New Keynesian Phillips Curve

Definition

A structural equation linking current inflation to expected future inflation and a measure of real economic activity (marginal costs or the output gap), derived from the optimal pricing decisions of firms facing nominal rigidities. The canonical purely forward-looking form is:

$p i_{t} = b e t a * E_{t} [p i_{t + 1}] + l amb d a * M C_{t} + x i_{t}$

where $p i_{t}$ is inflation, $b e t a$ is the discount factor, $l amb d a$ is the slope parameter (related to the frequency of price adjustment), $M C_{t}$ is real marginal costs, and $x i_{t}$ is a mark-up shock.

The hybrid specification adds a backward-looking term:

$p i_{t} = g amm a_{b} * p i_{t - 1} + g amm a_{f} * E_{t} [p i_{t + 1}] + l amb d a * M C_{t} + x i_{t}$

where $g amm a_{b}$ reflects dynamic price indexation.

Intuition

Firms face costs of adjusting nominal prices (Calvo’s random price-change opportunities, Rotemberg’s quadratic adjustment costs). Firms that can re-optimize set prices as a markup over the discounted sum of expected future marginal costs. Aggregation yields the NKPC: today’s inflation depends on today’s marginal costs plus the expected future path of marginal costs (captured by expected future inflation). The slope $l amb d a$ reflects how responsive inflation is to marginal costs — a steeper curve means inflation reacts more to real activity.

Formal notation

In the Calvo model with discount factor $b e t a$ and probability $x i$ of not re-optimizing prices each period:

lambda = (1 - xi)(1 - xi*beta) / xi

For the hybrid specification with dynamic indexation parameter $o m e g a$ :

gamma_b = omega / (1 + beta*omega) gamma_f = beta / (1 + beta*omega) lambda = (1 - xi)(1 - xi*beta) / (xi*(1 + beta*omega))

The purely forward-looking case obtains when $o m e g a = 0$ , giving $g amm a_{b} = 0, g amm a_{f} = b e t a$ .

If marginal costs are proportional to output (linear labor, no capital): MC_t = (1/tau) * Y_t, where $t a u$ is the intertemporal elasticity of substitution. The output coefficient is kappa = lambda / tau.

Variants

Purely forward-looking ( $g amm a_{b} = 0$ ): the baseline Calvo/ Rotemberg specification.
Hybrid NKPC ( $g amm a_{b} > 0$ ): accounts for inflation persistence through dynamic price indexation.
Open-economy NKPC: adds import prices or the real exchange rate to marginal costs.
Regime-dependent NKPC: in MS-DSGE models, $l amb d a$ or $g amm a_{b}$ may switch across regimes (Fernandez-Villaverde and Rubio-Ramirez 2007).
Truncated NKPC: in models with firm-specific capital, the slope depends on additional parameters.

Comparison

vs traditional Phillips curve (Phillips 1958, Samuelson-Solow 1960): the traditional curve is a reduced-form statistical relationship between inflation and unemployment. The NKPC is derived from micro- foundations and is forward-looking.
vs single-equation GMM/IV estimation (Gali-Gertler 1999): GMM estimates the NKPC in isolation. DSGE model-based estimation uses the full system, which helps identify $l amb d a$ but makes the estimate sensitive to the rest of the model.
vs state-dependent pricing (Golosov-Lucas 2007): Calvo pricing is time-dependent (random re-optimization probability); state-dependent models allow the probability to depend on the size of desired price changes. Different microfoundations but similar aggregate NKPC forms.

When to use

As a core building block in any New Keynesian DSGE model.
When modeling the inflation-output tradeoff for monetary policy analysis.
As the price-setting equation in macro-finance models that connect monetary policy to asset prices (e.g., the Leather-Sagi CRE model).

Known limitations

Slope identification is tenuous: Schorfheide (2008) documents that published DSGE estimates of $ka pp a$ range from <0.001 to 4.15, primarily due to differences in the implicit endogeneity correction. See nkpc-slope-identification-tenuous.
Backward-looking term vs persistent shocks: $g amm a_{b}$ and the autocorrelation of the mark-up shock $x i_{t}$ are negatively correlated in estimation and hard to disentangle without strong priors.
Marginal cost measurement: when marginal costs are latent (not observed via labor share), identification depends heavily on other model equations.
Structural stability: Fernandez-Villaverde and Rubio-Ramirez (2007) find that the Calvo parameter varies over time, suggesting the NKPC may not be a structural relationship in the usual sense.

Open problems

Reconciling micro evidence on price-change frequency (Bils-Klenow 2004: prices change every 4-7 months) with macro estimates of $x i$ that imply much longer durations (8+ quarters in some studies).
The role of strategic complementarities in price setting and their effect on the NKPC slope.
Nonlinear NKPC specifications for large shocks (the log-linear approximation may be inadequate during high inflation episodes).

Key papers

dsge-model-based-estimation-new-keynesian — comprehensive review of DSGE-based NKPC estimates; documents the wide range and identifies sources of variation
testing-indeterminacy-application-monetary-policy — estimates the NKPC in a small NK model that allows for indeterminacy

My understanding

The NKPC is the inflation equation in the 3-equation NK macro block (IS + NKPC + Taylor rule) that appears in the CRE asset pricing model. In the Leather-Sagi model, the kappa parameter governs the inflation- output tradeoff and enters the VAR system that drives the state dynamics under each regime. Schorfheide’s finding that kappa identification is sensitive to the rest of the model reinforces the importance of the full-system estimation (RBPF with cap rate data) rather than equation- by-equation approaches. The regime-switching extension means that kappa could in principle vary across regimes, though the current CRE model holds it constant.

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