1

Business Cycle
U.S. economy uctuates over time.
How can we build models to think about it?
Do we need di erent models than before to do so? Traditionally the answer was yes. Nowadays the answer is no.
We will focus on equilibrium models of the cycle.
2

Business Cycles and Economic Growth
How di erent are long-run growth and the business cycle?
Changes in Output per Worker Secular Growth Business Cycle
Due to changes in capital 1/3 0
Due to changes in labor 0
Due to changes in productivity 2/3
2/3
1/3
We want to use the same models with a slightly di erent focus.
3

Stochastic Neoclassical Growth Model
Cass (1965) and Koopmans (1965).
Brock and Mirman (1972).
Kydland and Prescott (1982).
Hansen (1985).
King, Plosser, and Rebelo (1988a,b).
4

References
King, Plosser, and Rebelo (1988a,b).
Chapter by Cooley and Prescott in Cooley's Frontier of Business Cycle
Research (in fact, you want to read the whole book).
Chapter by King and Rebelo ( Resurrection Real Business Cycle Models ) in Handbook of Macroeconomics .
Chapter 12 in Ljungqvist and Sargent.
5

Preferences
Preferences:
E
0 t
(1 + n ) t u c t s t t =0 for c t s t
0 ; l t s t
; l t s
2 (0 ; 1) t where n is population growth.
Standard technical assumptions (continuity, di erentiability, Inada conditions, etc...).
However, those still leave many degrees of freedom.
Restrictions imposed by economic theory and empirical observation.
6

Restrictions on Preferences
Three observations:
1. Risk premium relatively constant ) CRRA utility function.
2. Consumption grows at a roughly constant rate.
3. Stationary hours after the SWW ) Marginal rate of substitution between labor and consumption must be linear in consumption.
c t s t u c
= w t s t u l f l t s t
=
) w t s t
) t c
0 f l t s t
= t w
0
Explanation: income and substitution e ect cancel out.
7

Parametric Family
Only parametric that satisfy conditions (King, Plosser, and Rebelo,
1988a,b):
( cv ( l ))
1
1 if > 0, = 1
1 log c + log v ( l ) if = 1
Restrictions on v ( l ) :
1.
v 2 C
2
2. Depending on :
(a) If = 1, log v ( l ) must be increasing and concave.
(b) If < 1, v
1 must be increasing and concave.
3.
(c) If v ( l
>
) v
1,
00
( l v
)
1
> must be decreasing and convex.
(1 2 ) v
0
( l )
2 to ensure overall concavity of u:
8

Three Useful Examples
1. CRRA-Cobb Douglass:
1 l t s t
1
1
1
E
0 t =0 t
(1 + n ) t c t s t
2. Log-log (limit as !
1):
E
0 t =0 t
(1 + n ) t n log c t s t
+ log 1 l t s t o
3. Log-CRRA
E
0 t =0 t
(1 + n ) t
8
< log c t s t l t s t
1 +
1+
9
=
1
9

Household Problem
Let me pick log-log for simplicity:
E
0 t =0 t
(1 + n ) t n log c t s t
+ log 1 l t s t o
Budget constraint: c t s t
+ x t s t
= w t s t l t s t
+ r t s t k t s t 1
, 8 t > 0
Complete markets and Arrow securities.
We can price any security.
10

Problem of the Firm I
Neoclassical production function in per capita terms: y t s t
= e z t k t s t 1
(1 + ) t l t s t
1
Note: labor-augmenting technological change (Phelps, 1966).
We are setting up a model where the rm rents the capital from the household.
However, we could also have a model where rms own the capital and the households own shares of the rms.
Both environments are equivalent with complete markets.
11

Problem of the Firm II
By pro t maximization: e z t k t s t 1
(1 ) e z t k t s t 1
1
(1 + ) t l t s t
(1 + ) t l t s t
1
= r t s t
= w t s t
Investment x t induces a law of motion for capital:
(1 + n ) k t +1 s t
= (1 ) k t s t 1
+ x t s t
12

