- Email: [email protected]
The imperfect-common-knowledge Phillips curve: Calvo vs Rotemberg
Economics Letters 148 (2016) 45–47
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
The imperfect-common-knowledge Phillips curve: Calvo vs Rotemberg✩ Radek Šauer Research Center, Deutsche Bundesbank, Wilhelm-Epstein-Straße 14, 60431 Frankfurt, Germany
highlights • Imperfect-common-knowledge Phillips curves derived under Calvo and Rotemberg differ. • Inflation always depends on expectations of marginal costs and of future inflation. • Under Rotemberg inflation additionally depends on expectations of relative prices.
article
info
Article history: Received 31 May 2016 Received in revised form 8 September 2016 Accepted 19 September 2016 Available online 22 September 2016
abstract I derive the imperfect-common-knowledge Phillips curve under the assumption of Rotemberg pricing. The curve differs from the Calvo version in one important aspect. Expectations of future relative prices impact inflation. © 2016 Elsevier B.V. All rights reserved.
JEL classification: D83 E31 Keywords: Phillips curve Rotemberg Imperfect knowledge
1. Calvo and the imperfect common knowledge
derive the imperfect-common-knowledge Phillips curve:
Forecast surveys document that economic agents disagree about the future (e.g., Coibion and Gorodnichenko, 2012; Andrade et al., in press). Private information represents a natural explanation for such heterogeneous beliefs. Each economic agent has its specific information set and cannot access all relevant information. The heterogeneous beliefs can be formalized by models with imperfect common knowledge (i.a., Woodford, 2003). Nimark (2008) and Melosi (2013) impose imperfect common knowledge and Calvo (1983) pricing on firms. The authors then
πt = (1 − θ ) (1 − βθ)
http://dx.doi.org/10.1016/j.econlet.2016.09.021 0165-1765/© 2016 Elsevier B.V. All rights reserved.
∞
+ βθ
(1 − θ )k−1 πt(+k)1|t .
(1)
k=1
The variables in the Phillips curve have the following definitions. 1
(1)
j
Et mct (j) dj
mct |t = 0
(k+1)
reflect the views of the Deutsche Bundesbank. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. E-mail address: [email protected].
(1 − θ)k−1 mct(|kt)
k=1
mct |t ✩ The views expressed in this letter are those of the author and do not necessarily
∞
1
πt(+1)1|t =
(k)
j
=
Et mct |t dj
∀k = 1, 2, . . .
0 1
j
Et πt +1 dj 0
πt(+k+1|1t) =
1
j
(k)
Et πt +1|t dj ∀k = 1, 2, . . . . 0
46
R. Šauer / Economics Letters 148 (2016) 45–47
The Calvo pricing and the imperfect common knowledge cause inflation πt to depend on two sums. The first adds up average (k) higher-order expectations of marginal costs mct |t . The second adds (k) up average higher-order expectations of future inflation πt +1|t . The
parameter β is the discount factor, (1 − θ) the probability of price j reoptimization. The operator Et denotes expectations of firm j; mct (j) represents marginal costs of firm j. This is the first paper addressing the question whether the imperfect-common-knowledge Phillips curve looks similar under the Rotemberg (1982) pricing. I derive such a curve and show it differs from (1). 2. Profit maximization A continuum of firms [0; 1] populates the economy. Each firm produces a differentiated good. A bundler, which uses the Dixit–Stiglitz aggregator, combines the differentiated goods and supplies final good Yt . A firm j ∈ [0; 1] faces demand Yt (j) = (Zt (j))−ν Yt ,
(2)
where Zt (j) = Pt (j)/Pt denotes the relative price of firm j. The definition of the aggregate price level Pt implies
3. The Phillips curve I linearize the first-order condition (4). For convenience, I define πt (j) = Πt (j) − Π , mct (j) = (MCt (j) − MC (j)) /MC (j), zt (j) = (Zt (j) − Z (j)) /Z (j), and κ = (ν − 1) / (Ξ Π ). Given the informaj j tion set It the firm knows its today’s inflation (Et πt (j) = πt (j)). I can express the first-order condition as (see the online Appendix A)
πt (j) = κ Etj mct (j) + β Etj πt +1 (j) − κ Etj zt (j).
