Hotelling's lemma

As Hotelling's lemma is known in microeconomics and there, especially in the theory of the firm some properties of a profit function. It implies in particular that from the profit function directly the supply function of the produced goods ( output good), and the demand function with respect to the employed factors ( input goods ) effects: For optimum production, therefore, yields the partial derivative of the profit function after the goods price, the quantity sold, while the partial derivative with respect to the respective factor price of the (negative ) factor is used. In his assumptions Hotelling assumed that the prices of the goods produced by the market determines the amount of output but from producer.

Mathematically, it is an application of the envelope theorem. Named the lemma is according to the American statistician and economist Harold Hotelling.

Formal representation

Be the output price of a good that is produced from input goods. Production takes place using a given technology, which is represented by the production function; this indicates how much of the output good can be produced by means of a maximum of factor inputs ( so called, for example, the amount of input factor i used ). Further, let the vector of the corresponding factor prices ( so called, for example, the price for a unit of input factor i).

It is now called the profit function of the firm; it indicates for given prices of the output goods and the input goods, what profit a company can achieve maximum.

Hotelling's lemma ( Hotelling 1932): Let f be as usual steadily, monotonically increasing, strictly on the quasikonkav and applies. Furthermore, the usual conditions for the profit function are fulfilled, ie in particular and. Let f be beyond even strictly concave on the. Then:

Derivation

Take slightly simplistic immediately that the constraint in the optimization problem for the profit function is satisfied with equality, that is,. Of course, one could prove the lemma without this restriction, the result is each equivalent (for it would be shown anyway that the entire quantity produced is also available ).

Define. Consider the problem with the solution. The value function thereof and so. According Envelope theorem is therefore also

( the conditions of the theorem guarantee, for each of the required differentiability ), but this is just (like directly from the definition of g can be seen ) is equal, qed

Analogously, also, for all i,

Which again, q.e.d.