# 4.5. Calculating Demand Elasticities from Demand Functions

#cfa #cfa-level-1 #economics #microeconomics #reading-13-demand-and-supply-analysis-introduction #study-session-4

Although the concept of different elasticities of demand is helpful in sorting out the qualitative and directional effects among variables, the analyst will also benefit from having an empirically estimated demand function from which to calculate the magnitudes as well. There is no substitute for actual observation and statistical (regression) analysis to yield insights into the quantitative behavior of a market. (Empirical analysis, however, is outside the scope of this reading.) To see how an analyst would use such an equation, let us return to our hypothetical market demand function for gasoline in Equation 13 duplicated here:

Equation (27)

Qdx=8,400−400Px+60I−10Py

As we found when we calculated own-price elasticity of demand earlier, we need to identify “where to look” by choosing actual values for the independent variables, Px, I, and Py. We choose $3 for Px,$50 (thousands) for I, and $20 (thousands) for Py. By inserting these values into the “estimated” demand function (Equation 27), we find that quantity demanded is 10,000 gallons of gasoline per week. We now have everything we need to calculate own-price, income, and cross-price elasticities of demand for our market. Those respective elasticities are expressed in Equations 28, 29, and 30. Each of those expressions has a term denoting the change in quantity divided by the change in each respective variable: ∆Qx/∆Px, ∆Qx/∆I, and ∆Qx/∆Py. In each case, those respective terms are given by the coefficients on the variables of interest. Once we recognize this fact, the rest is accomplished simply by inserting values into the elasticity formulas. Equation (28) Edpx=(ΔQdxΔPx)(PxQdx)=[−400][310,000]=−0.12 Equation (29) EdI=(ΔQdxΔI)(IQdx)=[60][5010,000]=0.30 Equation (30) Edpy=(ΔQdxΔPy)(PyQdx)=[−10][2010,000]=−0.02 In our example, at a price of$3, the own-price elasticity of demand is –0.12, meaning that a 1 percent increase in the price of gasoline would bring about a decrease in quantity demanded of only 0.12 percent. Because the absolute value of the own-price elasticity is less than one, we characterize demand as being inelastic at that price, so an increase in price would result in an increase in total expenditure on gasoline by consumers in that market. Additionally, the income elasticity of demand is 0.30, meaning that a 1 percent increase in income would bring about an increase of 0.30 percent in the quantity demanded of gasoline. Because that elasticity is positive (but small), we would characterize gasoline as a normal good: An increase in income would cause consumers to buy more gasoline. Finally, the cross-price elasticity of demand between gasoline and automobiles is −0.02, meaning that if the price of automobiles rose by 1 percent, the demand for gasoline would fall by 0.02 percent. We would therefore characterize gasoline and automobiles as complements because the cross-price elasticity is negative. The magnitude is, however, quite small, so we would conclude that the complementary relationship is quite weak.

EXAMPLE 13

# Calculating Elasticities from a Given Demand Function

An individual consumer’s monthly demand for downloadable e-books is given by the equation Qdeb = 2 – 0.4Peb + 0.0005I + 0.15Phb, where Qdeb equals the number of e-books demanded each month, I equals the household monthly income, Peb equals the price of e-books, and Phb equals the price of hardbound books. Assume that the price of e-books is €10.68, household income is €2,300, and the price of hardbound books is €21.40.

1. Determine the value of own-price elasticity of demand for e-books.

2. Determine the income elasticity of demand for e-books.

3. Determine the cross-price elasticity of demand for e-books with respect to the price of hardbound books.

## Solution to 1:

Recall that own-price elasticity of demand is given by (∆Qeb/∆ Peb)(Peb/Qeb), and notice from the demand function that ∆Qeb/∆Peb = −0.4. Inserting the given variable values into the demand function yields Qeb = 2.088. So at a price of €10.68, the own-price elasticity of demand equals (–0.4)(10.68/2.088) = −2.046, which is elastic because in absolute value the elasticity coefficient is greater than one.

## Solution to 2:

Recall that income elasticity of demand is given by (∆Qeb/∆I)(I/Qeb). Notice from the demand function that ∆Qeb/∆I = 0.0005. Inserting in the values for I and Qeb yields income elasticity of (0.0005)(2,300/2.088) = 0.551, which is positive, so e-books are a normal good.

## Solution to 3:

Recall that cross-price elasticity of demand is given by (∆Qeb/∆Phb)(Phb/Qeb), and notice from the demand function that ∆Qeb/∆Phb = 0.15. Inserting in the values for Phband Qeb yields a cross-price elasticity of demand for e-books of (0.15)(21.40/2.088) = 1.537, which is positive, implying that e-books and hardbound books are substitutes.