial Market Equilibrium With Government
Recall that real private savings is increasing in the future real interest rate, rt+1. Since real
government expenditures and tax rates are assumed to be exogenous, real national savings is
then also increasing in the future real interest rate:
⇒ Financial market equilibrium occurs at the real interest rate rt+1∗ that equates the quan-
tity supplied of national savings snatt ∗ with the quantity demanded of investment invt∗.
EXAMPLE: Suppose that a government taxing consumption in a distortionary fashion tem-
porarily reduces the period t tax rate. Holding constant current government expenditures, there
will be a reduction in the budget surplus (or increase in the budget de�cit).
→ Since Ricardian Equivalence fails, there will be an increase in current consumption and
decrease in current saving at every possible future real interest rate:
• Crowding Out: The decrease in private investment that results from an increase in the
equilibrium real interest rate caused by government policy.
5
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1.2.3 Distortionary Taxes in General Equilibrium
Consider a government in the in�nite-period framework that levies a proportional tax on house-
holds' real wage income and real interest income at rates τnt and τ
i
t respectively. The represen-
tative household will then have the following intra- and intertemporal optimality conditions in
real terms:
∂u/∂lt
∂u/∂ct
= (1− τnt )wt
∂u/∂ct
β∂u/∂ct+1
= (1 + (1− τit+1)rt+1)
→ How do changes in these distortionary tax rates a�ect market activity?
EXAMPLE: Suppose that the government temporarily increases τnt and τ
i
t+1. Use the optimality
conditions to explain what happens in the labor and �nancial markets.
⇒ Labor Market: From the household's intratemporal optimality condition, the reduction in
the after-tax real wage rate will cause the household to reduce labor supply (assuming SE>IE).

⇒ Financial Market: From the household’s intertemporal optimality condition, the reduction

in the after-tax real interest rate will cause the household to increase current consumption and

reduce future consumption, thereby reducing savings (assuming SE>IE)

Graphically in the labor market, �nancial market, and goods market diagrams:

6

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Chapter 7: Intertemporal Fiscal Policy

Two-Period Partial Equilibrium Fiscal ModelNational Savings and Generalization to the Infinite-Period Framework

Macroeconomic Theory

1 Chapter 15: Monetary Policy in the Intertemporal Framework

Our dynamic framework has had no role for money, which acts as a medium of exchange, unit

of account, and store of value. We introduce money into the in�nite-period framework as a store

of value. This feature allows for us to use the model to study how the monetary policy of central

banks may in�uence real economic activity and con�ict with the classical dichotomy.

→ Introducing money requires us to consider an additional market, The Money Market.

• Money Demand: The nominal quantity demanded of money demanded by households

(as well as �rms and the government).

• Money Supply: The nominal quantity of money in circulation (e�ectively determined by

the banking system)

⇒ Equilibrium in the Money Market occurs when the nominal quantity of money supplied equals

nominal quantity of money demanded.

→ To generate money demand for households, we will use the Money-in-the-Utility (MIU)

approach, where demand for real money holdings is an explicit argument in the households’

utility function. For any given period t:

u

(

ct, lt,

MDt

Pt

)

(1)

where MDt is the nominal money demand (so that M

D

t /Pt is real money demand).

1.1 MIU Model

1.1.1 Households

max

{ct+s,lt+s,MDt+s}

∞

s=0

V =

∞∑

s=0

βsu

(

ct+s, lt+s,

MDt+s

Pt+s

)

(2)

subject to the nominal period-t budget constraint:1

Ptct + At + M

D

t = (1 + it)At−1 + M

D

t−1 + Wt(1− lt) (3)

→ The intratemporal and intertemporal conditions, along with a new `consumption-money’

optimality condition, can be derived by setting up the sequential Lagrangian, and taking the

�rst-order conditions with respect to ct,ct+1, lt,At, and M

D

t :

L =

∞∑

s=0

{βsu

(

ct+s, lt+s,

MDt+s

Pt+s

)

+λt+s(Pt+sct+s + At+s + M

D

t+s − (1 + it+s)At+s−1

−MDt+s−1 −Wt+s(1− lt+s))}

1

recall that for any nominal variable X, the real quantity is de�ned as x = X/P

1

Writing this out for s = 0,1:

L =u

(

ct, lt,

MDt

Pt

)

+ λt

(

Ptct + At + M

D

t − (1 + it)At−1 −M

D

t−1 −Wt(1− lt)

)

+ βu

(

ct+1, lt+1,

MDt+1

Pt+1

)

+ λt+1

(

Pt+1ct+1 + At+1 + M

D

t+1 − (1 + it+1)At −M

D

t −Wt+1(1− lt+1)

