So we\’ve Ritchie copying the Left Foot Forward mistake about the Laffer Curve. Something to which I\’ve already responded:

Umm, Alex, if you want to quote a research paper in support of your assertions it might be useful to read and understand the research paper to make sure that it does in fact support your assertions.

Specifically this one \”Other economists put the peak closer to 76 per cent.\”, the Diamond and Saetz paper. Now, as one of them has a Nobel and the other is highly likely to get one I do agree that this is an interesting and useful paper that we might want to consider.

Now, in that paper they are discussing the US taxation system. Which as you know is subtly different to ours. Most importantly, leaving the country doesn\’t get you off US taxation. Leaving the UK does get you off UK taxation. So whatever the peak is for the US the one for the UK will be a little lower. How much, don\’t know, but that you can flee the jurisdiction will lower the rate.

But much more important than this is that you\’ve not in fact understood what they were saying. The 76% rate is only if there are no allowances. No ability to shift income for example to capital gains. No charity credits, no pensions savings, no mortgage interest deductions.

As they point out, that isn\’t how the tax system works. So, within the constraints of the current tax system they say the peak of the curve is 54%. Do go read the paper to check, won\’t you?

They also point out that this is not in fact their estimation of the revenue maximising federal income tax rate. This is the revenue maximising \”all taxes\” rate on incomes. And something you\’ve obviously missed, for it seems like a detail in the US tax system, is that they are including the employer paid Medicaid tax in that top rate.

When we move over to the UK system this becomes very important though. The US has social security taxes, just like we have NI, employer and employee paid. However, in the US system, both are capped. No further contributions are made over a certain income level. The only tax that isn\’t capped is that Medicaid one. And Diamond and Saetz include it in their workings. And the important of this is that NI is not capped in the UK. Employers pay 13.8% however high the wages go. Employees pay 2% I think from this year? We must therefore include the NI rates in our calculations for the UK.

So, from the paper you yourself call into evidence we find that the top \”all tax\” revenue raising rate is 54%. In that we must include the two NI taxes, 13.8% and 2%. It\’s what is left after that that gives us our revenue maximising income tax rate.

That is, your very own evidence shows that 50 % income tax alone is over the revenue maximising rate.

As I say, you might find it useful to actually read the papers you\’re calling into evidence. For you might find out that if you don\’t you\’ll be calling into evidence something that entirely disproves rather than supports your assertions.

Not that that is all that unusual over here in leftyland where there are infinite money trees, unicorns poop rainbows and everyone does indeed have a pony.

So that\’s that part. And then Ritchie descends into hte depths of his own howling lunacy and ignorance.

I have – let’s be clear 50% only really applies at incomes of £1 million as an overall rate

So shall we get real?

Yes, let\’s get real. The Laffer Curve is about *marginal* tax rates. Ritchie has just there appealed to average tax rates. You\’d think that the whole marginalist revolution of the 1870s had entirely passed him by, wouldn\’t you?

As, apparently, it has in fact. You know, all this Alfred Marshall stuff, the Cambridge School, the influences upon and even teachers of Keynes himself?

“Christina Romer and David Romer have a new paper looking at evidence from the 1920s and 1930s and find that the revenue-maximizing rate on the highest earners is extremely high—over eighty percent.”

You’d think that even Murphy and the Left Foot Forward “evidence based” community would realise that the world today had changed a little since the 1920s.

Is there a Laffer Curve? The LC postulates a direct relationship between Tax Income and Tax Rates. Making up symbols, it is the assertion: I=f(R) where f is some computable function. Is this a valid assertion?

Suppose there were only one tax rate. All income is taxed at 20%. There is a stable money supply of, say, 1 billion groats. Our basic assertion that MV=PQ, removing the kludge factors of V and P(there is no inflation in our stable money system, and V is simply a multiplier to allow a primitive integration over time) becomes M=Q. (Any money “hoarded” is not in M). M becomes simply the national income over a single transaction cyle; all of it is spent on goods; the total money value of all the goods produced in the economy (Q) and the money spent on them is thus M. The tax income is thus easily shown to be 0.2 * M, or 200M groats over 1 transaction cycle (a “V period”). So our function is simply the tax rate itself. The line on the graph is a simple linear relationship.

Thus, any observed change in real economies that is non-linear to the tax rate must be due to something else. In particular, most tax systems are very complex. It is thus possible for wise persons to manipulate their tax liabilities and reduce their personal tax rate; setting up front companies, maximising allowances, employing the wife, and so on. It is not a relationship between “the tax rate” and tax revenue, then, but a very complex function of particular individual responses to particular taxation regimes, and their manipulability by wise persons, not a function of a notional aggregated tax rate.

How then can there be a Laffer Curve?

Ian,

Simple maths, as discussed here before. Tax at 0% raises zero income (obviously). Tax at 100% would be assumed to raise zero income (less obvious but difficult to argue against.) Countries with tax rates in the middle do indeed raise income.

So there is a function, from 0 to 100% with some shape and having some positive values. Therefor there must be at least one maximum inflection point. And, of the maximum inflection points, one or more must be at an absolute maximum value.

Hence the Laffer Curve, as basically described, must exist. What the shape is and where the maxima lie is very different.

All of your supposition, groats and notional aggregates are simply irrelevant.

Personally, I accept much of your argument and believe that the shape of the curve is too complex to be calculated or otherwise derived. But it must exist.

