How are we going to pay for it?

When you listen to all the discussion about balancing the budget, about how any increase in spending needs to be matched by an increase in tax, why does no one mention that an increase in spending automatically generates extra tax?
There is no need to raise tax rates!

In fact, unless people choose to save the extra spend then it all comes back to the government as extra tax. In other words the extra spending is self funding.

This seems miraculous. Surely anyone who argues this way must be crazy—a believer in the magic money tree? Actually no, this statement follows from a very solid piece of mathematics that we all learnt at school. Unfortunately most journalists and many politicians were either not very good at maths or prefer not to explain how it really works. As voters, we should demand better.

Before getting to the maths, let’s try to answer a specific question. How much does an extra teacher cost? Or an extra nurse or an extra Doctor?

The naive answer is just to take the teachers salary, but this is wrong because a fraction of this comes back immediately because the teacher pays tax. To work out the net cost we need to discount all the extra tax paid due to the extra worker. In addition to income tax, NI and council tax, the worker also pays VAT, duties on fuel etc. The part that is not immediately repaid in tax creates extra income for others in the economy who consequently pay more tax than they did before. Excluding savings and tax avoidance, and all other things being equal, the maths tells us that the extra cost of the extra worker to current taxpayers is exactly zero!

The piece of maths that proves this is known is a geometric sequence—it says the sum of all the tax paid is eventually equal to the initial spend. If we want to calculate the amount of government spend not returned in tax we use a geometrical sequence which works like this: For simplicity, let’s say that there is a 20 % tax on each exchange whether it is paying wages or VAT. For every £100 the government spends on the extra teacher, then £80 goes to the teacher and £20 goes back to the government. If the £80 is used to buy clothes £64 pounds goes to the shop, and another £16 back to the government. The £64 is mostly spent on wages for the employees, and another £12.80 goes back to the government. The geometrical progression tells us that after 40 such exchanges 99.99% of the money is returned to the government. The extra spend is self funding!

So if money is not the thing, what limits the number of public sector workers? The answer is supply and demand. If we try to create too many posts in public service we may drive up wages and drive down productivity by removing productive workers from other sectors. So the key is to decide if there is a demand, e.g., the economy is less productive than it could be because of crime, bad health or poor education. If the evidence is that we can improve things, then government spending is a simple win-win!

PS. For another take on this see here.


  1. Sean Danaher -

    Indeed Charles, you are absolutely correct. Its crazy that the understanding of economics are so poor. Of the many depressing things in the UK, is that economy is running at nowhere near full capacity; indeed productivity is dire. It is well known that Germany, The Netherlands and Ireland are at about 135% ahead of the UK in terms of productivity. What is less well known is that the UK is behind even Italy and Spain. There is ample spare capacity in the economy. We really need an “Economics 101” and even Labour doesn’t seem to under macroeconomics any more. If Diane Abbot discussed such things in these terms she might have had a triumph of an interview rather than a car crash. Fancy tackling multipliers next?

    1. Daniel Stone -

      Productivity is misunderstood too. It is actually a measure of the value contribution per hour or per worker to GDP.

      If we are less productive it is possibly because we have many more people on low wages. Or other similar factors.

      Full employment with low wages actually makes you look less productive.

      1. Sean Danaher -

        Indeed Daniel
        One of the issues in the UK is that so many of the newer jobs in the UK are poorly paid. so many of the newer jobs are in retail, the treatment of its workers by Sports Direct for example is a disgrace and the abuse of zero hours contracts is well known. The pay structure is increasingly looking like an hourglass and the bottom 50% of society is struggling more than for many decades. It is a one of the biggest challenges in the UK to up-skill the workforce. We are very poor here on Technical Education as compared the Germany or even Ireland for example.

  2. Peter May -

    So we have the mathematical certainty of all money ending up as tax unless it is saved – or put into a tax haven. We know the government spends and taxes and not the other way round and as Chris Dillow says here:
    “Negative real gilt yields mean that financial markets are paying the government to borrow: why not take up the offer?”
    And yet still the BBC persists in asking inane budget balancing questions and still politicians persist in answering them in that context.

    1. Daniel Stone -

      Too right. Why do the key media commentators not understand this?
      Evan Davis

      etc need to understand this stuff.

      How can we get them to read and understand?

      1. Andrew (Andy) Crow -

        It ain’t easy.

