<p>No this is not true. GDP (assuming for simplicity closed economy) is given by:</p>
<p><span class="math-container">$$Y = C + I +G \tag{*} $$</span></p>
<p>Where <span class="math-container">$Y$</span> is GDP, <span class="math-container">$C$</span> consumption spending, <span class="math-container">$I$</span> investment spending and <span class="math-container">$G$</span> government spending.</p>
<p>Now to see how taxes impact GDP we have to make some assumption on consumer behavior. A logical first approximation is as follows:</p>
<p><span class="math-container">$$C = c_0  +c_1(Y-T) \tag{**}$$</span></p>
<p>Where <span class="math-container">$c_0$</span> is autonomous consumption (consumption irrespective of income) <span class="math-container">$c_1$</span> 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 <span class="math-container">$T$</span> is a lump sum non-distortionary tax (e.g. tax that has no negative effects on the economy).</p>
<p>Now even with completely non-distortionary taxation we can see that when we substitute ** into * we get that:</p>
<p><span class="math-container">$$ Y  = \frac{1}{1-c_1}\left(c_0 +G +I - c_1T\right) \tag{***}$$</span></p>
<p>From the equation above we see that government spending multiplier is <span class="math-container">$\frac{1}{1-c_1}$</span> that means 1 dollar of spending increases GDP by <span class="math-container">$\frac{1}{1-c_1}$</span> but taxes have <strong>negative</strong> multiplier <span class="math-container">$-\frac{c_1}{1-c_1}$</span> so for every 1 dollar of non-distortionary taxation GDP shrinks by <span class="math-container">$\frac{c_1}{1-c_1}$</span>.</p>
<p>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).</p>
<p>In case you want to implement distortionary taxes such as 10% income tax this would actually reduce GDP since then:</p>
<p><span class="math-container">$$C = c_0  +c_1(1-t)Y  $$</span></p>
<p>Which substituting back to * gives us:</p>
<p><span class="math-container">$$ Y  = \frac{1}{1-c_1(1-t)}\left(c_0 +G +I \right) $$</span></p>
<p>Now it is easy to see that <span class="math-container">$Y$</span> in the equation above will be highest when <span class="math-container">$t=0$</span>, since <span class="math-container">$c_1$</span> must be between 0 and 1 and the higher <span class="math-container">$c_1$</span> the higher GDP will be.</p>
<p>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 <span class="math-container">$Y$</span> in future. But government cannot magically increase GDP by just taxing and spending money while keeping balanced budget.</p>

