maximum and minimum in economic models calculus and optimisation maximum and minimum in economics
TRANSCRIPT
Maximum and minimum in economic models
Calculus and OptimisationMaximum and minimum in
economics
Maximum and minimum in economics
We now move to the 2nd 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.
Maximum and minimum in economics
Maximum: The concept of ‘utility’
Minimum: production decisions
Calculus and optimisation tools
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
1 2, ,..., nsatisfaction f q q q
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
Maximum Calculus gives methods for finding this value.
Why is it possible to build such a function?
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.
satisfaction u cake
Maximum: The concept of ‘utility’
0
2
4
6
8
10
12
14
16
18
20
0 1 2 3 4 5 6
Extra U = 10
Extra U =5
Extra U =3
Extra U =1Extra U = -2
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!
Maximum and minimum in economics
Maximum: The concept of ‘utility’
Minimum: production decisions
Calculus and optimisation tools
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€)
0 0
1 100
2 140
3 150
4 155
5 158
6 165
7 175
8 190
9 215
10 260
11 345
12 470
13 650
Minimum: production decisions
Number of students (10)
Total cost (K€)
Cost /10 students
(K€)
0 0 -
1 100 100.0
2 140 70.0
3 150 50.0
4 155 38.8
5 158 31.6
6 165 27.5
7 175 25.0
8 190 23.8
9 215 23.9
10 260 26.0
11 345 31.4
12 470 39.2
13 650 50.0
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
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€)
0 0 - -
1 100 100.0 -
2 140 70.0 -30
3 150 50.0 -20
4 155 38.8 -11.2
5 158 31.6 -7.2
6 165 27.5 -4.1
7 175 25.0 -2.5
8 190 23.8 -1.2
9 215 23.9 0.1
10 260 26.0 2.1
11 345 31.4 5.4
12 470 39.2 7.8
13 650 50.0 10.8
What can we observe?
Why is that the case ?
Minimum: production decisions
0
20
40
60
80
100
120
140
160
180
200
1 2 3 4 5 6 7 8 9 10 11 12 13
Number of students (*10)
Cost per 10 students
Minimum point of average production costs
Maximum and minimum in economics
Maximum: The concept of ‘utility’
Minimum: production decisions
Calculus and optimisation tools
Calculus and optimisation tools
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
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
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