<p>In your comment you say you want the answer in context of the classic textbook model for goods market/output equilibrium.</p>

<p>Following the Blanchard macroeconomics textbook lets consider closed economy so the output will be given by:</p>

<p><span class="math-container">$$Y=C+I+G$$</span></p>

<p>where <span class="math-container">$Y$</span> is output/income (they must be always equal), <span class="math-container">$C$</span> consumption, <span class="math-container">$I$</span> investment and <span class="math-container">$G$</span> gov. spending.</p>

<p>The consumption will be given by: </p>

<p><span class="math-container">$$C = c_0 + c_1(Y-T)$$</span></p>

<p>where <span class="math-container">$c_0$</span> is the part of consumption that does not depend on income and <span class="math-container">$c_1$</span> is marginal propensity to consume (MPC) and <span class="math-container">$T$</span> are taxes.</p>

<p>In this basic model investment is just fixed <span class="math-container">$I=\bar{I}$</span></p>

<p>Solution to the model (i.e. autonomous spending) based on the set up above is given by (the solution is derived by substituting <span class="math-container">$C$</span> in second equation for its place in first one and then solving for <span class="math-container">$Y$</span>):</p>

<p><span class="math-container">$$Y = \frac{1}{1-c_1}[c_0+ \bar{I} +G -c_1T ]$$</span></p>

<p>Now since you say that we take it for granted that government runs balanced budget <span class="math-container">$G=T$</span> because borrowing is "taboo" we can substitute the equality between taxes and gov. spending so we have:</p>

<p><span class="math-container">$$Y = \frac{1}{1-c_1}[c_0+ \bar{I} +G -c_1G ] = \frac{1}{1-c_1}[c_0+ \bar{I} +(1-c_1)G ]$$</span></p>

<p>Now to clearly see the effect of G spending lets just set <span class="math-container">$c_0=\bar{I}=0$</span> (this is not necessary but it cleans the equation). So we have:</p>

<p><span class="math-container">$$Y =  \frac{1}{1-c_1}(1-c_1)G = G$$</span></p>

<p>Hence there is no multiplier on Government spending if budget is balanced. So without debt government cannot increase output in any way (at least not in this simple model). </p>

<p>The intuition is that of course if government just takes money from left pocket to the right there cannot be any increase in output. Since in this model all agents have the same MPC and hence all spending including the Gov. spending has the same multiplier. If you take 10 euros from consumers and just spend it as a government you are not creating any additional output. you are just reducing <span class="math-container">$C$</span> - reduction of which would also be affected by multiplier, by the same value you are increasing <span class="math-container">$G$</span> which is affected by the same multiplier. The effects must cancel each other out. Only if you would allow for heterogeneous agents where people have different MPC government could increase output in the short term by redistributing money from people with low MPC to people with high MPC, but the output cannot be raised "exponentially" and this would only be one time increase - i.e. increase in level or short run growth. I would recommend reading the full chapter 3 from Blanchard et all (2010). Macroeconomics an European perspective, that deals with this in greater detail.</p>