CC BY-SA 4.0
2022-09-27T15:16:42.363Z
"*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?
Giskard
CC BY-SA 4.0
2022-09-27T15:17:10.240Z
Also it seems like your treatment sidesteps the issue that $G$ is probably related to $t$ in the medium-run.
Giskard
CC BY-SA 4.0
2022-09-27T15:21:34.873Z
@Giskard because you cant increase size of a pie by cutting it into different pieces. BTW I am not sure if you read the full answer but I do mention explicitly government can do that over time through economic growth, but the question is about immediate change in GDP so any medium/long-term issues do not apply here and I clearly stated that in the long run government spending can do that through public investment
1muflon1
CC BY-SA 4.0
2022-09-27T15:28:36.793Z
@1muflon1 I don't mean growth via investments in infrastructure, I mean pyramid building/ditch digging a la Keynes.
Giskard
CC BY-SA 4.0
2022-09-27T15:28:41.330Z
@user253751 1. Inflation means that the value of dollar changes, that is not a downside of GDP. GDP does not doubles just value of currency changes. For example, if you use speedometer to measure speed, even though there will be measurement error, the speedometer is not worse if you just program there switch that changes units from km/h to m/h. Similarly inflation just means you changed measure of dollar from one measure to another 2. real GDP is unaffected by inflation. 3. GDP has a measurement error, but there is little reason to think that GDP measurement error is affected by taxes
1muflon1
CC BY-SA 4.0
2022-09-27T15:30:44.370Z
@Giskard but Keynes never claim GDP can be increased with **balanced** budget. You can do that by deficit spending.
1muflon1
CC BY-SA 4.0
2022-09-27T15:31:33.673Z
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?
Giskard
CC BY-SA 4.0
2022-09-27T15:32:39.253Z
@user253751 but that has little to do with taxes. Netherlands has private healthcare system like US, UK has nationalized healthcare yet, they both have almost the same healthcare spending relative to their economy sizes according to OECD. This is not to say that GDP is accurate measure of human welfare, but that is completely unrelated to this issue with taxes
1muflon1
CC BY-SA 4.0
2022-09-27T15:36:00.583Z
@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
CC BY-SA 4.0
2022-09-27T15:42:00.673Z
Huh, indeed you are right! I guess I should have read more macro. Is this true even when the MPC is not constant, i.e., $C$ is not a linear function of $Y-T$?
Giskard
CC BY-SA 4.0
2022-09-27T15:44:56.653Z
@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
CC BY-SA 4.0
2022-09-27T15:59:19.853Z
@Giskard wait I think we have some misunderstanding. When you asked if it is true even when C is not constant function of Y-T I meant something like C = c(1-t)Y etc. Of course it wont work for arbitrary function $C$ but note there are some physical restrictions on what C can be e.g. of course you cant consume negative quantities MPC **must** be between $0\leq MPC \leq 1$ or actually $0< MPC \leq 1$ since some consumption is necessary. If you look for examples with different (but still functions that satisfy the necessary restrictions) you can have look at the example and working problems
1muflon1
CC BY-SA 4.0
2022-09-27T16:00:39.157Z
in some textbook like Blanchard et al Macro. It will hold with different functional forms but of course basic assumptions have to be satisfied as for example you could easily violate it by just constructing some weird function C such as that for example C(Y-T) = 0 if T>0 etc
1muflon1
CC BY-SA 4.0
2022-09-27T16:33:47.550Z
Hey, I don't think it works out in the linear case either? Not sure anymore. Seems like (assuming $G=T$ and denoting $C$ by $f_C$ to emphasize the functional relationship) you will have $$Y-T = f_C(Y-T) + I.$$ If $f_C$ is strictly increasing this can have only one solution in $Y-T$. This is also true for the linear case.
Giskard
CC BY-SA 4.0
2022-09-27T16:36:02.503Z
However $Y$ is increasing as $T$ is increasing, because $Y = (Y-T) + T$? Substituting $G=T$ into your equation $$Y  = \frac{1}{1-c_1}\left(c_0 +G +I - c_1T\right) \tag{***}$$ we get
$$ Y  = \frac{1}{1-c_1}\left(c_0 +I + T - c_1T\right) = \frac{1}{1-c_1}\left(c_0 +I\right) + T.$$ Thus in this model $Y$ can be increased by increasing $T$ while holding a balanced budget?
Giskard
CC BY-SA 4.0
2022-09-27T16:50:40.627Z
@Giskard but for Y to be increased by T within the system you need to show result  that has outcome where xT with x>1 if you get that Y=T=G (assuming c_0=0) that does not mean that GDP is increased by taxes it is just tautology because GDP = C+I+G so of course 1 **exogenous** increase in G results in 1 increase in GDP but you can’t get more money or goods and services if we talk about real output than there already is in the system in the first place. For that you need result where you can show that Y=xT + … with x>1
