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A forward-looking Phillips Curve derived from Calvo pricing, linking current inflation to expected future inflation and the output gap.

Derivation

Step-by-step mathematical derivation with typeset equations and expandable detail.

Sections

Monopolistic competition and Calvo pricing

The New Keynesian Phillips Curve (NKPC) is derived from microfoundations in a general equilibrium model with two key ingredients: monopolistic competition and staggered price setting. Firms produce differentiated goods and face downward-sloping demand curves, giving them pricing power. Each firm $i$ sets its own price $p_i$ to maximize profits, taking aggregate demand and the prices of competitors as given. The CES demand structure implies that firm $i$ 's demand is $y_i = (p_i / P)^{-\varepsilon} Y$ , where $P$ is the aggregate price index, $Y$ is aggregate output, and $\varepsilon > 1$ is the elasticity of substitution across varieties.

Price stickiness is introduced via the Calvo mechanism: in each period, a fraction $1 - \omega$ of firms can reset their prices optimally, while the remaining fraction $\omega$ must keep their previous prices unchanged. The probability of being able to adjust is independent of time since the last adjustment. This random price-setting opportunity generates a distribution of prices across firms and creates a link between real activity and inflation, because firms that can adjust set prices based on expected future marginal costs, not just current conditions.

y_i = \left(\frac{p_i}{P}\right)^{-\varepsilon} Y

CES demand for firm $i$ 's variety: quantity demanded falls as the firm's relative price rises, with elasticity $\varepsilon$ .

P_t = \bigl[\omega P_{t-1}^{1-\varepsilon} + (1-\omega) p_t^{*\,1-\varepsilon}\bigr]^{\frac{1}{1-\varepsilon}}

Calvo price index: a CES aggregate of last period's price level (kept by fraction $\omega$ ) and the optimal reset price $p_t^*$ (set by fraction $1 - \omega$ ).

Deriving the NKPC from firm optimization

A firm that can reset its price at time $t$ chooses $p_t^*$ to maximize the expected discounted sum of profits over all future periods in which the price remains stuck. Because the price survives with probability $\omega$ per period, the firm's problem weights future periods by $\omega^j$ (times the stochastic discount factor). The first-order condition yields $p_t^* = \mu \cdot \sum_{j=0}^{\infty} (\beta \omega)^j E_t[mc_{t+j}]$ , where $\mu = \varepsilon/(\varepsilon - 1)$ is the desired markup and $mc_{t+j}$ is nominal marginal cost. The optimal price is a forward-looking weighted average of expected future marginal costs.

Log-linearizing the Calvo price index and the optimal pricing rule around a zero-inflation steady state produces the NKPC: $\pi_t = \beta E_t \pi_{t+1} + \kappa \hat{x}_t$ , where $\pi_t$ is inflation, $\hat{x}_t$ is the output gap (log deviation of output from its flexible-price level), and $\kappa = (1 - \omega)(1 - \beta\omega)/\omega \cdot \lambda$ with $\lambda$ depending on the slope of marginal cost. Current inflation depends on expected future inflation (because firms that set prices today know those prices may be stuck in the future) and on the current output gap (which drives marginal costs through the labor market).

p_t^* = \mu \sum_{j=0}^{\infty}(\beta\omega)^j E_t[mc_{t+j}]

Optimal reset price: a forward-looking markup over the discounted stream of expected future marginal costs, weighted by the probability $\omega^j$ of the price surviving $j$ periods.

\pi_t = \beta\, E_t\pi_{t+1} + \kappa\, \hat{x}_t

New Keynesian Phillips Curve: inflation is driven by expected future inflation and the current output gap, with slope $\kappa = \frac{(1-\omega)(1-\beta\omega)}{\omega}\lambda$ .

Divine coincidence: output gap and inflation stabilization

A remarkable property of the baseline NKPC is the 'divine coincidence': stabilizing inflation is equivalent to stabilizing the output gap. When $\pi_t = 0$ for all $t$ , the NKPC implies $\hat{x}_t = 0$ , meaning output is at its flexible-price level. Conversely, closing the output gap eliminates inflationary pressure. There is no trade-off between the two objectives. A central bank that perfectly stabilizes inflation automatically achieves the efficient level of output.

This result breaks down when cost-push shocks $u_t$ are introduced, modifying the NKPC to $\pi_t = \beta E_t \pi_{t+1} + \kappa \hat{x}_t + u_t$ . Now the central bank faces a genuine trade-off: a positive cost-push shock raises inflation even when the output gap is zero, forcing the policymaker to accept either above-target inflation or a negative output gap. The divine coincidence thus holds only in the absence of shocks to desired markups or other supply-side disturbances that create a wedge between the inflation-stabilizing and gap-stabilizing objectives.

\pi_t = 0 \;\Leftrightarrow\; \hat{x}_t = 0 \quad (\text{when } u_t = 0)

Divine coincidence: in the absence of cost-push shocks, zero inflation and a zero output gap are simultaneously achievable.

\pi_t = \beta\, E_t\pi_{t+1} + \kappa\, \hat{x}_t + u_t

NKPC with cost-push shocks: the shock $u_t$ breaks the divine coincidence and creates a policy trade-off between inflation and output stabilization.

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