Some Useful Mathematics for ECON10004
Many economic models start from the assumption that the economic decision maker is actively
pursuing a goal. For example, firms in the economy may be interested in maximizing profit,
governments may be interested in maximizing welfare, and individuals might be interested in
maximizing their own well‐being. Starting with the assumption that decision makers are optimizing
is useful for generating precise, solvable models, primarily because we can use a variety of
mathematical techniques to understand them. Models based on optimization also turn out to match
empirical behaviour well, and so they are viewed as a good foundation for models that seek to
describe real behaviour.
While we have tried to keep the mathematics simple throughout the course, our view is that you will
develop a better understanding of the material if you are exposed to the mathematics that underlies
the graphs in the lectures and book. The following guide will review some of the most common
ideas with a few worked examples.
Optimization
Suppose that you are a manager of a charity that resells tickets to the local sports events and uses
the proceeds to feed the homeless. The tickets are given to you free and the charity will receive any
revenue that you earn. Suppose also that the revenue that you receive depends only on the price
that you set.
In most circumstances, the quantity of tickets that you sell will be related to the price that you set. If
you set the price very low, you will sell all the tickets. If you raise the price, you will sell less tickets,
but each ticket that you sell will have a higher revenue per ticket.
Mathematically, we could represent this problem as
R(P) = Q(P)P,
where R(P) is the revenue that the charity receives if you choose a price P. This is equal to the
quantity of tickets that you sell, Q(P), multiplied by the price you sell each ticket, P. Note that Q(P)
is also a function that returns the number of tickets sold for a chosen price P.
Figure 1 shows a graphical representation of your revenue for different chosen prices. As you might
expect, if you set the price to zero, you would make no profit since you are giving the tickets away
for free. Likewise, if you set a price that is too high, you won’t be able to sell many tickets and your
profit will be low. Profits are thus hump shaped and the revenue maximizing price will be in the
middle at a price denoted as P*.
R(P)
R ( P* )
P*
P
Figure 1: Revenue as a Function of Price
As a manager of the charity, it is unlikely that you would know precisely where P* is without some
experimentation. For instance, suppose that in week 1 you choose a price P1 = $40 and observe that
the revenue is R(P1) = 100 and in week 2, you choose a price P2 = $41 and observe that revenue
increased to R(P2) = 120. Based on the experiment, you could conclude that there was a change in
revenue of $20 for a change in price of $1. Thus, revenue increased as price increased.
Mathematically, we could represent this by
R( P2 )  R( P1 )
R( P)
 0 or
 0,
P2  P1
P
where the delta is used to mean “the change in”. Having run the experiment, we know that the
price of $40 was probably too low and we are likely to improve welfare if we continue to raise our
price in small increments.
R(P)
Line Slope:
R ( P2 )
R ( P1 )
R( P2 )  R( P1 )
0
P2  P1
P
P1 P2
Suppose instead that you started at a price P3 =$80 in week 3 and observe that the revenue is R(P3) =
120. You then increase the price in week 4 to P4 = $81 and observe that revenue decreases to R(P4)
= 100. Based on the experiment, you could conclude that there was a change in revenue of ‐$20 for
a change in price of $1. This experiment would suggest that your prices are too high and that you
should lower them.
R(P)
Line Slope:
R( P4 )  R( P3 )
0
P4  P3
R ( P3 )
R ( P4 )
P3
P4
P
As can be seen by these examples, the slopes of the policy experiment can inform you about what to
do next. If the slope is positive, you want to increase prices. If the slope is negative, you want to
decrease prices. Maximization essentially boils down to finding a policy experiment where the slope
is zero for very small changes in P.
Derivatives
As you hopefully know, the limit of R( P ) for very small changes in P is called the derivative of the
P
dR
(
P
)
function, and is denoted by
or R '( P ) . More formally, the derivative of R(P) is given by
dP
dR ( P )
R( P   )  R( P )
 lim 0
,
dP
(P   )  P
where lim 0 just says we are evaluating the function as we take the difference between the two
prices,  , to zero. Note that this is equivalent to our slope equation above if we were to set
P2  P   and P1  P. An equivalent expression that is more commonly used is:
dR ( P )
R( P   )  R( P )
 lim   0
.