1muflon1
CC BY-SA 4.0
2022-09-27T17:48:22.243Z
"*of course 1 exogenous increase in G results in 1 increase in GDP*"  ||  "*But government cannot magically increase GDP by just taxing and spending money while keeping balanced budget.*" Magic and/or imprecise phrasing was achieved!
Giskard
CC BY-SA 4.0
2022-09-27T17:51:11.667Z
@Giskard but I meant by fiscal policy. Of course if country produces more so GDP is 11b instead of 10b GDP increases when there is balance budget. But the fiscal policy itself does not increase GDP because multiplier is equal to 1 even with case of lump sum tax
1muflon1
CC BY-SA 4.0
2022-09-27T17:51:36.077Z
Then you should probably write this instead of the false statement?
Giskard
CC BY-SA 4.0
2022-09-27T17:51:56.620Z
@Giskard which statement is false?
1muflon1
CC BY-SA 4.0
2022-09-27T17:53:03.990Z
I too was imprecise earlier: for the solution to the equation $Y-T = f_C(Y-T) + I$ to be unique it is not enough for $f_C$ to be strictly increasing; $0<f_C'()<1$ would suffice.
Giskard
CC BY-SA 4.0
2022-09-27T17:53:49.860Z
"*of course 1 exogenous increase in G results in 1 increase in GDP" || "But government cannot magically increase GDP by just taxing and spending money while keeping balanced budget.*"  You tell me which is false... obviously both statements cannot be true.
Giskard
CC BY-SA 4.0
2022-09-27T17:59:33.177Z
@Giskard ?? I do not understand what you mean. Maybe I used some word incorrectly so let me rephrase. Of course when economy produces more so the national income is 11 billion (or economy gets exogenous support like foreign aid) instead of 10 billion, then government can spend 1 billion more using extra 1 billion lump sum tax. but if national income was 10 billion as question asks if government can increase it from 10 billion to 11 by taxing, that is not possible. If output is 10 billion then 1 billion increase in lump sum tax just leaves it at ten billion
1muflon1
CC BY-SA 4.0
2022-09-27T18:49:20.797Z
I honestly you don't understand. Of course the government spending increases output. Does the government spend the money without spending it on anything in your answer? How does the economy not produce more if the total quantity demanded ($C+I+G$) is larger?
Giskard
CC BY-SA 4.0
2022-09-27T18:52:21.140Z
@Giskard but the demand is not larger, what makes you think that? When government levies the tax then C and I fall! Also aggregate demand is not equal to quantity demanded aggregate demand is expressed in \$ or in real terms it would be just unitless AD not quantities
1muflon1
CC BY-SA 4.0
2022-09-27T18:56:48.060Z
Okay, so are you claiming that $C+I+G$ is not increased as I have described [here](https://economics.stackexchange.com/questions/52884/does-applying-taxes-increase-the-gdp-by-the-same-amount/52887?noredirect=1#comment86207_52887)?
Giskard
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2022-09-27T19:00:36.747Z
Numerical example: $C = 2 + 0.5 \cdot (Y-T)$, $I = 6$. If $G=T = 10$, then $Y = 26$; but if ceteris paribus $G=T=20$ then $Y=36$. Thus output and aggregate demand $C+I+G$ are increased by 10 units, while the budget is kept balanced.
Giskard
CC BY-SA 4.0
2022-09-27T19:00:47.610Z
@Giskard it is exactly how it is described there, but when T increases other parts of the equation change as well. You cant increase T without I falling since I= -c_0 + (1-c_1)(Y-T) +(T-G) since we keep T=G we have just I=(1-c_1)(Y-T). Sure Y increases by T and then Y decreases by T via I
1muflon1
CC BY-SA 4.0
2022-09-27T19:01:47.327Z
@Giskard your example is impossible. I cannot be equal to 6 if Y=36 and c_1=0.5 that is impossible
1muflon1
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2022-09-27T19:02:23.240Z
[Why](https://economics.stackexchange.com/questions)?
Giskard
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2022-09-27T19:03:01.650Z
@Giskard I do not get it what does that mean?
1muflon1
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2022-09-27T19:09:11.423Z
The word "why"? Well, you made [this claim](https://economics.stackexchange.com/questions/52884/does-applying-taxes-increase-the-gdp-by-the-same-amount/52887?noredirect=1#comment86232_52887), and I don't see it as an axiom, I am wondering if you have some reasoning supporting it. If you are asking why it is a link to the main page: that is a way to subvert the 15 char minimum comment length.
Giskard
CC BY-SA 4.0
2022-09-27T19:13:32.937Z
@Giskard oh ok, I got confused by the link. I thought it is obvious. If marginal propensity is 0.5 and income is 36 then I simply cannot be 6 since I is given by portion of output not consumed or spent by gov.
1muflon1
CC BY-SA 4.0
