# 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.

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

__Solution__

- 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,

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.

- Production function, Q = Min (4L.K) is a Leontief production function.

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:

- The marginal product of labour and marginal product of capital
- Present that law of diminishing returns holds
- Present that if labour and capital are paid booty parity to their marginal products, aggregate product would be exhausted
- Compute the marginal rate of technical substitution of capital for labour
- 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,

** MP****A = ****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 = __MP____A__

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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**Other topics under Theory of Production and Cost analysis:**

- Break Even and Leverage Analysis
- Concept of Cost
- Cost Volume Profit Analysis for Accomplishing Target Profits
- Elasticity of Supply and Its Function
- Establishment of Cost Function Analysis
- Establishment of Short Run Cost Function
- Estimation of Returns To Scale
- Isoquants, Equal Product Curves
- Linearity Assumptions
- Linearity Assumptions and Choice of Product and Process
- Long Run average Cost Curve
- Optimum Input Combination
- Short Run Cost Function
- Survival Technique
- Theory of Production - Returns to One Variable Factor