Suppose you owned a premiership football club that was losing money and not winning many matches. You would face the following dilemma: should you spend more money on players, in the hope of improving the team’s performances, and therefore revenue, or should you save money by selling players but risk being relegated, which, if it happened, would mean that your club would lose out on huge amounts of money from lucrative TV deals?

Because your club is losing money, the current situation is unsustainable, so you are more or less forced to go one way or the other: either you try to spend your way to success or you play it safe. If you try the first strategy and it fails, then you end up in a much worse position than before — you have spent a lot of money and are still losing money. If you try the second strategy, then you will lose your best players, so that even if the club becomes financially sound it will probably face an extended period of not doing very well on the pitch.



In many countries at the moment, of which the UK is unfortunately one, there is a debate rather like this about the national finances. In the UK the national debt stands at somewhere between 900 billion and a trillion pounds. Those are the US billion and trillion, so I’m talking about pounds or so. Here are various facts to help put this figure in perspective.

1. It works out at about £16,000 for every person in the country.

2. Five years ago it was about half as much.

3. A graph of the national debt as a percentage of GDP looks something like this:

Note that it has sometimes been incredibly high — in the last century that was because of the cost of world wars.

4. The deficit (the rate of increase of the national debt per year) is running at around 140 billion.

So what do we do about this situation? There are two arguments, very similar to the arguments I discussed in connection with the hypothetical football club (but not entirely hypothetical — there have been actual examples, such as Leeds United, where attempting the spend-yourself-out-of-trouble strategy has had disastrous consequences). The first argument is often put forward by Ed Miliband and Ed Balls (leader of the opposition and shadow chancellor, respectively). They say that the best policy is to grow the economy and use the proceeds to reduce the deficit and eventually to start paying off the debt. The second is put forward by the government, who argue for reducing the deficit directly by means of deep spending cuts.

Who is right? More importantly, how do we decide? I don’t have good answers to these questions, but that is not my purpose in writing this post. Rather, I want to express my irritation at anybody who claims that the answer is obvious, which includes all politicians I have heard speaking about the subject.

To give a flavour of this, let me explain why it is “obvious” that spending cuts are a bad idea. By this I mean not that they cause a lot of misery — that is genuinely obvious — but that they are actually counterproductive, a claim that both Ed Miliband and Ed Balls have made.

As a preliminary, I shall need to dig out from the back of my brain some half-remembered ideas from an economics A’level I took about thirty years ago, which involved reading and understanding one textbook: Positive Economics by Richard Lipsey. He defined something called the circular flow of income, the rough idea of which is that we make money by providing goods and services and then go out and spend that money on other goods and services, and the money itself goes round and round. This can clearly be modelled by a flow on a graph (that varies over time), so the word “flow” isn’t too bad. (Lipsey tried to keep the maths to a minimum, so didn’t actually say this.) He then went on to talk about injections and withdrawals, which a mathematician might prefer to call sources and sinks. An example of a withdrawal is income tax, and an example of an injection is government spending. Another example of a withdrawal is personal saving, and another example of an injection is investment. And yet another withdrawal/injection pair is importing/exporting.

Keeping things qualitative, injections tend to cause the economy to expand and withdrawals make it contract. However, it isn’t quite as simple as that. Suppose, for example, that you inject money into the economy by printing it and using the printed money to finance a cut in income tax. Then a lot of people will have more money, so will be inclined to spend more. If production can increase then all is well and good, but if it can’t keep pace with the increased demand, then instead prices go up and the effect is inflation.

In general, injections create pressure in the direction of growth and inflation, whereas withdrawals create pressure in the direction of contraction and unemployment (the explanation for which is simple — if the economy is producing less then you need fewer people to do the producing).

Although I do not need to discuss it in the subsequent discussion, I should also mention monetary policy. Increasing the money supply has a growth/inflationary effect, as I have just said. Increasing interest rates has a shrinking effect, since the amount of money a company has to make to make more than the interest on a bank loan (or, if it already has the capital, to make more money than it would make simply by putting the capital in the bank and drawing the interest) goes up, making it harder to be profitable. And if your currency is strong, that has a shrinking effect because it tends to increase imports and decrease exports, thus increasing withdrawals and decreasing injections. And one final highly relevant effect: borrowing money is an injection, and paying it back is a withdrawal.

Obviously, there is much more to say than that little sketch, but it is enough for my purposes here. The argument given by the Eds is that spending cuts cause the economy to contract (or grow more slowly), that this causes unemployment to rise, and that this increases government spending because of the increased welfare bill, as well as decreasing the tax take. By contrast, if you increase public spending, then the effect is to make the economy grow, which increases the tax take and decreases the welfare bill, allowing the government to reduce the deficit.

On the face of it, this might seem a no-brainer: either you go for painful cuts that make the problem worse, or you go on a lovely spending spree that magically cures it. But actually it is a no-brainer in a different sense — it derives its plausibility from our temptation not to use our brains. The reason is that this argument is correct only if the amount of money spent stimulating the economy causes it to grow enough to increase revenue by enough to pay off what has been spent (and more). This might be the case, but there would have to be a significant gearing effect. In other words, what the Eds say is qualitatively correct (that is, when they say that A decreases if B increases, they are correct), but what matters is more quantitative: if the numbers don’t work out, then it’s more like the football team that splashes out on players who then don’t improve the results by enough to get the team out of financial trouble — which gets it into much worse trouble.

