Monetary Theory: Monopolistic Competition
Behzad Diba
University of Bern
March 2011
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
1 / 14
Di¤erentiated Products
To model price rigidity (later), we will need a setup with …rms setting
their prices
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
2 / 14
Di¤erentiated Products
To model price rigidity (later), we will need a setup with …rms setting
their prices
A tractable way to do this is the Dixit-Stiglitz framework with
di¤erentiated products
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
2 / 14
Di¤erentiated Products
To model price rigidity (later), we will need a setup with …rms setting
their prices
A tractable way to do this is the Dixit-Stiglitz framework with
di¤erentiated products
A continuum of intermediate goods are imperfect substitutes and the
…nal consumption good is a CES aggregate over these:
Ct =
8
<Z1
:
0
[Ct (i )]
e 1
e
di
9 e
=e 1
;
where e > 1 is the elasticity of substitution across intermediate goods
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
2 / 14
Di¤erentiated Products
To model price rigidity (later), we will need a setup with …rms setting
their prices
A tractable way to do this is the Dixit-Stiglitz framework with
di¤erentiated products
A continuum of intermediate goods are imperfect substitutes and the
…nal consumption good is a CES aggregate over these:
Ct =
8
<Z1
:
0
[Ct (i )]
e 1
e
di
9 e
=e 1
;
where e > 1 is the elasticity of substitution across intermediate goods
The producer of good i sets a price Pt (i ), and the representative
consumer decides how much of each good to buy
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
2 / 14
Households
Representative household solves
max E0
1
X
t
U (Ct; Nt)
t=0
where
Ct
Z
1
Ct(i)
1
1
1
di
0
subject to
Z
1
Pt(i) Ct(i) di + Qt Bt
0
for t = 0; 1; 2; ::: plus solvency constraint.
Bt
1
+ W t Nt
Tt
Optimality conditions
1. Optimal allocation of expenditures
Pt(i)
Pt
Ct(i) =
implying
Z
Ct
1
Pt(i) Ct(i) di = Pt Ct
0
where
Pt
Z
1
1
1
Pt(i)1
di
0
2. Other optimality conditions
Qt =
Un;t Wt
=
Uc;t
Pt
Uc;t+1 Pt
Et
Uc;t Pt+1
Retailers
An alternative representation is to consider perfectly competitive
retailers who buy the intermediate goods and produce a single …nal
good using the technology
Yt =
(Institute)
8
<Z1
:
0
[Yt (i )]
e 1
e
di
9 e
=e 1
(1)
;
Monetary Theory: Monopolistic Competition
March 2011
3 / 14
Retailers
An alternative representation is to consider perfectly competitive
retailers who buy the intermediate goods and produce a single …nal
good using the technology
Yt =
8
<Z1
:
0
[Yt (i )]
e 1
e
di
9 e
=e 1
(1)
;
Retailers sell the …nal good to consumers at a price Pt
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
3 / 14
Retailers
An alternative representation is to consider perfectly competitive
retailers who buy the intermediate goods and produce a single …nal
good using the technology
Yt =
8
<Z1
:
[Yt (i )]
0
e 1
e
di
9 e
=e 1
(1)
;
Retailers sell the …nal good to consumers at a price Pt
Retailers maximize their pro…ts
P t Yt
Z1
Pt (i )Yt (i )di
0
subject to (1)
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
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Retailers’Demand Functions
The FOCs for maximizing
8
9 e
<Z1
=e 1
e 1
Pt
[Yt (i )] e di
:
;
0
Z1
Pt (i )Yt (i )di
0
are
Pt ( i ) = Pt
(Institute)
8
<Z1
:
0
[Yt (j )]
e 1
e
dj
9 1
=e 1
;
[Yt (i )]
1
e
Monetary Theory: Monopolistic Competition
1
= Pt [Yt ] e [Yt (i )]
