**This post is now done.**

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I think control issues are a general problem with politicians and bureaucrats. I’m speculating, but I think that jobs where you can conceivably control some things attract people who believe controlling things is important.

A problem with that is if controllers believe they can control something that they can’t.

This comes up a lot in tax policy. There’s a real economic problem with tax labeling and tax incidence that no one should ignore. But political processes often focus on labeling exclusively, or as a matter of convenience. There’s also an issue that you can target a certain tax base, but you always need some element of buy-in from the parties being taxed. It’s perfectly legal for people to avoid being included in the tax base, and if buy-in is weak you may get some tax cheating on top of the avoidance.

Which brings us to the problem of multinational corporations. These don’t face one corporate tax rate. They face a bunch. And it’s reasonable and legal to expect them to pick and choose where to place the tax base they create.†

Here’s how that might work out. Country A has a corporate income tax rate of 10% and country B has a corporate income tax rate of 20%. Multinational firms from both countries then do everything they can to shift costs to country B (as a practical matter of tax planning, if not so much an ethical one, costs you measure internally are easier to move around than revenues that others might be able to measure externally to the firm) . Income is revenue minus costs, so the goal is to cancel out all the revenue from country B with costs from country B but also costs from country A that have been shifted over. If they do this perfectly, they end up with their income all taxed at 10%. This makes good financial sense, and going further, it is the fiduciary duty of corporate officers to make sure this happens. Now suppose the politicians in country B want to increase taxes. They raise their rate to 21%, and still collect nothing. Even worse, they may not be able to admit how it is possible for country A to raise their rates even more, say to 12%, and have their tax revenue go up by 20%.

Part of me does not believe that politicians can be that dumb.

But a different part of me is certain that sometimes they are that dumb. Here’s why. If country B really wants to increase tax revenues (instead of just labeling themselves as tough on business) they should undercut country A’s rate. Perhaps country B should try 9%. But if they do, country A should try 8%. That game ends with both of them charging a 0% rate. At first glance, this outcome might seem implausible, but governments do this all the time when they compete with each other to offer tax breaks to get firms to locate within their jurisdiction. In public finance there’s actually evidence of governments going past zero and offering packages that amount to negative tax rates on net. That’s what can happen if governments are not that dumb. If they are dumb, they might not go that far, and end up with a positive rate, but one that is different from other countries. This is actually what we observe in the real world, which makes me think there are a lot of politicians who really don’t get the economics involved in their job.

Which brings us to the U.S. corporate tax rate. The OECD says that, yes, the U.S. did have the highest statutory corporate tax rate (the right column at this site). Those are marginal rates, so there will be some weirdness to judging our system when all we’ll have is average rates from the data. Also, when we talk about the statutory rate, this is before deductions, exemptions, credits and so on. The effective tax rate might be much lower. The Congressional Budget Office (CBO) researched that for G-20 countries, and found that in 2012 … our marginal corporate rate was the highest, our average corporate tax rates were still the third highest, and our effective corporate rate was the fourth highest (see Table I here). That’s better, but not good. The change in positions also indicates that our system has a lot of complexity to it.

So yes, it is reasonable to think that high U.S. corporate tax rates were causing multinational firms to shift their costs around the world. How recent a phenomenon is that? Well, historical panel data sets on that are both complex and sketchy, but the OECD data for earlier periods shows the U.S. did not used to be at the top of the list. What’s happened is that many other large and/or rich countries have reduced their corporate tax rates over the last generation or so. This is because the wave towards lower tax rates that hit the U.S. starting in the 1980’s was a global phenomenon. It just did not hit our corporate income tax rate.

After all this, the U.S. was doing one more dumb thing. Well, smart from the perspective of collecting revenue, but dumb from the perspective of reducing tax distortions. The U.S. had been running a non-territorial tax system. In a territorial tax system, a U.S. multinational would pay tax on its income in each country that it operates, at the tax rate for that country … and then you were done. But this is not the system the U.S. had. Instead, if the tax owed in the foreign country was lower than what would have been paid if that operation was in the U.S., the firm owed the different too. Except that a U.S. firm could claim that the foreign income on which that extra tax was owed was still needed or active in that foreign country, say for future investments. In this case, that tax liability could be deferred indefinitely. And obviously, it could be invested in financial investments from the foreign country and earn even more income that could also be deferred.

