# PRODUCTION FUNCTION

Production process involves the transformation of inputs into output. The inputs could be land, labour, capital, entrepreneurship etc. and the output could be goods or services. In a production process managers take four types of decisions: (a) whether to produce or not, (b) how much output to produce, (c) what input combination to use, and (d) what type of technology to use.

Suppose we want to produce apples. We need land, seedlings, fertilizer, water, labour, and some machinery. These are called inputs or factors of production. The output is apples. In general a given output can be produced with different combinations of inputs. A production function is the functional relationship between inputs and output. It shows the maximum output which can be obtained for a given combination of inputs. It expresses the technological relationship between inputs and output of a product.

In general, we can represent the production function for a firm as:

Q = f (x1, x2, ….,xn)

Where Q is the maximum quantity of output, x1, x2, ….,xn are the quantities of various inputs, and f stands for functional relationship between inputs and output. For the sake of clarity, let us restrict our attention to only one product produced using either one input or two inputs. If there are only two inputs, capital (K) and labour (L), we write the production function as:

Q = f (L, K)

This function defines the maximum rate of output (Q) obtainable for a given rate of capital and labour input. It may be noted here that outputs may be tangible like computers, television sets, etc., or it may be intangible like education, medical care, etc. Similarly, the inputs may be other than capital and labour. Also, the principles discussed in this unit apply to situations with more than two inputs as well.

###### Economic Efficiency and Technical Efficiency

We say that a firm is technically efficient when it obtains maximum level of output from any given combination of inputs. The production function incorporates the technically efficient method of production. A producer cannot decrease one input and at the same time maintain the output at the same level without increasing one or more inputs. When economists use production functions, they assume that the maximum output is obtained from any given combination of inputs. That is, they assume that production is technically efficient.

On the other hand, we say a firm is economically efficient, when it produces a given amount of output at the lowest possible cost for a combination of inputs provided that the prices of inputs are given. Therefore, when only input combinations are given, we deal with the problem of technical efficiency; that is, how to produce maximum output. On the other hand, when input prices are also given in addition to the combination of inputs, we deal with the problem of economic efficiency; that is, how to produce a given amount of output at the lowest possible cost.

One has to be careful while interpreting whether a production process is efficient or inefficient. Certainly a production process can be called efficient if another process produces the same level of output using one or more inputs, other things remaining constant. However, if a production process uses less of some inputs and more of others, the economically efficient method of producing a given level of output depends on the prices of inputs. Even when two production processes are technically efficient, one process may be economically efficient under one set of input prices, while the other production process may be economically efficient at other input prices.

Let us take an example to differentiate between technical efficiency and economic efficiency. An ABC company is producing readymade garments using cotton fabric in a certain production process. It is found that 10 percent of fabric is wasted in that process. An engineer suggested that the wastage of fabric can be eliminated by modifying the present production process. To this suggestion, an economist reacted differently saying that if the cost of wasted fabric is less than that of modifying production process then it may not be economically efficient to modify the production process.

###### Short Run and Long Run

All inputs can be divided into two categories: i) fixed inputs and ii) variable inputs. A fixed input is one whose quantity cannot be varied during the time under consideration. The time period will vary depending on the circumstances. Although any input may be varied no matter how short the time interval, the cost involved in augmenting the amount of certain inputs is enormous; so as to make quick variation impractical. Such inputs are classified as fixed and include plant and equipment of the firm.

On the other hand, a variable input is one whose amount can be changed during the relevant period. For example, in the construction business the number of workers can be increased or decreased on short notice. Many ‘builder’ firms employ workers on a daily wage basis and frequent change in the number of workers is made depending upon the need. The amount of milk that goes in the production of butter can be altered quickly and easily and is thus classified as a variable input in the production process.

Whether or not an input is fixed or variable depends upon the time period involved. The longer the length of the time period under consideration, the more likely it is that the input will be variable and not fixed. Economists find it convenient to distinguish between the short run and the long run. The short run is defined to be that period of time when some of the firm’s inputs are fixed. Since it is most difficult to change plant and equipment among all inputs, the short run is generally accepted as the time interval over which the firm’s plant and equipment remain fixed. In contrast, the long run is that period over which all the firms’ inputs are variable. In other words, the firm has the flexibility to adjust or change its environment.

