15 Jan 2015

A Revised Brain Teaser on Tax Policy

Tax policy 18 Comments

OK you guys are getting too hung up on it being gasoline in the previous post. Let me start over. Now in the following, I’m not necessarily saying each arguments in any of the steps is correct. I’m just saying, you could easily see an economist going along with the chain of logic, and yet (as we’ll see) we wind up with a contradiction. So the point of this exercise is to pinpoint where the mistake comes in.

(0) Initially the government raises $1 trillion by levying a percentage tax on labor income. There are 100 consumer goods in the economy that initially have equal sales (by revenue).

(1) The government enacts a percentage tax on Consumer Good #1, and reduces the percentage tax on labor income, so that when the dust settles the Treasury takes in $10 billion from the tax on Good #1, and $990 billion from the (now lower) tax on labor. I.e. it was a revenue-neutral tax swap of $10 billion.

Many economists would probably go along with the claim that this move would increase the deadweight loss to the economy from the tax code. Clearly if the government did a complete tax swap–i.e. raised the entire trillion dollars by taxing the heck out of Consumer Good #1 while not taxing anything else–that would be very destructive compared to the status quo. (The intuition is that the tax base on Consumer Good #1 is much smaller than all of labor, so the percentage rate would have to be much higher on Good #1 than for labor.) So if doing a full tax-swap would be awful, then presumably a 1-percent tax swap would be bad too, just not as bad.

(2) Now if you buy the argument in step (1), then the case is even stronger that doing another $10 billion tax swap–this time for Consumer Good #2–would add to the deadweight loss. This is because the starting labor tax percentage here is lower than it was in step (1), since at this point the tax on labor only needs to bring in $990 billion. So on the margin, the benefit of cutting taxes on labor by $10 billion is lower than it was originally, while the harm of imposing new taxes of $10 billion on a consumer good is at least as high as it was before.

(3) If you bought the logic in Step (2), then it clearly applies for further $10 billion tax swaps for Consumer Goods #3…#100. Each $10 billion tax swap ought to impose incrementally higher deadweight losses on society from the new tax code.

==> However, what if we initially implemented a revenue-neutral flat consumer tax swap deal, to raise $1 trillion from a uniform levy on consumer goods, while eliminating the tax on labor altogether? This is the ideal of standard tax policy reform analysis. In a simplified model where there is no leisure or other forms of income, at worst this move would be a wash. But in a more realistic model, this implementation of a consumption rather than a labor tax would be a net positive.


==> So what the heck is going on here? If we do the sequential tax swap deals, one consumer good at a time, we end up with raising all of the government’s revenue from taxing the consumer goods while phasing out the labor tax, and yet we seemed to show that this would be awful. Yet we have a standard result that putting in a revenue-neutral consumption tax and eliminating labor taxes is great.


(Just to reiterate, I am deliberately feigning ignorance above, because I see at least some of the flaws in the breezy presentation. But it took me a minute to get my bearings, and it’s possible I’m still overlooking some subtleties.)

18 Responses to “A Revised Brain Teaser on Tax Policy”

  1. Chris says:

    This could be completely wrong, but I think the key point here is that the distortions are caused not just by making the taxed goods more expensive, but by changing their prices in relation to other goods. So when we tax the first good, we have distorted the economy by changing its relative price, which introduces deadweight loss. By putting the tax on the second good, we still distort the economy, but we actually reduce the deadweight loss caused by the first tax. So each subsequent tax simultaneously distorts the economy and partially fixes previous distortions. At some point, we get to a place where the post-tax relative prices of the goods are closer to the original relative prices then they were after the first distortion and then we see improved efficiency from further shifts and by the time we have completely shifted there is no longer any distortion in the relative prices (except for labor to consumption, but I guess we would rather encourage work than consumption if there has to be taxes on something). I’m sure I’m missing something here, but that’s my basic intuition about why this apparent contradiction arises

  2. Harold says:

    In your example, all goods initially have equal sales. After the first tax swap, the taxed product will have a lower share of the market. When we introduce the second tax swap, the market share of the first product will rise again by a small amount, so we would have to reduce the tax rate on it to keep the revenue neutral. Each new product brought into the tax swap reduces the tax rate on the already taxed products. At some point the rate on each product will be less than the previous income tax rate – that is the transition point.

    Say we have 10 products instead of 100. The first needs to be taxed at 50% to reduce the income tax by 10 billion. If we picked the second instead of the first, it would need to be taxed at 50% also. We are talking about after the dust has settled. Introducing the 50% tax on the second product makes the first more attractive, so sales go up, and we only need to tax it at 8% to raise the same money. The second item also only needs to be taxed at 8% because the tax on the first makes it more attractive. Each product we add reduces the rate on each taxed item.

    • Harold says:

      “and we only need to tax it at 8% to raise the same money.” Make that 45%.

    • Chris says:

      Right I think this is essentially the same point I was making above. Each additional tax decreases the relative cost of the already taxed goods, increasing market demand and lowering the rate we would need to charge to produce the same revenue

      • Harold says:

        Chris – I agree, your comment was not visible when I posted mine.

  3. Dirk says:

    I think Harold has got the main point.

