# Conditional factor demands

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Conditional factor demands

In economics, a conditional factor demand function specifies the cost-minimizing level of an input (factor of production) such as labor or capital, required to produce a given level of output, for given unit input costs (wage rate and rental rate) of the input factors.[1] The conditional portion of this phrase refers to the fact that this function is conditional on a given level of output, so output is one argument of the function. Since the optimal mix of input levels depends on the wage and rental rates, these rates are also arguments of the conditional demand functions for the inputs. This concept is similar to but distinct from the factor demand functions, which give the optimal demands for the inputs when the level of output is free to be chosen; since output is not fixed in that case, output is not an argument of those demand functions.

## Optimization problem

With two inputs, say labor and capital, the optimization problem is to

$\text{Minimize}\, wL + rK \, \, \text{with respect to}\,\, L \,\, \text{and} \,\, K,$
subject to
f(L,K) = q,

where L and K are the chosen quantities of labor and capital, w and r are the fixed unit costs of labor and capital respectively, f is the production function specifying how much output can be produced with any combination of inputs, and q is the fixed level of output required.

The resulting factor demand functions are of the general form

$L(w, r\,; q)$

and

$K(w, r\,; q).$

## References

1. ^ Varian, Hal., 1992, "Microeconomic Analysis" 3rd Ed., W.W. Norton & Company, Inc. New York.