Monetary Theory: Price Setting
Behzad Diba
University of Bern
March 2011
(Institute)
Monetary Theory: Price Setting
March 2011
1 / 10
Notation
Suppose the …rm producing intermediate good i, in our setup with
monopolistic competition, sets a price Pt (i ), and pays nominal
dividends
Dt (i ) = Pt (i )Yt (i ) Ψ[Yt (i )]
where Ψ[.] is the …rms nominal cost function
(Institute)
Monetary Theory: Price Setting
March 2011
2 / 10
Notation
Suppose the …rm producing intermediate good i, in our setup with
monopolistic competition, sets a price Pt (i ), and pays nominal
dividends
Dt (i ) = Pt (i )Yt (i ) Ψ[Yt (i )]
where Ψ[.] is the …rms nominal cost function
Recall that demand for this …rm’s output is related to aggregate
demand by
e
Pt ( i )
Yt ( i ) =
Yt
Pt
(Institute)
Monetary Theory: Price Setting
March 2011
2 / 10
Notation
Suppose the …rm producing intermediate good i, in our setup with
monopolistic competition, sets a price Pt (i ), and pays nominal
dividends
Dt (i ) = Pt (i )Yt (i ) Ψ[Yt (i )]
where Ψ[.] is the …rms nominal cost function
Recall that demand for this …rm’s output is related to aggregate
demand by
e
Pt ( i )
Yt ( i ) =
Yt
Pt
We saw that with ‡exible prices, the optimal price satis…es
(1
e) [Pt (i )]
where ψ[Yt (i )]
(Institute)
e
( P t ) e Yt =
e [Pt (i )]
e 1
(Pt )e Yt ψ[Yt (i )]
Ψ0 [Yt (i )] is nominal marginal cost
Monetary Theory: Price Setting
March 2011
2 / 10
One-period Price Rigidity
With ‡exible prices, as we saw, the optimal price is a markup over
marginal cost:
e
ψ[Yt (i )]
Pt ( i ) =
e 1
(Institute)
Monetary Theory: Price Setting
March 2011
3 / 10
One-period Price Rigidity
With ‡exible prices, as we saw, the optimal price is a markup over
marginal cost:
e
ψ[Yt (i )]
Pt ( i ) =
e 1
Now suppose the …rm sets Pt +1 (i ) at date t (there is one-period price
rigidity)
(Institute)
Monetary Theory: Price Setting
March 2011
3 / 10
The …rm sets Pt+1 (i) to maximize
t+1
Et
t
fPt+1 (i)Yt+1 (i)
[Yt+1 (i)]g
subject to
Pt+1 (i)
Pt+1
Yt+1 (i) =
Yt+1
The FOC is
Et
t+1
t
n
(1
) [Pt+1 (i)]
(Pt+1 ) Yt+1 + [Pt+1 (i)]
1
o
(Pt+1 ) Yt+1 [Yt+1 (i)] = 0
Multiply this by Pt+1 (i) (which is not random at date t) and divide by (1
to get
Et
t+1
Pt+1 (i)Yt+1 (i)
1
t
and
Pt+1 (i) =
1
Et f
1
Yt+1 (i) [Yt+1 (i)]
t+1 Yt+1 (i) [Yt+1 (i)]g
Et [ t+1 Yt+1 (i)]
=0
)
One-period Price Rigidity
With ‡exible prices, as we saw, the optimal price is a markup over
marginal cost:
e
ψ[Yt (i )]
Pt ( i ) =
e 1
Now suppose the …rm sets Pt +1 (i ) at date t (there is one-period price
rigidity)
Note the role of the markup and next-period’s marginal cost
(Institute)
Monetary Theory: Price Setting
March 2011
3 / 10
One-period Price Rigidity
With ‡exible prices, as we saw, the optimal price is a markup over
marginal cost:
e
ψ[Yt (i )]
Pt ( i ) =
e 1
Now suppose the …rm sets Pt +1 (i ) at date t (there is one-period price
rigidity)
Note the role of the markup and next-period’s marginal cost
If all …rms set their price in this way, a change in aggregate demand
will a¤ect output
(Institute)
Monetary Theory: Price Setting
March 2011
3 / 10
One-period Price Rigidity
With ‡exible prices, as we saw, the optimal price is a markup over
marginal cost:
e
ψ[Yt (i )]
Pt ( i ) =
e 1
Now suppose the …rm sets Pt +1 (i ) at date t (there is one-period price
rigidity)
