Monetary Theory: Price Setting - Harris Dellas competition, sets a price P t(i), ... Monetary...

Monetary Theory: Price Setting

Behzad Diba

University of Bern

March 2011

(Institute) Monetary Theory: Price Setting March 2011 1 / 10

Notation

Suppose the firm producing intermediate good i , in our setup withmonopolistic competition, sets a price Pt (i), and pays nominaldividends

Dt (i) = Pt (i)Yt (i)−Ψ[Yt (i)]

where Ψ[.] is the firms nominal cost function

Recall that demand for this firm’s output is related to aggregatedemand by

Yt (i) =[Pt (i)Pt

]−ε

We saw that with flexible prices, the optimal price satisfies

(1− ε) [Pt (i)]−ε (Pt )

ε Yt = −ε [Pt (i)]−ε−1 (Pt )

ε Ytψ[Yt (i)]

where ψ[Yt (i)] ≡ Ψ′[Yt (i)] is nominal marginal cost

Notation

where Ψ[.] is the firms nominal cost functionRecall that demand for this firm’s output is related to aggregatedemand by

Yt (i) =[Pt (i)Pt

]−ε

(1− ε) [Pt (i)]−ε (Pt )

ε Yt = −ε [Pt (i)]−ε−1 (Pt )

ε Ytψ[Yt (i)]

Notation

where Ψ[.] is the firms nominal cost functionRecall that demand for this firm’s output is related to aggregatedemand by

Yt (i) =[Pt (i)Pt

]−ε

(1− ε) [Pt (i)]−ε (Pt )

ε Yt = −ε [Pt (i)]−ε−1 (Pt )

ε Ytψ[Yt (i)]

One-period Price Rigidity

With flexible prices, as we saw, the optimal price is a markup overmarginal cost:

Pt (i) =(

ε− 1

)ψ[Yt (i)]

Now suppose the firm sets Pt+1(i) at date t (there is one-period pricerigidity)Note the role of the markup and next-period’s marginal costIf all firms set their price in this way, a change in aggregate demandwill affect outputAs a casual illustration, consider a binding cash-in-advance (CIA)constraint:

PtCt = Mt

Although one-period price rigidity implies non-neutrality of monetarypolicy, it cannot generate a persistent output response or sluggishprice response for several periods

Pt (i) =(

ε− 1

)ψ[Yt (i)]

Now suppose the firm sets Pt+1(i) at date t (there is one-period pricerigidity)

Note the role of the markup and next-period’s marginal costIf all firms set their price in this way, a change in aggregate demandwill affect outputAs a casual illustration, consider a binding cash-in-advance (CIA)constraint:

PtCt = Mt

The firm sets Pt+1(i) to maximize

[(Λt+1Λt

){Pt+1(i)Yt+1(i)−Ψ[Yt+1(i)]}

]subject to

Yt+1(i) =

[Pt+1(i)

]−εYt+1

The FOC is

[(Λt+1Λt

){(1− ε) [Pt+1(i)]

−ε(Pt+1)

εYt+1 + ε [Pt+1(i)]

−ε−1(Pt+1)

εYt+1ψ[Yt+1(i)]

Multiply this by Pt+1(i) (which is not random at date t) and divide by (1− ε)to get

[(Λt+1Λt

){Pt+1(i)Yt+1(i)−

ε− 1

)Yt+1(i)ψ[Yt+1(i)]

Pt+1(i) =

ε− 1

)Et {Λt+1Yt+1(i)ψ[Yt+1(i)]}

Et [Λt+1Yt+1(i)]

Pt (i) =(

ε− 1

)ψ[Yt (i)]

Now suppose the firm sets Pt+1(i) at date t (there is one-period pricerigidity)Note the role of the markup and next-period’s marginal cost

If all firms set their price in this way, a change in aggregate demandwill affect outputAs a casual illustration, consider a binding cash-in-advance (CIA)constraint:

PtCt = Mt

Pt (i) =(

ε− 1

)ψ[Yt (i)]

Now suppose the firm sets Pt+1(i) at date t (there is one-period pricerigidity)Note the role of the markup and next-period’s marginal costIf all firms set their price in this way, a change in aggregate demandwill affect output

As a casual illustration, consider a binding cash-in-advance (CIA)constraint:

PtCt = Mt

Pt (i) =(

ε− 1

)ψ[Yt (i)]

PtCt = Mt

Pt (i) =(

ε− 1

)ψ[Yt (i)]

