Saturday, 6 August 2016

Bulk Discounts and Progressive Income Tax

     I've written before in defence of the idea of progressive income tax, and I'll probably be writing more on it in the future. Today I want to advance an argument based on the buying power of money, and how it changes in a non-linear way with the total amount of money you have. In particular, I'm going to make the counterintuitive claim that the millionth dollar you earn is, in a very real way, worth more than the first dollar you earn.
     I realize that sounds crazy. After all, one of the fundamental features of money is that its units are complete fungible: one dollar is exactly equal to every other dollar. If I borrow a twenty from you today, and pay you back with two tens tomorrow, it simply makes no sense to say I didn't pay you back the same dollars; dollars have no independent identity. (Note that I'm talking about the twenty as money, not as a distinct artifact. You can distinguish one twenty dollar bill from another, but you cannot distinguish the dollars they represent.) But bear with me here.
     And I also realize that my claim will sound odd because it seems to be the exact opposite of another argument frequently used to justify progressive income tax, the idea that to a poor person, a 10% tax might be an unbearable burden while it would be scarcely noticeable to a millionaire. A single dollar is more valuable to a person with little money than it is to a millionaire, not less. So what gives?

     Neither of these two points is actually inconsistent with the argument I'll be making here. Yes, one dollar is exactly interchangeable with every other dollar. And yes, one dollar is a much more significant sum to someone with few dollar than it is to someone with many. But my claim is that the buying power per dollar increases the more dollars you have.

     Consider the idea of the volume discount. Say you can buy a chocolate bar for a dollar at the convenience store. If you're just buying one at a time, the price is the same for the rich person as it is for the poor person, and yes, $1 = $1. But if you go to a supermarket, you can probably buy a family-pak of 8 for $6, and if you go to a wholesale club, you can get two dozen for $12. When you buy in bulk, your per-unit price drops, and reciprocally, the number of chocolate bars per dollar spent goes up.
     This doesn't only hold for chocolate bars. It applies to almost all commodities, and even most services, thanks to economies of scale. And even for unique items like rare antiques or works of art or real estate, it is usually cheaper to obtain them if you have a lot of money than if you have just barely enough, because of financing costs; service charges are often waived on bank accounts if you keep above a certain minimum balance.
     Note that whatever the good or service in question, the price of a volume discounted item is always measured in dollars. Carrots become individually less valuable in dollars, the more of them you have, and so do chickens and cell phones and gallons of gasoline. They may do so at different rates, so if you're trading chickens for trucks, you might have to offer more chickens for the second truck than you did for the first. But dollars, as purely a unit of exchange with no intrinsic value, aren't subject to economies of scale and volume discounts, because they're what those volume discounted are measured in.

     So what does this mean for progressive income tax? What I want to suggest here is that in principle, it should be possible to come up with a good estimate of just how much more each marginal dollar of income can buy, and then to set a progressive income tax rate such that the after tax buying power of every dollar earned is equal.

     For example (and I’m just making these numbers up for illustration purposes), suppose one chocolate bar costs $1, 10 chocolate bars cost $9, 100 chocolate bars cost $75, and 1000 chocolate bars cost $500. That means that the average price for a chocolate bar is $1 if you buy one, 90 cents if you buy 10, 75 cents if you buy 100, and 50 cents if you buy a thousand. So, we would set the progressive tax brackets at 10% for income over $8, 25% for income over $74, and 50% for income above $499. Each additional dollar you earn, on this model, is approximately equivalent to one more chocolate bar you can buy.

     Now, in the real world, there are hundreds and hundreds of different things you might need to buy with your income, and the volume discounts for each come in at wildly different rates, so accurately measuring how much your overall buying power increases with income would be fiendishly difficult. I’m certainly not prepared to do the math here, but I hope I have shown that the actual, practical power of a dollar is not a constant; it depends on how many other dollars you have available to use with it, and the more dollars you have, the more powerful each of those dollars is. In practical terms, individual dollars are worth more en masse than they are alone, and I argue that this justifies taxing them at progressively higher rates as income rises. 


  1. A friend, for some reason unable to post a comment here, has emailed me the following:

    "As you know, Tom, I'm in favour of progressive taxation. But I have reached this conclusion based on the argument you allude to in your third paragraph. If I did not already believe that, then I doubt that your main argument would convince me. In fact, it might have the reverse effect.

    I support progressive taxation because a rich person feels less pain when forced to pay a hundred dollars than a poor person would. Progressive taxation is a fair way to distribute the burden equally among the population.

    But I'm not sure why you think it's automatically a good thing to take assets specifically from those people who are capable of getting the most benefit from them. If we had a barter economy, would you take meat away from the carnivores but leave it in the hands of the vegetarians?"

    1. I found myself writing a whole blog post in response to this comment, which is now posted, but I realize I haven't explicitly addressed the one point you raised here about meat and vegetarians.

      Note that in this analogy, you're treating money as an ordinary resource (meat) which is more useful to some people than to others. And yes, money is more useful to some people than to others (my whole point, in fact). But the difference is that while it makes sense to give more wood to the carpenter than to the tailor, because the carpenter can create more wealth from it than the tailor can, giving the money to the rich person instead of the poor one is different. The rich person can buy more stuff with the same money, but that doesn't CREATE more wealth; it just means the rich person can claim more of the already existing wealth.

  2. One of the fundamental ways that money becomes more valuable the more of it you have is flexibility. If I have two dollars and you have one, not only can I buy twice as much as you I can also buy things that you can't - that is anything from $1.01 to $2.00. That makes me more than twice as rich by any practical measure.

    1. That's true, but it's much harder to quantify the value of that flexibility to factor it into the escalating power of money.

      Another flexibility advantage is opportunism; if I have enough to buy twice as much toilet paper this month, I have a longer window in which to take advantage of the occasional sale on toilet paper, instead of being forced to buy it at a high price because I'm about to run out.

    2. The math is harder, yes, but must be acknowledged and in a practical system we can do this as an iterative process. You'd probably have to anyway.

      To the question raised in at the end of the first comment: Barter economies are very awkward for anyone not backed by super AIs that's why we have money. The question raised isn't 'why give meat to the vegetarians' but really 'why restrict the rich'. We restrict them because untrammeled power tends to overrun everything else in it's vicinity.

      Remember that money itself is a tool, not an intrinsic fact of the universe. It's a powerful tool, but that requires that it be regulated and controlled to limit the damage it can cause. Part of that regulation can include progressive taxation.

    3. I think you've understood my argument very well, Lachlan. I have not, however, in this argument talked about progressive tax as a way to "restrict the rich" (although in a limited sense you could describe it that way). And while reining in the power of the extremely wealthy is an important concern, it's not actually the argument I'm making HERE.

      There is nothing in the present argument to say that people shouldn't be able to become extremely wealthy. Rather, the argument is that from society's perspective, we should reward each person with the same amount of wealth for every dollar of income they earn. Without a progressive income tax system, rich people get far more wealth per dollar than poor people do.

    4. A solid point on the wealth per dollar issue.

      And I am not going to argue that people shouldn't be able to become very or even extremely wealthy.

      Now I'm going to think about this some, and pick at it before I take another swing at it. Possibly with some references, probably with book suggestions.

    5. OK, I'm back, having had thoughts.

      I actually didn't understand you. Not exactly.
      My concerns were not for the wealth inequality on a per dollar basis. I was worried about income inequality as a result of wealth, and that's a different but entangled problem.

      And with that I will recommend 'Systems of Survival' if you haven't read through it already and move on to the next post after I get home from work.

    6. I'll look for it. Thanks.

      You're right about income inequality itself being an entangled problem, but I kind of suspect that a progressive income tax that aims for constant purchasing power of after tax dollars would largely do away with the worst distortions of income inequality.