MATH 131-505 Spring 2015
6.7
c
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Liu
6.7 Applications to Economics and Biology
Consumer Surplus
Background: Suppose you need one pair of pants and are willing to pay $60 for it. You would
prefer to have another pair of pants and are willing to pay $50 for this second pair. Finally, you are
not sure just how badly you need a third pair, so you will pay only $40 for the third pair. You go
shopping and find that the dress pants you need are selling for $50 each. So you buy two pairs and
spend a total of $100. You were willing to pay $60 for the first one and $50 for the second one, for a
total of $110 for two of them. You then “saved” $110 − 100 = $10. Adding up all such savings of all
consumers is called the consumer surplus.
In general, let p(x) = D(x) be the demand equation, where p is
the unit price of a commodity and x is the quantity demanded
by the consumers at that price. Let p(x) = S(x) be the supply equation, where p is the unit price of a commodity and x
is the quantity made available by the producers at that price.
The intersection point (x0 , p0 ), is the equilibrium price. At the
equilibrium price, consumers will purchase the same number of
the commodity as the producers will supply.
As can be seen from the figure, D(x) > p0 when x < x0 . This means that there are consumers who
are willing to pay a higher price for the commodity than p0 . These consumers are then actually
experiencing a savings. The total amount of such “savings” is called the consumers’ surplus.
We also see from the figure that S(x) < p0 if x < x0 . This means that there are producers who are
willing to sell the commodity for less than the going price. For these producers the current price of
p0 represents a savings. The total of all such savings is called the producers’ surplus.
Definition:
• If p(x) = D(x) is the demand equation, p0 is the equilibrium price of the commodity, and x0 is
the equilibrium demand, then the consumers’ surplus is given by
Z x0
D(x) − p0 dx
0
• If p(x) = S(x) is the supply equation, p0 is the equilibrium price of the commodity, and x0 is
the equilibrium demand, then the producers’ surplus is given by
Z x0
p0 − S(x) dx
0
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MATH 131-505 Spring 2015
c
Wen
Liu
6.7
Example 1: A hot, wet summer is causing a mosquito population explosion in a lake resort area.
The number of mosquitos is increasing at an estimated given rate n(t) = 2900 + 10e0.5t per week
(where t is measured in weeks). By how much does the mosquito population increase between the
fifth and ninth weeks of summer?
Blood Flow
Poiseuille’s Law: The rate of blood flow, or flux (the volume of blood that passes a cross-section
per unit time), is
πP R4
F =
8ηl
where P is the pressure difference between the ends of a vessel, R is the radius of the blood vessel, l
is the length of the vessel, and η is the viscosity of the blood.
Example 2: Use Poiseuille’s Law to calculate the rate of flow in a small human artery where we
can take η = 0.028, R = 0.009cm, l = 2cm, and P = 3000dynes/cm2 .
Cardiac Output
The cardiac output of the heart is the volume of blood pumped by the heart per unit time, that
is, the rate of flow into the aorta. It is given by
F =Z
A
T
c(t)dt
0
where the amount of dye A is known and the integral can be approximated from the concentration
readings.
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MATH 131-505 Spring 2015
6.7
c
Wen
Liu
Example 3: (p. 479) A 5-mg bolus of dye is injected into a right atrium. The concentration of dye
(in milligrams per liter) is measured in the aorta at one-second intervals as shown below. Estimate
the cardiac output.
t
c(t)
0 1
2
3
4
5
6
7
8
9 10
0 0.4 2.8 6.5 9.8 8.9 6.1 4.0 2.3 1.1 0
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