RDP 2007-12: Dynamic Pricing and Imperfect Common Knowledge 1. Introduction

In standard new Keynesian models, firms set prices to equal a mark-up over expected marginal cost. The real marginal cost is determined by both exogenous and endogenous factors, where the exogenous factors are assumed to be common among all firms. In this paper I relax the assumption of only common exogenous factors by introducing an idiosyncratic component to firms' marginal costs. This does not only improve the realism of the model, but can also help reconcile two apparently conflicting stylised facts that the standard model cannot account for: aggregate inflation responds gradually and with inertia to shocks, while at the same time price changes of individual goods can be quite large.

The inability of the baseline new Keynesian model to match the inertia of inflation is well documented and has spurred economists to suggest explanations, often involving some type of mechanical indexation to past prices.^[1] For instance, Galí and Gertler (1999) suggest that a fraction of firms set the price of their own good equal to the previous period's average reset price plus the lagged inflation rate, while Christiano, Eichenbaum and Evans (2005) let a fraction of firms increase the price of their own good according to the lagged inflation rate. Both of these explanations of inflation inertia are attractive since they admit relatively parsimonious representations of realistic inflation dynamics, but they can be criticised as being ad hoc. In this paper the inertial behaviour of inflation is driven by optimising price-setters.

Private information is introduced into the price-setting problem of the firm through the idiosyncratic component of marginal costs. The optimal price of an individual good depends positively on a firm's own marginal cost and the price chosen by other firms, but individual firms cannot observe the marginal cost of other firms and, therefore, do not know the current price chosen by other firms with certainty.

This set-up may be referred to as firms having imperfect common knowledge in an environment with strategic complementarities.^[2] In such an environment, it is a well established result that agents tend to put too much weight on public relative to private information.^[3] In the present model this takes the form of firms ‘herding’ on the publicly observable lagged aggregate variables, inflation and output. This creates the appearance of inflation being partly backward-looking in spite of the fact that all firms are rational and forward-looking.

The idiosyncratic component in firms' marginal costs also helps to explain that individual price changes are significantly larger than average aggregate price changes. Obviously, increasing the variance of the idiosyncratic component of marginal costs will increase the variance of individual price changes, but this direct effect is not the only one. A firm's own marginal cost provides a signal about the marginal costs faced by other firms and a large idiosyncratic variance makes this signal less precise. The less precise signal mutes the response of prices to aggregate shocks, since more of a given shock will be attributed to idiosyncratic sources. Increasing the variance of the idiosyncratic component then unambiguously increases the relative magnitude of individual price changes as compared to aggregate price changes.

The idea that incomplete adjustment of prices to aggregate shocks can be explained by information imperfections is not new, but dates back to the Phelps-Lucas island model of the 1970s.^[4] Recently, this idea has had something of a revival. Mankiw and Reis (2002) and Woodford (2002) show how limited information availability, or limited information processing capacities, can produce persistent real effects of nominal disturbances.^[5] Sims (2003) and Mackowiak and Wiederholt (2007) use information processing capacity constraints to explain the inertial responses of aggregate time series to shocks. The model presented here differs from these studies in some important respects that are worth emphasising.

First, through the Calvo mechanism of price adjustment, the model presented here can be made consistent with observed average price durations.^[6] The importance of this assumption boils down to whether one believes that the price stickiness that can be observed in the data causes firms to be forward-looking in a quantitatively important way. In the present model, expectations of future inflation will play a prominent role in determining today's inflation since there is a positive probability that a firm's price may be effective for more than one period. In the papers by Mankiw and Reis (2002), Woodford (2002), and Mackowiak and Wiederholt (2007), the price-setting problem of the firm is a series of static decisions since there is no need for the firm to forecast the future when prices are changed in every period. The dynamic structure of the pricing problem in the present paper makes existing solution methods for models with private information and strategic interaction non-applicable, and we derive a new algorithm to solve the model.^[7] This may be of independent interest.

Second, the models of Mankiw and Reis, Woodford, and Mackowiak and Wiederholt are all closed by using a constant-velocity-of-money type of equation. Here, a richer (but still small) general equilibrium model where households choose how much to consume and how much labour to supply is presented. The model is also more explicit in terms of what firms observe. While the model is too simple to be used to quantify the degree of information imperfections, being explicit prevents us from treating the precision of firms' information as a completely free parameter.

Section 2 derives a Phillips curve under the assumptions of imperfect common knowledge and Calvo pricing. Section 2 also discusses two limit cases of marginal cost structures that preclude any private information, to illustrate how idiosyncratic components in firms' marginal costs can introduce delayed responses to aggregate shocks. Section 3 presents the general equilibrium model and defines the concept of hierarchies of expectations and the assumptions that will be imposed on these to solve the model. Section 3 also shows how the recursive structure of the Phillips curve and the IS equation can be exploited to find the solution of the model. Section 4 contains the main results of the paper and demonstrates that the model can explain the observed inertia of inflation as well as the observed, relatively large, changes of individual goods prices as compared to the average aggregate price changes, while matching the average duration of prices found in the data. Section 5 concludes.

Footnotes

See, for instance, Fuhrer and Moore (1995), Galí and Gertler (1999) and Galí, Gertler and Lopez-Salido (2001, 2005). [1]

For example, see Woodford (2002) and Adam (2006). [2]

See Morris and Shin (2002) and Chamley (2004). [3]

See Phelps (1970) and Lucas (1972, 1973, 1975). [4]

Variants of the Woodford (2002) framework include Hellwig (2005), Adam (2006) and Amato and Shin (2006). [5]

See Aucremanne and Dhyne (2004), Bils and Klenow (2004), Alvarez et al (2005), Klenow and Kryvtsov (2005) and Nakamura and Steinsson (2007). [6]

See Woodford (2002) or Morris and Shin (2002) for solutions of static decision models. [7]