Talking Past Each Other on Math in Economics

Trump’s recently proposed plan to cut corporate taxes has opened a debate about whether corporate tax cuts are good for workers. Opponents of the plan argue that it will only help corporations increase their profit while supporters believe a large portion of the benefits will accrue to workers through increased wages. I don’t want to comment on that debate. Instead, I want to discuss a point made by John Cochrane in his attempt to prove that a lower corporate tax can increase wages (responding to a post by Greg Mankiw). You can find his post here. Please read it before continuing.

Cochrane shows in a simple model that a decrease in taxes will cause an increase in wages of \frac{1}{1-\tau} where \tau is the current tax rate. In other words, if the tax rate is 1/3, a decrease in taxes of one dollar increases wages by $1.50. Cochrane then says something that I find incredibly misleading:

“This is also a lovely little example for people who decry math in economics. At a verbal level, who knows? It seems plausible that a $1 tax cut could never raise wages by more than $1. Your head swims. A few lines of algebra later, and the argument is clear. You could never do this verbally.”

There are two issues with Cochrane’s statement. The first is that it is pretty easy to prove that a $1 tax cut could raise wages by more than $1. Assume the only two inputs are labor and capital and profits are zero. Assume the rental price of capital is fixed. Any change in taxes must therefore cause a change in wages. If production doesn’t change, this change has to be the exact amount of the tax (otherwise profit would change). Now assume that the tax caused some deadweight loss so lowering it will also increase production. Again wages increase so they must have increased more than one for one with the increase in tax.

Now you might say those assumptions are a bit ridiculous and I would agree. But Cochrane actually used the exact same assumptions (and more). He just hid them behind some math. And that brings me to the second problem with Cochrane’s statement. A few lines of algebra later, the argument is actually not clear at all unless you already know what’s going on (even Greg Mankiw admits he doesn’t have intuition for the result in the post that Cochrane is expanding on).

Let’s take Cochrane’s “proof” piece by piece and outline the assumptions he needed to get his result.

He writes: the production technology is

    \[Y = F(K,L) = f(k)L ; k=K/L\]

To just write this line, we need three strong assumptions

Assumption 1: There are only two productive inputs, labor and capital

Assumption 2: We can represent this economy by an aggregate production function (which is almost certainly impossible)

Assumption 3: The aggregate production function exhibits constant returns to scale (multiplying each of its inputs by some factor also multiplies its output by that factor)

The last assumption is necessary to write the function in its f(k)L form and also ensures that firms have zero profit.

Next we have that firms maximize

    \[\max (1-\tau\)[F(K,L) - wL] - rK\]

Again, we are implicitly making more assumptions here

Assumption 4: Firms maximize profits every period (the setup of the problem guarantees that this behavior also maximizes lifetime profits, but another model might not have that property)

Assumption 5: All workers get paid the same wage, which is taken as given by individual firms (i.e. labor markets are perfectly competitive)

Assumption 6: The rental rate of capital is exogenously set. Mankiw set up the problem as a small open economy so that the interest rate (the price of capital) is constant. Note that the US is obviously not a small open economy.

Continuing, the firm’s first order conditions are

    \[(1-\tau)f'(k) = r\]

    \[f(k) - f'(k)k = w\]

Again, more assumptions

Assumption 7: Workers get paid their marginal product (technically this one follows from 4 and 5 above so maybe I shouldn’t count it).

Assumption 8: Firms know their production function as well as the marginal products of labor and capital.

Assumption 9:  Wages are fully flexible and can be changed at any time.

Assumption 10: Capital can move costlessly between countries.

I’ll stop there but I’m sure there are plenty more (including the assumptions of no involuntary unemployment, no money of any kind, and that the economy is always in equilibrium – assumptions common to many macro models). My point in doing this exercise is to demonstrate that in order to even begin to write an economic model using math you need to make strong assumptions. Without them the problem quickly becomes either impossible to solve or impossible to interpret. By hiding these assumptions (either intentionally or not) behind fancy equations, they often go unnoticed.

Nobody that criticizes math in economics is literally criticizing the use of algebra or calculus to provide intuition about an economic result. What we criticize are the restrictions that using math places on the economic problem. Mises described math in economics as a “vicious method, starting from false assumptions and leading to fallacious inferences. Its syllogisms are not only sterile; they divert the mind from the study of the real problems and distort the relations between the various phenomena.”

I can’t help but think that diverting the mind from the real problem is exactly what’s happening here. The corporate tax debate is really about the incidence of the tax. Do workers bear most of the burden, or does it primarily serve to prevent monopoly profits and rents? Cochrane’s example avoids this question by assumption. Rather than being a nail in the coffin for people who want less math in economics, it serves as a perfect example of why those criticisms exist. In some ways math provides clarity over verbal reasoning, but it can also be deceiving. Behind the formal logic and the proofs is a fragile set of assumptions that in many cases drive the results.

P.S. I don’t usually agree with Paul Krugman but I think he gets this one right in this post. He also shows which assumptions are driving the result, and that they are not ones that make much sense.

P.P.S. Larry Summers has a nice response to the debate as well

P.P.P.S. Casey Mulligan claims Krugman and Summers still get it wrong. I haven’t fully wrapped my head around his argument. What was Cochrane saying about algebra making everything clear?

P.P.P.P.S. I’m still in favor of cutting the corporate tax precisely because it is so hard to determine the incidence. Even if we want to stop monopoly profits (and I’m not sure that we do), it seems better to me to just focus on preventing monopolies.


Leave a Reply

Your email address will not be published. Required fields are marked *