4.7 Elasticity of Substitution

Thinking about the different kinds of production functions, one important feature concerns how substitutable capital and labor are. For example, with a linear production function, capital and labor are what we might call perfect substitutes: if $f(L,K) = L + K$, for example, you can always replace one hour of labor with one unit of capital. On the other hand, with a Cobb-Douglas production function, as we just saw, the MRTS changes as you move along an isoquant.

One measure of the substitutability of capital and labor is called the elasticity of substitution. A third production function, called CES (or “constant elasticity of substitution”), takes the form \(f(L,K) = (aL^\rho + bK^\rho)^{1 \over \rho},\)where the Greek letter $\rho$ is a parameter related to the elasticity of substitution. If you drag the blue dot along the isoquant in the graph below, you can see that the MRTS changes; and by using the slider, you can see how the elasticity of substitution affects the behavior of the MRTS:

[ See interactive graph online at https://www.econgraphs.org/graphs/firm/technology/ces_mrts ]

In this case, it diminishes, or gets smaller in absolute value, as you move to the right. This means that as you use more and more labor, the amount of capital it would take to replace one unit of labor gets smaller. The elasticity of substitution is defined as the percentage change in the MRTS due to a $1\%$ change in the ratio of capital to labor, $K/L$, as one moves along an isoquant.

Use the slider to change the elasticity of substitution. As you can see, if you drag the elasticity all the way to the left, the isoquants are extremely “bendy,” even to the point of being almost L-shaped; in the middle, the isoquants have the “bowed” shape of the Cobb-Douglas production function; and as it becomes perfectly elastic, the isoquants become linear, indicating that capital and labor are perfectly substitutable.

This substitutability of labor and capital has profound political and moral implications. A centerpiece of Andrew Yang’s 2020 Presidential campaign centered around solving the problem of structural job loss caused by automation — an example of companies shifting their production processes away from labor and toward capital. Semi-autonomous truck convoys may be able to deliver goods more safely, more quickly, and with lower emissions than human-driven trucks; but such automation threatens thousands of long-haul truck driving jobs, which are some of the best-paying jobs for workers without a college degree. So the elasticity of substitution isn’t just a dry mathematical formula: at its core, it’s a measurement of how much people’s jobs are at risk of being lost to automation. That doesn’t make the concept “bad,” any more than an earthquake is “bad;” but it does mean that if you’re interested in solving problems of income inequality, understanding the substitutability of labor and capital has got to figure prominently in your analysis.