2022-09-27T19:14:04.223Z
Marginal propensity **to consume** is 0.5.
Giskard
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2022-09-27T19:14:26.020Z
In your answer do you not treat $I$ as exogenous?
Giskard
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2022-09-27T19:15:26.933Z
@Giskard but marginal propensity **to consume** determines saving rate and hence   I= -c_0 + (1-c_1)(Y-T) +(T-G)
1muflon1
CC BY-SA 4.0
2022-09-27T19:16:12.580Z
So in this model both $Y$ and $I$ are endogenous? Seems like we have one equation with two unknowns? Or is there a second one?
Giskard
CC BY-SA 4.0
2022-09-27T19:19:02.257Z
@Giskard no, I is treated as having fixed value during derivation in order to solve the problem but this forces all other variables adjust. Alternatively you can hold Y fixed (as in OP question and then I has to adjust to it)
1muflon1
CC BY-SA 4.0
2022-09-27T19:22:06.747Z
"*I is treated as having fixed value during derivation in order to solve the problem*" okay, I will simply have to accept that I will never understand this. Mind you I got none of this from your answer, I feel like I am totally in to the dark about the assumptions of your mathematical model. Perhaps I am just exceptionally dim, but it is also possible that your answer skips some explanations/simplifies some things too much.
Giskard
CC BY-SA 4.0
2022-09-27T19:22:30.527Z
@Giskard For example lets use the OP assumed values (in billions) so Y=10, OP does not specify c_1 but lets assume 0.5, assume c_0=0 then OP assumes initially T=0. Hence, we have 10=2I and I=5. This implies that on the GDP identity statisticians would record Y=C+I=5+5=10. This is consistent with the consumption equation since $C=0.5Y=5$. | Next let us introduce the 1b T = 1bG. In this case we will have 10=2I+1 => I=4.5. Again we have consistent model since $C=0.5(Y-T) = 4.5$ so $Y=C+I+G$ we have 10=4.5+4.5+1. Again all equations are satisfied
1muflon1
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2022-09-27T19:24:25.457Z
Your argument is that all disposable income not consumed is saved, $Y-T-C = S = I$?
Giskard
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2022-09-27T19:26:22.387Z
@Giskard that is not my argument that is literally definition of saving in macroeconomics. Yes by definition $S\equiv Y-T - C$  this definition also implies that $S=I+T-G$ since $Y=C+I+G \implies Y-C=I+G$ (although with balanced budget it is just S=I as you say
1muflon1
CC BY-SA 4.0
2022-09-27T19:34:19.863Z
I don't see why you used the Keynesian model in this case; why not make your answer "by definition of $I = Y - T - C$. Thus if we assume $G = T$ (which is a consequence of this equation btw), then all effects of changes in $T$ are compensated for by changes in $I$: a unit increase in $T$ will cause a unit increase in $G$, but as we can see from the equation it will also directly cause a unit decrease in $I$. An increase in $T$ may also indirectly decrease $C$ through disposable income. However, any change in $C$ is also absorbed by $I$. Thus $Y$ cannot be changed by changes in $T$."
Giskard
CC BY-SA 4.0
2022-09-27T19:36:27.460Z
@Giskard 1. Well this is workhorse model of macro. I do not understand what you mean by saying why did you use Keynesian model? It is a way how we simply model macroeconomy. What else should I use? 2. Yep exactly what I tried to say in my answer.
1muflon1
CC BY-SA 4.0
2022-09-27T19:36:59.153Z
What is the point of bringing in MPC and autonomous consumption? $I = Y - T - C$ is not as fancy, but it seems like it literally answers everything that was asked here?
Giskard
CC BY-SA 4.0
2022-09-27T19:40:00.213Z
@Giskard 1. Well how else do you showcase the result holds?  2. Also in the second part when I show the stronger result for income tax it is necessary
1muflon1
CC BY-SA 4.0
2022-09-27T19:41:10.717Z
@Giskard okay you are right probably I=Y-T-C would be simpler, but still the second point holds you cant introduce income tax result that way
1muflon1
CC BY-SA 4.0
2022-09-27T19:43:09.247Z
2. I cannot comment on this part; this is the one where I no longer know the assumptions about which variables are exogenous and how many degrees of freedom + equations there are.
Giskard
CC BY-SA 4.0
2022-09-27T19:43:29.187Z
Anyway, thank you for the discussion (: Good night!
Giskard
CC BY-SA 4.0
2022-09-27T15:04:43.213Z
2022-09-27T15:10:07.697Z
2022-09-27T15:10:07.697Z
1muflon1
<p>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.</p>
<p>Tho its unlikely this would be effective in anything but the very short run. <a href="https://governology.substack.com/p/taxes" rel="nofollow noreferrer">Most taxes generally cause substantial deadweight losses</a> and so one would usually expect that taxes should reduce GDP - if not this year, then the next.</p>

