In these next few posts I extend the concepts developed in into the realm of the statistical estimation of the demand function. First, I consider the concept of elasticity of demand as a decision criterion. Then, after we have discussed the procedures for the specification of a demand function, I consider the implications of various specification problems for the elasticity of demand.

The slope of the own-price demand curve, DQ/DP, contains some information which may be useful to the management of the enterprise. It may be interpreted as the number of units by which quantity sold can be expected to change in response to a change in the price of the item, given all other determinants of demand. If the management is interested in nothing more than predicting the number of additional units which can be sold by changing price, the slope of the own-price demand curve is an entirely adequate decision criterion. However, if the management is concerned about profitability or one of its components, the revenue generated in selling the item, the simple slope of the own-price demand curve is an inadequate decision criterion. The reason for its inadequacy is that the slope of a linear own-price demand curve never changes over its entire positive-price range, but, as became apparent in the previous posts, total revenue does differ from one point to another along the demand curve.

Even if the slope of the demand curve does change because it is not linear, the simple slope still fails to convey information about how the revenue of the firm changes consequent upon a price change. A more useful revenue-oriented decision criterion can be constructed by computing the ratio of the percentage change of quantity demanded to the percentage change in the price which resulted in the quantity change, or %DQ / %DP.

Economists refer to this ratio as the own-price elasticity of demand, and they interpret it as a measure of the sensitivity (or responsiveness) of quantity demanded to a change in the item’s own price. The revenue-related importance of the own-price elasticity can be illustrated by demand equation Qx = 80 – 4Px. The linear demand curve has been divided into ranges indicated by brackets.

The upper portion of the demand curve, from price of $20 down to $10, is labeled the elastic range. It is characterized by positive marginal revenues and elasticity ratios (absolute values) greater than unity. The lower portion of the demand curve, from price $10 down to $0, is labeled the inelastic range. It is characterized by negative marginal revenues and fractional elasticity ratios (again, absolute values). The midpoint of the linear demand curve (at price $10) is labeled the unitarily elastic point because the absolute value of the demand elasticity ratio is precisely 1.0 at this point. Demand at the unitarily elastic point is also characterized by zero marginal revenue.

In the elastic range of the demand curve, any particular percentage decrease of price will result in a larger percentage increase of quantity demanded. Thus, what is lost to revenue by cutting price is more han made up for in increased quantity sold, so total revenue increases. For example, if price is lowered from $18 to $16, quantity demanded will increase from 8 units to 16 units, and total revenue will increase from $144 to $256. Price fell by 11 percent, but this was more than made up for by a 100 percent increase in quantity sold.

However, in the elastic range of the demand curve, any particular percentage increase of price will result in a larger-percentage decrease of quantity demanded, thus causing a decrease of total revenue. In this case, what is gained in raising the price is more than offset by the loss in quantity sold. Thus, if the manager can verify that the enterprise is presently selling the item at a point in the elastic range of the demand curve, the appropriate direction in which to change price in order to increase revenue is down.

The opposite conclusions emerge for the inelastic range of the demand curve. A price cut in the inelastic range will result in an increase of quantity demanded, albeit one of a smaller percentage magnitude, so that total revenue can be expected to decrease. If the objective is to increase revenue, price should be raised because a smaller-percentage decrease in quantity demanded will result. A price cut from $9 to $8 will result in an increase of quantity sold from 44 to 48 units, but a decrease of total revenue from $396 to $384.

Small percentage changes of price in the near neighborhood of the unitarily elastic point of the demand curve will be offset by the same percentage changes of quantity demanded (but in the opposite direction), thereby leaving revenue unchanged. In our example, if price is raised from $9 to $11, quantity demanded will fall from 44 to 36 units, leaving total revenue unchanged at $396.

If the enterprise were to progressively lower price, moving down the demand curve from its intercept with the price axis toward its intercept with the quantity axis, total revenue would increase to a maximum (at the unitarily elastic point), and then decrease; concurrently, marginal revenue would decrease from positive values, through zero (at the unitarily elastic point), to negative values. And, elasticity would fall from (absolute) values greater than unity to (absolute) values less than unity.