But wait! There’s one more big distortion. The “Americanness” of a firm was judged on the basis of its headquarters’ location. And some countries had territorial tax systems in place. This led to inversions: move the headquarters to a lower tax country with a territorial tax system and completely avoid rather than defer some U.S. corporate taxes.

This was considered a big problem under the Obama administration, and a motivation for tax reform for both countries.

So what we have are some symptoms: too much corporate profit parked overseas, and too many inversions. The causes are a corporate tax rate that's relatively high, and double taxation from a non-territorial tax system.

The Republican tax reform addresses this with three different measures. First is the reduction in the corporate income tax from 35% to 21%. Second is the conversion of U.S. corporate tax collection to a territorial system.

The third deals with the issue that corporations should not be (especially) rewarded for tax avoidance. This is exactly what would happen if old profits that were never taxed at the U.S. rate under the old system could be brought back to the U.S. and taxed at the new low rate. So, there's also a surcharge on repatriating the accumulation of past profits. It's set at about 15%; a little more than the decline in the main rate.

Of course, we’ve seen in the news over the last 2 months how many firms have already started repatriating those past profits parked overseas.

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All of this motivated a lot of angry discussion amongst macroeconomists last autumn. For someone like me, this was extraordinary: if economists ever bitch-slapped each other, this would be the occasion. On one side were non-political economists and conservative economists noting that a cut in the corporate tax rate theoretically should increase wages (this is how we know to go out and actually look for that sort of thing in the German data I posted about last month). On the other side were progressive/liberal economists saying something along the lines of "but not this time". Their arguments were pretty weak and everyone knew it.

John Cochrane might have summed it up the best:

[Cutting the corporate tax rate and having it help labor] is not the same as the Laffer curve [cutting tax rates and having tax revenue go up], which I think causes some of the confusion. The question is not whether one dollar of static tax cut produces more than a dollar of revenue. [I don't really see that as a confusion, but Cochrane must have run across people who made that argument). The question is whether it raises capital enough to produce more than a dollar of wages.

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.

Yep. This is why we do the math.

I also think, and it makes me a little sad to say this, but in this particular case the progressive/liberal argument was made strongly and verbally to audiences that don't do the math. And it showed. :(

This all started with Trump's Council of Economic Advisors releasing a position paper arguing for a lower corporate income tax. (This is not required, but is very readable for undergraduates). But, it's a serious economic summary, not a political hack's job, which harks back to Harberger's seminal 1962 paper showing that the corporate tax hit workers pretty hard.

First out of the gate was Paul Krugman, the Nobel-Prize winning economist from Princeton who now mostly writes progressive opinion pieces for the *The New York Times*, with a column with the subtle title "Lies, Lies, Lies, Lies, Lies, Lies, Lies, Lies, Lies, Lies" In it, he literally goes through 10 lies that he sees in the Republican position. Some of it's good stuff, but some of it is ... hmmm ... things I wouldn't be proud of. Most relevant to this post is his 4th one. It’s my job to help you filter these things, and I suggest you be very suspicious of an argument that starts out with the tax rate cut won’t help workers and ends by reasoning that this is because it will make the trade deficit go up. If that sounds like a dodge, it’s because it probably is. Next up was Larry Summers (Harvard economist, former Clinton Treasury secretary, and Obama advisor), who remarked more than once that he was in favor of cutting corporate tax rates, but that he didn’t agree with the forecasts. Then there was Jason Furman (a Harvard economist, and one of Obama’s economic advisors) who tweeted that he would not disagree with reducing the corporate tax rate to zero … but that he just didn’t like Trump’s numbers that didn’t reduce it that far. ‡ Sheesh. In the same thread, Brad DeLong got to the heart of the misgivings on the political left: none of the conservative/Republican justifications hold water if output per worker isn’t flexible. This is absolutely true, but of dubious import since the whole point of economic growth is that output per worker has been going up for about three centuries. That really is the heart of the matter: progressives/liberals/Democrats think there’s something stagnating worker productivity that can only be addressed by pursuing their policy agenda, and conservative/Republicans think that it’s that progressive/liberal/Democratic policy agenda itself that is what’s standing in the way. For my part, I think a lot of the “wages have stagnated over the last generation” argument is that compensation has not stagnated, but we’re just getting paid with a lot more coupons good for the healthcare of some lucky senior citizen chosen by some bureaucrat we don’t know.

