# Deriving the New Keynesian Phillips Curve (NKPC) with Calvo pricing

The Phillips Curve is an empirical observation that inflation and unemployment seem to be inversely related; when one is high, the other tends to be low.  It was identified by William Phillips in a 1958 paper and very rapidly entered into economic theory, where it was thought of as a basic law of macroeconomics.  The 1970s produced two significant blows to the idea.  Theoretically, the Lucas critique convinced pretty much everyone that you could not make policy decisions based purely on historical data (i.e. without considering that people would adjust their expectations of the future when your policy was announced).  Empirically, the emergence of stagflation demonstrated that you could have both high inflation and high unemployment at the same time.

Modern Keynesian thought – on which the assumed efficacy of monetary policy rests – still proposes a short-run Phillips curve based on the idea that prices (or at least aggregate prices) are “sticky.”  The New Keynesian Phillips Curve (NKPC) generally looks like this:

$\pi_{t}=\beta E_{t}\left[\pi_{t+1}\right]+\kappa y_{t}$

Where $y_{t}$ is the (natural) log deviation – that is, the percentage deviation – of output from its long-run, full-employment trend and $\beta$ and $\kappa$ are parameters.  Notice that (unlike the original Phillips curve), it is forward looking.  There are criticisms of the NKPC, but they are mostly about how it is derived rather than its existence.

What follows is a derivation of the standard New Keynesian Phillips Curve using Calvo pricing, based on notes from Kevin Sheedy‘s EC522 at LSE.  I’m putting it after this vile “more” tag because it’s quite long and of no interest to 99% of the planet.

The consumer, prices and aggregation

We first need firms to have pricing power.  We therefore use the monopolistic competition model of Dixit-Stiglitz, with a continuum of differentiated goods indexed by $i\in\left[0,1\right]$.

There is a constant and common elasticity of substitution between each good: $\varepsilon>1$

We aggregate across the different consumptions goods:

$C=\left(\int_{0}^{1}C\left(i\right)^{\frac{\varepsilon-1}{\varepsilon}}di\right)^{\frac{\varepsilon}{\varepsilon-1}}$

$P\left(i\right)$ is the price of good i, so the total expenditure on consumption is $\int_{0}^{1}P\left(i\right)C\left(i\right)di$

A representative consumer seeks to minimise their expenditure subject to achieving at least $C$ units of aggregate consumption.  Using the Lagrange multiplier method:

$L=\int_{0}^{1}P\left(i\right)C\left(i\right)di-\lambda\left(\left(\int_{0}^{1}C\left(i\right)^{\frac{\varepsilon-1}{\varepsilon}}di\right)^{\frac{\varepsilon}{\varepsilon-1}}-C\right)$

The first-order conditions are that, for every intermediate good, the first derivative of $L$ with respect to $C\left(i\right)$ must equal zero.  This implies that:

$P\left(i\right)=\lambda C\left(i\right)^{\frac{-1}{\varepsilon}}\left(\int_{0}^{1}C\left(j\right)^{\frac{\varepsilon-1}{\varepsilon}}dj\right)^{\frac{1}{\varepsilon-1}}$

Substituting back in our definition of aggregate consumption, replacing $\lambda$ with $P$ (since $\lambda$ represents the cost of buying an extra unit of the aggregate good $C$) and rearranging, we end up with the demand curve for each intermediate good:

$C\left(i\right)=\left(\frac{P\left(i\right)}{P}\right)^{-\varepsilon}C$

Note that if we raise this to the power of $\left(\frac{\varepsilon-1}{\varepsilon}\right)$ and integrate over $\left[0,1\right]$, we will produce the exact form of the aggregate price level:

$P=\left(\int_{0}^{1}P\left(i\right)^{1-\varepsilon}di\right)^{\frac{1}{1-\varepsilon}}$

[Side Note: If that Lagrangian looks odd to you or you’re wondering where the utility function is, you might appreciate this post on interpreting Lagrangian multipliers.]

The firm’s problem

Note: I am now using $C\left(\right)$ as the cost function (in real terms).  Quantities are now indicated as $Y\left(i\right)$.

