Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

Economists use a formula called the Cobb-Douglas Production Functions to model
the production levels of a company (or a country). Output Q at a factory is often
regarded as a function of the amount K of capital investment and the size L of the
labor force. Output functions of the form

( )

QKL AKL, = α^1 −α

where A and α are positive constants and 01 <α< have proved to be especially
useful in economic analysis. Such functions are known as Cobb-Douglas production
function.

Example 5

Suppose that the function represents the number of units

produced by a company with x units of labor and y units of capital.

Qxy(), = 500 x y0.3 0.7

a. How many units of a product will be manufactured if 300 units of labor and
50 units of capital are used?
b. How many units will be produced if twice the number of units of labor and
capital are used?
Solution

a. (^) ()()()
0.3 0.7
Q 300, 50 ==500 300 50 500 5.535 15.462 42, 791 units××=
b. If number of units of labor and capital are both doubled, then
x=× =2 300 600 and y=250 100×=
()()()
0.3 0.7
Q 600,100 ==500 600 100 500 6.815 25.119 85, 592 units××=
Thus we see that production is doubled if both labor and capital are doubled.

2 Partial Derivatives

Definition

Let zfxy= (),

a. The first parital derivative of f with respect to x is:

()

( ) ( )

0

,,

x ,lim

x

z f xxyfxy

fxy

xxΔ→

∂ +Δ −

==

∂Δ

b. The first partial derivative of f with respect to y is:

()

( ) ( )

0

,,

y ,limy

z f xy y f xy

fxy

yyΔ→

∂ +Δ −

==

∂Δ

2.1 Computation of Partial Derivatives

The function

z

x

∂

∂

or fxis obtained by differentiating f with respect to x, treating y as a

constant.

The function

z

y

∂

∂

or fy is obtained by differentiating f with respect to y, treating x as a

constant.

Example 1

For the functionf(xy,435)=−+x^2 xy y^2 , find
z
x

∂

∂

and

z

y

∂

∂

?

Solution