A Modern Equilibrium Model
Jesús Fernández-Villaverde
University of Pennsylvania
1
Household Problem
• Preferences:


1+γ
∞
 c1−σ

X
lt
t
t
β
max E
−ψ
1 − σ
1 + γ
t=0
• Budget constraint:
ct + kt+1 = wtlt + rtkt + (1 − δ) kt, ∀ t > 0
• Complete markets and Arrow-Debreu securities.
2
First Order Conditions
c−σ
= λt
t
βEtc−σ
t+1 = λt+1
γ
ψlt
= λtwt
λt = λt+1 (rt+1 + 1 − δ)
3
Problem of the Firm
• Neoclassical production function:
yt = Atktαlt1−α
• By profit maximization:
αAtktα−1lt1−α = rt
(1 − α) Atktαlt1−αlt−1 = wt
• Investment xt induces a law of motion for capital:
kt+1 = (1 − δ) kt + xt
4
Evolution of the technology
• At changes over time.
• It follows the AR(1) process:
log At = ρ log At−1 + zt
zt ∼ N (0, σ z )
• Interpretation of ρ.
5
A Competitive Equilibrium
• We can define a competitive equilibrium in the standard way.
• Conditions:
c−σ
= βEtc−σ
t
t+1 (rt+1 + 1 − δ)
γ
ψlt = λtwt
rt = αAtktα−1lt1−α
wt = (1 − α) Atktαlt1−αlt−1
ct + kt+1 = Atktα−1lt1−α + (1 − δ) kt
log At = ρ log At−1 + zt
6
Behavior of the Model
• We have an initial shock: productivity changes.
• We have a transmission mechanism: intertemporal substitution and
capital accumulation.
• We can look at a simulation from this economy.
• Why only a simulation?
7
Comparison with US economy
• Simulated Economy output fluctuations are around 75% as big as
observed fluctuations.
• Consumption is less volatile than output.
• Investment is much more volatile.
• Behavior of hours.
8
Assessment of the Basic Real Business Model
• It accounts for a substantial amount of the observed fluctuations.
• It accounts for the covariances among a number of variables.
• It has some problems accounting for the behavior of the hours worked.
• More important question: where do productivity shocks come from?
9
Negative Productivity Shocks I
• The model implies that half of the quarters we have negative technology shocks.
• Is this plausible? What is a negative productivity shocks?
• Role of trend.
10
Negative Productivity Shocks II
• s.d. of shocks is 0.01. Mean quarter productivity growth is 0.0048 (to
give us a 1.9% growth per year).
• As a consequence, we would only observe negative technological shocks
when εt < −0.0048.
• This happens in the model around 33% of times.
• Ways to fix it.
11
Some Policy Implications
• The basic model is Pareto-efficient.
• Fluctuations are the optimal response to a changing environment.
• Fluctuations are not a sufficient condition for inefficiencies or for government intervention.
• In fact in this model the government can only worsen the allocation.
• Recessions have a “cleansing” effect.
12
Extensions I
• We can extend our model in several directions.
• Examples we are not going to cover:
1. Fiscal Policy shocks (McGrattan, 1994).
2. Agents with Finite Lives (Rı́os-Rull, 1996).
3. Indivisible Labor (Rogerson, 1988, and Hansen, 1985).
4. Home Production (Benhabib, Rogerson and Wright, 1991).
13
Extensions We Study (Cumulative)
• Money (Cooley and Hansen, 1989).
14
Money
• Money in the Utility function (MIU).
• Comparison with Cash-in-Advance.
• You can show both approaches are equivalent.
• Are we doing the right thing (Wallace, 2001)?
15
Households
• Utility function:


Ã
!1−ξ
1+γ
1−σ
c

lt
mt
υ
t
t
E0
β
−ψ
.
+
1 − σ

1 − ξ pt
1+γ
t=0
∞
X
• Budget constraint:
mt bt+1
mt−1
bt
ct + xt +
+
= wtlt + rtkt +
+ Rt + Tt + Πt
pt
pt
pt
pt
16
First Order Conditions I
c−σ
= λt
t
βEtc−σ
t+1 = λt+1
γ
υ
Ã
ψlt
!−ξ
= λtwt
1
1
+ λt+1
pt
pt+1
1
1
λt
= λt+1Rt+1
pt
pt+1
λt = λt+1 (rt+1 + 1 − δ)
mt
pt
= λt
17
First Order Conditions II
−σ
c−σ
=
βE
{c
t
t
t+1 (rt+1 + 1 − δ)}
pt
−σ
=
βE
{c
R
}
c−σ
t t+1 t+1
t
pt+1
γ
ψlt = c−σ
t wt
υ
Ã
mt
pt
!−ξ
−σ
= c−σ
+
E
{c
t
t
t+1
18
pt
pt+1
}
The Government Problem
• The government sets the nominal interest rates according to the Taylor
rule:
µ ¶γ R µ ¶γ Ã !γ y
Rt
π t π yt
Rt+1
eϕt
=
R
R
π
y
• How? Through open market operations.
• How are those operations financed? Lump-sum transfers Tt such that
the deficit are equal to zero:
Tt =
mt mt−1 bt+1
bt
−
+
− Rt
pt
pt
pt
pt
19
Interpretation
• π: target level of inflation (equal to inflation in the steady state),
• y: the steady state output
• R: steady state gross return of capital.
• ϕt: random shock to monetary policy distributed according to N (0, σ ϕ).
• The presence of the previous period interest rate, Rt, is justified because we want to match the smooth profile of the interest rate over
time observed in U.S. data.
20
Advantages and Disadvantages of Taylor Rules
• Advantages:
1. Simplicity.
2. Empirical foundation.
• Problems:
1. Normative versus positive.
2. Empirical specification.
21
Behavior of the Model
• We have a second shock: interest shock.
• We have a transmission mechanism: intertemporal substitution and
capital accumulation.
• For a standard calibration, the second shock buys us very little.
22
Extensions We Study (Cumulative)
• Money (Cooley and Hansen, 1989).
• Monopolistic Competition (Blanchard and Kiyotaki, 1987, and Horstein,
1993).
23
Monopolistic Competition
• Final good producer. Competitive behavior.
• Continuum of intermediate good producers with market power.
• Alternative formulations: continuum of goods in the utility function.
• Otherwise, the model is the same as the model with money.
24
The Final Good Producer
• Production function:
yt =
ÃZ
1
0
!
ε−1
y ε di
it
ε
ε−1
where ε controls the elasticity of substitution.
• Final good producer is perfectly competitive and maximize profits,
taking as given all intermediate goods prices pti and the final good
price pt.
25
Maximization Problem
• Thus, its maximization problem is:
max ptyt −
yit
• First order conditions are:
ε
pt
ε−1
ÃZ
1
0
!
ε−1
y ε di
it
ε −1
ε−1
26
Z 1
0
pityitdi
ε − 1 ε−1
−1
− pit = 0
yitε
ε
∀i
Working with the First Order Conditions I
• Dividing the first order conditions for two intermediate goods i and j,
we get:
pit
=
pjt
Ã
yit
yjt
!− 1
ε
or:
pjt =
Ã
• Interpretation.
27
yit
yjt
!1
ε
pit
Working with the First Order Conditions II
• Hence:
1 ε−1
yε y ε
pjtyjt = pit it jt
• Integrating out:
Z 1
1
yε
Z 1
pjtyjtdj = pit it
0
0
28
ε−1
y ε dj
jt
1 ε−1
yε y ε
= pit it t
Input Demand Function
R1
• By zero profits (ptyt = 0 pjtyjtdj), we get:
1 ε−1
yε y ε
ptyt = pit it t
1 −1
yε y ε
⇒ pt = pit it t
• Consequently, the input demand functions associated with this problem are:
yit =
Ã
pit
pt
!−ε
• Interpretation.
29
yt
∀i
Price Level
R1
• By the zero profit condition ptyt = 0 pityitdi and plug-in the input
demand functions:
Z 1
Ã
!−ε
pt =
ÃZ
1
pit
pit
ptyt =
pt
0
• Thus:
ytdi ⇒ p1−ε
=
t
0
30
!
p1−ε
it di
1
1−ε
Z 1
0
p1−ε
it di
Intermediate Good Producers
• Continuum of intermediate goods producers.
• No entry/exit.
• Each intermediate good producer i has a production function
α l1−α
yit = Atkit
it
• At follows the AR(1) process:
log At = ρ log At−1 + zt
zt ∼ N (0, σ z )
31
Maximization Problem I
• Intermediate goods producers solve a two-stages problem.