Note the role of the markup and next-period’s marginal cost
If all …rms set their price in this way, a change in aggregate demand
will a¤ect output
As a casual illustration, consider a binding cash-in-advance (CIA)
constraint:
Pt Ct = Mt
(Institute)
Monetary Theory: Price Setting
March 2011
3 / 10
One-period Price Rigidity
With ‡exible prices, as we saw, the optimal price is a markup over
marginal cost:
e
ψ[Yt (i )]
Pt ( i ) =
e 1
Now suppose the …rm sets Pt +1 (i ) at date t (there is one-period price
rigidity)
Note the role of the markup and next-period’s marginal cost
If all …rms set their price in this way, a change in aggregate demand
will a¤ect output
As a casual illustration, consider a binding cash-in-advance (CIA)
constraint:
Pt Ct = Mt
Although one-period price rigidity implies non-neutrality of monetary
policy, it cannot generate a persistent output response or sluggish
price response for several periods
(Institute)
Monetary Theory: Price Setting
March 2011
3 / 10
Two-period Price Rigidity
Consider next a …rm setting the same price for two periods at date t
(Institute)
Monetary Theory: Price Setting
March 2011
4 / 10
The …rm sets Pt (i) to maximize
Pt (i)Yt (i)
t+1
[Yt (i)] + Et
fPt (i)Yt+1 (i)
t
[Yt+1 (i)]g
subject to
Yt (i) =
Pt (i)
Pt
Yt
Yt+1 (i) =
Pt (i)
Pt+1
Yt+1
and
The FOC is
0
=
(1
) [Pt (i)]
t+1
+ Et
t
1
(Pt ) Yt + [Pt (i)]
(Pt ) Yt [Yt (i)]
n
(1
) [Pt (i)] (Pt+1 ) Yt+1 + [Pt (i)]
Multiply this by Pt (i) and divide by (1
0
= Pt (i)Yt (i)
+ Et
1
t+1
1
X
k=0
k
Et
t+k
o
(Pt+1 ) Yt+1 [Yt+1 (i)]
) to get
Yt (i) [Yt (i)]
Pt (i)Yt+1 (i)
t
Or
1
Yt+k (i) Pt (i)
t
1
1
1
Yt+1 (i) [Yt+1 (i)]
[Yt+k (i)]
=0
Two-period Price Rigidity
Consider next a …rm setting the same price for two periods at date t
The optimal price depends on marginal cost at t and the expectation
of marginal cost at t + 1
(Institute)
Monetary Theory: Price Setting
March 2011
4 / 10
Two-period Price Rigidity
Consider next a …rm setting the same price for two periods at date t
The optimal price depends on marginal cost at t and the expectation
of marginal cost at t + 1
In the steady-state equilibrium with zero in‡ation , Λt is constant
(Institute)
Monetary Theory: Price Setting
March 2011
4 / 10
Two-period Price Rigidity
Consider next a …rm setting the same price for two periods at date t
The optimal price depends on marginal cost at t and the expectation
of marginal cost at t + 1
In the steady-state equilibrium with zero in‡ation , Λt is constant
In the neighborhood of a zero-in‡ation steady state, we get
p[
t (i ) =
1
1+β
[
ψ
t (i ) +
β
1+β
Et ψ\
t +1 (i )
linking the price deviation to a weighted average of the current
marginal-cost deviation and the current expectation of next-period’s
marginal-cost deviation
(Institute)
Monetary Theory: Price Setting
March 2011
4 / 10
2
Approximation
In a steady-state equilibrium with zero in‡ation, we have
P (i)Y (i) (1 + ) =
(i)Y (i) (1 + )
1
(implying that price is a markup over nominal marginal cost; and real marginal
cost is the inverse of the markup)
Approximating
Pt (i) Yt (i) + Et
t+1
Yt+1 (i) =
1
t
Yt (i)
t (i)
+ Et
t+1
Yt+1 (i)
t
near this state, we get
Y (i)(1+ ) [Pt (i)
P (i)] =
and
pd
t (i) =
1
1
1+
Y (i)[
d
t (i) +
2
t (i)
(i)] + Y (i)Et [
1+
d(i)
Et t+1
t+1 (i)
(i)]
t+1 (i)]
