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Production Function With Two Variable Inputs

 Production Function with Two Variable Inputs

Illustration 36

You are provided with the following functions where you are required to ascertain the type of production function it relates to.

  1. Q = 4L + 3K
  2. Q = Min(4L.k)

Solution

  1. Production function, Q = 4L + 3 K is a linear function. This is confirmed as under. Let L and K are enhanced by a definite number λ,
  2.                         Q’        =          4 λL + 3 λK

    λ can be featured outside and therefore,

                            Q’        =          λ (4L + 3K)     =          λQ.

    Therefore, increasing each input by λ, productivity also enhances by λ. This depicts this production function is linear homogenous.

  1. Production function, Q = Min (4L.K) is a Leontief production function.
  2. This is so termed as a noted American economist Wassily Leontief used this production function to describe the American economy.

    This a fixed proportion production function in which labour and capital are combined in the ratio of 4L and 1K.

Illustration 37

The following production is provided >> O = A^0.75 * B^0.25.

You are required to ascertain the following:

  1. The marginal product of labour and marginal product of capital
  2. Present that law of diminishing returns holds
  3. Present that if labour and capital are paid booty parity to their marginal products, aggregate product would be exhausted
  4. Compute the marginal rate of technical substitution of capital for labour
  5. Elasticity of Substitution

Solution

(1) MPA           =         dO       =          0.75A^0.75-1 * B^0.25
                                    dA

                                                =          0.75A^-0.25 * B^0.25

                                                =          0.75(B/A) ^0.25

            MPB     =         dO       =          0.25A^0.75 * B^0.25-1
                                   dB

                                                =          0.25A^0.75 * B^-0.75

                                                =          0.25(A/B) ^0.75

(2) Law of diminishing returns holds, if marginal product of labour,

            MPA    =          0.75(B/A) ^0.25 decreases with enhancement in the volume of labour. Now, provided the constant figures 0.75 and 0.25, if labour (A) enhances, capital (B) held constant, the term B/A will decrease and thus the marginal product of labour (=0.75(B/A) ^0.25) too will decrease.

(3) For the aggregate product to be exhausted if factors are paid their proceeds parity to their marginal products, the following equation which is called Euler’s theorem must be possessed,

                        O = MPA*A + MPB*B

Substituting the values of MPA and MPB procured from the provided production function in the above section (1) in Euler’s theorem we obtain the following:

                        O         =          0.75(B/A) ^0.25 * A      +          0.25(A/B) ^0.75 * B

                        O         =          0.75B^0.25 * A  +          0.25A^0.75 * B^0.25

                        O         =          A^0.75 * B^0.25 * (0.75 + 0.25)

                        O         =          A^0.75 * B^0.25 * 1

                                    =          A^0.75 * B^0.25

As A^0.75*B^0.25     =          O (that is provided)

Thus,              O         =          O

Therefore, product exhaustion of Euler’s theorem is applicable to provided production function.

(4) MRTSAB parities to the ratio of marginal products of labour and capital. Using the marginal products of labour and capital we have the following,

                        MRTSAB          =          MPA
                                                            MPB

                                                =          0.75(B/A) ^0.25
                                                            0.25(A/B) ^0.75

                                                =          3(B/A)

(5) Elasticity substitution        =         Δ (B/A) / B/A
                                                            Δ MRS / MRS

Substituting the value of MRS           =          3 (B/A), we procure,

Substitution Elasticity                       =       Δ (B/A) / (B/A)         =          1
                                                                        3 Δ (B/A) / 3 (B/A)

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