Ideally, the enterprise manager should predicate pricing decisions upon an accurately-formulated demand function for each item that the enterprise sells. This ideal is feasible only for enterprises which sell one or only a small number of items. The task of estimating a demand function is sufficiently arduous and costly that few firm managers are willing to devote the necessary resources to the task when demands for more than a few items must be estimated. In extreme cases, for example the grocery retailer or hardware wholesaler who stocks literally thousands of items, the task of estimating demand for all items becomes a physical and economic impossibility. We shall consider alternative approaches typically employed by such multi-item enterprises.
For the moment we shall focus upon the procedures for estimating the demand for a single item, for example dozens of grade A large eggs, or half-gallons of 2% butter-fat milk. The first step in specifying a demand function is to model the relationship between quantity demanded as dependent variable, and all demand determinants which the analyst thinks might affect quantity demanded as independent variables. The modeling process should follow the procedures to select the possible independent variables and the likely form of relationship of each (linear, polynomial) to the dependent variable and to other independent variables (additive, multiplicative). The modeling process should also hypothesize the expected sign of the coefficient of each independent variable, e.g., negative for own-price, cross-price of a complement, and income in the case of an inferior good; positive for cross-price of a substitute and income in the case of a normal good.
Once the function has been modeled, the content governance manager can estimate the parameters of the function in a manner similar to Fritz Machlup’s educated guess based upon a summing up of the situation compared to experience with similar situations in the past (“Marginal Analysis and Empirical Research,” in Essays in Economic Semantics, W. W. Norton & Company, 1967, p. 167). Or, he can engage in the more-formal process which we shall elaborate in the remainder of this section. The Machlup-like seat-of-the-pants method is likely what managers do most of the time and in regard to most of the items which they sell, especially when their enterprises sell large numbers of items. Although no explicit equation results from this process, an implicit demand equation does underlie each educated guess of the number of units salable, given various values of the relevant demand determinants. This informal approach may be the only one possible in the case of a new item for which no current information or historical data can be obtained.
The following discussion of the formal estimating procedure is important both because the manager may indeed wish to estimate the demand function for some of the items sold by the enterprise, and because knowledge of the formal procedure can be beneficial to the informal summing-up process even if the demand function is not explicitly specified.
The formal estimating procedure culminates in an explicit equation which can be used to compute (i.e., predict, estimate) the unit sales of the item under various demand-determinant conditions. The equation may be linear or polynomial, additive or multiplicative, and may include as many independent variables as the analyst deems significant to the explanation of unit sales. The typical form of such a linear, additive demand equation is
(12) Qd = a + b1X1 + b2X2 + … + bnXn,
where X1 through Xn are such demand determinants as own-price, cross-prices, incomes of prospective clients, advertising expenditures, etc.
If there is only one demand determinant, say own-price, the equation will be of the slope-intercept format. However, if there are more determinants than one in the equation, the constant can not be interpreted as an intercept parameter for any one of the demand determinants.
A typical second-order (or quadratic) demand equation including only one independent variable would take the following form,
(13) Qd = a + b1X1 + b2X12. Higher-ordered terms for X1 can be present, and terms for other variables (X2, X3, …) can be included to any order (squared, cubed) as deemed important.
Once the demand function has been modeled, and assuming that the item is already being sold so that pertinent data can be obtained, the usual procedure for estimating the parameters of the demand model is regression analysis. The associated inference statistics provide means for assessing the statistical significances of the estimated coefficients of the included variables. The analyst should attempt to include in the model as independent variables as many demand determinants as are likely to make significant contributions to explanation of the behavior of the quantity demanded. Then, any variables for which estimated coefficients are judged not to be statistically significant can be deleted from the model before it is respecified.
Occasionally the analyst will find data appropriate to the demand specification process published in industry or trade sources, or compiled by government or private agencies. More often than not, however, demand data for individual items in specific locales do not exist, and must be captured as a matter of original field research. The first field research decision which the analyst must make is whether to capture the data cross sectionally (i.e., across a number of subjects at a point in time), or as a time series (i.e., for the same subject over a period of time). A cross-sectional approach is preferred if it is thought that demand determinants not explicitly included in the model might change over time. However, a cross-sectional approach might require access to competitors’ demand information, which they are likely to be reluctant to provide voluntarily.