Evolution of the technology s t
= z t z t changes over time.
It follows the AR(1) process: z t
= z t 1
+ " t
" t
N (0 ; 1)
Interpretation of and :
13

Arrow-Debreu Equilibrium allocations f ^ t
( s t
) ; l b t
( s t
) ;
.
f b t
( s t
) ; b t
( s t
) ; b t
( s t
) g
1 t =0 ;s t 2 S t k t
( s t
) g
1 t =0 ;s t 2 S t such that: and
1. Given f ^ t
( s t
) g
1 t =0 ;s t 2 S t
; f ^ t
( s t
) ; l b t
( s t
) ; k t
( s t
) g
1 t =0 ;s t 2 S t solves f ^ t
( s t ) ; l b t
( s t max
) ; b t
( s t ) g
1 t =0 ;st 2 St
X s.t.
E
0 t =0 t
(1 + n ) t n log c t s t p t
( s t
) c t
( s t
) + (1 + n ) k t +1 t =0 s t 2 S t
X p t
( s t
) b t
( s t
) l t
( s t
) + b t
( s t
) + 1 t =0 s t 2 S t c t
( s t
) 0 for all t
+ log 1 l t s t s t k t +1 s t o
14

2. Firms pick f l b t
( s t
) ; k t
( s t
) g
1 t =0 ;s t 2 S t to minimize costs: e z t k t s t 1
(1 ) e z t k t s t 1
1
(1 + ) t l b t s t
(1 + ) t l b t s t
1
=
= b b t t s s t t
3. Markets clear: e z t k t s t 1 c t
( s t
) + (1 + n ) k t +1
(1 + ) t l b t s t
1 s t
=
+ (1 ) k t s t 1 for all t; all s t
2 S t
15

Sequential Markets Equilibrium I .
We introduce Arrow securities.
.
Household problem: c t
( s t
) ; l t
( s t
) ; k t
( s t
) ; n a t +1
( s t
; s t +1
) o s t +1
2 S solve
1 t =0 ;s t 2 S t max E
0 t
(1 + n ) t t =0 s.t.
c i t
( s t
) + (1 + n ) k t +1 n log c t s t s t
+
X s t +1 j s t
+ log 1 l t s t o
Q t
( s t
; s t +1
) a t +1
( s t
; s t +1
) w t
( s t
) l t
( s t
) + b t
( s t
) + 1 k t +1 s t
+ a t
( s t
) a t +1
( s t
; s c t
( s t t +1
)
) 0 for all
A t +1
( s t; s t +1 t
2 S t
) for all t; s t
2 S t
Role of A t +1
( s t +1
) :
16

Sequential Markets Equilibrium II c t
( s t
A SM equilibrium is prices for Arrow securities f allocations ^ i t
( s t
) ; l b t
( s t
) ; k t
( s t
) ; n
^ t
.
( s t
; s t +1
Q t
) o
( s t
; s t +1
) g
1 t =0 ;s
1 s t +1
2 S t =0 ;s t t 2 S
2 S t t ;s t +1
2 S and input prices f b t
( s t
) ; b t
( s t
) g
1 t =0 ;s t 2 S t
; such that:
1. Given f Q t
( s t
; s t +1
) g
1 t =0 ;s t
2 S t
;s t +1
2 S and f b t
( s t
) ; b t
( s t
) g
1 t =0 ;s t
2 S t
,
^ t
( s t
) ; l b t
( s t
) ; k t
( s t
) ; n a t +1
( s t
; s t +1
) o s t +1
2 S lem of the household.
2. Firms pick f l b t
( s t
) ; t
( s t
) g
1 t =0 ;s t 2 S t
1 t =0 ;s t
2 S t solve the probto minimize costs: e z t k t s t 1
1
(1 + ) t l b t s t
1
= b t s t
(1 ) e z t k t s t 1
3. Markets clear for all t; all s t
(1 + ) t l b t s t
2 S t
= b t s t
)+(1 + n ) k t +1 s t
= e z t k t s t 1
(1 + ) t l b t s t
1
+(1 ) t s t 1
17

