Editing Implicit function (section)

==Applications in economics==

===Marginal rate of substitution===

In [[economics]], when the level set {{math|1=''R''(''x'', ''y'') = 0}} is an [[indifference curve]] for the quantities {{mvar|x}} and {{mvar|y}} consumed of two goods, the absolute value of the implicit derivative {{math|{{sfrac|''dy''|''dx''}}}} is interpreted as the [[marginal rate of substitution]] of the two goods: how much more of {{mvar|y}} one must receive in order to be indifferent to a loss of one unit of&nbsp;{{mvar|x}}.

===Marginal rate of technical substitution===

Similarly, sometimes the level set {{math|''R''(''L'', ''K'')}} is an [[isoquant]] showing various combinations of utilized quantities {{mvar|L}} of labor and {{mvar|K}} of [[physical capital]] each of which would result in the production of the same given quantity of output of some good. In this case the absolute value of the implicit derivative {{math|{{sfrac|''dK''|''dL''}}}} is interpreted as the [[marginal rate of technical substitution]] between the two factors of production: how much more capital the firm must use to produce the same amount of output with one less unit of labor.

===Optimization===
{{Main|Mathematical economics#Mathematical optimization}}

Often in [[economic theory]], some function such as a [[utility function]] or a [[Profit (economics)|profit]] function is to be maximized with respect to a choice vector {{mvar|x}} even though the objective function has not been restricted to any specific functional form. The [[implicit function theorem]] guarantees that the [[first-order condition]]s of the optimization define an implicit function for each element of the optimal vector {{math|''x''*}} of the choice vector {{mvar|x}}. When profit is being maximized, typically the resulting implicit functions are the [[labor demand]] function and the [[supply function]]s of various goods. When utility is being maximized, typically the resulting implicit functions are the [[labor supply]] function and the [[demand function]]s for various goods.

Moreover, the influence of the problem's [[Parameter#Mathematical functions|parameters]] on {{math|''x''*}} — the partial derivatives of the implicit function — can be expressed as [[total derivative]]s of the system of first-order conditions found using [[Differential of a function#Differentials in several variables|total differentiation]].
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