Production processes of firms generally permit a variation in the proportion in which inputs are used. In the long run, input proportions can be varied considerably. For example, at Maruti Udyog Limited, an automobile dye can be made on conventional machine tools with more labour and less expensive equipment, or it can be made on numerically controlled machine tools with less labour and more expensive equipment i.e. the amount of labour and amount of equipment used can be varied. Later in this unit, this aspect is considered in more detail. On the other hand, there are very few production processes in which inputs have to be combined in fixed proportions. Consider, Ranbaxy or Smith-Kline-Beecham or any other pharmaceutical firm. In order to produce a drug, the firm may have to use a fixed amount of aspirin per 10 gm of the drug. Even in this case a certain (although small) amount of variation in the proportion of aspirin may be permissible. If, on the other hand, no flexibility in the ratio of inputs is possible, the technology is described as fixed proportion type. We refer to this extreme case later in this unit, but as should be apparent, it is extremely rare in practice.

#### PRODUCTION FUNCTION WITH ONE VARIABLE INPUT

Consider the simplest two input production process – where one input with a fixed quantity and the other input with is variable quantity. Suppose that the fixed input is the service of machine tools, the variable input is labour, and the output is a metal part. The production function in this case can be represented as:

Q = f (K, L)

Where Q is output of metal parts, K is service of five machine tools (fixed input), and L is labour (variable input). The variable input can be combined with the fixed input to produce different levels of output.

###### Total, Average, and Marginal Products

The production function given above shows us the maximum total product (TP) that can be obtained using different combinations of quantities of inputs. Suppose the metal parts company decides to know the output level for different input levels of labour using fixed five machine tools. Table -1 explains the total output for different levels of variable input. In this example, the TP rises with increase in labour up to a point (six workers), becomes constant between sixth and seventh workers, and then declines.

Two other important concepts are the average product (AP) and the marginal product (MP) of an input. The AP of an input is the TP divided by the amount of input used to produce this amount of output. Thus AP is the output-input ratio for each level of variable input usage. The MP of an input is the addition to TP resulting from the addition of one unit of input, when the amounts of other inputs are constant. In our example of machine parts production process, the AP of labour is the TP divided by the number of workers.

APL = Q/L

As shown in Table -1, the APL first rises, reaches maximum at 19, and then declines thereafter. Similarly, the MP of labour is the additional output attributable to using one additional worker with use of other input (service of five machine tools) fixed.

MPL = WQ/WL

Where W means ‘the change in’. For example, from Table-1 for MP4 (marginal product of 4th worker) WQ = 76–54 = 22 and WL = 4–3 =1. Therefore, MP4 = (22/1) = 22. Note that although the MP first increases with addition of workers, it declines later and for the addition of 8th worker it becomes negative (–4).

Figure-1: Relationship between TP, MP, and AP curves and the three stages of production

The graphical presentation of total, average, and marginal products for our example of machine parts production process is shown in Figure -1.

###### Relationship between TP, MP and AP Curves

Examine Table-1 and its graphical presentation in Figure-1. We can establish the following relationship between TP, MP, and AP curves.

1. a) If MP > 0, TP will be rising as L increases. The TP curve begins at the origin, increases at an increasing rate over the range 0 to 3, and then increases at a decreasing rate. The MP reaches a maximum at 3, which corresponds to an inflection point (x) on the TP curve. At the inflection point, the TP curve changes from increasing at an increasing rate to increasing at a decreasing rate.

b) If MP = 0, TP will be constant as L increases. The TP is constant between workers 6 and 7.

c) If MP < 0, TP will be declining as L increases. The TP declines beyond 7. Also, the TP curve reaches a maximum when MP = 0 and then starts declining when MP < 0.

2. MP intersects AP (MP = AP) at the maximum point on the AP curve.This occurs at labour input rate 4.5. Also, observe that whenever MP > AP, the AP is rising (upto number of workers 4.5) — it makes no difference whether MP is rising or falling. When MP < AP (from number of workers 4.5), the AP is falling. Therefore, the intersection must occur at the maximum point of AP. It is important to understand why. The key is that AP increases as long as the MP is greater than AP. And AP decreases as long as MP is less than AP. Since AP is positively or negatively sloped depending on whether MP is above or below AP, it follows that MP = AP at the highest point on the AP curve.

