Transcript Functions of Several Variables • Most goals of economic agents depend on several variables – Trade-offs must be made • The dependence of one.

Functions of Several Variables
• Most goals of economic agents depend
on several variables
– 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 (x1, x2 ,..., xn )
Partial Derivatives
• The partial derivative of y with respect
to x1 is denoted by
y
f
or
or fx or f1
x1
x1
1
• It is understood that in calculating the
partial derivative, all of the other x’s are
held constant.
Partial Derivatives
• Partial derivatives are the mathematical
expression of the ceteris paribus
assumption
– They show how changes in one variable
affect some outcome when other
influences are held constant
Calculating Partial Derivatives
1. If y  f ( x1, x 2 )  ax  bx1 x 2  cx , then
f
 f1  2ax1  bx 2 and
x1
f
 f2  bx1  2cx2
x 2
2
1
2
2
ax1  bx 2
2. If y  f (x1, x2 )  e
, then
f
f
ax  bx
ax  bx
 f1  ae
and
 f2  be
x1
x2
1
2
1
2
Calculating Partial Derivatives
3. If y  f (x1, x 2 )  a ln x1  b ln x2 , then
f
a
f
b
 f1 
and
 f2 
x1
x1
x 2
x2
Second-Order Partial Derivatives
• The partial derivative of a partial
derivative is called a second-order
partial derivative
(f / xi )
f

 fij
x j
xi x j
2
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
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 
dx1 
dx 2  ... 
dx n
x1
x 2
xn
dy  f1dx1  f2dx 2  ...  fndx 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
Second-Order Conditions
• The second-order partial derivatives
must meet certain restrictions for the
critical point to be a local maximum
• These restrictions will be discussed
later in this chapter
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
 2 x1  2  0
x1
y
 2 x 2  4  0
x 2
OR
x 1
x 2
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