The Hybrid New Keynesian Phillips Curve with Multiple Lags of Inflation

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1. Introduction

The hybrid new Keynesian Phillips curve (HNKPC) is generally expressed as an equation that relates current inflation to a real demand variable (usually either the output gap or real marginal cost), next period’s inflation, and last period’s inflation. A notable example of the HNKPC is the model of Galí and Gertler (1999). The HNKPC assumes Calvo (1983) price setting, in which a fraction of firms (randomly chosen) are able to reset their prices. Of the firms that can reset prices in a given period, a fraction set their price at a level that maximizes the present value of expected profits, while the remainder index their prices to the average price chosen by firms that reset prices in the previous period, adjusted for last period’s inflation rate. The former are considered forward-looking firms, and the latter are considered backward-looking firms. Microfoundations for the HNKPC are derived in Holmberg (2006), which presents a derivation of the HNKPC with one lag of inflation.

This study extends the previous literature by deriving an expression for the HNKPC in which firms index prices to an arbitrarily large number of lags of inflation. As a result, inflation depends on multiple lags of inflation, instead of just last period’s inflation rate, allowing for a more flexible specification in modeling inflation dynamics. Mavroeidis, Plagborg-Møller, and Stock (2014) estimate the HNKPC with both one lag and four lags of inflation (although they do not derive microfoundations for a specification that includes multiple lags of inflation), and they find significant differences in the estimated coefficients between these specifications, suggesting that specifications limiting the lag structure to one lag may miss important inflation dynamics. In addition, many empirical estimates of the Phillips curve (both purely backward looking and hybrid specifications) include multiple lags of inflation. Examples of studies that include multiple lags of inflation in estimating the Phillips curve include King and Watson (1994), Fuhrer (1995, 1997), Akerlof et al. (1996), Duca (1996), Gordon (1997), Murphy (2014), and Coibion and Gorodnichenko (2015).

Because there is evidence that incorporating more than one lag of inflation in the HNKPC affects the estimated coefficients and because previous estimates of the Phillips curve often include multiple lags of inflation, it is beneficial to provide microfoundations for a specification that includes multiple lags of inflation. The equation derived in the present study provides theoretical justification for empirical estimates of the HNKPC that include more than one lag of inflation.

2. Model of the HNKPC with Multiple Lags of Inflation

In Holmberg’s (2006) derivation of the HNKPC, it is assumed that backward-looking firms index their prices to the average price newly set last period plus last period’s inflation rate. This study extends Holmberg’s analysis by assuming that backward-looking firms index their prices to the average price newly set last period plus a weighted average of past inflation rates, allowing for an arbitrarily large number of lags of inflation. The derivation in this study is valid both when the demand variable is the output gap and when the demand variable is real marginal cost.

Following Holmberg (2006), let ${p}_{t}^{\ast}$ represent the price (in natural logs) set by firms that are able to adjust their prices in the current period and let $\theta $ represent the proportion of firms that are not able to reset their prices. Then, the average price ( ${p}_{t}$ ) is

${p}_{t}=\left(1-\theta \right){p}_{t}^{\ast}+\theta {p}_{t-1}.$ (1)

By subtracting ${p}_{t-1}$ from both sides of (1), the inflation rate can be expressed as

${\pi}_{t}={p}_{t}-{p}_{t-1}=\left(1-\theta \right)\left({p}_{t}^{\ast}-{p}_{t-1}\right).$ (2)

Let $\omega $ represent the proportion of firms resetting prices that index their prices to lagged inflation (i.e., backward-looking firms), ${p}_{t}^{b}$ represent the price set by backward-looking firms, and ${p}_{t}^{f}$ represent the price set by forward-looking firms. Then,

${p}_{t}^{\ast}=\omega {p}_{t}^{b}+\left(1-\omega \right){p}_{t}^{f}$. (3)

Solving (1) for ${p}_{t}^{\ast}$ and substituting this expression into (3) yields

$\frac{1}{1-\theta}{p}_{t}-\frac{\theta}{1-\theta}{p}_{t-1}=\omega {p}_{t}^{b}+\left(1-\omega \right){p}_{t}^{f}$. (4)

Subtracting ${p}_{t}$ from both sides of (4) and multiplying by $1-\theta $ results in the relationship,

$\theta {\pi}_{t}=\omega \left(1-\theta \right)\left({p}_{t}^{b}-{p}_{t}\right)+\left(1-\omega \right)\left(1-\theta \right)\left({p}_{t}^{f}-{p}_{t}\right)$. (5)

Let $\beta $ represent the discount factor and ${p}_{t+j}^{\ast ,e}$ represent firms’ expectation of the optimal price in period t + j when they set prices in period t. Then forward-looking firms set their prices equal to