The standard logic of price setting holds. If the firm expects higher marginal costs, it requires a higher price. The price-adjustment costs generate forward-looking firm-specific inflation. The firm does not want to set its price too high because the demand would be too low. Therefore, the expected relative price negatively affects the firm-specific inflation. The definition of the relative price leads to
(Zt (j))1−ν dj.
πt (j) =
(3)
0
The firm j knows all parameters and completely understands how the economy works. But it does not know realizations of all variables. Instead, the firm has the following information set: j It
= {Pτ (j), Pτ −1 , aτ (j)|τ = t , t − 1, t − 2, . . .} .
The firm knows the price it sets today and remembers its past prices. The firm learns the aggregate price level with one period lag. The firm additionally receives a noisy signal at (j) about a process at which is common to all firms: at (j) = at + ϵt (j).
j
Pt (j)
∞ τ =t
Λτ |t Pτ (j)Yτ (j) − Pτ MCτ (j)Yτ (j)
−Pτ
Ξ 2
Pτ (j) Pτ −1 (j)
−Π
2 Yτ
κΠ βΠ j j E mct (j) + E π t +1 (1 + β) Π + κ t (1 + β) Π + κ t βΠ + κ βΠ 2 j j E zt +1 (j) + E πt + (1 + β) Π + κ t (1 + β) Π + κ t (β Π + κ) Π j − E zt −1 (j). (1 + β) Π + κ t j
firm-specific inflations (πt = Appendix A)
1 0
πt (j) dj). I obtain (see the online
βΠ κΠ (1) mc + π (1) (1 + β) Π + κ t |t (1 + β) Π + κ t +1|t βΠ 2 βΠ + κ (1) + z + π (1) , (6) (1 + β) Π + κ t +1|t (1 + β) Π + κ t |t 1 j 1 j (1) (1) (1) where mct |t = 0 Et mct (j) dj, πt +1|t = 0 Et πt +1 dj, zt +1|t = 1 j 1 j (1) E z (j) dj, and πt |t = 0 Et πt dj. I repeatedly take expecta0 t t +1 j (1) (2) tions Et and integrate over (6) to get expressions for πt |t , πt |t , (1) (2) etc. Then in (6) I repeatedly substitute for πt |t , πt |t , etc. Finally,
I obtain the imperfect-common-knowledge Phillips curve under Rotemberg:
πt =
(ν − 1) Π (1 + β) Ξ Π 2 + ν − 1 k−1 ∞ βΞ Π 2 + ν − 1 (k) × mct |t 2 +ν −1 1 + β) Ξ Π ( k=1 + ×
I can write the first-order condition as j
1
The information set It ensures the firm knows its past relative j price (Et zt −1 (j) = zt −1 (j)). I integrate over all firms and apply two results which originate from (3). The relative prices sum to 1 zero ( 0 zt −1 (j) dj = 0); the aggregate inflation is the average of
s.t. (2).
0 = Et
πt (j) −
πt =
For instance, at (j) could be firm-specific productivity which consists of productivity at shared by all firms and an idiosyncratic component ϵt (j). Over all other variables the firm has to form j expectations Et . Richer information sets like in Nimark (2008) and Melosi (2013) would change nothing in the derivation of the Phillips curve. The production of the firm obeys constant returns to scale. Real marginal costs MCt (j) are firm-specific (e.g., because of firmspecific productivity). The firm has to pay price-adjustment costs, which depend on the steady-state gross inflation rate Π and the aggregate demand Yt . In every period the firm maximizes expected nominal profits discounted by Λτ |t . max Et
1
πt + zt −1 (j), Π Π where πt = Πt − Π . I use this relation and substitute for πt +1 (j) and zt (j) in (5): zt (j) =
1
1=
(5)
k=1
1 − ν + ν MCt (j) (Zt (j))−1 Yt (j)
− Ξ Πt (Πt (j) − Π ) (Zt −1 (j))
−1
+
Yt
+ Λt +1|t Ξ Πt +1 (Πt +1 (j) − Π ) Πt +1 (j) (Zt (j))−1 Yt +1 , where Πt = Pt /Pt −1 and Πt (j) = Pt (j)/Pt −1 (j).
