Hotelling's lemma is a result in microeconomics that relates the supply of a good to the maximum profit of the producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm.
Specifically, it states: The rate of an increase in maximized profits with respect to a price increase is equal to the net supply of the good. In other words, if the firm makes its choices to maximize profits, then the choices can be recovered from the knowledge of the maximum profit function.
Let
p
w
x:{R+} → X
X\subset{R+}
f:{R+} → {R+}
y(p)\triangleqf(x(p))
The maximum profit can be written by
\pi(p)=maxxp ⋅ y(p)-w ⋅ x(p).
\pi
p
y*(p)=
d\pi(p) | |
dp |
.
The lemma is a corollary of the envelope theorem.
Specifically, the maximum profit can be rewritten as
\pi(p,x*)=p ⋅ f(x*(p))-w ⋅ x*(p)
x*
y*
p
x*
d\pi | |
dp |
=
\partial\pi | |
\partialx |
| | |
x=x* |
\partialx | |
\partialp |
+
\partial\pi | |
\partialp |
=
\partial\pi | |
\partialp |
=f(x*(p))=y*(p)
Consider the following example.[1] Let output
y
p
x1
x2
w1
w2
y=
1/3 | |
x | |
1 |
1/3 | |
x | |
2 |
\pi(p,w1,w2,x1,x2)=py-w1x1-w2x2
\pi(p,w1,w2)=
1 | |
27 |
p3 | |
w1w2 |
Hotelling's Lemma says that from the maximized profit function we can find the profit-maximizing choices of output and input by taking partial derivatives:
\partial\pi(p,w1,w2) | |
\partialp |
=y=
1 | |
9 |
p2 | |
w1w2 |
\partial\pi(p,w1,w2) | |
\partialw1 |
=-x1=-
1 | |
27 |
p3 | ||||||
|
\partial\pi(p,w1,w2) | |
\partialw2 |
=-x2=-
1 | |
27 |
p3 | ||||||||||||
|
Note that Hotelling's lemma gives the net supplies, which are positive for outputs and negative for inputs, since profit rises with output prices and falls with input prices.
A number of criticisms have been made with regards to the use and application of Hotelling's lemma in empirical work.
C. Robert Taylor points out that the accuracy of Hotelling's lemma is dependent on the firm maximizing profits, meaning that it is producing profit maximizing output
y*
x*
. Akira Takayama (economist) . Mathematical Economics . New York . Cambridge University Press . 1985 . 978-0-521-31498-5 . 141–144 .
. Hal Varian . 1992 . Microeconomic Analysis . 3rd . New York . W. W Norton . 978-0-393-95735-8 . 43–45 . registration .