Staggered Price Setting
Taylor’s survey (summarized in Chapter 1) and subsequent work
(Institute)
Monetary Theory: Price Setting
March 2011
5 / 10
Staggered Price Setting
Taylor’s survey (summarized in Chapter 1) and subsequent work
Two-period Taylor contracts: half the …rms set a new price Pt and
the other half keep the price Pt 1 that they had set last period
(Institute)
Monetary Theory: Price Setting
March 2011
5 / 10
Staggered Price Setting
Taylor’s survey (summarized in Chapter 1) and subsequent work
Two-period Taylor contracts: half the …rms set a new price Pt and
the other half keep the price Pt 1 that they had set last period
The aggregate price level Pt follows
Pt =
8
<Z1
:
(Institute)
0
[Pt (i )]
1 e
di
9 1
=1 e
;
=
1
(P
2 t
Monetary Theory: Price Setting
1)
1 e
1
+ ( Pt ) 1
2
e
1
1 e
March 2011
5 / 10
Staggered Price Setting
Taylor’s survey (summarized in Chapter 1) and subsequent work
Two-period Taylor contracts: half the …rms set a new price Pt and
the other half keep the price Pt 1 that they had set last period
The aggregate price level Pt follows
Pt =
8
<Z1
:
0
[Pt (i )]
1 e
di
9 1
=1 e
;
1
(P
2 t
=
1)
1 e
1
+ ( Pt ) 1
2
e
1
1 e
In the neighborhood of a zero-in‡ation steady state
p
bt =
(Institute)
1
p
b
2 t
1
1
+ pbt
2
Monetary Theory: Price Setting
March 2011
5 / 10
Let Pt denote the new price set at t and
1
(Pt )
=
1
(P
2 t
1
1)
1
+ (Pt )1
2
In a deterministic steady state with zero in‡ation, we have P = P , and near
that steady state,
(1
) (P )
(Pt
P ) = (1
1
(P
2 t
) (P )
and
pbt =
1
pb
2 t
3
1
1
+ pbt
2
1
1
P ) + (Pt
2
P )
;
Staggered Price Setting
Taylor’s survey (summarized in Chapter 1) and subsequent work
Two-period Taylor contracts: half the …rms set a new price Pt and
the other half keep the price Pt 1 that they had set last period
The aggregate price level Pt follows
Pt =
8
<Z1
:
0
[Pt (i )]
1 e
di
9 1
=1 e
;
1
(P
2 t
=
1)
1 e
1
+ ( Pt ) 1
2
e
1
1 e
In the neighborhood of a zero-in‡ation steady state
p
bt =
Multi-period Taylor contracts
(Institute)
1
p
b
2 t
1
1
+ pbt
2
Monetary Theory: Price Setting
March 2011
5 / 10
Staggered Price Setting
Taylor’s survey (summarized in Chapter 1) and subsequent work
Two-period Taylor contracts: half the …rms set a new price Pt and
the other half keep the price Pt 1 that they had set last period
The aggregate price level Pt follows
Pt =
8
<Z1
:
0
[Pt (i )]
1 e
di
9 1
=1 e
;
1
(P
2 t
=
1)
1 e
1
+ ( Pt ) 1
2
e
1
1 e
In the neighborhood of a zero-in‡ation steady state
p
bt =
Multi-period Taylor contracts
1
p
b
2 t
1
1
+ pbt
2
Calvo contracts (discussed below)
(Institute)
Monetary Theory: Price Setting
March 2011
5 / 10
Calvo’s Model
A tractable way to model staggered price (or wage) setting with any
average duration
(Institute)
Monetary Theory: Price Setting
March 2011
6 / 10
Calvo’s Model
A tractable way to model staggered price (or wage) setting with any
average duration
In each period, each …rm gets to reset its price with a constant
probability (1 θ), regardless of when the current price was set
(Institute)
Monetary Theory: Price Setting
March 2011
6 / 10
Firms
Continuum of …rms, indexed by i 2 [0; 1]
Each …rm produces a di¤erentiated good
Identical technology
Yt(i) = At Nt(i)1
Probability of resetting price in any given period: 1
across …rms (Calvo (1983)).