Back to Cochrane: we do the math because it sorts through these things in a way that’s actually easier to dissect in the long-run (yeah … if you think about it, all math is tough in the short-run, and then you figure out how a technique works and use it repeatedly in the long-run because it seems so easy now).

The math was originally put out in a blog post by Greg Mankiw (another Harvard macroeconomist, who started out by founding the neo-Keynesian branch of macro and then ended up working in the Bush II White House). There was nothing new here. Most economists could have kicked this out, but Mankiw reminded everyone of it. Go read it, at least through the bold text. The simple math is in the Mankiw post. Something more formal, with a great discussion is in Cochrane. Mulligan chimes in too. An alternative way of thinking about it with graphs and Harberger triangles is offered by Landsberg. Browse the comments too: they are filled with chip-ins from top notch people.

Before we jump into the math, take a step back and think about how the world works. It’s a complex place, and everyone thinks about the world in simpler terms than the reality around them. The thing is, that’s a model. Even if you don’t write it down. Even if it isn’t mathematical, it’s still a model. We might call it naïve, or informal, or ocular to get the point across that it’s just our mental model and shouldn’t be taken too seriously. I think it’s fair to say that a conclusion from the mental model of most people is that the incidence of the corporate income tax on labor is zero.

What economists try to do is formalize some that. This means assigning some plausible functions to describe behavior, and then seeing how they interact. You may not find the math simple, but it is probably as simple as we can make it.

Mankiw’s model is algebraic. It uses the basic principle from algebra that you need an equation for every variable you want to solve for. There are three variables the wage, the capital-to-labor ratio, and the amount of tax revenue collected: w, k, and x. We can have extra variables in the equations, but we don’t solve for these … in the sense that we make statements about why they go up or down. Instead, with these extra variables, we presume that they go up or down as a reflection of outside forces or policies. Because they are coming from outside the 3 equations of the model, we call them exogenous variables. These are the tax rate, and the rate of return on capital: t and r. If these are exogenous, then the variables we solve for are called endogenous.

Mankiw starts with a production function, f(k). The notation f just means function. The k is the capital-to-labor ratio; basically tools per person. Economists presume that production functions share some features: they’re curved, they slope up, and they get flatter as k gets bigger. Because we’re not making a lot of assumptions, a production function is exceptionally flexible. All we’re assuming is that more tools always help you be more productive (that’s the upward slope), but that the initial tools are more important than the ones you added later (this is the getting flatter part, which economists call diminishing marginal productivity). We need a function, with a curve, to have our slope change. In terms of calculus, f’>0, and f’’<0. The first one says the slope is positive, and the second one is that the slope is getting smaller (or the curve is getting flatter).

Next up is the marginal product of capital (MPK). This is how much is produced by one additional unit of capital, and it is the slope of the production function, f’. (That’s a tiny bit of calculus: the first derivative of a function is another function which tells you its slope everywhere). For simplicity, we think of firms renting capital, in the same way that they hire labor. The MPK is then the very most we can afford to pay for capital: if you pay more than that you’ll lose money on every tool you rent. In sum, we produce f and we rent capital for f’.

Without taxes, this means that r=f’. Here r is the rental rate of capital: it functions that say way as w being the rate at which labor is paid.

But with taxes, part of that productivity of capital is skimmed off the top by taxes. So the rental rate is what’s left after you pay taxes, as in: r=(1-t)f’.

Hold that thought while we build a second equation. In its simplest form, with only two factors of production, all the value of production is paid out workers or to capital. So f=wl+rk (that’s an “el”, although it might look like an “eye” with this font). We can make things even simpler by assuming that there’s only one worker, and they use the k pieces of capital to produce f. In this case we get f=w+rk. Keep in mind that this equation is at the production stage, before the tax is paid. If I substitute in r=f’, and rearrange, we get w=f-kf’.

