Intermediate Math for Economics Anthony Tay Section 1 Page 1 Intermediate Mathematics for

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Intermediate Mathematics for Economics Anthony Tay

1. Introduction to mathematical economic models

The objective of this section is to introduce to you the elements of an economic model, and to provide a
few examples highlighting the type of mathematics you are likely to encounter in your undergraduate
economics experience.

Economists build “models” to explain various aspects of the economy. The models are in mathematical
form, although the mathematics is sometimes expressed or illustrated graphically.

It should not be surprising that economists choose to build mathematical models. Economics explains
human behavior as a manifestation of choices made with the intention of achieving various objectives, all
within the context of scarce resources which have alternate uses. The allocation of scarce resources to
achieve an objective is inherently a mathematical problem, and is most naturally expressed in that way.
Modeling economic problem mathematically also allows for mathematical techniques to be used to draw
out the logical conclusions and predictions of the model, and for statistical “econometric” techniques to
be used when taking the theories to data.

I will use a few examples of economic models
(i) as examples of the kind of models economists build;
(ii) to introduce the elements of mathematical economics models;
(iii) to motivate the mathematical topics included in this course.

The examples that follow contain mathematics that many of you are still unfamiliar with. They are not
intended to be an indication of what you should already know, but to ‘set the agenda’, and to give you a
look ahead to the type of mathematics we will be studying in this course.

Example 1.1 Suppose a firm chooses how much to produce in order to maximizes profit (total revenue
minus total costs). Let x denote a firm’s output quantity choice. Suppose there is ‘perfect competition’,
and the firm has to take price p as given. Then its revenue is R(x) px. Assume its total cost is given by
C(x)x2. The firm’s profit function is then
(x)  R(x)C(x)  pxx2.
This is sometimes called the firm’s objective function.

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What is the firm’s optimal choice of x? (That is, how much should the firm produce so that its profits are
maximized?) Let x* denote the firm’s optimal choice. You will learn, when we study optimization theory,
that the necessary and sufficient conditions for xx* to maximize profit (x) are (x*)0 and
(x)0 for all x. We have
(x*)  R(x*)C(x*)  p2x*  0
which gives x*  p/2. Furthermore, (x)  R(x)C(x)  2, so (x)0 is satisfied for all
values of x. The value x*  p/2 is a profit maximizing level of production.

This model begins with a postulate: that firm’s maximize profits. From this postulate, a relationship
between two economic variables is derived, output x and price p. Price in this model is an exogenous
variable (we don’t try to explain it, but simply treat it as ‘given’; in this case we justified fixing the price
by assuming perfect competition). The variable x is an endogenous variable, a variable whose behavior
we explain in relation to the exogenous variables. In this example, we derived x*  p/2 . More
importantly,
dx* 1
  0,
dp 2
i.e., the theory says that when price increases, firms choose to produce more.

The process of asking what happens (to the endogenous variable) when an exogenous variable or a
parameter of the model changes is called comparative statics. In this case, the model generates a
refutable proposition as a consequence of the assumption that firms maximize profits – that production
increases when prices increase. This proposition can then be compared with (‘tested against’) actual
observed behavior, a skill you will learn when you study econometrics.

A few remarks: all models are abstractions of the real-world, including whatever is needed for providing
an explanation for some observed behavior, and leaving out everything else. This is perhaps more so in
economics than in many other fields. There is obviously a tremendous amount of abstraction in our
example. Many firms produce many products, and even for the ‘same’ product there are often many
‘varieties’, but these have all been ‘lumped’ into one product type. Many variables one might expect to be
related to the issue have also been omitted, for example, labor costs, rental costs, employment, variables
related to available technology, etc. However, for the question that we asked, these are not relevant – in a
sense, all of these factors have all been summarized into the cost function.

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One skill you will have to develop is the ability to work with general functions instead of specific
functions. For instance, in dealing with the cost function, we would prefer to leave it as general as
possible, and make only the minimal assumptions about it. For instance, we might assume only that the
cost function is characterized by C(x)0 and C(x)0, instead of specifying a particular function
such as C(x)x2. Example 1.1 is recast in this more general form in Example 1.2.