CC BY-SA 4.0
2023-07-22T05:21:27.770Z
2023-07-22T05:21:27.770Z
Governology
<p>Before anything else, one needs to discuss</p>
<blockquote>
<p>&quot;...And why should the government tax the population?&quot;</p>
</blockquote>
<p>Would some of this tax revenue go towards covering certain needs of the population, that currently were (however) covered by private spending?</p>
<p>If some taxes would go towards better roads =&gt; less expenditure in maintaining/fixing private vehicles</p>
<p>If some taxes will go towards better security =&gt; less expenditure in private security companies (and maybe less expenditure to psychotherapy/medical expenses caused by the stress of feeling insecure)</p>
<p>etc</p>
<p>In other words, to discuss a realistic/interesting scenario, at least some amount of the tax revenue is expected to substitute for private spending.</p>
<p>Moreover, in an uncertain environment, it may be the case that the expected utility of the population may increase.</p>
<p>Say, <span class="math-container">$I$</span> is my gross income, <span class="math-container">$S$</span> is the expenses I incur for private security and <span class="math-container">$p$</span> is the probability that I do not get robbed (after spending <span class="math-container">$S$</span>).</p>
<p>My expected utility from consumption is
<span class="math-container">$$E(U) = p\cdot U(I-S)$$</span></p>
<p>Assume the government takes taxes me <span class="math-container">$T$</span> with which it improves security so a) I do not need to spend <span class="math-container">$S$</span> and b) the probability of not being robbed increases to <span class="math-container">$p' &gt; p$</span>. My new expected utility is
<span class="math-container">$$E(U') = p'\cdot U(I-T)$$</span></p>
<p>Because we expect <span class="math-container">$T&lt;S$</span> my expected utility has improved from two channels.</p>
<p>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).</p>

