Page 93 - DECO405_MANAGERIAL_ECONOMICS

Managerial Economics

Notes 6.2 Production Function with One Variable Input

A production function is a function that specifies the output of a firm, an industry, or an entire
economy for all combinations of inputs. In other words, it shows the functional relationship
between the inputs used and the output produced.
Mathematically, the production function can be shown as:
Q = f (X , X ..................X )
1 2 K
where Q = Output, X .............. X = Inputs used.
1 K
For purposes of analysis, the equation can be reduced to two inputs X and Y. Restating,
Q = f (X, Y)
where Q = Output
X = Labour

Y = Capital
A more complete definition of production function can be:
‘A production function defines the relationship between inputs and the maximum amount that
can be produced within a given period of time with a given level of technology’.

A production function can be stated in the form of a table, schedule or mathematical equation.
But before doing that, two special features of a production function are given below:
1. Labour and capital are both unavoidable inputs to produce any quantity of a good, and

2. Labour and capital are substitutes to each other in production.
A form of production functions is the Constant Elasticity of Substitution, CES function,
–h –1h
–h
Q = B[gL + (1 – g)K ]
where h > –1 and B, g and h are constants.
If h is assumed to be a variable, then the above function may be called the variable elasticity of
substitution, VES function.
Still another form is the fixed proportion production function also called the Leontief function.
It is represented by
K L
Q = minimum , , where a and b are constants and ‘minimum’ means that Q equals the
a b
smaller of the two ratios.
Finally there is a very simple linear production function. Assuming that the inputs are perfect
substitutes so that all factors may be reducible to one single factor, say, labour, L, than the linear
production function may be,
Q = aL, where ‘a’ is the constant term and L stands for labour.

In order to analyse the relationship between factor inputs and outputs, economists classify time
periods into short runs and long runs.

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