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Chapter 4 / Monday, September 30 | Production Functions

4.6 Cobb-Douglas Production Functions


In 1928, mathematician Charles Cobb and economist (and future politician!) Paul Douglas published an article in the American Economic Review titled “A Theory of Production.” In it, they attempted to write down a single production function which might capture the relationship between capital and labor in an economy. They suggested the functional form \(q = f(L,K) = AL^aK^b\) This wasn’t based on their observation of any particular production process, but rather because the mathematical properties of the function implied that (a) the shares of GDP accruing to labor and capital would remain relatively constant over time, even if the prices of those goods varied, and (b) it could be estimated using linear regression by taking the log of both sides: \(\ln q = \ln A + a \ln L + b \ln K\) Their work was so influential that it bears their name to this day — even though other economists such as Leon Walras had previously used the functional form.

Marginal products

The marginal products of labor and capital are given by \(\begin{aligned} MP_L &= aAL^{a - 1}K^b\\ MP_K &= bAL^aK^{b-1}\\ \end{aligned}\) Here, both the $MP_L$ and the $MP_K$ depend on the amount of labor and capital used. Think back to the simpler function we looked at last Wednesday, to describe the “production” of a grade from hours of studying: \(f(x) = x^a\) We showed that this has diminishing marginal returns when $a < 1$, constant marginal returns when $a = 1$, and increasing marginal returns when $a > 1$:

[ See interactive graph online at https://www.econgraphs.org/graphs/math/univariate_calculus/marginal_returns_to_studying ]

The marginal products of labor are similarly determined by the exponents on $L$ and $K$: the value of $a$ determines if the $MP_L$ is increasing, decreasing, or constant, while the value of $b$ determines this for $MP_K$.

Visual representation and isoquants

The following diagram allows you to play around with the parameters of $A$, $a$, and $b$, to see how they affect the isoquant map:

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

One counterintuitive aspect about isoquants is the fact that an increase in productivity leads to an inward shift of the isoquants. (You can see this if you raise $A$ in the diagram above.) This is because better technology means you can produce a given quantity with fewer inputs (less capital and labor); so the isoquant for any given quantity shifts in toward the origin when technology improves!

Marginal rate of technical substitution

Note that each of the isoquants is bowed in toward the origin: that is, the MRTS is decreasing as you move a long an isoquant by using more labor and less capital. This is true regardless of whether there are increasing, decreasing, or constant marginal returns to labor and capital. What’s the intution behind this?

Intuitively, what’s happening here is that the levels of labor and capital affect both the $MP_L$ and the $MP_K$: as you increase labor, you make capital more productive, and vice versa.

As you move down and to the right along an isoquant, you’re adding labor, which makes capital more productive. At the same time, you’re reducing capital, which makes labor less productive. So as you shed capital, you have to hire more and more labor to make up for the lower productivity.

We can also think of it in the opposite direction: suppose you have a lot of labor, and you’re thinking of firing some people and replacing their jobs with automation. As your labor force decreases, the people who remain are doing more and more work, so they become increasingly valuable; so as you continue reducing yoru labor force, you need to use more and more machines to do the work of the people you’ve just fired.

On Friday, when we calculate the exact expression for the MRTS, we’ll see how the values of $a$ and $b$ in the production function affect it.

Previous: Linear Production Functions
Next: Elasticity of Substitution
Copyright (c) Christopher Makler / econgraphs.org