CC BY-SA 4.0
2023-07-22T12:10:39.523Z
2023-07-22T12:10:39.523Z
Alecos Papadopoulos
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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</p>
<p>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?</p>

CC BY-SA 4.0
2022-09-27T15:04:16.893Z
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.
Hymns For Disco
<gdp><taxation>
Does applying taxes increase the GDP by the same amount?
Abanob Ebrahim
# Does applying taxes increase the GDP by the same amount?

(Asked by: username_0 on 2022-09-27)

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?

**Tags:** gdp, taxation

> **Comments:**
> **username_1**: 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.  

---

## 3 answers

### Answer by username_2 on 2022-09-27. Score: 1

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.

> **Comments:**
> **username_3**: "*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?  
> **username_3**: Also it seems like your treatment sidesteps the issue that $G$ is probably related to $t$ in the medium-run.  
> **username_2**: @username_3 because you cant increase size of a pie by cutting it into different pieces. BTW I am not sure if you read the full answer but I do mention explicitly government can do that over time through economic growth, but the question is about immediate change in GDP so any medium/long-term issues do not apply here and I clearly stated that in the long run government spending can do that through public investment  
> **username_3**: @username_2 I don't mean growth via investments in infrastructure, I mean pyramid building/ditch digging a la Keynes.  
> **username_2**: @user253751 1. Inflation means that the value of dollar changes, that is not a downside of GDP. GDP does not doubles just value of currency changes. For example, if you use speedometer to measure speed, even though there will be measurement error, the speedometer is not worse if you just program there switch that changes units from km/h to m/h. Similarly inflation just means you changed measure of dollar from one measure to another 2. real GDP is unaffected by inflation. 3. GDP has a measurement error, but there is little reason to think that GDP measurement error is affected by taxes  
> **username_2**: @username_3 but Keynes never claim GDP can be increased with **balanced** budget. You can do that by deficit spending.  
> **username_3**: 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?  
> **username_2**: @user253751 but that has little to do with taxes. Netherlands has private healthcare system like US, UK has nationalized healthcare yet, they both have almost the same healthcare spending relative to their economy sizes according to OECD. This is not to say that GDP is accurate measure of human welfare, but that is completely unrelated to this issue with taxes  
> **username_2**: @username_3 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  
> **username_3**: Huh, indeed you are right! I guess I should have read more macro. Is this true even when the MPC is not constant, i.e., $C$ is not a linear function of $Y-T$?  
> **username_2**: @username_3 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)  
> **username_2**: @username_3 wait I think we have some misunderstanding. When you asked if it is true even when C is not constant function of Y-T I meant something like C = c(1-t)Y etc. Of course it wont work for arbitrary function $C$ but note there are some physical restrictions on what C can be e.g. of course you cant consume negative quantities MPC **must** be between $0\leq MPC \leq 1$ or actually $0< MPC \leq 1$ since some consumption is necessary. If you look for examples with different (but still functions that satisfy the necessary restrictions) you can have look at the example and working problems  
> **username_2**: in some textbook like Blanchard et al Macro. It will hold with different functional forms but of course basic assumptions have to be satisfied as for example you could easily violate it by just constructing some weird function C such as that for example C(Y-T) = 0 if T>0 etc  
> **username_3**: Hey, I don't think it works out in the linear case either? Not sure anymore. Seems like (assuming $G=T$ and denoting $C$ by $f_C$ to emphasize the functional relationship) you will have $$Y-T = f_C(Y-T) + I.$$ If $f_C$ is strictly increasing this can have only one solution in $Y-T$. This is also true for the linear case.  
> **username_3**: However $Y$ is increasing as $T$ is increasing, because $Y = (Y-T) + T$? Substituting $G=T$ into your equation $$Y  = \frac{1}{1-c_1}\left(c_0 +G +I - c_1T\right) \tag{***}$$ we get $$ Y  = \frac{1}{1-c_1}\left(c_0 +I + T - c_1T\right) = \frac{1}{1-c_1}\left(c_0 +I\right) + T.$$ Thus in this model $Y$ can be increased by increasing $T$ while holding a balanced budget?  
> **username_2**: @username_3 but for Y to be increased by T within the system you need to show result  that has outcome where xT with x>1 if you get that Y=T=G (assuming c_0=0) that does not mean that GDP is increased by taxes it is just tautology because GDP = C+I+G so of course 1 **exogenous** increase in G results in 1 increase in GDP but you can’t get more money or goods and services if we talk about real output than there already is in the system in the first place. For that you need result where you can show that Y=xT + … with x>1  
> **username_3**: "*of course 1 exogenous increase in G results in 1 increase in GDP*"  ||  "*But government cannot magically increase GDP by just taxing and spending money while keeping balanced budget.*" Magic and/or imprecise phrasing was achieved!  
> **username_2**: @username_3 but I meant by fiscal policy. Of course if country produces more so GDP is 11b instead of 10b GDP increases when there is balance budget. But the fiscal policy itself does not increase GDP because multiplier is equal to 1 even with case of lump sum tax  
> **username_3**: Then you should probably write this instead of the false statement?  
> **username_2**: @username_3 which statement is false?  
> **username_3**: I too was imprecise earlier: for the solution to the equation $Y-T = f_C(Y-T) + I$ to be unique it is not enough for $f_C$ to be strictly increasing; $0<f_C'()<1$ would suffice.  
> **username_3**: "*of course 1 exogenous increase in G results in 1 increase in GDP" || "But government cannot magically increase GDP by just taxing and spending money while keeping balanced budget.*"  You tell me which is false... obviously both statements cannot be true.  
> **username_2**: @username_3 ?? I do not understand what you mean. Maybe I used some word incorrectly so let me rephrase. Of course when economy produces more so the national income is 11 billion (or economy gets exogenous support like foreign aid) instead of 10 billion, then government can spend 1 billion more using extra 1 billion lump sum tax. but if national income was 10 billion as question asks if government can increase it from 10 billion to 11 by taxing, that is not possible. If output is 10 billion then 1 billion increase in lump sum tax just leaves it at ten billion  