This relationship between MP and AP is not unique to economics. Consider a cricket batsman, say Sachin Tendulkar, who is averaging 50 runs in 10 innings. In his next innings he scores a 100. His marginal score is 100 and his average will now be above 50. More precisely, it is 54 i.e. (50 * 10 + 100)/(10+1) = 600/11. This means when the marginal score is above the average, the average must increase. In case he had scored zero, his marginal score would be below the average, and his average would fall to 45.5 i.e. 500/11 is 45.45. Only if he had scored 50 would the average remain constant, and the marginal score would be equal to the average.

###### The Law of Diminishing Marginal Returns

The slope of the MP curve in Figure-1 illustrates an important principle, the law of diminishing marginal returns. As the number of units of the variable input increases, the other inputs held constant (fixed), there exists a point beyond which the MP of the variable input declines. Table-1 illustrates this law. Observe that MP was increasing up to the addition of 4th worker (input); beyond this the MP decreases. What this law says is that MP may rise or stay constant for some time, but as we keep increasing the units of variable input, MP should start falling. It may keep falling and turn negative, or may stay positive all the time. Consider another example for clarity. Single application of fertilizers may increase the output by 50%, a second application by another 30% and the third by 20% and so on. However, if you were to apply fertilizer five to six times in a year, the output may drop to zero.

Three things should be noted concerning the law of diminishing marginal returns.

1. This law is an empirical generalization, not a deduction from physical or biological laws.
2. It is assumed that technology remains fixed. The law of diminishing marginal returns cannot predict the effect of an additional unit of input when technology is allowed to change.
3. It is assumed that there is at least one input whose quantity is being held constant (fixed). In other words, the law of diminishing marginal returns does not apply to cases where all inputs are variable.
###### Stages of Production

Based on the behaviour of MP and AP, economists have classified production into three stages:

Stage 1: MP > 0, AP rising. Thus, MP > AP.

Stage 2: MP > 0, but AP is falling. MP < AP but TP is increasing (because MP > 0).

Stage 3: MP < 0. In this case TP is falling.

These results are illustrated in Figure-1. No profit-maximising producer would produce in stages I or III. In stage I, by adding one more unit of labour, the producer can increase the AP of all units. Thus, it would be unwise on the part of the producer to stop the production in this stage. As for stage III, it does not pay the producer to be in this region because by reducing the labour input the total output can be increased and the cost of a unit of labour can be saved.

Thus, the economically meaningful range is given by stage II. In Figure-1 at the point of inflection (x), we saw earlier that MP is maximised. At point y, since AP is maximized, we have AP = MP. At point z, TP reaches a maximum. Thus, MP = 0 at this point. If the variable input is free then the optimum level of output is at point z where TP is maximized. However, in practice no input will be freely available. The producer has to pay a price for it. Suppose the producer pays Rs. 200 per worker per day and the price of a unit of output (say one apple) is Rs. 10. In this case the producer will keep on hiring additional workers as long as

(price of a unit of output) * (marginal product of labour) > (price of a unit of labour)

That is, marginal revenue of product (MRP) of labour > PL

On a similar analogy,

(price of a unit of output) * (marginal product of capital) > (price of a unit of capital)

That is, marginal revenue of product (MRP) of capital > PK

The left side denotes the increase in revenue and the right side denotes the increase in the cost of adding one more unit of labour. As long as the increment to revenues exceeds the increment to costs, the profit of the producer will increase. As we increase the units of labour, we see that MP diminishes. We assume that the prices of inputs and output do not change. In this case, as MP declines, revenues will start falling, and a point will come when the increase in revenue equals the increase in cost. At this point the producer will stop adding more units of input. With further addition, since MP declines, the additional revenues would be less than the additional costs, and the profit of the producer would decline.

Thus, profit maximization implies that a producer with no control over prices will increase the use of an input until—

Value of marginal product (MP) = Price of a unit of variable input