CHAPTER 6
1. Average Value
Given a set of data, we commonly find the average, which can give a good sense of the
data in a very simple, clear way. For a discrete set of data, like the number of fish in each
of 5 ponds, finding the average is simple. We simply add up the number in each pond and
divide by 5, the number of data points. This average might give us a way to predict the
number of fish in a 6th comparable pond. We often, however, have data or functions that
are continuous, like temperature. As a pot of room temperature water boils, it is at some
point every possible temperature between 50 deg C and 100 deg C, but there are infinite
temperatures between 50 and 100 degrees! We can’t average together infinite numbers in
the traditional way, but if we have a function, we can use integrals to find the average
value.
Example 1. In order for morphine to be effective in ambulatory patients, average plasma
concentrations must be 0.065 mg/L. In order to make sure a patient is on an appropriate
dose, one way to find the average concentration would be to take blood every hour and
average those values together the traditional way. Suppose we have the following values:
Time (Hours) Concentration (mg/L)
1
.1244
2
.0624
3
.0416
4
.0312
5
.0250
6
.0208
7
.0179
From this data, we have that the average plasma concentration is
.
Sampling every hour, however, may mean that we miss some information. It turns out
that morphine administered intravenously hits peak concentrations very quickly. Suppose
that C(t) = 4t2.5t
is the plasma concentration at time t. Graphing this function reveals
+.02
that sampling every hour may be insufficient.
1
2
CHAPTER 6
Sampling every 20 minutes (for the same time interval- the first 7 hours after the drug
was administered) gives us an average value of .064 mg/L, a significantly different value.
We could continue to sample more and more frequently, but that quickly becomes calculation
intensive and unreasonable. Instead, we can use the concentration function and calculus to
figure out the true average value.
Let’s first consider how we compute the average. First, we choose a set of equally
spaced times, which we’ll call t1 , t2 , ..., tn . When we measured plasma concentration every 20 minutes, we took
samples, so n =
. Then the average concentration was
C(t1 ) + C(t2 ) + ... + C(tn )
.
n
What is the relationship between the number of samples we take and the time between
each sample?
This relationship, however, is the same as the formula we used to find ∆x, the width
of each rectangle in a Riemann sum. If we solve for n, we get
Thus our formula becomes
C(t1 ) + C(t2 ) + ... + C(tn )
(b−a)
∆t
1
∗ (C(t1 ) ∗ ∆t + C(t2 ) ∗ ∆t + ... + C(tn ) ∗ ∆t)
b−a
As n increases to infinity, the expression C(t1 ) ∗ ∆t + C(t2 ) ∗ ∆t + ... + C(tn ) ∗ ∆t becomes
=
Therefore, as we take more and more samples, our formula for average value becomes
CHAPTER 6
3
Example 2. Using this formula, what is the average concentration for the first 7 hours
after the drug is administered?
Average values can also be seen graphically. Since the average value of f (x) from a to
b, A(f ) is
Z b
1
f (x)dx
b−a a
we have the following relationship:
We can draw this on the graph as a rectangle of width b − a and height
.
This rectangle as the same area as the region between f (x) and the x-axis, in other words,
the area of the rectangle is
.
Example 3. Find the average value of f (x) = 3x2 between x = 2 and x = 5. Then graph
R5
f (x) and draw in a rectangle with the same area as 2 f (x)dx.
4
CHAPTER 6
Example 4. Given the two graphs below, find the average values of f (x), g(x), f (x) + g(x)
and f (x) ∗ g(x) over the interval [0, 2].
Example 5. Using the last example, is it true that the average value of f (x) + the average
value of g(x) is the average value of f (x) + g(x)? What about the product?
CHAPTER 6
Example 6. Find the average value of f (x) =
out the integral without your calculator).
√
5
36 − x2 from x = 0 to x = 6 (Try figuring
2. Consumer and Producer Surplus
Recall that supply curves show the quantity of items the producers supply at different
prices. Demand curves show the quantity of the items that are bought at different prices.
A typical graph of supply and demand curves looks like
The place where the graphs cross is the equilibrium, which gives us the equilibrium price
p∗ and the equilibrium quantity q ∗ . At equilibrium, there are some consumers who would
have been willing to pay more and some suppliers who would be willing to produce the
item for less.
Definition 1. The consumer surplus is the total amount gained by consumers by buying
an item at the current price, rather than at the price they would have been willing to pay.
Definition 2. The producer surplus is the total amount gained by producers by selling
at the current price, rather than at the price they would be willing to accept.
6
CHAPTER 6
Both the consumer surplus and producer surplus measure how much consumers and
producers are saving. We’d like to find a way to identify consumer surplus and producer
surplus graphically, as areas under the supply and demand curves. First, suppose that all
goods are sold at equilibrium price, how can we illustrate the consumer’s expenditure?
But at some quantities, consumers were willing to pay a higher price than p∗ . If everyone
who was willing to pay more that p∗ paid their maximum price, the consumer expenditure
can be shown graphically
We call the shaded area the consumer expenditure; mathematically, it can be written
as
The consumer surplus is the amount consumers save by paying equilibrium price. Thus,
on the graph, it is the area between the demand curve and the horizontal line at p∗ .
Similarly, producers were willing to sell some quantities at a lower price than p∗ , so by
selling at equilibrium price, they save money. The amount saved is represented by the the
area between the supply curve and the horizontal line at p∗ .
Definition 3. The total gains from a trade is the consumer surplus + the producer surplus
CHAPTER 6
7
Thus, a completely labeled graph is:
Example 7. Suppose we have a supply curve p = 0.1q and the demand curve p = 240−0.2q.