March 2011
1
e
4 / 14
Retailers’Demand Functions
The FOCs for maximizing
8
9 e
<Z1
=e 1
e 1
Pt
[Yt (i )] e di
:
;
0
Z1
Pt (i )Yt (i )di
0
are
Pt ( i ) = Pt
8
<Z1
:
0
[Yt (j )]
e 1
e
dj
9 1
=e 1
;
[Yt (i )]
1
e
1
= Pt [Yt ] e [Yt (i )]
1
e
So, the retailers’demand functions are
Yt ( i ) =
(Institute)
Pt ( i )
Pt
e
Yt
Monetary Theory: Monopolistic Competition
March 2011
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Price Index
The zero-pro…t condition for retailers is
P t Yt =
Z1
0
(Institute)
Pt (i )Yt (i )di =
Z1
0
Pt ( i )
Pt ( i )
Pt
Monetary Theory: Monopolistic Competition
e
Yt di
March 2011
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Price Index
The zero-pro…t condition for retailers is
P t Yt =
Z1
Pt (i )Yt (i )di =
0
Z1
0
Pt ( i )
Pt ( i )
Pt
e
Yt di
This implies
( Pt )
1 e
=
Z1
[Pt (i )]1
e
di
0
and the Dixit-Stiglitz price index
Pt =
(Institute)
8
<Z1
:
0
[Pt (i )]1
e
di
9 1
=1 e
;
Monetary Theory: Monopolistic Competition
March 2011
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Consumers
Now, we can model consumers as maximizing
∞
E
∑ β t U ( C t , Nt )
t =0
subject to a solvency constraint and
Pt Ct + Qt Bt
Bt
1
+ Wt Nt
Tt
because the "expenditure minimization problem" is solved by retailers
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
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Consumers
Now, we can model consumers as maximizing
∞
E
∑ β t U ( C t , Nt )
t =0
subject to a solvency constraint and
Pt Ct + Qt Bt
Bt
1
+ Wt Nt
Tt
because the "expenditure minimization problem" is solved by retailers
We get our old FOCs for consumers
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
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Speci…cation of utility:
Ct1
U (Ct; Nt) =
1
Nt1+'
1+'
implied log-linear optimality conditions (aggregate variables)
wt
ct = Etfct+1g
pt =
1
ct + ' nt
(it
Etf
t+1 g
where it
log Qt is the nominal interest rate and
discount rate.
Ad-hoc money demand
mt
pt = yt
it
)
log
is the
Optimality conditions
1. Optimal allocation of expenditures
Pt(i)
Pt
Ct(i) =
implying
Z
Ct
1
Pt(i) Ct(i) di = Pt Ct
0
where
Pt
Z
1
1
1
Pt(i)1
di
0
2. Other optimality conditions
Qt =
Un;t Wt
=
Uc;t
Pt
Uc;t+1 Pt
Et
Uc;t Pt+1
Intermediate-goods Producers
Firm i sets Pt (i ) to maximize
P t ( i ) Yt ( i )
Wt Nt (i )
subject to
Yt (i ) = At [Nt (i )]1
and
Yt ( i ) =
(Institute)
Pt ( i )
Pt
α
e
Yt
Monetary Theory: Monopolistic Competition
March 2011
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Intermediate-goods Producers
Firm i sets Pt (i ) to maximize
P t ( i ) Yt ( i )
Wt Nt (i )
subject to
Yt (i ) = At [Nt (i )]1
and
Yt ( i ) =
Pt ( i )
Pt
α
e
Yt
For now, set α = 0 (just to lighten up the algebra) and write the
pro…t function as
[Pt (i )]1
(Institute)
e
( P t ) e Yt
Wt
At
[Pt (i )]
Monetary Theory: Monopolistic Competition
e
(Pt )e Yt
March 2011
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Price Setting
The FOC of …rm i, setting Pt (i ) to maximize
[Pt (i )]1
e
( P t ) e Yt
Wt
At
[Pt (i )]
e
( P t ) e Yt ,
is
(1
(Institute)
e) [Pt (i )]
e
( P t ) e Yt =
e
Wt
At
[Pt (i )]
Monetary Theory: Monopolistic Competition
e 1
( P t ) e Yt
March 2011
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Price Setting
The FOC of …rm i, setting Pt (i ) to maximize
[Pt (i )]1
e
( P t ) e Yt
Wt
At
[Pt (i )]
e
( P t ) e Yt ,
is
(1
e) [Pt (i )]
e
( P t ) e Yt =
Wt
At
e
[Pt (i )]
e 1
( P t ) e Yt
Since all …rms solve the same optimization problem, we use
