1. Maximum and minimum in economic models Calculus and Optimisation Maximum and minimum in economics

2. We now move to the 2 nd part of the course, which will focus on Calculus Calculus is the analysis of the properties of functions We will be re-using the algebra concepts The detail on the various methods will be covered in the coming weeks This week, we look at an introduction to how the concept of maximum and minimum are used in economics.

3. Maximum and minimum in economics Maximum: The concept of ‘utility’ Minimum: production decisions Calculus and optimisation tools

4. Maximum: The concept of ‘utility’ As was mentioned in the first few weeks, finding the best choice of a consumer means choosing the “best” outcome In other words, the satisfaction of consumers We also imagined a function f that gives satisfaction as a function of all the quantities of goods consumed

5. Maximum: The concept of ‘utility’ 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 MaximumCalculus gives methods for finding this value. Why is it possible to build such a function?

6. Maximum: The concept of ‘utility’ Lets use a practical example: Consumption of a single good Chocolate cake for example The function will be called the “utility” function This is the traditional name in economics for the satisfaction of an agent.

7. Maximum: The concept of ‘utility’ Extra U = 10 Extra U =5 Extra U =3 Extra U =1 Extra U = -2

8. Maximum: The concept of ‘utility’ Now of course, in reality, there is no function that can put a number on satisfaction But agents are able to say when their satisfaction increases or falls. This means that we can identify points where utility is maximum Methods in calculus allow us to find the maximum, even if the function itself is not defined!

9. Maximum and minimum in economics Maximum: The concept of ‘utility’ Minimum: production decisions Calculus and optimisation tools

10. Minimum: production decisions Imagine that the table gives the production costs of SciencesPo, given the size of the student population In order to plan for, the budget the director wants to have an idea of the cost per student of providing the lectures Lets work it out Number of students (  10) Total production costs (K€) 00 1100 2140 3150 4155 5158 6165 7175 8190 9215 10260 11345 12470 13650

11. Minimum: production decisions Number of students (  10) Total cost (K€) Cost /10 students (K€) 00- 1100100.0 214070.0 315050.0 415538.8 515831.6 616527.5 717525.0 819023.8 921523.9 1026026.0 1134531.4 1247039.2 1365050.0 What can we notice ? Why is this the case Additionally, the director would like to have an idea of the change in the costs per student when the student population increases

12. Minimum: production decisions Lets draw the cost per student Number of students (  10) Total cost (K€) Cost / 10 students (K€) Change in cost per student (K€) 00-- 1100100.0- 214070.0-30 315050.0-20 415538.8-11.2 515831.6-7.2 616527.5-4.1 717525.0-2.5 819023.8-1.2 921523.90.1 1026026.02.1 1134531.45.4 1247039.27.8 1365050.010.8 What can we observe? Why is that the case ?

13. Minimum: production decisions Cost per 10 students Minimum point of average production costs

14. Maximum and minimum in economics Maximum: The concept of ‘utility’ Minimum: production decisions Calculus and optimisation tools

15. For both cases, the maximum is the point where the function is neither increasing nor decreasing: Utility no longer increases but is not yet falling. Average costs are no longer falling but aren’t yet increasing. This is basically how you find maxima and minima in calculus. The methods may seem more ‘technical’, but the general stays the same

16. Calculus and optimisation tools First step: working on continuous functions The examples we have seen are discrete In other words, the functions are not smooth, so the tools of calculus cannot apply Second step : Partial/total derivatives Economics often uses functions of several variables, so we will have to take that into account when we look for the maximum

17. Calculus and optimisation tools Third step: constrained optimisation Today’s example shows cases of “free optimisation” We find a maximum/minimum regardless of anything else But in real life, we often have constraints to take into account Example: consumers have budgets they must respect So we have to take the constraint into account when looking for the minimum