Pricing II: Constant Elasticity

This module covers the relationships between price and quantity, elastic demand, inelastic demand, and optimal price under conditions of constant elasticity.

Authors: Paul Farris and Phil PfeiferMarketing Metrics Reference: Chapter 7

© 2010-14 Paul Farris, Phil Pfeifer and Management by the Numbers, Inc.

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YPrice Elasticity

People use elasticity loosely to mean how responsive is demand to changes in price:

• Elastic demand means quantity is very responsive.  • Inelastic demand is when quantity does not decrease (much) when

the price increases (and vice versa).  

At this level, most of us know what elasticity means. The challenge comes when we try to measure elasticity or responsiveness. 

The formal definition of elasticity has two important parts: • The unit change in quantity per small unit change in price (the slope)

and • The reference point for measuring those unit changes as percentage

changes.

As we saw in Pricing I, linear functions have constant slope and a different elasticity at each price point.  

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SConstant Elasticity Price-Quantity Functions

Unlike linear functions, constant elasticity functions exhibit a different slope, but the same elasticity at each point on the curve. This means the slope is changing at a very specific rate to keep elasticity constant.

A mathematical equation for this relationships is Q = kP℮ where e is the price elasticity of demand and k is a constant.* For constant elasticity functions we can use the elasticity to find the profit-maximizing price. If we know cost and elasticity, there are only two steps required.

*k is the quantity the firm would sell at a price = $1.

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Definition

Under conditions of constant elasticity, Q = kP℮ and k = Q / P℮

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To calculate the optimal price using cost and elasticity there are two steps:

1. Calculate the absolute value of the reciprocal of elasticity. The resulting number tells us what margin is optimal (and, by extension, what price is optimal). For example, a -2 elasticity would result in an optimal margin of 50%., a -4 elasticity would result in an optimal margin of 25%, and so on.

2. Use cost and the optimal margin to calculate the optimal price = Cost / (1- Margin). For a cost of $0.75 per unit and an elasticity of -2, the profit-maximizing price would be $0.75 / (1 - .5) = $1.50.

Constant Elasticity Price-Quantity Functions

Definition

With constant elasticity, Optimal margin = ABS (1 / elasticity)

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Exercise 1

Question 1

For a constant elasticity function with an elasticity of -4 and a unit cost of $2.00, what is the profit-maximizing price?

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Answer:

Step 1: Optimal Margin = absolute value of (1 / -4) = 25%

Step 2: Optimal Price = $2.00 / (1 - .25) = $2.67

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SHigher Elasticity Means Lower Margins

The more “elastic” the curve,the greater the % change in quantity for a given % change in price.

Compare the curve with the -4 elasticity with the curve with a -2 elasticity (displayed at right).

Higher elasticity mean lower optimal margins. The more “responsive” is demand, the more aggressive the firm should be in its pricing.

Note: As we saw in the last section, at the optimal price, elasticity is always greater than 1 (or less than -1, depending on how you like to think about it). Margin can never be more than 100%!! And if it is less than 0%, you have bigger pricing problems than finding the optimal.

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Spreadsheet Calculation of Profit-Maximizing Price

Note: Profit-maximizing price occurs at 25% margin = 1 / (-4)

Q = 500*P-4

Cost $1.25Price Quantity $Margin %Margin Profit

$1.00 500.00 -0.25 -25% -125.00$1.10 341.51 -0.15 -14% -51.23$1.20 241.13 -0.05 -4% -12.06$1.30 175.06 0.05 4% 8.75$1.40 130.15 0.15 11% 19.52$1.50 98.77 0.25 17% 24.69$1.60 76.29 0.35 22% 26.70$1.70 59.87 0.45 26% 26.94$1.80 47.63 0.55 31% 26.20$1.90 38.37 0.65 34% 24.94$2.00 31.25 0.75 38% 23.44$2.10 25.71 0.85 40% 21.85$2.20 21.34 0.95 43% 20.28$2.30 17.87 1.05 46% 18.76$2.40 15.07 1.15 48% 17.33$1.67 64.28 0.42 25% 27.00

InsightIf your percentage margin is less than 1 / elasticity, you should seriously consider raising your price.

Profit Maximizing Price

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Calculating Profit-Maximizing Price

If you are only given two observations of price and quantity, it is impossible to say whether the best function to use is linear or constant elasticity. For example, if a business determined the following:

Price Quantity $10 100 $15 50

If demand is linear: slope = 50 / 5 = 10 and MRP = 20.For a cost of $5, the optimal price would be: ½ * (Cost + MRP) = ½ * (5 + 20) = $12.50

For constant elasticity demand, elasticity is calculated using Excel or a scientific calculator as ln (50 / 100) / ln (15 / 10) = -1.71.The optimal margin would be 1 / 1.71 = 58.5% and the optimal price would be cost / (1 - margin) = $5 / (1 - .585) = $12.05

Note: ln = natural log. So, ln(.5) would be read as natural log of .5

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For constant elasticity demand functions you will need to use Excel or a scientific calculator to predict unit sales resulting from a new unit price. The formula is: Q = K*Pe.

Q = K*Pe where e is the price elasticity of demand and K is a constant

Question 2: With two points of price and quantity, the elasticity is estimated by calculating ln (Q2 / Q1) / ln (P2 / P1), but how do you find K?

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Answer:

Use either Q2 and P2 or Q1 and P1 in the equation. For example, using quantity of 100 and price of $10 from the previous example and the elasticity of -1.71, we have 100 = K * 10 ^ -1.71. 100 =K*.0195, K = 5,128.

To “predict” sales at a price of $15, Q= 5,128 * 15 ^ -1.71 = 50.

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Further Reference

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Marketing Metrics by Farris, Bendle, Pfeifer and Reibstein, 2nd edition, chapter 7.

- And the following MBTN modules -

Pricing I: Linear Demand - This module explores pricing under the assumption of demand curves that are not linear.

Profit Dynamics - This module is a more basic module that examines price-volume trade-offs.

Promotion Profitability - This module examines the role of promotion in pricing decisions.