James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 8.4 Applications to Economics and Biology 569
In this section we consider some applications of integration to economics (consumer
surplus) and biology (blood flow, cardiac output). Others are described in the exercises.
p
P
0
X
x
FIGURE 1
A typical demand curve
p
p=p(x)
(X, P)
Consumer Surplus
Recall from Section 4.7 that the demand function psxd is the price that a company has to
charge in order to sell x units of a commodity. Usually, selling larger quantities requires
lowering prices, so the demand function is a decreasing function. The graph of a typical
demand function, called a demand curve, is shown in Figure 1. If X is the amount of the
commodity that can currently be sold, then P − psXd is the current selling price.
At a given price, some consumers who buy a good would be willing to pay more; they
benefit by not having to. The difference between what a consumer is willing to pay and
what the consumer actually pays for a good is called the consumer surplus. By finding
the total consumer surplus among all purchasers of a good, economists can assess the
overall benefit of a market to society.
To determine the total consumer surplus, we look at the demand curve and divide the
interval f0, Xg into n subintervals, each of length Dx − Xyn, and let x i * − x i be the right
endpoint of the ith subinterval, as in Figure 2. According to the demand curve, x i21 units
would be purchased at a price of psx i21 d dollars per unit. To increase sales to x i units,
the price would have to be lowered to psx i d dollars. In this case, an additional Dx units
would be sold (but no more). In general, the consumers who would have paid psx i d dollars
placed a high value on the product; they would have paid what it was worth to them.
So in paying only P dollars they have saved an amount of
ssavings per unitdsnumber of unitsd − fpsx i d 2 Pg Dx
Considering similar groups of willing consumers for each of the subintervals and adding
the savings, we get the total savings:
P
p(x )-P i
Îx
(X, P)
0 ⁄ x x
i_1 x i X
o n
fpsx i d 2 Pg Dx
i−1
(This sum corresponds to the area enclosed by the rectangles in Figure 2.) If we let
n l `, this Riemann sum approaches the integral
FIGURE 2
1
y X
fpsxd 2 Pg dx
0
p
p=p(x)
which economists call the consumer surplus for the commodity.
The consumer surplus represents the amount of money saved by consumers in purchasing
the commodity at price P, corresponding to an amount demanded of X. Figure 3
shows the interpretation of the consumer surplus as the area under the demand curve and
above the line p − P.
P
(X, P)
0
X
x
FIGURE 3
consumer
surplus
p=P
Example 1 The demand for a product, in dollars, is
p − 1200 2 0.2x 2 0.0001x 2
Find the consumer surplus when the sales level is 500.
SOLUTION Since the number of products sold is X − 500, the corresponding price is
P − 1200 2 s0.2ds500d 2 s0.0001ds500d 2 − 1075
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