Mathematics in economics
Lecture 4
Mgr. Jiří Mazurek, Ph.D.
Mathematics in Economics/PMAT

Lecture 4
Differential calculus of two real variables




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Many economic functions contain more then  one variable.
For example, Cobb-Douglas function includes labour L and
capital K as well as the technological parameter A:
We will limit ourselved to functions of two variables. A graph
of a function of two real variables is a plane in 3D space,
See the next slide.

Lecture 4
A graph of Cobb-Douglas function




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Cobb_douglas

Lecture 4
The domain of a function of two real variables




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By the domain of a function              we mean all ordered pairs              for which the
function f is defined.
,
The domain is usually a subspace of Cartesian coordinate
system (CCS).
In the problems solved later, we will depict the domain by
shadowing an appropriate part of CCS.

Lecture 4
Finding a domain of a function – Problem 1




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Find the domain of the function                               .
Solution:                       ,                  (1)
Now we change the inequality into the equality:
This is the quation of a linear function (a line), which we draw
in CCS (see the picture on the next slied). Then, we must find
whether the domain lies on the left or right hand side to the
line via one suitable point. We know that point [0,0] is on the left.
We substitute this point into (1). Because (1) is untrue, the
domain is on the right side of the line.

Lecture 4
Problem 1 – cont.




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graf 16

Lecture 4
Finding a domain of a function – Problem 2




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Find the domain of the function:                           .
Solution:
From the square root we get:                            (1).
Transforming the inequality into equality yields:
This is the equation of the circle with a radius r = 3
and a centre [0,0]. Again, one point will decide whether the
domain is inside or outside of the circle. We can use [0,0]
again:                       is untrue, so the domain is outside
of the circle, including the circle.

Lecture 4
Finding a domain of a function – Problem 2




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Graf 17b

Lecture 4
A derivative of a function of two real variables




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Let f (x,y) be a function of two real varibles. Then, this function
can be differentiated with regard to x and/or y. These derivatives
are called partial derivatives and are denoted as follows:


Definition:

Lecture 4
Partial derivatives – Problem 3




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Find partial derivatives of the function:                           .
Solution: when differentiating with regard to x, we consider
x to be a variable while y is considered a constant, and
vice versa.

Lecture 4
Second derivatives of a function of two real variables




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If first derivatives are differentiable again, we obtain
second derivatives:
The latter two derivatives are called mixed derivatives. It can be
prooved they are equal:

Lecture 4
Second derivatives – Problem 4




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Find all four second derivatives of a function:
Solution:
We start with the first derivatives:
Then, we compute the second derivatives:

Lecture 4
Cobb – Douglas function




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C-D function:                    .
Usually, we assume that                .
Then, C-D:
Marginal product of labour:
Marginal product of capital:

Lecture 4
A utility function
 Let n be the number of different types of good. Let Q1, Q2, …. Be the amount of the good 1, 2,
etc.
Then a function                         is called the utility function. Typically, a utility
function is concave:



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graf 43

Lecture 4
Marginal utility
 Marginal utility is defined as follows:
                                                                      etc.
Example:
Find marginal utilities of the utility function                     .
Solution:



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Lecture 4
A tangent plane




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                  be a differentiable function at a point                  .
Then the tangent plane to the function f at the point C is:

The norm vector is given as follows:

Lecture 4
A tangent plane – Problem 5




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Find the tangent plane to the function                           at the
point C [2,1,10].
Solution:
We compute the first derivatives:
Then, we substitute point C:
Therefore, we obtain:
And finally:

Lecture 4
A total differential




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Let              be differentiable at a point                     .
Then, the total differential is given as:
The total differential expresses (approximately) an increment of a function, which depends on
increments of x (dx) and y (dy).

Lecture 4
A total differential – Problem 6




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Find the total differential of the function                              , at
the point C [1,1,9] and for dx = 0,1, dy = 0,2 .
Solution:
We start with the first derivatives:
Substituting into general form yields:
Substituting C and differentials:

Lecture 4
Problems to solve 1 (Assignment 6)




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Find the domain of the following functions:

Lecture 4
Problems to solve 2 (Assignment 6)




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Find the partial derivatives of the first and second order:

Lecture 4
Problems to solve 3 (Assignment 6)




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Find marginal products of labour and capital:
Find the total differential of the function                                  .

Lecture 4
Problems to solve 4 (skip it)




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Find the tangent plane of the function                      at the
point C [–2,2, ?].
Find the marginal utilities of the function                    at the
point Q1 = 9 and Q2 = 8.

Lecture 4
Problems to solve 5 (skip it)




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A mathematical model of revenue R based on a price P and
advertisement costs A has the following form:
Find:
a) change of R with regard to the change of p,
b) change of R with regard to the change of A,
c) write a total differential of R.

Lecture 4
Thank you for your attention




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