        !1 I recommend finding a website that publishes intelligent commentary. (Not unlike this one)

        2) Chose carefully from the blog output items which you understand. And forward them to people of like mind so they might come back to source, to find some more. BUT

        3) More important forward them preferably in targeted fashion (by specific pertinent issue on an individual by individual basis) to people spouting BS. Done with tact this is not trolling. It’s debating.

        I KNOW this works. It’s a well-slow process, but I see my words quoted, sometimes with my permission sought. And sometimes just slipped into the flow of the other person’s new perception of the world.

        It is essential (IMO) to engage with people whom you disagree with. And to assiduously avoid arguing with them (Never fight with a pig because you both end-up covered in shit and the pig loves it)

        Try to ensure that your contributions are concise and don’t ramble on like this one.

  3. David Howdle -

    I’ve just stumbled upon this website. I’m not an economist. I’m not a mathematician. But I’m hooked! More please!

    1. Dave O'Neill -

      Its over 40 years since I studied basic economics but if I recall there was variation in multipliers due to imports. Money paying for imports was lost out of the progression. Of course if exports balanced imports then things would even out. So balance of trade and tariffs come into it and we get back to the single market I suppose. Presumably there are other sources of leakage from and contributions to the progression? Is it really quite as simple as suggested above?

      1. Charles Adams -

        Yes, you are right that overseas payments are a leak but eventually this money comes back – you cannot spend pounds in China so the Chinese buy UK companies and university degrees with their pounds. In the longer term, the balance of payments is stabilised via the exchange rate. Eventually if we do not make or supply enough stuff that others want we get relatively poorer.

        A much more pernicious leak is a tax haven where the rich can escape the money circuit and avoid contributing their fair share. A tolerance of tax havens is a clear indicator that politicians are not serious about social justice.

      2. Sean Danaher -

        Indeed. Can I plug the Progressive Pulse Book of the Month “Dirty Secrets – How Tax Havens Undermine the Economy”? I suspect when Dave O’Neill gave up learning Economics (40+ years ago) for Physics, Tax havens were not the monster they have now become.

      3. Daniel Stone -

        Doesn’t the ONS etc use tax receipts vs expenditure metrics anyway?

        Clearly they don’t balance – probably since some of the return is deferred and yes, leakage.

      4. Andrew (Andy) Crow -

        Aren’t multi-pliers those things you get at a hardware store that fold out into pliers and various screwdriver tips and for all I know a thing for taking stones out of horses hooves?

        Can’t find a smiley icon anywhere. What sort of webpage is this? 🙂

      5. Andrew (Andy) Crow -

        Ah! A clever one , I see.

  4. Brian Batey -

    As a nation we are hemorrhaging money. The current government is outsourcing major projects and services to company’s based offshore. Therefore they remove great swathes of our combined wealth.
    This is in my opinion is a criminal and will hopefully be rectified after the election.
    Great article by the way. I’d never looked at tax and earnings in this way.

  5. George S Gordon -

    Even innumerate journalists should be capable of understanding this. You could just show a few cycles of the progression without depending on the (simple) maths. The trend would be clear.

    They might say – what happens if people do save part of their salary or put it in a pension?

    I guess profiling savings is a whole other topic, but a simple example might help. Suppose everyone in the chain saves x% of their income after tax, where x is perhaps somewhat above the present low figure. You could assume that savings interest is negligible, so no tax is paid on savings interest. How much tax is then returned to the government?

    Including pension contributions would presumably be a bit more complex because of tax allowances.

    Shall I dust off my brain cells, or fire up Excel, or has someone already got the answer?

    1. Charles Adams -

      You are right that private sector savings creates a public sector deficit (excluding overseas transfers, this is an accounting identity called sectorial balances – I will try to do a post on this at some point).

      One interesting point is that that austerity may be counter productive by encouraging people to save more. One reason we save is uncertainty about the future. Cutting services, health, pension, social care etc. makes people more insecure, want to save more, increasing the deficit.

      It would be wrong to think that the government can perfectly control the deficit, but they could at least plug all the leaks in the money circuit.

      1. Andrew (Andy) Crow -

        Additional to the point you raise about the perverse effect of austerity encouraging people to save against uncertainty, I’ve also seen it suggested that, also perversely (perhaps) people are inclined to save more when savings interest rates are low because their savings are growing more slowly.