Pt = Pt ( i ) =
(Institute)
e
e
1
Wt
At
Monetary Theory: Monopolistic Competition
March 2011
8 / 14
Price Setting
The FOC of …rm i, setting Pt (i ) to maximize
[Pt (i )]1
e
( P t ) e Yt
Wt
At
[Pt (i )]
e
( P t ) e Yt ,
is
(1
e) [Pt (i )]
e
( P t ) e Yt =
Wt
At
e
[Pt (i )]
e 1
( P t ) e Yt
Since all …rms solve the same optimization problem, we use
Pt = Pt ( i ) =
e
e
1
Wt
At
Note that Wt /At is nominal marginal cost here
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
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Monopoly Distortion
The equilibrium real wage is below the marginal product of labor:
Wt
=
Pt
(Institute)
1
e
e
At
Monetary Theory: Monopolistic Competition
March 2011
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Monopoly Distortion
The equilibrium real wage is below the marginal product of labor:
Wt
=
Pt
1
e
e
At
With our iso-elastic utility (used just for illustration), we got
( Ct )
σ
Wt
= ( Nt ) ϕ
Pt
which, in equilibrium (with Ct = At Nt ), gives
( Nt ) σ + ϕ =
1
e
e
( At ) 1
σ
illustrating that the monopoly distortion reduces employment, output,
and consumption
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
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Multi-period Firms
With ‡exible prices, our …rms solve static pro…t-maximization
problems
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
10 / 14
Multi-period Firms
With ‡exible prices, our …rms solve static pro…t-maximization
problems
But once we introduce price rigidity, the …rms will be solving dynamic
optimization problems
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
10 / 14
Multi-period Firms
With ‡exible prices, our …rms solve static pro…t-maximization
problems
But once we introduce price rigidity, the …rms will be solving dynamic
optimization problems
We need to specify the objective function of multi-period …rms
(decide at what rate they should discount future pro…ts)
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
10 / 14
Multi-period Firms
With ‡exible prices, our …rms solve static pro…t-maximization
problems
But once we introduce price rigidity, the …rms will be solving dynamic
optimization problems
We need to specify the objective function of multi-period …rms
(decide at what rate they should discount future pro…ts)
A natural way to do this is to introduce a stock market and assume
…rms maximize their stock-market value.
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
10 / 14
Multi-period Firms
With ‡exible prices, our …rms solve static pro…t-maximization
problems
But once we introduce price rigidity, the …rms will be solving dynamic
optimization problems
We need to specify the objective function of multi-period …rms
(decide at what rate they should discount future pro…ts)
A natural way to do this is to introduce a stock market and assume
…rms maximize their stock-market value.
Let Dt denote nominal dividends, Qts the stock price, and st the
number of shares
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
10 / 14
Multi-period Firms
With ‡exible prices, our …rms solve static pro…t-maximization
problems
But once we introduce price rigidity, the …rms will be solving dynamic
optimization problems
We need to specify the objective function of multi-period …rms
(decide at what rate they should discount future pro…ts)
A natural way to do this is to introduce a stock market and assume
…rms maximize their stock-market value.
