Wellesley College
Economics 201
Mathematical Concepts and Methods for Economics
The major mathematical concept that will interest us in Economics 201 is the
optimum
of a function.
(Optima are either maxima or minima.)
Most individual decisions in economics involve either
maximization (utility, profits) or minimization (cost, expenditure) and these decisions also often
involve constraints that must be taken into consideration (certain amount of income to spend, cost
levels).
We will also consider the concept of equilibrium, which involves the solution of a number
of equations simultaneously (i.e. finding values for several variables which solve several equations
at once).
The most obvious example of solving for an equilibrium in economics is finding the price
and quantity at which supply equals demand.
Optimization: Functions of One Variable
Start with a function like y = f(x) or
A
= f(Q).
The derivative, which will be denoted as dy/dx or
d
A
/dQ or f
N
(x), measures the slope of the function at a certain point (i.e. for a particular value of x
or Q).
When d
A
/dQ > 0,
A
is an
increasing
function of Q;
A
moves in the same direction as Q.
When d
A
/dQ < 0,
A
is a
decreasing
function of Q;
A
moves in the opposite direction from Q.
When d
A
/dQ = 0, the function has a stationary point (can be a maximum or a minimum or neither.)
This last condition is the
first order condition
for an optimum for a function of one variable.
EXAMPLE:
A
=
A
(Q) = 2 Q
2
+ 24 Q + 5
d
A
/dQ = 4 Q + 24 so function is increasing when  4 Q + 24 > 0 or when Q < 6.
The function is decreasing for  4 Q + 24 < 0 or when Q > 6.
The function has a stationary point when  4 Q + 24 = 0 or when Q = 6.
Stationary Points: Maxima or Minima?
How can you tell whether a stationary point is a maximum or a minimum of your function?
You
need to look at how the
derivative
is
changing
at the stationary point.
You need to look at the
derivative of the derivative, or the
second
derivative of the function.
The second derivative is denoted: d
2
A
/ dQ
2
or
d/dQ (d
A
/dQ) or
A
O
(Q).
When
A
O
(Q) > 0, then
A
N
(Q) is moving in the same direction as Q (the slope of the function
is
increasing
as Q increases).
When
A
O
(Q) < 0, then
A
N
(Q) is moving in the opposite direction from Q (the slope is
decreasing
as Q increases).
When
A
O
(Q) = 0, then
A
N
(Q) is switching from negative to positive (or positive to negative)
as you increase Q and there is an inflexion point in
A
(Q).
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second order condition
for determining whether you have a maximum or a minimum consists
of checking the sign of the second derivative of the function at that value for which you found the
stationary point. If f
O
< 0, the stationary point is a maximum.
If f
O
> 0, it is a minimum.
(Functions
which have maxima look like hills; they are
concave
functions.
Functions which have minima look
like valleys; they are
convex
functions.)
If f
O
= 0, the stationary point is an inflexion point.
EXAMPLE:
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 Fall '08
 JOHNSON
 Economics, Derivative, Optimization, Convex function, Stationary point, OPTIMA

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