        The ‘economist’ apparently expects people to reduce their savings and go out on a spending binge and boost the high street economy because savings are not worth holding.

        Some economists apparently have a rather rigid view of what constitutes ‘rational’ behaviour.

      1. George S Gordon -

        Thanks Steve, that’s clear and very well explained.

        Since businesses are involved in the loop you describe, that presumably means their savings contribute to the government deficit. I assume business savings equate to lack of investment. The media often report that, by “holding onto cash” and not spending it, business is holding back productivity improvements. What the media generally don’t recognise is the perhaps more immediate effect – that lack of business investment contributes directly to the deficit (unless I’m missing something here?).

        Are there accepted current average figures for the savings factor for both individuals and businesses that could be fed into an extended version of your analysis?

  6. Matt -

    Thanks gr8 article.

  7. Tyler -

    Problem is that it is simply not true in practice, and definitely not in an open economy. You can look through history and see how many times increased spending simply has not led to a big enough increase in tax revenue to make it spending neutral – let alone the fact that this leaves the country that does it with a structural deficit in the case of a recession – just look a the UK and Gordon Brown post 2008.

    There are a number of logical fallacies in the above argument, but the main ones being that it assumes productivity greater than 1, assumes no money is saved, assumes zero inflation and assumes all the money stays in the economy (which it won’t given we import more than we export) and assumes the cost of the debt financing (which would be needed certainly to start with) is negligible. I could go on, but it’s yet another example of poor economic understanding. If we could solve all our economic problems simply by spending more, why hasn’t it ever happened, despite various countries trying it over the years?

    Next I guess we’ll be hearing about Richard Murphy’s magic money tree. Sorry – I mean Green QE.

    1. Charles Adams -

      Savings were mentioned in the original post and overseas payments in the comments. Nobody is pretending that you can solve ALL problems. The question is whether you think there are problems that could be solved.

      1. Tyler -

        And the various other problems with your thesis?

        If what you say is true, why did the UK end up with such a huge budget deficit post the 2008 crash? Public spending increased every year from 1980 to 2008 (and beyond), so by your theory we should not have ended up with such a large post crash deficit. So why did we?

      2. Charles Adams -

        It comes back to your point about savings. Ignoring overseas transfer, the public sector deficit is equal to private sector saving. In 2007-8, default on private debt threatened the whole private banking sector house of cards, credit dried up, people stopped spending, i.e., what income they had was saved or used to pay down debt, less tax was paid, deficit went up. And on top of that, there was a huge transfer to the banking sector.

        There are loads of books on the global financial crisis, not all good. I like this talk by Richard Koo.

        It’s long but quicker than reading a book!

    2. Maris Piper -

      Well said. The headline claim does not accord with reality. Take the handful of occasions when government is running a sustainable surplus, and then decides to increase spending. The theory expounded should mean permanent surpluses as everything extra spent will eventually come back.
      Also not mentioned is the loss to society of what that teacher might otherwise do, and the loss to pupils of what they might otherwise learn in the time allocated. That doesn’t have to mean there’s a net loss, but that the benefit of the extra teacher is the extra education given less what would have happened anyway.

      1. Charles Adams -

        There is no such thing as a sustainable surplus! Assuming overseas transfer balance over the longer term, a public sector surplus means that the private sector are in deficit, i.e., they are depleting their savings or taking on more debt which cannot last. In contrast, governments can run a sustainable deficit as long as the economy grows. The difference is that states go on working forever whereas people only work for 40 years.

        I will try to address the limits (when is the extra teacher one too many) in a future post. You are right that the limit are about what gets done, not about money! As Ann Pettifor say: We can afford what we can create – it’s how a debt based money system works.

      2. Maris Piper -

        My choice of the phrase ‘sustainable surplus’ was to avoid conflation with a surplus caused by a one-off event, the sale of a major public asset for example, or a tax windfall.
        It is though notionally possible to run a surplus of £100 a year for a timescale that exceeds human history – that is impractical but is sustainable by any reasonable standard.
        It still doesn’t get away from the point that surpluses have occurred in UK public spending before, not due to one-off events, and government upped spending and taxed the extra spending, and re-taxed it as it was recycled. The surplus pretty quickly became a deficit.

    3. paula stanley -

      What ever happened to Richard Murphy’s QE ?
      BTW Very grateful for all of this . I’m so lacking in knowledge in this area. Altho quite beyond me at the moment I feel a need to try to understand at least a basic understanding of economics and how to understand government spending..I am astounded that key journalists do not ask the most pressing questions on the economy.
      More please. Thank You.