PtCt = Mt

Two-period Price Rigidity

Consider next a firm setting the same price for two periods at date t

The optimal price depends on marginal cost at t and the expectationof marginal cost at t + 1

In the steady-state equilibrium with zero inflation , Λt is constant

In the neighborhood of a zero-inflation steady state, we get

p̂t (i) =(

11+ β

)ψ̂t (i) +

)Et ψ̂t+1(i)

linking the price deviation to a weighted average of the currentmarginal-cost deviation and the current expectation of next-period’smarginal-cost deviation

The firm sets Pt(i) to maximize

Pt(i)Yt(i)−Ψ[Yt(i)] + βEt

[(Λt+1Λt

){Pt(i)Yt+1(i)−Ψ[Yt+1(i)]}

]subject to

Yt(i) =

[Pt(i)

]−εYt

Yt+1(i) =

[Pt(i)

]−εYt+1

The FOC is

0 = (1− ε) [Pt(i)]−ε

(Pt)εYt + ε [Pt(i)]

−ε−1(Pt)

εYtψ[Yt(i)]

[(Λt+1Λt

){(1− ε) [Pt(i)]

−ε(Pt+1)

εYt+1 + ε [Pt(i)]

−ε−1(Pt+1)

εYt+1ψ[Yt+1(i)]

}]Multiply this by Pt(i) and divide by (1− ε) to get

0 = Pt(i)Yt(i)−(

ε− 1

)Yt(i)ψ[Yt(i)]

[(Λt+1Λt

){Pt(i)Yt+1(i)−

ε− 1

)Yt+1(i)ψ[Yt+1(i)]

1∑k=0

[(Λt+kΛt

)Yt+k(i)

{Pt(i)−

ε− 1

)ψ[Yt+k(i)]

p̂t (i) =(

11+ β

)ψ̂t (i) +

)Et ψ̂t+1(i)

p̂t (i) =(

11+ β

)ψ̂t (i) +

)Et ψ̂t+1(i)

p̂t (i) =(

11+ β

)ψ̂t (i) +

)Et ψ̂t+1(i)

2 Approximation

In a steady-state equilibrium with zero inflation, we have

P (i)Y (i) (1 + β) =

ε− 1

)ψ(i)Y (i) (1 + β)

(implying that price is a markup over nominal marginal cost; and real marginalcost is the inverse of the markup)Approximating

[Yt(i) + βEt

(Λt+1Λt

)Yt+1(i)

ε− 1

){Yt(i)ψt(i) + βEt

(Λt+1Λt

)Yt+1(i)ψt+1(i)]

}near this state, we get

Y (i)(1+β) [Pt(i)− P (i)] =

ε− 1

){Y (i)[ψt(i)− ψ(i)] + βY (i)Et[ψt+1(i)− ψ(i)]

p̂t(i) =

1 + β

)ψ̂t(i) +

1 + β

)Et ̂ψt+1(i)

Staggered Price Setting

Taylor’s survey (summarized in Chapter 1) and subsequent work

Two-period Taylor contracts: half the firms set a new price P∗t andthe other half keep the price P∗t−1 that they had set last period

The aggregate price level Pt follows

[Pt (i)]1−ε di

11−ε

{12(P∗t−1)

1−ε +12(P∗t )

1−ε

} 11−ε

In the neighborhood of a zero-inflation steady state

p̂t =12p̂∗t−1 +

12p̂∗t

Multi-period Taylor contracts

Calvo contracts (discussed below)

[Pt (i)]1−ε di

11−ε

{12(P∗t−1)

1−ε +12(P∗t )

1−ε

} 11−ε

p̂t =12p̂∗t−1 +

12p̂∗t

[Pt (i)]1−ε di

11−ε

{12(P∗t−1)

1−ε +12(P∗t )

1−ε

} 11−ε

p̂t =12p̂∗t−1 +

12p̂∗t

[Pt (i)]1−ε di

11−ε

{12(P∗t−1)

1−ε +12(P∗t )

1−ε

} 11−ε

p̂t =12p̂∗t−1 +

12p̂∗t

Let P ∗t denote the new price set at t and

(Pt)1−ε

2(P ∗t−1)

1−ε +1

2(P ∗t )1−ε

In a deterministic steady state with zero inflation, we have P = P ∗, and nearthat steady state,