Lastly, we put together an equation to describe how tax revenue, x, is related to tax rate, t. Now, we’re not talking about rates of return, but rather amounts paid out. The revenue reduction is then dx = –kf’dt. Starting on the right of that, dt is the change in the tax rate. In this case that’s negative. That tax cut is applied to the amount that’s paid out to the owners of capital, kf’. The negative sign in front is there so that we can talk about a tax cut as a positive number, like we cut taxes by $100. We don’t say we changed taxed by –100$, even though that would mean the same thing. If we described tax cuts as a negative number, we wouldn’t need the minus sign in the equation, but since we describe them as positive numbers we do need the minus in there to change the sign.

So the question is, what’s dw/dx? That is the change in wages for a certain change in tax revenue no longer paid by owners of capital?

Before we go there, we have to think about the whole algebra problem. And we ought to make it match up with what we know about the real world. So we’ve got 3 equations:

r=(1-t)f’

w=f-kf’

dx=-kf’dt

Three equations means we can solve for three unknowns. Recall that f is a function of the variable k, and not a variable in and of itself; d is the same way, it’s shorthand for a Greek letter delta denoting change in a number. The three unknowns that we solve for are k, w, and x. The thing coming from the outside that makes those 3 change is the tax rate t.

But if you look, there’s one variable we haven’t accounted for: the rate of return on capital, r. We have two choices: 1) make it endogenous by adding another equation so we can solve for it, or 2) assume that it’s also exogenous. The first one sounds like more work, so we want to avoid that. The cool thing is, making that assumption is good economics too!

How can that be? Well, which is more different across countries: r or w? We’ve discussed in class how differences in incomes often differ by 50:1 around the world. Do rates of return? No; you don’t hear about people investing in, say, China, because they can earn a rate of return of 100% per year rather than 10 % per year … but you do hear them say they’re investing in China because the labor is a tenth the cost there.

There’s good reason for that. Capital moves much more readily from place to place than labor does. In fact, a defining feature of the financial world over the last generation has been rapid movement of capital around the world for very small differences in rates of return.

OK. Don’t lose sight of the big picture here. We’re trying to make a little, formal, model to tell us how a corporate tax rate cut affects workers. It doesn’t have to be perfect. All it has to be is a step forward from the presumption from the mental model that the incidence of the corporate tax on labor just has to be zero. So, we’re going to assume that r is exogenous too. It can move, but it doesn’t move because of what we described in the model, but rather because we might want to see how it changes other stuff.

Now, back to the three equations. Solving a three equation system is something most students do in 9th grade algebra. This is a doable thing if the equations are all lines. We have a problem here that they are not lines. So we need to linearize the (first two) equations, because of the f, and f’. In mathematics, you linearize by taking total differentials. And we’ll have to use the chain rule because we have multiplication. This is one of the few instances in this course where we’ll have to use calculus. Here are the three equations, with the first two now linearized:

dr = (1-t)f’’dk – f’dt

dw = f’dk – f’dk – kf’’dk

dx = –kf’dt

Saying the r and t are exogenous means they won’t change unless we’re exploring what happens when they do change. So dr = 0, and:

0 = (1-t)f’’dk – f’dt

dw = f’dk – f’dk – kf’’dk

dx = –kf’dt

Also note that we can cancel in the RHS of the second equation, so:

0 = (1-t)f’’dk – f’dt

dw = – kf’’dk

dx = –kf’dt

Rearrange the last equation to get:

-f’dt = dx/k

Substitute that into the first equation to get:

0 = (1-t)f’dk +dx/k

Solve that one for dk to get:

dk = –dx/[(1-t)kf’]

Lastly, substitute that into the middle equation to get:

dw = kf’dx/[(1-t)kf’]

We can cancel the k’s to get:

dw = f’dx/[(1-t)f’]

And now, this is really cool from an economic perspective. Recall that f is a production function. But it's a generic one: anything that slopes up and gets flatter, right? We have f’ in both the numerator and denominator, so we can cancel them. What’s so cool is this means no matter what details or complexity we want in our model of production to make it more realistic, it will not affect the final conclusion about dw and dx:

dw = dx/(1-t)

We’re interested in dw/dx, the response of wages to a change in tax revenue. We get this by division, so:

dw/dx = -1/(1-t)

If the corporate tax rate is about 1/3 (which is roughly what it was in the U.S.) we get dw/dx = 1.5. Thus, if we increase corporate income tax revenue by 2 dollars, wages go down by 3. The works out to an incidence of 60% on workers (3 is 60% of the total 2+3 lost by the owners of capital and the owners of labor).

