2.4 The Marginal Rate of Technical Substitution

Fundamentally, the isoquant illustrates a tradeoff. Suppose Chuck needs to catch 20 fish a day to survive, and he’s currently using some amount of labor and capital $(L,K)$. He might consider fashioning a stronger spear in order to have to spend a bit less time fishing, while keeping his output at 20 fish. To do so, he would need to figure out how much more capital he would need ($\Delta K$) to reduce his time fishing by some amount ($\Delta L$). The rate of additional capital needed per labor reduced, $\Delta K / \Delta L$, is called his marginal rate of technical substitution between labor and capital. (Note: Some textbooks refer to this as the “Technical Rate of Substitution.”)

Visually, the MRTS is represented by the magnitude of the slope of an isoquant:

How do we calculate the MRTS? We can use implicit function theorem to simply assert that the slope along an isoquant is given by \(\left.{dK \over dL}\right|_{f(L,K) = q} = - {\partial f/\partial L \over \partial f/\partial K}\) Since the MRTS is the magnitude of the slope, it’s therefore given by the formula \(MRTS = \left|- {\partial f/\partial L \over \partial f/\partial K}\right| = {MP_L \over MP_K}\) There’s a good economic interpretation of this formula. Let’s think about this in terms of the marginal products of labor and capital. The marginal product of labor says that the change in output due to a small change in labor is given by \(MP_L = {\Delta q \over \Delta L}\) Suppose Chuck were to reduce his labor by just enough to drop his production by $\Delta q$. Solving the above equation for $\Delta L$, this would mean he would need to reduce his labor by \(\Delta L = -{\Delta q \over MP_L}\) To stay along the isoquant, Chuck then needs to add capital to bring his output back up by $\Delta q$. By the same argument, the additional amount of capital he would need to use to raise his output by $\Delta q$ would be \(\Delta K = {\Delta q \over MP_K}\) Therefore \(MRTS = \left|{\Delta K \over \Delta L}\right| = \left|\frac{\Delta q / MP_K}{-\Delta q / MP_L}\right| = {MP_L \over MP_K}\)

Different functional forms

So far we’ve analyzed the features of a production function: given $f(L,K)$, we can find the marginal products, plot the isoquants, and calculate the MRTS. We now face our first modeling choice: which production function best suits a particular production process? While there are an infinite number of possible functional forms, let’s look at a few that are commonly used to model production technologies: linear, Leontief, and Cobb-Douglas production functions.