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

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 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 Why economics contains so much maths The Scientific approach 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 Negative (or decreasing): Variations happen in opposite directions

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 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) 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 Often the most convenient way of representing a function... a (accidents /year) r (cm/m 2 ) Car accidents as a function of rainfall

What is a function ?... 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/m 2 ) Car accidents as a function of rainfall q 1 >q a (accidents /year)

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).

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 q MaximumGraphically, that’s easy! But generally, how do you find this maximum ?

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 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

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 x Δy = y 2 – y 1 Δx = h y 1 = f(x) xx + h y 2 = f(x + h) y = f(x)

The derivative of a function y x Problem: Different points give different slopes Which one gives the best measurement of the slope of the curve?

The derivative of a function 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.

The derivative of a function 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.

The derivative of a function y x The slope of the tangent is equal to the change in y following an infinitesimal variation in x. Δy Δx

The derivative of a function y x As we saw, a maximum is reached when the slope of the tangent is equal to zero.

The derivative of a function y x B A s 1 C 1 -s Slope > 0 Slope < 0 Slope = 0

The derivative of a function 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”.

The derivative of a function y x Δy = y 2 – y 1 Δx = h y 1 = f(x) xx + h y 2 = 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 The derivative of a function

Example k (constant)0 f(x) = 3  f ’ (x)=0 x1 f(x) = 3x  f ’ (x)=3 f(x) = 5x ²  f ’ (x)=10x

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: The maximum of f(x) occurs when f’(x)=0 So we set the derivative equal to zero and solve for x.

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 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

Constrained optimisation y x We know how to find the “top of the hill” Its the point where both partial derivatives are equal to zero

Constrained optimisation 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)

Constrained optimisation 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

Constrained optimisation 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

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)

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

The Lagrangian method Let’s look at the Lagrangian a bit closer 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: The lagrangian equation for this problem is:

The Lagrangian method Taking partial derivatives:

The Lagrangian method Setting the partial derivatives equal to zero: