Conditional factor demands

Typically this concept arises in a long run context in which both labor and capital usage are choosable by the firm, so a single optimization gives rise to conditional factor demands for each of labor and capital.

In the simplest mathematical formulation of this problem, two inputs are used (often labor and capital), and the optimization problem seeks to minimize the total cost (amount spent on factors of production, say labor and physical capital) subject to achieving a given level of output, as illustrated in the graph.

Each of the convex isoquants shows various combinations of labor and capital usage all of which would allow a given amount of output to be produced.

At the tangency the marginal rate of technical substitution between the factors (the absolute value of the slope of the isoquant at the optimal point) equals the relative factor costs (the absolute value of the slope of the isocost curve).

This optimization can be formalized as follows: where L and K are the chosen quantities of labor and capital, w and r are the fixed unit costs of labor (wage rate) and capital (rental rate) 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.