Economic applications



Economic applications of derivatives
 The elasticity of a function y = f(x):
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 The price elasticity of demand:
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 The price elasticity of supply:
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 Marginal product of labour:
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 Marginal revenue:
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  Marginal cost:
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Solved problems
 Find marginal revenue MR (x) of the total revenue
                                     and marginal costs of the total costs
                               .
Solution:
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Solved problem
Find extremes of the production function                     . Draw its graph.

Solution:
 The first derivative is                   , we find roots of the first derivative: L = 0 and L =
1. By the use of the second derivative, or by checking signs of the first derivative, we obtain
that L = 0 is a local minimum and L = 1 is a local maximum. Therefore, the highest (optimal)
production is achieved when L = 1.
The graph is provided on the next slide.
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graf 44b

Assignment

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Find the maximum of total revenue function                                       .
Find the minimum of total cost function:                             .
Find the maximum of the profit function:                               .
Find the maximum of total revenue function:                                 .

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.

A graph of Cobb-Douglas function




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Cobb_douglas

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:

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

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



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