## Raising tax can raise GDP

Mon 13 June 2016

I've been improving my grounding in macroeconomics recently and have noticed something that isn't stated explicitly in textbooks, nor mentioned in public political debate, but I do think it's correct and liable to provoke some thought.

The main point of this blog post is expressed in the title's 5 words: raising tax can raise GDP. I'll present a plain English version first and then do it more thoroughly with mathematics.

Students of macroeconomics might well shout RUBBISH! at this on instinct. Let me explain why with an example and then explain why it's not rubbish (hint: 'can' not 'will'). I'll use ballpark figures to keep the arithmetic simple and the point clear.

Consider a household with an income of £50,000. Let's say it pays £10,000 of tax and that it chooses to save 20% of its income after tax. Then of its £40,000 post-tax income, £8,000 will be saved and £32,000 will be spent.

If the government raised taxes so that the household paid another £5000 in tax then the after-tax income would drop to £35,000, of which £7000 is saved and £28,000 will be spent.

The hike in taxes causes this household's spending to drop from £32,000 to £28,000 and the difference will be equal to revenue lost to businesses, be they supermarkets, cinemas or car dealers. Since the tax hike affects all households to some degree, the net result, all else being equal, is that GDP will drop. This is the perfectly reasonable justification behind the cry of RUBBISH!

But this scenario leaves an important question unanswered: what is the government going to do with the extra taxes it's raised? The RUBBISH! criticism stands if the government hoards the extra tax for a surplus, or indeed if it's intent on reducing a deficit. But let's consider what happens if the government decides to spend its extra tax income.

In macroeconomics, GDP can be expressed as the sum of all household spending (called consumption in macro-speak) plus government spending plus other amounts which we'll take as constant here which is what is meant by "all else being equal" (these other amounts are investment and net exports). The key point to take from this is that £1 of government spending and £1 of household spending both contribute to GDP equally (even if they can play quite different roles in shaping our society).

Before the tax rise, our example household spends £32,000 and gives £10,000 to the government. So the total private plus public spending we can attribute to that household, and which contributes to GDP, is £42,000.

After the tax rise, the household spends £28,000 and gives £15,000 to the government. So the total spending is now £43,000 and so £1000 more is contributed to GDP than before the tax rise.

This household is earning well above the median income, but with around 27 million households in the UK, it's clear that a boost of several hundred pounds per household is going to lead to a boost to the economy of several billion pounds, perhaps even tens of billions.

So, all else being equal, raising taxes can boost GDP if a government spends the extra amount raised. The reason is simple in our example scenario: households save some of what they earn whereas the government spends all it raises in tax.

In reality, not every household saves money and some do not earn enough to save anything at all, but, on average, households do save some money, and households on higher incomes save more and pay more in tax than those on lower incomes.

The UK government is almost always in deficit, meaning that it spends more than it receives in taxes. However, it's highly unlikely the current Conservative government will raise taxes, and even if it did, it would not use the extra revenue to increase spending, but instead would celebrate a reduction in the deficit.

The scenario we've considered is much more relevant to proposals made during the 2016 Scottish parliament election. Three of the four opposition parties did propose raising taxes and pledged to use the extra money raised to increase spending on education and other vital services. If they had done so, the end result of such modest tax increases would have given Scotland's GDP a small boost, which is certainly needed given the drag placed on it by recession in North Sea activity due to the low oil price. Unfortunately, those parties all remain in opposition and the largest opposition party — the Conservatives — along with the SNP Government have ruled out any significant tax rises.

There are two points worth bearing in mind. Firstly, an argument about whether public or private sector spends money more efficiently is a separate point to the one being made. As long as someone is spending the money into the domestic economy then there's a boost to GDP. Similarly, it's also a separate issue as to whether you think increasing GDP is in itself a worthy goal.

Secondly, the point I'm making is a variant on a Keynesian stimulus but one I felt it necessary to articulate because I heard a few folk claim that raising taxes in the context of austerity will lower GDP. If austerity is taken to mean simply cutting spending then this may be true, but a tax rise can offset the cut's impact on GDP. That said, a rise in GDP is small comfort if the spending cuts fall on services that people depend on.

## Mathematical version

Let's start from a well known equation from macroeconomics for GDP $$Y$$:

$$Y = C + G + I + X - M \label{income-full}$$

This equation is an identity. In other words, it's true by definition and makes no assumptions.

Before I define the terms, let me sketch its meaning intuitively. The left hand side represents total income in an economy and the right hand side is a total of various kinds of spending in an economy. Consider a very simple economy that consists of just you and one shop-owner. You go into the shop and spend £10. You give a £10 note to the shop-owner and she has an income of £10. What the above equation expresses is conceptually no more profound than that: your spending is another's income. The added complexity comes from adding up various kinds of spending and deciding what kind of transactions to include, and what not to include. That latter decision is mainly about avoiding double counting.

