Phillips Curve Derivation in New Keynesian Models

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Core Idea

The New Keynesian Phillips curve shows that inflation depends on expected future inflation, the output gap, and marginal costs. Unlike the traditional Phillips curve (which posits a stable unemployment-inflation tradeoff), the NKPC is forward-looking and depends on real variables. This microfounded derivation from Calvo pricing explains why monetary policy affects inflation through demand pressure on costs, not backward-looking wage adjustment, and why supply shocks can cause stagflation.

Explainer

From the traditional Phillips curve, you know the empirical observation that inflation and unemployment tend to move inversely — tight labor markets push wages and prices up. From Calvo pricing, you know that firms do not adjust prices continuously; instead, each period only a random fraction of firms get the opportunity to reset their prices, while the rest are stuck with their existing prices. The New Keynesian Phillips Curve (NKPC) derives the inflation-output relationship from these microfoundations, producing a relationship that is fundamentally forward-looking rather than backward-looking.

The derivation begins with a firm that gets the chance to reset its price. Because it knows it may be stuck with this price for several periods (the Calvo lottery may not select it again soon), it does not simply set price equal to current marginal cost. Instead, it sets a price that is optimal *on average* over the expected duration it will be locked in — a weighted average of current and expected future marginal costs. When you aggregate across all firms (some resetting, most stuck at old prices), the overall price level evolves as a blend of newly set prices and inherited prices. The resulting equation for inflation takes a remarkably clean form: π_t = βE_t[π_{t+1}] + κx_t, where π_t is current inflation, E_t[π_{t+1}] is expected future inflation, x_t is the output gap (or equivalently, real marginal cost), β is the discount factor, and κ is a slope parameter that depends on how frequently firms reset prices and how sensitive marginal costs are to output.

The forward-looking nature of this equation is its most important feature and its sharpest departure from the traditional Phillips curve. In the old framework, inflation was driven by past inflation through adaptive expectations — inflation had inertia because people expected tomorrow's inflation to look like yesterday's. In the NKPC, inflation today depends on what firms expect inflation to be *tomorrow*. If a central bank credibly commits to lowering future inflation, firms that reset prices today will choose lower prices in anticipation, and current inflation falls — even before the policy has fully taken effect. This is why central bank credibility and communication matter enormously in the New Keynesian framework: expectations of future policy feed directly into today's pricing decisions.

The NKPC also clarifies why supply shocks cause stagflation — the simultaneous appearance of rising inflation and falling output that the traditional Phillips curve could not accommodate. An adverse supply shock (like an oil price spike) raises marginal costs directly, pushing inflation up through the κx_t term. But because the shock also reduces potential output, the output gap may turn negative even as inflation rises. The traditional Phillips curve, which linked inflation only to the unemployment gap, could not separate demand-driven from cost-driven inflation. The NKPC, by grounding inflation in real marginal costs, naturally accounts for both channels. This microfounded structure is what makes the NKPC the inflation equation at the heart of modern DSGE models used by central banks worldwide.

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Counting to 10Counting to 20Understanding ZeroThe Number ZeroCounting to FiveOne-to-One CorrespondenceCombining Small Groups Within 5Addition Within 10Addition Within 20Two-Digit Addition Without RegroupingTwo-Digit Addition with RegroupingAddition Within 100Repeated Addition as MultiplicationMultiplication Facts Within 100Division as Equal SharingDivision as Grouping (Measurement Division)Division: Grouping (Repeated Subtraction) ModelDivision: Fair Sharing ModelDivision as Equal SharingDivision as GroupingBasic Division FactsDivision Facts Within 100Two-Digit by One-Digit DivisionDivision with RemaindersRemainders and Quotients in DivisionDivision Word ProblemsIntroduction to Long DivisionFactors and MultiplesPrime and Composite NumbersEquivalent FractionsRelating Fractions and DecimalsDecimal Place ValueReading and Writing DecimalsComparing and Ordering DecimalsAdding and Subtracting DecimalsMultiplying DecimalsDividing DecimalsDividing FractionsMixed Number ArithmeticOrder of OperationsInteger Order of OperationsVariable ExpressionsCombining Like TermsOne-Step EquationsTwo-Step EquationsSolving Multi-Step EquationsEquations with Variables on Both SidesAngle Pairs: Complementary, Supplementary, and VerticalParallel Lines and TransversalsCorresponding AnglesAlternate Interior AnglesTriangle Angle Sum TheoremExterior Angle TheoremTriangle Inequality TheoremSimilar Triangles: AA SimilaritySimilar Triangles: SSS and SAS SimilarityProportions in Similar TrianglesRight Triangle Trigonometry IntroductionTrigonometric Ratios ReviewRadian MeasureConverting Between Degrees and RadiansThe Unit CircleGraphing Sine and CosineGraphing Tangent and Reciprocal Trigonometric FunctionsDerivatives of Trigonometric FunctionsAntiderivativesIndefinite IntegralsBasic Integration RulesRiemann SumsDefinite Integral DefinitionFundamental Theorem of Calculus Part 1Fundamental Theorem of Calculus Part 2U-SubstitutionIntegration by PartsSeparable Differential EquationsIntegrating Factor Method for First-Order Linear ODEsFirst-Order Linear Ordinary Differential EquationsSecond-Order Linear Homogeneous Differential EquationsCharacteristic Equation Method for Linear ODEsComplex Roots and Oscillatory SolutionsSpring-Mass Systems and Mechanical VibrationsResonance and Damping in Forced VibrationsRLC Circuit Applications of Differential EquationsIntroduction to Differential EquationsSolow Growth ModelReal Business Cycle TheoryNew Keynesian Economics FrameworkCalvo Pricing and Sticky PricesPhillips Curve Derivation in New Keynesian Models

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