(1− ε) (P )−ε

(Pt − P ) ∼= (1− ε) (P ∗)−ε[

2(P ∗t−1 − P ∗) +

2(P ∗t − P ∗)

andp̂t =

2p̂∗t−1 +

2p̂∗t

[Pt (i)]1−ε di

11−ε

{12(P∗t−1)

1−ε +12(P∗t )

1−ε

} 11−ε

p̂t =12p̂∗t−1 +

12p̂∗t

[Pt (i)]1−ε di

11−ε

{12(P∗t−1)

1−ε +12(P∗t )

1−ε

} 11−ε

p̂t =12p̂∗t−1 +

12p̂∗t

Calvo’s Model

A tractable way to model staggered price (or wage) setting with anyaverage duration

In each period, each firm gets to reset its price with a constantprobability (1− θ), regardless of when the current price was set

Focusing on a symmetric equilibrium, all the firms that get to set anew price at time t, choose the same price P∗tThe evolution of the aggregate price level is governed by

[Pt (i)]1−ε di

11−ε

θ(Pt−1)1−ε + (1− θ)(P∗t )1−ε} 11−ε

Calvo’s Model

[Pt (i)]1−ε di

11−ε

θ(Pt−1)1−ε + (1− θ)(P∗t )1−ε} 11−ε

� 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 � �, independentacross �rms (Calvo (1983)).

� � 2 [0; 1] : index of price stickiness� Implied average price duration 1

1��

Calvo’s Model

Focusing on a symmetric equilibrium, all the firms that get to set anew price at time t, choose the same price P∗t

The evolution of the aggregate price level is governed by

[Pt (i)]1−ε di

11−ε

θ(Pt−1)1−ε + (1− θ)(P∗t )1−ε} 11−ε

Optimal Price Setting

maxP �t

�k Et

�Qt;t+k

�P �t Yt+kjt � t+k(Yt+kjt)

�subject to

Yt+kjt = (P�t =Pt+k)

�� Ct+k

for k = 0; 1; 2; :::where

Qt;t+k � �k�Ct+kCt

��PtPt+k

Optimality condition:1Xk=0

�k Et

�Qt;t+k Yt+kjt

�P �t �M t+kjt

�= 0

where t+kjt � 0t+k(Yt+kjt) andM� ��1

Calvo’s Model

[Pt (i)]1−ε di

11−ε

θ(Pt−1)1−ε + (1− θ)(P∗t )1−ε} 11−ε

Evolution of Calvo Prices

In a deterministic steady state with zero inflation,

(Pt )1−ε = θ(Pt−1)1−ε + (1− θ)(P∗t )

1−ε

implies P = P∗, and we get

(1− ε) (P)−ε (Pt − P) ∼= (1− ε) (P)−ε [θ(Pt−1 − P)+(1− θ)(P∗t − P∗)]

andp̂t ∼= θp̂t−1 + (1− θ)p̂∗t

So the inflation deviations (from the steady-state value of zero) satisfy

π̂t ≡ p̂t − p̂t−1 ∼= (1− θ)(p̂∗t − p̂t−1)

Evolution of Calvo Prices

In a deterministic steady state with zero inflation,

(Pt )1−ε = θ(Pt−1)1−ε + (1− θ)(P∗t )

1−ε

implies P = P∗, and we get

(1− ε) (P)−ε (Pt − P) ∼= (1− ε) (P)−ε [θ(Pt−1 − P)+(1− θ)(P∗t − P∗)]

andp̂t ∼= θp̂t−1 + (1− θ)p̂∗t

So the inflation deviations (from the steady-state value of zero) satisfy

π̂t ≡ p̂t − p̂t−1 ∼= (1− θ)(p̂∗t − p̂t−1)

Aggregate Price Dynamics

Pt =�� (Pt�1)

1�� + (1� �) (P �t )1�� 1

1��

Dividing by Pt�1 :

�1��t = � + (1� �)

�P �tPt�1

�1��

Log-linearization around zero in�ation steady state

�t = (1� �) (p�t � pt�1) (1)

or, equivalentlypt = � pt�1 + (1� �) p�t

Implications of Staggered Price Setting

Cutting the nominal interest rate lowers the expected real interestrate and increases aggregate demand

With our iso-elastic utility function, this relationship is

ct ∼= Et [ct+1]−1σ[it − Et (πt+1)− ρ]

Some firms react to an increase in aggregate demand by raising prices

Other firms (with rigid prices) end up increasing production

The difference in responses across firms raises interesting normativequestions (discussed in Chapter 4)

ct ∼= Et [ct+1]−1σ[it − Et (πt+1)− ρ]