Here are the definitions:

• $$Y$$ is the income of everyone resident in this country arising from economic activity that takes place here — for present purposes you can think of it as GDP.
• $$C$$ is consumption — it represents all household spending on groceries, on a meal at a restaurant, on clothes, on fuel or on paying for a month's IT internet, and so on.
• $$G$$ is government spending — money spent by the government on goods and services. This includes buying tables and chairs for schools or government offices, paying nurses. teachers and judges for services they provide, and so on. It excludes benefit and pension payments.
• $$I$$ is investment — this is mainly money spent by businesses and as such is expected to bring a return in the future, for example on factory machinery or in installing or upgrading an office's IT system. It also includes inventories, which are goods of some value held in stock by businesses. Investment also includes some non-business spending, such as the purchase of a newly built house.
• $$X$$ is exports — money paid by customers abroad in exchange for exported goods or services provided here.
• $$M$$ is imports — money paid by customers in this country for goods and services provided from abroad. This is the only term to be subtracted because it is removing spending that is included in $$C$$, $$G$$ or $$I$$; such spending will be someone's income in, say, China, but not here so it shouldn't be included in $$Y$$.

But, for present purposes, we do not need all this detail, so I'll tidy the last three terms into a term called $$R$$ (the Rest) to keep the maths simpler:

$$Y = C + G + R \label{income}$$

Next, let's define tax as being a fraction $$\gamma$$ of income:

$$T = \gamma Y \label{tax}$$

We can define a standard consumption function:

$$C = a + b (Y-T) \label{consumption}$$

where $$a$$ represents the absolute minimum consumption in an economy that might occur when everyone is nervous of some future calamity and saving every penny earned. $$Y-T$$ represents disposable income, that is income after tax has been deducted. A fraction $$b$$ of it is spent on consumption. In economic good times when people are optimistic about their future incomes, it's likely that $$b$$ will increase; conversely, it'll be lower in a recession.

If we use (\ref{tax}) to substitute $$T$$ in (\ref{consumption}) we get:

$$C = a + b (1-\gamma) Y \label{consumption-tax}$$

Next, let's say that government spending $$G$$ is made of two parts:

$$G = T + F= \gamma Y + F \label{gov}$$

This expresses government spending as the sum of tax revenue $$T$$ and non-tax revenue $$F$$. We'll consider what $$F$$ represents later on.

So let's substitute $$C$$ from (\ref{consumption-tax}) and $$G$$ from (\ref{gov}) into (\ref{income}):

$$Y = a + b (1-\gamma) Y + \gamma Y + F + R$$

Then collect terms with $$Y$$ on the left

$$Y (1 - b - \gamma +b\gamma) = a + F + R$$

which gives us the final result

$$Y = \frac{a + F + R}{(1-b)(1-\gamma)} \label{result}$$

## Interpretation

Firstly, note that $$\gamma$$ is not the same thing as income tax as it is a fraction of total income (GDP) in the economy, not just what people receive as pay. However, a lowering of the income tax rate will likely cause a lowering of $$\gamma$$ because income tax is the single largest source of tax; it makes up about one fifth of the total tax take.

To interpret this result, let's consider that $$\gamma$$ changes, and all quantities stay the same:

• If $$\gamma$$ is reduced then $$(1-\gamma)$$ will increase and $$Y$$ will decrease.
• If $$\gamma$$ increases then $$(1-\gamma)$$ will decrease and so $$Y$$ will increase.

In other words, this model is telling us that increasing tax rates with all other variables staying constant will raise GDP. To understand this we need to consider the two effects of changing $$\gamma$$ in the above equations.

Increasing taxation reduces consumer spending $$C$$ and we can see that the reduction is $$b \gamma Y$$ in (\ref{consumption-tax}). The other effect of increasing taxation is to increase government spending $$G$$ due to the term $$\gamma Y$$ in (\ref{gov}) which represents the bit of government spending funded by tax. Since $$b$$ is a fraction and so less than one, that means the increase in government spending is larger than the decrease in consumer spending and so all else being equal raising the tax rate $$\gamma$$ means that $$Y$$ is increased.

In other words, a government will spend 100% of the money it raises in tax. But if the government were not to take that tax and it was left in consumer hands, then they would chose to only spend a fraction $$b$$ of it.

What if all else is not equal? That is, what if other variables on the right hand side of (\ref{result}) don't stay constant and, most significantly, what if they change in response to $$\gamma$$ changing, i.e. they are functions of $$\gamma$$? I can't hope to address this properly in a blog post, but I'll sketch my thoughts on it.