)

+ …

Taking FOCs:

∂L

∂ct

= 0 −→

∂u

∂ct

+ λtPt = 0 −→

∂u

∂ct

1

Pt

= −λt (4)

∂L

∂ct+1

= 0 −→ β

∂u

∂ct+1

+ λt+1Pt+1 = 0 −→ β

∂u

∂ct+1

1

Pt+1

= −λt+1 (5)

∂L

∂lt

= 0 −→

∂u

∂lt

+ λtWt = 0 −→

∂u

∂lt

1

Wt

= −λt (6)

∂L

∂At

= 0 −→ λt − (1 + it+1)λt+1 = 0 −→−λt = −(1 + it+1)λt+1 (7)

∂L

∂MDt

= 0 −→

∂u

∂MDt

1

Pt

+ λt −λt+1 = 0 −→

∂u

∂MDt

1

Pt

+ λt = λt+1 (8)

→ Intratemporal Optimality Condition (Nominal terms): Using Equations (4) and (6):

∂u

∂ct

1

Pt

=

∂u

∂lt

1

Wt

⇒

∂u/∂lt

∂u/∂ct

=

Wt

Pt

(9)

→ Intertemporal Optimality Condition (Nominal terms): Using Equations (4)-(5) into (7):

−λt = −(1 + it+1)λt+1

∂u

∂ct

1

Pt

= (1 + it+1)β

∂u

∂ct+1

1

Pt+1

⇒

∂u/∂ct

β∂u/∂ct+1

=

Pt

Pt+1

(1 + it+1) (10)

2

→ Consumption-Money Optimality Condition: Using Equation (7) into (8):

∂u

∂MDt

1

Pt

+ λt = λt+1

∂u

∂MDt

1

Pt

+ λt =

λt

(1 + it+1)

∂u

∂MDt

1

λtPt

+ 1 =

1

(1 + it+1)

∂u

∂MDt

1

λtPt

=

1

(1 + it+1)

−1

∂u

∂MDt

1

λtPt

=

1−1− it+1

(1 + it+1)

∂u

∂MDt

1

λtPt

=

−it+1

(1 + it+1)

Using Equation (4) into the above:

(

∂u

∂MDt

) −Pt(∂u

∂ct

)

Pt

= −it+1(1 + it+1)

⇒

∂u/∂MDt

∂u/∂ct

=

it+1

(1 + it+1)

(11)

⇒ Equation (11) is the new consumption-money optimality condition.

→ Intuition: The household desires to have real money holdings such that the MRS of money

for consumption is equal to the relative price of holding money to consumption, each in terms of

their opportunity cost (i.e. savings in interest bearing asset).

⇒ This optimality condition characterizes the households’ demand for money:

→ An increase in the nominal interest increases the right-hand side of Equation (11), which

implies that the rate of exchange of money for consumption increases.

→ Assuming a dominant substitution e�ect, the household responds by reducing their money

holdings relative to their consumption.

→ Graphically, the household’s money demand function is expressed in (Mt, it+1) space:

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1.1.2 Central Bank

⇒ The central bank (exogenous) is assumed to be able control the nominal money supply2 with

the intent of achieving a speci�c nominal interest rate in the money market.

→ Buys (sells) nominal quantities of interest bearing assets from households with money to

increase (decrease) the money supply as desired.

→ The money supply is assumed to be exogenous absent a monetary policy shock.

→ Graphically, the money supply function is shown in (Mt, it+1) space:

1.1.3 Firms

In nominal terms, the �rm’s problem is:

max

{nt+s,kt+s+1}∞s=0

Profit =

∞∑

s=0

(1+it+s)

−s{Pt+sf(kt+s,nt+s)−Pt+s(kt+1+s−(1−δ)kt+s)−Wt+snt+s}

(12)

→ Taking the �rst-order condition with respect to labor and future capital yields the usual

labor demand and investment demand optimality conditions for the �rm:

∂f/∂nt =

Wt

Pt

(13)

∂f/∂kt+1 + (1−δ) =

Pt

Pt+1

(1 + it+1) (14)

2

Reality is of course much more complex. The banking system as a whole is thought to determine the

outstanding nominal quantity of money through the creation of bank deposits. Since we are not modeling

commercial banks explicitly, it is common to assume for simplicity that the central bank can attain any desired

quantity of money supply.

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1.1.4 General Equilibrium

With the addition of the money market, we have four equilibrium conditions: Labor Market,

Financial Market, Goods Market, and the Money Market.

