# How common are discontinuous tax rates? And why would any nation have them?December 6, 2012 1:38 PM   Subscribe

I had previously thought nonflat taxes were all marginal, but to my surprise the UK Stamp Duty Land Tax doesn't use marginal rates at all. Are there other taxes like this? And why would a nation implement taxes this way?

Marginally-implemented tax rates are widely misunderstood but seem to be the most common way to implement nonflat taxes. I teach mathematics, and have for years used the construction of income tax as a peicewise linear function to illustrate both continuity as a concept and as a desirable property for taxes (since you don't want to create perverse incentives to fall on one or the other side of an otherwise meaningless line). Since I took it as axiomatic that this was a good thing, I was shocked to learn that the UK's land transfer tax doesn't use marginal rates at all, and just has sharp discontinuities at £125,000, £250,000, £500,000, £1,000,000, and £2,000,000. This apparently affects real estate prices in the UK in the way you might expect, with prices piling up on the lower side of these lines. So, seeing as how this has been an element of my teaching, I have two big questions about this practice:

* Is it used anywhere other than in this one particular UK tax? I'd really never heard of any modern nation doing this.

* Is there a particular goal it seeks to achieve -- e.g. does the government of the United Kingdom have some sort of compelling interest in disincentivizing particular real-estate prices?
posted by jackbishop to Work & Money (10 answers total) 3 users marked this as a favorite

income tax as a peicewise linear function to illustrate both continuity as a concept and as a desirable property for taxes

AFAIK, this isn't really true. At least with Federal Income Tax, there's a table of discrete values showing tax liability for different amounts of income. It's close, but not exactly equal to actually computing the percentages for that exact amount of income. Basically it's done on a "slab basis", but the slabs are pretty small.

Here's the 2011 table. For someone making \$76,699, they are in the 76,650-76,700 bracket so their tax liability is \$15,294; if they make \$2 more then they are in the next bracket higher, and their tax liability increases to \$15,306. That extra \$2 thus costs them \$12.

The brackets are small enough that they don't significantly distort or create any perverse incentives, though. Nobody actually gives up income in order to reduce their tax liability (at least if they understand the tax code; some people may do that because they fail at understanding marginal taxation, but that's their problem), even though in some really rare edge cases it might happen.

It would be mathematically cleaner if they just published an algorithm and let you punch in the values and come out with your exact tax liability, but that's not how it works. I think even the tax prep software packages use the IRS tables under the hood, not continuous functions.
posted by Kadin2048 at 2:06 PM on December 6, 2012 [1 favorite]

Stamp Duty provides a huge chunk of cash to the UK Govt, so there's little incentive for it to scrap it. But buyers have to factor into their purchase the cost of stamp duty (it's paid by the buyer) and this affects their ability to move into a property that falls into the next tax band. In the South-East of England £500,000 is barely enough for an adequate family home, so we're not talking about extravagant properties or millionaires.

Here's a study (.pdf) on the effects of stamp duty on economic mobility. If people can't move home, they can't move to another part of the country for work.
posted by essexjan at 2:07 PM on December 6, 2012

What I never understood is why, if tax is technically based off these tables, why the tables have to be an approximation to a piecewise linear function.
posted by madcaptenor at 2:23 PM on December 6, 2012

Yes, as someone looking right around the 250,000 mark in the London housing market, it creates a bizarre effect where there are lots of places at 249,900 and then nothing until 270,000 or so. There's the odd 255,000 that's clearly looking to be bargained down.

The only reason I can see for the existence of this is that the UK government doesn't give a shit. But perhaps that's colored by my general feelings towards them (yes I am aware the tax bands are not the creation of the current coalition).
posted by crabintheocean at 2:39 PM on December 6, 2012

Here is a history of stamp duty for the last 30 years, you can see it was effectively progressive back in 1984, in that you paid nothing below a limit then 1% above it. This changed from the late 1990s as the Labour Government pushed up the rate and introduced higher levels of tax on more expensive housing, i suspect they felt that this was not something that would impact on their base and did not thus need to be progressive. I imagine this changed over time as the housing bubble grew and more voters had higher priced houses, but that there was never enough of an issue there to push them to cut the income, at least not until the bubble started to fade and Gordon Brown wanted to prop it up a bit and the economy along with it. Now George Osborne can't afford to bring it down and doesn't seem to have the motivation to introduce marginal rates.
posted by biffa at 3:00 PM on December 6, 2012 [2 favorites]

AFAIK, this isn't really true. At least with Federal Income Tax, there's a table of discrete values showing tax liability for different amounts of income.