Each firm’s profits, in real terms, are given by:

$\Pi\left(i\right)=\frac{P\left(i\right)Y\left(i\right)}{P}-C\left(Y\left(i\right)\right)$

The firm sets its own price, $P\left(i\right)$, to maximise profits, taking the demand function and aggregate prices as given (because they are too small to influence other firms or the whole economy directly).  This gives us the optimal price as a fixed mark-up over marginal cost:

$\frac{P\left(i\right)^{\ast}}{P}=\left(\frac{\varepsilon}{\varepsilon-1}\right)C^{\prime}\left(\frac{P\left(i\right)}{P}^{-\varepsilon}Y\right)=\left(\frac{\varepsilon}{\varepsilon-1}\right)C^{\prime}\left(Y\left(i\right)\right)$

Prices above or below this optimal level will result in a reduction in profits.  Let $Z\left(i\right)\equiv C^{\prime}\left(Y\left(i\right)\right)$ and define lower-case variables as being natural log deviations (i.e. percentage deviations) from their long-run trend values, so that:

• $p\equiv \ln P-\ln\overline{P}$
• $p\left(i\right)\equiv \ln P\left(i\right)-\ln\overline{P}$
• $z\left(i\right)\equiv \ln Z\left(i\right)-\ln\overline{Z}$

We can then approximate the profit function around $p\left(i\right)^{\ast}$ with a quadratic function (note that if we were doing a Taylor-series expansion, the first-order term would drop out due to the first-order condition of the optimisation):

$\Pi\left(i\right)\simeq-\frac{c}{2}\left(p\left(i\right)-p-z\left(i\right)\right)^{2}$

… with the approximate optimal price:

$p\left(i\right)^{\ast}=p+z\left(i\right)$

Making assumptions about the cost function

We assume a constant elasticity of marginal cost with respect to output.  That is, $Z\left(i\right)=dY\left(i\right)^{\gamma}$ for some constant $d$, so that  we can say:

$z\left(i\right)=\gamma y\left(i\right)$

Again in (natural) log deviations, the demand function can be written as:

$y\left(i\right)=-\varepsilon\left(p\left(i\right)-p\right)+y$

Where:

• $y\simeq\int_{0}^{1}y\left(i\right)di$;
• $p\simeq\int_{0}^{1}p\left(i\right)di$; and
• $z\simeq\int_{0}^{1}z\left(i\right)di$

Side note: These aggregates for the price-level and output assume away Jensen’s inequality.  The left-hand side is the log of a summation and the right-hand side is the sum of logs.  From one of my lecturers here at LSE:

The formula is correct as a first-order approximation in log deviations around a steady state in which the nominal price level is constant or the inflation rate is zero.

However […], things are different if the steady-state rate of inflation is non-zero. Here, even if the Jensen’s inequality terms (2nd order and above) are neglected, the first-order approximation needs to be modified.

This question has been looked at in several papers by Guido Ascari (for example […], his 2004 paper in the Review of Economic Dynamics).

While in principle this is important, in practice what matters is the average inflation rate. For inflation rates less than 5%, the modification turns out not to be that important quantitatively.

Moving on.  If we substitute the log-deviation form of the demand function into the formula for the optimal price and use the assumption of constant elasticity of marginal cost, we end up with:

$p\left(i\right)^{\ast}=p+\left(\frac{\gamma}{1+\varepsilon\gamma}\right)y$

So each firm’s optimal price increases with the average price of other firms and with aggregate output.  Defining $\alpha\equiv\left(\frac{\gamma}{1+\varepsilon\gamma}\right)$ we get the familiar:

$p\left(i\right)^{\ast}=p+\alpha y$

Note that $\alpha\in\left[0,1\right]$ since both $\varepsilon$ and $\gamma$ are greater than one.

Aggregate demand

The simplest specification of aggregate demand, in (natural) log deviation terms, is:

$y = m-p$

(equivalent to $MV=PY$ with $V=1$)

Substituting this into our equation for the optimal price gives:

$p\left(i\right)^{\ast}=\left(1-\alpha\right)p+\alpha m$

Note that $\alpha$ is an inverse measure of strategic complementarity – it pays more for firms to set their prices together when $\alpha$ is smaller.  Strategic complementarity goes up ($\alpha$ goes down) when the elasticity of substitution ($\varepsilon$) goes up or when the elasticity of marginal cost w.r.t. output ($\gamma$) goes down.

Price rigidity

We now need to introduce some rigidity into aggregate prices.  The natural and most common way to achieve this is to suppose that there is price rigidity at the level of the individual firm.  There are two broad categories of pricing model:  time-dependent and state-dependent.  Under time-dependent pricing, the timing of any given update is determined exogenously, whereas under state-dependent pricing both the timing and the magnitude of price changes are endogenous.  There are a variety of time-dependent price-setting models, including those by Fischer (1977), Taylor (1980) and Calvo (1983).

