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Production means transformation of physical inputs into physical output. All those goods and services that a firm uses in the process of production may be termed as 'inputs.' For example a firm may be using labour, capital and land but in addition to these it may be using fuel, raw materials, steel, dyes and number of other things.

A Production function expresses a relationship between input and output under conditions of given technology. Thus, given the technical conditions of production, the output of a particular commodity depends on the quantity of inputs. The production function is expressed as follows-

Here Q stands for production or output of a commodity.

a, b, c, d, etc. stand for inputs or productive resources which are used for producing the commodity.

This function show the quantity of output Q depends on the quantities of the factors of production A, B, C, D. This shows that there exists some relationship between output Q and quantities of inputs a, b, c, d etc. but it does not tell us the specific form of relationship. This unspecified relationship is denoted by f.

If the form of the function is given, we can find by using mathematical methods of quantitative nature of relationship.

For eg:- homogeneous production of the first degree, enjoys wide popularity among the economists, implies that if factors of production are increased in given proportion, output also increases in the same proportion.

Hence, linear homogeneous production function represents the case of a constant returns to scale. If there are only two factors x and y, then homogeneous production function of the first degree can be mathematically expressed as-

Where m is any real number.

The above function means that if factors x and y increased by m-times, output Q also increases m-times.

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