Here is another silly argument against spending cuts that one often hears. Every time new figures come out that show that the economy is shrinking, or growing disappointingly slowly, you can guarantee that there will be a Labour politician on the radio claiming that this shows that the policy of cuts is not working. This is silly because, as I’ve already discussed, it is just very basic economics that cutting spending tends to slow growth. And it’s not just Labour politicians who are silly in this way: I’ve also heard coalition politicians attempting to defend cuts by denying that they are a threat to growth, or even somehow saying that they are designed to create sustainable growth. (It’s just conceivable that something like that could be true. For instance, I have sometimes seen suggestions that shrinking and regrowing an economy can cause weaker and less efficient businesses to go under, leaving the economy leaner and fitter. But any effect there might be along these lines will be a long-term one.) So the entire argument is happening in the wrong place, so to speak. The true argument should not be about whether or not spending cuts suppress growth — of course they do — but about whether or not this unfortunate suppression of growth is actually necessary.

What about the argument that spending cuts reduce the government’s tax revenues and increase its welfare spending? Again, this is true, and again the conclusion one draws from it depends critically on those nasty little things that politicians don’t like to discuss publicly: numbers. Suppose, for instance, that a further spending cut of £1 billion per year will have knock-on effects that reduce tax revenues by £200m and increase the welfare bill by £150m. (I have no idea whether these numbers are even remotely plausible but I’m making an abstract point here.) Then the net effect is not counterproductive, since there is a total reduction to the deficit of £650m — it’s just that that reduction is less than the £1 billion. What that boils down to is the depressing fact, if you do decide to go for the cutting option, that in order to reduce the deficit by £1 billion you have to make cuts of more than £1 billion. Of course, it is logically possible that the knock-on effects of cuts might be so severe that they actually cancel out the cuts completely (which seems to be what certain Labour politicians are claiming), but to demonstrate that would be far more difficult than it is to use qualitative arguments about the direction of monotonicity of certain functions.

So far I’ve mainly argued against some common anti-cuts arguments, but the pro-cut arguments are not exactly watertight either, especially because of another phenomenon that politicians often (but not always) like to ignore: there is a continuum of possibilities. [To give another example, I always felt that the most powerful argument against going to war in Iraq was not that it was clearly unnecessary but that it was clearly unnecessary then, an argument that Robin Cook tried and failed to make the main one. In that case the continuum arose because we could decide not just on the yes/no question but also on the timing.] It isn’t just a question of whether to cut, but of how much to cut. This is a continuum that isn’t completely ignored, since Labour is not actually saying that it wouldn’t cut at all — just that it would cut less. But the debate does often seem to slip and become discretized into a yes/no question.

But if we stick with the how-much question, then the wisdom of what the Conservatives are doing becomes completely non-obvious. I do not see any compelling argument against the view that the country would be better off with a less severe package of cuts, and I also do not see any compelling argument that it would be worse off. Even at the current levels it seems that we will be running a deficit for several more years, so the plan is presumably to wait until the global economy improves (if it does, which is another major uncertainty — some have suggested that the banking crisis was a game-changer that has interrupted the normal economic cycle), at which point we can go into surplus. If that is the plan, how do we decide what the optimal deficit is for now? Should we go for the minimum level of cuts that will be sufficiently convincing to the international money markets that we can keep the interest rates on government debt at their current low levels?

I don’t know. But as I said early in this post, that’s the point: it is very hard indeed to know. My gut tells me that growing our way out of trouble is too risky a strategy — I don’t want the UK to be another Leeds United — but unlike our dear prime minister I don’t trust my gut if I don’t have the arguments to back it up. And not even my gut tells me how deep the cuts should be if we do have them. (There is also the question of which cuts to make if you do decide to cut by a certain amount, but that question is orthogonal to the one I am discussing here.)

I also do not believe anybody else who says that they have a clear (and correct, with all the numbers included) argument in favour of one particular level of cuts. But we have to make decisions, so what do we do? The answer seems to be to decide on ideological grounds (if you’re left wing then you want to cut less and if you’re right wing then you want to cut more) and then to go off and find some professional economists who can back up your point of view, which is of course easy because there will be highly reputable economists on both sides. Is there any other way of operating? Probably not, but in my dreams I like to imagine politicians actually admitting the truth, which is that the current circumstances are full of huge uncertainties, and the result of those uncertainties is that they are forced to take hugely risky decisions that might turn out to be disastrously wrong. (Just to be clear, I’m claiming here that any response to this situation is hugely risky and potentially disastrously wrong. It might seem that cutting is the safer option, but if the price is a significant contraction of the economy that wasn’t actually necessary, then I call that disastrously wrong.)

PS I haven’t suddenly decided to turn this into a political blog. In fact, I started writing this post long before the AV referendum but wasn’t happy with it. I plan to get back to some more purely mathematical posts before long.

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