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2020-01-15T17:19:31.980Z
I'm confused. In my example if I tax \$100 and then spend them, then GDP increases by \$100. What am I doing wrong? Is there a model where said actions would lead to \$100 increase of GDP?
KarmaPeasant
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2020-01-15T17:24:08.273Z
Well if you tax 100 and spend them GDP always remains constant - taxing means taking money from people (their income and hence their consumption). So taxing 100\$ means you reduce GDP by 100\$. Unless the taxation is on some external entity - some tribute from foreign conquest then GDP cant just change through taxing and spending if budget balanced. You could increase GDP by running deficit though (at least in short run - although to deal with long and short run I would have to go to full AD-AS IS-LM model).
1muflon1
CC BY-SA 4.0
2020-01-15T17:31:30.117Z
"means you reduce GDP by \$100" Not in my calculations. After taxation GDP will be \$999966.67 (decreased by \$33.33, not by \$100. Remember, the population is stingy). After that if we spend all tax money GDP will be  \$1000100 (increased by \$133.33)
KarmaPeasant
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2020-01-15T17:36:22.807Z
And by the way, can you please elaborate how you got solution formula in its first form?
KarmaPeasant
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2020-01-15T17:42:18.953Z
@user161005  1.The formula is derived by substitution of second equation into the first one (I will update text after this comment). 2. This is just nitpicking but if MPC is 0.25 then in no simple undergraduate model you can get multiplier 1.33 in this case multiplier would be 1.25 as multiplier = 1/(1-MPC) 3. the point is that since taxing reduces income and hence consumption you will also get negative multiplier. In fact every tax dollar raised will have negative multiplier MPC/(1-MPC). If G=T positive and negative multipliers cancel. You have to have some debt  (G>T) there to raise GDP
1muflon1
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2020-01-15T17:51:02.910Z
2.The expenditure multiplier is 1/(1-MPC), I agree. But if we will substitue MPC with 0.25 then it will be 1.33, not 1.25 https://www.wolframalpha.com/input/?i=1%2F%281-0.25%29   3. The taxation multiplier is 0.33 https://www.wolframalpha.com/input/?i=0.25%2F%281-0.25%29   They don't seem to be canceling out each other in my example
KarmaPeasant
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2020-01-15T17:57:45.090Z
@user161005 oh sorry you are actually right about multiplier =1.33  my bad... once you start dealing with abstract high level math you forget how to do simple fractions. Anyway that did not mattered for my answer at all. And yes they will cancel out let’s parametrize  the model we have Y=1/0.75*(100-0.25*100)=100 (I rewritten 1.33 as 1/0.75 to get precise result) hence there is no multiplier there anymore if you raise 100\$ through tax and have balanced budget(cancel out is maybe unfortunate wording because in math it implies subtraction, here it means that there no longer is any multiplier)
1muflon1
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2020-01-15T18:02:32.497Z
Then how can you explain that by consequently adjusting GDP for increased taxation and then for government spending I get +\$N for GDP? What is my mistake?
KarmaPeasant
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2020-01-15T18:06:59.160Z
@user161005 ? I don’t understand i consequently adjusted GDP for taxation and government spending and you saw that there was no multiplier. Can you explain what you mean? Or show me your math
1muflon1
CC BY-SA 4.0
2020-01-15T18:29:03.253Z
At first I reduce GDP by Tax_Multiplier\*AddionalAmountTaxed. In this case -0.33\*100=33  After that I increase GDP by ExpenditureMultiplier\*AddionalAmountTaxed, in this case it's 1.33\*100=133. 133>33, meaning that now GDP increased by 100
KarmaPeasant
CC BY-SA 4.0
2020-01-15T18:34:36.593Z
@user161005 but you forgot about minus sign also taxes reduce GDP so it’s actually 133-33=100. Yes thanks to multiplier 100 government spending (where the 100 is transfer from the original output/income) increases the original output from 100 to 133 but at the same time it decreases the output by additional 33 due to taxes.
1muflon1
CC BY-SA 4.0
2020-01-15T18:48:47.207Z
"so it’s actually 133-33=100" Please notice that 100 isn't GDP, but addition to GDP. If GDP=X (before taxation), then after taxation and spending it will become GDP=X+100.  " but at the same time it decreases the output by additional 33 due to taxes" I agree, but we will still have 100 as addditon to our GDP
KarmaPeasant
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2020-01-15T18:55:57.253Z
@user161005 where do you think taxation comes from? That would only work if taxes are some kind of tribute from abroad, like when past one nation created a client state which was forced to pay taxes to the home state. If there is no foreign country to tax then the tax have to be applied to the income/output so we have Y-T (see the second equation)... Sure if you can take the tax out of thin air GDP will increase by 100 but then it would no longer be tax but some exogenous Government production - like government having magic box that gives 100
1muflon1
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2020-01-16T04:46:48.793Z
"tax have to be applied to the income/output so we have Y-T" In my example I tax wealth that people have, not their income or production. So if you're, say, a preper living in a bunker and who has money, but produces nothing and doesn't have any income, then you will be taxed nonetheless.
KarmaPeasant
CC BY-SA 4.0
2020-01-16T04:49:14.253Z
"where do you think taxation comes from?" From pockets of people, of course. "That would only work if taxes are some kind of tribute from abroad" What makes you think so?
KarmaPeasant
CC BY-SA 4.0
2020-01-16T07:13:44.710Z
@user161005 1. In this model and in almost all undergraduate macro model there is not any wealth. 2. Wealth is saved income so under certain models there is actually not much difference between taxing wealth and income (however this is not even domain of macroeconomic which u used as a tag this would be in some public economics model. 3. You said you want explanation given in context of standard undergraduate goods/output equilibrium model. 4. Because ultimately taxes are transfers of output they don’t magically create output so at some point they are subtracted from GDP if they are domestic
1muflon1
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2020-01-16T07:27:28.673Z
Expanding on my point 4. Even if at time t there is let’s say tax on investment (let’s be sloppy and just call it wealth) 100\$ then at time t investment would be reduced by 100 + the effect of multiplier (since from above equations it’s clear it’s applied to investment as well) so GDP would be lower at time t and then government could not spend it run surplus at time t and then spend the money at time t+1 where it would get multiplied but again the sum of GDPs would still be the same (if multiplier does not change). The tax just in this case reduces GDP in one period at expense of other
1muflon1
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2020-01-16T07:44:03.037Z
Sorry last sentence was supposed to say increases GDP in one period at expense of other