βΞ Π 2 (1 + β) Ξ Π 2 + ν − 1 ∞ βΞ Π 2 + ν − 1
(4)
×
(1 + β) Ξ Π 2 + ν − 1
βΞ Π 3 (1 + β) Ξ Π 2 + ν − 1 ∞ βΞ Π 2 + ν − 1 k=1
k−1
(1 + β) Ξ Π 2 + ν − 1
k−1
πt(+k)1|t
(k)
zt +1|t ,
(7)
R. Šauer / Economics Letters 148 (2016) 45–47
(k+1)
where mct |t (k+1) zt +1|t
=
=
j (k) Ez 0 t t +1|t
1
1 0
j
(k)
(k+1)
Et mct |t dj, πt +1|t
=
1 0
j
(k)
Et πt +1|t dj, and
dj for k = 1, 2, . . . .
4. The difference between Calvo and Rotemberg Now it becomes clear how the Phillips curves in (1) and (7) differ. Under the Rotemberg pricing inflation still depends on expectations of marginal costs and on expectations of future inflation. But a new term appears. Inflation, in addition, depends on (k) average higher-order expectations of future relative prices zt +1|t . In models with perfect information the Phillips curves under Calvo and Rotemberg easily resemble each other up to the first order. One just needs to assume zero trend inflation or full price indexation (i.a., Ascari and Rossi, 2012). Here, under the imperfect common knowledge, the Phillips curves differ despite of the full indexation. The difference results from the heterogeneous beliefs. Each firm has private information in its information set and forms its own beliefs. Therefore, the average higher-order expectations of future relative prices do not equal zero, and the Phillips curves differ. If the firms were still imperfectly informed, but information was not private, the average higher-order expectations of future relative prices would equal zero. In consequence, the Phillips curves under Calvo and Rotemberg would become identical.
47
Acknowledgment I am grateful to an anonymous referee for helpful suggestions and comments. Appendix A. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.econlet.2016.09.021. References Andrade, P., Crump, R.K., Eusepi, S., Moench, E., 2016. Fundamental disagreement. J. Monetary Econ. http://dx.doi.org/10.1016/j.jmoneco.2016.08.007, (in press). Ascari, G., Rossi, L., 2012. Trend inflation and firms price-setting: Rotemberg Versus Calvo. Econom. J. 122, 1115–1141. http://dx.doi.org/10.1111/j.14680297.2012.02517.x. Calvo, G.A., 1983. Staggered prices in a utility-maximizing framework. J. Monetary Econ. 12, 383–398. http://dx.doi.org/10.1016/0304-3932(83)90060-0. Coibion, O., Gorodnichenko, Y., 2012. What can survey forecasts tell us about information rigidities? J. Polit. Econ. 120, 116–159. http://dx.doi.org/10.1086/ 665662. Melosi, L., 2013. Signaling Effects of Monetary Policy. PIER Working Paper Archive 13-029. Penn Institute for Economic Research, Department of Economics, University of Pennsylvania. Nimark, K., 2008. Dynamic pricing and imperfect common knowledge. J. Monetary Econ. 55, 365–382. http://dx.doi.org/10.1016/j.jmoneco.2007.12.008. Rotemberg, J.J., 1982. Monopolistic price adjustment and aggregate output. Rev. Econom. Stud. 49, 517–531. http://dx.doi.org/10.2307/2297284. Woodford, M., 2003. Imperfect common knowledge and the effects of monetary policy. In: Aghion, P., Frydman, R., Stiglitz, J., Woodford, M. (Eds.), Knowledge, Information, and Expectations in Modern Macroeconomics: In Honor of Edmund Phelps. Princeton University Press.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
The imperfect-common-knowledge Phillips curve: Calvo vs Rotemberg✩ Radek Šauer Research Center, Deutsche Bundesbank, Wilhelm-Epstein-Straße 14, 60431 Frankfurt, Germany
highlights • Imperfect-common-knowledge Phillips curves derived under Calvo and Rotemberg differ. • Inflation always depends on expectations of marginal costs and of future inflation. • Under Rotemberg inflation additionally depends on expectations of relative prices.
