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without changing the price of soft drinks, sales have increased to 3,000 bottles per week. The advertising elasticity of demand tells you how responsive your vending machine soft drink sales are to the change in advertising expenditures.
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To determine the advertising elasticity of demand, follow the customary steps:
1. Because $400 and 2,000 are the initial advertising expenditures and quantity sold, put $400 into A0 and 2,000 into Q0. 2. Because $500 and 3,000 are the new spending on advertising and sales, put $500 into A1 and 3,000 into Q1. 3. Divide the expression on top of the equation.
(Q1 – Q0) equals 1,000 and (Q1 + Q0) equals 5,000. Dividing 1,000 by 5,000 equals 1⁄5.
4. Divide the expression in the bottom of the equation.
(A1 – A0) equals $100, and (A1 + A0) equals $900. Dividing $100 by $900 equals 1⁄9.
5. Divide the top result, 1⁄5, by the bottom result, 1⁄9.
You get the advertising elasticity of demand equal to 9⁄5 or 1.8. Thus, the advertising elasticity of demand for soft drinks equals
You can conclude that a 1 percent increase in advertising expenditures increases demand 1.8 percent.
Calculating Elasticity with Calculus (If You Must)
The formulas throughout this chapter determine average elasticities for a range of values. For example, in the section “Determining the price elasticity of demand: Formulas are your friend,” you calculate the price elasticity of demand for the range of values between P0 and P1. Similarly, you calculate the income elasticity of demand for the income range between I0 and I1 in the section “Determining the income elasticity of demand: Yet another formula friend.” However, sometimes you need a more precise elasticity value. In these cases, you need to determine what is called the point elasticity, and calculus comes to your rescue.
The most important point elasticity is the point price elasticity of demand. This value is used to calculate marginal revenue, one of the two critical components in profit maximization. (The other critical component is
Chapter 4: Using the Elasticity Shortcut
marginal cost, which I introduce in Chapter 8.) As you can see in Chapter 9, profits are always maximized when marginal revenue equals marginal cost.
The formula to determine the point price elasticity of demand is
In this formula, ∂Q/∂ P is the partial derivative of the quantity demanded taken with respect to the good’s price, P0 is a specific price for the good, and Q0 is the quantity demanded associated with the price P0.
The following equation represents soft drink demand for your company’s vending machines:
In the equation, Q represents the number of soft drinks sold weekly, P is the price per bottle from the vending machine in dollars, I is weekly income in dollars, PC is the price at a convenience in dollars, and A is weekly advertising expenditures in dollars. Assume initially that P is $1.50, I is $600, PC is $1.25, and A is $400. Substituting those values into the demand equation indicates that 2,000 bottles will be sold weekly.
To determine the point price elasticity of demand given P0 is $1.50 and Q0 is 2,000, you need to take the following steps:
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1. Take the partial derivative of Q with respect to P, ∂ Q/∂ P.
For your demand equation, this equals –4,000.
2. Determine P0 divided by Q0.
Because P is $1.50, and Q is 2,000, P0/Q0 equals 0.00075. 3. Multiply the partial derivative, –4,000, by P0/Q0, 0.00075.
The point price elasticity of demand equals –3.
Therefore, at this point on the demand curve, a 1 percent change in price causes a 3 percent change in quantity demanded in the opposite direction (because of the negative sign).
In order to maximize profits, you need to know how much each additional unit you sell adds to your revenue, or in other words, you need to know marginal revenue. If you know the point price elasticity of demand, η, the following formula can enable you to quickly determine marginal revenue, MR, for any given price
Assume your company charges a $1.50 per bottle of soft drink, and the point price elasticity of demand is –3. To determine how much revenue you add by selling an additional bottle:
1. Determine (1 + 1/η).
Substituting –3 for η gives (1 + 1/[–3]) or (1 – 1⁄3) or 2⁄3.
2. Multiply the price, $1.50, by 2⁄3.
The marginal revenue equals $1.00.
So the marginal revenue received when an additional bottle is sold is
If your cost of providing the extra bottle is less than $1.00, you will increase your profits by selling it.
Similarly, you can calculate point elasticities for the income elasticity of demand, cross-price elasticity of demand, and advertising elasticity of demand using the following formulas:
✓ The point income elasticity of demand:
In this formula, ∂Q/∂I is the partial derivative of the quantity taken with respect to income, I is the specific income level, and Q is the quantity purchased at the income level I. ✓ The point cross-price elasticity of demand:
In this formula, ∂Qx/∂P y is the partial derivative of good x’s quantity taken with respect to good y’s price, Py is a specific price for good y, and Qx is the quantity of good x purchased given the price Py
. ✓ The point advertising elasticity of demand:
In this formula, ∂Q/∂A is the partial derivative of the quantity demanded taken with respect to advertising expenditures, A is the specific amount spent on advertising, and Q is the quantity purchased.
