Editing Legendre transformation (section)

===Microeconomics===
Legendre transformation arises naturally in [[microeconomics]] in the process of finding the ''[[supply (economics)|supply]]'' {{math|''S''(''P'')}} of some product given a fixed price {{math|''P''}} on the market knowing the [[cost curve|cost function]] {{math|''C''(''Q'')}}, i.e. the cost for the producer to make/mine/etc. {{math|''Q''}} units of the given product.

A simple theory explains the shape of the supply curve based solely on the cost function. Let us suppose the market price for a one unit of our product is {{math|''P''}}. For a company selling this good, the best strategy is to adjust the production {{math|''Q''}} so that its profit is maximized. We can maximize the profit
<math display="block">\text{profit} = \text{revenue} - \text{costs} = PQ - C(Q)</math>
by differentiating with respect to {{math|''Q''}} and solving
<math display="block">P - C'(Q_\text{opt}) = 0.</math>

{{math|''Q''<sub>opt</sub>}} represents the optimal quantity {{math|''Q''}} of goods that the producer is willing to supply, which is indeed the supply itself:
<math display="block">S(P) = Q_\text{opt}(P) = (C')^{-1}(P).</math>

If we consider the maximal profit as a function of price, <math>\text{profit}_\text{max}(P)</math>, we see that it is the Legendre transform of the cost function <math>C(Q)</math>.