Optimal Price Adjustment : Tests of a Price Equations in U.S. Manufacturing

This paper presents an equation of the dynamic path of prices in a monopolistically competitive market in which firms sell to both old and new customers. Both types are able to search for the lowest price, given search costs, where the expected number of searches is given by the inverse of the probability of receiving a given price p. New customers engage in search until it yields a price that is at least as low as their reservation price. Old customers are motivated to search only if the incumbent firm raises its price by more than a calculated reservation price change. The assumption of a linear homogeneous production function in labor and materials implies a static equilibrium monopoly price that the firm predicts using an exponential forecasting rule. This dynamic setup motivates firms to maximize the integral of the expected discounted cashflow function of price and its rate of change. The solution of the implied Hamiltonian generates an optimal shadow price for the cost of changing price, and an Euler equation having the form of a second-order differential equation in the price of the good, in which the price follows a dynamic path toward an optimally moving equilibrium price. In the implied optimal control law, the firms adjusts to equilibrium with the entire planning period in view, discounting future stationary points with the positive root and weighting past mark-up points using the negative root. A discrete-time approximation of this dynamic pricing rule is a distributed lag equation with time-varying weights related to the stable and unstable roots of the Euler equation. The weights are specified as functions of the firm's discount rate, approximated by the corporate bond rate, and of the change in the ratio of unfilled orders to capacity, representing demand pressures. The variable weights are estimated using an orthogonal power series approximation. The variable-weight schedules indicate that (1) the moving equilibrium path and adjustments toward it are governed by business conditions, specifically, the ratio of unfilled orders to capacity. This is the first study to derive and estimate dynamic price behavior based on a theory of the firm with dynamic optimization in which prices adjust optimally toward an optimally derived moving target