    One way to reduce (but not eliminate the loss) would be to figure out what the tax percentage should be for each product at the end of the complete swap. Because we are swapping based on tax revenue (10 billion at a time) we ignore the fact the were are taxing the consumer product at a much higher rate for each consumer product during the interim.

    A better way would be to:
    Calculate what the final consumer good tax percentage would be for all products (say 8%).
    Figure out the revenue that would be generated by swapping the taxes for consumer good 1.
    Calculate the reduced tax rate for labor income to make it revenue neutral.
    Make the swap for consumer good 1.
    repeat for all consumer goods.

    Granted I’m not taking into account the losses caused when taxing some goods and not others. There will be some misallocations in the economy when some goods are more expensive due to the new taxes than others

  4. Transformer says:

    Here is my take on this:

    1. We start with 100% of revenue raised by a labor tax
    2. We end with 100% of revenue raised by a fixed-rate sales tax on all 100 goods.

    Deadweight Loss (DWL) at 1 is assumed to be greater than DWL at 2.

    For this to be true, as we move from 1>2, DWL on labor must decrease at a faster rate than DWL on consumer goods increases.

    As each good starts to be taxed (and the tax on labor reduced) the reduction in deadweight loss from the reduced labor tax must exceed the increased deadweight loss from the increased sales tax.

    So there will not be a particular good that tips the balance – it a linear change between the 2 extremes.

    • Transformer says:

      I also quite like Chris’s idea above that as you levy the sales tax on an additional good, it slightly reduces the DWL on the already taxed goods. In this case you have to factor in both the reduced DWL from the lower labor tax and on the other taxed goods. There may indeed be a tipping point beyond which the DWL loss of taxing an additional good will be outweighed by the reduced DWL on already taxed goods.

  5. khodge says:

    isn’t this just a generalized version of the Landsburg multiplier:

  6. khodge says:

    Will a deadweight loss carry past period one? By period two the marginal suppliers will be out of business and the market place will look very different as people adjust their priorities and as certain consumer goods are simply no longer available.

  7. guest says:

    I think this exercise mistakenly treats labor and goods 1-100 as aggregates, and that you can’t determine how well the economy is doing by tweaking them.

    For example, what if, by reducing taxes on labor, and raising them on labor-saving devices, you’re merely making it more expensive for one laborer to satisfy his preference for leisure?

    It’s individual preferences that matter, not the aggregates.

    • Harold says:

      “you’re merely making it more expensive for one laborer to satisfy his preference for leisure?” I am not with you here. Do you mean that the only effect is on one laborer?

      • guest says:

        No, what I’m saying is that, since value is subjective to the individual, you can’t aggregate the effects of tax policy.

        You can only know how much you are benefitting or destroying the economy by reference to the specific individuals who are affected by it, and by what means *aren’t* chosen by these individuals to satisfy their preferences given that their or their neighbor’s opportunity costs have been violently raised.

        And since you can’t know what means would have been chosen in the absense of violently imposed opportunity costs (taxation), the tax numbers aren’t going to tell you anything useful (presuming the goal of taxes isn’t to make government more rich).

        Also, a tax on one good is effectively cronyism for the manufacturers of other goods.

  8. Transformer says:

    Are you aiming to provide the answer to your brainteaser, or was making us think about this an end in itself ?

  9. Silas Barta says:

    So for being so late to this party; it’s one of my key interests, and I detailed my objections to this kind of analysis:


    Specifically, things *are* different when the consumption good throws off a negative externality (or follows a generalization of this phenomenon I can’t quite articulate at the moment.)

    The intuition is this: imagine we’re instead that we’re deciding between:

    Policy A: the government manufactures cell phones and gives them out for free to anyone who asks, with a wait list.
    Policy B: the government manufactures cell phones and sells them at some (pseudo-)market clearing price (perhaps with a little boost for the poor to abstract away from the wealth effects here).

    Obviously, both policies are really bad compared to a real free market. But which is less bad?

    Intuitively, the second, because at least then, people economize on their cell phone usage. The prices match up (more closely) with costs. Pricing signals are more accurate. This remains true even though the second policy is equivalent to a hyper-narrow high-DWL tax, compared to the broad-based tax.

    My intuition on this is that it’s more important for cell phone users to pay *a* price than to “over incentivize” them to avoid cell phone usage, analogous to a carbon tax/TIE “over penalizing” of carbon usage.

    Though I don’t currently have a more formal model.

    • Bob Murphy says:

      OK Silas the TIE stuff is tricky, and I definitely see what you’re saying… But for the brain teaser, I really *was* trying to abstract away from negative externalities.

      BTW I think collectively, the group hit on the main problems with the argument in the OP, which is why I didn’t post a solution.

    • Harold says:

      I don’t think anyone is suggesting that a “correct” carbon tax would not provide benefits from reduced externalities of carbon. It is only that some seem to suggest that reducing income tax would be a boost to the economy *on top of* any such effect. The TIE shows why there would be no such boost.

      If taxing things to reduce income tax were effective, then a cheese tax would be a boost to the economy, as long as the revenue were used to reduce income tax. If such a boost were real, the we should implement a carbon tax even if there is no negative externality from fossil carbon.

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