Calvo Staggered Pricing Model

The Calvo staggered pricing model is a model of nominal price rigidity in which a firm may adjust its price in a given period only with a fixed probability, independent of how long its price has been in place or how far it has drifted from the optimal level. Guillermo Calvo introduced it in the 1983 Journal of Monetary Economics article "Staggered Prices in a Utility-Maximizing Framework". The device keeps aggregation simple, because the fraction of firms that adjust each period is constant, and it became the standard way to introduce sticky prices into the New Keynesian models used in monetary policy analysis. Its most important product is the New Keynesian Phillips curve, which links inflation to expected future inflation and to real marginal cost.

Consider a continuum of monopolistically competitive firms. In each period a firm receives a signal, with probability 1 − θ, allowing it to reset its price. With probability θ it must keep its previous price. The signal is independent across firms and time. Two implications follow.

• The expected duration of a price is 1/(1 − θ) periods. If θ = 0.75 in a quarterly model, prices last four quarters on average.
• The hazard rate of price adjustment, the probability of adjusting given that the price has lasted a number of periods, is constant at 1 − θ. This memoryless property is the source of the model's analytical convenience, because the distribution of prices across firms evolves according to a single state variable.

The assumption is a reduced-form representation of menu costs, information costs, or contracting frictions, and it sidesteps the state-dependent decisions of firms that would be needed to derive adjustment from first principles.

A firm that can reset its price chooses the reset price p* to maximize the expected discounted sum of profits over the period in which the price will remain in place. Because the chance that the price survives k periods is θ^k, future profits are weighted by θ^k times the household's stochastic discount factor. The first-order condition gives a reset price that is a weighted average of current and expected future nominal marginal costs, multiplied by a constant markup:

p*_t = μ + (1 − βθ) Σ (βθ)^k E_t[mc_t+k],

where β is the discount factor, μ the log desired markup, and mc nominal marginal cost in logs. The aggregate price index is a weighted average of the prices of the fraction θ that did not adjust and the fraction 1 − θ that reset. Combining the two relations and log-linearizing around a zero-inflation steady state, one obtains

π_t = β E_t[π_t+1] + κ x_t,

with κ = (1 − θ)(1 − βθ)/θ, where x_t is the real marginal cost gap, proportional to the output gap under standard preferences. This relation is the New Keynesian Phillips curve, popularized by Roberts (1995), Yun (1996), and Woodford (2003), and given empirical form by Galí and Gertler (1999). Inflation is determined by expected future inflation and current slack, so it is forward-looking: current inflation reflects what firms expect their costs to be while their prices remain fixed.

The slope κ decreases with θ. The stickier prices are, the less inflation responds to slack. If firms adjust frequently, θ is small and κ is large, so inflation moves rapidly with demand.

Taylor (1980) modeled staggered time-dependent contracts of a fixed length, for example two periods, in which each cohort sets a price that applies for exactly the contract length. Both models are time-dependent, so the timing of adjustment is exogenous, but they differ in the hazard function. In Taylor's setup the hazard is zero until the contract expires and then equals one, while Calvo's is flat. The Taylor structure keeps track of a cohort distribution and needs more state variables, and it generates a more persistent response of output to a monetary shock in some calibrations. The Calvo structure, with a long tail of firms that happen not to adjust, yields a more tractable but somewhat less realistic distribution. A third class, state-dependent pricing in the style of menu-cost models, allows the probability of adjustment to depend on the gap between actual and desired price. In such models the "selection effect" lowers the real effects of monetary shocks relative to Calvo, because the firms that adjust are those furthest from the optimum.

Micro price data have been used to evaluate the model's assumptions. Bils and Klenow found that, in US consumer price data, prices change on average about every four to six months, more often than aggregate macro estimates of nine to twelve months suggest. Nakamura and Steinsson (2008) showed that once sales and product substitutions are excluded, the median regular price duration is around eight to eleven months. Klenow and Malin surveyed the micro evidence and noted large heterogeneity in adjustment frequency across goods, a hazard function that is not flat, and the presence of many small and large price changes, patterns that sit uneasily with the simple Calvo hazard. Heterogeneous-Calvo models, in which each sector has its own θ, give aggregate dynamics that are less persistent than a homogeneous model with the same average duration, because the sectors with flexible prices dominate the response of inflation.

The Calvo model gives the supply side of the standard three-equation New Keynesian framework, together with a dynamic IS curve and an interest rate rule. Within it, monetary policy has real effects because prices cannot adjust at once. The model supports the analysis of optimal policy under commitment and discretion, in which the central bank's inability to precommit generates an inflation bias like that in the Barro–Gordon monetary policy model, and under which price dispersion generates welfare costs of inflation. Critics raise several objections. The assumption of exogenous adjustment violates the Lucas critique, since the probability θ does not respond to the policy regime, and at high inflation firms plainly adjust more often. The model has trouble producing the inflation persistence seen in data without adding indexation or rule-of-thumb pricing. It also abstracts from the information frictions explored in the literature on information asymmetry and rational inattention. Even so, its simplicity is the reason it has remained the baseline representation of price stickiness in a general equilibrium setting for four decades.

1. Calvo, G. A. (1983). Staggered Prices in a Utility-Maximizing Framework. Journal of Monetary Economics, 12(3), 383–398. https://doi.org/10.1016/0304-3932(83)90060-090060-0)
2. Taylor, J. B. (1980). Aggregate Dynamics and Staggered Contracts. Journal of Political Economy, 88(1), 1–23. https://doi.org/10.1086/260845
3. Galí, J., & Gertler, M. (1999). Inflation Dynamics: A Structural Econometric Analysis. Journal of Monetary Economics, 44(2), 195–222. https://doi.org/10.1016/S0304-3932(99)00023-900023-9)
4. Nakamura, E., & Steinsson, J. (2008). Five Facts about Prices: A Reevaluation of Menu Cost Models. The Quarterly Journal of Economics, 123(4), 1415–1464. https://doi.org/10.1162/qjec.2008.123.4.1415