A Primer on Tax Incidence Analysis (Part 1) — How taxes lower your salary and the number of widgets you can buy.

After reading some comments, I thought it would be helpful to go back to the basics of tax incidence analysis. Once again, it’s important to keep in mind the difference between tax incidence and tax collection. If we mix up those two, we won’t get anywhere.

Look at the economy right before and right after the IRS cashes the tax check. Whose pools of money have shrunk? That’s tax collection. A corporation’s income properly belongs to its shareholders and so when the corporate income gets taxed, the collection is against the shareholders.

Now, compare two economies. One in which the tax exists and the other where the tax does not exist. The difference between the two is the tax incidence. If I choose to work less because the higher tax makes it less worthwhile for me, that’s tax incidence. This is what we are looking at here. And ultimately, when making tax policy, this is what we are interested in.

So let’s look at a very simple market for “widgets”. The supply for widgets is S = 0.5P where S is how many widgets firms are ready to produce at a price P. The demand for widgets is D = 100 – 0.5P where D is the number of widgets people are ready to buy at a price P.

If the quantity of widgets people are willing to buy is greater than the quantity of widgets people are willing to sell, the buyers will bid up the price until the quantities are equal. If the quantity of widgets people are willing to sell is greater than the quantity of widgets people are willing to buy, the sellers will bid down the price until the quantities are equal. That’s the equilibrium where S=D.

So we can solve for this easily:

0.5P = 100 – 0.5P

P = 100 and S = D = 50.

50 widgets are produced and sold at $100 each. So now let’s calculate the surplus. The consumer surplus is how much the item was valued at minus how much it was purchased for. We get that from the demand curve. Consumers were willing to pay $198 for the first widget, but they paid $100 for each widget, so the surplus from the first widget is $98 and so on and so forth. (Actually, we assume continuous quantities, so we integrate the area under the demand curve and subtract the price paid times the quantity. Look at the diagrams for the consumer surplus and producer surplus triangles if the math-speak is confusing you) The total consumer surplus is $2,500. You can do the same for the producer surplus. Producers were willing to be paid $2 for the first widget, but they were paid $100 for every widget so the producer surplus on the first widget was $98. As it turns out, the total producer surplus is also of $2,500. So we have $5,000 of economic value that has been produced through these trades.

Now let’s introduce a $20 tax per widget. We will say that the tax is collected on the buyers of widgets. In other words, the quoted price is a before-tax price. So now we have to adjust the demand function. After all, buyers don’t care about the before-tax price. They care about the money that enters and leaves their wallet whether it’s because of taxes or any other reason. Consumers are now ready to pay $20 less per widget to the producer since they then pay an extra $20 to the government. So we have to adjust the demand curve.

D = 100 – 0.5(P + 20) we can simplify this D = 90 – 0.5P

[Updated Thanks Michael. The math is correct, I simply made a typo in the above-equation]

Solving for the new equilibrium

0.5P = 90 – 0.5P

P = 90 S = D = 45. In other words, now, only 45 widgets are made and the producers are paid $90 for each widget only. Of course, the consumer has to pay the $20 on top which means the consumer is now paying $110 per widget. The total tax collected is $900.

So now once again let’s calculate the surplus for consumers and producers. Well, for the first widget, the consumer was willing to pay $198, but they paid $110 including the tax, so their surplus for that first widget is of $88. The total consumer surplus is $2,025. For the first widget, the producers were willing to accept $2 but they got $90 for every widget so they got a surplus of $88 on the first widget. The total producer surplus is $2,025 for a total economic value created of $4,050. That’s $950 less than the $5,000 created without the tax.

Note that in this simple example, the seller and buyer share equally in the loss from the tax despite the consumer writing the check to the IRS. You could re-do the math having the producer send the check. The result is the same. I will follow up with another post detailing how in different markets, the burden might be shared differently between producers and consumers.

Note that this works the same way whether we are selling widgets to consumers or selling work to our employers. If you tax something, it will cost more to those who buy it, bring in less to those who produce it and less of it will be bought, sold and produced.

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40 Responses to A Primer on Tax Incidence Analysis (Part 1) — How taxes lower your salary and the number of widgets you can buy.

