# Short Notes: Production # Production Function Short Run and Long Run

It is a functional relationship between physical output and physical inputs. It shows how output responds to different levels of inputs. It is a technical relationship, not any economic relationship. It may be expressed as under:

Qx = f ( X1, X2, X3 ….) i.e output depends on various inputs ( factor as well as non factors)

On assumption of input to be only labour and capital, Production function can be expressed as Qx = f( L, K )

Often, L is taken as a variable factor and K as a fixed factor, and throughout our analysis we shall follow this practice.

On the basis of time period, production function is categorised as :

(i) Short Run Production Function, and

(ii) Long Run Production Function.

Details are as under:

Short Run Production Function

Short run is the time period in which output can be varied only by changing the variable inputs, like labour. The inputs that remain fixed are called fixed inputs or fixed factors.

Short run production function is a relation between inputs and output for a given technology in which output can be varied by changing one factor (say labour) only. All other factors remain fixed. Mathematically Q = f (L) i.e., output is a function of labour.

Long Run Production Function

Long run is the time period in which distinction between fixed and variable inputs disappears. All inputs are variable. So output can be varied by changing all the inputs simultaneously.

Long run production function is a relation between inputs output for a given technology in which output can be varied by altering all factor inputs simultaneously. Here, factor ratio remains constant.

Distinction between Short Run Production Function (SRPF) and Long Run Production Function (LRPF)

Following are the points of distinction:

1. In SRPF, output changes are accompanied by changes in factor proportions whereas in LRPF, factor proportion does not change whatever may be the level of output.

2. In SRPF, output is varied by altering one factor of production whereas in LRPF, output is sought to be increased by altering all the inputs.

3. The scale of output does not change with change in level of output in case of SRPF. However in case of LRPF, the scale of output changes when there is change in level of output.

Note : Scale of output refers to production capacity of the firm. Often it is measured in terms of size of the plant/machinery installed by the firm. It is a fixed factor during the short period. Short period is a period of time during which fixed factors (or size of the plant or scale of production) cannot be changed. In the long period, all factors are variable including the plant /machinery. Accordingly, during the long period, scale of production tends to change.

# Concepts of TP (Total Product), MP (Marginal Product) and AP (Average Product)

Total Product (TP)

TP refers to output corresponding to all the units of the variable factor used in production. We are taking L (labour) as a variable factor. For different levels of employment of labour, we get different levels of output. Upon adding all these different levels of output, we get total product. Example: if 5 units of labour are used and contribution of each in total output is 8, 10, 13, 11 and 10 respectively then TP is 8+10+13+11+10 = 52.

Average Product (AP)

Average product is estimated as total product divided by number of units of variable factor. Thus, AP = TP/Q

AP is the output per unit of the variable factor.

Marginal Product (MP)

Marginal product is the change in TP per unit change in the variable factor. Thus, MP =ΔTP

It is also estimated as :

MPL = TPn – TPn-1

Note that:

Since input (labour here) can’t take negative values, marginal product is undefined at zero level of input (of labour). However, MP may be negative under certain situations.

Example: Using the following data, find AP and MP.

 Labour 0 1 2 3 4 5 TP 0 12 26 42 52 60

Ans. Estimating AP and MP, given TP

 Labour TP MP AP 0 0 – 0 1 12 12 – 0 = 12 12 2 26 26 – 12 = 14 13 3 42 42 – 26 = 16 14 4 52 52 – 42 = 10 13 5 60 60 – 52 = 8 12

Since AP = TP/L,

TP = AP x L.

If 4 units of labour are employed and AP = 13, then TP = 13 x 4 = 52.

As noted earlier, TP is also equal to ΣMP i.e., TP = ΣMP.

TP = 12+14+16+10+8 = 60

# Returns to a Factor : Law of Diminishing Returns or The Law of Variable Proportions

Let us assume that a firm?s production function is of the following type : Where, Q = Output

L = Labour, a variable factor

K = Capital, a fixed factor.

Now the firm can increase output by changing labour only. When the firm expands output by increasing labour, it alters the proportion between the fixed factor and variable factor. As a result marginal product of labour  may initially rise when more of labour is employed. But after reaching a certain level of employment, it must start falling. Why? Because increasing application of labour is bound to cause a situation of over-exploitation of capital (the fixed factor), owing to which productivity of labour must fall. This defines the law of variable proportions. It states: when output is increased by employing additional units of the variable factor (fixed factor remaining constant) MP of the variable factor must ultimately start declining, even when it tends to rise initially. This is also known as law of diminishing marginal product. This law is explained with the help of Table 1 and Fig. 1.

Table 1.
Law of Variable Proportions :
Behaviour of TP and MP, and Stages of Production

 Units of Land Units of Labour Total Product Marginal Product Stage of Production 1 1 2 2 Stage I:Stage of increasing MP. Also called a stage of increasing returns to a factor Stage II:Stage of diminishing MP. Also called a stage of diminishing returns to a factor. Stage III: Stage of negative returns. Also called a stage of negative returns to a factor. 1 2 5 3 1 3 9 4 1 4 12 3 1 5 14 2 1 6 15 1 1 7 15 0 1 8 14 -1

Behaviour of TP and MP, and Stages of Production Fig. 1

The Table 1 shows that as more and more units of labour are combined with fixed amount of land, MP (marginal product) tends to rise till 3 units of labour are employed. In this situation, TP (total product) increases at the increasing rate.

