02 Aug 2012

My Two Cents on Caplan’s “Proof”

Bryan Caplan, Economics 26 Comments

[UPDATE below.]

I saw this post from Bryan Caplan when I was jet-setting to Vegas, and thought something was really wrong with it. But I haven’t had time until now to write it up. Here’s Bryan:

When economists explain marginalism, students often object, “But surely no one ever changes his behavior over a single penny.” However, they’re provably wrong. If “no one ever changes his behavior over a single penny,” raising your price by a penny automatically increases your profit. So does raising your price by another penny. And another penny. And another penny… Any firm could earn infinite profit by sequentially raising its price, one penny at a time. Since this absurd, the premise must be wrong. People can, do, and must occasionally change their behavior over a single penny.

Wait, there’s more. On the typical day, most firms don’t raise their prices. Given the plausible assumption that firms want to make as much money as possible, we can infer that every firm expects that raising any price by a penny would lead to lower profits. This is only possible if every firm expects that raising any price by a penny would change some customers’ behavior.

The lesson: Behavior that responds to a one-penny change isn’t just a theoretical curiosity; unless price-setters are deeply mistaken about their own markets, behavior that responds to a one-penny change is all around us, always has been, and always will be.

I don’t know who originated this one-penny proof. I suspect authorship is lost in the sands of time. No matter. The author, whoever he may be, created something great: A simple, timeless, and virtually bullet-proof argument about all human behavior and pricing decisions. How many papers in the latest AER can claim anything remotely comparable?

Sorry Bryan, but not only am I not impressed by the rigor of this demonstration…I don’t even think it’s right.

Bryan is assuming his conclusion, without apparently realizing it: Critics challenge the notion that people (they have in mind consumers) might care about changes of one penny, and Bryan responds with an argument that assumes people (he has in mind producers) care about changes of one penny.

Look, the quickest way for me to make my point is to replace every occurrence of “one penny” in Bryan’s argument with “one-quadrillionth of a penny.” Did I just prove to you that everyone in the economy is acutely sensitive to price, down to the one-quadrillionth of a penny? Are market prices what they are right now, and not one-quadrillionth of a penny higher, because there is at least one person who would lower his purchases at that higher price?

If my point above seems too cute for you, switch to a more straightforward claim: I personally round off to the nearest dollar when I fill in the tip at a restaurant. In other words, I will calibrate the tip such that the final tally is a whole dollar amount. It’s definitely not the case that I would have ordered more or less food, depending on if the price of the entree is $12.99 versus $12.98.

In some markets, pennies matter. For example, when people are shopping for gasoline, they definitely pay attention to the posted price per gallon, down to the penny. But when someone is in the market for a new SUV, I don’t think pennies come into play at all.

Note that I’m not challenging Bryan’s basic conclusion; I’m not making an argument against marginalism per se. I’m just saying I don’t find Bryan’s “one-penny proof” very compelling.

UPDATE: Kavram in the comments write:

Skylien got it right. If marginal pennies were truly inconsequential the fact that so many market prices (notably installment payments for infomercial products) end with .99 would be one of the greatest coincidences known to mankind.

Actually, this is evidence in my favor, and massive evidence against Bryan. If prices were set by “rational” forces of supply and demand, you would expect an uniform distribution of trailing digits. In other words, 1% of the goods and services should end with a price of .00, another 1% should end with .01, etc.

The fact that in some markets, prices end with .99 means that something besides Caplan’s mechanism is at work. Yes, in a tautologous sense, the producers of infomercial products charge $19.99 because they think that if they charge $20.00 their sales will go down. But, if they charged $19.98 their sales would go down too! Worse yet, if they charged $19.72666 (and they expressed it like that, in the ad), then they would probably sell zero units because the average late-night viewer would be freaked out by that.

So, did I just prove that demand curves slope upward? Is this a Giffin good, perhaps? I mean, here we have a case where the seller makes more profit, by charging a higher price.

