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Several types of mathematical functions are commonly employed in the measurement of production function but in applied research, four types have had the widest use. These are linear functions, power functions, quadratic functions, and cubic functions.
1. Linear Function:
A linear production function would take the form –
Total product, Y = a + bX
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From this function, equation for average product will be –
Y/X = a/X + b
The equation for the marginal product will be –
ΔY/ΔX = b
2. Power Function:
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A power function expresses output, Y, as a function of input X in the form –
Y = aXb
Some important special properties of such power functions are mentioned below:
1. The exponents are the elasticities of production. Thus, in the above function, the exponent b represents the elasticity of production.
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2. The equation is linear in the logarithms, that is, it can be written –
Log Y = log a + b log X
When the power function is expressed in logarithmic form as above, the coefficient b represents the elasticity of production.
3. If one input is increased while all others are held constant, marginal product will decline.
3. Quadratic Function:
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The production function may be quadratic, taking the following form-
Y = a + bX – cX2
where the dependent variable, Y, represents total output and the independent variable, X, denotes input. The small letters are parameters; their probable values, of course, are determined by a statistical analysis of the data.
The special properties of the quadratic production function are as under:
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(i) The minus sign in the last term denotes diminishing marginal returns.
(ii) The equation allows for decreasing marginal product but not for both increasing and decreasing marginal products.
(iii) The elasticity of production is not constant at all points along the curve as in a power function, but declines with input magnitude.
(iv) The equation never allows for an increasing marginal product.
(v) When X = O, Y = a. This means that there is some output even when no variable input is applied.
(vi) The quadratic equation has only one bend as compared with a linear equation which has no bends.
4. Cubic Function:
The cubic production function takes the following form –
4 = a + bx + cX2 – dX3
Some important special properties of a cubic production function are:
(i) It allows for both increasing and decreasing marginal productivity.
(ii) The elasticity of production varies at each point along the curve.
(iii) Marginal productivity decreases at an increasing rate in the later stages.
A very popular production function which deserves special mention is the Cobb-Douglas function. It relates output in American manufacturing industries from 1899 to 1922 to labor and capital inputs, taking the form –
P = bLαC1-α
where P = Total output
L = Index of employment of labor in manufacturing
C = Index of fixed capital in manufacturing
The exponents α and 1 – α are the elasticities of production that is, α and 1 – α measure the percentage response of output to percentage changes in labor and capital, respectively. The function estimated for the USA by Cobb and Douglas is –
P = 1.01 L.75 C25
R2 = 0.9409
The production function shows that a 1 per cent change in labor input, capital remaining constant is associated with a 0.75 per cent change in output. Similarly, a 1 per cent change in capital, labor remaining constant, is associated with a 0.25 per cent change in output. The coefficient of determination (R2) means that 94 per cent of the variations on the dependent variable (p) were accounted for by the variations in the independent variables (L and C).
An important point to note is that the Cobb-Douglas function indicates constant returns to scale. That is, if factors of production are each raised by 1 per cent. The output will increase by 1 per cent. This indicates that no economies or diseconomies or large scale are evident- on the average, large or small-scale plant may be equally profitable in the US manufacturing industry. In other words, one can assume constant, average and marginal production costs for the US industries during period.
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