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Mathematics for Economists

Note 3. Marginal production of production function is always moves downward viz
2
2
w P 0 and w P 0
wM 2 wC 2

Task What is the merit of C.E.S. Production function?

7.4.2 Advantages of CES Production Function over Cobb-Douglas
Production Function

1. Compared to Cobb-Douglas Production Function, CES Production Function brings more
general technique. Under CES Production Function, elasticity of substitution is constant and
it is not necessary that this elasticity is equal to a unit.
2. Compared to Cobb-Douglas Production Function, CES Production Function keeps more
important parameters. This way it has a wide area in substitutability and efficiency.
3. Cobb-Douglas Production Function and CES Production Function have a special form. If in
CES function a = 0 then we will get Cobb-Douglas Production Function.
4. It is easy to find out the parameters under CES Production Function. Besides this function has
removed all the problems and unrealistic assumptions of Cobb-Douglas Production Functions.

7.4.3 Limitation of CES Production Function

Although CES production function has removed all the problems and unrealistic assumptions of
Cobb-Douglas Production Function and is widely used in economics, but yet it is criticized:
1. Like Cobb-Douglas Production Function, this function also considers only two factors of
production (Labor and Capital). It does not apply to other factors of production. Prof H.
th
Uzawa says it is difficult to apply this function on n factor of production.
For example: if A ab is production function, find out the demand of factor a and b, where
their prices are constant at P and P . If demand curve x = b– ap, what is the factor demand in
b
a
term s of prices and constants.
Solution: Given production function
x =$ ab
b
= Aa 1/2 1/2 ...(i)
Differentiating a and b from equation (i) separately
wx 1 Aa 1/2 1/2
b
wa = 2
wx 1 1/2 1/2
And = Aa b ...(ii)
wb 2
From equation (i) and (ii)

1 Aa 1/2 1/2 x
b
MP = xA ab
a 2 a 2a
b
1 Aa 1/2 1/2 x
MP = 2 b 2b
b

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