Recursive Competitive Equilibrium rium concept: Recursive Competitive Equilibrium (RCE).
Developed by Mehra and Prescott (1980).
RCE emphasizes the idea of de ning an equilibrium as a set of functions that depend on the state of the model.
Two interpretation for states:
1. Pay-o relevant states: capital, productivity, .....
2. Other states: promised utility, reputation, ....
Recursive notation: x and x
0
.
18

Value Function for the Household
.
Individual state: k:
Aggregate states: K and z:
Recursive problem: v ( k; K; z ) = max c;x;l n log c + log (1 l ) + (1 + n ) E v k
0
; K
0
; z
0 s:t: c + x = r ( K; z ) k + w ( K; z ) l
(1 + n ) k
0
= (1 ) k + x
(1 + n ) K
0
= (1 ) K + X ( K; z ) z
0
= z + "
0 j z o
19

De nition of Recursive Competitive Equilibrium
A RCE for our economy is a value function v ( k; K; z ), households policy functions, c ( k; K; z ) ; x ( k; K; z ) ; .
l ( k; K; z ), aggregate policy functions C ( K; z ) ; X ( K; z ), and L ( K; z ), and price functions r ( K; z ) and w ( K; z ) such that those functions satisfy:
1. Recursive problem of the household.
2. Firms maximize: e z
K
1
((1 + ) L ( K; z ))
1
(1 ) e z
K ((1 + ) L ( K; z ))
= r ( K; z )
= w ( K; z )
3. Consistency of individual and aggregate policy functions, c ( k; K; z ) =
C ( K; z ), x ( k; K; z ) = X ( K; z ), l ( k; K; z ) = L ( K; z ) ; 8 ( K; z ) :
4. Aggregate resource constraint:
C ( K; z ) + X ( K; z ) = e z
K ((1 + ) L ( K; z ))
1
; 8 ( K; z )
20

Equilibrium Conditions
1 c t
( s t )
= r t s t
E t c t +1
1 s t +1 r t +1 s t +1
= e z t c t s t
1 l t
( s t )
= w t s t k t s t 1
1
(1 + ) t l t
+ 1 s t
1 w t e z t s t k t s
= (1 ) e z t k t t 1 s t 1
(1 + )
(1 ) t l t s t c t s t
+ (1 + n ) k t +1
1
(1 + ) t l t s t s t
=
+ (1 ) k t s t 1 z t
= z t 1
+ " t
21

Scaling the Economy I
Economy has long-run growth rate equal to ( n + ).
Per capita terms, the economy grows at a rate .
Hence, the model is non-stationary and we need to rescale it.
General condition: transform every non-stationary variable into a stationary one by dividing it by (1 + ) t x t s t
= x t s t
(1 + ) t
22

Scaling the Economy II
We can rewrite the preferences (and adding a suitable constant):
= E
0
E
0 t =0 t =0 t
(1 + n ) t c t s t v l t s t
1 t
(1 + n ) t
(1 + ) t c t s t
1 v l t s t
1
E
0 t =0 t
(1 + n ) t
(1 + ) t (1 ) c t s t v l t s t
1
1
1
1
1
)
1
23

Scaling the Economy III
We can rewrite the preferences (and adding a suitable constant):and
= E
0
E
0 t
(1 + n ) t n log c t s t
+ log v l t s t t =0 t
(1 + n ) t n log (1 + ) t c t s t
+ log v l t o s t t =0
E
0 t
(1 + n ) t n log e t s t
+ log v l t s t o t =0 o
)
24