1muflon1
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2020-01-16T07:58:04.613Z
Forget models and intuition, what is wrong with my calculations? Please show me your two-step calculations of increasing taxes by \$100 with consequent spending of said \$100.
KarmaPeasant
CC BY-SA 4.0
2020-01-16T08:22:17.173Z
@user161005 well to do any calculation you have to have some model. But I will interpret your comment as forget the formalism and show me simple calculation in ill defined model. 1 step you tax 100e from economy in addition this tax will have further 33e negative effect on GDP due to multiplier effect so you are taking together 100e from economy. 2. You inject 100e to economy which thanks to multiplier are multiplied to 133 so you reduce GDP by 133 and then increase by 133. To avoid this you have to assume the tax does not extract the original 100e which would mean it is not tax in economic
1muflon1
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2020-01-16T08:24:28.673Z
Terms as definition of tax is involuntary transfer. Or it has to come from different time periods which you on purpose exclude (although that would still violate T=G). Or the taxes are not domestic so the original product is not transferred from GDP at home but it comes as subtraction from GDP abroad in some other country
1muflon1
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2020-01-16T08:30:18.817Z
Sorry in my 1. Comment it should be taking together 133 that was typo
1muflon1
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2020-01-16T08:47:10.770Z
So it seems I found the root of our disagreement. For you taxing people by \$100 will decrease GDP by \$133, while for me it will decrease GDP only by $33. My calculations are based on this video: https://youtu.be/UsJmBBtwzYY?t=215     Or in other words, suppose we taxed a preper living in bunker, who has money but doesn't spend them. In this case taxing them won't reduce GDP by even a single cent because they wouldn't spend their money anyway. The only things that maters is how much we decreased consumption by our taxes. 100*0.33, meaning we decrease it by \$33.
KarmaPeasant
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2020-01-16T09:17:59.230Z
But the video actually proves my point. If you watch it all way till the end you will see that in the end tax decreases output by MPC/(1-MPC) exactly like in my answer (if you expand the terms u will see T is multiplied by c_1/(1-c_1). So I don’t understand what’s your point here you see you get exactly to zero here 1.33*100- 1.33*0.25*100. So multipliers exactly cancels out. I think the confusion stems for the fact that you only focus too much on the first part that change in the tax is multiplied only by 0.25 which ignores that in equilibrium 1.33 is also applied
1muflon1
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2020-01-16T09:38:35.097Z
Also again when I say cancels out I mean multiplier is 1 i.e there is no multiplier GDP is just what it was 100GDP gets multiplied by 1. And the 100 cannot be just gotten from thin air. It can be from past as I mentioned so it’s possible to decrease GDP yesterday by 100 to increase it today by 100 etc. also in more complex models again if MPC is different between time periods or where you have heterogeneous agents there could be some positive effect but you won’t even find those models at Khan Academy. Actually based on the video I am 100% sure that video uses the mode I gave you in my answer
1muflon1
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2020-01-16T09:48:03.343Z
Furthermore, to further clarify of course in this simple model there is no explicit production so you can say Gov. just produced 100GDP out of thin air but then you can’t call it taxation because definition of word taxation in economics is involuntary **transfer** of resources
1muflon1
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2020-01-16T10:15:12.293Z
1."tax decreases output by MPC/(1-MPC)". Actually it's decreased by Δt*MPC/(1-MPC). When Δt=100 and MPC=0.25 it all equals to 33.(3). And if you say that your answer algebraically says the same then how can you even claim that tax will decrease GDP by 133? 2."1.33*100- 1.33*0.25*100" How is it zero?? Wolfram Alpha says it's 99.75  3."tax is multiplied only by 0.25 which ignores that in equilibrium 1.33 is also applied " What are you talking about? We don't use Expenidutre multiplier in taxes. And we don't multiply money by MPC, we multiply by Tax Multiplier
KarmaPeasant
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2020-01-16T10:36:24.563Z
But I never said GDP will be zero that implies that GDP would be destroyed. My point is that the tax which allowed you to get the government spending is 100 (T=G) So this is just transfer. Forget about the multiplier for a second. If you issue 100 tax without any multiplier the effect on original GDP 100 would be 100-100+100. Where first 100 is some original GDP that was there in the economy (again tax must come from somewhere - **tax is not production** by definition.
1muflon1
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2020-01-16T10:41:01.500Z
Now adding in the multiplier the changes can be summarized as follows: $\Delta T = -100$, $\Delta G = + 100$ but you also have 2 canceling $\Delta$ on GDP as if you would not issue the taxes the money would still be in the economy (again unless they are from past or external source). Applying the multipliers means that you have positive change of 1/(1-0.25) 100 - 0.25/(1-0.25)100 hence the G just changes by 100 the (99.75 is rounding error since you dont use exact values) but where does that G = 100 comes from? Thin air? No it was originally subtracted from some other part of economy
1muflon1
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2020-01-16T10:56:01.350Z
Also I mean this is just accounting if you have some government credit account G=100 then there must be some corresponding government debit (in this case T=100). Solving the model that I presented in answer and which is also the model Khan clearly uses simply tells you that GDP (ignoring all irrelevant parts like investment for this discussion) $\Delta Y= 1/(1-MPC)(1-MPC) \Delta G$ so GDP increases 1 dollar for every dollar spent. But where does those dollars spent come from? If they come from present domestic economy then GDP wont increase
1muflon1
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2020-01-16T11:03:09.177Z
Also what I mean is not that the spending G does not increases Y but it only increases Y for every dollar spent $\Delta Y= \Delta G$ but there is no additional multiplier effect here on this government spending and since to get any government spending at all you have to get the dollars from somewhere if you get the dollars from Gross output then GDP must remain constant. Of course if you can get the G spending outside the model and outside the GDP then GDP increases that’s just asking does GDP increase if G increases well yes! But this is not some magical addition to gross product
1muflon1
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2020-01-16T11:11:55.767Z