This is a situation of increasing returns to the factor. But, with the application of 4th unit of labour, situation of diminishing returns sets in: MP starts decreasing and TP increases only at the decreasing rate.

Diminishing MP reduces to zero. Total output is maximum (= 15), when marginal output is zero.

Eventually, MP may be negative. Now output (TP) starts declining as from 15 to 14 when 8th unit of the variable factor is employed.

Fig. 1 shows that:

(i) MP tends to rise till OL units of labour are used with the constant application of land. This corresponds to point E on the MP curve. This is a situation of increasing returns to a factor.

(ii) When MP is rising, TP tends to rise at an increasing rate. This occurs till point K on the TP curve. This corresponds to the situation of increasing returns to a factor.

(iii) Beyond OL units of labour, MP tends to decline, and TP increases only at diminishing rate. This occurs between E and S on MP curve and between K and T on TP curve. This corresponds to a situation of diminishing returns to a factor.

(iv) When employment of labour exceeds OS units, MP becomes negative. Accordingly, TP starts declining. This is a situation of negative returns to a factor, occurring beyond point T on TP curve and beyond point S on MP curve.

Three Stages of Production

Law of variable proportions suggests three stages of production, as under:

(i) Stage 1, when MP is increasing, called the stage of increasing returns,

(ii) Stage 2, when MP is diminishing, called the stage of diminishing returns, and

(iii) Stage 3, when MP is negative, called the stage of negative returns.

Now the moot point : In which stage a producer would like to operate?

‘Stage-II’ is the answer. Why?

A producer will not like to remain in Stage-I although his MP is rising and TP is rising at an increasing rate. The firm, in this stage, always has incentive to expand and earn more profit. So it is always willing to come out of this stage and earn more profit through expansion. In Stage-III, TP starts declining and MP becomes negative. No firm would like to employ such doses of labour which may decrease its TP with negative MP. This stage is also ruled out as a stage of producer’s equilibrium. It is in Stage-II that a producer will like to operate. It is in this stage that the firm reaches the point of equilibrium where its profits are maximised. It is Stage-II which satisfies an important condition of producer’s equilibrium : that in a state of equilibrium, firm’s MP must be rising. Falling MP in Stage-II implies a situation of rising MC.

# Relation between (i) TP and MP, (ii) AP and MP

The relationship between these is studied with reference to Fig. 2 and in two parts as under :

(a) Relation between TP and MP

(i) When MP is increasing, TP increases at increasing rate. This happen till OL1 labour is employed.

(ii) When MP is decreasing, TP increases at a diminishing rate. This happens from OL1 onward, till OL2 amount of labour is applied.

(iii) When MP = 0, TP becomes maximum. Total product is maximum when OL2  labour is employed.

(iv) If MP is negative, TP starts declining. It happens when more than OL2  units of labour are employed. Fig. 2

Relation between AP and MP

(i) AP increases so long as MP > AP. It happens in Fig. 2 till point ‘a’ where AP is at its top.

(ii) AP decreases when MP < AP. It happens in Fig. 2 beyond point ‘a’ where AP is at its top.

(iii) AP is at its maximum when AP = MP. It is exactly at point ‘a’ in Fig. 2. Thus, MP curve cuts AP from its top.

(iv) MP may be zero or negative, but AP continues to be positive.

(v) Returns to Scale

This concept relates to long run when all the factors are variable. It refers to behaviour of output when all inputs are changed simultaneously in same proportion.

Changing both L and K in the same proportion, a producer may encounter the following situations :

Constant Returns to Scale

When total output increases in exactly the same proportion in which all the factors of production are increased, it is called constant returns to scale. Example: If labour and capital in a firm are increased by 30% and output also increases by 30%, it is a situation of constant returns to scale.

Mathematically

Q = f (L, K)

If both L and K are increased by 10% and output also increases by 10%, then the new production function with constant returns to scale will be:

10%.Q = f (10% _ L, 10% _ K)

i.e., if L and K are increased ‘t’ times, the output ‘Q’ will also increase ‘t’ times

t.Q = f (t . L, t . K)

Increasing Returns to Scale

When total output increases more than proportionately than the increase in all factor inputs, it is called increasing returns to scale. Example: If L and K are increased by 20% and output increases by 30% increasing returns to scale are said to be applying

There are various reasons for the application of increasing returns to scale. Importantly, when the scale of production is increased, there is greater division of work and as a result of specialisation, there is increase in productivity. Consequently output increases in greater proportion than the increase in all inputs.

Diminishing Returns to Scale

Diminishing returns to scale occur when increase in output is in smaller proportion than the increase in all factor inputs. Example : When all inputs are increased by (say) 15% and output increases by (say) 10%, diminishing returns to scale are applying.

Diminishing returns to scale occur owing largely to managerial constraints of a large business. Strikes, look-outs, leakage and pilferage become common events when the scale of production grows beyond manageable limits. Consequently productivity suffers and diminishing returns start operating.