The significance of own-price elasticity of demand is that it is indicative of what is likely to happen to the enterprise’s revenues when it changes the price of an item which it sells. Enterprise managers may not explicitly compute elasticity ratios, and they may not use the term elasticity of demand. However, we may infer that they must have employed an elasticity thought process in making a rational decision to change price if their enterprises have survived and are profitable.

To this point we have assumed that the demand curve is linear, but only for purposes of simplicity. Since only one point on a demand curve exists at any moment (all other points are only hypothetical or virtual), it can be argued that the shape of the demand curve away from the extant point (whether curved, bent, kinked, etc.) is really a non-issue. But, even if the demand curve is curvilinear, all of the elasticity formulas introduced in this post can be applied to compute or estimate the elasticity of demand. Graphically, whether demand is elastic or inelastic at a particular point on a non-linear demand curve can be discerned by observing the characteristics of a tangent drawn to the curve at the point.

The demand curve for an item may take any slope, although a negative slope is expected according to the law of demand. Negatively-sloped demand curves which approach the vertical are often said to be “inelastic demand curves.”

It would be more accurate to note that the relevant price range spans only the inelastic portion of the demand curve. There is an elastic range (off the vertical axis scale), but it is irrelevant under current pricing conventions. I should also note that the marginal revenues associated with points on the visible portion of this demand curve are all negative. This point is important because it is not possible for an enterprise to reach a profit maximizing equilibrium in the inelastic range of its demand curve since its marginal cost can never be negative.

A similar consideration should also be noted in regard to demand curves with very shallow slopes, approaching the horizontal. While such demand curves are often described as being “highly elastic,” it would be more accurate to say that the relevant quantity range encompasses only the elastic range of the demand curve. There is an inelastic range, but only at quantities which are unattainable under current market and supply conditions. Marginal revenues associated with elastic points on such shallowly sloped demand curves will be positive, and thus can accommodate a profit-maximizing equilibrium solution for the enterprise.

Our discussion of elasticity to this point has focused on the own-price elasticity of demand, but elasticity is a more general concept not restricted exclusively to own-price. The demand elasticity ratio can be computed with respect to any relevant demand determinant, including own-price. Letting the symbol “X” refer to an unspecified demand determinant, its elasticity can be computed by any of the following formulas, given the requisite information:

(1) X elasticity = %DQ / %DX = (DQ/Q) / (DX/X).

A simple algebraic rearrangement of this elasticity formula yields

(2) X elasticity = DQ/Q . X/DX = DQ/DX . X/Q.

If the limit concept of the calculus is applied,

(3) X elasticity = DQ/DX . X/Q = dQ/dx . X/Q,

at the limit as DX approaches 0.

The net of this formula development process is that X elasticity can be computed as the product of the derivative of the demand function with respect to X, and the ratio of the amount of X to the quantity Q. If there are other relevant demand determinants than X, the X elasticity ratio should be computed as a partial (rather than a simple) derivative, i.e.,

(4) X elasticity = dQ/dX . X/Q.

This formula is referred to as the point elasticity formula because it can be computed from information about one point on the X demand curve if the equation of the demand curve is known.

Alas, this latter condition, i.e., that the equation of the demand curve must be known, may constitute a serious barrier to the employment of the elasticity ratio as a decision criterion because the equation often is not known, or cannot be satisfactorily estimated. However, an approximation to point elasticity, known as arc elasticity can be computed if information about two points along the X demand curve are known, even if the equation of the X demand curve is not known. Formula (5) can be constructed from formula (2):

(5) X elasticity = %DQ/%DX = (DQ/Q) / (DX/X)

= ((Q2 – Q1)/Q1) / ((X2 – X1) / X1).

where the subscripts refer to the two points identified as points 1 and 2. The expression (Q2 – Q1) constitutes “DQ,” and the expression (X2 – X1) is “DX.” I note that point 1 is taken as the base or starting point for the computation. However, this particular formulation exhibits a deficiency in that different values for the computed X elasticity ratio emerge if the identities of points 1 and 2 are reversed. This deficiency can be relieved by estimating the average elasticity over the arc of the demand curve between points 1 and 2 using formula

(6) X elasticity = [(Q2-Q1) / ((Q2+Q1)/2)] / [(X2-X1) / ((X2+X1)/2)]

where (Q2+Q1)/2) constitutes the average of Q2 and Q1, and ((X2 + X1)/2) is the average of X2 and X1. Finally, formula (6) can be simplified because the 2s in the denominators of the ratio cancel each other, resulting in

(7) X elasticity = [(Q2-Q1) / (Q2+Q1)] / [(X2-X1) / (X2+X1)].

