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Transcript of 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)[Yt (i)]where [.] is the firms nominal cost function

    Recall that demand for this firms output is related to aggregatedemand by

    Yt (i) =[Pt (i)Pt

    ]eYt

    We saw that with flexible prices, the optimal price satisfies

    (1 e) [Pt (i)]e (Pt )e Yt = e [Pt (i)]e1 (Pt )e Yt[Yt (i)]

    where [Yt (i)] [Yt (i)] is nominal marginal cost

    (Institute) Monetary Theory: Price Setting March 2011 2 / 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)[Yt (i)]where [.] is the firms nominal cost functionRecall that demand for this firms output is related to aggregatedemand by

    Yt (i) =[Pt (i)Pt

    ]eYt

    We saw that with flexible prices, the optimal price satisfies

    (1 e) [Pt (i)]e (Pt )e Yt = e [Pt (i)]e1 (Pt )e Yt[Yt (i)]

    where [Yt (i)] [Yt (i)] is nominal marginal cost

    (Institute) Monetary Theory: Price Setting March 2011 2 / 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)[Yt (i)]where [.] is the firms nominal cost functionRecall that demand for this firms output is related to aggregatedemand by

    Yt (i) =[Pt (i)Pt

    ]eYt

    We saw that with flexible prices, the optimal price satisfies

    (1 e) [Pt (i)]e (Pt )e Yt = e [Pt (i)]e1 (Pt )e Yt[Yt (i)]

    where [Yt (i)] [Yt (i)] is nominal marginal cost

    (Institute) Monetary Theory: Price Setting March 2011 2 / 10

  • One-period Price Rigidity

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

    Pt (i) =(

    e

    e 1

    )[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-periods 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

    (Institute) Monetary Theory: Price Setting March 2011 3 / 10

  • One-period Price Rigidity

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

    Pt (i) =(

    e

    e 1

    )[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-periods 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

    (Institute) Monetary Theory: Price Setting March 2011 3 / 10

  • The firm sets Pt+1(i) to maximize

    Et

    [(t+1t

    ){Pt+1(i)Yt+1(i)[Yt+1(i)]}

    ]subject to

    Yt+1(i) =

    [Pt+1(i)

    Pt+1

    ]Yt+1

    The FOC is

    Et

    [(t+1t

    ){(1 ) [Pt+1(i)] (Pt+1) Yt+1 + [Pt+1(i)]1 (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+1t

    ){Pt+1(i)Yt+1(i)

    (

    1

    )Yt+1(i)[Yt+1(i)]

    }]= 0

    and

    Pt+1(i) =

    (

    1

    )Et {t+1Yt+1(i)[Yt+1(i)]}

    Et [t+1Yt+1(i)]

    1

  • One-period Price Rigidity

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

    Pt (i) =(

    e

    e 1

    )[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-periods 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

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

    (Institute) Monetary Theory: Price Setting March 2011 3 / 10

  • One-period Price Rigidity

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

    Pt (i) =(

    e

    e 1

    )[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-periods 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

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

    (Institute) Monetary Theory: Price Setting March 2011 3 / 10

  • One-period Price Rigidity

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

    Pt (i) =(

    e

    e 1

    )[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-periods 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

    (Institute) Monetary Theory: Price Setting March 2011 3 / 10

  • One-period Price Rigidity

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

    Pt (i) =(

    e

    e 1

    )[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-periods 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

    (Institute) Monetary Theory: Price Setting March 2011 3 / 10

  • 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 , t is constantIn the neighborhood of a zero-inflation steady state, we get

    pt (i) =(

    11+

    )t (i) +

    (

    1+

    )Et t+1(i)

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

    (Institute) Monetary Theory: Price Setting March 2011 4 / 10

  • The firm sets Pt(i) to maximize

    Pt(i)Yt(i)[Yt(i)] + Et[(

    t+1t

    ){Pt(i)Yt+1(i)[Yt+1(i)]}

    ]subject to

    Yt(i) =

    [Pt(i)