Yes, and if you look at the last two pages of that document you will find the piecewise linear functions that the tables are close approximations to. They only have the discrete tables because they lose less on the resulting rounding errors than they would by needing to chase down people who screw up the calculations.
posted by flabdablet at 7:39 PM on December 6, 2012

Is it used anywhere other than in this one particular UK tax? I'd really never heard of any modern nation doing this.

HST works this way on fast food in Ontario. The provincial (8%) portion is waived if the entire amount of the order is \$4 or less. So, if you spend \$4 before taxes, it's \$4.20; if you spend \$4.01 before taxes, it's \$4.53.
posted by one more dead town's last parade at 6:10 AM on December 7, 2012

Pakistan's income tax has discontinuous jumps; see "Behavioral Responses to Notches: Evidence from Pakistani Tax Records" by Henrik Kleven and Mazhar Waseem, which exploits this fact rather nicely:

"Using administrative tax records from Pakistan, we investigate behavioral responses to notches created by discontinuous jumps in income tax liability. Notches introduce very strong incentives for bunching on the low-tax side and density holes on the high-tax side of cutoﬀ points. We develop a method for estimating structural earnings elasticities with respect to the marginal tax rate using moments of the density distribution around notch points, and show that notches oﬀer an ideal opportunity to estimate structural elasticities in a world where optimization frictions are important. We provide evidence of large and sharp bunching below notch points along with missing mass above notch points. Moreover, observed bunching is strongly attenuated by frictions as a large share of individuals are unresponsive even in strictly dominated regions above notches. The evidence of bunching and frictions is used to estimate long-run responses not attenuated by frictions. The implied earnings responses to notches are very large, but the underlying structural elasticities driving those responses are small. This ﬁnding shows the strongly distortionary nature of notched incentive schemes. We also present evidence on income shifting between wage income and self-employment income using notches, and consider a tax reform that facilitates a comparison between notches and kinks."
posted by L0 at 3:11 AM on December 9, 2012

Oh, something analogous to the HST happens for sales tax exemptions and holidays in many places as well. For instance, New York State has an exemption for clothing and footwear costing less than \$110. Above that, you get hit with the 4% sales tax (and many localities have adopted the same exemption, so the local rate kicks in at that point as well).

(You're quite right that this sort of thing isn't sensible, but there are more tax rules being designed than there is sense to go around.)
posted by L0 at 3:14 AM on December 9, 2012

Thanks to all for the other examples! I particularly didn't know about the Pakistani income tax, although that one seems to have changed, albeit in ways which still don't completely solve the problem. AFAICT from looking at an accounting firm's summary of tax law the Finance Act of 2008 provided a discontinuity-removing kludge which effectively introduces several new short high-rate marginal brackets to smooth over the gaps (for instance, the "marginal relief" rate, as I understand it, gives the completely wonky marginal tax schedule of 0% to 350K, 1.5% to 400K, 20% to approximately 423K, 2.5% to 450K, 20% again to about 477K, then back to 3.5% to 550K, and so forth). So at least now it's a continuous function (and piecewise linear, even), but it's not exactly a traditional progressive tax with those high-rate margins in arbitrary places.

Broadly, the things mentioned above seem to divide into three classes:

* Discontinuous by implementation: e.g. the US tax tables. These are clearly designed to be continuous with marginal rates, and it's only as a matter of implementation (in this case not trusting the American people with arithmetic) that they're discontinuous. In fact technically all financial functions are discontinuous, if only because money has a degree of granularity. I generally don't regard these systems as falling under the heading of the "discontinuous-design" idiosyncrasy I'm finding so interesting here, because clearly the actual underlying design is avoiding this problem, even if it can't be translated completely into the implementation, and the perverse incentives are minimized.

* Discontinuous by binary presence/absence: The Ontario HST and similar small-transaction tax holidays fall into this category, and I suspect there are other binary fixed exemptions as well, where income below a certain point qualifies one for a specific, fixed tax credit. In the Ontario case it's clearly a matter of convenience and provides fairly minimal perverse incentive; perhaps other tax credits are, for some reason, indivisible? This kind of design seems somewhat fraught with danger, but I can at least see the appeal.

* Discontinuous by apparent ignorance/conscious avoidance of marginal rates: This is where the SDLT and former Pakistani income tax fall -- and that's kind of incomprehensible to me, so seeing major world governments doing this is kind of a shock.
posted by jackbishop at 7:37 AM on December 9, 2012

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