Example 1.2 Suppose a firm chooses how much to produce in order to maximize profits, which are
computed as Total Revenue minus Total Costs. Let x denote a firm’s output quantity choice. If the firm
is a price taker, i.e., it has to take price p as given, then its revenue is R(x) px. Denote its total costs
by C(x) where C(x)0 and C(x)0 for all x. The firm’s profit function is then
(x)  R(x)C(x)  pxC(x) , C(x)0 and C(x)0 for all x
What is the firm’s optimal choice of x? Again letting x* denote the firm’s optimal choice, the necessary
and sufficient conditions for xx* to maximize profit are
(x*)  R(x*)C(x*)  pC(x*)  0
and (x)  R(x)C(x)  C(x)  0 for all x.
That is, x* must satisfy
R(x*)  C(x*) (Marginal Revenue = Marginal Cost)
which in our specific example says that C(x*) p, and
(x)R(x)C(x)0 (MC cuts MR from below)
The latter condition says that marginal costs C(x) cuts marginal revenue R(x) from below, because
C(x)R(x): the slope of C(x) is greater than the slope of R(x).

The cost function has been kept very general, with only two characteristics specified. This keeps the
analysis as general as possible, and helps to identify the assumptions that drive the results. One can argue
about the reasonableness of the assumptions made, but this a good thing about the analysis – we have
focused the argument on to specific assumptions, and efforts to improve the analysis can focus on these.
The analysis also highlights general principles rather than specific results. Here, it is that profit
maximizing firms will operate where ‘marginal cost equals marginal revenue, as long as the marginal
revenue curve cuts the marginal cost curve from below’. Such a result should be familiar to those who
have studied introductory economics; you may have already drawn the corresponding curves in an earlier
course. If so, you have already carried out the analysis above! Graphs, however, are difficult to apply in
more complicated modeling situations. It is easier to do so with mathematical analysis

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One question students sometimes raise is whether we can reasonably assume that a firm is able to employ
optimization theory to get to the maximum. Again, this is not an issue. We are usually not interested in
how the firm gets to the maximum, but in how the firm behaves at the maximum. In our example, we
want to characterize the firm’s optimal choice as a function of the price given to the firm. The firm might
get there in a variety of ways. We will get there by optimization theory.

Apart from characterizing behavior, we also want to know what happens to the endogenous variable when
the exogenous variable changes. But how do we do this with general functions? In a later section you will
learn how to compute comparative statics results using implicit differentiation with general functions.

Writing the optimal production level x* as a function of p, x* x*(p), we have
p  C(x*(p)).
Implicit differentiation then gives
dx*(p)
1  C(x*(p)) .
dp
On the left hand side, the derivative of p with respect to itself is simply 1. On the right hand side, we have
a composite function: Cis a function of x*, which in turn is a function of p. Applying the chain rule –
we differentiate C wrt to x*, and multiply the result by the derivative of x* with respect to p, to give
d dx*(p)
C(x*(p))  C(x*(p))
dp  dp
dx*(p)
Rearranging 1  C(x*(p)) gives
dp
dx*(p) 1
  0,
dp C(x*(p))
since C(x)0 for all x, by assumption. As price increases, the firm increases production (supply curve
slopes upwards).

You have done implicit differentiation before, but probably not with general functions. We will do so in
this course. Note also that in working with general functions, you will have to be comfortable working
with the function notation, and working with the function-prime notation for derivatives.

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Another mathematical procedure we use a lot of is solving simultaneous equations.

Example 1.3 Demand and Supply Suppose the supply and demand functions for a good are
(supply) qs a a p, a 0
1 2 2
(demand) qd a a pa r, a 0, a 0
3 4 5 4 5
where qs is quantity supplied, qd is quantity demanded, p is the price of the good, and r is the price of
a substitute good. Suppose that the equilibrium price p*, and quantity bought and sold q*, is the level at
which the market clears: qd qs(q*). Then p* satisfies
a a p*  a a p*a r,
1 2 3 4 5
a a a
which gives p*  3 1  5 r.
a a a a
2 4 2 4
Substituting into either the demand or supply equation (supply is easier) gives
 a a a  a a aa   a a 
q*  a  a  3 1  5 r   2 3 1 4    2 5 r
1 2
a2 a4 a2 a4   a2 a4  a2 a4 
In this model, we are explaining the behavior of the price and quantity sold of a good: p and q are the
endogenous variables. The behavior of these two goods are explained in relation to the price of the
substitute good, r, which we take as exogenous; we do not attempt to explain its behavior. An increase in
r results in both the equilibrium price and the equilibrium quantity (since a and a are both positive).
2 5

The ‘constants’ a , a , a , a , a are called parameters. There is actually little to distinguish a
1 2 3 4 5
parameter from an exogenous variable. We often also ask how equilibrium p and q change when one of
the parameters change. For instance, we a increases, then both p* and q* become more sensitive to
5
changes in r.

