Chapter 2
THE MATHEMATICS OF
OPTIMIZATION
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.
The Mathematics of Optimization
• Many economic theories begin with the
assumption that an economic agent is
seeking to find the optimal value of some
function
– Consumers seek to maximize utility
– Firms seek to maximize profit
• This chapter introduces the mathematics
common to these problems
Maximization of a Function of
One Variable
• Simple example: Manager of a firm
wishes to maximize profits
  f (q)

Maximum profits of
* occur at q*
*
 = f(q)
q*
Quantity
Maximization of a Function of
One Variable
• The manager will likely try to vary q to see
where the maximum profit occurs
– An increase from q1 to q2 leads to a rise in 


0
q
*
2
 = f(q)
1
q1
q2
q*
Quantity
Maximization of a Function of
One Variable
• If output is increased beyond q*, profit will
decline
– An increase from q* to q3 leads to a drop in 


0
q
*
 = f(q)
3
q*
q3
Quantity
Derivatives
• The derivative of  = f(q) is the limit of
/q for very small changes in q
d df
f (q1  h )  f (q1 )

 lim
dq dq h0
h
• Note that the value of this ratio depends
on the value of q1
Value of a Derivative at a Point
• The evaluation of the derivative at the
point q = q1 can be denoted
d
dq q  q
1
• In our previous example,
d
0
dq q q
1
d
0
dq q q
3
d
0
dq q  q *
First Order Condition for a
Maximum
• For a function of one variable to attain
its maximum value at some point, the
derivative at that point must be zero
df
dq
0
q q*
Second Order Conditions
• The first order condition (d/dq) is a
necessary condition for a maximum, but
it is not a sufficient condition

If the profit function was u-shaped,
the first order condition would result
in q* being chosen and  would
be minimized
*
q*
Quantity
Second Order Conditions
• This must mean that, in order for q* to
be the optimum,
d
d
 0 for q  q * and
 0 for q  q *
dq
dq
• Therefore, at q*, d/dq must be
decreasing
Second Derivatives
• The derivative of a derivative is called a
second derivative
• The second derivative can be denoted
by
d 
d f
or
or f ' ' (q)
2
2
dq
dq
2
2
Second Order Condition
• The second order condition to represent
a (local) maximum is
d 
0
2
dq q  q *
2
Rules for Finding Derivatives
db
1. If b is a constant, then
 0.
dx
2. If a and b are constants and b  0,
b
dax
b 1
then
 bax .
dx
d ln x 1
3.

dx
x
x
da
4.
 a x ln a for any constant a.
dx
Rules for Finding Derivatives
d[f (x)  g(x)]
5.
 f ' (x)  g ' (x)
dx
d[f (x)  g(x)]
6.
 f (x)g ' (x)  f ' (x)g(x)
dx
 f (x) 
d

g(x)  f ' (x )g(x )  f (x)g ' (x)

7.

provided
2
dx
[g(x )]
that g(x)  0.
Rules for Finding Derivatives
8. If y  f (x) and x  g(z) and if both f ' (x)
dy
dy dx
df dg
and g' (z) exist, then




dz dx dz dx dz
This is called the chain rule. The chain rule
allows us to study how one variable (z) affects
another variable (y) through its influence on
some intermediate variable (x).
Example of Profit Maximization
Suppose that the relationship between
profit and output is
 = 1,000q - 5q2
The first order condition for a maximum is
d/dq = 1,000 - 10q = 0
q* = 100
Since the second derivative is always -10,
q=100 is a global maximum.