# Production Functions

A production function is an equation that establishes relationship between the factors of production (i.e. inputs) and total product (i.e. output). There are three main types of production functions: (a) the linear production function, (b) the Cobb-Douglas production and (c) fixed-proportions production function (also called Leontief production function).

The linear production function and the fixed-proportion production functions represent two extreme case scenarios. The linear production function represents a production process in which the inputs are perfect substitutes i.e. one, say labor, can be substituted completely with the capital. The fixed fixed-proportion production function reflects a production process in which the inputs are required in fixed proportions because there can be no substitution of one input with another. The Cobb-Douglas production function represents the typical production function in which labor and capital can be substituted, if not perfectly.

## Linear Production Function

A linear production function is of the following form:

$$ \text{P}\ =\ \text{a}\times \text{L}+\text{b}\times \text{K} $$

Where P is total product, a is the productivity of L units of labor, b is the productivity of K units of capital.

Let’s consider A1A Car Wash. A worker working in 8-hour shift can wash 16 cars and an automatic wash system can wash 32 cars in 8 hours. The owner of A1A Car Wash is faced with a linear production function. If she must cater to 96 motorists, she can either use zero machines and 6 workers, 4 workers and 1 machine or zero workers and 3 machines.

A linear production function is represented by a straight-line isoquant.

## Fixed-Proportion (Leontief) Production Function

The fixed proportion production function is useful when labor and capital must be furnished in a fixed proportion. The equation for a fixed proportion function is as follows:

$$ \text{Q}=\text{min}(\text{aK} \text{,} \ \text{bL}) $$

Where Q is the total product, a and b are the coefficient of production of capital and labor respectively and K and L represent the units of capital and labor respectively.

The total product under the fixed proportions production function is restricted by the lower of labor and capital.

Let’s consider A1A Car Wash which is open for 16 hours each day. It has 3 wash bays and 4 workers. If a car wash takes 30 mins of worker time and 30 mins of wash bay occupancy, the total number of washes possible will depend on which factor is the limiting factor i.e. which one runs out first as shown below:

$$ \ \text{Q}=\text{min}\left(\frac{\text{16}}{\text{0.5}}\times\text{3} \text{,} \ \frac{\text{8}}{\text{0.5}}\times\text{4}\right)=\text{min}\left(\text{96,64}\right)=\text{64} $$

It is because due to lower number of workers available, some wash bays will stay redundant.

A fixed-proportion production function corresponds to a right-angle isoquant.

## Cobb-Douglas Production Function

The Cobb-Douglas production function allows for interchange between labor and capital. It represents the typical convex isoquant i.e. an isoquant in which labor and capital can be substituted with one another, if not perfectly.

The Cobb-Douglas production function is represented by the following formula:

$$ \text{Q}=\text{A}\times \text{K}^\text{a}\times \text{L}^\text{b} $$

Where Q is the total product, K represents the units of capital, L stands for units of labor, A is the total factor productivity, and a and b are the output elasticities of capital and labor respectively.

by Obaidullah Jan, ACA, CFA and last modified on