OPMT 5701
Applications of Lagrangian
Utility Maximization with a simple rationing constraint
Consider a familiar problem of utility maximization with a budget constraint:
Maximize
U = U (x, y )
subject to B = Px x + Py y
and
x>x
But where a ration on x ha
OPMT 5701: Calculus for Utility Problems
Kevin Wainwright
October 14, 2003
1
Using Calculus For Utility Maximization Problems
1.1
Review of Some Derivative Rules
1. Partial Derivative Rules:
U = xy
U = xa y b
a
U = xa y b = xb
y
U = ax + by
U = ax1/2 + by
OPMT 5701
Two Variable Optimization
Using Calculus For Maximization Problems
One Variable Case
If we have the following function
y = 10x x2
we have an example of a dome shaped function. To nd the maximum of the dome, we
simply need to nd the point where t
OPMT 5701
Multivariable Calculus
Partial Derivatives
Single variable calculus is really just a special case of multivariable calculus. For the
function y = f (x), we assumed that y was the endogenous variable, x was the exogenous
variable and everything e
OPMT 5701
Optimization with Constraints
The Lagrange Multiplier Method
Sometimes we need to to maximize (minimize) a function that is subject to some sort of
constraint. For example
Maximize
z = f (x, y )
subject to the constraint
x + y 100
For this kind
1
1.1
Derivatives: The Five Basic Rules
Nonlinear Functions
The term derivative means slope or rate of change. The ve rules we are about
to learn allow us to nd the slope of about 90% of functions used in economics,
business, and social sciences.
Suppose
OPMT 5701
Notes on Natural Logarithm and the Exponential e
1. The Number e
dy
= ex
dx
dy
then
= ef (x) f 0 (x)
dx
if y
= ex then
if y
= ef (x)
2. Examples
(a)
= e3x
y
dy
dx
= e3x (3)
(b)
y
dy
dx
3
= e7x
3
= e7x (21x2 )
(c)
y
dy
dt
= ert
= rert
3. Logarith
OPMT 5701 Lecture Notes
1
The Chain Rule
Of all the basic rules of derivatives, the most challenging one is the chain rule.
However, like the other rules, if you break it down to simple steps, it too is quite
manageble. There are a couple of approaches to
OPMT 5701 Lecture Notes
1
1.1
Natural Logarithm and the Exponential e
1. The Number e
dy
= ex
dx
dy
then
= ef (x) f 0 (x)
dx
if y
= ex then
if y
= ef (x)
2. Examples
1. (a)
= e3x
y
dy
dx
= e3x (3)
(b)
y
dy
dx
3
= e7x
3
= e7x (21x2 )
(c)
y
dy
dt
= ert
= re
OPMT 5701 Lecture Notes
1
Matrix Algebra
1. Gives us a shorthand way of writing a large system of equations.
2. Allows us to test for the existance of solutions to simultaneous systems.
3. Allows us to solve a simultaneous system.
DRAWBACK: Only works for
OPMT 5701 Lecture Notes
Implicit Dierentiation
This section assumes the students have read the section on implicit dierentiation in Chapter 13 of the text book.
Suppose we have the following:
2y + 3x = 12
we can rewrite it as
2y
y
= 12 3x
3
= 6 x
2
Now we
The dierence between dy and
y
dy is an approximation found by moving along the tangency.
y is the
dierence between two points on the actual function y = f (x). Given the
function
y = x2
the dierential is
dy = 2xdx
suppose x = 2 and dx = :01 then the diere
OPMT 5701 Lecture Notes
One Variable Optimization
October 10, 2006
Critical Points
A critical point occurs whenever the rst derivative of a function is equal to zero, i.e. if
y = f (x)
then
dy
= f 0 (x) = 0
dx
is a critical point. A critical is a stationa