Computing Elasticity

One of economists’ and businessmen most useful tools for the purpose of sensitivity analysis is the use of elasticity. It is not difficult to underline the contribution of the concept of elasticity to managerial decision-making as it measures the responsiveness of a number of variables to the implementation of potential strategies. The ex ante application of the concept of the price elasticity of demand, for example, resides in its contribution to the understanding of the likely response of customers to price changes. The ex post application of the price elasticity of demand, in contrast, permits the evaluation of the effectiveness of price changes concerning customer response which, coupled with the study of competitors’ reactions, constitutes the foundation for the analysis of future price changes. While there are a large number of elasticity measures, our focus is on demand measures of price elasticity. This is the reason for the inclusion of the concept of elasticity in a chapter dedicated to consumer theory.

The concept of elasticity

Elasticity is defined as a measure of the responsiveness of one variable to changes in another variable. One could measure elasticities for a large number of pairs of variables. Usually in economics, measures of elasticity study the response of a quantity variable to changes in another variable. Thus, for instance, the price elasticity of demand measures the response of the quantity demanded to changes in prices, the cross price elasticity of demand measures the response of the quantity demanded of one good to changes in the price of another good or the price elasticity of supply measures the response of the quantity supplied to changes in prices. Our focus is on the price elasticity of demand, the income elasticity of demand and the cross price elasticity of demand.

The price elasticity of demand

The price elasticity of demand measures the response of the quantity demanded to changes in prices. This responsiveness is measured in percentage terms. Expression below summarises the computation of the price elasticity of demand (ε^p):

where %Δq^d is the percentage change in the quantity demanded and %Δp is the percentage change in price.

When the price change results in a more than proportionate change in the quantity demanded, the term %Δq^d is larger than %Δp which implies an absolute value of the price elasticity of demand (|ε^p|) greater than one. The demand function is considered to be ‘elastic’. In contrast when the price increase or decrease results in a less than proportionate change in the quantity demanded, the term %Δq^d is smaller than %Δp which results in an absolute value of the price elasticity of demand |ε^p| smaller than one. The demand function is considered ‘inelastic’. Only when the price change results in an equally proportional quantity change, the absolute value of the price elasticity of demand |ε^p| is one.

Because we only consider negatively sloped demand curves, the relationship between the price and the quantity demanded is negative. This results in a negative price elasticity value. For simplicity, nevertheless, a large number of economists consider the absolute value of the price elasticity of demand (|ε^p|). Thus, the formula can take the values summarised by Table 1.

In the extreme when any change in the price does not effect the quantity demanded, the |ε^p|=0 and the demand curve is illustrated as a vertical line. In contrast, when the price change results in an infinite change in the quantity demanded, |ε^p|=∞ , the demand function is plotted as a horizontal line. The demand functions are ‘perfectly inelastic’ and ‘perfectly elastic’ respectively.

Computation of the price elasticity of demand

There are two main methods used in the computation of the price elasticity of demand. The method used depends on the purpose of the study and the data set available. Thus, given two points on a demand curve, one can only compute the price elasticity of demand on the ‘arc’ defined between the two points. In contrast, a demand function, expressed in its mathematical form, permits the computation of the price elasticity of demand at any one point on the demand curve. While the first is known as the ‘arc’ measure of the price elasticity of demand, the latter is known as the ‘point price elasticity of demand’. The usefulness and selection of the measure depends on the study the analyst wishes to perform on the demand function.

The ‘arc price elasticity of demand’ measures the value of the price elasticity between two points on a demand function. The formula for the price elasticity of demand is thus developed to consider the two points.

where Δq^d is the change in the quantity demanded from the original to the new point on the demand curve, Δp is the change in price, p⁰ is the original price, and q⁰ is the original quantity demanded.

Note that while we have based our percentages from the original point with values p0 and q0, one could base the percentage on the middle point or any point as long as the calculations used were consistent through the analysis.

Figure 1 illustrates graphically the calculation of the ‘arc’ price elasticity of demand.

Suppose, for example, two points on a demand function are given: p⁰=9, q⁰=98, and p¹=39, q¹=10. The absolute value of the arc price elasticity of demand would be:

Because the absolute value is smaller than 1, the demand curve between the two points is considered to be inelastic.

In contrast to the ‘arc’ measure, the ‘point price elasticity of demand’ measures the value of the price elasticity of demand at one point on the demand function. This measure results from reducing the arc existing between two points on a demand function so much that the difference becomes infinitesimally small. The intuition explaining this reduction is that the price elasticity of demand is computed at one point. Mathematically, this can be expressed with the use of the derivative of the demand function at that particular point:

where dq^d/dp is the derivative of the demand function with respect to the price.

Suppose a demand function is given as q^d = 39 - 8p, the absolute value of the point price elasticity of demand at a point where p=3 and q=15 is calculated below

Because the absolute value of the point price elasticity of demand is greater than 1, the demand curve at this point is elastic.