CHAPTER 8
Unconstrained Optimisation Problems with One or More
Variables
In Chapter 6 we used differentiation to solve optimisation problems. In
this chapter we look at more economic applications of this technique:
profit maximisation
in
perfect competition
and
monopoly
; and
strategic optimisation
problems that arise in
oligopoly
or when there
are
externalities
. Then we show how to
maximise or minimise a
function of more than one variable
, and again look at applications.
—
—
1. The Terminology of Optimisation
Suppose that an economic agent wants to choose some value
y
to maximise a function Π(
y
).
For example,
y
might be the quantity of output and Π(
y
) the profit of a firm. From Chapters
5 and 6 we know how to solve problems like this, by differentiating.
The agent’s optimisation problem is:
max
y
Π(
y
)
Π(
y
) is called the agent’s
objective function
 the thing he wants to optimise.
To solve the optimisation problem above, we can look for a
value of
y
that satisfies two conditions:
The firstorder condition:
d
Π
dy
= 0
and the secondorder condition:
d
2
Π
dy
2
<
0
But remember that if we find a value of
y
satisfying these conditions it is not necessarily the
optimal choice, because the point may not be the
global
maximum of the function.
Also,
some functions may not have a global maximum, and some may have a maximum point for
which the second derivative is zero, rather than negative. So it is always important to think
about the shape of the objective function.
An optimisation problem may involve minimising rather than maximising – for example
choosing output to minimise average cost:
min
q
AC
(
q
)
The first and secondorder conditions are, of course,
dAC
dq
= 0 and
d
2
AC
dq
2
>
0, respectively.
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8. UNCONSTRAINED OPTIMISATION PROBLEMS
2. Profit Maximisation
Suppose a firm has revenue function
R
(
y
) and cost function
C
(
y
), where
y
is the the quantity
of output that it produces. Its profit function is:
Π(
y
) =
R
(
y
)

C
(
y
)
The firm wants to choose its output to maximise profit. Its optimisation problem is:
max
y
Π(
y
)
The firstorder condition for profit maximisation is:
d
Π
dy
=
R
(
y
)

C
(
y
) = 0
which is the familiar condition that
marginal revenue
equals
marginal cost
.
Examples
2.1
:
A monopolist has inverse demand function
P
(
y
) = 35

3
y
and cost function
C
(
y
) = 50
y
+
y
3

12
y
2
.
(i) What is the profit function?
Π(
y
)
=
yP
(
y
)

C
(
y
)
=
(35

3
y
)
y

(50
y
+
y
3

12
y
2
)
=

y
3
+ 9
y
2

15
y
(ii) What is the optimal level of output?
The firstorder condition is:
d
Π
dy
=

3
y
2
+ 18
y

15 = 0
There are two solutions:
y
= 1 or
y
= 5.
Check the secondorder condition:
d
2
Π
dy
2
=

6
y
+ 18
When
y
= 1,
d
2
Π
dy
2
= 12 so this is a minimum point.
When
y
= 5,
d
2
Π
dy
2
=

12 so this is the optimal level of output.
(iii) What is the market price, and how much profit does the firm make?
The firm chooses
y
= 5. From the market demand function the price is
P
(5) = 20.
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 Spring '10
 Vines
 Economics, Perfect Competition, Supply And Demand, max Π, unconstrained optimisation problems

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