Maths refresher course
for Economics
Part 1:
Why economics contains so much maths
Part 1
Why economics contains so much
maths
The Scientific approach
What is a model ?
Why economics contains so much maths

Economics tries to understand the behaviour
of agents




Resources are limited, therefore agents have
to make choices.
These depend on the incentives faced by the
agent
Because agents are different, they can benefit
from exchange
Producers and consumers therefore meet on
markets that ensure an efficient use of these
resources
Why economics contains so much maths

The aim of economics if to answer the
following questions:





What to consume ? How much ?
What to produce ? (Which good ?)
How to produce it ? (Which technology ?)
Why are some countries richer than others?
Why do some countries have high
unemployment ? High inflation?
Why economics contains so much maths

In order to answer these questions and
understand how agents make their
decisions, economics models:



The decision making process of the agent
The flows of goods and services in the economy
We use models because it is impossible to
understand directly the depth and
complexity of human behaviour

This is what we’ll examine in the next 2 sections,
and look at how models are used in science
Why economics contains so much maths

In other words, we use maths because we’re
not intelligent enough to do without !!

Example of Sir Arthur Lewis (Nobel
prize 1979) on Dani Rodriks’ Blog
 http://rodrik.typepad.com/dani_rodriks_weblog/2007/09/whywe-use-math.html

There is an apparent paradox:
 « Maths are complicated »
 In fact, it’s a simplification !!
Why economics contains so much maths

Mathematics allows us:



To use a symbolic notation for all the variables in
a given problem.
To develop notations and rules that define
logical relations (logical operations are
“codified”)
As a result one can carry out a complex
series of logical operations without making
mistakes, forgetting variables, etc.
Why economics contains so much maths

Which types of mathematics do
economists use ?

Algebra and Calculus, mainly analysing
the properties of functions
 Part 2

Statistics
 Which you will see with Evens Salies on Friday
Part 1
Why economics contains so much
maths
The Scientific approach
What is a model ?
The Scientific approach

What makes something “Scientific” ?



A lab coat ?
Laboratory
equipment?
The capacity to run
experiments ?
The Scientific approach

The central objective of science : explain the
phenomena that we observe

In other words we try and understand the
causal links of a problem


Understanding a problem is finding its cause !
Practical aspect: How do we do this in a
complex world?


Several different explanations are possible
They can also interact!
The Scientific approach


You need a systematic method to evaluate all the
possible explanations, and eliminate those that are
not valid.
In particular you need to be able to impose a
« ceteris paribus » condition (all other things being
equal)


Example of the thermometer and temperature
Therefore you need to be able to create a simplified
representation of reality to be able to isolate these
effects!

We will see that this is where the maths come into
play
The Scientific approach

The scientific method:
Reality
Theory
You observe a
phenomenon
You identify
variables that
can explain it
No: you start
over again!
You write a
model (simplify
the problem)
Do the
predictions fit
the data?
You get a set of
predictions
Yes: you have a
valid theory
Part 1
Why economics contains so much
maths
The Scientific approach
What is a model ?
What is a model?

What is a model ?



“A simplified representation of reality”
In other words, a representation which removes
the unnecessary complexity of reality to focus on
the key mechanisms of interest
“A model’s power stems from the
elimination of irrelevant detail, which allows
the economist to focus on the essential
features of economic reality.” (Varian p2)
What is a model?


It is important to understand that models are
central to how humans perceive reality
Human understanding of the world (not just in
economics !) comes from understanding
simplified versions of a complex world.


The role of the scientific process is to separate
good and valid simplifications from invalid ones.
“One must simplify to the maximum, but no
more” Albert Einstein
What is a model?

Illustration of a general, simple “model”
You are in Nice
 You don’t know your way around, and you
get lost.
 You ask a passerby where you are
 This person gives you two possible
answers as to your location


Which is the more useful (i.e. instructive
model) ?
What is a model?
You are here
What is a model?
You are here
What is a model?

Modelling in economics

We assume a simplified agent and
environment
 even if you know that this is unrealistic !!
We try and understand how things work in
this ideal situation.
 Then we try and relax the simplifying
assumptions one by one and see how the
mechanisms change

What is a model?

The simplified agent used is typically called
the “Homo œconomicus”




Has complete knowledge of his objectives
(preferences or production quantities)
Has complete knowledge of the conditions on all
the markets (perfect information)
Has a very large “computational capacity” to work
out all the possible alternatives and their
payoffs.
These simplifications can be relaxed
Maths refresher course
for Economics
Part 2:
Basic Calculus
Part 2
What is a function ?
Calculus and optimisation
The derivative of a function
Constrained maximisation
What is a function ?


“ Many undergraduate majors in
economics are students who should know
calculus but don’t – at least not very well”
(Varian, preface)
So before starting on the models and the
theory, it is important to understand the
components of models : functions
What is a function ?