Scaling the Economy IV
The resource constraint, diving both sides by (1 + ) t c t s t e z t k t s t 1
+ (1 + n ) (1 + ) l t s t
1
+ (1 t +1
) s t
= k t s t 1
Input prices: w t r t s s t t
= e z t k t
= (1 ) e s t 1 z t t
1 l t s t 1 s l t t
1 s t
25

A New Competitive Equilibrium
We can de ne a competitive equilibrium in the rescaled economy.
Equilibrium conditions (log case): c t s t
(1 + )
= c t
( s t ) r t s t
E t c t +1
1 s t +1 r t +1 c t s t
1 l t
( s t )
= e z t k t
= e t s t s t 1
1 l t s t +1 s t
1
+ 1
+ (1 + w n t s t
= (1
) (1 + )
) e z t t s t 1 l t t +1 z t
= s t z
= t 1 e
+ z t k t s t 1
" t s t l t s t
1
+ (1 ) k t s t 1
26

Existence and Welfare Theorems
There is a unique equilibrium in this economy once we impose the right transversality condition.
Both welfare theorems hold.
We can move back and forth between the market equilibrium and the social planner's problem.
27

Behavior of the Model
We want to characterize behavior of the model.
Three type of dynamics:
1. Balanced growth path.
2. Transitional dynamics (Cass, 1965, and Koopmans, 1965).
3. Ergodic behavior.
28

Stochastic Behavior
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?
To simulate the model we need:
1. To select parameter values.
2. To compute the solution of the model.
29

Selecting Parameter Values
How do we determine the parameter values?
Two main approaches:
1. Calibration.
2. Statistical methods: Methods of Moments, ML, Bayesian.
Advantages and disadvantages.
30

Calibration as an Empirical Methodology
Emphasized by Lucas (1980) and Kydland and Prescott (1982).
Two sources of information:
1. Well accepted microeconomic estimates.
2. Matching long-run properties of the economy.
Problems of 1. and 2.
References:
1. Browning, Hansen and Heckman (1999) chapter in Handbook of
Macroeconomics .
2. Debate in Journal of Economic Perspectives, Winter 1996: Kydland and Prescott, Hansen and Heckman, Sims.
31

Calibration of the Standard Model
Parameters: , , , , , n , , : n : population growth in the data.
: per capita long run growth.
: capital income. Proprietor's income?
32

Balanced Growth Path
Equilibrium conditions in the BGP: c + (1 + n
1 + r
=
1 c c
1 l
( r + 1 )
= e
=
1 l
1 w = (1 )
) (1 + ) k =
1
+ (1 )
A system of 5 equations on 5 unknowns.
33

First:
Three Conditions of the Balanced Growth Path r = =
1 +
1 +
Also:
(1 + n ) (1 + ) k = (1 ) k + x )
= + 1 (1 + n ) (1 + )
Finally,
1 l
= (1 ) l
) =
1 1 l l
34

Using the Three Conditions to Calibrate the Model
First, we use data on hours of work to nd
= (1 )
1 l l
Second, give data and
= + 1 (1 + n ) (1 + ) we determine .
Finally, we get :
1
= (1 + ) + 1
35

Frisch Elasticity I
De ne the Frisch Elasticity as: d log l d log w c constant
For our parametric family:
1
1.
c (1 l )
1
1
1
:
1 l l
:
2. log c + log (1 l ):
1 l l
:
3. log c l
1+
1+
: 1 = :
36

Frisch Elasticity II
Empirical evidence is that l 1 = 3 (Ghez and Becker, 1975).
Then, our Frisch Elasticity is 2.
Empirical evidence:
1. Traditional view: MaCurdy (1981), Altonji (1986), Browning, Deaton and Irish (1985) between 0 and 0.5.
2. Revisionist view: between 0.5 and 1.6 (Browning, Hansen, and
Heckman, 1999). Some estimates (Imai and Keane, 2004) even higher (3.8).
37