article
info
Article history: Received 31 May 2016 Received in revised form 8 September 2016 Accepted 19 September 2016 Available online 22 September 2016
abstract I derive the imperfect-common-knowledge Phillips curve under the assumption of Rotemberg pricing. The curve differs from the Calvo version in one important aspect. Expectations of future relative prices impact inflation. © 2016 Elsevier B.V. All rights reserved.
JEL classification: D83 E31 Keywords: Phillips curve Rotemberg Imperfect knowledge
1. Calvo and the imperfect common knowledge
derive the imperfect-common-knowledge Phillips curve:
Forecast surveys document that economic agents disagree about the future (e.g., Coibion and Gorodnichenko, 2012; Andrade et al., in press). Private information represents a natural explanation for such heterogeneous beliefs. Each economic agent has its specific information set and cannot access all relevant information. The heterogeneous beliefs can be formalized by models with imperfect common knowledge (i.a., Woodford, 2003). Nimark (2008) and Melosi (2013) impose imperfect common knowledge and Calvo (1983) pricing on firms. The authors then
πt = (1 − θ ) (1 − βθ)
http://dx.doi.org/10.1016/j.econlet.2016.09.021 0165-1765/© 2016 Elsevier B.V. All rights reserved.
∞
+ βθ
(1 − θ )k−1 πt(+k)1|t .
(1)
k=1
The variables in the Phillips curve have the following definitions. 1
(1)
j
Et mct (j) dj
mct |t = 0
(k+1)
reflect the views of the Deutsche Bundesbank. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. E-mail address: [email protected].
(1 − θ)k−1 mct(|kt)
k=1
mct |t ✩ The views expressed in this letter are those of the author and do not necessarily
∞
1
πt(+1)1|t =
(k)
j
=
Et mct |t dj
∀k = 1, 2, . . .
0 1
j
Et πt +1 dj 0
πt(+k+1|1t) =
1
j
(k)
Et πt +1|t dj ∀k = 1, 2, . . . . 0
46
R. Šauer / Economics Letters 148 (2016) 45–47
The Calvo pricing and the imperfect common knowledge cause inflation πt to depend on two sums. The first adds up average (k) higher-order expectations of marginal costs mct |t . The second adds (k) up average higher-order expectations of future inflation πt +1|t . The
parameter β is the discount factor, (1 − θ) the probability of price j reoptimization. The operator Et denotes expectations of firm j; mct (j) represents marginal costs of firm j. This is the first paper addressing the question whether the imperfect-common-knowledge Phillips curve looks similar under the Rotemberg (1982) pricing. I derive such a curve and show it differs from (1). 2. Profit maximization A continuum of firms [0; 1] populates the economy. Each firm produces a differentiated good. A bundler, which uses the Dixit–Stiglitz aggregator, combines the differentiated goods and supplies final good Yt . A firm j ∈ [0; 1] faces demand Yt (j) = (Zt (j))−ν Yt ,
(2)
where Zt (j) = Pt (j)/Pt denotes the relative price of firm j. The definition of the aggregate price level Pt implies
3. The Phillips curve I linearize the first-order condition (4). For convenience, I define πt (j) = Πt (j) − Π , mct (j) = (MCt (j) − MC (j)) /MC (j), zt (j) = (Zt (j) − Z (j)) /Z (j), and κ = (ν − 1) / (Ξ Π ). Given the informaj j tion set It the firm knows its today’s inflation (Et πt (j) = πt (j)). I can express the first-order condition as (see the online Appendix A)
πt (j) = κ Etj mct (j) + β Etj πt +1 (j) − κ Etj zt (j).
