Mathematics for Economists
[Section A – Part 1 / 4]
Instructor: Annika M. Mueller, Ph.D., The Wang
Yanan Institute for Studies in Economics
Main Reference: C. P. Simon and L. Blume
(1994), Mathematics for Economists
Further Reading: W. Nicholson and C. Snyder
(2011), Microeconomic Theory
Optimization

Economic models often based on
assumption that
economic agents
(individuals, firms) are seeking to find the
optimal value of some function, for
example



consumers maximize utility
firms maximize profit
Cover mathematics common to these
problems
Maximization of a Function of One
Variable

Classic example: Profit maximization of
a firm   f(q)

Profit maximum
(*) occurs at
quantity q*
*
 = f(q)
q*
Quantity
Profit Maximization (cont'd)

q* will likely be reached through an
adjustment process

an increase from q1 to q2 leads to a rise in
profits


0
q
*
2
 = f(q)
1
q1
q2
q*
Quantity
Profit Maximization (cont'd)

If output is increased beyond q*, profit will
decline

an increase from q* to q3 leads to a decline of
profits


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
(
q

h
)

f
(
q
)
1
1

lim
h

0
dq
dq
h
• 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
dqqq
1
d
0
dqqq
3
d
0
dqq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

The first order condition is a necessary
condition for a maximum, but not a
sufficient condition
c
If the function was u-shaped (think: cost
function), the first order condition results
in q* being chosen, minimizing c.
c*
q*
Quantity
Second Order Conditions

In order for q* to be the optimum for a
profit function,
d


0
for
q

q
* and
dq
d


0
for
q

q
*
dq
• Therefore, at q*, d/dq must be zero.
Second Derivatives


The derivative of a derivative is called a
second derivative
The second derivative can be denoted
by
2
2
d
 d
f
or 2or
f"
(
q
)
2
dq dq
Second Order Condition

The second order condition to
represent a (local) maximum is
d


f"
(
q
)

0
2
q

q
*
dq
q

q
*
2
Functions of Several Variables

Most functions that economic agents
are trying to optimize depend on more
than just one variable


trade-offs must be made
The dependence of one variable (y) on
a series of other variables (x1,x2,…,xn)
is denoted by
y
f(
x
,
x
,...,
x
)
1
2
n
Partial Derivatives

The partial derivative of y with respect
to x1 is denoted by

y 
f
or or
f
f
xor
1

x

x
1
1
1
• It is understood that in calculating the
partial derivative, all of the other x’s are
held constant
“How do changes in one variable affect some
outcome when other influences are held
constant (ceteris paribus)”
Partial Derivatives

A more formal definition of the partial
derivative is

f
f
(
x

h
,
x
,...,
x
)

f
(
x
,
x
,...
x
)
2
n
2
n
1
1

lim
h

0

x
h
1
x
,...,
x
2
n
Partial Derivatives

Units of measurement matter:

if q represents the quantity of gasoline
demanded (measured in billions of
gallons) and p represents the price in
dollars per gallon, then q/p will measure
the change in demand (in billions of
gallons per year) for a dollar per gallon
change in price
Second-Order Partial Derivatives

The partial derivative of a partial
derivative is called a second-order
partial derivative
2

(

f/
x
)

f
i


f
ij

x

x
j
jx
i
Young’s Theorem

Under general conditions, the order in
which
partial
differentiation
is
conducted to evaluate second-order
partial derivatives does not matter
fij  f ji
Use of Second-Order Partials


Second-order partials play an important
role in many economic theories
One of the most important is a
variable’s own second-order partial, fii


shows how the marginal influence of xi on
y (y/xi) changes as the value of xi
increases
a value of fii < 0 indicates diminishing
marginal effectiveness
Total Differential


Suppose that y = f(x1,x2,…,xn)
If all x’s are varied by a small amount,
the total effect on y will be

f

f

f
dy
dx
dx

...
dx
1
2
n

x

x

x
1
2
n
dy

f
dx

f
dx

...

f
dx
1
1
2
2
n
n
First-Order Condition for a
Maximum (or Minimum)


A necessary condition for a maximum (or
minimum) of the function f(x1,x2,…,xn) is
that dy = 0 for any combination of small
changes in the x’s
The only way for this to be true is if
f1
f2
...

fn
0
• A point where this condition holds is
called a critical point
Finding a Maximum

Suppose that y is a function of x1 and
x2
y = - (x1 - 1)2 - (x2 - 2)2 + 10
y = - x12 + 2x1 - x22 + 4x2 + 5

First-order conditions imply that
y
 2x1 2  0
x1
y
 2x2 4  0
x2
OR
x1*  1
x2*  2
The Envelope Theorem


The envelope theorem concerns how the
optimal value for a particular function
changes when a parameter of the function
changes
This is easiest to see by using an example
The Envelope Theorem

Suppose that y is a function of x
y = -x2 + ax


For different values of a, this function
represents a family of inverted parabolas
If a is assigned a specific value, then y
becomes a function of x only and the value
of x that maximizes y can be calculated
The Envelope Theorem
Optimal Values of x and y for alternative values of a
V
a
lu
eo
fa
0
1
2
3
4
5
6
V
a
lu
eo
fx*
0
1
/2
1
3
/2
2
5
/2
3
V
a
lu
eo
fy*
0
1
/4
1
9
/4
4
2
5
/4
9
The Envelope Theorem
y*
10
9
8
7
As a increases,
the maximal value
for y (y*) increases
6
5
4
3
2
1
a
0
0
1
2
3
4
5
6
7
The Envelope Theorem

To calculate the slope of the function, we
must solve for the optimal value of x for
any value of a
dy/dx = -2x + a = 0
x* = a/2

Substituting, we get
y* = -(x*)2 + a(x*) = -(a/2)2 + a(a/2)
y* = -a2/4 + a2/2 = a2/4
The Envelope Theorem

Therefore,
dy*/da = 2a/4 = a/2 = x*

But, we can save time by using the
envelope theorem
– for small changes in a, dy*/da can be
computed by holding x at x* and calculating
y/ a directly from y
The Envelope Theorem
y/ a = x

Holding x = x*
y/ a = x* = a/2

This is the same result found earlier
The Envelope Theorem

The envelope theorem states that the
change in the optimal value of a function
with respect to a parameter of that function
can be found by partially differentiating the
objective function while holding x (or
several x’s) at its optimal value
dy
*
y
{
x

x
*
(
a
)}
da

a
The Envelope Theorem

The envelope theorem can be extended to
the case where y is a function of several
variables
y = f(x1,…xn,a)

Finding an optimal value for y would
consist of solving n first-order equations
y/xi = 0
(i = 1,…,n)
The Envelope Theorem

Optimal values for theses x’s would be
determined that are a function of a
x1* = x1*(a)
x2* = x2*(a)
.
.
.
xn*= xn*(a)
The Envelope Theorem

Substituting into the original objective
function yields an expression for the
optimal value of y (y*)
y* = f [x1*(a), x2*(a),…,xn*(a),a]

Differentiating yields
dy
*

f
dx
f
dx

fdx

f
1
2
n

 
 
...

 
da

x
da

x
da

x
da

a
1
2
n
The Envelope Theorem


Because of first-order conditions, all terms
except f/a are equal to zero if the x’s are
at their optimal values
Therefore,
dy
*
f
{
x

x
*
(
a
)}
da

a