  1. Joe Seydl says:

    “The total producer surplus is $2,025 for a total economic value created of $4,050. That’s $950 less than the $5,000 created without the tax.”

    Right, but now the government has $950, which can be used to add economic value in some other way — that $950 doesn’t just evaporate. So long as the $950 is used to provide a public good or service that is useful to society and that otherwise would not be provided by the private sector, one could argue that there would be no deadweight loss in your above example. Furthermore, how do you know that the $950 in revenue is not used in such a way that lowers production costs for the widget producer? Maybe the $950 is used to build a new road that lowers transportation costs for the widget producer, thereby shifting the supply curve for the widget producer downward.

    • PrometheeFeu says:

      Actually, the government has only collected $900 while it has remove $950 worth of value. I should have highlighted the deadweight loss of $50, but I wanted to focus on the fact that the burden of the tax is not necessarily born by the person who writes the check to the IRS.

      Now, it is true that the government could potentially use that money to do all sorts of great things. But we can’t know that from the tax. What I said above is accurate whether they spend the money on wonderful or horrible things. Either way, it has a cost and the cost is divided between consumers and producers.

      • Joe Seydl says:

        Sorry, you’re right — $900 in tax revenue, w/ $50 of deadweight loss. But here’s the whole issue I have with your argument, as well as the arguments in your prior posts: You constantly say that determining tax incidence is very hard, but then you turn around and say that all taxes create distortions and thus should be avoided.

        For example, you just said that “it is true that the government could potentially use that money to do all sorts of great things. But we can’t know that from the tax.” Fine. But this, then, does not imply that the $20 tax per widget reduces economic efficiency. We can’t determine this without knowing what the government plans to do with the $900 in tax revenue it collected from the widget tax. The above example might reduce efficiency if the government spends the $900 on something useless, such as a grant for an uncompetitive firm like Solyndra. On the other hand, the above example might increase efficiency if the $900 is spent in such a way that lowers production costs for the widget maker — for example, maybe the $900 is spent on upgrading technical school infrastructure, so that students graduating from technical school and entering the widget production industry have a more competitive skill set and are able to produce widgets more efficiently.

        Similarly, in your prior examples, you state that determining tax incidence on someone like Warren Buffet is very difficult, since we don’t really know what portion of the corporate tax burden is actually born by Buffet. Then you turn around and say: “Keep capital gains taxes low!” You haven’t provided a sufficient argument to suggest that capital gains taxes should be kept low. Maybe they should be, maybe the shouldn’t be. What I’m basically saying is that if you’re going to claim that determining tax incidence is very hard, you can’t then pick a side of the argument to be on with conviction; there are likely cases in which capital gains taxes should be lower, just as there are likely cases in which capital gains taxes should be higher.

        • PrometheeFeu says:

          “You constantly say that determining tax incidence is very hard, but then you turn around and say that all taxes create distortions and thus should be avoided.”

          When I say that determining the tax incidence is hard, I mean that it is hard to take a real-life situation and say: “Given all taxes that are paid, here is the economic burden born by Mr. Buffett and here is the economic burden born by his secretary.” However, we can easily determine that the incidence of the corporate tax will be split between shareholders, employees and vendors. We also know that some distortion occurs and for a particular tax in a particular market, we may be able to get an estimate of the incidence. So tax incidence analysis can bring something useful to the policy debate.

          “But this, then, does not imply that the $20 tax per widget reduces economic efficiency. We can’t determine this without knowing what the government plans to do with the $900 in tax revenue it collected from the widget tax.”

          I see what you are saying but I don’t think that’s a particularly useful level of analysis. All taxes do create distortions. In other words, the production of the revenue does have a cost. Just because the cost is worth paying in a particular instance doesn’t mean we shouldn’t see it as a cost.

          “Similarly, in your prior examples, you state that determining tax incidence on someone like Warren Buffet is very difficult, since we don’t really know what portion of the corporate tax burden is actually born by Buffet. Then you turn around and say: “Keep capital gains taxes low!” You haven’t provided a sufficient argument to suggest that capital gains taxes should be kept low.”