Equivalence between Utility Functions
With log c t
+ log (1 l t
), the static FOC is: c t
1 l t
= w t while with log c t l
1+ t
1+
, the static FOC is
c t l t
= w t
Loglinearize both expressions: c
1 l cl c t
+
(1 l )
2 l b t
= w w t
) c t
+ l
1 l l b t
= b t
38

cl b t
+ c t
+ l b t
= w b t
) l b t
= w t
If we calibrate the model to l 1 = 3: c t
+
1
2 l b t
= w t and hence, both utility functions are equivalent if we make = l
1 l
.
In the case l 1 = 3, = 1 = 2 :
39

Solow Residual
Last step is to calibrate z t
= z t 1
+ " t
Obtain the Solow residual after a time trend has been removed.
Estimate and by OLS.
Problems of estimate.
40

Solution Methods
Value function iteration.
Projection.
Perturbation:
1. Generalization of linearization.
2.
Dynare .
41

General Structure of Linearized System
There are many linear solvers. Fundamental equivalence.
\A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models Easily" by Harald Uhlig.
Given m states x t
; n controls y t
; and k exogenous stochastic processes z t +1
, we have:
Ax t
+ Bx t 1
+ Cy t
+ Dz t
= 0
E t
( F x t +1
+ Gx t
+ Hx t 1
+ J y t +1
+ Ky t
+ Lz t +1
+ M z t
) = 0
E t z t +1
= N z t where C is of size l n; l n and of rank n; that F is of size
( m + n l ) n; and that N has only stable eigenvalues.
42

Policy Functions I
We guess policy functions of the form: x t
= P x t 1
+ Qz t y t
= Rx t 1
+ Sz t where P; Q; R; and S are matrices such that the computed equilibrium is stable.
43

Policy Functions I
For simplicity, suppose l = n: See Uhlig for general case (I have never be in the situation where l = n did not hold).
Then:
1.
P satis es the matrix quadratic equation:
F J C
1
A P
2
J C
1
B G + KC
1
A P KC
1
B + H = 0
The equilibrium is stable i max ( abs ( eig ( P ))) < 1.
2.
R is given by:
R = C
1
( AP + B )
44

3.
Q satis es:
N
0
F J C
= vec
1
A + I k
J C
J R + F P +
1
D L N + KC
G
1
KC
D M
1
A vec ( Q )
4.
S satis es:
S = C
1
( AQ + D )
45

How to Solve Quadratic Equations
To solve for the m m matrix P :
P
2
P = 0
1. De ne the 2 m 2 m matrices:
" #
=
I m
0 m
; and =
"
0 m
0 m
I m
#
2. Let s be the generalized eigenvector and be the corresponding generalized eigenvalue of with respect to : Then we can write s
0
= x
0
; x
0 for some x 2 < m
:
46

3. If there are m generalized eigenvalues
1
;
2
; :::; m together with generalized eigenvectors s
0
= x
0 i
; x
0 i s for some
1
; :::; s x i m
2 < of m with respect to and if ( x
1
; :::; x m
; written as
) is linearly independent, then:
P =
1 is a solution to the matrix quadratic equation where = [ x
1
; :::; x m
] and = [
1
; :::; m
]. The solution of P is stable if max j i j < 1.
Conversely, any diagonalizable solution P can be written in this way.
47

Comparison with US economy
Simulated Economy output uctuations are around 70% as big as observed uctuations.
Consumption is less volatile than output.
Investment is much more volatile.
Behavior of hours.
48

Assessment of the Basic Real Business Model
It accounts for a substantial amount of the observed uctuations.
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?
49

Negative Productivity Shocks ogy shocks.
Is this plausible? What is a negative productivity shocks?
Role of trend: negative shocks also include growth of technology below the trend.
s.d. of shocks is 0.007. Mean quarter productivity growth is 0.0047
(to give us a 1.9% growth per year).
As a consequence, we would only observe negative technological shocks when " t
< 0 : 0047.
This happens in the model around 25% of times. Comparison with the data.
50