${p}_{t}^{f}=\left(1-\beta \theta \right){\displaystyle \underset{j=0}{\overset{\infty}{\sum}}{\beta}^{j}{\theta}^{j}}{E}_{t}{p}_{t+j}^{\ast}$, (6)

which can be expressed as,

${p}_{t}^{f}=\left(1-\beta \theta \right){p}_{t}^{\ast}+\beta \theta {E}_{t}{p}_{t+1}^{f}$ ^{1}. (7)

In Roberts (1995: p. 977), a firm’s optimal price can be expressed as
${p}_{t}^{*}={p}_{t}+\kappa {y}_{t}$, where
${y}_{t}$ represents the output gap^{2}. A similar relationship is provided in Galí and Gertler (1999), in which a firm’s optimal price equals its nominal marginal cost (
$m{c}_{t}^{n}$ ). The relationship between nominal marginal cost and real marginal cost (
$m{c}_{t}$ ) is
$m{c}_{t}^{n}={p}_{t}+m{c}_{t}$, so that
${p}_{t}^{*}={p}_{t}+m{c}_{t}$. This is the same equation as in Roberts, but with
${y}_{t}=m{c}_{t}$ and
$\kappa =\text{1}$. Thus,
${y}_{t}$ can represent either real marginal cost or the output gap. The price set by forward-looking firms is therefore equal to

${p}_{t}^{f}=\left(1-\beta \theta \right)\left({p}_{t}+\kappa {y}_{t}\right)+\beta \theta {E}_{t}{p}_{t+1}^{f}$, (8)

and the difference between ${p}_{t}^{f}$ and ${p}_{t}$ is

${p}_{t}^{f}-{p}_{t}=\left(1-\beta \theta \right)\kappa {y}_{t}-\beta \theta {p}_{t}+\beta \theta {E}_{t}{p}_{t+1}^{f}$. (9)

In Galí and Gertler (1999) and Holmberg (2006), when backward-looking firms set prices, they set them equal to the average price set last period by firms that adjusted prices, plus last period’s inflation rate. In the present study, when these firms set prices, they set them equal to the average price set last period by firms that adjusted prices plus a weighted average of lagged inflation. Thus, the price set by backward-looking firms can be expressed as

${p}_{t}^{b}={p}_{t-1}^{\ast}+{\displaystyle \underset{i=1}{\overset{T}{\sum}}{\lambda}_{i}}{\pi}_{t-i}$ with $\underset{i=1}{\overset{T}{\sum}}{\lambda}_{i}}=1$, (10)

where the $\lambda $ ’s represent the weight given to each lag of inflation in the indexation process.

Combining (1) and (10) yields

$\begin{array}{c}{p}_{t}^{b}-{p}_{t}={p}_{t-1}^{\ast}+{\displaystyle \underset{i=1}{\overset{T}{\sum}}{\lambda}_{i}}{\pi}_{t-i}-\left(1-\theta \right){p}_{t}^{\ast}-\theta {p}_{t-1}\\ =\theta \left({p}_{t-1}^{\ast}-{p}_{t-2}\right)+{\displaystyle \underset{i=1}{\overset{T}{\sum}}{\lambda}_{i}}{\pi}_{t-i}-\left(1-\theta \right)\left({p}_{t}^{\ast}-{p}_{t-1}\right).\end{array}$ ^{3} (11)

Since ${\pi}_{t}=\left(1-\theta \right)\left({p}_{t}^{\ast}-{p}_{t-1}\right)$ and ${\pi}_{t-1}=\left(1-\theta \right)\left({p}_{t-1}^{\ast}-{p}_{t-2}\right)$,

${p}_{t}^{b}-{p}_{t}=-{\pi}_{t}+\frac{\theta}{1-\theta}{\pi}_{t-1}+{\displaystyle \underset{i=1}{\overset{T}{\sum}}{\lambda}_{i}}{\pi}_{t-i}$. (12)

By solving (5) for ${p}_{t}^{f}$, the price set by forward-looking firms is

${p}_{t}^{f}=\frac{\theta}{\left(1-\theta \right)\left(1-\omega \right)}{\pi}_{t}-\frac{\omega}{1-\omega}\left({p}_{t}^{b}-{p}_{t}\right)+{p}_{t}$, (13)

which, shifted one period forward, implies that

${E}_{t}{p}_{t+1}^{f}=\frac{\theta}{\left(1-\theta \right)\left(1-\omega \right)}{E}_{t}{\pi}_{t+1}-\frac{\omega}{1-\omega}\left({E}_{t}{p}_{t+1}^{b}-{E}_{t}{p}_{t+1}\right)+{E}_{t}{p}_{t+1}$. (14)