Let Dt denote nominal dividends, Qts the stock price, and st the
number of shares
Set st = 1 in equilibrium (just a normalization)
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
10 / 14
Stock Market
Consumers maximize
∞
Et
∑ βj
t
U ( C j , Nj )
j =t
subject to a solvency constraint and
Pt Ct + Qt Bt + Qts st
(Institute)
Bt
1
+ (Qts + Dt )st
Monetary Theory: Monopolistic Competition
1
+ Wt Nt
Tt
March 2011
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Stock Market
Consumers maximize
∞
Et
∑ βj
t
U ( C j , Nj )
j =t
subject to a solvency constraint and
Pt Ct + Qt Bt + Qts st
Bt
1
+ (Qts + Dt )st
1
+ Wt Nt
Tt
Letting Λt denote the marginal value of nominal income (the
multiplier on the budget constraint), the FOCs for Ct and st are
Uc ,t
Qts Λt
(Institute)
= Pt Λ t
= βEt [Λt +1 (Qts+1 + Dt +1 )]
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March 2011
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Stock Prices
Iterate forward on
Qts Λt = βEt [Λt +1 (Qts+1 + Dt +1 )]
to get
Qts Λt = Et
∞
∑ β k Λ t + k Dt + k
k =1
and
Qts = Et
∞
∑ βk
k =1
(Institute)
Λ t +k
Λt
Dt + k
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March 2011
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Stock Prices
Iterate forward on
Qts Λt = βEt [Λt +1 (Qts+1 + Dt +1 )]
to get
Qts Λt = Et
∞
∑ β k Λ t + k Dt + k
k =1
and
Qts = Et
∞
∑ βk
k =1
Λ t +k
Λt
Dt + k
Note that this extends the simple "discounted dividends" model of
stock prices
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
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Firm’s Objective
So a …rm maximizing its market value (Qts + Dt ) st has to maximize
∞
Et
∑ βk
k =0
(Institute)
Λ t +k
Λt
Dt + k
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March 2011
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Firm’s Objective
So a …rm maximizing its market value (Qts + Dt ) st has to maximize
∞
Et
∑ βk
k =0
Λ t +k
Λt
Dt + k
Note that the "stochastic discount factor for nominal payo¤s,"
βk
Λ t +k
Λt
comes from the way consumers (the owners of …rms) discount
uncertain nominal ‡ows from the future to the present
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
13 / 14
Firm’s Objective
So a …rm maximizing its market value (Qts + Dt ) st has to maximize
∞
Et
∑ βk
k =0
Λ t +k
Λt
Dt + k
Note that the "stochastic discount factor for nominal payo¤s,"
βk
Λ t +k
Λt
comes from the way consumers (the owners of …rms) discount
uncertain nominal ‡ows from the future to the present
With our iso-elastic utility function, we have
βk
(Institute)
Λ t +k
Λt
= βk
Ct + k
Ct
σ
Monetary Theory: Monopolistic Competition
Pt
Pt + k
March 2011
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Stochastic Discount Factor
Although the nominal formulation above will serve us in Chapter 3, a
more common formulation is to express the real stock price as the
expected present value of real dividends, discounted at the consumers’
intertemporal marginal rate of substitution (which is the stochastic
discount factor for real payo¤s)
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
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Stochastic Discount Factor
Although the nominal formulation above will serve us in Chapter 3, a
more common formulation is to express the real stock price as the
expected present value of real dividends, discounted at the consumers’
intertemporal marginal rate of substitution (which is the stochastic
discount factor for real payo¤s)
Letting λt denote the multiplier on the real budget constraint (the
marginal value of real income), we get
Uc ,t = λt
and the stochastic discount factor
βk
λ t +k
λt
for real dividends (payo¤s) at date t + k
(Institute)
Monetary Theory: Monopolistic Competition
March 2011
14 / 14
Stochastic Discount Factor
Although the nominal formulation above will serve us in Chapter 3, a
more common formulation is to express the real stock price as the
expected present value of real dividends, discounted at the consumers’
intertemporal marginal rate of substitution (which is the stochastic
discount factor for real payo¤s)
Letting λt denote the multiplier on the real budget constraint (the
marginal value of real income), we get
Uc ,t = λt
and the stochastic discount factor
βk
λ t +k
λt
for real dividends (payo¤s) at date t + k
This general approach to asset pricing is called "the
consumption-based capital-asset-pricing" model (CCAPM)
(Institute)
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March 2011
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