• First, given wt and rt, they rent lit and kit in perfectly competitive
factor markets in order to minimize real cost:
min {wtlit + rtkit}
lit,kit
subject to their supply curve:
α l1−α
yit = Atkit
it
32
First Order Conditions
• The first order conditions for this problem are:
α l−α
wt = % (1 − α) Atkit
it
α−1 1−α
rt = %αAtkit
lit
where % is the Lagrangian multiplier or:
α wt
kit =
lit
1 − α rt
• Note that ratio capital-labor only is the same for all firms i.
33
Real Cost
• The real cost of optimally using lit is:
µ
¶
α
wtlit +
wtlit
1−α
• Simplifying:
µ
¶
1
wtlit
1−α
34
Marginal Cost I
• The firm has constant returns to scale.
• Then, we can find the real marginal cost mct by setting the level of
labor and capital equal to the requirements of producing one unit of
α l1−α = 1
good Atkit
it
• Thus:
α l1−α = A
Atkit
t
it
Ã
α wt
lit
1 − α rt
!α
35
1−α
lit
= At
Ã
α wt
1 − α rt
!α
lit = 1
Marginal Cost II
• Then:
µ
¶
Ã
1
1
α wt
wt
mct =
1−α
At 1 − α rt
!−α
µ
¶1−α µ ¶α
1
1
1 1−α α
=
w
rt
1−α
α
At t
• Note that the marginal cost does not depend on i.
• Also, from the optimality conditions of input demand, input prices
must satisfy:
α wt
kt =
lt
1 − α rt
36
Maximization Problem II
• The second part of the problem is to choose price that maximizes
discounted real profits, i.e.,
max
pit
(Ã
!
pit
∗
− mct yit
pt
)
subject to
∗
yit+τ
=
• First order condition:
Ã
1 pit
pt pt
!−ε
Ã
Ã
pit
pt
!−ε
pit
yt+τ − ε
− mct
pt
37
yt+τ ,
!Ã
pit
pt
!−ε−1
1
yt+τ = 0
pt
Mark-Up Condition
• From the fist order condition:
1−ε
Ã
!Ã
!−1
pit
pit
− mct
=0⇒
pt
pt
pit = ε (pit − mctpt) ⇒
ε
pit =
mctpt
ε−1
• Mark-up condition.
• Reasonable values for ε.
38
Behavior of the Model
• Presence of monopolistic competition is, by itself, pretty irrelevant.
• Why? Constant mark-up.
• Similar to a tax.
• Solutions:
1. Shocks to mark-up (maybe endogenous changes).
2. Price rigidities.
39
Extensions We Study (Cumulative)
• Money (Cooley and Hansen, 1989).
• Monopolistic Competition (Blanchard and Kiyotaki, 1987, and Horstein,
1993).
• Price rigidities (Calvo, 1983).
40
The Baseline Sticky Price Model
• The basic structure of the economy is as before:
1. A representative household consumes, saves, holds money, and
works.
2. There is a monetary authority that fixes the one-period nominal
interest rate through open market operations with public debt.
3. Final output is manufactured by a final good producer, which uses
as inputs a continuum of intermediate goods manufactured by monopolistic competitors.
• Additional constraint: the intermediate good producers face the constraint that they can only change prices following a Calvo’s rule.
41
Maximization Problem II
• The second part of the problem is to choose price that maximizes
discounted real profits, i.e.,
max Et
pit
∞
X
(βθp)τ vt+τ
τ =0
(Ã
!
pit
∗
− mct+τ yit+τ
pt+τ
subject to
∗
yit+τ
=
Ã
pti
pt+τ
!−ε
yt+τ ,
• Think about different elements of the problem.
42
)
Pricing I
• Calvo pricing.
• Parameter θp.
• What if θp = 0?
43
Pricing II
• Alternative pricing mechanisms:
1. Time-dependent pricing: Taylor and Calvo.
2. State-dependent pricing.
• Advantages and disadvantages:
1. Klenow and Kryvstov (2005): “intensive” versus “extensive” margin of price changes.
2. Caplin and Spulber (1987), Dotsey et al. (1999).
44
Discounting
• vt is the marginal value of a dollar to the household, which is treated
as exogenous by the firm.
• We have complete markets in securities,
• Then, this marginal value is constant across households.
• Consequently, β τ vt+τ is the correct valuation on future profits.
45
First Order Condition
• Substituting the demand curve in the objective function, we get:
Ã


