# Theory Of Production - Returns To One Variable Factor

__Illustration 32__

You are provided with following production function.

Q = L^0.50 K^0.50

- Determine the marginal product of labour

- If the fixed volume of capital in the short run parities 1000 units, what is the short run production function?

- Depict the marginal product of labour MPL is less than average product of labour AP in the short run production function in (2).

__Solution__

- To obtain marginal product of labour, we distinguish the provided production function with respect to labour. Therefore

MPL = __dQ__ = 0.50L^-0.50
K^0.50

dL

= 0.50(K/L) ^ 0.50

- Note that in the short run production function, one factor is variable with the quantity of other fixed factor. Therefore,

Q = L^0.50 K ^0.50

Q = L^0.50 (900)^0.50

Since the square root of 900 is 30,

Q = L^0.50 * 30

= 30L^0.50

Therefore, short run production function is Q = 30L^0.50

MPL =__dQ__

dL

= 30 * 0.50L^-0.50

= __ 15 __

L^0.50

APL = Q
/ L

= __30L^0.50__

L

= 30L^0.50-1

= __ 30 __

L^0.50

Relating the values of MPL and APL we determine that MPN < APL

__Illustration 33__

Let us assume the following production function of an industry,

O = 3L^2 – 0.2L^3

Where O is output and L is the volume of Labour used.

- Ascertain the MPL
- Value of L that optimises output O
- Value of L at which its Average Product is optimum

__Solution__

It is noted that the above function is short run production function as there is no fixed term in it; all terms in it contain the variable factor, Labour (L).

(1) MPL = __dO__

dL

= 2 * 3L – 3 * 0.2L ^ 2

= 6L – 0.6L^2

To determine the average product AP of labour, we must divide the aggregate output by L, therefore,

APL = __O__ = __3L^2 – 0.2L^3__

L L

= 3L – 0.2L^2

- We can determine the value of the variable factor L that optimises output O and also the value of labour L at which its average product is optimum.

The value of variable input L optimises output O that ca be procured by setting marginal product function of the variable input equal to null. We have obtained above that MPL = 6L – 0.6L^2. Setting it equal, to zero we have

6L – 0.6L^2 = 0

0.6L^2
= 6L

__0.6L^2__ = 6

L

0.6L = 6

L = 6 / 0.6 = 10

At 10 units of labour, value of O will be optimum.

- Value of L at which its AP is optimum:

Value of average product function will be optimised where its first derivative equals zero.

AP of labour procured above = 3L – 0.2L^2

= __dAP__ = 3 – 0.4L = 0

dL

0.4L = 3

L = 3 / 0.4 = 7.5

Therefore, when 7.5 units of labour are used its average product will be optimum.

__Illustration 34__

Presume a firm producing cotton cloth has the following production function.

O = 4K^ ½ * 2L^ ½

Ascertain the marginal product of labour and capital.

__Solution__

This is Cobb-Douglas production function specific values of exponents.

MPL = __dO__

dL

= 4 * (½) K^ ½ * 2L ^ ½ – 1

= 2 √ K * 2 -√L

= __K__

L

MPK = __dO__ = 4
* (1/2) K^ ½ - 1 * 2 L^ ½

dK

= 2K^- ½ * 2 √ L

= __2L__

K

__Illustration 35__

Let us assume the following production function

O = 1.50 A^0.75 B^0.25

Determine the elasticity of productivity O with respect to A – Labour and B – Capital. Provide an economic interpretation of this productivity elasticity.

__Solution__

Provided the production function is

O = 1.50 A^0.75 B^0.25

Productivity elasticity of labour EA = __MP____A__

APA

MPA = __dO__ = 0.75
* 1.50L^-0.25 * B^0.25

dA

APA = __O__ = 1.125A^0.75
* B^0.25

A

= __1.50A
^0.75 * B^0.25__

A

= 1.50A ^-0.25 * B^0.25

EA = __MP____A__ = __1.125A^-0.25
* B^0.25__

APA 1.50A
^-0.25 * B^0.25

= 0.75

Productivity elasticity of capital EB = __MP____B__

APB

MPB = __dO__ = 0.25
* 1.50 A^0.75 * B^-0.75

dB

= 0.375 L^0.75 * K^-0.75

APB = __O__ = __1.50A^0.75
* B^0.25__

B B

= 1.50A^0.75 * B^-0.75

EB = __MP____B__ = __0.375
A^0.75 * B^-0.75__

APB 1.50A^0.75
* B^-0.75

= 0.25

From the value of elasticity of labour equal to 0.75 it follows that 1 percent enhancement in employment of labour causes 0.75 enhancements in productivity, which is less than ration. Likewise, productive elasticity of capital being equal to 0.25 entails that one percent enhancement in capital causes 0.25 percent enhancement in productivity of the product.

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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
- Production Function with Two Variable Inputs
- Short Run Cost Function
- Survival Technique