Substituting (12) shifted one period forward into (14) yields

$\begin{array}{c}{E}_{t}{p}_{t+1}^{f}=\frac{\theta}{\left(1-\theta \right)\left(1-\omega \right)}{E}_{t}{\pi}_{t+1}-\frac{\omega}{1-\omega}[-{E}_{t}{\pi}_{t+1}+\left(\frac{\theta}{1-\theta}+{\lambda}_{1}\right){\pi}_{t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\displaystyle \underset{i=1}{\overset{T-1}{\sum}}{\lambda}_{i+1}}{\pi}_{t-i}]+{E}_{t}{p}_{t+1},\end{array}$ (15)

which can be expressed as,

$\begin{array}{c}{E}_{t}{p}_{t+1}^{f}=\frac{\theta +\left(1-\theta \right)\omega}{\left(1-\theta \right)\left(1-\omega \right)}{E}_{t}{\pi}_{t+1}-\frac{\omega}{1-\omega}\left(\frac{\theta}{1-\theta}+{\lambda}_{1}\right){\pi}_{t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\omega}{1-\omega}{\displaystyle \underset{i=1}{\overset{T-1}{\sum}}{\lambda}_{i+1}}{\pi}_{t-i}+{E}_{t}{p}_{t+1}.\end{array}$ (16)

By substituting (16) into (9), the difference between ${p}_{t}^{f}$ and ${p}_{t}$ is

$\begin{array}{c}{p}_{t}^{f}-{p}_{t}=\left(1-\beta \theta \right)\kappa {y}_{t}-\beta \theta {p}_{t}+\beta \theta [\frac{\theta +\left(1-\theta \right)\omega}{\left(1-\theta \right)\left(1-\omega \right)}{E}_{t}{\pi}_{t+1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\omega}{1-\omega}\left(\frac{\theta}{1-\theta}+{\lambda}_{1}\right){\pi}_{t}-\frac{\omega}{1-\omega}{\displaystyle \underset{i=1}{\overset{T-1}{\sum}}{\lambda}_{i+1}}{\pi}_{t-i}+{E}_{t}{p}_{t+1}]\end{array}$

$\begin{array}{c}{p}_{t}^{f}-{p}_{t}=\left(1-\beta \theta \right)\kappa {y}_{t}+\beta \theta [\frac{\theta +\left(1-\theta \right)\omega}{\left(1-\theta \right)\left(1-\omega \right)}{E}_{t}{\pi}_{t+1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\omega}{1-\omega}\left(\frac{\theta}{1-\theta}+{\lambda}_{1}\right){\pi}_{t}-\frac{\omega}{1-\omega}{\displaystyle \underset{i=1}{\overset{T-1}{\sum}}{\lambda}_{i+1}}{\pi}_{t-i}+{E}_{t}{\pi}_{t+1}]\end{array}$

$\begin{array}{c}{p}_{t}^{f}-{p}_{t}=\left(1-\beta \theta \right)\kappa {y}_{t}+\beta \theta [\frac{1}{\left(1-\theta \right)\left(1-\omega \right)}{E}_{t}{\pi}_{t+1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\omega}{1-\omega}\left(\frac{\theta}{1-\theta}+{\lambda}_{1}\right){\pi}_{t}-\frac{\omega}{1-\omega}{\displaystyle \underset{i=1}{\overset{T-1}{\sum}}{\lambda}_{i+1}}{\pi}_{t-i}].\end{array}$ (17)

Substituting (12) and (17) into (5) results in the relationship,

$\begin{array}{l}\theta {\pi}_{t}=\omega \left(1-\theta \right)\left(-{\pi}_{t}+\frac{\theta}{1-\theta}{\pi}_{t-1}+{\displaystyle \underset{i=1}{\overset{T}{\sum}}{\lambda}_{i}}{\pi}_{t-i}\right)+\left(1-\omega \right)\left(1-\theta \right)\{\left(1-\beta \theta \right)\kappa {y}_{t}\begin{array}{c}\\ \end{array}\\ \text{\hspace{1em}}\text{\hspace{1em}}+\beta \theta \left[\frac{1}{\left(1-\theta \right)\left(1-\omega \right)}{E}_{t}{\pi}_{t+1}-\frac{\omega}{1-\omega}\left(\frac{\theta}{1-\theta}+{\lambda}_{1}\right){\pi}_{t}-\frac{\omega}{1-\omega}{\displaystyle \underset{i=1}{\overset{T-1}{\sum}}{\lambda}_{i+1}}{\pi}_{t-i}\right]\}\end{array}$