Economists are also often interested in how consumers and producers are affected when the parameters of
the model change. For instance, the area between the demand curve and the line p p* from q0 to
qq* is a measure of consumer surplus, whereas the area between the supply curve and the line p p*
from q0 to qq* is producer surplus. Computing areas between curves involves computing integrals.

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The following is another example of an analysis using simultaneous equations.

Example 1.4 Let C represent consumption and Y total output. Let G represent government
expenditure, and assume a0, and 0b1. Suppose
CabY
Y CG
For a fixed level of government expenditure, the equilibrium value of C and Y are the values where
both equations are satisfied. What are these values?

Here G is exogenous, and C and Y are endogenous. Subsituting CabY into Y CG, we get
Y abYG, so
a G
Y   .
1b 1b

 a G 
Substituting this into CabY gives Cab  , i.e.,
1b 1b
a bG
C   .
1b 1b
Models of this sort (but substantially more sophisticated) might be useful for policy analysis. This
particular model says that a one unit increase in G increases Y by more than one unit, if 0b1, since
dY /dG  1/(1b).

A quick remark: in examples 1.3 and 1.4, the expressions involve slightly involved expressions, but the
mathematical manipulations there are exactly the same as in the following simple problem: what values of
x and y solve the two equations
2xy3
x3y1 Ans: x8/5 and y1/5
Expressions in economics (and all applied mathematical sciences) tend to be more complicated than we
are accustomed to in mathematics classes. It is useful to bear in mind that the mathematical principles
involved are often no deeper than what you already know.

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The topics covered in this course are those that are used in the above examples and their extensions to
multivariable situations, and in statistics/econometrics:
- differential and integral calculus (univariate and multivariate)
- optimization (univariate and multivariate, constrained and unconstrained)
- matrix algebra.

In covering the topics listed in course, there will be some emphasis on helping the student develop a
deeper understanding of the mathematical concepts involved. This is harder than it sounds. All of you
have studied quite a bit of calculus in terms of how to compute derivatives, but many of you have learnt
this mechanically, or learnt about differentiation and derivatives at a very superficial “heuristic” level.
Getting to the foundations of calculus will require ‘unlearning’, and then ‘re-learning’, which for many of
you will be challenging. It is important to go through the process, however. More advanced work in
economics and econometrics use mathematics that go well beyond what we cover. To develop the ability
to acquire more advanced material, on your own if necessary, will require a proper foundational
understanding of calculus.

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Exercises
1. Take example 1.1
(i) Plot the firm’s cost function. Describe the important features of the function in your own words.
(ii) The firm’s marginal revenue is the derivative of its revenue function, its marginal cost is the
derivative of its cost function. Plot in a single diagram, the firm’s marginal cost curve, and it’s marginal
revenue curve for different values of p (say, p4,8,12, 16). Explain intuitively why it is optimal for
the firm to be producing at the intersection of the marginal revenue and the marginal cost curves. If p
increases, will the firm produce more, or less?
(iii) Plot the firm’s profit function for the same values of p as in part (ii). As p increases, does the
firm’s optimized profit levels increase, or decrease? Plot the firm’s maximized profit levels against p;

2. In example 1.1, suppose the firm’s cost function is
x3
C(x)  2x2 4x2.
3
(i) Plot the firm’s marginal cost curve.
(ii) Show that for 0 p4, that the firm’s MC = MR at two points, x2 p and x2 p .
Explain intuitively why the firm’s profit is maximized at x2 p , but not at x2 p.
(iii) Show that the firm’s MC curve cuts the MR curve from above at x2 p, whereas it cuts the
MR curve from below at x2 p .

3. Plot the two equations CabY and Y CG on the C -Y plane; use arbitrary values for
a0 and b(0,1), and plot three different versions of the second equation, corresponding to three
different values of G.