    Pt

    ]Yt

    and

    Yt+1(i) =

    [Pt(i)

    Pt+1

    ]Yt+1

    The FOC is

    0 = (1 ) [Pt(i)] (Pt) Yt + [Pt(i)]1 (Pt) Yt[Yt(i)]

    +Et

    [(t+1t

    ){(1 ) [Pt(i)] (Pt+1) Yt+1 + [Pt(i)]1 (Pt+1) Yt+1[Yt+1(i)]

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

    0 = Pt(i)Yt(i)(

    1

    )Yt(i)[Yt(i)]

    +Et

    [(t+1t

    ){Pt(i)Yt+1(i)

    (

    1

    )Yt+1(i)[Yt+1(i)]

    }]Or

    1k=0

    kEt

    [(t+kt

    )Yt+k(i)

    {Pt(i)

    (

    1

    )[Yt+k(i)]

    }]= 0

    1

  • 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 , t is constantIn the neighborhood of a zero-inflation steady state, we get

    pt (i) =(

    11+

    )t (i) +

    (

    1+

    )Et t+1(i)

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

    (Institute) Monetary Theory: Price Setting March 2011 4 / 10

  • 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 , t is constant

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

    pt (i) =(

    11+

    )t (i) +

    (

    1+

    )Et t+1(i)

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

    (Institute) Monetary Theory: Price Setting March 2011 4 / 10

  • 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 , t is constantIn the neighborhood of a zero-inflation steady state, we get

    pt (i) =(

    11+

    )t (i) +

    (

    1+

    )Et t+1(i)

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

    (Institute) Monetary Theory: Price Setting March 2011 4 / 10

  • 2 Approximation

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

    P (i)Y (i) (1 + ) =

    (

    1

    )(i)Y (i) (1 + )

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

    Pt(i)

    [Yt(i) + Et

    (t+1t

    )Yt+1(i)

    ]=

    (

    1

    ){Yt(i)t(i) + Et

    (t+1t

    )Yt+1(i)t+1(i)]

    }near this state, we get

    Y (i)(1+) [Pt(i) P (i)] =(

    1

    ){Y (i)[t(i) (i)] + Y (i)Et[t+1(i) (i)]

    }and

    pt(i) =

    (1

    1 +

    )t(i) +

    (

    1 +

    )Et t+1(i)

    2

  • Staggered Price Setting

    Taylors survey (summarized in Chapter 1) and subsequent work

    Two-period Taylor contracts: half the firms set a new price Pt andthe other half keep the price Pt1 that they had set last period

    The aggregate price level Pt follows

    Pt =

    10

    [Pt (i)]1e di

    11e

    =

    {12(Pt1)

    1e +12(Pt )

    1e} 1

    1e

    In the neighborhood of a zero-inflation steady state

    pt =12pt1 +

    12pt

    Multi-period Taylor contracts

    Calvo contracts (discussed below)

    (Institute) Monetary Theory: Price Setting March 2011 5 / 10

  • Staggered Price Setting

    Taylors survey (summarized in Chapter 1) and subsequent work

    Two-period Taylor contracts: half the firms set a new price Pt andthe other half keep the price Pt1 that they had set last period

    The aggregate price level Pt follows

    Pt =

    10

    [Pt (i)]1e di

    11e

    =

    {12(Pt1)

    1e +12(Pt )

    1e} 1

    1e

    In the neighborhood of a zero-inflation steady state

    pt =12pt1 +

    12pt

    Multi-period Taylor contracts

    Calvo contracts (discussed below)

    (Institute) Monetary Theory: Price Setting March 2011 5 / 10

  • Staggered Price Setting

    Taylors survey (summarized in Chapter 1) and subsequent work

    Two-period Taylor contracts: half the firms set a new price Pt andthe other half keep the price Pt1 that they had set last period

    The aggregate price level Pt follows

    Pt =

    10

    [Pt (i)]1e di

    11e

    =

    {12(Pt1)

