Laws of Returns to Scale | Production

In this article we will discuss about the laws of returns to scale

If the quantities of all the inputs used by a firm are increased by a particular proportion, then it is said that the scale of production of the firm has increased. For example, if all input quantities have increased three times, then we say the scale of production has also increased three times.

Now, the increase in the quantity of output produced by the firm in response to a change in the scale of production is called the Returns to Scale. Economists have told us about three laws of the returns to scale.

These laws are:

(i) Law of constant returns to scale,

(ii) Law of increasing returns to scale, and

(iii) Law of decreasing returns to scale. If the firm’s quantity of output increases by the same proportion as its scale of production, we say the law of constant returns to scale operates. In this case, if all the input quantities increase three times (say), then the scale of production would increase three times, and output also would increase three times.

On the other hand, if in response to an increase in the scale of production by a certain proportion, the firm’s output increases by a larger proportion, then we obtain the operation of the law of increasing returns to scale. In this case, if all the input quantities increase three times, and so, the scale of production increases three times, output would increase by more than three times.

Lastly, if in response to an increase in scale of production by a certain proportion, output increases by a smaller proportion, then we obtain the operation of the law of decreasing returns to scale. In this case, if all the input quantities increase three times leading to an increase in the scale of production by three times, output would increase by less than three times.

Of the three laws of returns, which one would operate in a particular field of production depends upon the nature of the production function. Let us suppose that a firm uses only two inputs, X and Y, and produces only one output, Q. Here if the firm’s production function is

q = f(x, y) = x + +y                                  (8.92)

the law of constant returns to scale would operate. This is because we obtain from (8.92) that if the firm uses the input combination (x = 2, y = 2), then its output would be q = 2 + 2 + 2 = 6 units.

Now, if the firm increases the use of both the inputs twice as much, i.e., if it uses the input combination (x = 4, y = 4), then its output would be q = 4 + 4 + 4=12 units. That is, if the firm’s scale of production becomes twice as much, the firm’s production quantity also becomes twice as much, and we obtain the law of constant returns to scale.

Again, if the production function of the firm is

q = f(x, y) = x² + xy + y²                                   (8.93)

we would obtain the law of increasing returns to scale. This is because (8.93) gives us that if the firm uses any input combination (x = 1, y = 2), then its output would be q = 1 + 2 + 4 = 7 units. But when the firm increases the use of both the inputs three times as much, i.e., when it uses the combination (x = 3, y = 6), its output would be q = 9 + 18 + 36 = 63 units.

That is, here we see, if the firm’s scale of production becomes thrice as much, its output would increase by more than that proportion, it would become 9 times as much, and we obtain the law of increasing returns to scale.

Lastly, let us suppose that the production function of the firm is

q = f(x, y) = x^1/2 +(xy)^1/4 + y^1/2                    (8.94)

Here we would obtain the law of decreasing returns to scale. This we may verify in this way. If the firm uses any input combination (x = 4, y = 4), the production function (8.94) gives us q = 2 + 2 + 2 = 6 units. But when the firm increases the use of both the inputs 9/4 times and uses the combination (x = 9, y = 9), then its output would be q = 3 + 3 + 3 = 9 units.

Therefore, here the firm’s scale of production has increased 9/4 times, but its output has not increased 9/4 times to become 6 x 9/4 = 13½ units. Here output has been only 9 units—it has increased less than proportionately. So here we obtain the law of decreasing returns to scale.

We have discussed above the laws of returns to scale. Since an increase in the scale of production requires, by definition, an increase in the quantities of all the inputs by a certain proportion and since quantities of all the inputs can be increased only in the long run, the laws of returns to scale always refer to the long run.