Under identical conditions, 10 workers move more bricks than 2 workers. A larger firm produces more quantity of a product than what a smaller firm does. Using the term ‘output’ for the quantity of a product produced and ‘inputs’ for the amount of factors of production employed for the purpose, we can say that output varies directly with inputs.
A situation may arise later when increasing an input, other things remaining the same, may lead to a fall in the output. A mathematical relationship between units of output and units of inputs is known as a production function.
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In the example above, let the number of bricks moved be Q and the number of workers, L. Assuming that one worker moves 500 bricks per day on an average, the number of bricks that would be moved per day can be mathematically expressed as
Q = 500L
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Equation (4.1) expresses output Q in terms of input L and shows how Q varies with the variations in L. If L is 1, Q is 500; if L is 2, Q is 1,000 and if L is 10, Q is 5,000. A mathematical relationship such as this in Equation (4.1), provides an example of the production function. In general, it can be expressed as
Q = f (L) = α L
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Where, α is a symbolic representation of 500. Equation (4.2) represents a production function of a firm employing a single variable factor, labour.
The production function of a firm employing all the four factors of production, in like manner, can be expressed as
Q = f (B, L, K, E)
Where, B = units of factor land and building
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L = units of factor labour
K = units of factor capital
E = units of factor entrepreneur
Likewise, the production function for a firm employing two factors of production, namely, L and K, can be expressed as
Q = f (L, K)
A typical production function involving two variable factors of production is the Cobb Douglas production function which is commonly used in the literature on theory of production
Q =A Lα Kβ
Where, A = a constant, called efficiency parameter
L = quantity of labour
K = quantity of capital
α, β = positive constants, the ratio α / β = factor intensity
For the purpose of numerical illustration, let A be 100, a be 1 / 3 and β be 2 / 3 Further
Let L be 1,000 units of labour and K be 216 units of capital. The quantity of output then is given as
Q = 100 × (1000)1/3 × (216)2/3 = 100 × (10) × (36)
= 36,000 units of output
Factor intensity, α / β = (1 / 3) / (2 / 3) = 1 / 2. The higher this ratio, the more labour intensive is the technique of production. Similarly, the lower the ratio, the more capital intensive is the technique of production.
Likewise, the higher the value of the efficiency parameter, the higher is the efficiency of production. For instance, if A = 200 in this illustration, Q = 72,000 units of output despite same quantities of labour and capital inputs.
You will learn more about this production function in higher classes. Production functions for firms employing only three factors of production can be expressed in the same manner. Production, from the viewpoint of variability of the factors of production, is classified into the following four categories
(i) Very Short Run Production:
This refers to that period in which no factor of production can be increased to increase output if so desired.
(ii) Short Run Production:
This refers to that period in which not more than one factor of production can be increased to increase output if so desired. Usually, it is labour that can be increased in short run.
(iii) Long Run Production:
This refers to that period in which not more than two factors of production can be increased to increase output if so desired. Usually, it is labour and capital which can be increased in this case.
(iv) Very Long Run Production:
This refers to that period in which it is possible to increase more than two factors of production with a view to increasing output. Increasing factors other than labour and capital generally requires a period long enough to increase all the factors of production.
Very short run production is too static to require any analysis. In like manner, very long run production too is too dynamic to require any analysis. In latter case, everything changes including technology, and above all, the tastes and preferences of consumers.
This might render the product completely obsolete and out of demand. On the basis of these arguments, it is only short run and long run productions that deserve analysis.