2 [0; 1] : index of price stickiness
Implied average price duration
1
1
, independent
Calvo’s Model
A tractable way to model staggered price (or wage) setting with any
average duration
In each period, each …rm gets to reset its price with a constant
probability (1 θ), regardless of when the current price was set
Focusing on a symmetric equilibrium, all the …rms that get to set a
new price at time t, choose the same price Pt
(Institute)
Monetary Theory: Price Setting
March 2011
6 / 10
Optimal Price Setting
max
Pt
subject to
1
X
k
t+k (Yt+kjt )
Et Qt;t+k Pt Yt+kjt
k=0
Yt+kjt = (Pt =Pt+k )
Ct+k
for k = 0; 1; 2; :::where
Qt;t+k
Optimality condition:
1
X
k
where
t+kjt
k
Ct+k
Ct
Et Qt;t+k Yt+kjt Pt
k=0
0
t+k (Yt+kjt )
and M
1
Pt
Pt+k
M
t+kjt
=0
Calvo’s Model
A tractable way to model staggered price (or wage) setting with any
average duration
In each period, each …rm gets to reset its price with a constant
probability (1 θ), regardless of when the current price was set
Focusing on a symmetric equilibrium, all the …rms that get to set a
new price at time t, choose the same price Pt
The evolution of the aggregate price level is governed by
Pt =
8
<Z1
:
0
(Institute)
[Pt (i )]1
e
di
9 1
=1 e
;
= θ ( Pt
1)
Monetary Theory: Price Setting
1 e
+ (1
θ )(Pt )1
e
March 2011
1
1 e
6 / 10
Evolution of Calvo Prices
In a deterministic steady state with zero in‡ation,
( Pt ) 1
e
= θ ( Pt
1)
1 e
+ (1
θ )(Pt )1
e
implies P = P , and we get
(1
e ) (P )
e
( Pt
P ) = ( 1 e ) ( P ) e [ θ ( Pt 1
+(1 θ )(Pt P )]
P)
and
p
bt = θb
pt
(Institute)
1
+ (1
Monetary Theory: Price Setting
θ )p
bt
March 2011
7 / 11
Evolution of Calvo Prices
In a deterministic steady state with zero in‡ation,
( Pt ) 1
e
= θ ( Pt
1)
1 e
θ )(Pt )1
+ (1
e
implies P = P , and we get
(1
e ) (P )
e
( Pt
P ) = ( 1 e ) ( P ) e [ θ ( Pt 1
+(1 θ )(Pt P )]
P)
and
p
bt = θb
pt
1
θ )p
bt
+ (1
So the in‡ation deviations (from the steady-state value of zero) satisfy
bt
π
(Institute)
p
bt
p
bt
1
= (1
θ )(p
bt
Monetary Theory: Price Setting
p
bt
1)
March 2011
7 / 11
Aggregate Price Dynamics
Pt =
Dividing by Pt
1
(Pt 1)
1
+ (1
) (Pt )
:
1
t
= + (1
)
Pt
Pt 1
1
1
1
1
Log-linearization around zero in‡ation steady state
t
= (1
) (pt
pt 1 )
+ (1
) pt
or, equivalently
pt =
pt
1
(1)
Implications of Staggered Price Setting
Cutting the nominal interest rate lowers the expected real interest
rate and increases aggregate demand
(Institute)
Monetary Theory: Price Setting
March 2011
8 / 10
Implications of Staggered Price Setting
Cutting the nominal interest rate lowers the expected real interest
rate and increases aggregate demand
With our iso-elastic utility function, this relationship is
ct = Et [ct +1 ]
(Institute)
1
[ it
σ
Et ( π t + 1 )
Monetary Theory: Price Setting
ρ]
March 2011
8 / 10
Implications of Staggered Price Setting
Cutting the nominal interest rate lowers the expected real interest
rate and increases aggregate demand
With our iso-elastic utility function, this relationship is
