1
Lecture Notes on Monotone Comparative Statics and Producer Theory
Susan Athey
updated Fall 2002
These notes summarize the material from the Athey, Milgrom, and Roberts monograph
which is most important for basic producer theory.
It is not presented in the same order
as the monograph, so I have not preserved theorem numbers, etc., but I don’t think it will
be too hard to find the relevant sections.
Much of the material is drawn from the end of
Chapter 2.
These notes, as well as the monograph, may have typos and suffer from “copy-and-paste”
mistakes as well as notational inconsistencies.
We apologize in advance.
I.
Comparative Statics and Producer Theory
Consider a firm with production function
F
(
k
,
l
), where
k
is capital and
l
is labor.
For the
moment, let consider the cost-minimization problem.
The firm’s problem is as follows:
,
.. ( ,)
min
kl
stF kl q
rk wl
=
+
In the case where output is strictly increasing in each input and the production function
F
is suitably well-behaved, we can define the isoquant function
L
(
k
,
q
) according to the
following implicit function:
F
(
k
,
L
(
k
,
q
)) =
q
.
Thus, we can rewrite the firm’s problem as follows:
max
k
−
rk
−
wL
(
k
,
q
)
(L1)
Questions
:
Does firm’s choice of capital (
k
) increase or decrease monotonically with the
input prices (
r
,
w
) and the level of output required (
q
)?
Do the answers depend on any
properties of the production function?
Notice that we in general want conditions that will allow us to answer these questions
independent of the particular parameter values.
In other words, we want conditions on
the production function
F
that will be sufficient for comparative statics conclusions, and
we do not want to rule out
ex ante
any particular values of
q
,
r
, and
w
.