4. In example 1.3, suppose the supply and demand curves are
(supply) qs 3p,
(demand) qd 102pr,
Suppose r2. Plot the supply and demand functions with the horizontal axis representing qs and qd ,
and the vertical axis representing p.
(i) The equilibrium price is occurs when qs qd . What is the equilibrium price?
(ii) What happens to the demand function when r increases from r2to r3? From r3 to
r6? Does anything happen to the supply curve? What happens to the equilibrium price?

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Intermediate Mathematics for Economics Anthony Tay

Section 5 Functions of One Variable

You have already worked with equations such as
yax2 bxc
that relate a ‘dependent variable’ y to an ‘independent variable’ x. In such cases, we often say ‘ y is
a function of x’ and write something like y f(x), where in this example, f(x)ax2 bxc. The
main objective of this section is to give you a deeper understanding of a function as a mathematical
concept, to get familiarized with the related terminology and the f(.) notation, and to review the
properties of some important functions.

Before starting, we should go through some notation: I will assume you are familiar with the usual set
theory notation, and their meanings: ,,,, etc.. “” means “implies”. I will use the symbol
to represent the set of all natural numbers {1, 2, 3, …}. I will use the symbol to represent the set of
all real numbers. Thus  . Some of you may have encountered complex numbers before. The set
of complex numbers will be denoted by , thus  . We will not be using complex numbers in
this course.

To represent the set of all pairs of real numbers, I will use the symbol 2. The symbol 3 will
represent the set of all triplets of real numbers.

The symbol “(a,b)” where a and b are real numbers, will sometimes be used to represent the set of
all real numbers between a and b (noninclusive). Thus (a,b) . To say that x0, we may write
x(0,). However, sometimes the symbol “(a,b)” is also used to represent the pair of numbers a
and b. In this context, (a,b) 2. Which interpretation is used depends on context. Please be aware
of this.

5.1 Definition and Notation for Functions of One Variable

If x and y are variables that take numerical values, and if the value of y depends on what the value
of x is, then we say that “y is a function of x”, and we may write y f(x) where f(.) indicates
the form of the association. Often, these can be represented by an equation, e.g., if the association
between
yabx for some constants a and b,
then y f(x)abx. The notation f(x) in this example represents the expression abx.
Occasionally you will find authors who use the variable name and the function name synonymously,
and write something like y y(x).

There is more that must be said to fully describe the relationship between y and x. For what values
of x are we applying this association? If x represents something that cannot take negative values,
then we should specify yabx, x0. If no such restriction needs to applied, we might simply say
yabx, x (that is, “for all real numbers x”).

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Anthony Tay
Example 5.1.1 If A represents the area of a circle, and r its radius, then Ar2, r0.

Note that we could have defined Ar2 for all r . Mathematically this is fine. We could even
have defined Ar2 for all r , i.e., for complex numbers. But for the given application of the
function – to relate the radius of a circle to its area – it makes sense to restrict r to positive real
numbers.

Definition The values of x over which a function f(x) is to be defined is called the domain of
the function. The values taken by the function over its domain is its range.

Example 5.1.1 continued The range of the function Ar2, r0 is (0,).

Remember, the domain of a function is an important part of the definition of a function. The
appropriate domain to use depends on a number of things such as the regions over which x and y
make sense given what the function is being used for. Sometimes the domain of a function also has to
be limited because the function cannot be defined mathematically over certain regions:

Example 5.1.2 f(x)(x1)/(x1), then f(x) cannot be defined for x1 because we cannot
divide by zero. The largest possible domain (limiting ourselves to real numbers) is the set of all real
numbers x1. We will simply say
f(x)(x1)/(x1), x1
If, in addition, our application requires that f(x)0, then the domain must be further restricted to
x(,1)(1,). If negative values of x have no meaning, then we should specify x(1,).

Example 5.1.3 What is the largest possible domain of f(x) x ? If we require f(x) to take
real values only, then this is x0. However, if we allow f(x) to take complex values, then this
restriction is unnecessary, e.g., 1i is perfectly valid.

In this course, we will restrict both the domain and range to real numbers.

The full, general, definition of a function is:

Definition. A function f from a set A to another set B is a rule that associates each element of A
with an element of the set B.

We write f :AB. “ f(x)” denotes that element of B which is associated with the element xA.
The set A is called the domain of the function f .

Example 5.1.4 Sequences are merely functions whose domain are the natural numbers
f :  .

In the notation f :AB, the set B is not necessarily the range, any set encompassing the range will
be fine.

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