A function is a relation between:



A given variable that we are trying to explain
A set of explanatory variables
A variable is a quantity:



That varies with time,
That can be measured on a given scale
Examples: Temperature, pressure, income,
wealth, age, height
What is a function ?


The relation between two variables X and Y
can be:
Positive (or increasing):

Variations happen in the same direction
X


Y
X

Y
Negative (or decreasing):

Variations happen in opposite directions
X

Y
X

Y
What is a function ?


Practical Example : Let’s use a road safety
example
You are asked by the Ministry of the Interior
to identify a cost-effective way of reducing
the number of deaths on the road due to car
accidents.



Which are the important variables?
What measures are associated ?
Is the direction of the relation?
What is a function ?

The same function can have different “faces”


The same relation between variables can be
expressed in different ways
1: “Literary” representation


This is the one from the previous slide, and
involves just mentioning the variables that enter
the function
“The number of accidents is a positive function
of average rainfall, the speed of driving and the
quantity of alcohol consumed.”
What is a function ?

2: Symbolic representation

A bit more “rigorous”, this uses symbols to
represent the relation between variables


a  f  r , s, q 
  
Mathematical symbol meaning “function of”


Where a is the number of accidents, r is rainfall,
s is the speed and q is alcohol consumption.
But... When read out, this just corresponds
to the literary version !!
What is a function ?

3: Algebraic representation

This is the “scary” one, because it involves
“maths” (algebra, actually)
a  10  0.9r  0.5s 2  eq


The problem is that to express a function this
way, you need to know exactly:
 The “functional form” (Linear, quadratic,
exponential)
 The values of the parameters
Finding these is often part of the work of an
economist
What is a function ?

4: Graphical representation

a
(accidents
/year)
Often the most convenient way of representing a
function...
Car accidents as a function of
rainfall
a  f  r , s, q 
  
r
(cm/m2)
What is a function ?


a
(accidents
/year)
... But a diagram can only represent a link
between two variables (a and r here)
If alcohol consumption q increases, then a whole
new curve is needed to describe the relation
Car accidents as a function of rainfall
q1>q
a  f  r , s, q1 
   
a  f  r , s, q 
  
r
(cm/m2)
Part 2
What is a function ?
Calculus and optimisation
The derivative of a function
Constrained maximisation
Calculus and optimisation

The economic approach often models the
decision of an agent as trying to choose the
“best” possible outcome



The highest “satisfaction”, for consumers
The highest profit, for producers
Imagine a function f that gives satisfaction
(or profits) as a function of all the quantities
of goods consumed (or produced).
satisfaction  f  q1 , q2 ,..., qn 
Calculus and optimisation

In terms of modelling, finding the “best
choice” is effectively like trying to find the
values of the quantities of goods for which
function f has a maximum
satisfaction
Maximum
Graphically, that’s easy!
But generally, how do
you find this maximum ?
q
Calculus and optimisation

For both examples, the optimum is the
point where the function is neither
increasing nor decreasing:



Satisfaction no longer increases but is not yet
falling.
Road deaths are no longer falling but aren’t yet
increasing.
This is basically how you find maxima and
minima in calculus.

The methods may seem ‘technical’, but the
general idea is simple
Part 2
What is a function ?
Calculus and optimisation
The derivative of a function
Constrained maximisation
The derivative of a function

Imagine that we want to find the maximum of a
particular function of x
y  f  x

Example
y  2 x 2  1

How do we find out at which point it has a
maximum without having to use a graph?

We need to introduce the concept of a derivative
The derivative of a function

We will use the following approach:



We will introduce the concept of a tangent
This will allow us to introduce the concept of
a derivative using the graphical approach,
which is more intuitive.
You are on a given point on a function, and
you want to calculate the slope at that point
The derivative of a function
y
y2 = f(x + h)
y = f(x)
Δy = y2 – y1
y1 = f(x)
Δx = h
x
y f  x  h   f  x 
Slope 

x
h
x+h
x
The derivative of a function
y
Problem: Different points give
different slopes
Which one gives the best
measurement of the slope of the
curve?
x
The derivative of a function
y
Mathematically, the measurement of
the slope gets better as the points get
closer.
The best case occurs for an
infinitesimal variation in x. The
resulting line is called the tangent.
x
The derivative of a function
y
The tangent of a curve is the straight
line that has a single contact point with
the curve, and the two form a zero
angle at that point.
x
The derivative of a function
y
Δy
The slope of the tangent is equal to the
change in y following an infinitesimal
variation in x.
Δx
x
The derivative of a function
y
As we saw, a maximum is reached
when the slope of the tangent is equal
to zero.
x
The derivative of a function
y
Slope = 0
B
A
C
Slope > 0
1
Slope < 0
1
-s
s
x
The derivative of a function