    1e +12(Pt )

    1e} 1

    1e

    In the neighborhood of a zero-inflation steady state

    pt =12pt1 +

    12pt

    Multi-period Taylor contracts

    Calvo contracts (discussed below)

    (Institute) Monetary Theory: Price Setting March 2011 5 / 10

  • Staggered Price Setting

    Taylors survey (summarized in Chapter 1) and subsequent work

    Two-period Taylor contracts: half the firms set a new price Pt andthe other half keep the price Pt1 that they had set last period

    The aggregate price level Pt follows

    Pt =

    10

    [Pt (i)]1e di

    11e

    =

    {12(Pt1)

    1e +12(Pt )

    1e} 1

    1e

    In the neighborhood of a zero-inflation steady state

    pt =12pt1 +

    12pt

    Multi-period Taylor contracts

    Calvo contracts (discussed below)

    (Institute) Monetary Theory: Price Setting March 2011 5 / 10

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

    (Pt)1

    =1

    2(P t1)

    1 +1

    2(P t )

    1

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

    (1 ) (P ) (Pt P ) = (1 ) (P )[

    1

    2(P t1 P ) +

    1

    2(P t P )

    ],

    andpt =

    1

    2pt1 +

    1

    2pt

    3

  • Staggered Price Setting

    Taylors survey (summarized in Chapter 1) and subsequent work

    Two-period Taylor contracts: half the firms set a new price Pt andthe other half keep the price Pt1 that they had set last period

    The aggregate price level Pt follows

    Pt =

    10

    [Pt (i)]1e di

    11e

    =

    {12(Pt1)

    1e +12(Pt )

    1e} 1

    1e

    In the neighborhood of a zero-inflation steady state

    pt =12pt1 +

    12pt

    Multi-period Taylor contracts

    Calvo contracts (discussed below)

    (Institute) Monetary Theory: Price Setting March 2011 5 / 10

  • Staggered Price Setting

    Taylors survey (summarized in Chapter 1) and subsequent work

    Two-period Taylor contracts: half the firms set a new price Pt andthe other half keep the price Pt1 that they had set last period

    The aggregate price level Pt follows

    Pt =

    10

    [Pt (i)]1e di

    11e

    =

    {12(Pt1)

    1e +12(Pt )

    1e} 1

    1e

    In the neighborhood of a zero-inflation steady state

    pt =12pt1 +

    12pt

    Multi-period Taylor contracts

    Calvo contracts (discussed below)

    (Institute) Monetary Theory: Price Setting March 2011 5 / 10

  • Calvos 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 ), regardless of when the current price was setFocusing on a symmetric equilibrium, all the firms that get to set anew price at time t, choose the same price PtThe evolution of the aggregate price level is governed by

    Pt =

    10

    [Pt (i)]1e di

    11e

    ={

    (Pt1)1e + (1 )(Pt )1e} 11e

    (Institute) Monetary Theory: Price Setting March 2011 6 / 10

  • Calvos 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 ), 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 PtThe evolution of the aggregate price level is governed by

    Pt =

    10

    [Pt (i)]1e di

    11e

    ={

    (Pt1)1e + (1 )(Pt )1e} 11e

    (Institute) Monetary Theory: Price Setting March 2011 6 / 10

  • Firms

    Continuum of rms, indexed by i 2 [0; 1] Each rm produces a dierentiated 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 11

  • Calvos 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 ), regardless of when the current price was setFocusing on a symmetric equilibrium, all the firms that get to set anew price at time t, choose the same price Pt

    The evolution of the aggregate price level is governed by

    Pt =

    10

    [Pt (i)]1e di

    11e

    ={

    (Pt1)1e + (1 )(Pt )1e} 11e

    (Institute) Monetary Theory: Price Setting March 2011 6 / 10

  • Optimal Price Setting

    maxP t

    1Xk=0

    k EtQt;t+k

    P t Yt+kjt t+k(Yt+kjt)

    subject to

    Yt+kjt = (Pt =Pt+k)