!
Ã
!
1−ε
−ε
∞


X
p
p
it
ti
τ
∗
max Et
(βθp) vt+τ 
−
mct+τ  yit+τ
pit


pt+τ
pt+τ
τ =0
• The solution p∗it implies the first order condition:

µ ∗ ¶1−ε

pit

∗−1

∞


(1
−
ε)
p
X
it
pt+τ

µ ∗ ¶−ε
(βθp)τ vt+τ 
Et


pit

∗−1

τ =0
+ε
p

it mct+τ
pt+τ
46







 ∗
 yit+τ = 0





Working on the Expression I
• Then:
!
)
∗
pit ∗−1
τ
Et
(βθp) vt+τ
(1 − ε)
pit + εp∗−1
= 0⇒
it mct+τ yit+τ
p
t+τ
τ =0
(Ã
!
)
∞
∗
X
pit
ε
τ
Et
(βθp) vt+τ
−
= 0
mct+τ yit+τ
p
ε
−
1
t+τ
τ =0
∞
X
(Ã
• Thus, in a symmetric equilibrium, in every period, 1 − θp of the intermediate good producers set p∗t as their price policy function, while
the remaining θp do not reset their price at all.
47
Working on the Expression II
• Note, if θp = 0 (all firms move), all terms τ > 0 are zero:
vt
• Then
(Ã
p∗it
ε
−
mct
pt
ε−1
!
yit
)
=0
ε
mctpt
ε−1
the result we had from the basic monopolistic case.
p∗it =
48
Price Level
• The price index evolves:
h
i 1
1−ε
∗1−ε 1−ε
pt = θppt−1 + (1 − θp) pt
• Dividing by pt−1:
πt =
p∗t
∗
where π t = p .
t−1
pt
pt−1
h
i 1
∗1−ε 1−ε
= θp + (1 − θp) π t
49
Aggregation I
• Now, we derive an expression for aggregate output.
• Remember that:
kit
α wt
=
lit
1 − α rt
• Since this ratio is equivalent for all intermediate firms, it must also be
the case that:
α wt
kt
=
lt
1 − α rt
50
Aggregation II
• We get:
kit
kt
=
lit
lt
• If we substitute this condition in the production function of the interα l1−αwe derive:
mediate good firm yit = Atkit
it
yit = At
Ã
kit
lit
!α
51
lit = At
Ã
kt
lt
!α
lit
Aggregation III
• The demand function for the firm is:
yit =
Ã
pit
pt
!−ε
yt
• Thus, we find the equality:
Ã
pit
pt
!−ε
yt = At
52
Ã
∀i,
kt
lt
!α
lit
Aggregation IV
• If we integrate in both sides of this equation:
yt
Z 1 Ã !−ε
pit
0
pt
di = At
• Then:
yt =
where
vt =
Ã
³R
´− 1
−ε
1
and jt = 0 pit di ε .
lt
0
litdi = Atktαlt1−α
At α 1−α
kt lt
vt
Z 1 Ã !−ε
pit
0
!α Z
1
kt
pt
53
jt−ε
di = −ε = jt−εpεt
pt
Interpretation of jt
• What is jt?
• Measure of price disperssion.
• Measure of efficiency losses implied by inflation.
• Important to remenber for welfare analysis.
54
Equilibrium
• A definition of equilibrium in this economy is standard.
• The symmetric equilibrium policy functions are determined by the following blocks:
1. The first order conditions of the household.
55
The First Order Conditions of the Household
−σ
c−σ
=
βE
{c
t
t
t+1 (rt+1 + 1 − δ)}
−σ
−σ Rt+1
ct = βEt{ct+1
}
π t+1
γ
ψlt = c−σ
t wt
Ã
!−ξ
mt
−σ
−σ 1
υ
= ct + Et{ct+1
}
pt
π t+1
mt bt+1
m
bt
+
= wtlt + rtkt + t−1 + Rt + Tt + Πt
ct + xt +
pt
pt
pt
pt
56
Equilibrium
• A definition of equilibrium in this economy is standard.
• The symmetric equilibrium policy functions are determined by the following blocks:
1. The first order conditions of the household.
2. Pricing decisions by firms.
57
Pricing Decisions by Firms.
The firms that can change prices set them to satisfy
Et
∞
X
(βθp)τ vt+τ
τ =0
(Ã
p∗t
pt+τ
where:
∗
yit+τ
=
Ã
−
pti
pt+τ
!
ε
mct+τ yit+τ
ε−1
!−ε
yt+τ ,
and
µ
¶1−α µ ¶α
1
1
1 1−α α
wt rt
mct =
1−α
α
At
58
)
=0
Equilibrium
• A definition of equilibrium in this economy is standard.
• The symmetric equilibrium policy functions are determined by the following blocks:
1. The first order conditions of the household.
2. Pricing decisions by firms.
3. Pricing of inputs and outputs.
59
Input and Output Prices
• Input prices satisfy:
α wt
kt =
lt
1 − α rt
• The price level evolves:
πt =
pt
pt−1
h
i 1
∗1−ε 1−ε
= θp + (1 − θp) π t
Equilibrium
60
• A definition of equilibrium in this economy is standard.
• The symmetric equilibrium policy functions are determined by the following blocks:
1. The first order conditions of the household.
2. Pricing decisions by firms.
3. Pricing of inputs and outputs.
4. Taylor rule and market clearing.
61
Taylor Rule and Market Clearing
• Taylor rule
Rt+1
=
R
µ
¶γ R µ ¶γ Ã !γ y
Rt
π t π yt
eϕt
R
π
• Markets clear:
ct + xt = yt =
y
At α 1−α
kt lt
vt
R 1 ³ pit ´−ε
where vt = 0 p
di and kt+1 = (1 − δ) kt + xt.
t
62
Further Extensions
• Investment-specific technological change.
• Sticky wages.
• Open economy.
63