Theory of the Firm
Sold Out

This paper presents an overview of the application of the mathematical theory of "high-low" search, to firms' pricing and production decisions. We show how this methodology can be used to determine an optimal sequence of price-quantity decisions by a firm through time.

We suppose that the firm chooses a sequence of prices and quantities supplied over time not only with a view to earning current profit (given the current information about the demand curve) but also in order to acquire information about the demand curve by observing its inventory stocks as a result of these price and quantity decisions.
We compare and contrast the high-low model with the conventional microeconomic model of pricing and production. We show how the firm uses its pricing and production decisions to partition the uncertainty interval it faces and thereby influence the value of the information which it receives.

In traditional microeconomic theory, the firm is assumed to maximize its profit, given a known demand curve for its product (the price) and a known total cost curve. In practice firms have little information on their demand curves and whatever information they do have is usually gleaned from selling their products at varying prices and observing the resulting inventories.

This paper aims to capture the idea that the firm learns about its demand curve through its pricing, production and inventory holding decisions. With this in mind, we outline a new methodology, based on the mathematical theory of high-low search (developed by Alpern Baston and Bostock) for determining an optimal sequence of price-quantity decisions through time.

In previous work we have applied this methodology to models in which a firm faces a fixed price and an unknown quantity demanded. By contrast, this paper deals with the more general problem of formulating joint pricing and production decisions on the basis of what is currently known about the demand curve, and learning about the demand curve from the outcomes of these decisions.

We suppose that the firm chooses a sequence of prices and quantities over time with the purpose of earning current profit (given the current information about the demand curve) and acquiring information about the demand curve by observing its inventories. The novel feature of our analysis is that a firm's pricing and production decisions may be made with a view to the information it thereby gains about its demand curve. In contrast the conventional microeconomic theory of the firm assumes that the demand curve is known.

If this is the case the firm has no incentive to hold inventories, provided that the demand function and the cost function remain unchanged through time. When the demand function is unknown, however, we show that inventory holdings can arise as the result of optimal pricing and production decisions.

We assume that the firm's initial information about its demand is only that price and quantity demanded lie in known intervals and that demand function does not vary over time.

At the beginning of each period t, the firm makes a decision about its price and its quantity put up for sale; we call this decision its "guess" since the firm is trying to find the price- quantity combination which maximizes the present value of profits under its unknown demand function. When making the guess the firm has an information set consisting all previous guesses and all previous inventory stocks. Once a new guess has been made, the firm observes its new inventory stock and from this observation it makes inferences about its demand function. In particular, if inventories are zero it infers that demand at the current price must be at least as large as the quantity put up for sale. If inventories are positive then the firm learns the exact level of demand at the current price.

In our framework the firm chooses a price-quantity strategy which is a sequence of rules, for determining its guess based on the current information available. This general model in which demand is unknown and prices and quantities are chosen is too complex for an analytical solution.

If, however, the product is assumed to have a fixed price, so that the only unknown is the amount demanded at this price, then more tractable results can be obtained. The general model does, however, illuminate a tradeoff which faces the firm: on the one hand, it may expect more current profit by reducing production and thereby reducing its chances of holding inventories; on the other, it can expect more future profit by increasing production, increasing its chances of holding inventories and thereby increasing its chances of gaining more information about its demand curve. In this sense inventories are held on account of the information they convey.