In economics, a conditional factor demand function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 specifies the cost

Cost

In production, research, retail, and accounting, a cost is the value of money that has been used up to produce something, and hence is not available for use anymore. In business, the cost may be one of acquisition, in which case the amount of money expended to acquire it is counted as cost. In this...

-minimizing level of an input (factor of production) such as labor or capital

Capital (economics)

In economics, capital, capital goods, or real capital refers to already-produced durable goods used in production of goods or services. The capital goods are not significantly consumed, though they may depreciate in the production process...

, required to produce a given level of output

Output (economics)

Output in economics is the "quantity of goods or services produced in a given time period, by a firm, industry, or country," whether consumed or used for further production.The concept of national output is absolutely essential in the field of macroeconomics...

, for given unit input costs (wage rate and rental rate) of the input factors. 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

Labor demand

In economics, labor demand refers to the number of hours of hiring that an employer is willing to do based on the various exogenous variables it is faced with, such as the wage rate, the unit cost of capital, the market-determined selling price of its output, etc...

 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

subject to

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

Production function

In microeconomics and macroeconomics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs...

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


and