    4. AMcG -

      All of the factors you have mentioned would exist whether or not the government spend was made. They are not driven by the spending. So they are certainly NOT logical fallacies contained in the original equation. They are attendant circumstances to the spend of course and if you choose to examine them you are free to do so. You could expand your purview still further if you wanted and even make intelligent remarks about other factors. But if you address the original equation It Is not only consistent but is based on an accounting identity that is always true. Read some Warren Mosley

  8. Peter May -

    Where on earth did the £435 billion of QE come from? The warehouse out the back? Wasn’t that spending more?
    And the cost of debt financing is negligible – at the moment at least. If money is leaking into tax havens – it is – we should stop it.
    Savings would be used to support investment if the state gave a market leading interest rate.
    More than zero inflation suggests that we’d actually get back more than was spent!
    But accept your point about importing more than we export. That is our only problem.
    But surely you cannot deny that Britain has a Sovereign currency so can create Sterling for ever if required. In various situations it may not be wise or even useful but that is certainly what a Sovereign currency allows.

    1. David Penn -

      is the assumption of productivity being greater than 1 an issue? there seems to be diametrically opposite views and I don’t have the knowledge to judge which is more reliable

  9. Alan M -

    Thanks Charles. The problem was compounded when McDonnell said ‘There is no magic money tree’ because that of course gave The Media the tree to belt him with a la Basil Fawlty.

    I can’t paste links on this machine but please have a google at ‘The Deficit’s a Good Thing. A Necessary Thing’ on YouTube. The Louis Armstrong soundtrack is a bonus. I think it, or a UK version of it, should be compulsory viewing (but I know Richard’s not a fan!)

  10. Bob Edwards -

    One interesting aspect of this debate is the average rate of tax applied to recover government expenditure via the tax multiplier. For the sake of simplicity I am ignoring leaks in the chain due to imports/exports and other factors discussed in the thread above.

    At an average tax rate of 10% it takes approx. 59 cycles to recover £100 of government expenditure, and the total gross income created amounts to £998.

    At an average tax rate of 20% it takes approx. 21 cycles to recover £100 of government expenditure, and the total gross income created amounts to £495.

    At an average tax rate of 30% it takes approx. 30 cycles to recover £100 of government expenditure, and the total gross income created amounts to £333.

    So, the lower the average rate of tax, the more income is created, and the longer it takes government to recover expenditure via taxation.

    Would anyone like to speculate on the length of the average transfer cycle?

    My concern is that corporations (especially smaller companies), have become cash cows since the banking crisis, larger corporations have shifted liquidity to tax havens, and wealthy individuals have done the same, or are burying gold ingots in their gardens.

    Combined with the import/export, pensions and other savings, repayment of debt etc. these tendencies must extend the recovery of expenditure by government? And any extension must surely increase government borrowing?

    Plugging the leaks seems an obvious, beneficial objective. Otherwise, we will continue to have the present government exhorting lower taxes, to create more wealth, when in actual fact it will merely swell offshore reserves, small company bank accounts and buried treasure chests.

    1. Charles Adams -

      Good point! The actual tax rate does not need to be that high if you can capture every exchange equally, which of course is not easy. The leaks disproportionately favour the rich. It is much easier to tax labour than capital which feeds increasing inequality, and whatever your politics, inequality will come back and bite you at some point.

      1. Jeremy Smythe -

        Professor Adams

        Brilliant article.

        On cycles, if it was me, I would reduce the number of cycles to 1 (or 2). Then there would be no leakage at all and we would have the upside of a much fairer world?

        However, unfortunately the article has proved (scientifically) that tax does not need to be raised? Hence, the problem I have with Bob’s explanation on cycles, is that I am worried that people might suggest that tax could therefore be reduced? Once the problem of leakage is solved, by getting rid of tax evasion, what’s to stop say a neo-liberal then suggesting: “OK, that’s great, let’s reduce the average tax rate to 10%, or even 5%. Because it will all come back, we now know that for certain.”

        What am I missing here?

      2. Charles Adams -

        I do not think one or two is a good idea. You need to the money to flow around doing useful work before returning, but the longer it takes to return the greater the leakage, so the optimum tax rate depends on the leakages. Currently leaks are a designer part of the system. The return on capital – house price appreciation, capital gains in pensions, ISAs, inheritance are scarcely taxed. Consequently the taxes on labour have to be set quite high to compensate.