ct = Et [ct +1 ]
1
[ it
σ
Et ( π t + 1 )
ρ]
Some …rms react to an increase in aggregate demand by raising prices
(Institute)
Monetary Theory: Price Setting
March 2011
8 / 10
Implications of Staggered Price Setting
Cutting the nominal interest rate lowers the expected real interest
rate and increases aggregate demand
With our iso-elastic utility function, this relationship is
ct = Et [ct +1 ]
1
[ it
σ
Et ( π t + 1 )
ρ]
Some …rms react to an increase in aggregate demand by raising prices
Other …rms (with rigid prices) end up increasing production
(Institute)
Monetary Theory: Price Setting
March 2011
8 / 10
Implications of Staggered Price Setting
Cutting the nominal interest rate lowers the expected real interest
rate and increases aggregate demand
With our iso-elastic utility function, this relationship is
ct = Et [ct +1 ]
1
[ it
σ
Et ( π t + 1 )
ρ]
Some …rms react to an increase in aggregate demand by raising prices
Other …rms (with rigid prices) end up increasing production
The di¤erence in responses across …rms raises interesting normative
questions (discussed in Chapter 4)
(Institute)
Monetary Theory: Price Setting
March 2011
8 / 10
Price Setting in Calvo’s Model
Firms that get to set a new price at time t set it to maximize the
expect present value of pro…ts over all future states in which this price
prevails
(Institute)
Monetary Theory: Price Setting
March 2011
9 / 10
Optimal Price Setting
max
Pt
subject to
1
X
k
t+k (Yt+kjt )
Et Qt;t+k Pt Yt+kjt
k=0
Yt+kjt = (Pt =Pt+k )
Ct+k
for k = 0; 1; 2; :::where
Qt;t+k
Optimality condition:
1
X
k
where
t+kjt
k
Ct+k
Ct
Et Qt;t+k Yt+kjt Pt
k=0
0
t+k (Yt+kjt )
and M
1
Pt
Pt+k
M
t+kjt
=0
Price Setting in Calvo’s Model
Firms that get to set a new price at time t set it to maximize the
expect present value of pro…ts over all future states in which this price
prevails
Note the similarities (and di¤erences) with the FOC for our 2-period
price setting problem:
1
∑ β k Et
k =0
(Institute)
Λ t +k
Λt
Yt + k ( i ) P t ( i )
Monetary Theory: Price Setting
e
e
1
ψ[Yt +k (i )]
March 2011
=0
9 / 10
Price Setting in Calvo’s Model
Firms that get to set a new price at time t set it to maximize the
expect present value of pro…ts over all future states in which this price
prevails
Note the similarities (and di¤erences) with the FOC for our 2-period
price setting problem:
1
∑ β k Et
k =0
Λ t +k
Λt
Yt + k ( i ) P t ( i )
e
e
1
ψ[Yt +k (i )]
=0
The link between new prices and real marginal cost
(Institute)
Monetary Theory: Price Setting
March 2011
9 / 10
Equivalently,
1
X
k
Et Qt;t+k Yt+kjt
k=0
where M Ct+kjt
t+kjt =Pt+k
and
Pt
Pt 1
M M Ct+kjt
t 1;t+k
Pt+k =Pt
=0
t 1;t+k
1
Perfect Foresight, Zero In‡ation Steady State:
Pt
=1 ;
Pt 1
t 1;t+k
=1 ;
Yt+kjt = Y
;
Qt;t+k =
k
;
MC =
1
M
Log-linearization around zero in‡ation steady state:
1
X
pt pt 1 = (1
)
( )k Etfmc
c t+kjt + pt+k
k=0
where mc
c t+kjt
mct+kjt
pt 1 g
mc.
Equivalently,
pt =
+ (1
)
1
X
k=0
where
log
1.