@user161005 yes I agree that when we change G by 100 Y also increases by 100 but where did the G change in 100 come from? Thin air? The point is that of course government spending increases GDP by government spending of taxes are balanced. But where do you think government spending in this simple model comes from? You would get some extra GDP here only if G>T when government runs deficit otherwise you can get 1 dollar of GDP for 1 dollar spent by government. There was no contention about that but as long as government does not produce Gov spending must come from output
1muflon1
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2020-01-16T15:56:46.103Z
By the way, do you agree that no matter how hard we will tax a preper living in a bunker (who also has no income, no production, but has some cash on hands) we WON'T harm our GDP even by a single cent?
KarmaPeasant
CC BY-SA 4.0
2020-01-16T16:06:05.780Z
@user161005 Are you asking in this particular model? In this model it would not decrease GDP. Here tax has no deadweight loss as it is lump sum tax. Its just transfer from one part of economy into another so if GDP is 100 you tax the whole output and give it to G=100 then GDP will be still 100 (thats btw the point). So as long as you use lump sum taxes you can tax as much as you want without hurting the economy but that does not mean you can create something from nothing the maximum tax is the whole GDP and since G=Y, GDP wont change only all output transferred from private to public hands
1muflon1
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2020-01-16T16:47:03.970Z
Suppose we gave our preper ability to avoid tax by saying that tax will be applied only to people who eat meat (no matter if they are prepers or not), with $100 per person, so it's NOT a lumpt tax anymore. Will we hurt GDP by when taxing this bunker-living meat-eating preper? For me it seems like not, their spending=0, they don't keep their money in a bank, don't invest them, thus for economy their cash is as good as non-existent. And consequently, if we will spend dollars that we took from the preper, then we will start chain of spendings that will boost GDP.
KarmaPeasant
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2020-01-16T16:55:18.163Z
@user161005 1. What do you mean by hurting? The discussion was never whether tax hurts someone in this model or not. The discussion was wether you can increase GDP by spending if you run balanced budget. That can’t happen in any simple mode that does not have some complex hysteresis effect. 2. At the same time I never claimed that taxes in this model hurt anyone. 3. If it’s not lump sum tax then how do you tax him if he has no economic activity? Non-lump sum tax has to be applied to something. 4. The only case that this would not enter the GDP calculations would be if that something you
1muflon1
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2020-01-16T16:59:00.963Z
Tax was never produced. If it’s some production then it has to be in GDP at one point or another and in this simple models there is no inheritance from some foreign country or things like that. All GDP is produced at some point so to get the tax you have to subtract it from some part of GDP no matter where you put in terms of national account. The only way how what you describe would work is if those resources were not recorded on GDP maybe due to criminal activity so the first time they get registered in accounts is when they get taxed, but that’s whole different topic of measurement error
1muflon1
CC BY-SA 4.0
2020-01-16T17:06:19.993Z
1."What do you mean by hurting?" I didn't mean about hurting a person, I mean hurting GDP, i.e. decreasing GDP.  2." If it’s not lump sum tax then how do you tax him if he has no economic activity?" What is so difficult about it? We just go and tax them for meat-eating, no matter if they are economically active or not. As for tax, we apply it to cash that they put under their mattress. The cash itself was accumulated before they became living in a bunker.
KarmaPeasant
CC BY-SA 4.0
2020-01-16T17:12:10.437Z
@user161005 1. I knew you mean something with GDP but what do you mean by hurting GDP? If you mean by hurting if it lowers GDP then it depends on type of tax non-lump sum taxes would actually hurt GDP, and would have to be modeled differently than above (decrease it in this model). Any non lump sum and non-piguvian tax hurts output (but I assume no externalities in this discussion). 2. Eating (consumption) is economic activity! I got confused because you don’t use proper terminology. 3. Of course it affects GDP. What are the money under the mattress where those money come from?
1muflon1
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2020-01-16T17:16:47.057Z
1."it’s some production then it has to be in GDP" The cash that the preper saved was indeed in GDP. But it was in **past** GDP. In the current year these money aren't invested/loaned/spent/put in a bank, so fail to see how we would account for preper's under-mattress money in current GDP. 2.Wait, how is preper's eating is an economic activity? They are practically living outside of economy, unless we tax them!
KarmaPeasant
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2020-01-16T17:16:49.160Z
The last sentence in your paragraph is important yes if you assume that all that money was gotten from elsewhere so it never was part of GDP then sure, that’s the same as having tribute from abroad - but the money will get subtracted from some output. If we turn of foreign sector the output must be domestic
1muflon1
CC BY-SA 4.0
2020-01-16T17:32:44.540Z
@user161005 1 consumption is economic activity- pls just google it. 2. In this simple model that the khan academy video and my answer is based on a model that does not have multiple time periods. In a multi period model multipliers would look different - as I prefaced in my answer in some models taxing and spending could increase output but it does not generally happens when budget is balanced. In such model if it’s a tax on saving then it will affect incentives to save and even to consume since savings is just future consumption. You can have some situations like liquidity trap where this
1muflon1
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2020-01-16T17:38:21.997Z
government could increase output by this kind of taxation and spending in the short run. However, in the long run it’s simply impossible to raise output above its natural or full employment level. Also again these questions are outside the simple goods market equilibrium model - I prefaced in my answer so I absolutely don’t get the point here.
1muflon1
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2020-01-16T18:14:47.470Z
3. if you assume any non-lump sum taxes in this model the multipliers won’t be anymore 1/(1-MPK) non lump sum taxes generally affect multipliers as well so actually that’s even assumption that would break your assumptions on size of multipliers etc. so there we won’t even get exactly the above result. Generally speaking non-lump sum taxes make the multiplier worse, some could even make it negative so you could even find here that GDP is actually lower but it would also depend exactly what kind of non-lump sum taxes we are talking about. In principle there is infinite number of tax schedules
1muflon1
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2020-01-15T17:05:41.777Z
2020-01-15T18:13:53.940Z
2020-01-15T18:13:53.940Z
1muflon1
<p>Suppose (it's an example) GDP= $1 000 000, Marginal Propencity to Consume(MPC)=0.25, that <strong>borrowing money is an absolute taboo for our government</strong> and that said government wants to increase GDP.</p>

