Demand Analysis and Optimal Pricing 93The figure depicts a useful result. Any linear demand curve can be divided
into two regions. Exactly midway along the linear demand curve, price elasticity
is unity.To the northwest (at higher prices and lower quantities), demand is
elastic. To the southeast (at lower prices and greater quantities), demand is
inelastic. For example, consider a point on the inelastic part of the curve
such as B: P $100 and Q 1,200. Here the point elasticity is EP
(dQ /dP)(P/Q) (4)(100/1,200) .33. Conversely, at a point on the elas-
tic portion of the demand curve such as A (P $300 and Q 400), the point
elasticity is EP(4)(300/400) 3.0.CHECK
STATION 3Compute the price elasticity at point M. Show that the elasticity is unity. This result holds
for the midpoint of any linear demand curve.Figure 3.3b depicts the firm’s total revenue curve for different sales volumes.
It displays the familiar shape of an upside-down U. Total revenue increases as
quantity increases up to the revenue peak; at still higher quantities, revenue falls.
Let’s carefully trace the relationship between price elasticity and changes
in revenue. Suppose that management of the software firm is operating at point
A on the demand curve in Figure 3.3a. Its price is $300, it sells 400 copies of the
software program, and it earns $120,000 in revenue per week. Could the firm
increase its revenue by cutting its price to spur greater sales? If demand is elas-
tic, the answer is yes. Under elastic demand, the percentage increase in quan-
tity is greater than the percentage fall in price. Thus, revenue—the product of
price and quantity—must increase. The positive change in quantity more than
compensates for the fall in price. Figure 3.3b shows clearly that starting from
point A, revenue increases when the firm moves to greater quantities (and
lower prices). Starting from any point of elastic demand, the firm can increase
revenue by reducing its price.
Now suppose the software firm is operating originally at point B, where
demand is inelastic. In this case, the firm can increase revenue by raising its
price. Because demand is inelastic, the percentage drop in quantity of sales is
smaller than the percentage increase in price. With price rising by more than
quantity falls, revenue necessarily increases. Again, the revenue graph in Figure
3.3b tells the story. Starting from point B, the firm increases its revenue by
reducing its quantity (and raising its price). As long as demand is inelastic, rev-
enue moves in the same direction as price. By raising price and reducing quan-
tity, the firm moves back toward the revenue peak.
Putting these two results together, we see that when demand is inelastic or
elastic, revenue can be increased (by a price hike or cut, respectively).
Therefore, revenue is maximized when neither a price hike nor a cut will help;
that is, when demand is unitary elastic, EP1. In the software example, the
revenue-maximizing quantity is Q 800 (Figure 3.3b). This quantity (along
with the price, P $200) is the point of unitary elasticity (in Figure 3.3a).c03DemandAnalysisAndOptimalPricing.qxd 8/18/11 6:48 PM Page 93