        Economists like Piketty say that you really need very progressive taxation or taxes on wealth to prevent rising inequality.

      3. Tyler -

        The above poster has made a point I was going to follow up with – if all the money spent by the government comes back to it eventually in the form of tax, why do tax rates need to go up (which is certainly what Richard Murphy wants) at all – especially given the assumption you are making that debt servicing costs are very low, so the time for that money to recycle back to government is fairly immaterial?

        I fear that what you are erroneously trying to argue for here is the economic version of the perpetual motion machine.

      4. Peter May -

        To put in my threepenn’worth I’d suggest taxes do not have to go up, but you might want to increase them to help reduce inflation or change behaviour – improve equality or just stop people drinking too much or to put a damper on the housing market. You could also reduce taxes as Labour did with the reduction in VAT and car scrappage schemes to boost demand.

        If we didn’t have a balance of payments deficit, then government debt would be entirely voluntary. The government could choose to just create money itself (as it did with QE) but chooses instead to borrow because the financial sector likes government debt because it is 100% secure.

  11. Chris Carter -

    How come we have a deficit (and such large govt debt) in the first place then?

    1. AMcG -

      Because the banks crashed the economy in 2008 and we bailed them out. Then the Tory govt initiated policies which tanked the economy again and made it take longer to reduce the deficit.

      And whether you think the deficit or debt are large or not is a matter of opinion. The US has a much larger debt and we had a much larger debt post WW2. They don’t seem to be worried about their debt. Why are we worried about ours?

  12. Ian Ryder -

    Great satire. The nonsense in this article is exactly the kind of absurd reasoning that economic illiterates use.

    1. Peter May -

      Novel opinion. You’ll have to improve your reasoning if you want to post again.

    2. AMcG -

      But it is largely maths isn’t it? And the maths is sound. You may well argue other forces are at work on the economy than are encompassed in this progression. Economics is more than Maths (although some people would say less than) but that doesn’t change the maths.

      1. Charles Adams -

        Yes, money is just maths but what something is worth is a matter of opinion. Both the quantity and the value of money are not conserved.

  13. JOHN HALL -

    The large government debt is due to the BANKs and The rich not being regulated (Margret Thatcher)
    The reserves of mutual building societies were plundered to enable banks to create a crisis in the housing market. Perhaps they should be aloud to join the ARMS trade.

  14. John Carlisle -

    Much to commend these exchanges. Sounds like we need to go back to Alfred Marshall and the Cambridge economists, i.e. the velocity of circulation is what creates wealth – or not. Certainly we have a huge imbalance of hoarding money by the really rich people and companies and debt by the poor.
    Higher income tax, capital gains (especially) and corporation tax for the rich is part of reducing inequality; but not when we have an economically illiterate and neo-liberal treasury. So do we go back to the Roosevelt/Keynes years of price increases to drive up the lowest wages, or even apply Minsky’s government being the employer of last resort paying a living wage? But this would require a level of organisational thinking that certainly would defy the incompetents in 10 and 11 Downing Street.
    As for productivity, why (Sean Danaher) always start with upskilling the workforce? It is the system of management that is stuffing up productivity most everywhere in the UK, making it almost impossible, esp. in the public sector, to do a good job.

    1. Sean Danaher -

      I suppose I have a natural bias towards education but you make very good points. The public sector in particular has been treated very shabbily. I think the Torys are intent on shrinking the state and returning us to the 1930s and Brexit will remove many of the brakes on this.

  15. Jersey Royal -

    If the outcome of your theory does not accord with reality, then you need to either check your results or check your assumptions. Richard Feynman will have said that better. But as others have said, reality is that since 2002 we have had deficits, and real aggregate government spending per head in the 2010-2015 period was the highest of any 5 year period in British history.
    So check your assumptions, Charles ( there’s nothing wrong with the way you sum a geometrical progression – that is taught in 6th form ). It’s the premises that are wrong – they assume that all the tax income remains from what would otherwise be raised if the government did not intervene and increase that spending.
    There is no such thing as a free lunch. As Fournier and Johansson found in their 2016 OECD paper, there really are government spending programmes that can not only shrink tax revenue, but can shrink an economy to boot. The headline ones being subsidies and pensions.

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