    Ct+k

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

    Qt;t+k kCt+kCt

    PtPt+k

    Optimality condition:1Xk=0

    k EtQt;t+k Yt+kjt

    P t M t+kjt

    = 0

    where t+kjt 0t+k(Yt+kjt) andM 1

  • Calvos 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 ), regardless of when the current price was setFocusing on a symmetric equilibrium, all the firms that get to set anew price at time t, choose the same price PtThe evolution of the aggregate price level is governed by

    Pt =

    10

    [Pt (i)]1e di

    11e

    ={

    (Pt1)1e + (1 )(Pt )1e} 11e

    (Institute) Monetary Theory: Price Setting March 2011 6 / 10

  • Evolution of Calvo Prices

    In a deterministic steady state with zero inflation,

    (Pt )1e = (Pt1)1e + (1 )(Pt )1e

    implies P = P, and we get

    (1 e) (P)e (Pt P) = (1 e) (P)e [(Pt1 P)+(1 )(Pt P)]

    andpt = pt1 + (1 )pt

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

    t pt pt1 = (1 )(pt pt1)

    (Institute) Monetary Theory: Price Setting March 2011 7 / 11

  • Evolution of Calvo Prices

    In a deterministic steady state with zero inflation,

    (Pt )1e = (Pt1)1e + (1 )(Pt )1e

    implies P = P, and we get

    (1 e) (P)e (Pt P) = (1 e) (P)e [(Pt1 P)+(1 )(Pt P)]

    andpt = pt1 + (1 )pt

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

    t pt pt1 = (1 )(pt pt1)

    (Institute) Monetary Theory: Price Setting March 2011 7 / 11

  • Aggregate Price Dynamics

    Pt = (Pt1)

    1 + (1 ) (P t )1 11

    Dividing by Pt1 :

    1t = + (1 )P tPt1

    1

    Log-linearization around zero ination steady state

    t = (1 ) (pt pt1) (1)

    or, equivalentlypt = pt1 + (1 ) pt

  • 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]1[it Et (t+1) ]

    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)

    (Institute) Monetary Theory: Price Setting March 2011 8 / 10

  • 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]1[it Et (t+1) ]

    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)

    (Institute) Monetary Theory: Price Setting March 2011 8 / 10

  • 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]1[it Et (t+1) ]

    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)

    (Institute) Monetary Theory: Price Setting March 2011 8 / 10

  • 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]1[it Et (t+1) ]

    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)

    (Institute) Monetary Theory: Price Setting March 2011 8 / 10

  • 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]1[it Et (t+1) ]

    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)

    (Institute) Monetary Theory: Price Setting March 2011 8 / 10

  • Price Setting in Calvos 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:

    1

    k=0

    kEt

    [(t+k

    t

    )Yt+k (i)

    {Pt (i)

    (e

    e 1

    )[Yt+k (i)]

    }]= 0

    The link between new prices and real marginal cost

    (Institute) Monetary Theory: Price Setting March 2011 9 / 10

  • Optimal Price Setting

    maxP t

    1Xk=0

    k EtQt;t+k

    P t Yt+kjt t+k(Yt+kjt)

    subject to

    Yt+kjt = (Pt =Pt+k)

    Ct+k

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

    Qt;t+k kCt+kCt

    PtPt+k

    Optimality condition:1Xk=0

    k EtQt;t+k Yt+kjt

    P t M t+kjt

    = 0

    where t+kjt 0t+k(Yt+kjt) andM 1

  • Price Setting in Calvos 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:

    1

    k=0

    kEt

    [(t+k

    t

    )Yt+k (i)

    {Pt (i)

    (e

    e 1

    )[Yt+k (i)]

    }]= 0

    The link between new prices and real marginal cost

    (Institute) Monetary Theory: Price Setting March 2011 9 / 10

  • Price Setting in Calvos 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:

    1

    k=0

    kEt

    [(t+k

    t

    )Yt+k (i)

