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.

 

A Different Kind of Economic Modeling

In macroeconomics, research almost always follows a similar pattern. First, the economist comes up with a question. Maybe they look at data and generate some stylized facts about some aspect of the economy. Then they set out to “explain” these facts using a structural economic model (I put explain in quotes because this step usually involves stripping away everything that made the question interesting in the first place). Using their model, they can then make some predictions or do some policy analysis. Finally, they write a paper describing their model and its implications.

There is nothing inherently wrong with this approach to research. But there are some issues. The first is that every paper looks exactly the same. Every paper needs a model. Sometimes papers adapt existing models, but they need enough difference to be a contribution on their own (but not so much difference that you leave the narrow consensus of modern macroeconomic methodology). Rarely, if ever, is there any attempt to compare models, to evaluate their failures and successes. It’s always: previous papers missed this and that feature while mine includes it.

This kind of iterative modeling can give the illusion of progress, but it really just represents sideways movement. The questions of macroeconomics haven’t really changed much in the last 100 years. What we have done is develop more and more answers to those questions without really making any progress on figuring out which of those answers is actually correct. Thousands of answers to a question is in many ways no better than none at all.

I don’t think it’s too much of a mystery why macroeconomics looks this way. Everybody already knows how an evaluation of our current answers to macroeconomic questions would go. The findings: we don’t know anything and all our models stink. I’d be surprised if even 10% of economists would honestly suggest a policymaker to carry out the policy that their papers suggest.

Academics still need to publish of course so they change the criteria that describes good macroeconomic research. Rarely is a paper evaluated on how well it answers an economic question. Instead, what matters is the tool used to answer the question. An empirical contribution without a model will get yawns in a macro seminar. A new mathematical contribution that uses a differential equations derived from a heat diffusion equation from physics? Mouths will be watering.

The claim is that these tools can then be used by other researchers as we continue to get closer and closer to the truth. The reality is that they are used by other researchers, but they only use the tools to develop their own slightly “better” tool in their own paper. In other words, the primary consumer of economic papers is economists who want to write papers. Widely cited papers are seen as better. Why? Because they helped a bunch of other people write their own papers? When does any of this research start to actually be helpful to people who aren’t responsible for creating it? Should we measure the quality of beef by how many cows it can feed?

Again there is an easy explanation for why economists are the only ones who can read economics papers. They would be completely unintelligible to anybody else. Reading and understanding the mechanism behind a macroeconomic paper is often a herculean task even for a trained economist. A non-expert has no chance. There could be good reasons for this complexity. I don’t expect to be able to open a journal on quantum mechanics and get anything out of it. But there is one enormous difference between physics and economics models. The physics ones actually work.

Economics didn’t always look this way. Read a paper by Milton Friedman or Armen Alchian. Almost no equations, much less the giant dynamic systems in models today. Does anybody think modern economic analysis is better than the kind done by those two?

The criticism of doing economics in words rather than math is that it is harder to be internally consistent. An equation has fewer interpretations than a sentence. I’m sympathetic to that argument. But I think there are better ways to add transparency to economics than by writing everything out in math that requires 20 years of school to understand. There are ways to formalize arguments other than systems of equations, ways to explain the mechanism that generates the data other than structural DSGE models.

The problem with purely verbal arguments is that you can easily lose your train of logic. Each sentence can make sense on its own but completely contradict another piece of the argument. Simultaneous systems of equations can prevent this kind of mistake. They are just one way. Computer simulations can provide the same discipline. Let’s say I have some theory about the way the world works. If I can design a computer simulation that replicates the kind of behavior I described in words, doesn’t that prove that my argument is logically consistent? It of course doesn’t mean I am right, but neither does a mathematical model. Each provides a complete framework that an outside observer can evaluate and decide whether its assumptions provide a useful view of the world.

The rest of the profession doesn’t seem to agree with me that a computer simulation and a system of equations serve the same purpose. I’m not exactly sure why. One potential worry is that it’s harder to figure out what’s actually happening in a computer simulation. With a system of equations I can see exactly which variables affect others and the precise channel of each kind of change. In a simulation, outcomes are emergent. Maybe I develop a simulation where an increase in taxes causes output to fall. Simply looking at the rules I have given each agent of how to act might not tell me why that fall occurred. It might be some complex interaction between these agents that generates that result.

That argument makes sense, but I think it only justifies keeping mathematical models rather than throwing out computer models. They serve different purposes. And computer models have their own advantages. One, which has yet to be explored in any serious way, is the potential for visual results. Imagine that the final result of an economic paper was not a long list of greek symbols and equals signs, but rather a full moving mini economic world. Agents move around, trade with each other. Firms set prices, open and close. Output and unemployment rise and fall. A simple version of such a model is the “sugarscape” model of Axtell and Epstein which creates a simple world where agents search for and trade sugar in order to survive.

