Math 142 Lecture Notes for Section 6.7
Section 6.7 -
1
Additional Applications of the Integral
Recall that in supply and demand we have two equations for price, the supply equation
p = S(x) and the demand equation p = D(x) where x is the number of units supplied
or demanded respectively. The point of market equilibrium is the value (x0 , p0 ) where
S(x0 ) = p0 = D(x0 ).
Definition 6.7.1:
The consumers’ surplus represents the total savings to consumers who are willing to
pay more than the price p0 for the product but are still willing to pay p0 . This can be
expressed as
Z
x0
(D(x) − p0 )dx.
0
Definition 6.7.2:
The producers’ surplus represents the total gain to producers who are willing to supply
units at a lower price than p0 but are still able to supply units at p0 . This can be
expressed as
Z
x0
(p0 − S(x))dx.
0
Example 6.7.3:
Find the consumers’ surplus at a price level of $120 for the price-demand equation
p = 350 − 0.75x.
Math 142 Lecture Notes for Section 6.7
2
Example 6.7.4:
Find the producers’ surplus when 400 items are sold if the supply equation is given by
p = e0.05x .
Example 6.7.5:
If p = D(x) = 50e−0.05x and p = S(x) = 40e0.05x , find the following:
(a) Equilibrium Point:
(b) Consumers’ surplus at the equilibrium point:
(c) Producers’ surplus at the equilibrium point:
Section 6.7 Suggested Homework: 15-25(odd)