          My point regarding taxes on capital gains is not that they are born by Buffett or someone else in particular. My point is more generally that when you tax an activity, you get less of it. So if we tax capital gains, capital formation/investment is made comparatively less attractive than spending on consumer goods. Since capital formation gives us more economic growth, we want to avoid making it relatively less attractive than consumption.

          • steve2 says:

            Does it necessarily follow that this capital will be used in a way that will translate into positive public welfare? Do these gains necessarily translate into more employment?

            Steve

          • Viking Vista says:

            Economic growth comes from capital formation, but capital formation does not always lead to economic growth.

            Forgetting the latter leads to all sort of coercive redistributive malinvestment schemes from the government like fiat money central banking and “stimulus” spending. With real (private) investment, nonproductive hairbrained schemes occur, but are less likely because people are using their own money, and when they occur, the investors’ funds limit the loss, and result in less capital under their control. With government “investment” of the funds of unwilling others, losses fequently lead to greater confiscations.

            Capital formation only tends to lead to economic growth when growth is selected for by the profit and loss system, not the politically motivated dreams of looters.

  2. Joe Seydl says:

    “We also know that some distortion occurs and for a particular tax in a particular market, we may be able to get an estimate of the incidence. So tax incidence analysis can bring something useful to the policy debate.”

    I agree. So then specifically show the estimates of the incidence on someone like Buffet — or at least link to a study that does. If you do this, and if the incidence analysis suggests that Buffet, as a shareholder, is indeed bearing a relatively large portion of the corporate tax burden, then this would reinforce all of your arguments. Without this analysis, I cannot interpret your argument as anything but mere speculation.

    “Just because the cost is worth paying in a particular instance doesn’t mean we shouldn’t see it as a cost.”

    Right. But you shouldn’t leave out the case in which the costs might be worth paying for! When you do this, obviously the uninformed reader is going to ascribe to the view that taxes are always and everywhere efficiency reducing. I’d rather like to see you give your readers both sides of the debate, letting them determine for themselves where to stand on the issue.

    “My point is more generally that when you tax an activity, you get less of it. So if we tax capital gains, capital formation/investment is made comparatively less attractive than spending on consumer goods. Since capital formation gives us more economic growth, we want to avoid making it relatively less attractive than consumption.”

    This is the point that frustrates me the most. First, the evidence showing that low capital gains taxes encourages more investment is spotty at best — I don’t normally link to someone as partisan as Jared Bernstein, but he highlights what I’m talking about here: http://jaredbernsteinblog.com/taxing-capital-gains-at-ordinary-rates-evidence-says-do-it-so-does-buffet/. Second, what kind of capital formation/investment are you talking about? Yes, many investments do give us more economic growth. For example, if a company invests more in basic R&D and those investments pay dividends in the form of lowering production costs for whatever good or service that company is producing, then economic growth will be stronger and living standards will be higher in the medium to long run. But what if the capital formation/investment from lowering capital gains taxes leads to wasteful rent-seeking activities? What if it spawns a whole industry of accountants devoted to sidestepping taxation, investors devoted to channeling funds overseas to opaque tax havens, and speculators devoted to mispricing risk and thus fueling bubbles in the economy in search of short-term profits? In the latter case, I highly doubt that more capital formation/investment would lead to stronger economic growth.

    • PrometheeFeu says:

      “So then specifically show the estimates of the incidence on someone like Buffet — or at least link to a study that does. If you do this, and if the incidence analysis suggests that Buffet, as a shareholder, is indeed bearing a relatively large portion of the corporate tax burden, then this would reinforce all of your arguments.”

      Actually it wouldn’t. My point in my first post was specifically that Buffett and Obama were not talking about tax incidence, but they were instead talking about tax collection. So when Mankiw was correcting their tax collection numbers, it didn’t make sense to complain that he wasn’t doing incidence analysis.

      If you want to see what sort of results we get with empirical work, look at this meta-study by the CBO (http://www.cbo.gov/ftpdocs/122xx/doc12239/06-14-2011-CorporateTaxIncidence.pdf) or this study by the Kansas City Fed (http://www.kansascityfed.org/Publicat/RegionalRWP/RRWP07-01.pdf). The estimates range from shareholders more than fully bearing the cost of the tax to labor more than fully bearing the cost of the tax. This is still an open question.