<p>What can the government do to increase GDP of its country? Well, it can increase government spendings. The expenditures multiplier is equal to 1.33, meaning that for each \$1 spent by the government the GDP increases by \$1.33. But with borrowing out of question our government will have to tax population first in order to raise the money. But additonal taxation will decrease our GDP. Will our expenditure multiplier be big enough to compensate for effects of our tax multiplier? Yes, because the absolute value of the tax multiplier will always be less than the absolute value of expenditure multiplier, no matter what MPC we will have (except if we will have MPC=1, but in this case we will get divizion by zero error), consequently meaning that EM-TM always >0. In our particular example we will lower GDP by 0.33 for each dollar that we taxed, but at the same time increase by 1.33 for each dollar that we spent, meaning that taxation and consequent spending of N dollars will give us +N dollars to GDP, 1.33x-0.33=x(1.33-0.33)=x*1=x</p>

<p>1.Is my understanding of given model correct?</p>

<p>2.If it's correct, then why don't government tax and spend tax money much much more actively than they are doing today? It looks like it would be a "cheat code" for unlimited exponential growth. Maybe there are other factors that prevent it, but that our simple model ignores?</p>

CC BY-SA 4.0
2020-01-15T16:34:02.917Z
In your question you keep referencing some model, but your question does not include example of any model you just post examples of parameters like MPC that could be applied to any kind of model. The answer to this question will vary depending on which "model" you want to use. For example 99% undergraduate macro models are not dynamic so there cant even be any growth. I feel that you have in mind the macro goods market model like here:https://courses.lumenlearning.com/boundless-economics/chapter/introducing-aggregate-expenditure/ is that correct or do you want to use different model?
1muflon1
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2020-01-15T16:40:21.620Z
@1muflon1 My bad, I thought that it's already constitutes model. But okay, suppose we work with the model that you linked, now what?
KarmaPeasant
<macroeconomics><gdp><taxation>
Do I understand correctly that taxation-based government expenditures allow GDP to grow?
KarmaPeasant
# Do I understand correctly that taxation-based government expenditures allow GDP to grow?

(Asked by: username_0 on 2020-01-15)

Suppose (it's an example) GDP= $1 000 000, Marginal Propencity to Consume(MPC)=0.25, that **borrowing money is an absolute taboo for our government** and that said government wants to increase GDP.

What can the government do to increase GDP of its country? Well, it can increase government spendings. The expenditures multiplier is equal to 1.33, meaning that for each \$1 spent by the government the GDP increases by \$1.33. But with borrowing out of question our government will have to tax population first in order to raise the money. But additonal taxation will decrease our GDP. Will our expenditure multiplier be big enough to compensate for effects of our tax multiplier? Yes, because the absolute value of the tax multiplier will always be less than the absolute value of expenditure multiplier, no matter what MPC we will have (except if we will have MPC=1, but in this case we will get divizion by zero error), consequently meaning that EM-TM always >0. In our particular example we will lower GDP by 0.33 for each dollar that we taxed, but at the same time increase by 1.33 for each dollar that we spent, meaning that taxation and consequent spending of N dollars will give us +N dollars to GDP, 1.33x-0.33=x(1.33-0.33)=x\*1=x

1.Is my understanding of given model correct?

2.If it's correct, then why don't government tax and spend tax money much much more actively than they are doing today? It looks like it would be a "cheat code" for unlimited exponential growth. Maybe there are other factors that prevent it, but that our simple model ignores?

**Tags:** macroeconomics, gdp, taxation
> **Comments:**
> **username_1**: In your question you keep referencing some model, but your question does not include example of any model you just post examples of parameters like MPC that could be applied to any kind of model. The answer to this question will vary depending on which "model" you want to use. For example 99% undergraduate macro models are not dynamic so there cant even be any growth. I feel that you have in mind the macro goods market model like here:https://courses.lumenlearning.com/boundless-economics/chapter/introducing-aggregate-expenditure/ is that correct or do you want to use different model?  
> **username_0**: @1muflon1 My bad, I thought that it's already constitutes model. But okay, suppose we work with the model that you linked, now what?  

---

## 1 answers

### Answer by username_1 on 2020-01-15. Score: 2

In your comment you say you want the answer in context of the classic textbook model for goods market/output equilibrium.

Following the Blanchard macroeconomics textbook lets consider closed economy so the output will be given by:

$$Y=C+I+G$$

where $Y$ is output/income (they must be always equal), $C$ consumption, $I$ investment and $G$ gov. spending.

The consumption will be given by:

$$C = c\_0 + c\_1(Y-T)$$

where $c\_0$ is the part of consumption that does not depend on income and $c\_1$ is marginal propensity to consume (MPC) and $T$ are taxes.

In this basic model investment is just fixed $I=\bar{I}$

Solution to the model (i.e. autonomous spending) based on the set up above is given by (the solution is derived by substituting $C$ in second equation for its place in first one and then solving for $Y$):

$$Y = \frac{1}{1-c\_1}[c\_0+ \bar{I} +G -c\_1T ]$$

Now since you say that we take it for granted that government runs balanced budget $G=T$ because borrowing is "taboo" we can substitute the equality between taxes and gov. spending so we have:

$$Y = \frac{1}{1-c\_1}[c\_0+ \bar{I} +G -c\_1G ] = \frac{1}{1-c\_1}[c\_0+ \bar{I} +(1-c\_1)G ]$$

Now to clearly see the effect of G spending lets just set $c\_0=\bar{I}=0$ (this is not necessary but it cleans the equation). So we have:

$$Y = \frac{1}{1-c\_1}(1-c\_1)G = G$$

Hence there is no multiplier on Government spending if budget is balanced. So without debt government cannot increase output in any way (at least not in this simple model).

