What is the difference between diminishing returns and constant returns to scale

The difference between economies of scale and returns to scale is that economies of scale show the effect of an increased output level on unit costs, while the return to scale focus only on the relation between input and output quantities. If production increases by more than the proportional change in factors of production, this means there are increasing returns to scale.

If production increases by the same proportional change as all factors of production are also changing, then there are constant returns to scale. If production increases by less than that proportional change in factors of production, there are decreasing returns to scale.

Increasing returns to scale happen when all the factors of production are increased; at this point, the output increases at a higher rate. Decreasing or decreasing returns to scale are taking place when all the factors of production increase in a given proportion, but the output increases at a lesser rate than that of the increase in factors of production.

To compare this to increasing returns to scale: for decreasing returns to scale, increasing inputs leads to smaller increases in output; for increasing returns to scale, increasing inputs leads to the opposite—larger increases in output.

For example, if the factors of production are doubled, then the output will be less than doubled. Constant returns to scale occur when the output increases in exactly the same proportion as the factors of production. In other words, when inputs i. As an example of constant returns to scale, if the factors of production are doubled, then the output will also be exactly doubled. Note that returns to scale take place over the long run, during which time labor and capital are typically variable.

Thus, the elasticity of scale e measures the responsiveness of output to an increase in all inputs by the scalar factor l , normalized here to 1. One of the interesting results of the elasticity of scale is that e is actually the sum of the output elasticities of the different inputs.

Let e i denote the elasticity of output with respect to factor x i , i. We assert, then, that:. The relationship between the elasticity of scale and the output elasticities tell us that, indeed, there is a relationship between returns to scale and marginal productivities.

However, it is important not to get confused between the two and assume that, say, diminishing marginal productivity is somewhow related to decreasing returns to scale.

This is not true. Constant returns or increasing returns to scale are compatible with diminishing marginal productivity. For instance, examine Figure 3. Thus, increasing both capital and labor by the amount l leads to an increase in output by l. But increasing capital employment only by the amount l leads to an increase in output of merely m.

We can see the non- relationship between returns to scale and marginal productivity more clearly if we take a specific functional form for the production function. Consider the famous Cobb-Douglas production function Wicksell , p. This is the following:. Increasing both capital and labor by the scalar l , then we obtain:. Now, the marginal products of capital and labor are:. To see diminishing marginal productivity, we must show that marginal products decline as the relevant factors rise.

Pursuing this, we see that:. However, this does not necessarily imply what kind of returns to scale we will obtain. Thus, while diminishing marginal productivity may be implied by decreasing or constant returns to scale, it does not itself necessarily imply any particular kind of returns to scale. One final word may be in order. Recall that diminishing marginal productivity was guaranteed by the concavity of the production function. However, we only sought to make the assumption that the production function was quasi-concave but, as we saw, this was not enough to guarantee diminishing marginal productivity.

However, it can be shown that under constant returns to scale, quasi-concavity does imply diminishing marginal productivity. To see this, consider the following. As a result, we can rewrite this as:. Thus substituting:. Finally, by constant returns to scale once again, the terms on the right hand side can be rewritten as follows:.

Thus, quasi-concavity combined with constant returns to scale yields concavity. We know from before that concavity is sufficient for diminishing marginal productivity. Thus, in the presence of constant returns to scale, quasi-concavity actually implies diminishing marginal productivity.

One of the interesting implications of the different returns to scale is obtained via Euler's Theorem on homogenous functions. These properties will be very useful later when considering the marginal productivity theory of distribution.

For the moment, it is worthwhile noting one interesting property of constant returns to scale that is obtained via Euler's Theorem. Now, earlier on we showed that the average product of labor is higher than marginal product only when there is diminishing marginal productivity of labor. By our discussion of the ridge lines of the isoquant topography, we know that negative marginal products are ruled out as irrelevant.

Consequently, we see that in the special case of constant returns to scale, when we rule out negative marginal productivity of capital, we are simultaneously ruling out increasing marginal productivity of labor. In terms of our earlier Figure 2. We can do this exercise in reverse and show that ruling out negative marginal productivity of labor is equivalent to ruling out increasing marginal productivity of capital.

Thus, the conclusion that imposes itself is that when we are using constant returns to scale production functions of two inputs, ruling out negative marginal products is equivalent to imposing diminishing marginal productivity on all factors.

Effectively, these conditions from Inada, are the following limiting conditions:. In other words, i states that the marginal product of a factor approaches zero as the amount of the factor approaches infinity and that the marginal product approaches infinity and that amount of that factor approaches zero. These conditions are ssufficient to guarantee that for all finite amounts of factors used, the marginal products are positive and diminishing, i.

It is noticeable that Cobb-Douglas production functions fulfill the Inada conditions automatically. Properly speaking, "pure scale" effects should always be considered along a ray from the origin. Increasing and decreasing returns should, in its purest definition, be a mere relabelling of the isoquants or, if we wish, a change in the spacing between the isoquants. However, the economic justification of increasing and decreasing returns to scale often relies on changing factor proportions as scale changes.

The murkiness that often acompanies the justification of increasing or decreasing returns thus stems in good part from an imperfect definition of returns to scale in the work of the early Neoclassicals. Two of the major properties of constant returns to scale production functions is that both the average productivities and the marginal productivities of factors are independent of the scale of production, i.

Let us turn to marginal products. Differentiating with respect to one of the inputs, say K:. A similar exercise achieves the same result for the marginal product of labor. Consequently, as the marginal rate of technical substitution MRTS is simply the ratio of marginal products, then it follows that the MRTS is unaffected by scale. We can see the implication for increasing or decreasing returns. In sum, under constant returns, increasing scale does not affect marginal products; under increasing returns, an increase in scale increases all marginal products and under decreasing returns, an increase in scale decreases all marginal products.

However, it is important to note, that a change in scale under any returns to scale, will not affect the marginal rate of technical substitution MRTS.

This is true under increasing, decreasing or, as we saw, constant returns. This is because all marginal products change by the same proportion. Thus, along any expansion ray from the origin, the MRTS is the same for every isoquant, whether we have increasing, decreasing or constant returns.

Increasing the scale of production always means an expansion along one of these rays. The constancy of isoquant slopes along any expansion ray is the meaning of our statement that the marginal rate of technical substitution is unaffected by scale.

Homogeneous functions whatever the degree are special cases of a more general class of functions known as homothetic functions Shephard, A function F is homothetic if it is itself a monotonic transformation of a homogeneous function.

It is an increase in long run average costs resulting from an increase in scale of production and output. This relates to increasing fixed, not variable factors of production e. An example of decreasing returns to scale could be if a company becomes too large, they may become inflexible and slow to respond to the market conditions.

This means that long run average cost will increase if output increases. Both laws show that increasing a variable or fixed factor, productivity initially will increase but then start to decline. The main difference between the two is time scale and therefore the factors which are increased variable or fixed. Both of these laws are useful for firms decision making especially if they are looking to profit maximize and work at the most cost efficient level.