      “Right. But you shouldn’t leave out the case in which the costs might be worth paying for!”

      I’m leaving out the way the revenue is spent entirely. That is not the focus of this particular post. If you read this post, not once do I say that the tax was worth it or was not worth it. An earlier draft featured the phrase: “Note that the loss due to the tax is greater than the revenue collected” but I removed it specifically because I didn’t want to confuse the reader with the issue of how the tax revenue will be spent. The model is easy if you’ve seen it before, but if you haven’t it’s probably best to spend a couple minutes thinking about it rather than starting to wonder whether you think Congress can be trusted to spend money efficiently or not. There will be plenty of opportunities to discuss how to spend the tax in the future.

    • Viking Vista says:

      “…you shouldn’t leave out the case in which the costs might be worth paying for…”

      You think that there are readers here who don’t know that governments spend tax dollars?

      You are talking about a different issue regarding the value of having politicians confiscate people’s money to spend on things the people demonstrably (because of the force required) do not want. Even then, I think the readers are well acquainted with the track record for government spending. The point here is that taxes reduce trade and result in DEFINITE, if incalculable, economic losses due to that effect. Whatever gain that you think a politician’s vote buying adds to the economy must overcome that loss.

      Even then, this discussion is about a measure of collective loss or gain, so you have to decide if it matters to you that one man’s judgement of collective gain comes at the self-percieved loss of many unwillingly participating individuals.

      • Joe Seydl says:

        Look, the only point I was trying to make is that the widget maker would not be in business with certain public goods, such as roads to transport widgets on and a police force to ensure that criminals do not loot the widget maker’s supply warehouses. That’s all. After re-reading the whole debate, I acknowledge that I overstepped a bit. The whole purpose of PrometheeFeu’s post was to provide a very simple example of how the burden of a tax is always shared, without involving any government spending decisions. I respect that. And as I stated earlier, I will look forward to PrometheeFeu’s future posts in which government spending decisions are assessed.

        • Joe Seydl says:

          *without certain public goods

        • Viking Vista says:

          You posted a question that I’m sure was on the mind of some readers, and I see your “fair enough” post where you seem to concede that your question was beside the point. I’m not implying that you are detracting from the discussion. I’m just pointing out a deficiency in your (not unique, not unintelligent, not offensive) line of thinking.

          For example, in your reply to me, you find it important to point out that the widget company would not be in business without the availability of public services (which as I previously said, everyone is immediately aware is one type of tax expenditure). But the discussion is about government and taxation. So what you are really saying is that the widget company would not be in business if the government didn’t monopolize certain services and force people to purchase them. You believe that if the widget company, its customers, its suppliers, its ancillary industries, people in its neighborhood or territory or country were permitted to keep their tax dollars to instead spend according to their own preferences, that the widget company would not exist.

          The implication is that a public good is not something that people actually prefer, but is a lesser choice forced upon them by agents of government. Either that or the implication is that people must be forced to purchase what they want. Or perhaps it is that there is one group of people who force a second group of people to give the first group what they want. At any rate, your belief that economic gain comes, in an essential part at least, from forcing people to do what they would not choose to do, carries with it some dubious assumptions.

          But now I am guilty of prolonging your tangent. Perhaps we can continue in a more related blog post in the future.

  3. DPG says:

    PrometheeFeu: I would like to commend you for not getting drawn into the normative debates of Joe Seydl. You lay out the positive economics very clearly. Good luck in the future.

  4. Greg says:

    mhh, sorry to rain on your parade but this is stupid. this is comparative-statics. you are totally missing out more crucial dynamic impact of taxes obviously taxes lower you disposable income for consumer goods, but you also have to understand how they affect labour market decisions and capital market decisions (savings) and therefore production decisions. all this talk about whose burden it is to pay the tax is irrelevant in general equilibrium, what matters is the overall loss/gain of the system as a whole.

    i am very classically liberal but since i am doing economics at graduate school i have to say that the main point is not whether purchases of goods falls because of taxes (obviously they do fall) but whether what the taxes are spent on, on the whole, bring in more utility than the taxes take away.

    if you want my semi-expert opinion: generally taxes are bad, mostly because they allow rent seekers to do stupid things which messes up the system in general equilibrium, but also because government spending crowds out stuff.