The intuition is that of course if government just takes money from left pocket to the right there cannot be any increase in output. Since in this model all agents have the same MPC and hence all spending including the Gov. spending has the same multiplier. If you take 10 euros from consumers and just spend it as a government you are not creating any additional output. you are just reducing $C$ - reduction of which would also be affected by multiplier, by the same value you are increasing $G$ which is affected by the same multiplier. The effects must cancel each other out. Only if you would allow for heterogeneous agents where people have different MPC government could increase output in the short term by redistributing money from people with low MPC to people with high MPC, but the output cannot be raised "exponentially" and this would only be one time increase - i.e. increase in level or short run growth. I would recommend reading the full chapter 3 from Blanchard et all (2010). Macroeconomics an European perspective, that deals with this in greater detail.

> **Comments:**
> **username_0**: I'm confused. In my example if I tax \$100 and then spend them, then GDP increases by \$100. What am I doing wrong? Is there a model where said actions would lead to \$100 increase of GDP?  
> **username_1**: Well if you tax 100 and spend them GDP always remains constant - taxing means taking money from people (their income and hence their consumption). So taxing 100\$ means you reduce GDP by 100\$. Unless the taxation is on some external entity - some tribute from foreign conquest then GDP cant just change through taxing and spending if budget balanced. You could increase GDP by running deficit though (at least in short run - although to deal with long and short run I would have to go to full AD-AS IS-LM model).  
> **username_0**: "means you reduce GDP by \$100" Not in my calculations. After taxation GDP will be \$999966.67 (decreased by \$33.33, not by \$100. Remember, the population is stingy). After that if we spend all tax money GDP will be  \$1000100 (increased by \$133.33)  
> **username_0**: And by the way, can you please elaborate how you got solution formula in its first form?  
> **username_1**: @user161005  1.The formula is derived by substitution of second equation into the first one (I will update text after this comment). 2. This is just nitpicking but if MPC is 0.25 then in no simple undergraduate model you can get multiplier 1.33 in this case multiplier would be 1.25 as multiplier = 1/(1-MPC) 3. the point is that since taxing reduces income and hence consumption you will also get negative multiplier. In fact every tax dollar raised will have negative multiplier MPC/(1-MPC). If G=T positive and negative multipliers cancel. You have to have some debt  (G>T) there to raise GDP  
> **username_0**: 2.The expenditure multiplier is 1/(1-MPC), I agree. But if we will substitue MPC with 0.25 then it will be 1.33, not 1.25 https://www.wolframalpha.com/input/?i=1%2F%281-0.25%29   3. The taxation multiplier is 0.33 https://www.wolframalpha.com/input/?i=0.25%2F%281-0.25%29   They don't seem to be canceling out each other in my example  
> **username_1**: @user161005 oh sorry you are actually right about multiplier =1.33  my bad... once you start dealing with abstract high level math you forget how to do simple fractions. Anyway that did not mattered for my answer at all. And yes they will cancel out let’s parametrize  the model we have Y=1/0.75*(100-0.25*100)=100 (I rewritten 1.33 as 1/0.75 to get precise result) hence there is no multiplier there anymore if you raise 100\$ through tax and have balanced budget(cancel out is maybe unfortunate wording because in math it implies subtraction, here it means that there no longer is any multiplier)  
> **username_0**: Then how can you explain that by consequently adjusting GDP for increased taxation and then for government spending I get +\$N for GDP? What is my mistake?  
> **username_1**: @user161005 ? I don’t understand i consequently adjusted GDP for taxation and government spending and you saw that there was no multiplier. Can you explain what you mean? Or show me your math  
> **username_0**: At first I reduce GDP by Tax_Multiplier\*AddionalAmountTaxed. In this case -0.33\*100=33  After that I increase GDP by ExpenditureMultiplier\*AddionalAmountTaxed, in this case it's 1.33\*100=133. 133>33, meaning that now GDP increased by 100  
> **username_1**: @user161005 but you forgot about minus sign also taxes reduce GDP so it’s actually 133-33=100. Yes thanks to multiplier 100 government spending (where the 100 is transfer from the original output/income) increases the original output from 100 to 133 but at the same time it decreases the output by additional 33 due to taxes.  
> **username_0**: "so it’s actually 133-33=100" Please notice that 100 isn't GDP, but addition to GDP. If GDP=X (before taxation), then after taxation and spending it will become GDP=X+100.  " but at the same time it decreases the output by additional 33 due to taxes" I agree, but we will still have 100 as addditon to our GDP  
> **username_1**: @user161005 where do you think taxation comes from? That would only work if taxes are some kind of tribute from abroad, like when past one nation created a client state which was forced to pay taxes to the home state. If there is no foreign country to tax then the tax have to be applied to the income/output so we have Y-T (see the second equation)... Sure if you can take the tax out of thin air GDP will increase by 100 but then it would no longer be tax but some exogenous Government production - like government having magic box that gives 100  
> **username_0**: "tax have to be applied to the income/output so we have Y-T" In my example I tax wealth that people have, not their income or production. So if you're, say, a preper living in a bunker and who has money, but produces nothing and doesn't have any income, then you will be taxed nonetheless.  
> **username_0**: "where do you think taxation comes from?" From pockets of people, of course. "That would only work if taxes are some kind of tribute from abroad" What makes you think so?  
> **username_1**: @user161005 1. In this model and in almost all undergraduate macro model there is not any wealth. 2. Wealth is saved income so under certain models there is actually not much difference between taxing wealth and income (however this is not even domain of macroeconomic which u used as a tag this would be in some public economics model. 3. You said you want explanation given in context of standard undergraduate goods/output equilibrium model. 4. Because ultimately taxes are transfers of output they don’t magically create output so at some point they are subtracted from GDP if they are domestic  