Some Policy Implications
The basic model is Pareto-e cient.
Fluctuations are the optimal response to a changing environment.
Fluctuations are not a su cient condition for ine ciencies or for government intervention.
In fact in this model the government can only worsen the allocation.
Recessions have a \cleansing" e ect.
51

Asset Market Implications I
We will have the fundamental asset pricing equation:
Q t
( s t
; s t +1
) = s t +1 j s t u
0 c t +1 s t +1
; l t +1 s u 0 ( c t
( s t ) ; l t
( s t )) t +1
If utility is separable and log in consumption:
Q t
( s t
; s t +1
) = s t +1 j s t c c t s t t +1 s t +1
Now, c t s t is the equilibrium consumption.
Since c t s t is smooth in the model, Q t
( s t
; s t +1
) will also be smooth.
Hence, we will have the standard equity premium puzzle.
52

Asset Market Implications II
E t c c t s t t +1 s t +1
R b t s t
Return to invest in capital:
E t c c t s t t +1 s t +1 r t +1 s t +1
+ 1
By non-arbitrage:
E t c c s t t +1 s t +1
R b t s t
= E t c c t s t t +1 s t +1 r t +1 s t +1
+ 1
Presence of capital ties down returns.
53

Further Extensions
Two objectives:
1. Fix empirical problems.
2. Address additional questions.
Examples:
1. Indivisible labor supply.
2. Capacity utilization.
3. Investment Speci c technological change.
4. Monopolistic Competition.
54

Lotteries
Our
.
Hansen (1985).
General procedure to deal with non-convexities.
For example, an agent can either work 0 hours or l hours. Why?
Extensive versus intensive margin.
Then, expected utility: pu ( c
1
; l ) + (1 p ) u ( c
2
; 0)
Resource constrain in the economy (law of large numbers): pc
1
+ (1 p ) c
2
= c
55

Aggregation
First order condition: u c
( c
1
; l
.
u c
( c
2
; 0).
For our log-log utility function log c + log (1 l ) ; we have c = c
1
= c
2
Also, In the aggregate, we have that l = pl .
Then, expected utility is log c + p log (1 l ) + (1 p ) log 1 ) log c + Al where A = log(1 l ) l
:
Note that this utility function belongs to the class log c l
1+
1+
= 0, i.e., with in nite Frisch elasticity.
56 with

Capacity Utilization
In benchmark model, the short run elasticity of capital is zero while in the long run is in nite.
Empirical evidence of use of machinery, number of shifts, or electricity consumption.
Modi ed production function: y t s t
= e z t u t s t k t where u t is the utilization rate.
s t 1
Depreciation:
(1 + n ) k t +1 s t
= 1 u t s t
(1 + ) t l t s t
1 k t s t 1
+ x t s t
57

Combining Both Extensions
We can generate 70 percent of aggregate uctuations with a s.d. of
0.003.
How do we look at the Solow residual in this model?
This implies negative technological growth in around 5 percent of quarters, roughly observation in the data.
58

Investment-Speci c Technological Change
Greenwood, Herkowitz, and Krusell (1997 and 2000): importance of technological change speci c to new investment goods for understanding postwar U.S. growth and aggregate uctuations.
Observation: fall in the relative price of capital.
Implications for NIPA.
A simple way to model it:
(1 + n ) k t +1 s t
= (1 ) k t s t 1
+ v t x where v t is the inverse of the relative price of capital.
t s t
Two di erent technological shocks with di erent implications.
59

Monopolistic Competition
Final good producer with 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 standard RBC model.
60

The Final Good Producer
Production function: y t
( s t
) =
Z
0
1
( y it
( s t
))
" 1
" di
!
"
" 1 where " controls the elasticity of substitution.
Final good producer is perfectly competitive and maximize pro ts, taking as given all intermediate goods prices p ti
( s t
) and the nal good price p t
( s t
).
61