If the analysis must be restricted to a time-series approach, the analyst should take care to include within the model all demand determinants which are likely to change over time. The analyst may also include so-called “dummy variables” (values of 0 and 1) as a means of quantifying such qualitative conditions as type of outlet (e.g., convenience store vs. full-line grocery store) or whether or not a special promotion is in effect. A student’s demand specification research paper is included as Appendix C2 to illustrate the procedures and some of the problems that were encountered in estimating a demand function.
A so-called specification error is often indicated by a coefficient of multiple determinations (R2) which is substantially below unity. The specification error occurs either because one or more important determinants of demand were omitted from the model, or because an included variable was raised to the wrong power (e.g., linear instead of quadratic or cubic).
Another type of specification error, an identification problem, may not be indicated by any inference statistic. The best indicator that an identification problem may have occurred is a sign on an estimated regression coefficient which is different than expected, e.g., a positive sign on the own-price regression coefficient. The cause of an identification error is a simultaneous relationship between the dependent variable (quantity demanded) and some determinant (e.g., consumer income) which was omitted from the model.
As an illustration of the problem, let us suppose that a cross-sectional data capture process yielded the quantity and price data in columns (1) and (2). The row-wise pairs of observations in the quantity and price columns serve as coordinates for plotting points A, B, C, D, E and F. Because these points scatter around an upward-sloping curve, D1, the coefficient of determination is quite low, 0.07, so a specification error may be indicated. Also, the slope of the curve D1 is positive, thus suggesting a violation of the law of demand. The problem is that points A, B, C, D, E, and F do not lie along a common own-price demand curve. Each point lies on a separate own-price demand curve which differs from the others because another determinant which has not been included in the model, income, has varied from observation to observation. These separate own-price demand curves.
Additional data, shown in column (3), were subsequently obtained for the average incomes of the clients at stores where the original price and quantity data were collected. Column (4) of contains an index of the income data; a convenient income observation was selected to serve as the index base. The column (4) index numbers were then used to adjust the column (1) quantity data (the adjustment is analogous to the process of deflating a money-value series by a price index) to remove the effects of the income variations. The adjusted quantity data are recorded in column (5). Now, when the adjusted quantity data are plotted against the price data from column (2), points F, G, H, I, J, and K. A curve D2 has been fitted through these points which yield a coefficient of determination of 0.86. Also, the slope of D2 is negative as expected from the law of demand.
The curve D2 in can be interpreted as an income-constant demand curve. Curve D1 exhibits an identification problem because, since the points lie on different demand curves due to the income variation, the true locus of the own-price demand curve could not be identified. The identification problem occurred because of the simultaneous change between own-price (included in the model) and income (not included in the model).
Suppose that an analyst estimates a demand function like curve D1, but fails to recognize the presence of an identification problem. If the own-price elasticity of demand is computed at any point along D1, it will have a positive sign, which indicates perverse own-price demand elasticity. However, if the identification problem is eliminated using a procedure similar to that described in the previous section so that own-price elasticity can be computed for curve D2, the expected negative own-price elasticity ratio will result.
In many cases, however, an unexpected sign of a computed elasticity ratio or the wrong slope of the estimated demand curve may not occur as a clue to the presence of an identification problem.
The path traced out by points K, L, and M constitute not a true demand curve, but rather a demand expansion path. If a demand equation is estimated from data for points K, L, and M, it will have the expected negative slope, but the slope will be shallower than that of a the true demand curve at any of its loci. Also, if average arc elasticity is computed from information about points K and L, the elasticity ratio will be negative as expected, but will imply that demand is far more elastic than it truly is along any locus of the true income-constant own-price demand curve. In fact, the implication may be that own-price demand is elastic when it truly is inelastic. A price decision maker who bases a price change decision on such an erroneously computed elasticity ratio will likely lower price when it should be raised.
The moral of the story is that the analyst should take great care to be sure that the demand function is being estimated from points along a common, fixed-locus, own-price demand curve (i.e., one which exhibits no identification problem). Great care should be taken to ascertain that the two points from which the average arc elasticity ratio is computed do lie along the same demand curve.
The concept of the demand function provides the economist with a formalized vehicle for analyzing the demand for an item. Enterprise managers usually function to “size-up” and estimate the quantity demanded of an item without formally estimating a demand function. But more sophisticated managers who require more accuracy in their demand estimates may be willing to devote the necessary time and effort to the formal estimation procedures.