Price-setting under Calvo

Unlike the Fischer and Taylor models that assume that each firm updates every N periods, the Calvo model assumes that every firm faces a constant probability, $1 - \phi$, of updating their price in each period.  This means that for every firm, the probability that today’s price will still be used in $j$ period’s time is given by $\phi^{j}$.

When deciding on the (natural log) price, we denote that choice as $x_{t}=p_{t}\left(i\right)$ and call it the “reset price.”  Note that a) because of the symmetry of the model, every firm who gets to update in a given period will have the same reset price; and b) the reset price need not necessarily be the optimal price for that period, since firms will be taking into account the profit that they expect to receive in future periods between price updates.

We once again use a quadratic approximation of the per-period deviation from maximum-possible profit.  Supposing that we reset the firm’s price in period $t$, the expected deviation for period $t+j$ will be given by:

$-\frac{c}{2}E_{t}\left[\left(x_{t}-p^{\ast}_{t+j}\right)^{2}\right]$

We introduce $\beta$ as the discount factor and thus have the firm’s objective function in period $t$ when determining their reset price:

$-\frac{c}{2}\sum_{j=0}^{\infty}\beta^{j}\phi^{j}E_{t}\left[\left(x_{t}-p^{\ast}_{t+j}\right)^{2}\right]$

The first-order conditions give:

$x_{t}=\left(1-\beta\phi\right)\sum_{j=0}^{\infty}\left(\beta\phi\right)^{j}E_{t}\left[p^{\ast}_{t+j}\right]$

We can break the sum and express this in a recursive form:

$x_{t}=\beta\phi E_{t}\left[x_{t+1}\right]+\left(1-\beta\phi\right)p^{\ast}_{t}$

The aggregate price level will then be a weighted sum of all previous reset prices:

$p_{t}=\sum_{j=0}^{\infty}\left(1-\phi\right)\phi^{j}x_{t-j}$

Or, in recursive form:

$p_{t}=\phi p_{t-1}+\left(1-\phi\right)x_{t}$

Combining the two recursive-form equations, we substitute out the reset prices to obtain:

$p_{t}-\phi p_{t-1}=\beta\phi E_{t}\left[p_{t+1}-\phi p_{t}\right]+\left(1-\phi\right)\left(1-\beta\phi\right)p^{\ast}_{t}$

We now deploy a trick.  We add-and-subtract $\phi p_{t}$ both on the LHS and within the expectation on the RHS to get:

$\phi\left(p_{t}- p_{t-1}\right)+\left(1-\phi\right)p_{t}=\beta\phi E_{t}\left[p_{t+1}-p_{t}+\left(1-\phi\right)p_{t}\right]+\left(1-\phi\right)\left(1-\beta\phi\right)p^{\ast}_{t}$

Defining inflation as $\pi_{t}\equiv p_{t}-p_{t-1}$, we can rearrange this to give:

$\pi_{t}=\beta E_{t}\left[\pi_{t+1}\right]+\left(\frac{\left(1-\phi\right)\left(1-\beta\phi\right)}{\phi}\right)\left(p^{\ast}_{t}-p_{t}\right)$

Finally, using our previous equation for the optimal price ($p_{t}\left(i\right)^{\ast}=p_{t}+\alpha y_{t}$), we have:

$\pi_{t}=\beta E_{t}\left[\pi_{t+1}\right]+\left(\frac{\alpha\left(1-\phi\right)\left(1-\beta\phi\right)}{\phi}\right)y_{t}$

Or, defining $\kappa\equiv\left(\frac{\alpha\left(1-\phi\right)\left(1-\beta\phi\right)}{\phi}\right)$:

$\pi_{t}=\beta E_{t}\left[\pi_{t+1}\right]+\kappa y_{t}$

Which is the New Keynesian Phillips Curve.  Note that both more strategic complementarity (lower $\alpha$) and more price stickiness (higher $\phi$) lead to a lower $\kappa$ (a flatter slope), meaning that large deviations of output from trend will result in only low levels of inflation in the short-run.

#### 2 Responses to “Deriving the New Keynesian Phillips Curve (NKPC) with Calvo pricing”

• drew green

John, thanks for this. Textbooks are not always concise and can take a long time to paint a picture. Your posts are pretty easy to follow, and thus, helpful.

drew

• No worries – glad to help!