    {Pt (i)

    (e

    e 1

    )[Yt+k (i)]

    }]= 0

    The link between new prices and real marginal cost

    (Institute) Monetary Theory: Price Setting March 2011 9 / 10

  • Equivalently,1Xk=0

    k Et

    Qt;t+k Yt+kjt

    P tPt1

    M MCt+kjt t1;t+k

    = 0

    where MCt+kjt t+kjt=Pt+k and t1;t+k Pt+k=Pt1

    Perfect Foresight, Zero Ination Steady State:

    P tPt1

    = 1 ; t1;t+k = 1 ; Yt+kjt = Y ; Qt;t+k = k ; MC =

    1

    M

  • Log-linearization around zero ination steady state:

    pt pt1 = (1 )1Xk=0

    ()k Etfcmct+kjt + pt+k pt1gwhere cmct+kjt mct+kjt mc.Equivalently,

    pt = + (1 )1Xk=0

    ()k Etfmct+kjt + pt+kg

    where log 1.Flexible prices ( = 0):

    pt = +mct + pt

    =) mct = (symmetric equilibrium)

  • Inflation Dynamics in Calvos Model

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

    In this case, real marginal cost equals

    WtPtAt

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

    With diminishing returns to labor (0 < < 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

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

  • Particular Case: = 0 (constant returns)

    =) MCt+kjt =MCt+kRewriting the optimal price setting rule in recursive form:

    pt = Etfpt+1g + (1 ) cmct + (1 )pt (2)Combining (1) and (2):

    t = Etft+1g + cmctwhere

    (1 )(1 )

  • Inflation Dynamics in Calvos Model

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

    In this case, real marginal cost equals

    WtPtAt

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

    With diminishing returns to labor (0 < < 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

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

  • Inflation Dynamics in Calvos Model

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

    In this case, real marginal cost equals

    WtPtAt

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

    With diminishing returns to labor (0 < < 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

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

  • Generalization to 2 (0; 1) (decreasing returns)Dene

    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 (pt pt+k) (3)

    Implied ination dynamics

    t = Etft+1g + cmct (4)where

    (1 )(1 )

    1 1 +

  • Inflation Dynamics in Calvos Model

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

    In this case, real marginal cost equals

    WtPtAt

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

    With diminishing returns to labor (0 < < 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

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

  • Real Marginal Cost and Inflation Dynamics in CalvosModel

    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 in

    t = Ett+1 + mc t

    is inversely related to the degree of price rigidity

    Iterating forward,the bounded solution for inflation is

    t = Et

    j=0

    j mc t+j

    (Institute) Monetary Theory: Price Setting March 2011 11 / 11

  • Real Marginal Cost and Inflation Dynamics in CalvosModel

    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 in

    t = Ett+1 + mc t

    is inversely related to the degree of price rigidity

    Iterating forward,the bounded solution for inflation is

    t = Et

    j=0

    j mc t+j

    (Institute) Monetary Theory: Price Setting March 2011 11 / 11

  • Real Marginal Cost and Inflation Dynamics in CalvosModel

    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 in

    t = Ett+1 + mc t

    is inversely related to the degree of price rigidity

    Iterating forward,the bounded solution for inflation is

    t = Et

    j=0

    j mc t+j

    (Institute) Monetary Theory: Price Setting March 2011 11 / 11

  • Real Marginal Cost and Inflation Dynamics in CalvosModel

    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 in

    t = Ett+1 + mc t

    is inversely related to the degree of price rigidity

    Iterating forward,the bounded solution for inflation is

    t = Et

    j=0

    j mc t+j

    (Institute) Monetary Theory: Price Setting March 2011 11 / 11

    Monetary Theory: Price SettingNotationOne-period Price RigidityTwo-period Price RigidityStaggered Price SettingCalvo's ModelEvolution of Calvo PricesImplications of Staggered Price SettingPrice Setting in Calvo's ModelInflation Dynamics in Calvo's Model