If people trusted the government tax rise was going to lead to increased spending which benefited the economy, perhaps for the reason argued here, or else because they believed, say, spending on improved education might bring benefits in the longer term, then their optimism may be reflected in increased consumption (larger $$b$$) and this will boost the economy further. But, if the populous is not so enlightened, and are spooked about profligate government spending or mounting public debt, or if they simply don't trust the government to manage the economy, then this may dent their optimism and reduce consumption (lower $$b$$), cancelling some or all of the tax-and-spend rise boost to GDP.

Notice the use of the word 'believe' here. People do not react to reality itself, but what they believe to be true of reality. Despite the apparent certainty of starting with a mathematical identity, we've ended up having to consider not only whether $$b$$ is a function of $$\gamma$$ but also if it's a function of mass psychology! I'm not sure how you might express such a function — perhaps some stochastic Ito calculus is needed.

Another interesting point is the extent to which government can influence the economy. Let's start with the easy bit: variables which a government really has no control over. The minimum level of consumption $$a$$ required by the society does not obviously seem to be under government control, nor does the fraction of income $$b$$ spent on consumption.

Next, consider the $$R$$ term which is composed of investment $$I$$ and net exports $$X-M$$.

Investment can be influenced by interest rates, but in the UK the last Labour government gave the Bank of England the responsibility for setting them to keep inflation close to the 2% target. The government may have some indirect control here and can enact policies that encourage investment in specific areas, such as feed-in tariffs for solar energy generation (recently cut). But investment is prone to psychological views of the present and future in the same way as consumption. In fact, perhaps even moreso. It was one of John Maynard Keynes's major insights that although $$I$$ may be much smaller than $$C$$ in size, it can vary suddenly and trigger unpredictable swings in GDP $$Y$$.

A government may be able to influence exports on longer timescales with trade deals and can make the price of exported goods lower by devaluing Sterling, but it has to balance that with the above mentioned setting of domestic rates and the possibility of capital flowing out of the country.

So, that leaves us with $$F$$ and $$\gamma$$ that appear to be most directly under government control. Recall that $$F$$ is the bit of government spending that is not funded by tax revenue. Let's rewrite (\ref{tax}) as:

$$F = G - T \label{deficit}$$

Written like this, it's obvious that $$F$$ represents a budget deficit: government spending minus its tax revenues. Equation (\ref{result}) makes it pretty clear that reducing $$F$$ will reduce GDP. If austerity is to work (meaning that it will one day raise GDP) then it must do so by boosting $$a$$ or $$b$$ or $$I$$ or $$R$$. I cannot think of a credible way this could happen. But there's an interesting point here. Even if the government is reducing $$F$$ by cutting spending, there's still the possibility that a tax rate rise, i.e. an increase in $$\gamma$$, can offset it, and potentially hold GDP constant or even allow it to increase. I can't see the current Conservative government doing this and so I think this will remain an untested theoretical point for the foreseeable future and besides, the more sensible thing to do according to equation (\ref{result}) is not try to reduce $$F$$ in the first place.

There is also another interpretation of $$F$$ in the context of Scotland — a fiscal transfer. I won't get into details here, as this deserves a blog post to itself, but since $$F$$ is supported at the UK level, and because the Scottish government is compelled by law to balance its budget with only modest borrowing powers, any rise in tax will necessarily be matched by a rise in spending. This means an increase in $$\gamma$$ will lead to a boost in GDP. Again, for the political reasons mentioned above (at the end of the non-mathematical section), I can't see this happening.

## Final thoughts

The strength of the above model is that it starts out from a mathematical identity that must be true. The first weakness was introduced when we assumed a linear form of the consumption function $$C$$ in (\ref{consumption}). Even if $$C$$ is not linear in $$Y$$ and if $$b$$ varies a little with $$\gamma$$ then the result will retain some validity so long as $$C$$ increases with $$Y$$. But of course it has absolutely no relevance to reality in the circumstances of 2008 when the financial system seized up and confidence fell through the floor. I think it can be used to think about the situation of the economy outside of such crisis times and so might well be relevant to present circumstances.

The second weakness is when we used it to solve for $$Y$$ in (\ref{result}). This result describes an equilibrium, but as all quantities on the right hand side are continually changing no true equilibrium can ever be reached. This could be relevant if changes in variables such as $$a$$, $$b$$ and $$\gamma$$ are gradual enough so that we can imagine the economy going through a series of equilibria where (\ref{result}) is close to being true. However, if $$I$$ dropped suddenly, as it would at the onset of recession, then a proper dynamic model is required and it would somehow need to factor in the above mentioned mass psychology.

I believe this kind of simple model has some use but only so long as the assumptions it makes are clearly understood and are relevant to real circumstances.

I would welcome increased government spending on education and health because both will benefit society and the economy in the long term. If this is funded by raising progressive taxes then we can also tackle the problem of income inequality and, if the assumptions made above hold, we can also give the economy a wee boost too.