Hotelling's Lemma
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Hotelling's lemma is a result in
microeconomic Microeconomics is a branch of economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics focuses on the ...
s that relates the supply of a good to the maximum profit of the producer. It was first shown by
Harold Hotelling Harold Hotelling (; September 29, 1895 – December 26, 1973) was an American mathematical statistician and an influential economic theorist, known for Hotelling's law, Hotelling's lemma, and Hotelling's rule in economics, as well as Hotelling ...
, and is widely used in the
theory of the firm The theory of the firm consists of a number of economic theories that explain and predict the nature of the firm, company, or corporation, including its existence, behaviour, structure, and relationship to the market. Firms are key drivers in eco ...
. 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.


Formal statement

Let p denote a variable price, and w be a constant cost of each input. Let x:\rightarrow X be a mapping from the price to a set of feasible input choices X\subset . Let f:\rightarrow be the
production function In economics, a production function gives the technological relation between quantities of physical inputs and quantities of output of goods. The production function is one of the key concepts of mainstream economics, mainstream neoclassical econ ...
, and y(p)\triangleq f(x(p)) be the net supply. The maximum profit can be written by :\pi (p) = \max_ p\cdot y(p) - w \cdot x(p). Then the lemma states that if the profit \pi is differentiable at p, the maximizing net supply is given by :y^*(p) = \frac .


Proof for Hotelling's lemma

The lemma is a corollary of the
envelope theorem In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, in ...
. Specifically, the maximum profit can be rewritten as \pi (p, x^*) = p\cdot f(x^*(p)) - w \cdot x^*(p) where x^* is the maximizing input corresponding to y^*. Due to the optimality, the first order condition gives By taking the derivative by p at x^*, :\frac = \frac\bigg, _ \frac + \frac = \frac = f(x^*(p)) = y^*(p) where the second equality is due to (). QED


Application of Hotelling's lemma

Consider the following example. Let output y have price p and inputs x_1 and x_2 have prices w_1 and w_2. Suppose the production function is y = x_1^x_2^. The unmaximized profit function is \pi(p, w_1, w_2, x_1, x_2) = p y - w_1x_1 - w_2x_2. From this can be derived the profit-maximizing choices of inputs and the maximized profit function, a function just of the input and output prices, which is \pi(p, w_1, w_2 ) = \frac \frac 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: \frac = y = \frac \frac \frac = -x_1 = -\frac \frac \frac = -x_2 = - \frac \frac 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.


Criticisms and empirical evidence

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^* and cost minimizing input x^*. If a firm is not producing at these optima, then Hotelling's lemma would not hold.


See also

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Hotelling's law Hotelling's law is an observation in economics that in many markets it is Rationality, rational for producers to make their products as similar as possible. This is also referred to as the principle of minimum differentiation as well as Hotellin ...
*
Hotelling's rule Hotelling's rule defines the net price path as a function of time while maximizing economic rent in the time of fully extracting a non-renewable natural resource. The maximum rent is also known as Hotelling rent or scarcity rent and is the maxi ...
*
Supply and demand In microeconomics, supply and demand is an economic model of price determination in a Market (economics), market. It postulates that, Ceteris_paribus#Applications, holding all else equal, the unit price for a particular Good (economics), good ...
*
Shephard's lemma Shephard's lemma is a result in microeconomics having applications in the theory of the firm and in consumer choice. The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost-minimizing point of a ...


References

* * * * {{cite book , last=Varian , first=H. , authorlink=Hal Varian , year=1992 , title=Microeconomic Analysis , edition=3rd , location=New York , publisher=W. W Norton , isbn=978-0-393-95735-8 , page
43–45
, url=https://archive.org/details/microeconomicana00vari_0/page/43 , url-access=registration Eponyms in economics Lemmas