maths refresher course for economics part 1: why economics contains so much maths

Maths refresher course for Economics

Part 1:

Why economics contains so much maths

Part 1


The Scientific approach

What is a model ?


Economics tries to understand the behaviour of agentsResources are limited, therefore agents

have to make choices.These depend on the incentives faced by the

agentBecause agents are different, they can

benefit from exchangeProducers and consumers therefore meet on

markets that ensure an efficient use of these resources


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?


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


Example of Sir Arthur Lewis (Nobel prize 1979) on Dani Rodriks’ Blog

http://rodrik.typepad.com/dani_rodriks_weblog/2007/09/why-we-use-math.html

There is an apparent paradox: « Maths are complicated » In fact, it’s a simplification !!

In other words, we use maths because we’re not intelligent enough to do without !!


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.


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



What is a model ?


What makes something “Scientific” ?

A lab coat ?Laboratory

equipment?The capacity to run

experiments ?


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!


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 method:Reality

Theory

You observe a phenomenon

You identify variables that can explain it

You write a model

(simplify the problem)

You get a set of predictions

Do the predictions fit

the data?

Yes: you have a valid theory

No: you start over again!

Part 1



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 NiceYou don’t know your way around, and you

get lost.You ask a passerby where you areThis 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?

Modelling in economicsWe 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


“ 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


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


The relation between two variables X and Y can be:

Positive (or increasing):Variations happen in the same direction

Negative (or decreasing):Variations happen in opposite directions

X Y

X Y

X Y

X Y


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?


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.”


2: Symbolic representation A bit more “rigorous”, this uses symbols to

represent the relation between variables

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 !!

, ,a f r s q


3: Algebraic representation This is the “scary” one, because it involves

“maths” (algebra, actually)

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

210 0.9 0.5 qa r es


4: Graphical representation Often the most convenient way of representing

a function...

a (accidents

/year)

r (cm/m2)

, ,a f r s q

Car accidents as a function of rainfall


... 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

r (cm/m2)

1, ,a f r s q

Car accidents as a function of rainfall

q1>q

, ,a f r s q

a (accidents

/year)

Part 2






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). 1 2, ,..., nsatisfaction f q q q


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

q

Maximum Graphically, that’s easy!

But generally, how do you find this maximum ?


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






Imagine that we want to find the maximum of a particular function of x

Example

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

y f x

22 1y x


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


y

x

Δy = y2 – y1

Δx = h

f x h f xy

Slopex h

y1 = f(x)

x x + h

y2 = f(x + h)y = f(x)


y

x

Problem: Different points give different slopes

Which one gives the best measurement of the slope of the curve?


y

x

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.


y

x

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.


y

x

The slope of the tangent is equal to the change in y following an infinitesimal variation in x.

Δy

Δx


y

x

As we saw, a maximum is reached when the slope of the tangent is equal

to zero.


y

x

B

A

s

1

C

1

-s

Slope > 0 Slope < 0

Slope = 0


With continuous functions, each curve is made up of an infinite number of points This is because points on the curve are separated

by infinitely small steps (infinitesimals) There is an infinite number of corresponding

tangents This is not the case for discrete curves

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”.


y

x

Δy = y2 – y1

Δx = h

f x h f xy

Slopex h

y1 = f(x)

x x + h

y2 = f(x + h)y = f(x)

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 :

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


0

( ) ( )( ) lim

h

f x h f xf x

h


Example

k (constant) 0 f(x) = 3 f’(x)=0

x 1 f(x) = 3x f’(x)=3

f(x) = 5x² f’(x)=10x

f (x)

nx

x

f (x)

n 1n x

1

2 x


This means that in order to find the extreme point of a function of a single variable, we first take the 1st derivative:

The maximum of f(x) occurs when f’(x)=0

So we set the derivative equal to zero and solve for x.

24 4 6y f x x x

8 4 0x

8 4dy

f x xdx

1

2x

Part 2





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


Formally, what is a constraint (i.e. In terms of mathematics)? You are trying to maximise/minimise a function

However, there are some limits on x and y: there are certain values that cannot be exceeded.

Typical examples in economics x and y have to be positive x and y represent resources in limited supply

yxf ,

kyxg ,


y

x

We know how to find the “top of the hill”

Its the point where both partial derivatives are equal to zero


y

x

Let’s imagine that we aren’t allowed to climb to the top of the hill...

What is the highest point you can reach ?

We are restricted to a road that travels on the hill (in red)


y

x

Here it is

Can you see something special about the constrained maximum?

Starting from the y-axis, the road is climbing

Past that point the road starts going down


This example illustrates the general idea:

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

yxf ,

The Lagrangian method

A constrained maximisation problem is often presented like this:

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)

0,:s.t.

,max

yxg

yxf


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 yxgyxfyxL ,,,,


Let’s look at the Lagrangian a bit closer

It integrates the constraint to the original function by multiplying it with λThis is called the “lagrangian multiplier” It translates the units used in the constraint

g into the units used in the function fBut we don’t worry about its value: when

the constraint is satisfied, it disappears!

yxgyxfyxL ,,,,


Let’s use an example:

The lagrangian equation for this problem is:

102:s.t.

max 22

yx

yx

102,, 22 yxyxyxL


Taking partial derivatives:

102,, 22 yxyxyxL

102

,,

22,,

2,,

yxyxL

yy

yxL

xx

yxL


Setting the partial derivatives equal to zero:

0102

022

02

yx

y

x

102

2

yx

y

x

4

4

2

y

x

maths refresher course for economics part 1: why economics contains so much maths

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