14
Further Integration Techniques
and Applications of the Integral
Copyright © Cengage Learning. All rights reserved.
14.4
Applications to Business and Economics: Consumers’
and Producers’ Surplus and Continuous Income Streams
Copyright © Cengage Learning. All rights reserved.
Consumers’ Surplus
3
Consumers’ Surplus
Consider a general demand curve presented, as is
traditional in economics, as p = D(q), where p is unit price
and q is demand measured, say, in annual sales
(Figure 15).
Thus, D(q) is the price at which the demand will be q units
per year. The price p0 shown on the graph is the highest
price that customers are willing to pay.
Figure 15
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Consumers’ Surplus
Suppose, for example, that the graph in Figure 15 is the
demand curve for a particular new model of computer.
When the computer first comes out and supplies are low
(q is small), “early adopters” will be willing to pay a high
price.
This is the part of the graph on the left, near the p-axis. As
supplies increase and the price drops, more consumers will
be willing to pay and more computers will be sold.
We can ask the following question: How much are
consumers willing to spend for the first units?
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Consumers’ Surplus
Consumers’ Willingness to Spend
We can approximate consumers’
willingness to spend on the first
units as follows.
Figure 16
We partition the interval [0, ] into
n subintervals of equal length, as
we did when discussing Riemann sums.
Figure 16 shows a typical subinterval, [qk − 1, qk].
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Consumers’ Surplus
The price consumers are willing to pay for each of units
qk – 1 through qk is approximately D(qk – 1), so the total that
consumers are willing to spend for these units is
approximately D(qk – 1)(qk – qk – 1) = D(qk – 1)q, the area of
the shaded region in Figure 16.
Thus, the total amount that consumers are willing to spend
for items 0 through is
which is a Riemann sum.
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Consumers’ Surplus
The approximation becomes better the larger n becomes,
and in the limit the Riemann sums converge to the integral
This quantity, the area shaded in Figure 17, is the total
consumers’ willingness to spend to buy the first units.
Figure 17
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Consumers’ Surplus
Consumers’ Expenditure
Now suppose that the manufacturer simply sets the price
at ,with a corresponding demand of , so D( ) = .
Then the amount that consumers will actually spend to buy
these is
, the product of the unit price and the
quantity sold.
This is the area of the rectangle
shown in Figure 18.
Notice that we can write
as suggested
by the figure.
Figure 18
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Consumers’ Surplus
The difference between what consumers are willing to pay
and what they actually pay is money in their pockets and is
called the consumers’ surplus.
Consumers’ Surplus
If demand for an item is given by p = D(q), the selling price
is , and is the corresponding demand [so that
D( ) = ], then the consumers’ surplus is the difference
between willingness to spend and actual expenditure:
10
Consumers’ Surplus
Graphically, it is the area between the graphs of
p = D(q) and p = , as shown in the figure.
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Example 1 – Consumers’ Surplus
Your used-CD store has an exponential demand equation
of the form
p = 15e−0.01q
where q represents daily sales of used CDs and p is the
price you charge per CD. Calculate the daily consumers’
surplus if you sell your used CDs at $5 each.
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Example 1 – Solution
We are given D(q) = 15e−0.01q and
By definition,
D( ) =
= 5. We also need
.
or 15e−0.01 = 5
which we must solve for
:
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Example 1 – Solution
cont’d
We now have
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Example 1 – Solution
cont’d
= $450.69 per day
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Producers’ Surplus
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Producers’ Surplus
We can also calculate extra
income earned by producers.
Consider a supply equation of
the form p = S(q), where S(q)
is the price at which a supplier
is willing to supply q items
(per time period).
Figure 19
Because a producer is generally willing to supply more
units at a higher price per unit, a supply curve usually has a
positive slope, as shown in Figure 19.
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Producers’ Surplus
The price p0 is the lowest price that a producer is willing to
charge.
Arguing as before, we see that the minimum amount of
money producers are willing to receive in exchange for
items is
.
On the other hand, if the producers charge
items, their actual revenue is
per item for
The difference between the producers’ actual revenue and
the minimum they would have been willing to receive is the
producers’ surplus.
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Producers’ Surplus
Producers’ Surplus
The producers’ surplus is the extra amount earned by
producers who were willing to charge less than the selling
price of per unit and is given by
where
Graphically, it is the area of the
region between the graphs of
p = and p = S(q) for 0 ≤ q ≤ ,
as in the figure.
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Example 2 – Producers’ Surplus
My tie-dye T-shirt enterprise has grown to the extent that I
am now able to produce T-shirts in bulk, and several
campus groups have begun placing orders. I have informed
one group that I am prepared to supply
T-shirts
at a price of p dollars per shirt. What is my total surplus if
I sell T-shirts to the group at $8 each?
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Example 2 – Solution
We need to calculate the producers’ surplus when
= 8.
The supply equation is
but in order to use the formula for producers’ surplus, we
need to express p as a function of q.
First, we square both sides to remove the radical sign:
q2 = 400(p − 4)
so
giving
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Example 2 – Solution
We now need the value of
corresponding to
cont’d
= 8.
Substituting p = 8 in the original equation, gives
Thus,
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Example 2 – Solution
cont’d
Thus, I earn a surplus of $106.67 if I sell T-shirts to the
group at $8 each.
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Continuous Income Streams
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Example 4 – Continuous Income
An ice cream store’s business peaks in late summer; the
store’s summer revenue is approximated by
R(t) = 300 + 4.5t – 0.05t2 dollars per day (0 ≤ t ≤ 92)
where t is measured in days after June 1. What is its total
revenue for the months of June, July, and August?
Solution:
Let’s approximate the total revenue by breaking up the
interval [0, 92] representing the three months into n
subintervals [tk − 1, tk ], each with length t .
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Example 4 – Solution
cont’d
In the interval [tk − 1, tk ] the store receives money at a rate of
approximately R(tk − 1) dollars per day for t days, so it will
receive a total of R(tk − 1)t dollars.
Over the whole summer, then, the store will receive
approximately
R(t0)t + R(t1)t + · · · + R(tn − 1)t dollars.
As we let n become large to better approximate the total
revenue, this Riemann sum approaches the integral
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Example 4 – Solution
cont’d
Substituting the function we were given, we get
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Continuous Income Streams
Total Value of a Continuous Income Stream
If the rate of receipt of income is R(t) dollars per unit of
time, then the total income received from time t = a to t = b
is
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Continuous Income Streams
Future Value of a Continuous Income Stream
If the rate of receipt of income from time t = a to t = b is R(t)
dollars per unit of time and the income is deposited as it is
received in an account paying interest at rate r per unit of
time, compounded continuously, then the amount of money
in the account at time t = b is
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Continuous Income Streams
Present Value of a Continuous Income Stream
If the rate of receipt of income from time t = a to t = b is R(t)
dollars per unit of time and the income is deposited as it is
received in an account paying interest at rate r per unit of
time, compounded continuously, then the value of the
income stream at time t = a is
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Continuous Income Streams
We can derive this formula from the relation
FV = PVer(b − a)
because the present value is the amount that would have
to be deposited at time t = a to give a future value of FV at
time t = b.
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