> **username_1**: Expanding on my point 4. Even if at time t there is let’s say tax on investment (let’s be sloppy and just call it wealth) 100\$ then at time t investment would be reduced by 100 + the effect of multiplier (since from above equations it’s clear it’s applied to investment as well) so GDP would be lower at time t and then government could not spend it run surplus at time t and then spend the money at time t+1 where it would get multiplied but again the sum of GDPs would still be the same (if multiplier does not change). The tax just in this case reduces GDP in one period at expense of other  
> **username_1**: Sorry last sentence was supposed to say increases GDP in one period at expense of other  
> **username_0**: Forget models and intuition, what is wrong with my calculations? Please show me your two-step calculations of increasing taxes by \$100 with consequent spending of said \$100.  
> **username_1**: @user161005 well to do any calculation you have to have some model. But I will interpret your comment as forget the formalism and show me simple calculation in ill defined model. 1 step you tax 100e from economy in addition this tax will have further 33e negative effect on GDP due to multiplier effect so you are taking together 100e from economy. 2. You inject 100e to economy which thanks to multiplier are multiplied to 133 so you reduce GDP by 133 and then increase by 133. To avoid this you have to assume the tax does not extract the original 100e which would mean it is not tax in economic  
> **username_1**: Terms as definition of tax is involuntary transfer. Or it has to come from different time periods which you on purpose exclude (although that would still violate T=G). Or the taxes are not domestic so the original product is not transferred from GDP at home but it comes as subtraction from GDP abroad in some other country  
> **username_1**: Sorry in my 1. Comment it should be taking together 133 that was typo  
> **username_0**: So it seems I found the root of our disagreement. For you taxing people by \$100 will decrease GDP by \$133, while for me it will decrease GDP only by $33. My calculations are based on this video: https://youtu.be/UsJmBBtwzYY?t=215     Or in other words, suppose we taxed a preper living in bunker, who has money but doesn't spend them. In this case taxing them won't reduce GDP by even a single cent because they wouldn't spend their money anyway. The only things that maters is how much we decreased consumption by our taxes. 100*0.33, meaning we decrease it by \$33.  
> **username_1**: But the video actually proves my point. If you watch it all way till the end you will see that in the end tax decreases output by MPC/(1-MPC) exactly like in my answer (if you expand the terms u will see T is multiplied by c_1/(1-c_1). So I don’t understand what’s your point here you see you get exactly to zero here 1.33*100- 1.33*0.25*100. So multipliers exactly cancels out. I think the confusion stems for the fact that you only focus too much on the first part that change in the tax is multiplied only by 0.25 which ignores that in equilibrium 1.33 is also applied  
> **username_1**: Also again when I say cancels out I mean multiplier is 1 i.e there is no multiplier GDP is just what it was 100GDP gets multiplied by 1. And the 100 cannot be just gotten from thin air. It can be from past as I mentioned so it’s possible to decrease GDP yesterday by 100 to increase it today by 100 etc. also in more complex models again if MPC is different between time periods or where you have heterogeneous agents there could be some positive effect but you won’t even find those models at Khan Academy. Actually based on the video I am 100% sure that video uses the mode I gave you in my answer  
> **username_1**: Furthermore, to further clarify of course in this simple model there is no explicit production so you can say Gov. just produced 100GDP out of thin air but then you can’t call it taxation because definition of word taxation in economics is involuntary **transfer** of resources  
> **username_0**: 1."tax decreases output by MPC/(1-MPC)". Actually it's decreased by Δt*MPC/(1-MPC). When Δt=100 and MPC=0.25 it all equals to 33.(3). And if you say that your answer algebraically says the same then how can you even claim that tax will decrease GDP by 133? 2."1.33*100- 1.33*0.25*100" How is it zero?? Wolfram Alpha says it's 99.75  3."tax is multiplied only by 0.25 which ignores that in equilibrium 1.33 is also applied " What are you talking about? We don't use Expenidutre multiplier in taxes. And we don't multiply money by MPC, we multiply by Tax Multiplier  
> **username_1**: But I never said GDP will be zero that implies that GDP would be destroyed. My point is that the tax which allowed you to get the government spending is 100 (T=G) So this is just transfer. Forget about the multiplier for a second. If you issue 100 tax without any multiplier the effect on original GDP 100 would be 100-100+100. Where first 100 is some original GDP that was there in the economy (again tax must come from somewhere - **tax is not production** by definition.  
> **username_1**: Now adding in the multiplier the changes can be summarized as follows: $\Delta T = -100$, $\Delta G = + 100$ but you also have 2 canceling $\Delta$ on GDP as if you would not issue the taxes the money would still be in the economy (again unless they are from past or external source). Applying the multipliers means that you have positive change of 1/(1-0.25) 100 - 0.25/(1-0.25)100 hence the G just changes by 100 the (99.75 is rounding error since you dont use exact values) but where does that G = 100 comes from? Thin air? No it was originally subtracted from some other part of economy  
> **username_1**: Also I mean this is just accounting if you have some government credit account G=100 then there must be some corresponding government debit (in this case T=100). Solving the model that I presented in answer and which is also the model Khan clearly uses simply tells you that GDP (ignoring all irrelevant parts like investment for this discussion) $\Delta Y= 1/(1-MPC)(1-MPC) \Delta G$ so GDP increases 1 dollar for every dollar spent. But where does those dollars spent come from? If they come from present domestic economy then GDP wont increase  
> **username_1**: Also what I mean is not that the spending G does not increases Y but it only increases Y for every dollar spent $\Delta Y= \Delta G$ but there is no additional multiplier effect here on this government spending and since to get any government spending at all you have to get the dollars from somewhere if you get the dollars from Gross output then GDP must remain constant. Of course if you can get the G spending outside the model and outside the GDP then GDP increases that’s just asking does GDP increase if G increases well yes! But this is not some magical addition to gross product  
