Rotemberg Pricing

Rotemberg pricing is one of the two workhorse microeconomic formulations of nominal price rigidity in modern New Keynesian macroeconomics and business and economics theory, introduced by Argentine-American economist Julio J. Rotemberg in his 1982 Journal of Political Economy article "Sticky Prices in the United States" and his companion 1982 Review of Economic Studies article "Monopolistic Price Adjustment and Aggregate Output." Rather than assuming fixed lump-sum menu costs (which produce lumpy (S, s) state-dependent price jumps) or exogenous Poisson adjustment arrivals as in Calvo pricing (Guillermo Calvo, 1983), the Rotemberg model postulates that every monopolistically competitive firm can adjust its nominal price in every period t, but faces a convex quadratic adjustment cost proportional to the squared percentage change in its nominal price, (φ / 2)(P_i,t / P_i,t−1 − 1)². Because all firms face identical technologies and initial prices and adjust smoothly in every period, Rotemberg pricing preserves a symmetric equilibrium (P_i,t = P_t) with zero cross-sectional price dispersion, yielding a log-linearized New Keynesian Phillips Curve (NKPC) that is observationally equivalent at first order to Calvo pricing while differing in its higher-order nonlinear and trend-inflation dynamics.

Household Demand Aggregation

In the standard New Keynesian DSGE environment (and monetary extensions of the Ramsey–Cass–Koopmans model), a continuum of monopolistically competitive firms indexed by i ∈ [0, 1] produces differentiated intermediate varieties Y_i,t. A representative household or competitive final-good bundler aggregates these varieties via a Dixit–Stiglitz (1977) Constant Elasticity of Substitution (CES) technology:

Cost minimization by the bundler implies that each intermediate firm i faces a downward-sloping isoelastic demand curve for its product as a function of its relative price P_i,t / P_t:

where P_t = [ ∫₀¹ P_i,t^1−ε di ]^1/(1−ε) is the aggregate CES price index. In the absence of nominal rigidities, firm i sets its frictionless optimal price P_i,t^* in every period as a constant gross markup μ ≡ ε / (ε − 1) > 1 over its nominal marginal cost MC_tⁿ = P_t · mc_t (where mc_t is real marginal cost), implying a steady-state real marginal cost mc^* = (ε − 1) / ε = 1 / μ.

The Rotemberg Quadratic Price Adjustment Cost Function

Rotemberg (1982) departed from the frictionless markup benchmark by assuming that changing a nominal price from P_i,t−1 to P_i,t imposes a real convex cost scaled by aggregate real output Y_t:

where φ ≥ 0 is the Rotemberg price-stickiness parameter (measured in units of output lost per squared percentage price change). Whereas a physical menu cost of printing a new catalog is independent of the size of the price change, Rotemberg (1982, 2005) motivated convex adjustment costs through customer anger, reputational goodwill loss, and B2B contract renegotiation costs: customers tolerate small, gradual price adjustments as routine cost pass-through, whereas large, sudden price hikes provoke customer resentment and search for rival suppliers. Just as convex installation costs in Tobin's q cause firms to smooth physical investment over time, convex price-adjustment costs cause firms to smooth nominal price changes across quarters.

First-Order Euler Equation and Symmetric Equilibrium

Each firm i ∈ [0, 1] chooses its sequence of nominal prices {P_i,t}_t=0^∞ to maximize the expected present discounted value of real profits (discounted using the household's stochastic discount factor Λ_t,t+k = β^k(U′(C_t+k) / U′(C_t))):

Substituting the CES demand function Y_i,t = (P_i,t / P_t)^−εY_t and differentiating with respect to P_i,t yields the firm's intertemporal Euler equation. Because every firm i ∈ [0, 1] starts with the same initial price P_i,−1 = P₋₁ and faces the same marginal cost mc_t and concave objective function, every firm chooses the same price in every period (P_i,t = P_t and Y_i,t = Y_t for all i ∈ [0, 1]).

Defining gross inflation Π_t ≡ P_t / P_t−1 = 1 + π_t and imposing P_i,t = P_t in the firm's first-order condition yields the nonlinear Rotemberg Phillips curve:

Unlike Calvo pricing—which requires tracking an auxiliary state variable for cross-sectional price dispersion S_t = ∫₀¹ (P_i,t / P_t)^−εdi ≥ 1—the nonlinear Rotemberg Phillips curve involves zero backward-looking state variables, allowing nonlinear global projection methods (such as Chebyshev collocation at the Zero Lower Bound on nominal interest rates under a Taylor rule) to solve the model with minimal state dimensionality.