    • Ghengis Khak says:

      Yes. A blog post titled “A Primer on Tax Incidence Analysis (Part 1)” should include all sorts of dynamic and second-order effects that you learned about in grad school rather than being, you know, a primer on tax incidence analysis.

    • cmprostreet says:

      “the main point is not whether purchases of goods falls because of taxes (obviously they do fall) but whether what the taxes are spent on, on the whole, bring in more utility than the taxes take away.”

      Once a positive NPV spending program has been found, it should be funded by whatever means destroys the least value (which may include borrowing and taxing later). Unless this tax destroys the least value compared to any other possible tax, it shouldn’t be used to fund ANY spending program, no matter how wonderful.

      Once you have a positive NPV spending program in mind, you can hunt for the proper funding mechanism. It’s backward to point out that if you raised the tax first, you then might be able to find a good way to spend the money. There’s a huge difference between identifying something good and pursuing it through the path of least destruction, vs. destroying things and hoping something good comes out of it. Since most tax discussion revolves around a tax proposal not tied to any particular spending program, I don’t see the problem with pointing out that breaking stuff and hoping for the best is not a good policy. Before we go ahead and destroy something, the burden is on whoever insists it will be worthwhile to demonstrate why it will turn out well, and why this is the best means to do so.

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  6. Eduardo Weisz says:

    I agree with you that any kind of taxation has this kind of problem and agree with you when you say that factors like kinds of taxation and demand and supply elasticityhave an efect. But, when talking about it we must also talk about public goods like security once you are suposing there is efective property rights in your simple economy. I mean, in a real world discussion it would be absurd to consider that this is not an issue …. What do you think?

    As for welfare policies, I believe the 1929 crisis have shown that at some level they are necessary for the entire system survival. It is interesting to note that if the Czar had one platoon in 1917, one single platoon!, the Russian Revolution would have never happened. I believe that there is a novel that shows this point. It is called “The grapes of Wrath” and was writen by a guy named John Steinback. It’s good reading and shows the problem with your view.

    • PrometheeFeu says:

      See my answer to Joe Seydl above. I deliberately stayed away from discussing any benefit or harm that could come from the later expenditure of the tax revenue. If you read the post, you will note that no-where did I say that the tax was worth it, or not worth it. I’m simply focusing of what the cost is in this post and there will be plenty of future opportunities to discuss government expenditures and its effects.

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  8. jimbino says:

    If there are market distortions resulting in economic inefficiency, a tax that reduces the distortion will be a good thing.

    A $2 gasoline tax is but one example. A big tax on breeding is another.

  9. Bill Stepp says:

    Government spending on anything, including infrastructure, is waste consumption, as Rothbard pointed out in Power and Market (the best book on the economics of government). And I’m not just talking about a $600 toilet seat, a $16 donut, a bridge to nowhere, or 800 military bases outside the US. Anything G does that would be done on the private sector (not the drug war) can be done meeting consumer demand on the free market and by free market institutions.
    G spending is done to meet the political demand of politicians, not the economic demand of consumers.

  10. Michael says:

    Thanks for the post!

    Shouldn’t a “+” sign belong in the parenthesis in this line:

    D = 100 – 0.5(P – 20) we can simplify this D = 90 – 0.5P

    I’m no economist, but I do know my arithmetic!

    Best regards.

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  12. Vblanco says:

    “A Primer on Tax Incidence Analysis (Part 1) — How taxes lower your salary and the number of widgets you can buy.”

    You did a good job of proving your point, but what puzzles me is how any one can argue against such an obvious truth.It is a truth that hardly needs detailed analysis and graphs to demonstrate. I can accomplish the same thing by looking at how much I would have in my wallet if no taxes are paid, as compared to how much I have left because taxes were paid.

    If I pay taxes, I don’t have the money to buy the number of widgets I’d like; but, if no taxes are paid, I can buy more of the widgets I’d like.