To reiterate, I am not claiming that market prices are random. Yes, they are what they are for a reason; human beings consciously agree to terms, and so market prices must “come from” humans somehow. What I am claiming is that these prices aren’t the simple product of intro-level profit and utility maximization, in the way Caplan believes. If he tells his students, “All prices in a market economy are set at that exact level, such that a one-penny increase in price would cause at least one buyer to fall out of the market and thus lower profits,” then I agree with any student who says, “Are you nuts?!”

26 Responses to “My Two Cents on Caplan’s “Proof””

  1. Sealander says:

    Is this different than when Krugman asked if John Wheeler-Dealer actually produces $10,000 more if he works an hour more? Seems analogous, which you resolved with calculus.

    • Bob Murphy says:

      Sealander, Krugman had made a claim invoking textbook price theory, which itself uses calculus. So that’s why my initial response to him involved calculus.

      Then, by reader request, I later did a post with discrete units.

  2. skylien says:

    Prices very often are set at psychological important points. This advantage is lost if you raise it even by one penny even if people wouldn’t mind one penny. I mean 4.99 is more appealing than 5.00 and 5.01 etc…

  3. Richard Williamson says:

    Seems to me the argument is pretty simple: It is indisputable people will buy more gasoline (say) at $1 than $10 a gallon. If you had a lot of time on your hands and started increasing the price in increments of one penny and see how much gas gets sold, there must therefore be at least one point where a price movement induces people to buy less gas. But strictly speaking all the argument proves is that ‘some pennies must matter’, it is not necessary (and almost certainly untrue) that all pennies matter.

  4. kavram says:

    Skylien got it right. If marginal pennies were truly inconsequential the fact that so many market prices (notably installment payments for infomercial products) end with .99 would be one of the greatest coincidences known to mankind. Also marginal pennies on stock prices can mean a great deal, from what I know they’re sometimes measured down to the 1/8th of a penny.

    Pennies don’t matter for *all* transactions, but they certainly matter for some.

    • Bob Murphy says:

      Pennies don’t matter for *all* transactions, but they certainly matter for some.

      So I am right, and Caplan is totally wrong. Thanks for the support, kavram.

      (But, your argument about 99 cent pricing proves the opposite of what you think it does, vis-a-vis Caplan. Details forthcoming…)

  5. Strat2131 says:

    There are several faults with his line of thought.

    Businesses face 1) bounded rationality, and 2) menu costs.

    Meaning 1) businesses could know that raising prices by a penny will improve there revenues, but don’t find it in their interest to change their menus, update there accounts etc.

    2.) There also unaware of the marginal buyers corner decision (To buy or not to buy,) and do to the extremely limited increase in returns, it becomes risky (offending a customer etc, losing a client.)

    To wrap it up; most individuals do not care about a penny, that is because very few have their corner decision on a penny, and that is just based on probability. Although its hard to put down due to the fuzziness of being “indifferent” I guarantee you a penny will matter at 2.99, 3.01 and 3.51 for certain objects (gas, vending machines, tolls etc.)

    Play the penny game long enough and demand slowly falls to zero. (usually in jagged increments, where 3.01 3.24 have little impact and 3.24 > 3.25 has a large impact. etc. Again, raising prices by a penny seems like a full proof plan because the sample size would need to be sooo large to recieve a signal.

  6. Ben Kennedy says:

    What about minimum wage? In these circles, the general argument is “sure, nobody loses their job immediately over a wage increase of X. But over time, the market notices and these people are no longer empliyable”. Also, we generally hear the same reducto ad abusdum, “If an increase of N is good and doesn’t distort markets or create unemployment, then surely an increase of 100 X N would be even better”. Are these standard arguments regarding the effects of marginal wage increases distinguishable from Bryan’s argument concerning marginal prices?

    • Major_Freedom says:

      That’s a good point Ben. But the “surely an increase of 100 X N would be even better” argument is not the main premise “in these circles” for why minimum wage causes unemployment and why we should abolish it.

      The “surely an increase of 100 X N would be even better” is more a rhetorical jab at minimum wage supporters, to put them in an uncomfortable position, hoping that they will be convinced otherwise. It isn’t an official Austrian based argument, even though you see quite a few Austrians making that jab. It’s a flawed means of getting to the same conclusion “Minimum wage laws BOO!”