With continuous functions, each curve is made up
of an infinite number of points



This is not the case for discrete curves


This is because points on the curve are separated by
infinitely small steps (infinitesimals)
There is an infinite number of corresponding
tangents
There is only a finite number of points (no
infinitesimals)
What we need is a recipe for calculating the slope of
a function for any given point on it

Luckily, even though there are an infinite number of
points, this allows us to derive such “recipes”.
The derivative of a function
y
y2 = f(x + h)
y = f(x)
Δy = y2 – y1
y1 = f(x)
Δx = h
x
y f  x  h   f  x 
Slope 

x
h
x+h
x
The derivative of a function

General rule:

Let f be a continuous function defined at point.
The derivative of the function f’(x) is the
following limit of function f at point x :
f ( x  h)  f ( x )
f ( x)  lim
h 0
h

In other words, it is literally the calculation of
the slope as the size of the step become
infinitely small.
 This is done using limits, but we will use specific rules
that don’t require calculating this limit every time
The derivative of a function
f (x)
f (x)
Example
k (constant)
0
f(x) = 3  f’(x)=0
x
1
f(x) = 3x  f’(x)=3
n x n 1
1
2 x
f(x) = 5x²  f’(x)=10x
x
n
x
The derivative of a function

This means that in order to find the extreme point
of a function of a single variable, we first take the
1st derivative:
y  f  x   4x2  4x  6
dy
 f   x   8x  4
dx

The maximum of f(x) occurs when f’(x)=0

So we set the derivative equal to zero and solve for x.
8x  4  0
1
x
2
Part 2
What is a function ?
Calculus and optimisation
The derivative of a function
Constrained maximisation
Constrained optimisation

What is a constraint? How does it affect the
maximum?




Let’s take an example:
Imagine that money was no issue (You’ve just
won the lottery). What purchase would satisfy
you the most?
Now imagine you only have 50€. What do you
buy?
Constrained optimisation means that you try
and do you best given the aspects of the
situation that cannot be changed
Constrained optimisation

Formally, what is a constraint (i.e. In terms
of mathematics)?

You are trying to maximise/minimise a function
f x, y 

However, there are some limits on x and y: there
are certain values that cannot be exceeded.
g x, y   k

Typical examples in economics
 x and y have to be positive
 x and y represent resources in limited supply
Constrained optimisation
y
We know how to
find the “top of the
hill”
Its the point where
both partial
derivatives are
equal to zero
x
Constrained optimisation
y
Let’s imagine that
we aren’t allowed
to climb to the top
of the hill...
We are restricted to
a road that travels
on the hill (in red)
What is the highest
point you can
reach ?
x
Constrained optimisation
y
Here it is
Starting from the
y-axis, the road is
climbing
Past that point the
road starts going
down
Can you see
something special
about the
constrained
maximum?
x
Constrained optimisation

This example illustrates the general idea:
f x, y 



When looking for a free maximum, you set the
partial derivatives (the slopes) equal to zero and
solve the resulting system
When looking for a constrained maximum, you
set the partial derivatives (the slopes) equal to
the slopes of the constraints, and solve the
resulting system
The Lagrangian method allows us to do this
The Lagrangian method

A constrained maximisation problem is
often presented like this:
max f x, y 

s.t. : g x, y   0

Where:


f(x,y) is the function that you have to maximise
g(x,y) is the set of constraints on x and y (which
has to give zero)
The Lagrangian method

As we saw in the example, the general idea
is to set the partial derivatives of f(x,y) equal
to the partial derivatives of g(x,y)


We then solve the resulting system to find the x,y
that maximise f(x,y) whilst still satisfying g(x,y)
This is done by using the Lagrangian equation,
which converts a constrained maximisation
problem into a free maximisation problem
Lx, y,    f x, y   g x, y 
The Lagrangian method

Let’s look at the Lagrangian a bit closer
Lx, y,    f x, y   g x, y 

It integrates the constraint to the original
function by multiplying it with λ



This is called the “lagrangian multiplier”
It translates the units used in the constraint g
into the units used in the function f
But we don’t worry about its value: when the
constraint is satisfied, it disappears!
The Lagrangian method

Let’s use an example:
max x 2  y 2

s.t. : x  2 y  10

The lagrangian equation for this problem is:
Lx, y,    x 2  y 2   x  2 y  10
The Lagrangian method
Lx, y,    x  y   x  2 y  10
2

2
Taking partial derivatives:
Lx, y,  
 2x  
x
Lx, y,  
 2 y  2
y
Lx, y,  
 x  2 y  10

The Lagrangian method

Setting the partial derivatives equal to zero:
2 x  

y  
 x  2 y  10

2 x    0

 2 y  2  0
 x  2 y  10  0

x  2

y  4
  4