> **username_3**: I honestly you don't understand. Of course the government spending increases output. Does the government spend the money without spending it on anything in your answer? How does the economy not produce more if the total quantity demanded ($C+I+G$) is larger?  
> **username_2**: @username_3 but the demand is not larger, what makes you think that? When government levies the tax then C and I fall! Also aggregate demand is not equal to quantity demanded aggregate demand is expressed in \$ or in real terms it would be just unitless AD not quantities  
> **username_3**: Okay, so are you claiming that $C+I+G$ is not increased as I have described [here](https://economics.stackexchange.com/questions/52884/does-applying-taxes-increase-the-gdp-by-the-same-amount/52887?noredirect=1#comment86207_52887)?  
> **username_3**: Numerical example: $C = 2 + 0.5 \cdot (Y-T)$, $I = 6$. If $G=T = 10$, then $Y = 26$; but if ceteris paribus $G=T=20$ then $Y=36$. Thus output and aggregate demand $C+I+G$ are increased by 10 units, while the budget is kept balanced.  
> **username_2**: @username_3 it is exactly how it is described there, but when T increases other parts of the equation change as well. You cant increase T without I falling since I= -c_0 + (1-c_1)(Y-T) +(T-G) since we keep T=G we have just I=(1-c_1)(Y-T). Sure Y increases by T and then Y decreases by T via I  
> **username_2**: @username_3 your example is impossible. I cannot be equal to 6 if Y=36 and c_1=0.5 that is impossible  
> **username_3**: [Why](https://economics.stackexchange.com/questions)?  
> **username_2**: @username_3 I do not get it what does that mean?  
> **username_3**: The word "why"? Well, you made [this claim](https://economics.stackexchange.com/questions/52884/does-applying-taxes-increase-the-gdp-by-the-same-amount/52887?noredirect=1#comment86232_52887), and I don't see it as an axiom, I am wondering if you have some reasoning supporting it. If you are asking why it is a link to the main page: that is a way to subvert the 15 char minimum comment length.  
> **username_2**: @username_3 oh ok, I got confused by the link. I thought it is obvious. If marginal propensity is 0.5 and income is 36 then I simply cannot be 6 since I is given by portion of output not consumed or spent by gov.  
> **username_3**: Marginal propensity **to consume** is 0.5.  
> **username_3**: In your answer do you not treat $I$ as exogenous?  
> **username_2**: @username_3 but marginal propensity **to consume** determines saving rate and hence   I= -c_0 + (1-c_1)(Y-T) +(T-G)  
> **username_3**: So in this model both $Y$ and $I$ are endogenous? Seems like we have one equation with two unknowns? Or is there a second one?  
> **username_2**: @username_3 no, I is treated as having fixed value during derivation in order to solve the problem but this forces all other variables adjust. Alternatively you can hold Y fixed (as in OP question and then I has to adjust to it)  
> **username_3**: "*I is treated as having fixed value during derivation in order to solve the problem*" okay, I will simply have to accept that I will never understand this. Mind you I got none of this from your answer, I feel like I am totally in to the dark about the assumptions of your mathematical model. Perhaps I am just exceptionally dim, but it is also possible that your answer skips some explanations/simplifies some things too much.  
> **username_2**: @username_3 For example lets use the OP assumed values (in billions) so Y=10, OP does not specify c_1 but lets assume 0.5, assume c_0=0 then OP assumes initially T=0. Hence, we have 10=2I and I=5. This implies that on the GDP identity statisticians would record Y=C+I=5+5=10. This is consistent with the consumption equation since $C=0.5Y=5$. | Next let us introduce the 1b T = 1bG. In this case we will have 10=2I+1 => I=4.5. Again we have consistent model since $C=0.5(Y-T) = 4.5$ so $Y=C+I+G$ we have 10=4.5+4.5+1. Again all equations are satisfied  
> **username_3**: Your argument is that all disposable income not consumed is saved, $Y-T-C = S = I$?  
> **username_2**: @username_3 that is not my argument that is literally definition of saving in macroeconomics. Yes by definition $S\equiv Y-T - C$  this definition also implies that $S=I+T-G$ since $Y=C+I+G \implies Y-C=I+G$ (although with balanced budget it is just S=I as you say  
> **username_3**: I don't see why you used the Keynesian model in this case; why not make your answer "by definition of $I = Y - T - C$. Thus if we assume $G = T$ (which is a consequence of this equation btw), then all effects of changes in $T$ are compensated for by changes in $I$: a unit increase in $T$ will cause a unit increase in $G$, but as we can see from the equation it will also directly cause a unit decrease in $I$. An increase in $T$ may also indirectly decrease $C$ through disposable income. However, any change in $C$ is also absorbed by $I$. Thus $Y$ cannot be changed by changes in $T$."  
> **username_2**: @username_3 1. Well this is workhorse model of macro. I do not understand what you mean by saying why did you use Keynesian model? It is a way how we simply model macroeconomy. What else should I use? 2. Yep exactly what I tried to say in my answer.  
> **username_3**: What is the point of bringing in MPC and autonomous consumption? $I = Y - T - C$ is not as fancy, but it seems like it literally answers everything that was asked here?  
> **username_2**: @username_3 1. Well how else do you showcase the result holds?  2. Also in the second part when I show the stronger result for income tax it is necessary  
> **username_2**: @username_3 okay you are right probably I=Y-T-C would be simpler, but still the second point holds you cant introduce income tax result that way  
> **username_3**: 2. I cannot comment on this part; this is the one where I no longer know the assumptions about which variables are exogenous and how many degrees of freedom + equations there are.  
> **username_3**: Anyway, thank you for the discussion (: Good night!  

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### Answer by username_4 on 2023-07-22. Score: 0

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](https://governology.substack.com/p/taxes) and so one would usually expect that taxes should reduce GDP - if not this year, then the next.


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### Answer by username_5 on 2023-07-22. Score: 0

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).


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