2.7 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 and the MRTS

The marginal products of labor and capital are given by \(\begin{aligned} MP_L &= aAL^{a - 1}K^b\\ MP_K &= bAL^aK^{b-1}\\ \end{aligned}\) This may seem convoluted, but when we take the ratio to form the MRTS it gets much simpler: \(MRTS = {MP_L \over MP_K} = {aAL^{a - 1}K^b \over bAL^aK^{b-1}} = {a \over b}\times{K \over L}\) This has some important qualities:

• The MRTS is increasing in $a$ and decreasing in $b$. Notice that $a$ is the exponent on $L$; the higher $a$ is, the more productive each unit of labor is. If labor is more productive, the MRTS is greater (i.e., the isoquant is steeper), because it would take more capital to make up the production lost by reducing a given labor input. Likewise, $b$ measures the productivity of capital: the more productive capital is, the less capital you would need to make up for a reduction in labor, so the lower the MRTS.
• The MRTS is increasing the capital/labor ratio $(K/L)$. The more capital you use relative to labor, the steeper the isoquant. One way of thinking about this is to realize that labor reinforces capital: the value of $MP_K$ is increasing in $L$. So adding a little bit of labor is extremely valuable if you have lots of capital relative to labor, and less valuable if you have lots of labor already and very little capital. Visually, this means that the isoquants are bowed in toward the origin.

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

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!