Mathematics for Economists
Coordinator: dr. Yolanda Grift
•
•
•
•
•
•
First course in the first year
Fundamentals you need for economics
Entrance test in the Introduction week
Lectures and tutorials
Digital math environment for practicing
Lot of practice
2
Why do you need mathematics?
Economists specify, analyze and quantify relationships among
economic variables:
-the relationship between prices and quantities
-or between national income and consumption.
Economists use verbal, graphical, mathematical and statistical
tools.
Mathematics will focus on the third tool. Together with verbal ability,
an economist should possess all these tools.
3
What is the aim of Mathematics for Economists?
The central issue in Mathematics is constrained optimisation:
Utility maximising behaviour of consumers
• Subject to a budget constraint
Minimising cost by producers
• Subject to an output constraint
Maximising welfare by the government
• Subject to a cost constraint
4
Constrained optimisation: short introduction
On the next slides you will find:
-
The founding father, Joseph-Louis Lagrange
A graphical representation
The general mathematical representation and solution
An example
5
Joseph-Louis Lagrange
Turijn, 25 januari 1736 – Parijs, 10 april 1813
Tomb in the crypt of the Pantheon
Name on the Eifel Tower (72 in total)
Mathematician, worked with Euler
1790s: new standard units of measurement:
-meter and kilogram,
-and decimal subdivision
Method of Constrained optimisation
known as Lagrange multiplier method
6
Figure 1. Find x and y to maximize f(x, y) subject
to a constraint (shown in red) g(x, y) = c.
7
Figure 2: Contour map of Figure 1. The red line shows the constraint g(x, y) = c. The
blue lines are contours of f(x, y). The point where the red line tangentially touches a
blue contour is our solution. Since d1 > d2, the solution is a maximization of f(x, y).
8
Constraint optimisation: Lagrange Multiplier
Maximise
subject to a constraint
Lagrange function:
Maximise
𝑓𝑓 = 𝑓𝑓(𝑥𝑥, 𝑦𝑦)
𝑔𝑔 𝑥𝑥, 𝑦𝑦 = 𝑐𝑐
𝐺𝐺 𝑥𝑥, 𝑦𝑦, 𝜆𝜆 = 𝑓𝑓 𝑥𝑥, 𝑦𝑦 − 𝜆𝜆 𝑐𝑐 − 𝑔𝑔 𝑥𝑥, 𝑦𝑦
𝜕𝜕𝜕𝜕
=0
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕 �⇒𝑦𝑦=𝑦𝑦
=0
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
=0
𝜕𝜕𝜕𝜕
𝑥𝑥
∗
� ⇒ �𝑥𝑥
𝑦𝑦 ∗
9
Constraint optimisation: an example
0.6
𝐾𝐾
1
0.4 −2
.
𝐿𝐿
A cost minimising firm has the production function 𝑄𝑄 𝐾𝐾, 𝐿𝐿 = 100
+
The prices of
capital and labour can be regarded as fixed and equal to 27 and 8 respectively. Obtain
expressions for K and L.
Minimise 27𝐾𝐾 + 8𝐿𝐿 s.t. 𝑄𝑄 𝐾𝐾, 𝐿𝐿 = 100
0.6
𝐾𝐾
+
1
0.4 −2
𝐿𝐿
Lagrange function: G 𝑥𝑥, 𝑦𝑦, 𝜆𝜆 = 27𝐾𝐾 + 8𝐿𝐿 − 𝜆𝜆 𝑄𝑄 − 100
𝜕𝜕𝜕𝜕
=0
𝜕𝜕𝐾𝐾
𝜕𝜕𝜕𝜕
=0
𝜕𝜕𝐿𝐿
3
𝜕𝜕𝜕𝜕 𝐾𝐾,𝐿𝐿,𝜆𝜆
0.6 0.4 −2
=27+𝜆𝜆 50
+
𝐾𝐾 𝐿𝐿
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
=0
𝜕𝜕𝜕𝜕
3
𝜕𝜕𝜕𝜕 𝐾𝐾,𝐿𝐿,𝜆𝜆
0.6 0.4 −2
=8+𝜆𝜆 50
+
𝐾𝐾 𝐿𝐿
𝜕𝜕𝐿𝐿
−0.6 𝐾𝐾 −2 =0
−0.4 𝐿𝐿 −2 =0
� ⇒𝐿𝐿 = 1.5𝐾𝐾
𝜕𝜕𝜕𝜕 𝐾𝐾, 𝐿𝐿, 𝜆𝜆
0.6 0.4
= 𝑄𝑄 − 100 𝐾𝐾 + 𝐿𝐿
𝜕𝜕𝜕𝜕
1
−
2
1
0.6
0.4 −2
+
𝐾𝐾
𝐿𝐿
=0
1.3𝑄𝑄 2
15,000
⇒
1.3𝑄𝑄 2
𝐿𝐿∗ = 1.5
15,000
𝐾𝐾 ∗ =
10
Learning objectives Mathematics for Economists
At the end of the course the student is able to:
•
•
•
•
•
•
•
understand, control and apply elementary notions of mathematics;
use mathematics to specify, analyze and quantify relationships
among economic variables;
recognize the economic meaning from mathematical notions and
models;
describe clear and structured solutions of mathematical problems
solve unconstrained optimization problems of multivariate functions
solve constrained optimization problems of multivariate functions
(the Lagrange multiplier method)
has introductory knowledge of growth models, dynamics, matrices
and integration.
11
Questions?
[email protected]