Mathematical Economics
dr Wioletta Nowak
Lecture 5
• Production Function
• The Marginal Productivity of i-th Factor (the Marginal
Product of Capital and the Marginal Product of Labour),
Marginal Rate of Technical Substitution, Output Elasticities,
• Returns to Scale: Constant, Increasing, Decreasing,
• Elasticity of Substitution between Two Factors of
Production,
• CES (Constant Elasticity of Substitution) Production
Function,
• The Cobb-Douglas Production Function,
• The Leontief-Koopmans Production Function
Production Function
• The production function f  x1 , x2  describes
the maximum level of output that can be
obtained for a given vector of inputs  x1 , x2 .
Isoquant
• Input combinations x1 , x2 
that yield the
same (maximum) level of output y define an
isoquant.
• x2  x1  is the function that tells us how
much of x2
it takes to produce y if we are
using x1 units of the other input.
Isoquant
The technical rate of substitution (TRS)
between two inputs is a slope of the isoquant.
The technical rate of substitution
• TRS measures how one of the inputs must be adjust
in order to keep output constant when another input
changes.
Output Elasticities
Elasticity of Substitution
Returns to Scale
• Constant returns to scale (CRS): Technology
exhibits constant returns to scale if
f t  x1 , t  x2   t  f  x1 , x2 
• If all inputs increase by a factor 2 then the
new output is twice the previous output given.
Returns to Scale
• Increasing returns to scale (IRS):
Technology exhibits increasing returns to scale
if
f t  x1 , t  x2   t  f  x1 , x2 
• If all inputs increase by a factor 2 then the
new output is more than twice the previous
output given.
Returns to Scale
• Decreasing returns to scale (DRS):
Technology exhibits decreasing returns to
scale if
f t  x1 , t  x2   t  f  x1 , x2 
• If all inputs increase by a factor 2 then the
new output is less than twice the previous
output given.
Elasticity of Scale
• It may well happen that a technology exhibits
increasing returns to scale for some values of x
and decreasing returns to scale for other
values.
• Thus a local measure of returns to scale is
useful.
• The elasticity of scale measures the percentage
increase in output due to a percentage increase
in all inputs.
Elasticity of Scale
Exercise 1
• Check the returns to scale for the following
technologies:
The CES production function
• The constant elasticity of substitution (CES)
production function has the form:
The CES Function
• Marginal product of capital
• Marginal product of labour
• Output elasticity of capital
• Output elasticity of labour
• Elasticity of Substitution
• Returns to Scale,
• Elasticity of Scale
The CES Function


 
y  A a  x1  (1  a ) x2


1
y  A ax

1
 (1  a ) x

1



2
A  0, a  (0,1), 0    1


 
y  A a  x1  (1  a) x2
1 
y
 y 
 aA  
x1
 x1 
 x1 
 1  aA  
 y



1
1 
y
 y 
 (1  a ) A  
x2
 x2 
 x2 
 2  (1  a ) A  
 y

1   2  1




1
 
y  A a  x1  (1  a) x2
1 
a  x2 
 
TRS  
1  a  x1 


 
y  A a  x1  (1  a) x2
t  1


CRS

1
 
lim A a  x1  (1  a ) x2
 0

1
a 1 a
1 2
 Ax x
Mathematical Economics
dr Wioletta Nowak
Lecture 6
• Neoclassical producer theory. Perfectly
competitive firms.
• The profit function and profit maximization
problem.
• Properties of the input demand and the output
supply.
• Cost minimization problem. Definition and
properties of the conditional factor demand
and the cost function.
• Profit maximization with the cost function.
Long and short run equilibrium.
Perfect Competition
• Perfect competition describes a market in
which there are many small firms, all
producing homogeneous goods.
• No entry/exit barriers.
• The firm takes prices as a given in both its
output and factor markets.
• Perfect information.
• Firms aim to maximize profits.
Profit Maximization in the Long Run
The first-order condition can be exhibited graphically
Cost Minimization in the Long Run
The cost minimization problem can be exhibited
graphically
Profit Maximization with the Cost Function
Example 1
Profit Maximization and Cost Minimization in the
Short Run