Production Function in Managerial Economics

Definition of Production Function

The technological relationship between inputs and output of a firm is generally referred to as the production function. The production function shows the functional relationship between the physical inputs and the physical output of a firm in the process of production.

According to Samuelson, “The production function is the Technical relationship telling the maximum amount of output capable of being produced by each and every set of specified inputs. It is defined for a given set of technical knowledge.”

According to Stigler, “The production function is the name given to the relationship between the rates of input of productive services and the rate of output of product. It is the economist’s summary of technical knowledge.

In fact the production function shows the maximum quantity of output. Q, that can be produced as a function of the quantities of inputs X1, X2, X3…Xn.

In equation form the production function can be presented as :

Q = f(X1, X2, X3…Xn, T)


  • Q: Stands for the physical quantity of output produced.
  • f: represents the functional relationship.
  • X1, X2, X3…Xn: indicate the quantities used of factors X1, X2, X3…Xn
  • T  stands for a given State of Technology; Technology held constant.

Production function, thus expresses the technological functional relationship between inputs and output. It shows that output is the function of several inputs. Besides, the Production function must be considered with reference to a particular period of time and for a given state of technology.

It may be remembered that the Production function shows only the physical relationship between inputs and the output. It is basically an engineering concept; whereas selecting an optimal input combination is an economic decision which requires additional information with respect to prices of the factor inputs and the market demand for the output.

Read: Cobb-Douglas Production Function

Short-Run versus Long-Run Production Function

The short run and the long run have no calendrical specificity. These are only functional and analytical period-wise classification. The Short-run is that period of time in which at least one of the factors of production remains fixed. Whereas, the Long-run is that period of time in which all factors are variable. The major determinant of the short-run or long-run time periods is the existence or non-existence of fixed input. When one or more inputs remain constant we consider that period of time as short period; whereas when all inputs are capable of being varied that period is regarded as the long-period.

If we consider a simple production function with two inputs labor (l) and capital (k) and only one output (Q) then we can summarize the short-run production function as :

Q = f (l,k) or Q = f (l, k)

The k or l shows that the amount of that input is fixed.

The long-run production function may be summarized as

Q = f (l, k)

Where both labor and capital are variable inputs. Since in short-run, not all inputs can be varied simultaneously, the proportions in which inputs are combined go on varying. Therefore the analysis of input-output relation depicted by the short-run production function is called the Returns to Variable Proportions. It takes shape in the Laws of Returns. Whereas the long-run production function gives the input-output relationship when all inputs are varied. In fact economists are particularly interested in finding out as to what happens to the output when all inputs are varied proportionately. This analysis of relationship between proportionate change in inputs and the resulting output gives rise to proportionate change in inputs and the resulting output gives rise to Returns to Scale.

Read More: Duality between Product Function and Cost Function