# Fixed-Proportions Production Functions

by Gerald Hanks; Updated September 26, 2017A production function represents the mathematical relationship between a business's production inputs and its level of output. Production capital includes the equipment, facilities and infrastructure the business uses to create the final product, while production labor quantifies the number of man-hours needed to complete the process from start to finish. A fixed-proportions production function is a function in which the ratio of capital (K) to labor (L) does not fluctuate when productivity levels change.

### Examples of Fixed-Proportions Production Functions

In a fixed-proportions production function, both capital and labor must be increased in the same proportion at the same time to increase productivity. When the production function is displayed on a graph, with capital on the horizontal axis and labor on the vertical axis, the function appears as a straight line with a constant slope. For instance, a factory requires eight units of capital and four units of labor to produce a single widget. The factory must increase its capital usage to 40 units and its labor usage to 20 units to produce five widgets.

### Fixed-Proportions and Substitutions

The production function identifies the quantities of capital and labor the firm needs to use to reach a specific level of output. The measure of a business's ability to substitute capital for labor, or vice versa, is known as the elasticity of substitution. In a fixed-proportions production function, the elasticity of substitution equals zero. This means that adding an additional unit of capital without adding additional labor will have no effect on increasing productivity. Both factors must be increased in the same proportion to increase output.

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