Theory of the Firm
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The paper presents a model of "high-low search" under uncertainty, in which a "conservative" firm "searches" for an unknown product demand by making a sequence of production decisions. After each production decision and the concomitant sales, the firm infers whether its supply is "too high" or "too low". We show how the production decision reduces the firm's demand uncertainty interval and how this reduced uncertainty (in turn) affects its future production decisions.

This paper is concerned with the way in which firms learn about the demand for their products and introduces a new approach to the analysis of production decisions under demand uncertainty. In conventional analyses, firms are assumed to set their levels of production so as to maximize their profits, given the probability distribution of product demands which they face. In the Bayesian literature, firms begin with a probability distribution which reflects their "prior" beliefs concerning the behaviour of product demand shocks. Firms base their production decisions on this distribution, after which they observe sales and update their prior distribution. In this framework the process of search for new information concerning demand is a sampling problem. By contrast, firms in our analysis do more than take random samples from the distribution of product demands. Instead, they use their production decisions to determine what sort of information about demands they will receive. Each production decision may be interpreted as an "experiment" which is designed to generate new information concerning demand.

We assume that the firm does not produce to order and that the product demand which the firm faces is constant through time, so that a firm's current observation about product demand is relevant to its future production decisions. We analyse the firm's production decisions within a framework which captures the effects of past supply decisions on the information currently available about demand. We consider a very simple model in which prices do not play a role: demand is a given quantity which the firm' attempts to determine. Initially the firm knows only that the demand for its product lies within a specified interval. We assume that the firm faces uncertainty rather than risk, in the sense that it does not know even the probability distribution of demand for its product. We picture the firm learning about the demand for its product by supplying output and observing how much of this output remains unsold. We call this behaviour "high-low search" since the firm learns that its supply is "too high" when the quantity sold falls short of the quantity put up for sale and inventories remain, and that its supply is "too low" when its inventories are exhausted. When the supply is "too high", the firm can infer the exact level of demand: in particular demand is equal to the quantity sold. When the supply is "too low", the firm is unable to infer the exact level of demand. In short, the information which the firm receives is asymmetric: positive inventories yield quantitative information about product demand whereas stock-outs yield only qualitative information is available when inventories are exhausted.

We examine a minimax production strategy for the firm, one which minimizes the costs in the current and future time periods of over production and under the most adverse demand underproduction conditions. The firm chooses a "supply strategy", a sequence of quantities supplied in each time period. The quantities supplied follow this sequence until demand is known (because there is excess supply) and thereafter supply is set equal to demand. Firms make their production decisions not only with a view to maximizing their current profits given their current information, but also with a view to improving their knowledge of demand so as to increase their profits in the future.

We derive a formula and numerical for simulations which illustrate how the optimal sequence of supply decisions depends on the rate at which the firm discounts costs in future time periods and on the relative costs of overproduction and underproduction.

In the traditional microeconomic theory of production, the firm's information about demand is assumed to be independent of its supply decisions. By contrast, this paper argues that in the presence of uncertainty this independence can not in general be upheld. When demand is uncertain, information about this demand is revealed by the activities of supplying goods and observing how many of them are sold. If firms are aware of this, production decisions will be made with a view to revealing such information.