This final formulation, the so-called “average arc elasticity” formula, can be computed if only two points on the X demand curve are known, but it must be recognized as only an approximation to the true elasticity at either known point, or any point on the arc between the known points. Depending upon the shape (i.e., concavity) of the X demand curve, the average arc elasticity ratio may be an over- or understatement of true point elasticity. The reader is invited to explore the conditions resulting in over- or understatement.

I now explore the conceptual sense of several specific X-demand elasticity ratios. The so-called income elasticity of demand may be computed if information about the clientele’s income is known and other demand determinants remain unchanged. Any of the elasticity formulas elaborated in the previous section may be employed simply by substituting income for X in the selected formula, e.g.,

(8) income elasticity = [%DQ] / [%DI].

The computed income elasticity of demand for a normal good is expected to be positive, while that for an inferior good is expected to be negative. But conclusions should never be assumed before conducting empirical research. From this perspective, a computed positive income elasticity of demand ratio may be taken as the basis for an inference that the item is a normal good; likewise, a computed negative income elasticity ratio implies that the item is an inferior good. But even if the item is deemed normal, the value of the elasticity ratio may contain useful information. Positive income elasticities less than unity imply that the demand for the item is relatively inelastic (i.e., unresponsive) with respect to income changes. Income elasticities greater than unity suggest that the demand for the item is relatively elastic with respect to income changes.

The management of the firm may have chosen to take an aggressive approach to the demand for its item by mounting a promotional effort. The relevant question in this regard is whether the demand for the item is elastic with respect to the promotional expenditure (e.g., the advertising budget for a particular medium). The relevant elasticity formula can be expressed as

(9) advertising elas = %DQ / %D Advertising Budget.

The management might be pleased to find a positive advertising elasticity ratio greater than unity. The reader is left to imagine the management’s reactions to advertising elasticity ratios less than unity (or, heaven forbid, negative!).

Finally, we consider the sense of the so-called cross (or cross-price) elasticity of demand ratio. This term may be understood in comparison with the term “own-price” elasticity of demand. A cross-price demand curve shows the graphic relationship between the quantity demanded of one item, say x, relative to the price of another item, y or z. As noted earlier, y and z may refer to substitutes and complements, respectively, for x. The cross-elasticity of demand for the substitute good y can be expressed as

(10) substitute cross elasticity = %DQx / %DPy,

and that for complement good z as

(11) complement cross elasticity = %DQx / %DPz.

The sign of the substitute cross elasticity ratio is expected to be positive: when the price of y rises, less of y will be demanded, but more of its substitute x will be demanded. And the sign of the complement cross elasticity ratio is expected to be negative. Again, neither substitutability nor complementarities should be assumed; empirical evidence should be compiled to reveal whether two goods appear to be substitutes or complements, and the magnitudes of the computed ratios (in absolute value, greater or lesser than unity) should indicate how good or strong is the relationship. The substitute cross elasticity ratio has been proposed as an index of the ability of a competitor to penetrate the market of the enterprise by cutting price (or the ability of the enterprise to insulate itself from competitor’s price changes).

I have now considered the content governance implications of own-price, cross-price, income, and advertising elasticity of demand. The principles underlying these concepts are applicable to demand elasticities of yet other determinants which have not been specified. Each item can be expected to have a set of demand determinants which are specific to it, and which may not be pertinent to many or any other items.

While demand elasticity can be computed for any demand determinant for which sufficient information is available, I do not mean to suggest that successful content governance managers must make explicit demand elasticity computations before each and every demand decision. Rather, the concept of demand elasticity is the economist’s explanation of a thought process through which the successful content governance managers must have passed in making the decisions which resulted in the success of the enterprise. Whether or not any demand elasticity ratio has been computed, the content governance manager has to have asked the question of whether a contemplated demand determinant change is likely to result in a larger or smaller percentage change in the quantity demanded of the item. I shall subsequently discover that the elasticity concept can be extended into the realm of supply, production, and cost.

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