The production function is the functional relationship between the quantity of goods produced (output) and the factor of production (inputs) used to produce them. 

A production function describes the relationship between the quantities of productive factors (labour and capital) employed and the amount of output produced.

The production function may also be defined as a mathematical relationship that defines the maximum amount of output that can be produced with a given set of input, assuming that the most available efficient methods of production are used.

Given two inputs, the production function shows the maximum output that can be produced using different combinations of capital and labour.

Mathematically, the production function is denoted as the maximum amount of output that can be produced with K units of capital and L units of Labor.

 $$Q=F(K, L  )$$

where Q is the level of output

             K is the quantity of capital input

             L is the quantity of labour input.

Characteristics of production functions.

1. Connects inputs and output: The production function depicts the functional relationship between the quantities of inputs used and the outputs produced

It shows the maximum output that can be obtained from a given quantity of inputs.

2.  No monetary significance: A production function has no monetary significance as it only shows the physical relationship between inputs and outputs.

That is, neither the monetary cost of inputs nor the price of products sold are shown in the production function. 

It only shows the functional relationship between physical inputs and physical outputs produced.

3. Affected by a change in technology: A technological change can allow a firm to produce more output with a given set of inputs, which can affect the production function.

The production function is determined by the state of technology. Any technology change will affect production function.

However, production functions require that the state of technology is given and held constant. 

4. Related to time:  Production functions, like demand functions, are always considered in terms of a specific period. It depicts a flow of output as a result of a flow of inputs over a set period.

Types Of Production Functions

1. Linear homogeneous production function: This indicate that a proportionate change in input will result in an equal proportionate change in output.

In this type of production function, the factor inputs are perfect substitutes.

The linear production function is usually in the form

$$F(K, L)=aK+bL$$

Where $a$ and $b$ are constant.

One feature of linear homogeneous production function is that the degree of production function is equal to one, suggesting that the production function has a constant return to scale.

A production function is said to exhibit constant returns to scale when doubling of inputs yields in a precise doubling of outputs. This is the case with linear homogeneous production functions.

The isoquant of linear homogeneous production is a straight line, implying a constant marginal rate of technical substitution.

2. Cobb-Douglas production function: This is a long-run production function named after Paul Douglas, an American economist, and Charles Cobb, a mathematician.

A Cobb-Douglas production function describes the relationship between a specific amount of output and a set of inputs that are substitute but not perfect substitutes, allowing for the substitution of factor inputs to a certain extent. 

In other words, a cobb-Douglas production is a production function in which inputs can be substituted by another, but only to a limited extent.

That is, capital and labour can be substituted to a limited extent.

The relationship depicted by Cobb-Douglas Isn’t exactly linear as the variables representing factor inputs aren’t raised to 1.

They are raised to some other quantities that indicate their elasticities. 

Mathematically, the Cobb-Douglas production function is represented in the form:

$$Q = AK^aL^b$$


A = positive constant

a and b =positive fractions

b = 1–a

Because $b=1-a$, the Cobb-Douglas production function can also be written as:

$$Q = AK^aL^{1-a}$$

3. Constant elasticity of substitution production function: This production function displays a homogeneity of one. 

This indicates that a change in the factor inputs will result in an exact corresponding change in the output level. 

CES production function is represented as:

$Q = A [aK^{β}+ (1-a)L^{-β}]^{-1/β}$

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4. Leontief production functions: This type of production function, introduced by Wassily Leontief, does not allow for any substitution of factor inputs.

That is, no substitution of labour and capital is possible.

The Leontief production function is also called the fixed-proportions production function because it implies that inputs are used in fixed proportions to one another.

In other words, the Leontief production function implies that Capital and Labor must be used in a fixed proportion.

The isoquant of leontief production function is L-shaped indicating that two inputs are perfect complements, as there cannot be substituted for one another.

It is represented mathematically as:

$$q= min (\frac{z_1}{a}, \frac{z_2}{b})$$

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