This is why we do the math: a naïve view is that the incidence is zero, but a little math shows that 60% is a much better second guess. (Do keep in mind that this may seem like a lot of math to a student doing it the first time, but it is mostly just me laying out every single step in detail). Don’t believe me: here’s where I worked it out originally:

Yeah. Read it on my phone while sitting on the bed waiting for my family to get ready to go out, and knocked out the math on one of those cheap pads they leave by the phone. I am not saying you should be able to do this right now, but I am saying that you should be learning that a minute or two of math can help you readily dismiss naïve viewpoints.

An additional point to keep in mind is how marginal behavior works. This model made an initial inclusion of economics and math into a mental model. The marginal returns to that (in terms of changing what we thought the answer would be) were huge. We can add a lot of bells and whistles onto this model: more equations, more variables, more complex behaviors … but they’re unlikely to change that 60% answer too much because we’ve already made the biggest marginal contribution to our thinking.

The way that theory and econometrics work is that you assume away as much as you can in the theory to get to the heart of the matter. This gives you an idea of what to look for in the econometrics. Then you go get as much good data as you can, and hopefully you find something that confirms the theory. That is why I posted about that paper with the German data in January: it’s an empirical confirmation that 60% is a good starting point.

How could this possibly be so? It just seems like it can’t be right. To see why, go back to the three equations.

0 = (1-t)f’’dk – f’dt

dw = – kf’’dk

dx = –kf’dt

Suppose we cut the corporate tax rate, so dt < 0. The first equation still has to hold. But the only thing in there that can change in response is k. So if dt < 0, that whole last term on the RHS is negative. We’re subtracting that, so it means the term on the left must be positive too. But if 0 < t < 1, and f’ > 0, then dk must be positive. Not surprisingly, a tax cut increases the amount of capital in the worker’s hands.

But in the second equation, an increase in capital, times the negative f’’ gets us a positive value for dw. Not surprisingly, more capital increase the MPL of workers and they get paid more.

It’s worth discussing how both the casual and more formal complaints from liberal/progressive/Democrats fits into this.

The casual response is that firms just won’t invest the extra money they have after the tax cut back into capital. That would be the case if capital didn’t flow around the world to where returns are the highest. But this is one of the major international macroeconomic problems of the last 50 years: capital does exactly that. S it might be OK to surmise that your employer wouldn’t be that smart, but somebody else must be or we wouldn’t have all that capital movement in the first place.

The formal complaint, stated most clearly by DeLong, is the dw is just zero. Wages are stagnant. End of story. As an assumption, that’s OK. But that assumption would mean the w is the exogenous variable in the model, and not r. I’ll leave it to you to resolve. But, in short, if all the economic pressure from the tax cut goes into rates instead of wages, then rates would go up instead of wages. That’s possible, but not plausible: everything we know about the data for the last several decades is that rates are far more equal around the world than wages are. Because capital movements are so elastic with respect to rates, capital would pour into the U.S. in even huger amounts than envisioned in the model as set out by Mankiw. And that would put pressure on wages to go up. But if you’re going to assume that wages are fixed, then the high rates will remain, and capital will pour in even faster. The *reductio ad absurdum* is that all capital from everywhere flows into the U.S. That just does not seem realistic.

† Thomas Piketty’s *Capital in the 21st Century *made a big splash a few years ago. There’s evidence that many non-economists didn’t read much past the first chapter. But economists did (Piketty is a big name guy on our field’s political left). Piketty wants big government. He doesn’t hide that. But big government requires big taxes. Multi-national corporations can short-circuit that by moving their income around to lower tax jurisdictions. Not surprisingly, Piketty suggested that we need a single global tax system and rate to apply to corporations, with a bureaucracy to distribute the proceeds down to individual countries.

‡ Furman should watch out, since Casey Mulligan snarkily dubbed the first pass result showing that the wages paid to labor are elastic with respect to corporate tax rate cuts a “Furman ratio”. It gets worse: Furman made a fairly common initial dodge, assertingd that the result was from a simpler static model, and that a more complex dynamic would be different. Except that Erlingsson called his bluff and found that a “dynamic Furman ratio” supports the conservative/Republican position more strongly. Mulligan also pointed out that Summers argued for corporate tax rates cuts in one of his most heavily cited papers using a model very much like presented here.

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