Price Setting in Calvo’s Model

Firms that get to set a new price at time t set it to maximize theexpect present value of profits over all future states in which this priceprevails

Note the similarities (and differences) with the FOC for our 2-periodprice setting problem:

∑k=0

[(Λt+k

)Yt+k (i)

{Pt (i)−

ε− 1

)ψ[Yt+k (i)]

The link between new prices and real marginal cost

Optimal Price Setting

maxP �t

�k Et

�Qt;t+k

�P �t Yt+kjt � t+k(Yt+kjt)

�subject to

Yt+kjt = (P�t =Pt+k)

�� Ct+k

for k = 0; 1; 2; :::where

Qt;t+k � �k�Ct+kCt

��PtPt+k

Optimality condition:1Xk=0

�k Et

�Qt;t+k Yt+kjt

�P �t �M t+kjt

�= 0

where t+kjt � 0t+k(Yt+kjt) andM� ��1

∑k=0

[(Λt+k

)Yt+k (i)

{Pt (i)−

ε− 1

)ψ[Yt+k (i)]

∑k=0

[(Λt+k

)Yt+k (i)

{Pt (i)−

ε− 1

)ψ[Yt+k (i)]

Equivalently,1Xk=0

�k Et

�Qt;t+k Yt+kjt

�P �tPt�1

�M MCt+kjt �t�1;t+k

��= 0

where MCt+kjt � t+kjt=Pt+k and �t�1;t+k � Pt+k=Pt�1

Perfect Foresight, Zero In�ation Steady State:

P �tPt�1

= 1 ; �t�1;t+k = 1 ; Yt+kjt = Y ; Qt;t+k = �k ; MC =1

Log-linearization around zero in�ation steady state:

p�t � pt�1 = (1� ��)

(��)k Etfcmct+kjt + pt+k � pt�1g

where cmct+kjt � mct+kjt �mc.

Equivalently,

p�t = � + (1� ��)1Xk=0

(��)k Etfmct+kjt + pt+kg

where � � log ��1.

Flexible prices (� = 0):p�t = � +mct + pt

=) mct = �� (symmetric equilibrium)

Inflation Dynamics in Calvo’s Model

With a linear production function (α = 0), all the firms have the samemarginal cost

In this case, real marginal cost equals

and an increase in aggregate demand increases real marginal cost byincreasing the real wage

With diminishing returns to labor (0 < α < 1), firms with differentprices have different marginal costs; but aggregation across firms isfacilitated by the Calvo structure

In this case, an increase in aggregate demand increases real marginalcost through two channels: by increasing the real wage and reducingthe marginal product of labor

Particular Case: � = 0 (constant returns)

=) MCt+kjt =MCt+k

Rewriting the optimal price setting rule in recursive form:

p�t = �� Etfp�t+1g + (1� ��) cmct + (1� ��)pt (2)

Combining (1) and (2):

�t = � Etf�t+1g + � cmctwhere

� � (1� �)(1� ��)

Generalization to � 2 (0; 1) (decreasing returns)De�ne

mct � (wt � pt)�mpnt

� (wt � pt)�1

1� �(at � �yt)� log(1� �)

Using mct+kjt = (wt+k � pt+k)� 11�� (at+k � �yt+kjt)� log(1� �),

mct+kjt = mct+k +�

1� �(yt+kjt � yt+k)

= mct+k ��

1� �(p�t � pt+k) (3)

Implied in�ation dynamics

�t = � Etf�t+1g + � cmct (4)

� � (1� �)(1� ��)

1� �

1� � + ��

Real Marginal Cost and Inflation Dynamics in Calvo’sModel

An increase in real marginal cost increases inflation because it erodesthe monopoly markup

To protect their markups, firms setting new prices choose a higherprice, which leads to inflation

The response coeffi cient λ in

πt = βEtπt+1 + λm̂c t

is inversely related to the degree of price rigidity θ

Iterating forward,the bounded solution for inflation is

πt = λEt∞

∑j=0

βj m̂c t+j

πt = λEt∞

∑j=0

βj m̂c t+j

πt = λEt∞

∑j=0

βj m̂c t+j

πt = λEt∞

∑j=0

βj m̂c t+j

Monetary Theory: Price Setting - Harris Dellas competition, sets a price P t(i), ... Monetary...

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