(
)k Etfmct+kjt + pt+k g
Flexible prices ( = 0):
pt =
=)
mct =
+ mct + pt
(symmetric equilibrium)
In‡ation Dynamics in Calvo’s Model
With a linear production function (α = 0), all the …rms have the same
marginal cost
(Institute)
Monetary Theory: Price Setting
March 2011
10 / 10
Particular Case:
= 0 (constant returns)
=) M Ct+kjt = M Ct+k
Rewriting the optimal price setting rule in recursive form:
pt =
) mc
c t + (1
Etfpt+1g + (1
Combining (1) and (2):
t
where
=
Et f
(1
t+1 g
)(1
+
mc
ct
)
)pt
(2)
In‡ation Dynamics in Calvo’s Model
With a linear production function (α = 0), all the …rms have the same
marginal cost
In this case, real marginal cost equals
Wt
P t At
and an increase in aggregate demand increases real marginal cost by
increasing the real wage
(Institute)
Monetary Theory: Price Setting
March 2011
10 / 10
In‡ation Dynamics in Calvo’s Model
With a linear production function (α = 0), all the …rms have the same
marginal cost
In this case, real marginal cost equals
Wt
P t At
and an increase in aggregate demand increases real marginal cost by
increasing the real wage
With diminishing returns to labor (0 < α < 1), …rms with di¤erent
prices have di¤erent marginal costs; but aggregation across …rms is
facilitated by the Calvo structure
(Institute)
Monetary Theory: Price Setting
March 2011
10 / 10
Generalization to
De…ne
2 (0; 1) (decreasing returns)
mct
Using mct+kjt = (wt+k
(wt
pt )
(wt
pt )
pt+k )
mpnt
1
(at
1
1
1
yt )
(at+k
mct+kjt = mct+k +
= mct+k
log(1
yt+kjt)
(yt+kjt
1
(pt
1
log(1
)
),
yt+k )
pt+k )
(3)
Implied in‡ation dynamics
t
where
=
(1
Et f
)(1
t+1 g
+
)
mc
ct
1
1
+
(4)
In‡ation Dynamics in Calvo’s Model
With a linear production function (α = 0), all the …rms have the same
marginal cost
In this case, real marginal cost equals
Wt
P t At
and an increase in aggregate demand increases real marginal cost by
increasing the real wage
With diminishing returns to labor (0 < α < 1), …rms with di¤erent
prices have di¤erent marginal costs; but aggregation across …rms is
facilitated by the Calvo structure
In this case, an increase in aggregate demand increases real marginal
cost through two channels: by increasing the real wage and reducing
the marginal product of labor
(Institute)
Monetary Theory: Price Setting
March 2011
10 / 10
Real Marginal Cost and In‡ation Dynamics in Calvo’s
Model
An increase in real marginal cost increases in‡ation because it erodes
the monopoly markup
(Institute)
Monetary Theory: Price Setting
March 2011
11 / 11
Real Marginal Cost and In‡ation Dynamics in Calvo’s
Model
An increase in real marginal cost increases in‡ation because it erodes
the monopoly markup
To protect their markups, …rms setting new prices choose a higher
price, which leads to in‡ation
(Institute)
Monetary Theory: Price Setting
March 2011
11 / 11
Real Marginal Cost and In‡ation Dynamics in Calvo’s
Model
An increase in real marginal cost increases in‡ation because it erodes
the monopoly markup
To protect their markups, …rms setting new prices choose a higher
price, which leads to in‡ation
The response coe¢ cient λ in
π t = βEt π t +1 + λmc
ct
is inversely related to the degree of price rigidity θ
(Institute)
Monetary Theory: Price Setting
March 2011
11 / 11
Real Marginal Cost and In‡ation Dynamics in Calvo’s
Model
An increase in real marginal cost increases in‡ation because it erodes
the monopoly markup
To protect their markups, …rms setting new prices choose a higher
price, which leads to in‡ation
The response coe¢ cient λ in
π t = βEt π t +1 + λmc
ct
is inversely related to the degree of price rigidity θ
Iterating forward,the bounded solution for in‡ation is
∞
π t = λEt
c t +j
∑ βj mc
j =0
(Institute)
Monetary Theory: Price Setting
March 2011
11 / 11