Maximization Problem
Thus, its maximization problem is: max y it
( s t
) p t
( s t
) y t
( s t
)
Z
0
1 p it
( s t
) y it
( s t
) di
First order conditions are for 8 i :
" p t
" 1
Z
0
1
( y it
( s t
))
" 1
" di
!
"
" 1
1
" 1
( y it
"
( s t
))
" 1
"
1 p it
( s t
) = 0
62

Working with the First Order Conditions
Dividing the rst order conditions for two intermediate goods i and j , we get:
.
p it
( s t
) p jt
( s t
)
= y it
( s t
)
!
1
" y jt
( s t
) or: p jt
( s t
) = y it
( s t
)
!
1
" p it
( s t
) y jt
( s t
)
Hence: p jt
( s t
) y jt
( s t
) = p it
( s t
) y it
( s t
)
1
" y jt
( s t
)
" 1
"
Integrating out:
Z
0
1 p jt
( s t
) y jt
( s t
) dj = p it
( s t
) y it
( s t
)
1
"
Z
0
1 " 1 y jt
" dj = p it
( s t
) y it
( s t
)
1
" y jt
( s t
)
" 1
"
63

Input Demand Function
By zero pro ts ( p t
( s t
) y t
( s t
) =
R
0
1 p jt
( s t
) y jt
( s t
) dj ), we get: p t
( s t
) y t
( s t
) = p it
( s t
) y it
( s t
)
1
" y jt
( s t
)
" 1
"
) p t
( s t
) = p it
( s t
) y it
( s t
)
1
" y t
( s t
)
1
"
Consequently, the input demand functions associated with this problem are: y it
( s t
) = p it
( s t
)
!
p t
( s t
)
" y t
( s t
) 8 i
Interpretation.
64

Price Level
By the zero pro t condition p t
( s t
) y t
( s t
) = plug-in the input demand functions:
R
0
1 p it
( s t
) y it
( p t
( s t
) y t
( s t
) =
Z
0
1 p it
) p t
( s t
)
( s t
)
1 " p it
( s t
= p t
( s t
)
Z
1 p it
0
)
!
( s t
)
" y t
1 "
( s t
) di di s t
) di and
Thus: p t
( s t
) =
Z
0
1 p it
( s t
)
1 " di
!
1
1 "
65

Intermediate Good Producers
Continuum of intermediate goods producers.
No entry/exit.
Each intermediate good producer i has a production function y it
( s t
) = A t k it
( s t
) l it
( s t
)
1
A t follows the AR(1) process: log A t
= log A t 1
+ z t z t
N (0 ; z
)
66

Maximization Problem I
Intermediate goods producers solve a two-stages problem.
First, given w t and r t
, they rent l it and k it in perfectly competitive factor markets in order to minimize real cost: l it
( s t min
) ;k it
( s t
) f w t
( s t
) l it
( s t
) + r t
( s t
) k it
( s t
) g subject to their supply curve: y it
= A t k it
( s t
) l it
( s t
)
1
67

First Order Conditions
The rst order conditions for this problem are: w r t t
( s t
) = % (1 ) A t k it
( s t
) l it
( s t
)
( s t
) = % A t k it
( s t
)
1 l it
( s t
)
1 where % is the Lagrangian multiplier or: k it
( s t
) =
1 w t r t
( s t
) l it
( s t
)
( s t
)
Note that ratio capital-labor only is the same for all rms i:
68

Real Cost
The real cost of optimally using l it is: w t
( s t
) l it
( s t
) +
1 w t
( s t
) l it
( s t
)
Simplifying:
1
1 w t
( s t
) l it
( s t
)
69

Marginal Cost I
The rm has constant returns to scale.
Then, we can nd the real marginal cost mc t
( s t
) by setting the level of labor and capital equal to the requirements of producing one unit of good A t k it
( s t
) l it
( s t
)
1
= 1
Thus:
A t k it
( s t
) l it
( s t
)
1
= A t
1
= A t
1
70 w r t t
( s t
) l it
( s t
) w t
( s t
)
!
( s t
) l it
!
( s t
) = 1 r t
( s t
) l it
( s t
)