⇒ Money Market Equilibrium: Occurs at a nominal interest rate it+1∗ such that MDt =

MSt , where M

D

t is determined by the households’ consumption-money optimality condition and

MSt is determined by Central Bank policy. Graphically:

⇒ Financial Market Equilibrium: Occurs at real interest rate rt+1∗. such that st = invt.

As usual, it’s characterized using households’ and �rms intertemporal optimality conditions to

obtain:

∂u/∂ct

β∂u/∂ct+1

= 1 +

∂f

∂kt+1

−δ

⇒ Labor Market Equilibrium: Occurs at wt∗ such that nSt = nDt . As usual, it’s charac-

terized using the households’ and �rms’ intratemporal optimality condition to obtain:

∂u/∂lt

∂u/∂ct

=

∂f

∂nt

⇒ Goods Market Equilibrium: Occurs when ADt = ASt. As usual without a federal

government:

f(kt,nt) = ct ∗+invt∗

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1.2 Short-run Monetary Policy Analysis with the MIU model

1.2.1 A Closer Look at Money Demand

Note that real money balances MDt /Pt appear in the households’ utility function while the house-

hold chooses nominal money balances.

→ This implies that the price level will appear in the consumption-money optimality condition.

EXAMPLE: Suppose that the sub-utility function takes the natural log form so that:

u

(

ct, lt,

MDt

Pt

)

= log ct + log lt + log

MDt

Pt

Evaluated at this utility function, the consumption-money optimality condition becomes:

∂u/∂MDt

∂u/∂ct

=

it+1

1 + it+1

1/(MDt /Pt)

1/ct

=

it+1

1 + it+1

ct

(MDt /Pt)

=

it+1

1 + it+1

⇒ MDt =

(

1 + it+1

it+1

)

Ptct

⇒ Implications of the Money Demand Condition: Since the money market equilibrium condition

MSt = M

D

t =

(

1 + it+1

it+1

)

Ptct always holds in general equilibrium changes to M

S

t must change

one or more of it+1, Pt, and ct.

→ The extent to which real variables are a�ected depends on the speed which Pt adjusts within

a given period.

1.2.2 Monetary Policy Shocks

• A Monetary Policy Shock will unexpectedly increase or decrease MSt .

EXAMPLE: Suppose the Central Bank increases MSt . Use a money market and goods market

diagram to graphically show how output and/or the price level is a�ected when Aggregate Supply

is Classical, and when it is Keynesian.

⇒ Classical Aggregate Supply: No speci�ed relationship between Pt and qt

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• Monetary Neutrality: monetary policy shocks have no e�ect on real economic activity.

→ Occurs when goods prices and wages are fully �exible

⇒ Intuition: Despite the initial reduction in it+1, which acts to increase desired ct given the

households’ intertemporal optimality condition, the within-period increase in Pt exactly o�sets

the e�ect of the decrease in it+1 on ct.

→ Since consumption is una�ected, the �nancial market equilibrium remains unchanged.

→ Since there are no changes to the supply or demand schedules, nominal wages adjust so

that the labor market equilibrium remains unchanged.

⇒Keynesian Aggregate Supply: Assumed that Pt may not change within a given period.

• Monetary Non-Neutrality: monetary policy shocks a�ect real economic activity.

→ Occurs when goods prices and/or wages are `sticky’.

⇒ Intuition: The initial reduction in it+1 increases desired ct given the households’ intertem-

poral optimality condition, which is not o�set by an increase in Pt.

→ Since current consumption changes, there will be a decrease in supply schedule in the �-

nancial market and an increase in the labor supply schedule in the labor market (both resulting

from the households’ inter- and intratemporal optimality conditions).

→ There will be a change to real economic variables in general equilibrium.

1.3 Limits to Conventional Monetary Policy

• The Lower Bound: Nominal interest rates cannot fall below some lower bound i.

→ As nominal interest rates approach i, typically to be a rate close to zero, the household becomes

indi�erent between holding money or interest-bearing assets as a store of value.

⇒ At the lower bound, increases in the money supply cannot e�ect real economic activity

through changes to the nominal interest rate.

→ Consumption-money condition becomes

∂u/∂MDt

∂u/∂ct

=

i

(1 + i)

, which implies that money

demand becomes horizonal over i.

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Chapter 15: Monetary Policy in the Intertemporal Framework

MIU ModelShort-run Monetary Policy Analysis with the MIU modelLimits to Conventional Monetary Policy

Macroeconomic Theory

1 Interlude: General Equilibrium Macroeconomics

All of the models that we have used thus far have been partial equilibrium, as we have looked at the

behavior of agents one side of each market in isolation. We now move to a general equilibrium

setting where we study states of the model where each market has reached a `market-clearing’

state where quantity supplied equals quantity demanded.