Find the equilibrium price and quantity, the consumer surplus, and the producer surplus.
8
CHAPTER 6
We know how to express these quantities graphically; how can we do it mathematically?
For the following mathematical expressions, determine what they represent on the graph
and explain the economic meaning.
(1) p∗ q ∗
(2)
R q∗
(3)
R q∗
(4)
R q∗
0
0
0
D(q)dq
S(q)dq
D(q)dq − p∗ q ∗
(5) p∗ q ∗ −
(6)
R q∗
0
R q∗
0
S(q)dq
(D(q) − S(q))dq
Sometimes, outside forces prevent prices from reaching the equilibrium price. Suppose that
instead, the price is set at p+ . We can find the consumer and producer surplus and total
gains, using the following figure.
CHAPTER 6
9
3. Present and Future Value
Definition 4. The future value, B, of a payment, P, is the amount to which the P would
have grown if deposited today in an interest-bearing bank account.
Definition 5. The present value, P, of a future payment, B, is the amount that would
have to be deposited in a bank today to produce excactly B in the account at the relevent
time in the future.
For the purposes of this section, we’ll assume interest is compounded continuously. Recall
that when interest is compounded continuously, if the initial investment is $A, then after
t years, the account contains $Aert , where r is the interest rate. Using this, we have the
following two equations:
P =
B=
Example 8. Suppose you were going to be paid $100,000.00 a year from now. What is the
present value, assuming a 2.5% interest rate? What will the future value be 3 years after
you’re paid?
These definitions and formulas work for single payment, but sometimes we think about
payments as happening continuously. In this case, we can define a continuous function that
represents the income stream, S(t), which has units of dollars/year. (Notice that S(t)
is a rate). For an income stream, the future value represents the amount of money you
would have if you continuously deposited the income stream into an account until some
future date. The present value represents the amount of money you’d have to deposit
today in order to match what you would get from the income stream by the future date.
Since money is being deposited continously, computing the future and present value is a
little more complicated. We’ll figure out a formula for present value first.
Suppose we have some income stream, S(t) and we’re considering the time period from
now until T , some number of years in the future. For any one deposit, B, made t years in
10
CHAPTER 6
the future, its present value is given by P =
. If we wanted to estimate
the present value of S(t), we could divide it up into many small intervals (maybe one for
each day). If we assumed that S(t) didn’t change much over each little interval, we could
measure S(t) once during that interval and multiply it by the length of that interval to
estimate how much money came in during the entire interval.
Example 9. Suppose my income stream is given by S(t) and I choose to make my intervals
1
one day each. t is measured in years, so each interval is 365
years. At 3am on May 1s t,
S(t) = 560, 235 dollars per year. Using these values, we’d estimate that on May 1s t, I
made
Taking this estimate, represented mathematically by
the present value for each interval by
, we can estimate
Then, over the entire time period from now until T , we can estimate the present value
of the income stream by adding together the estimates for each interval. Graphically, this
will look like
This, of course, is just a Riemann sum! Our estimate will get better when we take smaller
intervals. As the length of the intervals goes to zero, we get the following expression of the
present value:
CHAPTER 6
11
Using this, we can define the future value of an income stream to be
Future Value = Present Value ∗ erM
Example 10. Suppose you have a constant income stream of $50,000 per year for 10
years. Assume that your interest rate is 6% and compute the present and future values of
the income stream.
Example 11. I predict that my (hypothetical) child will be 18 in 26 years. By then, I want
to have $50,000 in a college account that earns 3% interest. If I wanted to make a single
lump sum deposit right now, how much will I need to deposit?
Since that’s not possible, suppose I want to deposit money continously for 26 years at
a constant rate of $S/year. What rate should I deposit at?
12
CHAPTER 6
4. Relative Growth Rates
Usually in Calculus, a rate is a derivative. When we talk about populations, however,
we often use confusing terminology. If the population of Binghamton, NY is growing at a
rate of 1.1% per year, that’s the relative rate of change or the relative growth rate, not
the derivative. The derivative is sometimes called the absolute rate of change. They are,
however, related: If some population is given by the function P (t), then the absolute rate
of change, the derivative, is written dP
dt and the relative rate of change of P is
relative growth rate =
The relative growth rate of a population is not enough information to find the actual change
in a population, but it is enough to find the percent change in the population. First we’ll
compute a derivative that initially seems random:
d
(ln(P (t))) =
dt
CHAPTER 6
13
This, however, is just the
. Now, recall
0
the fundamental theorem of calculus: If f (t) is a function with derivative f (t), then
Z b
f 0 (t)dt =
a
Using this,
Z b
a
P 0 (t)
dt =
P (t)
R b 0 (t)
dt is the total amount ln(P (t)) changes from t = a to t = b.
The interpretation of a PP (t)
Thus by finding the integral of the relative growth rate, we can find the factor by which a
population changes between a and b (e.g. increasing by a factor of 2 would mean that the
population doubled).
Example 12. Given the following graph of the relative growth rate, find the factor by which
the population increased over 50 years.
Example 13. Suppose some population P (t) has a constant relative growth rate r.
First, graph the relative growth rate
14
CHAPTER 6
Now, interpret
Rt
o
rdt in terms of area:
0
(t)
Remember that r = PP (t)
. Using that, interpret
population increased from 0 to t years.
Rt
o
rdt to find the factor by which the
Now, set those quantities equal to each other and solve for P (t)
Example 14. Consider the following graph of the relative growth rate of a population:
(1) On what interval is the population increasing? By what percentage does the population increase during this interval?
(2) On what interval is it decreasing? By what percentage does the population decrease?
CHAPTER 6
(3) By what percentage does the population change during the entire period?
15