# Does applying taxes increase the GDP by the same amount?

Let's say we have a country with a population of 1 million that are all working. The GDP per capita is 10,000 dollars so the GDP of this hypothetical country is $10 billion. Now let's say that no taxes are applied and the people are covering 100% of their needs with the amount they are making, but since there is no tax revenue, there is no government spending, infrastructure maintenance, security.. etc. and the country is deteriorating. If the government now applies 10% taxes over the income of the population, then the tax revenue will be$1 billion and the population will be left with 9 billion which would now be enough to cover just 90% of their needs.

Now in order for them to earn more to cover this 10% drop in their income and consequently their needs, they will need to work for the government to re-earn this $1 billion. If they do, then this population GDP becomes$11 billion instead of 10 billion and so applying 10% taxes increased the GDP by the same amount

From the way I explained this, it is obvious that I might not have a good understanding of how this works, but this is how I could think about it. So is this by any means correct, or am I making wrong assumptions?

• People have not only needs but also discretionary spending. Commented Sep 27, 2022 at 14:07
• There are not necessarily "wrong assumptions", as you are describing a hypothetical of your own device. However, there are impractical and inapplicable assumptions you make, such as the people's income being exactly equal to their needs. Commented Sep 27, 2022 at 15:04

No this is not true. GDP (assuming for simplicity closed economy) is given by:

$$Y = C + I +G \tag{*}$$

Where $$Y$$ is GDP, $$C$$ consumption spending, $$I$$ investment spending and $$G$$ government spending.

Now to see how taxes impact GDP we have to make some assumption on consumer behavior. A logical first approximation is as follows:

$$C = c_0 +c_1(Y-T) \tag{**}$$

Where $$c_0$$ is autonomous consumption (consumption irrespective of income) $$c_1$$ is marginal propensity to consume (MPC) which tells you what share of income is consumed and what saved. E.g. MPC=0.7 means people consume 70% of their income, finally $$T$$ is a lump sum non-distortionary tax (e.g. tax that has no negative effects on the economy).

Now even with completely non-distortionary taxation we can see that when we substitute ** into * we get that:

$$Y = \frac{1}{1-c_1}\left(c_0 +G +I - c_1T\right) \tag{***}$$

From the equation above we see that government spending multiplier is $$\frac{1}{1-c_1}$$ that means 1 dollar of spending increases GDP by $$\frac{1}{1-c_1}$$ but taxes have negative multiplier $$-\frac{c_1}{1-c_1}$$ so for every 1 dollar of non-distortionary taxation GDP shrinks by $$\frac{c_1}{1-c_1}$$.

These two multipliers cancel each other out when you want to run balanced budget (which you imply in your question but saying everything is just funded by taxes with no mention of debt).

In case you want to implement distortionary taxes such as 10% income tax this would actually reduce GDP since then:

$$C = c_0 +c_1(1-t)Y$$

Which substituting back to * gives us:

$$Y = \frac{1}{1-c_1(1-t)}\left(c_0 +G +I \right)$$

Now it is easy to see that $$Y$$ in the equation above will be highest when $$t=0$$, since $$c_1$$ must be between 0 and 1 and the higher $$c_1$$ the higher GDP will be.

However, note this does not mean that taxes can't increase GDP over time through economic growth. For example, building infrastructure today might allow us to produce more in the future, so such investment would increase people's $$Y$$ in future. But government cannot magically increase GDP by just taxing and spending money while keeping balanced budget.

• "But government cannot magically increase GDP by just taxing and spending money while keeping balanced budget." Why not? Is this not one of the issues with GDP as an indicator? Commented Sep 27, 2022 at 15:16
• In your model, if I increase taxes and government spending by \$10 each, then the balance is not affected and consumption changes by less than \$10, right? Thus $C + G$ increases? Commented Sep 27, 2022 at 15:31
• @Giskard 1. it is not my model it is the standard Keynesian cross model. 2. No where did you got that? If you do the math the multipliers perfectly cancel each other since $\frac{1}{1-c_1}-\frac{c_1}{1-c_1}$ so the resulting multiplier on government spending is 1 - that is to say there is no multiplier government spending does increase GDP exactly by 1 but then that spending is funded by subtracting 1 from GDP
– 1muflon1
Commented Sep 27, 2022 at 15:36
• @Giskard yes it is still true. It can be violated when you have different people with different $c_1$ then you could actually have a result where $Y$ increases by redistributing from people with low MPC to people with high MPC, but note OP asked about infrastructure spending not transfers between people. Also transfers between people are not part of G but they would change individualized Y by negative T (negative tax is transfer)
– 1muflon1
Commented Sep 27, 2022 at 15:44
• Anyway, thank you for the discussion (: Good night! Commented Sep 27, 2022 at 19:43

There may be scenarios where what you're saying would be true. If the tax really put people in dire straights and forced them to work more than usual and trade more than usual to survive, presumably the GDP could be increased this way. However, it should be noted that while GDP might go up, the wellbeing of the society would surely go down. Inducing people to work harder when they otherwise wouldn't is most certainly a negative stressor as well as a cause for hasty actions and reduced investment.

Tho its unlikely this would be effective in anything but the very short run. Most taxes generally cause substantial deadweight losses and so one would usually expect that taxes should reduce GDP - if not this year, then the next.

Before anything else, one needs to discuss

"...And why should the government tax the population?"

Would some of this tax revenue go towards covering certain needs of the population, that currently were (however) covered by private spending?

If some taxes would go towards better roads => less expenditure in maintaining/fixing private vehicles

If some taxes will go towards better security => less expenditure in private security companies (and maybe less expenditure to psychotherapy/medical expenses caused by the stress of feeling insecure)

etc

In other words, to discuss a realistic/interesting scenario, at least some amount of the tax revenue is expected to substitute for private spending.

Moreover, in an uncertain environment, it may be the case that the expected utility of the population may increase.

Say, $$I$$ is my gross income, $$S$$ is the expenses I incur for private security and $$p$$ is the probability that I do not get robbed (after spending $$S$$).

My expected utility from consumption is $$E(U) = p\cdot U(I-S)$$

Assume the government takes taxes me $$T$$ with which it improves security so a) I do not need to spend $$S$$ and b) the probability of not being robbed increases to $$p' > p$$. My new expected utility is $$E(U') = p'\cdot U(I-T)$$

Because we expect $$T my expected utility has improved from two channels.

In general, it is ambiguous what will happen. The general principle of decreasing returns is expected to hold here, so if we look at an almost non-existent state, then the initial tax revenue is expected to have large benefits for the population (essentially exploiting scale economies and positive externalities of various publicly provided goods).