    Anyone taking your presentation beyond that simple point is either extremely confused about the simplicity of how taxes deprive the tax payer of his own profits, or they are working on an agenda. An agenda that attacks anyone suggesting government has no right to your profits, and an agenda that denies what we all know from observation…..government can’t do anything as well or as cost effective as the private sector.

    • Viking Vista says:

      There are a great many self-proclaimed free marketeers who say things like “customers pay all corporate taxes” and “all business taxes are passed on to the customers”. Tax incidence analysis theoretically helps them see why they are mistaken. Even then, I find many cannot grasp it.

      Of course, there is one cost this analysis misses. If my financial loss due to taxes is the same either way, I would much rather not be the party who must interact with the taxman. To some degree, I suppose, this particular preference will be reflected by, and the cost mitigated by, my behavior to choose to be one or the other party.

      • Viking Vista says:

        Note that the taxman may choose to split his interaction for the tax upon both parties, leaving you only the choice of not trading, if you find the interaction so disagreeable.

        • Vblanco says:

          Good to see you’re alive and well. VV, I see nothing in your two replies that make the obvious of this:
          “How taxes lower your salary and the number of widgets you can buy.”
          any more obvious.

          Without graphs and math, I think it is plain as the egg/freckles/hair/zits on your face that if taxes take some of your money, then you have less money with which to buy product, and all one has to do is pull out his balance sheet, wallet, piggy bank, or look under his mattress to verify that it is obvious.

          The fact that supply and demand are affected by taxes is also not a surprise to any businessman who understands that with less money he can buy less widgets, but with more money he can buy more widgets. The businessman understands that if taxes affect him, then taxes must affect the businesses with which he deals. Well, if he doesn’t understand that then he qualifies for a spot in the OWS crowd, and I suppose there are some out there that just might be that ignorant. But, VV, you’re not one of them. Salud!

    • PrometheeFeu says:

      There is a little more to that. The point is that the tax raises the price of widgets which lowers the number of widgets you can afford AND lowers the receipts per widgets by producers which lowers the number of widgets that they are willing to receive.

      • Vblanco says:

        Isn’t saying it your way here, the same as holding a mirror to my understanding of the point. We see the same thing, only in reverse.

        You say (increased) taxes on widgets, means I can afford to buy less widgets for the same amount of money I was spending on widgets. Agreed. Taxes affected the sale of widgets which affects the production of widgets, etc. et. al.

        I say (increased) taxes on my money, means (hold up that mirror) I can afford to buy less widgets for the amount of money I have left.

        Taxes on your widget or taxes on my money achieves the same result, less widgets sold, which inevitably means less widgets produced.

        I’d say the problem is that typically taxes are raised on the widgets, while at the same times taxes are raised on the customer’s income. Government 2 – People 0. (if you’re keeping score).

        • Viking Vista says:

          Widget trading is taxed. Labor trading is taxed. The proportion of the tax paid by the trading parties, and the amount of dead weight loss, is independent of which party the tax is directly levied upon, and is instead a function of elasticities. This is the point of tax incidence.

          However, there is a difference between being and not being the direct object of the taxman’s attention–compliance costs, legal risks, etc. This factors into one’s preferred price, but risk is very subjective with people in similar circumstances having greatly different feeling about having to deal with legal authorities.

    • Joshua Lyle says:

      There’s a significant of difference between “taxes lower your salary and the number of widgets you can buy” and “HOW taxes lower your salary and number of widgets you can buy”.

      • Vblanco says:

        And, the widget seller sees less widgets going out the door, and he doesn’t care how that came to be, he only knows he ain’t selling the same amount of widgets.

        So how does the “how” make the widget producer and seller feel better or more confident that he is going to stay in business making and selling widgets? You may think the “how” his customers came to have less money is of some kind of startling importance; but, I can’t see it being of primary concern to the widget producer and seller.. His customers have less money. His customers now buy less widgets.

        Rocket science………not.

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  15. You did a good job of proving your point, but what puzzles me is how any one can argue against such an obvious truth.It is a truth that hardly needs detailed analysis and graphs to demonstrate. I can accomplish the same thing by looking at how much I would have in my wallet if no taxes are paid, as compared to how much I have left because taxes were paid.

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