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COBB-DOUGLAS PRODUCTION FUNCTION
The C-D production function is based on the empirical study of the American manufacturing industry made by Paul H Douglass and C.w Cobb during the period 1899 to 1922. It is a linear homogenious production Function of degree one which takes into the account of two inputs that is, Labour and Capital, for a entire output of the manufacturing industry. The general form of Cobb -Douglass production function can be expressed as:- Q = AL αKβ Here:- Q = Output L = Labour input K = Capital input α&β = Possitive parametors ( α>0, β > 0 , α+ β=1) A = Technical change ( it assumed to be constant ) The equation tells that output depends directly on L &K ,and that part of output which cannot be explained by L and K is explained by A which is the Residual often called Technical Change. And A is assumed to be constant.
Properties The following are the important properties of Cobb-Douglass production function:- 1.Constant Returns to Scale C – D production function exhibits constant returns to scale . To prove it , let us increase the quantities of L & K by λtimes and output must also increased by λtimes . Then the increased output (Q*) will be ; Q* = λQ • α+β=1 Constant returns to scale • α+β>1 Increasing returns to scale • α+β<1 Diminishing returns to scale
2. The Average Product (AP) and Marginal Product (MP) of factors The C-D production function tells that the AP & MP of factors is a function of the ratio of the factors . Q = ALαKβ APL = A (K/L)β APK = A (L/K)α MPL = α A (K/L)β MPK = β A (L/K )α 3. Marginal Rate of Substitution between Capital and Labour (MRS LK) The MRS LK can be derived from the C-D production function. MRS LK = ∂Q/∂L ÷ ∂Q /∂K
4. Elasticity of factors substitution The Elasticity of factors substitution of the C-D production function is equal to unity. Its proof, the elasticity of substitution (es) between K&L is defined as; σ = ( f1 × f2 )/(f12 × Q) Here ; • f1 =MPL • f2 = MPK • f12 = cross partial derivative of L& K ∂Q = ∂L × ∂K σ = 1 when the elasticity of substitution is unity the production function is homogenious of degree one. That is constant returns to scale
5. Euler’s theorum • The application of Euler’s theorum to distribution in an other property of the C-D production function . If the production function • Q = f(K/L) is homogenious of degree one , then according to Euler’s theorum • Q = L (∂Q/∂L) + K ( ∂Q/∂K ) • Apply the general form of C-D production function we get Q = Q
6. Factor intensity • In the C-D production function Q = ALαKβ , the factor intensity is measured by the ratio ( α /β ). Higher the ratio , the production function is more labour intensive and lower the ratio , the production function will be capital intensive . • 7. Efficiency of production • The coefficient ‘A’ in the C-D production function helps in measuring the efficiency in the organisation of the factors of production. If two firms have the same α , β , L and K but produce different quantities of output , this difference may be due to the superior organisation of more efficient firm as against the other. The more efficient firm will have a larger ‘ A ‘ than the other firm
8. Multiplicative Function • The C-D production function is a multiplicative function . It means that if an input has zero value , the output will also be zero . This property highlights the fact that all inputs are necessary for production in a firm. • 9. Output elasticity • It can be defined as the proportionate change in output with a given change in input . The output elasticity of L & K can be calculated with the help of C-D production function :- • Output elasticity of labour = ( ∂Q / ∂L ) ÷ ( L / Q ) • Output elasticity of capital = ( ∂Q / ∂K ) ÷ (K / Q )
Criticisms • The C-D production function considers only two inputs , labour and capital and neglects some important inputs like raw materials , which are used in production. • In the C-D production function , the problem of measurement of capital arises because it takes only the quantity of capital available for production . But the full use of the available capital can be made only in periods of fullemployment. This is unrealistic situation. • The C-D production function is criticised because it shows constant returns to scale. But constant returns to scale are not an actuality , for either increasing or decreasing returns to scale are applicable to production.
The C-D production function is based on the assumption of substitutability of factors and neglects the complimentarity of factors. • This function is based on the assumption of perfect competitionin the factor market which is unrealistic. • One of the weakness of C-D production function is the aggregation problem.
Importance • The C-D production function has been used widely in empirical studies of manufacturing industries and in inter-industry comparisons. • It is used to determine the relative shares of labour and capital in total output. And it is also used to prove Euler’s theorum. • Its parametors α and β represents elasticity coefficients that are used for inter-sectoral comparisons. • This production function is linear homogenious of degree one which shows constant returns to scale. • If α + β >1 - increasing returns to scale • α + β <1 - decreasing returns to scale • This production function is more than two variables.
Conclusion • Thus the practicability of the C-D production function in the manufacturing industry is a doubtful proposition. This is not applicable to agriculture where for intensive cultivation , increasing the quantity of inputs will not raise output proportionately.
