Profit function

The profit function (also: profit function ) is a mathematical function in microeconomics and especially in the theory of the company that indicates for a given price of the production factors as well as for a given sales price of the goods produced how high the maximum profit that a company can achieve is.

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One denotes the quantity of a good produced by a company, whereby the good arises from various production factors. Let it now be the associated production function ; this is a real-valued function that outputs the maximum achievable output quantity for a given factor input. is the vector of the factor inputs ( accordingly denotes the amount of production factor used ). Let further be the vector of the associated factor prices ( is accordingly the price of a unit of production factor ). Then the maximally attainable profit of the enterprise is given by the profit function ${\ displaystyle y}$ ${\ displaystyle f (\ mathbf {x})}$ ${\ displaystyle \ mathbf {x} = (x_ {1}, \ ldots, x_ {n})}$ ${\ displaystyle x_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle \ mathbf {w} = (w_ {1}, \ ldots, w_ {n})}$ ${\ displaystyle w_ {i}}$ ${\ displaystyle i}$

{\ displaystyle \ pi (p, \ mathbf {w}) \ equiv \ max _ {(\ mathbf {x}, y) \ geq \ mathbf {0}} py- \ mathbf {w} \ cdot \ mathbf {x }}

under the secondary condition

{\ displaystyle \; f (\ mathbf {x}) \ geq y}

The profit function of a company is consequently a maximum value function, applied to the difference between revenue and the factor costs incurred for its production under the secondary condition that the productivity limit of the company is adhered to.

properties

It can be shown that , provided that the underlying production function is continuous , strictly monotonically increasing and strictly quasi-concave on the , and that , among other things, it fulfills the following properties: ${\ displaystyle \ mathbf {\ pi} (p, \ mathbf {w})}$ ${\ displaystyle f}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle f (\ mathbf {0}) = 0}$

Monotonously increasing in . ${\ displaystyle p}$
Monotonously falling in . ${\ displaystyle \ mathbf {w}}$
Grade one homogeneity in . ${\ displaystyle (p, \ mathbf {w})}$ and. ${\ displaystyle \ pi (\ alpha p, \ alpha \ mathbf {w}) = \ alpha \ pi (p, \ mathbf {w}) \; \ forall p, \ mathbf {w}}$ ${\ displaystyle \ alpha> 0}$
Convex in. ${\ displaystyle (p, \ mathbf {w})}$
Differentiable in(, ""). ${\ displaystyle (p, \ mathbf {w})}$ ${\ displaystyle p> 0}$ ${\ displaystyle \ mathbf {w}> 0}$