SE-

That’s what I’m questioning. I don’t think there is any meaningful “curve” with aggregate tax rate on one axis and aggregate tax income on another axis. The fact that you can plot two variables doesn’t mean they are related. Any deviation from the linear- the interesting thing about the purported Laffer Curve- is not a function of tax rates, but of other factors related to the particular form of the particular tax system.

That’s why I went back to that very basic model. It tells us that a Laffer Curve as described does not appear derivable from first principles. Tax income differentials are, for instance, dependent on the differential tax rates in the system, not the aggregate “tax level”. That is; if I can switch my profit tax burden from a 40% rate to a 30% rate by accounting ingenuity, that kind of thing.

There simply doesn’t seem to be the asserted function of R.

You because you cannot assert the function doesn’t mean that you cannot model it. You might not be able to model it very well (cue huge digression in to climate change) and, I agree, “R” is itself hugely complex.

However, as we have quite so many different tax systems in the world, whereas we only have one climate, and politicians tinker endlessly and at least annually with different values and constructions of R, I bet it would be possible to get a range of curve shapes and find out if there is any underlying meta-information that we could use to best understand how to set our R.

From the point of view that having decided govt spending is x and x + d (debt repayment) = I, we want to get I with as low a value of R as possible (slightly modified by the different growth effects of some taxes). This is not the point that Ritchie is generally making. He wants to maximise x and R, largely regardless of the effects on I.

Ian B: I too am sympathetic to your argument. However…you don’t seem to quite understand the mathematical ideas you’re trying to use.

A function is, to quote Wikipedia, “a correspondence that associates each input with exactly one output”. So as long as any given tax rate x yields a *single* revenue number y, then we can say that y is a function of x such that y = f(x).

You go on to say that “That’s what I’m questioning. […] The fact that you can plot two variables doesn’t mean they are related. Any deviation from the linear- the interesting thing about the purported Laffer Curve- is not a function of tax rates, but of other factors related to the particular form of the particular tax system.”

But if you can plot two variables, it does mean that one is a function of the other. That’s the definition! (Assuming, of course, that any line perpendicular to the y-axis crosses the curve at most once. Which in this case, it does.) So that rather makes a nonsense of your first comment.

I think what you’re trying to say is that tax rates don’t *directly* change tax revenue; they change people’s behaviours (for example, by causing avoidance, or driving people into the grey market, or whatever), and that changes tax revenue. Which is true, but also precisely the idea behind the Laffer curve!

As for your model…you assume that V is unchanging, but this is not a good assumption. V may be effectively constant for most real world tax rates, but imagine a tax rate of 100%. Faced with a 100% tax rate, people would not make transactions (“I’ll give the government $2, if you give me a can of coke.”), so V would be zero. Presto, your simple model now demonstrates the Laffer curve.

Martin Gardner drew a nice picture of a “neo-laffer curve” – http://www.flashq.org/math.pdf page 5.

The Diamond and Saez paper has been abused by both sides of the argument. Here (again) are my thoughts – http://pb204.blogspot.com/2011/12/optimum-tax-rates.html

Cody, yes I agree my language was poorly conveyed. I got into this trouble last time on this issue. I am not a mathematician. I’ll try again.

The assertion I made was that the tax take is simply the same variable as the tax rate, simply mutliplied throughout the economy; just as the average height of the population is the same variable as the individual heights of the population, aggregated. Thus, a 20% tax rate will produce a 20% tax income. A 30% tax rate will produce a 30% tax income. They aren’t a function of each other. They are the same variable, expressed differently. Thus, you cannot expect anything but a linear “relationship” between X and Y, if as I assert X and Y are

the same thing. There is no actual relationship at all; if you plot a graph with one on each axixunder the illusion that they are differentyou will never get anything but a straight line, from 0,0 to 100,100.The actual reason that State tax income can vary with tax rates is that people shift their tax liabilities as best they can; the variability will thus entirely be a function of the flexibility of the tax regime. The income isn’t a function of the aggegate tax rate; the aggregate tax rate and tax income

being the same thing expressed differentlyare both functions of theparticular tax regime. If we were to create the simplest possible tax regime- everyone pays X% of all income, no other rules at all; the purported Laffer Curve would not exist at all other than that straight line.In the I=f(R), all the variability is in the function f, and not any of it in the R. Surrpetitious Evil above mentioned climate change; it is the equivalent of all the variability being in the feedbacks, and nothing at all in the CO2 level; in which situation it is meaningless to plot a graph of CO2 on one axis and T on the other.

NB I possibly introduced a red herring at the end thar since T and CO2 are of course different variables.

Ian B of course the Laffer Curve doesn’t exist if you assume away changes in the money supply and price level. Those are the transmission mechanism by which the Laffer Curve works!! If you’re a closed economy, P increases so that money revenue increases but real revenue falls. If you’re an open economy on a floating exchange rate standard, P can’t change because that’s the world price level; what changes is the exchange rate, which depreciates, causing the same result (a fall in the purchasing power of tax revenue). An open economy on the gold standard sees gold flow out of the economy to purchase goods elsewhere; since MV = PY and P can’t change (world prices again) then if gold flows out then M falls and therefore Y, real GDP, falls.

This is called the Mundell-Fleming model.

Ah, so it’s a bugger-thy-neighbour thing then?

What happens if those foreign johnnies don’t play cricket, and instead do the same thing?

It’s not a “beggar-my-neighbour” thing because the effect is of the same magnitude in a closed economy, where there is no “neighbour”. If the rest of the world raises taxes as well (in the open economy case) then the world price level will rise. In the open economy case the world price can’t change

as a result of changes in the domestic economy. If the rest of the world changes then the world price will change.