The standard logic of price setting holds. If the firm expects higher marginal costs, it requires a higher price. The price-adjustment costs generate forward-looking firm-specific inflation. The firm does not want to set its price too high because the demand would be too low. Therefore, the expected relative price negatively affects the firm-specific inflation. The definition of the relative price leads to
(Zt (j))1−ν dj.
πt (j) =
(3)
0
The firm j knows all parameters and completely understands how the economy works. But it does not know realizations of all variables. Instead, the firm has the following information set: j It
= {Pτ (j), Pτ −1 , aτ (j)|τ = t , t − 1, t − 2, . . .} .
The firm knows the price it sets today and remembers its past prices. The firm learns the aggregate price level with one period lag. The firm additionally receives a noisy signal at (j) about a process at which is common to all firms: at (j) = at + ϵt (j).
j
Pt (j)
∞ τ =t
Λτ |t Pτ (j)Yτ (j) − Pτ MCτ (j)Yτ (j)
−Pτ
Ξ 2
Pτ (j) Pτ −1 (j)
−Π
2 Yτ
κΠ βΠ j j E mct (j) + E π t +1 (1 + β) Π + κ t (1 + β) Π + κ t βΠ + κ βΠ 2 j j E zt +1 (j) + E πt + (1 + β) Π + κ t (1 + β) Π + κ t (β Π + κ) Π j − E zt −1 (j). (1 + β) Π + κ t j
firm-specific inflations (πt = Appendix A)
1 0
πt (j) dj). I obtain (see the online
βΠ κΠ (1) mc + π (1) (1 + β) Π + κ t |t (1 + β) Π + κ t +1|t βΠ 2 βΠ + κ (1) + z + π (1) , (6) (1 + β) Π + κ t +1|t (1 + β) Π + κ t |t 1 j 1 j (1) (1) (1) where mct |t = 0 Et mct (j) dj, πt +1|t = 0 Et πt +1 dj, zt +1|t = 1 j 1 j (1) E z (j) dj, and πt |t = 0 Et πt dj. I repeatedly take expecta0 t t +1 j (1) (2) tions Et and integrate over (6) to get expressions for πt |t , πt |t , (1) (2) etc. Then in (6) I repeatedly substitute for πt |t , πt |t , etc. Finally,
I obtain the imperfect-common-knowledge Phillips curve under Rotemberg:
πt =
(ν − 1) Π (1 + β) Ξ Π 2 + ν − 1 k−1 ∞ βΞ Π 2 + ν − 1 (k) × mct |t 2 +ν −1 1 + β) Ξ Π ( k=1 + ×
I can write the first-order condition as j
1
The information set It ensures the firm knows its past relative j price (Et zt −1 (j) = zt −1 (j)). I integrate over all firms and apply two results which originate from (3). The relative prices sum to 1 zero ( 0 zt −1 (j) dj = 0); the aggregate inflation is the average of
s.t. (2).
0 = Et
πt (j) −
πt =
For instance, at (j) could be firm-specific productivity which consists of productivity at shared by all firms and an idiosyncratic component ϵt (j). Over all other variables the firm has to form j expectations Et . Richer information sets like in Nimark (2008) and Melosi (2013) would change nothing in the derivation of the Phillips curve. The production of the firm obeys constant returns to scale. Real marginal costs MCt (j) are firm-specific (e.g., because of firmspecific productivity). The firm has to pay price-adjustment costs, which depend on the steady-state gross inflation rate Π and the aggregate demand Yt . In every period the firm maximizes expected nominal profits discounted by Λτ |t . max Et
1
πt + zt −1 (j), Π Π where πt = Πt − Π . I use this relation and substitute for πt +1 (j) and zt (j) in (5): zt (j) =
1
1=
(5)
k=1
1 − ν + ν MCt (j) (Zt (j))−1 Yt (j)
− Ξ Πt (Πt (j) − Π ) (Zt −1 (j))
−1
+
Yt
+ Λt +1|t Ξ Πt +1 (Πt +1 (j) − Π ) Πt +1 (j) (Zt (j))−1 Yt +1 , where Πt = Pt /Pt −1 and Πt (j) = Pt (j)/Pt −1 (j).
