RDP 2007-12: Dynamic Pricing and Imperfect Common Knowledge 2. Idiosyncratic Marginal Costs

In most (and perhaps all interesting) economies, one agent's optimal decision depends on the decisions of others. In an economy where all firms and agents are symmetric and all exogenous disturbances are common across firms and agents, knowing the actions of others is a trivial task. An agent can, by observing his own exogenous disturbance, infer the disturbances faced by everybody else and take action based on that information knowing that, in equilibrium, all agents will choose the same action. This is not possible in an economy with idiosyncratic exogenous shocks. Instead, each agent has to form an expectation of the other agents' actions based on what he can observe directly and on collected information. The expectation will be imperfect if the collection process adds noise to the observation or if it takes time.

In this paper, these ideas are applied to the price-setting problem of firms that are subject to idiosyncratic marginal cost shocks and where the aggregate price level is only observable with a lag. Individual firms care about the aggregate price level since demand for their own good depends on its price relative to other goods, but due to the idiosyncratic marginal cost shocks, firms cannot infer the aggregate price level perfectly by observing their own marginal cost. The lagged observation then becomes important as a source of information that individual firms use to form expectations about the aggregate price level. The positive correlation between the optimal current price and the lagged price level causes inflation to appear to react to shocks with inertia.

Idiosyncratic marginal cost shocks introduce private information into the price-setting problem of firms and it is demonstrated below how this forces firms to form higher-order expectations, that is, expectations of others' expectations, about marginal cost and future inflation. The variance of the idiosyncratic component of marginal cost determines how accurate a firm's own marginal cost is as an indicator of the average economy-wide marginal cost. By studying the model under two extreme assumptions about this variance, this section also demonstrates analytically how idiosyncratic marginal cost shocks can introduce delayed responses of inflation to aggregate shocks.

2.1 The Optimal Reset Price with Imperfect Common Knowledge

Apart from the introduction of the idiosyncratic marginal cost component, the framework below is a standard new Keynesian set-up with sticky prices and monopolistic competition. As in Calvo (1983), there is a constant probability (1 − θ) that a firm, indexed by j ∈ (0,1), resets its price in any given period and firms operate in a monopolistically competitive environment. In what follows, all variables are in log deviations from steady-state values. The price level follows

where is the average price chosen by firms resetting their price in period t

Firm j's optimal reset price is the familiar discounted sum of firm j's expected future nominal marginal costs

where β is the firm's discount factor and E_t(j)[·] ≡ E[· | I_t(j)] is an expectations operator conditional on firm j's information set at time t, which, in this section, is given by:

(for a derivation of the optimal reset price, Equation (3), see Woodford 2003 and the references therein). The structural parameters and the lagged price level p_s-1 are common knowledge. The actual economy-wide marginal cost cannot be directly observed (not even with a lag), but firm j can observe its own marginal cost mc_t(j) which is a sum of the economy-wide component mc_t and the firm-specific component ε_t(j)

Since the common and the idiosyncratic components are not distinguishable by direct observation, firm j cannot know with certainty the economy-wide average marginal cost mc_t. The average marginal cost matters for the optimal price of firm j though, since average marginal cost partly determines the current price level. If the average marginal cost process is persistent, then current average marginal cost will also be informative about future marginal costs, and future price levels. To set the price of its good optimally, firm j thus has to form an expectation of the average marginal cost.

The filtering problem faced by the individual firm is thus similar to that faced by the inhabitants of the market ‘islands’ in the well-known Lucas (1975) paper, but with some differences. In Lucas's model, information is shared among agents between periods so that all agents have the same prior expectation about the aggregate price change, while in this model no such information sharing occurs. This means that since all firms solve a similar signal extraction problem before they set prices, it also becomes relevant for each firm to form higher-order expectations, that is, expectations of average expectations, and so on. By repeatedly substituting in the expression for the price level (1) and the expression for the average reset price (2) into the optimal reset price (3), current inflation can be written as a function of average higher-order expectations of the current average marginal cost and future inflation:

(The Phillips curve (6) is derived in Appendix A.) The following notation is used to denote higher-order expectations:

In the Phillips curve (6), estimates of order k are weighted by (1 − θ)^k. Since (1 − θ) is smaller than unity, the impact of expectations is decreasing as the order of expectation increases. This fact is exploited later in order to find a finite dimensional representation of the state of the model. Also, note that (1 − θ) is decreasing in θ, that is, higher-order expectations are less important when prices are very sticky; when fewer firms change their prices in a given period (when θ is large), average expectations are less important for the firms that actually do change prices.