dP
Just like our example above, the derivative is simply the slope of the function at a given point. Thus,
the derivative of dR( P1 ) is very different from the derivative at dR( P3 )
dP
dP
First‐order condition for a maximum
As we saw in our example, if the derivative of revenue was positive, we would want to increase
price. Vice versa, if the derivative of revenue was negative, we would want to decrease price.
This intuition leads to a result that is quite general: for a function to attain its maximum value at a
point, it must be that the derivative of the function has a slope of zero. Hence, in cases where we
know the functional form, an easy way to find an optimum is to take the derivative and set this value
equal to zero. Formally, we are looking for the optimal P* such that
dR ( P * )
0
dP
One danger you will have to look out for later in your studies is that the slope of a function at a
minimum is also zero. Thus, blindly following this rule can get you the lowest possible value. There
are formal ways to test for this, but I find the easiest is to recognize that at a maximum, the slope
just to the left of P* should be positive and the slope just to the right of P* should be negative. You
can check for these by just plugging in slightly smaller and larger numbers and making sure the
slopes are correct.
Some Useful Derivatives
The formula for the derivative can seem rather daunting. Luckily, if you know a few simple patterns,
you will be able to derive most things without using it. Here, I will use f(x), g(x), and h(x) since these
are the functions used most often in textbooks. I will also use the shorthand f’(x) instead of the
more formal df(x)/dx, when expressing the derivative of a function f(x). Finally, I will use the large
brackets [] to denote parenthesis.
1. If f(x) and g(x) are functions:
d [ f ( x )  g  x ) ]
 f '( x )  g '( x )
dx
2. If c is a constant, then
dc
0
dx
3. If c is a constant, then
d [cf ( x )]
d [ f ( x )]
c
 cf '( x )
dx
dx
4. If n is a constant,
d [ xn ]
 nx n 1
dx
5. Note that if n=1, rule (3) and (4) implies:
d [cx]
c
dx
6. Product Rule: If f(x) and g(x) are functions:
d [ f ( x ) g  x ) ]
 f '( x ) g ( x )  f ( x ) g '( x )
dx
7. Chain Rule: If z=f(y) and y=g(x) and if both f’(y) and g’(x) exist, then:
d [ f ( g ( x ))]
 [ f '( g ( x ))]g '( x )
dx
Discussion: The first of these rules show that if we have a “+” or “‐“ sign anywhere in a function, we
can treat each part separately when taking the derivative. This is extremely useful in economics
because we are often interested in the relationship between profits, revenue, and costs:
 ( q)  R( q)  C ( q)
If we are interested in maximizing profits, we are interested in finding the q* where  '( q*)  0. Rule
one says we can just treat R(q) and C(q) separately and thus we are looking for the place where
R '(q )  C '(q)  0,
Or, equivalently
R '(q )  C '(q ).
This is the marginal revenue equals marginal cost rule that you will see many times this year.
Rules number 2 and 3 help us to deal with constants. If, for instance, C(q) = 10q+5,
C '( q )  10
d [ q] d [5]

 10  0  10
dq
dq
Finally, the product rule helps us take derivatives of more complex functions and ones where we do
not have explicit formulas. Recall, for instance, that in our example at the beginning we had
R(P) = Q(P)P
If we wanted to take the derivative of this, we could use the product rule to find that:
R '( P )  Q '( P ) P  Q ( P ).
We will see in class that this equation has a nice economic interpretation when discussing
monopolies.
Partial Derivatives
In the later part of the class, we will start to deal with problems where an outcome depends on
more than one variable. For example, we may be interested in understanding what happens to the
profit of a firm in an environment where the market price depends both on their own quantity q1
and the quantity of a second firm q2:
 ( q1 , q2 )  100q1  q12  q1q2  cq1  50q2
In these cases, we would want to consider how profit changes with q1 holding q2 fixed. The partial
derivative is exactly this operation. Formally, the partial derivative of profit is
 ( q1 , q2 )
 ( q1   , q2 )   ( q1 , q2 )
 lim  0
q1

where q2 means that we are holding q2 fixed at a certain amount. Notice that this is exactly the
same as a single‐variable derivative where we treat all the other variables as constants. For the
profit function above we would have
 ( q1 , q2 )
 100  2 q1  q2  c
q1
To find the maximum q1 we would again find the place where this slope is equal to zero. Here it will
be an implicit function that depends on the quantity chosen by the other firm:
q1 
100  q2  c
2