> **username_1**: @user161005 yes I agree that when we change G by 100 Y also increases by 100 but where did the G change in 100 come from? Thin air? The point is that of course government spending increases GDP by government spending of taxes are balanced. But where do you think government spending in this simple model comes from? You would get some extra GDP here only if G>T when government runs deficit otherwise you can get 1 dollar of GDP for 1 dollar spent by government. There was no contention about that but as long as government does not produce Gov spending must come from output  
> **username_0**: By the way, do you agree that no matter how hard we will tax a preper living in a bunker (who also has no income, no production, but has some cash on hands) we WON'T harm our GDP even by a single cent?  
> **username_1**: @user161005 Are you asking in this particular model? In this model it would not decrease GDP. Here tax has no deadweight loss as it is lump sum tax. Its just transfer from one part of economy into another so if GDP is 100 you tax the whole output and give it to G=100 then GDP will be still 100 (thats btw the point). So as long as you use lump sum taxes you can tax as much as you want without hurting the economy but that does not mean you can create something from nothing the maximum tax is the whole GDP and since G=Y, GDP wont change only all output transferred from private to public hands  
> **username_0**: Suppose we gave our preper ability to avoid tax by saying that tax will be applied only to people who eat meat (no matter if they are prepers or not), with $100 per person, so it's NOT a lumpt tax anymore. Will we hurt GDP by when taxing this bunker-living meat-eating preper? For me it seems like not, their spending=0, they don't keep their money in a bank, don't invest them, thus for economy their cash is as good as non-existent. And consequently, if we will spend dollars that we took from the preper, then we will start chain of spendings that will boost GDP.  
> **username_1**: @user161005 1. What do you mean by hurting? The discussion was never whether tax hurts someone in this model or not. The discussion was wether you can increase GDP by spending if you run balanced budget. That can’t happen in any simple mode that does not have some complex hysteresis effect. 2. At the same time I never claimed that taxes in this model hurt anyone. 3. If it’s not lump sum tax then how do you tax him if he has no economic activity? Non-lump sum tax has to be applied to something. 4. The only case that this would not enter the GDP calculations would be if that something you  
> **username_1**: Tax was never produced. If it’s some production then it has to be in GDP at one point or another and in this simple models there is no inheritance from some foreign country or things like that. All GDP is produced at some point so to get the tax you have to subtract it from some part of GDP no matter where you put in terms of national account. The only way how what you describe would work is if those resources were not recorded on GDP maybe due to criminal activity so the first time they get registered in accounts is when they get taxed, but that’s whole different topic of measurement error  
> **username_0**: 1."What do you mean by hurting?" I didn't mean about hurting a person, I mean hurting GDP, i.e. decreasing GDP.  2." If it’s not lump sum tax then how do you tax him if he has no economic activity?" What is so difficult about it? We just go and tax them for meat-eating, no matter if they are economically active or not. As for tax, we apply it to cash that they put under their mattress. The cash itself was accumulated before they became living in a bunker.  
> **username_1**: @user161005 1. I knew you mean something with GDP but what do you mean by hurting GDP? If you mean by hurting if it lowers GDP then it depends on type of tax non-lump sum taxes would actually hurt GDP, and would have to be modeled differently than above (decrease it in this model). Any non lump sum and non-piguvian tax hurts output (but I assume no externalities in this discussion). 2. Eating (consumption) is economic activity! I got confused because you don’t use proper terminology. 3. Of course it affects GDP. What are the money under the mattress where those money come from?  
> **username_0**: 1."it’s some production then it has to be in GDP" The cash that the preper saved was indeed in GDP. But it was in **past** GDP. In the current year these money aren't invested/loaned/spent/put in a bank, so fail to see how we would account for preper's under-mattress money in current GDP. 2.Wait, how is preper's eating is an economic activity? They are practically living outside of economy, unless we tax them!  
> **username_1**: The last sentence in your paragraph is important yes if you assume that all that money was gotten from elsewhere so it never was part of GDP then sure, that’s the same as having tribute from abroad - but the money will get subtracted from some output. If we turn of foreign sector the output must be domestic  
> **username_1**: @user161005 1 consumption is economic activity- pls just google it. 2. In this simple model that the khan academy video and my answer is based on a model that does not have multiple time periods. In a multi period model multipliers would look different - as I prefaced in my answer in some models taxing and spending could increase output but it does not generally happens when budget is balanced. In such model if it’s a tax on saving then it will affect incentives to save and even to consume since savings is just future consumption. You can have some situations like liquidity trap where this  
> **username_1**: government could increase output by this kind of taxation and spending in the short run. However, in the long run it’s simply impossible to raise output above its natural or full employment level. Also again these questions are outside the simple goods market equilibrium model - I prefaced in my answer so I absolutely don’t get the point here.  
> **username_1**: 3. if you assume any non-lump sum taxes in this model the multipliers won’t be anymore 1/(1-MPK) non lump sum taxes generally affect multipliers as well so actually that’s even assumption that would break your assumptions on size of multipliers etc. so there we won’t even get exactly the above result. Generally speaking non-lump sum taxes make the multiplier worse, some could even make it negative so you could even find here that GDP is actually lower but it would also depend exactly what kind of non-lump sum taxes we are talking about. In principle there is infinite number of tax schedules  

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