βΞ Π 2 (1 + β) Ξ Π 2 + ν − 1 ∞ βΞ Π 2 + ν − 1
(4)
×
(1 + β) Ξ Π 2 + ν − 1
βΞ Π 3 (1 + β) Ξ Π 2 + ν − 1 ∞ βΞ Π 2 + ν − 1 k=1
k−1
(1 + β) Ξ Π 2 + ν − 1
k−1
πt(+k)1|t
(k)
zt +1|t ,
(7)
R. Šauer / Economics Letters 148 (2016) 45–47
(k+1)
where mct |t (k+1) zt +1|t
=
=
j (k) Ez 0 t t +1|t
1
1 0
j
(k)
(k+1)
Et mct |t dj, πt +1|t
=
1 0
j
(k)
Et πt +1|t dj, and
dj for k = 1, 2, . . . .
4. The difference between Calvo and Rotemberg Now it becomes clear how the Phillips curves in (1) and (7) differ. Under the Rotemberg pricing inflation still depends on expectations of marginal costs and on expectations of future inflation. But a new term appears. Inflation, in addition, depends on (k) average higher-order expectations of future relative prices zt +1|t . In models with perfect information the Phillips curves under Calvo and Rotemberg easily resemble each other up to the first order. One just needs to assume zero trend inflation or full price indexation (i.a., Ascari and Rossi, 2012). Here, under the imperfect common knowledge, the Phillips curves differ despite of the full indexation. The difference results from the heterogeneous beliefs. Each firm has private information in its information set and forms its own beliefs. Therefore, the average higher-order expectations of future relative prices do not equal zero, and the Phillips curves differ. If the firms were still imperfectly informed, but information was not private, the average higher-order expectations of future relative prices would equal zero. In consequence, the Phillips curves under Calvo and Rotemberg would become identical.
47
Acknowledgment I am grateful to an anonymous referee for helpful suggestions and comments. Appendix A. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.econlet.2016.09.021. References Andrade, P., Crump, R.K., Eusepi, S., Moench, E., 2016. Fundamental disagreement. J. Monetary Econ. http://dx.doi.org/10.1016/j.jmoneco.2016.08.007, (in press). Ascari, G., Rossi, L., 2012. Trend inflation and firms price-setting: Rotemberg Versus Calvo. Econom. J. 122, 1115–1141. http://dx.doi.org/10.1111/j.14680297.2012.02517.x. Calvo, G.A., 1983. Staggered prices in a utility-maximizing framework. J. Monetary Econ. 12, 383–398. http://dx.doi.org/10.1016/0304-3932(83)90060-0. Coibion, O., Gorodnichenko, Y., 2012. What can survey forecasts tell us about information rigidities? J. Polit. Econ. 120, 116–159. http://dx.doi.org/10.1086/ 665662. Melosi, L., 2013. Signaling Effects of Monetary Policy. PIER Working Paper Archive 13-029. Penn Institute for Economic Research, Department of Economics, University of Pennsylvania. Nimark, K., 2008. Dynamic pricing and imperfect common knowledge. J. Monetary Econ. 55, 365–382. http://dx.doi.org/10.1016/j.jmoneco.2007.12.008. Rotemberg, J.J., 1982. Monopolistic price adjustment and aggregate output. Rev. Econom. Stud. 49, 517–531. http://dx.doi.org/10.2307/2297284. Woodford, M., 2003. Imperfect common knowledge and the effects of monetary policy. In: Aghion, P., Frydman, R., Stiglitz, J., Woodford, M. (Eds.), Knowledge, Information, and Expectations in Modern Macroeconomics: In Honor of Edmund Phelps. Princeton University Press.