2
LECTURE #24: TUE, NOV 20, 2007
PRINT DATE: AUGUST 21, 2007
(COMPSTAT1)
Example
π
(
p, q
*
(
p
)); for each
p
, pick the
q
that maximizes profits for that
p
; call this function
q
*
(
p
).
Now ask how profit adjust as price changes and the producer adjusts quantity.
Other examples of value functions in economics are the expenditure function and the indirect utility
function.
Answer is given by the envelope theorem which says that in this case,
df/db
=
∂f/∂b
. (i.e., you’ve
learnt to tell the difference between
df
and
∂f
; now you find that in this case, there isn’t any
difference.)
The envelope theorem: Varian
: Given
f
:
R
2
→
R
1
(differentiable) and a function
x
*
:
R
1
→
R
1
(differentiable) defined by the condition that for each
b
,
x
*
(
b
) maximizes
f
(
b,
·
).
Then the total
derivative of the function
f
(
b, x
*
(
b
)) with respect to
b
is
df
(
·
, x
*
(
·
))
/db
=
∂f
(
·
, x
*
(
·
))
/∂b
.
Mathematical proof is trivial
d
f
(
b, x
*
(
b
))
d
b
=
f
b
(
b, x
*
(
b
)) +
f
x
(
b, x
*
(
b
))
d
x
*
(
b
)
d
b
Necessary condition for
x
to maximize
f
(
b,
·
) is that
f
x
(
b, x
) = 0; this is how
x
*
(
b
) is defined; hence
f
x
(
b, x
*
(
b
)) = 0 by definition of
x
*
(
b
).
The picture is much more important; note that in the picture, the golden rule is broken, for display
purposes: the first component of the function is pictured on the horizontal axis.
•
The line in the domain
x
*
(
b
) has the property that vertically above points on this line, the
function
f
(
b,
·
) is maximized in the
x
direction.
•
Now as you move out along the line
x
*
(
b
) there are in general two contributors to the change
in
f
;
f
changes because
b
changes AND because
x
changes.
•
In this particular case,
f
doesn’t change when
x
changes, but it does change when
b
changes.