2.2 Two Limit Cases without Private Information

By the argument presented above, an individual firm needs to form an expectation of the economy-wide average marginal cost (and higher-order estimates of the average marginal cost) to set the price of its own good optimally. To do so, the individual firm uses its knowledge of the structure of the economy, the observations of the lagged price level and of its own marginal cost. The size of the variance of the idiosyncratic component relative to the size of the variance of the average marginal cost innovation determines how accurate firms' estimates will be. Two limiting cases of this variance ratio can help provide some intuition. When the variance of the idiosyncratic component is set to zero, Equation (6) reduces to the standard new Keynesian Phillips curve. In the second, and opposite case, the variance of the idiosyncratic component is assumed to be very large, and this will demonstrate how imperfect information introduces a link between past and current inflation. Both cases preclude any private information, and hence admit analytical solutions. In this section, the simplifying assumption is also made that average marginal cost is driven by the exogenous AR(1) process

This will facilitate the exposition, and the next section presents a simple general equilibrium model where marginal costs are determined by both exogenous and endogenous factors.

2.2.1 Common marginal costs

If the variance of the idiosyncratic component of firms' marginal costs is equal to zero, that is, , it follows that:

Since firms know the structure of the economy, Equation (8) implies that there is no uncertainty of any order. Formally,

Since all orders of current marginal cost expectations coincide, so do all orders of future inflation expectations and Equation (6) is reduced to the standard new Keynesian Phillips curve:

where inflation is completely forward-looking, with marginal cost as the driving variable. By repeated forward substitution, Equation (10) can be written as

which shows that inflation is only as persistent as marginal cost when the individual firm's own marginal cost is a perfect indicator of the economy-wide average.

2.2.2 Large variance of idiosyncratic marginal cost component

This section illustrates the consequences for inflation dynamics when the observation of a firm's own marginal cost holds no information about the economy-wide average. This is strictly true only when the variance of the idiosyncratic marginal cost component reaches infinity, but shocks with infinite variance prevent the law of large numbers from being evoked to calculate the average marginal cost. For illustrative purposes I will temporarily give up on some mathematical rigour. In the following example, the variance of the idiosyncratic component of a firm's marginal cost is ‘large enough’ for the firm to discard its own marginal cost as an indicator of the economy-wide average. Instead, each firm uses only the common observation of the lagged price level to form an imperfect expectation of the economy-wide average marginal cost. In this setting, it can be shown that the observation of the lagged price level p_t−1 perfectly reveals lagged average marginal cost mc_t−1. As there is no other source of information available about current average marginal cost, the first-order expectation is simply given by ρmc_t−1. This structure is common knowledge and implies that there is some first-order uncertainty about average marginal costs, that is, , but no higher-order uncertainty, so that for k,l > 0.

Current inflation can be written as a function of actual and first-order expectation of current marginal cost by exploiting the fact that such an expression must nest the solved full information Phillips curve (11) if, by chance, actual and first-order expectation of marginal cost coincide so that . From the Phillips curve (6) we know that the coefficient on the actual marginal cost is (1 − θ) (1 − βθ). To find the coefficient on the first-order expectation of marginal cost, we simply subtract (1 − θ)(1 − βθ) from the coefficient in the full information solution (11) to get

Using the fact that and that , we can re-arrange Equation (12) into a moving-average representation in the innovations ν_t

The impulse response to a shock to the average marginal costs will then be hump-shaped if the coefficient on the current innovation ν_t is smaller than the coefficient on the lagged innovation ν_t−1. This will be the case when the persistence parameter ρ is sufficiently large. Thus, the MA representation (13) tells us that the lagged price level will appear to have a positive impact on current inflation only if the average marginal cost follows a persistent process, since only then will lagged inflation hold any information about the marginal costs currently faced by other firms and of future marginal costs. If there is no persistence in marginal costs (ρ = 0), lagged inflation does not hold any information relevant to the price-setting problem of the firm and inflation becomes a white noise process.

The assumption of very large idiosyncratic marginal cost shocks, and that it is common knowledge that all firms condition on the same information, made an analytical expression for inflation possible. In the general case, when , neither the lagged price level nor the observation of a firm's own marginal cost completely reveal the average marginal cost or other firms' estimates of average marginal cost. Both the firm's own marginal cost as well as the lagged price level will then be needed to form optimal higher-order expectations of marginal costs and, due to the Calvo mechanism, higher-order expectations of future inflation.