⇒ Each of our three macro markets � labor market, �nancial market, and goods market �

will have an associated Equilibrium Condition that describes a situation where markets clear

such that quantity supplied equals quantity demanded.

1.1 Labor Market Equilibrium

Representative household supplies labor and representative �rm demands labor, the interaction

of which determines market-clearing equilibrium real wage rate wt∗ and labor nt*.

Household Labor Supply Optimality Condition:

∂u/∂lt

∂u/∂ct

= wt (1)

Firm Labor Demand Optimality Condition:

∂f

∂nt

= wt (2)

→ Setting Equation (2) and (1) equal, yields the labor market equilibrium condition.

∂u/∂lt

∂u/∂ct

=

∂f

∂nt

(3)

Economic Intuition: When the representative household and �rm face the same real wage rate,

quantity of labor supplied and demanded are equal because the labor supply function is strictly

increasing in the wage rate and that the labor demand function is strictly decreasing in the wage

rate, i.e. they intersect at that wage rate. In the labor market equilibrium, households are paid

the marginal product of their labor.

→ Graphically, labor market equilibrium in (nt,wt) space:

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1.2 Financial Market Equilibrium

Representative household supplies savings and representative �rm demands savings for invest-

ment, the interaction of which determines market-clearing equilibrium real interest rate rt+1∗

and supply of savings st* and investment invt∗.

Household Savings Supply Optimality Condition:

∂u/∂ct

β∂u/∂ct+1

= 1 + rt+1 (4)

Firm Capital Demand Optimality Condition:

∂f

∂kt+1

= rt+1 + δ (5)

→ Using Equation (4) and (5) to eliminate the real interest rate, yields the �nancial market

equilibrium condition:

∂u/∂ct

β∂u/∂ct+1

= 1 +

∂f

∂kt+1

−δ (6)

Economic Intuition: When the representative household and �rm face the same real interest rate,

quantity of savings supplied is equal to the quantity of savings demanded for investment because

the savings supply function is strictly increasing in the real interest rate and that the investment

demand function is strictly decreasing in the real interest rate, i.e. they intersect at that real

interest rate. In the �nancial market equilibrium, households receive an interest rate on their

principal equal to the marginal product of capital less the marginal value of capital depreciated.

→ Graphically, �nancial market equilibrium in ({st, invt},rt+1) space:

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1.3 Goods Market Equilibrium

The interaction aggregate demand and aggregate supply of goods and services, which determines

market-clearing equilibrium goods price Pt∗ and level of total real output qt ∗.

• Aggregate Demand (AD): The desired quantity demanded for all goods and services by

households, �rms, and government.

ADt ≡ ct ∗+invt∗ (7)

Note that if there were taxes in the model to fund government spending, gt, we would add gt to

the expression for aggregate demand.

→ How does ADt vary with Pt?

Household Savings Supply Nominal Optimality Condition:

∂u/∂ct

β∂u/∂ct+1

=

Pt

Pt+1

(1 + it+1)

→ If Pt ↑, then ct∗ ↓ if substitution e�ect dominates income e�ect

Firm Investment Demand Nominal Optimality Condition:

∂f

∂kt+1

=

Pt

Pt+1

(1 + it+1)

→ If Pt ↑, then kt+1∗ and invt∗ ↓

⇒ Because ct∗ and invt∗ decrease as Pt increase, ADt is downward sloping in (qt,Pt) space:

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• Aggregate Supply (AS): The desired quantity supplied by �rms.

ASt ≡ f(kt∗,nt∗) (8)

→ In our current framework, the AS function has no relationship with Pt, and therefore is

vertical in (qt,Pt) space. This is called the `classical’ AS function.

→ Setting ADt = ASt as in Equations (7) and (8) yields the goods market equilibrium condition:

f(kt∗,nt∗) = ct ∗+invt ∗ (9)

⇒ When the goods market equilibrium condition (9), the labor market equilibrium condition

(3), and the �nancial market equilibrium condition (6) simultaneously hold, the model is said

to be in general equilibrium.

→ Along with budget constraints of all economic agents, these conditions completely char-

acterize the general equilibrium solution.

→ In other frameworks the AS function is completely horizontal (Keynesian AS) or upwards

sloping (New-Keynesian AS) over Pt. The slope of the AS curve in these models depends on

how quickly Pt adjusts relative to other prices (wt and rt+1). If Pt does not change at our model

frequency (referred to as `sticky prices’) then the AS function is completely horizontal. The

implication of the classical AS function is therefore that Pt adjusts instantaneously in response

to changes to costs of production.

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Interlude: General Equilibrium Macroeconomics

Labor Market Equilibrium

Financial Market Equilibrium

Goods Market Equilibrium

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