Now imagine a much more complicated version of that that looked a lot more like a real economy. Rather than being able to “see” the relationships between variables in an equation, I could literally see how agents act and interact visually. My ideal world of economic research would not be writing papers, but creating apps. I want to download your model on my computer and play with it. Change the parameter values, apply different shocks, change the number and types of agents. And then observe what happens. Will this actually tell us anything useful about the economy? I’m not sure. But I think it’s worth a try. (I’m currently trying to do it myself. Hopefully I can post a version of it here soon)

Keynesian Economics Part 2 Investment and Output

In my last post on Keynesian economics I outlined a simple example that I think captures the core of Keynes’s economics. It will help to understand this post if you read that one first.

Keynes’s key insight was that an attempt to save by an individual does not always lead to an increase in aggregate saving. I showed how using a simple example in the last post, but we can also generalize the problem. Imagine that each consumer consumes only a fraction of their income (it does not have to be the same across individuals, but I will assume it is for simplicity). Then total consumption spending is given by

    \[C = bY\]

Where C is consumption, Y is income (and total output), and b is the fraction of income spent on consumption (the marginal propensity to consume).

Let’s say that the only spending in the economy is consumption spending. You might already be able to see that we have a problem. Total spending must always equal total income in the economy so that

    \[Y = C = bY\]

Which can only be true if Y=0, so the economy breaks down. Perhaps this scenario is easiest to see if we imagine the case where there is one worker and one firm. The worker works for the firm and gets paid Y. He then decides to buy bY of the output he just produced. The firm realizes he made too much stuff, so he cuts back on production. But this means he reduces his demand for the worker’s labor and cuts his hours. But now the worker makes less so he spends even less and the process continues until no production is carried out at all. The only way we could sustain production through consumption alone would be if nobody wanted to save at all.

If consumption spending isn’t enough to keep firm production positive, we need some demand from another source. One source could be other firms in the form of investment. If we fix income at Y and assume again households only want to consume bY, it is still possible that firms can make up the additional spending by investing (1-b)Y. Keynes argued that there is no reason to expect that investment would always exactly fill gap. If desired investment by firms is less than the difference between consumption and income, they won’t be able to sell all of their product and will cut back on production. We can see that if we write out our equation again, now with investment, it becomes

    \[Y = C + I = bY + I\]

And solving for Y gives

    \[Y = \frac{I}{1-b}\]

So the level of investment determines the level of income. It was through this logic that Keynes concluded that it was the “animal spirits” of firms that determined the state of the economy. It’s possible that the level of investment exactly corresponds to the full employment level of output of an economy, but there is nothing that guarantees that it will.

There are still a few subtleties we need to consider. The first is the role of interest rates. In the classical view of the economy, when people try to save more, they increase the supply of loanable funds, which pushes down interest rates (think of banks having excess money to lend and the only way they can get rid of it is by lowering the interest rate). That lower interest rate then makes previously unprofitable investment projects become profitable and investment rises. If the interest rate falls enough, it’s possible that the increase in investment would be enough to offset the decrease in consumption.

Keynes didn’t deny this possibility. However, he argued (I think correctly), that interest rates are certainly not the only, and likely not even the primary, factor that goes into a firms investment decision. If a firm expects demand to be low due to a recession, there is no interest rate where it will be profitable for them to make that investment. And, as we saw in the last post, by failing to make those investments, firms’ expectations become self-fulfilling and their pessimism is proven correct. Interest rate adjustments alone therefore cannot save us from a Keynesian recession.

Another potential question comes from the assumptions of the Keynesian consumption function. It is obviously unrealistic to assume that each household wants to consume the same constant fraction of their income. People like Milton Friedman have argued that what people really care about when making consumption decisions is their permanent income. If my income falls today, but I expect it to return to its previous level tomorrow, I will borrow in the bad times to keep a constant level of consumption. I think this criticism is valid, but I don’t think it stops Keynes’s story. As long as aggregate consumption is less than total output (which it almost certainly will be), we still need investment to fill the gap. We still rely on expectations of firms to be correct regarding their future demand.

By focusing on the case where investment was exactly enough to move the economy to full employment, Keynes argued that “classical” economists implicitly restricted the economy to a special case. Keynes set out to correct that theory by proposing a “general theory” where investment fluctuated unpredictably and could (and often is) less than the level that would sustain full employment. I think this contribution is extremely valuable and unfortunately often overlooked. Even modern “New Keynesian” models bear little resemblance to the economy Keynes described. Models with money at all are rare and ones that allow the type of monetary disequilibrium in Keynes’s theory are all but nonexistent.

What has been emphasized instead have been the policy implications of Keynes’s work. In my next post I will provide an argument that Keynesian policies do not solve the problem Keynes described.