Marginal Cost II
Then: mc t
( s t
) =
=
1
1
1
1
1
1 w t
( s t
)
A t
1 w r t t
1 1
A t w t
( s t
)
1
(
( s s t t
)
)
!
r t
( s t
)
Note that the marginal cost does not depend on i .
Also, from the optimality conditions of input demand, input prices must satisfy: k t
( s t
) =
1 w t r t
( s t
) l t
( s t
)
( s t
)
71

Maximization Problem II
The second part of the problem is to choose price that maximizes discounted real pro ts, i.e.,
( p it p max it
( s t
) p t
(
( s s t t
)
) mc t
( s t
)
!
y it
( s t
)
) subject to y it
( s t
) = p it
( s t
)
!
p t
( s t
)
" y t
( s t
) ; p
First order condition: it p t
( s t
)
!
( s t
)
" y p t t
( s t
)
( s t
)
" p it p t
( s t
( s t
)
) mc t
( s t
)
! p it p t
( s t
( s t
)
)
!
" 1 y t
( s t
) p t
( s t
)
= 0
72

Mark-Up Condition
From the st order condition:
1 " p it
( s t p t
( s t
)
) mc t
( s t
)
! p it
( s t p t
( s t
)
)
!
1
= 0 p it
( s t
) = " ( p it
( s t
) mc t
( s t
) p t
( s t
)) )
" p it
( s t
) =
" 1 mc t
( s t
) p t
( s t
)
)
Mark-up condition.
Reasonable values for ":
73

Aggregation I
To derive an expression for aggregate output, remember that: k l it it
(
( s s t t
)
)
=
1 w t
( s t
) r t
( s t
)
Since this ratio is equivalent for all intermediate rms, it must also be the case that: k l it it
(
( s s t t
)
)
= k t
( s t
) l t
( s t
)
=
1 w t
( s t
) r t
( s t
)
If we substitute this condition in the production function of the intermediate good rm A t k it
( s t
) l it
( s t
)
1 we derive:
!
!
y it
= A t k l it it
( s t
)
( s t
) l it
( s t
) = A t k t
( s t
) l t
( s t
) l it
( s t
)
74

Aggregation II y it
( s t
) = p it
( s t
)
!
p t
( s t
)
" y t
( s t
) 8 i;
Thus, we nd the equality: p it p t
(
( s t
) s t
)
!
" y t
( s t
) = A t k t
( s t
)
!
l t
( s t
) l it
( s t
) y t
(
If we integrate in both sides of this equation: s t
)
Z
0
1 p it
( s t
)
!
p t
( s t
)
" di = A t k t
( s t
)
! Z l t
( s t
) 0
1 l it
( s t
) di = A t k t
( s t
) l t
( s t
)
1
75

Aggregation III
Then: where y t
( s t
) =
A t
.
v t
( s t
) k t
( s t
) l t
( s t
)
1 v t
( s t
) =
Z
0
1 p it
( s t
)
!
p t
( s t
)
" di = j t
( s t
) p t
( s t
)
"
"
But note that: p t
( s t
) =
Z
0
1 p it
( s t
)
1 " di
!
1
1 "
= p it
( s t
) since all intermediate good producers charge the same price.
Then: v t
( s t
) =
R
0
1 p it
( s t
) p t
( s t
)
" di = 1 and: y t
= A t k t
( s t
) l t
( s t
)
1
76

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.
77

Further Extensions
We can extend our model in many other directions.
Examples we are not going to cover:
1. Fiscal Policy shocks (McGrattan, 1994).
2. Agents with Finite Lives (R os-Rull, 1996).
3. Home Production (Benhabib, Rogerson, and Wright, 1991).
78