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in the good’s price, all other factors held constant. In algebraic terms,

we have

[3.7]

where P 0 and Q 0 are the initial price and quantity, respectively. For example,

consider the airline’s demand curve as described in Equation 3.4. At the cur-

rent $240 fare, 100 coach seats are sold. If the airline cut its price to $235, 110

seats would be demanded. Therefore, we find

In this example, price was cut by 2.1 percent (the denominator), with the result

that quantity increased by 10 percent (the numerator). Therefore, the price

elasticity (the ratio of these two effects) is 4.8. Notice that the change in quan-

tity was due solely to the price change. The other factors that potentially could

affect sales (income and the competitor’s price) did not change. (The require-

ment “all other factors held constant” in the definition is essential for a mean-

ingful notion of price elasticity.) We observe that there is a large percentage

quantity change for a relatively small price change. The ratio is almost fivefold.

Demand is very responsive to price.

Price elasticity is a key ingredient in applying marginal analysis to

determine optimal prices. Because marginal analysis works by evaluating

“small” changes taken with respect to an initial decision, it is useful to measure

elasticity with respect to an infinitesimally small change in price. In this

instance, we write elasticity as

[3.8a]

We can rearrange this expression to read

[3.8b]

In words, the elasticity (measured at price P) depends directly on dQ /dP, the

derivative of the demand function with respect to P (as well as on the ratio of

P to Q).

Epa

dQ

dP

ba

P

Q

b.

Ep

dQ /Q

dP/P

.

Ep

(110100)/100

(235240)/240



10.0%

2.1%

4.8.



¢Q/Q

¢P/P



(Q 1 Q 0 )/Q 0

(P 1 P 0 )/P 0

Ep

% change in Q

% change in P

84 Chapter 3 Demand Analysis and Optimal Pricing

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