$\begin{array}{l}\theta {\pi}_{t}=-\left(1-\theta \right)\omega {\pi}_{t}+\theta \omega {\pi}_{t-1}+\left(1-\theta \right)\omega {\displaystyle \underset{i=1}{\overset{T}{\sum}}{\lambda}_{i}}{\pi}_{t-i}+\left(1-\omega \right)\left(1-\theta \right)\left(1-\beta \theta \right)\kappa {y}_{t}\\ \text{\hspace{1em}}\text{\hspace{1em}}+\beta \theta {E}_{t}{\pi}_{t+1}-\beta \theta \omega \left(\theta +\left(1-\theta \right){\lambda}_{1}\right){\pi}_{t}-\beta \theta \omega \left(1-\theta \right){\displaystyle \underset{i=1}{\overset{T-1}{\sum}}{\lambda}_{i+1}}{\pi}_{t-i}\end{array}$

$\begin{array}{l}\left[\theta +\omega \left(1-\theta \right)+\beta \theta \omega \left(\theta +\left(1-\theta \right){\lambda}_{1}\right)\right]{\pi}_{t}\\ =\left(1-\omega \right)\left(1-\theta \right)\left(1-\beta \theta \right)\kappa {y}_{t}+\beta \theta {E}_{t}{\pi}_{t+1}+\theta \omega {\pi}_{t-1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}+\omega \left(1-\theta \right){\displaystyle \underset{i=1}{\overset{T}{\sum}}\left({\lambda}_{i}-\beta \theta {\lambda}_{i+1}\right){\pi}_{t-i}}.\end{array}$ (18)

Dividing both sides of (18) by the coefficient on ${\pi}_{t}$ yields the HNKPC curve equation,

${\pi}_{t}=\frac{\beta \theta {E}_{t}{\pi}_{t+1}+\theta \omega {\pi}_{t-1}+\omega \left(1-\theta \right){\displaystyle \underset{i=1}{\overset{T}{\sum}}\left({\lambda}_{i}-\beta \theta {\lambda}_{i+1}\right){\pi}_{t-i}}+\left(1-\omega \right)\left(1-\theta \right)\left(1-\beta \theta \right)\kappa {y}_{t}}{\theta +\omega \left[1-\theta +\beta \theta \left(\theta +\left(1-\theta \right){\lambda}_{1}\right)\right]}$

(19)

in which
${y}_{t}$ can represent either the output gap or real marginal cost^{4}.

When wages are indexed to only one lag of inflation, so that ${\lambda}_{1}=1$ and ${\lambda}_{2}={\lambda}_{3}={\lambda}_{4}=\cdots ={\lambda}_{\infty}=0$, the HNKPC simplifies to the formulation in Galí and Gertler (1999) and Holmberg (2006) in which

${\pi}_{t}=\frac{\beta \theta {E}_{t}{\pi}_{t+1}+\omega {\pi}_{t-1}+\left(1-\omega \right)\left(1-\theta \right)\left(1-\beta \theta \right)\kappa {y}_{t}}{\theta +\omega \left[1-\theta \left(1-\beta \right)\right]}$. (20)

Thus, when prices are indexed to one lag of inflation, the model yields the same equation as in Galí and Gertler (1999) and Holmberg (2006). However, the model in the present study extends the previous literature by allowing firms to index their prices to more than one lag of inflation, instead of just a single lag, resulting in a HNKPC that can include any number of lags of inflation.

3. Conclusion

In recent years, there has been an increased emphasis on developing microfoundations for macroeconomic models. While some empirical estimates of the Phillips curve include multiple lags of inflation, they are included on an ad hoc basis. This study provides microeconomic justification for a HNKPC with more than one lag of inflation, and it derives an equation that researchers can use in empirically estimating the HNKPC with multiple lags of inflation. In addition, dynamic stochastic general equilibrium (DSGE) models, such as Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2007), generally assume that a fraction of wages and prices that are reset are indexed to last period’s inflation rate. The present study derives an equation in which these variables can be indexed to multiple lags of inflation, possibly enabling these DSGE models to fit macroeconomic data more closely.

NOTES

^{1}See, for example, Equations (7.56) and (7.57) on p. 328 of Romer (2019).

^{2}The equation in Roberts (1995) is
${p}_{t}^{*}={\stackrel{^}{p}}_{t}+\beta {y}_{t}+{\epsilon}_{t}$, where
${p}_{t}$ and
$\kappa $ in the present study correspond to
${\stackrel{^}{p}}_{t}$ and
$\beta $ in Roberts and where the error term in Roberts is ignored.

^{3}The relationship,
${p}_{t-1}^{\ast}-{p}_{t-1}=\theta \left({p}_{t-1}^{\ast}-{p}_{t-2}\right)$, can be derived by lagging (1) by one period.

^{4}In the term involving the summation sign,
${\lambda}_{T+1}=0$, so the last term of the summation is
$\omega \left(1-\theta \right){\lambda}_{T}{\pi}_{t-T}$.

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