Production involves transformation of inputs into outputs. The output is a function of input. The functional relationship between physical inputs and physical output of a firm is called production function. The word ‘function’ in mathematics means the precise relationship that exists between one dependent variable and a number (or one) of independent variables.

The production function states the maximum quantity of output that can be produced from any given quantities of various inputs during a given period of time. In brief, the production function is a catalogue of different output possibilities. Alternatively, it states the minimum quantity of inputs necessary to produce a given quantity of output. Algebraically, a production function can be stated as

Q = f (a, b, c………… n)

The above production function tells as the quantity of the output ‘Q’ which is produced by the given quantities of inputs of a, b, c……. n. Thus production function expresses the technological relationship between the quantity of output and the quantities of the various inputs used for the production. If the state of technology changes, the production function also changes. If a carpenter produces wooden tables in a day, its production function consists of maximum number of tables that can be produced from a given quantities of various inputs such as wood, varnish, labour time, machine time and floor space. It is flow of inputs resulting in a How of output during a specified period of time.

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**I. It is a technical relation.**

The engineer sees that the various combinations of inputs are applied and the output resulting from them by using a particular process of production. There are many processes of production and for each process there is a relationship between various combinations of inputs and resulting output.

**2- It has economic importance.**

Production function has got an economic importance for the entrepreneurs. It helps the entrepreneurs to minimize the output form a given combination of inputs.

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**3. Production functions differ from firm to firm:**

Each firm has its own production function. This production function is determined by the state of technology. If the state of technology changes the old production function is disturbed.

**Assumptions of production function**

1. It is associated with specified period of time.

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2. The state of technology is constant during the period of time.

3. The producer is expected to use the best and the most efficient technique.

4. The factors of production are divisible.

Production function is stated with reference to a particular period of time. In economics we are concerned with two types of production function :(i) the production function when the quantities of some inputs are constant and the quantity of one input is varied. This type of input-output relationship forms the subject-matter of the law of variable proportion. Secondly the productions function with all factors variable. This type of input-output relationship forms the subject-matter of the law of returns to scale.

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In case of short-run production function (variable proportion) some factors held constant and other factors are combined with varied proportion. The ratio of variable factor to that of the fixed factor goes on increasing on the quantity of the variable factor is increased. When all factors are increased in the same proportion the increase in output so obtained represents returns to scale. In the long run all factors are varied.