The Marginal Rate of Technical Substitution

From the isoquants, one can derive the marginal rate of technical substitution, MRTS. The MRTS corresponds to the marginal rate of substitution, MRS, from consumer theory. MRTS can approximately be calculated as

Since the curves slope downwards, if ΔK is positive then ΔL must be negative, and vice versa. That means that MRTS is a negative number. By convention, however, the minus sign is often omitted.

The Marginal Rate of Technical Substitution and the Marginal Products

There is an important relation between MRTS and the marginal products of labor and capital. As we have seen, MPL = Δq/ΔL, which means that if we increase the amount of hours worked with ΔL, production will increase with Δq = MPL * ΔL. Similarly, for capital we have that MPK = Δq/ΔK, so if we use one unit less of capital, production decreases with Δq = MPK * ΔK. For instance, if the marginal product of labor, MPL, is 3 and we add 1 more hour of labor, we will produce 3*1 = 3 units more of the good. Let us combine these two observations in the following way. If you increase labor with ΔL, production will increase with Δq = MPL * ΔL. However, suppose that you at the same time decrease your use of capital exactly so much that you still produce the same quantity as you did in point B. Then the total change in q must be zero. We can express this as

If we rearrange the last expression, we can get an alternative expression for MRTS

The marginal rate of technical substitution, MRTS, which, by definition, equals (minus) the change in capital divided by the change in labor, also equals one marginal product divided by the other. Note that on the left-hand side, we have the marginal product of labor in the numerator but in the middle, we have ΔL in the denominator.

Returns to Scale

Suppose that, using labor L and capital K, we produce the quantity q of a good. If we would double both L and K, we would probably increase the quantity produced as well, but by how much? If q is also doubled, we have constant returns to scale (we can think of this as that the scale is the same for (L,K) and q). If instead q increases by less than two times, we have decreasing returns to scale, and if it increases by more we have increasing returns to scale. More generally, we increase L and K by a factor t, and then check if q increases by more, less, or by the same factor. We can express this as

Look again at the expressions above: f(L,K) is the quantity produced from the start. . f(t*L,t*K) is the quantity produced if you increase both K and L by the factor t. Then you ask if you get more, less, or the same as t times the production you had from the beginning, i.e. t*f(L,K).
There can be different reasons why we get different returns to scale. For instance:

Constant returns to scale. Suppose we have a factory that produces a certain quantity of a good. Then we build another factory that has the same size and that uses the same number of workers, so that we now have two factories. It seems reasonable to assume that the second factory produces the same quantity as the first one does. This means that, as we double the inputs we also double the output.
Decreasing returns to scale. Suppose it becomes more and more difficult to coordinate the production as the size increases, so that we get higher and higher costs for administration. Then the costs will increase proportionally more than the production, and the production will grow by less than the inputs.
Increasing returns to scale. Oftentimes, large firms are more efficient than small firms are. This is called large-scale advantages.

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