Log-linearizing the nonlinear Rotemberg Phillips curve around a zero-inflation steady state (Π̄ = 1, π_t = Π_t − 1, Λ̄ = β, and steady-state real marginal cost mc^* = (ε − 1) / ε) by substituting mc_t = mc^*(1 + m̂c_t) yields the canonical New Keynesian Phillips Curve (NKPC):

Compare this to the log-linearized NKPC derived from Guillermo Calvo's (1983) staggered pricing model, in which each firm faces a constant probability θ ∈ (0, 1) of being unable to reset its price in any quarter:

First-Order Observational Equivalence Theorem (Roberts, 1995): Up to a first-order log-linear approximation around a zero-inflation steady state, Rotemberg quadratic adjustment cost pricing and Calvo pricing generate the same aggregate inflation and output dynamics when the Rotemberg curvature parameter φ is calibrated to match the Calvo non-adjustment probability θ via the structural mapping:

For example, in a quarterly calibration with elasticity of substitution ε = 6 (implying a 20% markup μ = 1.20), discount factor β = 0.99, and Calvo stickiness θ = 0.75 (average price duration of 4 quarters), the NKPC slope is κ ≈ 0.0858, which corresponds to a Rotemberg parameter of φ = 5 / 0.0858 ≈ 58.25.

Although Rotemberg pricing and Calvo pricing coincide at first-order and second-order approximations around zero inflation (Nisticò, 2007), the two models diverge in nonlinear simulations, positive trend inflation (Π̄ > 1), and microeconomic data:

1. Aggregate Resource Accounting vs. Productive Efficiency Wedge: Under Rotemberg pricing, all firms charge the same price (P_i,t = P_t), so aggregate output is produced without cross-firm misallocation (Y_t = A_tN_t), but the quadratic adjustment cost absorbs final goods via the resource constraint C_t = [1 − (φ / 2)π_t²]Y_t. In contrast, under Calvo pricing, no goods are consumed by price changes (Y_t = C_t), but staggered prices create relative price dispersion S_t > 1, reducing aggregate output per unit of labor (Y_t = A_tN_t / S_t).
2. Divergence Under Positive Trend Inflation (Ascari and Rossi, 2012): Guido Ascari and Lorenza Rossi (2012) proved that when steady-state trend inflation is positive (e.g., 2%–4% annually under a Taylor rule) and prices are not indexed, Calvo pricing causes relative price dispersion S_t to rise steeply with trend inflation and shrinks the determinacy region, whereas Rotemberg pricing lowers the average markup distortion without generating cross-sectional dispersion.

Why do macroeconomists use Rotemberg pricing if microeconomic scanner data show lumpy price changes?

Microeconomic CPI and retail scanner datasets (Bils and Klenow, 2004; Nakamura and Steinsson, 2008) show that individual store prices remain unchanged for several months and then jump by 8%–10%, which resembles menu cost or Calvo pricing at the micro level. Nevertheless, macroeconomists widely use Rotemberg pricing for aggregate DSGE modeling because (1) by Roberts's (1995) equivalence theorem, its first-order and second-order aggregate dynamics match Calvo pricing, and (2) in nonlinear global models (such as studying the Zero Lower Bound or financial crises), Rotemberg pricing eliminates the backward-looking price-dispersion state variable S_t.

How is the Rotemberg parameter φ calibrated in empirical DSGE models?

In empirical Bayesian DSGE estimation or calibration, researchers calibrate φ via the structural equivalence formula φ = (ε − 1)θ / [(1 − θ)(1 − βθ)], where θ is the quarterly probability of non-adjustment inferred from microeconomic price-duration data. If the average duration between price changes is 4 quarters (θ = 0.75), β = 0.99, and the demand elasticity is ε = 6 to 11, φ lies between 58 and 116.

Why are both first-order and second-order welfare losses identical between Rotemberg and Calvo pricing around zero inflation?

Under Calvo pricing, a small inflation rate π_t creates cross-sectional price variance across the fraction 1 − θ of adjusting firms and θ of non-adjusting firms, which reduces productive efficiency by (ε / 2) Var_i(ln P_i,t) ≈ (ε / 2)[θ / ((1 − θ)(1 − βθ))]π_t². Under Rotemberg pricing with φ = (ε − 1)θ / [(1 − θ)(1 − βθ)], the resource cost subtracted from consumption is (φ / 2)π_t². Evaluated inside the household's second-order utility Taylor expansion around the efficient steady state, both expressions produce the same relative weight κ⁻¹ on inflation variance π_t² relative to output-gap variance ŷ_t².

1. Rotemberg, J. J. (1982). "Sticky Prices in the United States." Journal of Political Economy, 90(6), 1187–1211. https://doi.org/10.1086/261117
2. Rotemberg, J. J. (1982). "Monopolistic Price Adjustment and Aggregate Output." The Review of Economic Studies, 49(4), 517–531. https://doi.org/10.2307/2297284
3. Roberts, J. M. (1995). "New Keynesian Economics and the Phillips Curve." Journal of Money, Credit and Banking, 27(4), 975–984. https://doi.org/10.2307/2077783
4. Ascari, G., & Rossi, L. (2012). "Trend Inflation and Firms Price-Setting: Rotemberg Versus Calvo." The Economic Journal, 122(563), 1115–1141. https://doi.org/10.1111/j.1468-0297.2012.02517.x
5. Rotemberg, J. J. (2005). "Customer Anger at Price Increases, Changes in the Frequency of Price Adjustment and Monetary Policy." Journal of Monetary Economics, 52(4), 829–852. https://doi.org/10.1016/j.jmoneco.2005.03.004