Functions, variables and models What is a function? What is a variable?

Functions, variables and models Functions and variables Using them in economic models

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 the road safety example you had to prepare for today 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)

Functions, variables and models Functions and variables Using functions in economic models

1. Combine functions to create a model The central variable of interest is described by a function We already have an example with the number of car accidents Next step: Each explanatory variable is in turn described by another function We then get a system of equations Example: last week’s model of global warming

Using functions in economic models The interaction of the variable brings a better understanding of the overall phenomenon However: not all variables are described by functions You need to simplify (goal of a model) You choose to explain only the important variables These choices can then be checked by experiment/ observation.

Using functions in economic models 2. A second approach is maximisation or minimisation This can be combined with a system of equations Number of driving licences revoked per year Minimum In our example: the minister is going to try and minimise the number of accidents per year. a (accidents /year)

Using functions in economic models 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).

Using functions in economic models 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 ?