    • Bob Murphy says:

      Ben Kennedy wrote:

      Also, we generally hear the same reducto ad abusdum, “If an increase of N is good and doesn’t distort markets or create unemployment, then surely an increase of 100 X N would be even better”. Are these standard arguments regarding the effects of marginal wage increases distinguishable from Bryan’s argument concerning marginal prices?

      It is amazing to me how you guys are assuming the very thing under dispute…

      If the government says, “We want to raise the minimum wage by one-quadrillionth of a penny per hour,” I would say, “That will have no effect on the unemployment rate.”

      Then Ben, you come along and say, “But Bob, that means you can help workers with no downside!”

      To which I respond, “No, workers don’t care about getting an extra one-quadrillionth of a penny per hour, either. There is no ‘help’ to speak of, here.”

      Now maybe an actual penny increase will matter, maybe it won’t. But Bryan’s “proof” certainly isn’t the way to go, in establishing this. It definitely doesn’t hold in general.

  7. Daniel Kuehn says:

    Exact;y right. Glad there’s a Bob post up I can agree on.

    And as for the point about not abandoning marginalism, all you really have to do is incorporate search costs and psychic costs into the costs considered by the consumer. Once you do that you’re pretty safe going back to that naive version of marginalism that says every penny matters everywhere. The point is, a penny change in the price may not matter everywhere if search costs are sufficiently high (which, to bring it back to Bryan, does gives firms some monopoly power that can increase their profits).

    But there’s obviously a limit to that dictated by search costs. It doesn’t allow for this obviously ridiculous “and another penny, and another penny” stuff.

  8. RPLong says:

    Bob,

    The reason it’s not a compelling proof is because it’s merely a reapplication of the Paradox of the Heap, which Daniel and I were discussing the other day.

    I think the key problem is that pennies are merely fractions of consumption in this example. Nobody changes his mind over a penny, unless that penny makes the difference between consuming the next best opportunity cost or not. For most of us, a single penny doesn’t determine whether we can afford that downtown condo or that nice sportscar, so who cares?

    But for some people, that penny “rounds off the whole.” It makes the difference between a whole sportscar and not-quite-a-sportscar. Here I think Rothbard’s view of it is useful: The opportunity cost is a consumptive whole unit, not just a fraction of a unit. I think that’s the core problem with assuming perfect mathematical continuity in economics models.

    • Major_Freedom says:

      Ah, RPLong made this comment as I was typing my comment. He is thinking the same thing as me.

  9. Major_Freedom says:

    I think the “one penny proof” is not even a one penny proof, but a particular collection of pennies proof, or a dollars and cents proof.

    For just look at the form of the argument being made:

    “When economists explain marginalism, students often object, “But surely no one ever changes his behavior over a single penny.” However, they’re provably wrong. If “no one ever changes his behavior over a single penny,” raising your price by a penny automatically increases your profit. So does raising your price by another penny. And another penny. And another penny… Any firm could earn infinite profit by sequentially raising its price, one penny at a time. Since this absurd, the premise must be wrong. People can, do, and must occasionally change their behavior over a single penny.”

    But if we assume a seller did this, then we’re no longer in the context of a one penny increase in price, we’re in a context of prices being raised by multiple pennies!

    The context of those who say people don’t care about paying one more penny, are those cases where the price is raised by only one penny…and no more. As soon as Caplan said “So does raising your price by another penny”, he switched context to “What would happen if you raise your price by two pennies?”. As soon as Caplan said “And another penny”, he switched the context to “What would happen if you raise your price by three pennies?”. As soon as Caplan added the second “And another penny”, he switched the context to “What would happen if you raise your price by four pennies?”.

    It is not wrong to say that at the margin, someone can be apathetic towards paying one more penny. Murphy’s restaurant tip scenario is a good example. If the bill comes out to $45.27, then I would tip so that the bill ends up at $50.00. If the bill came in at one penny more, at $45.28, then I would still tip so that the bill come out at $50.00. This is the context at hand. It would be wrong for Caplan to then say “Oh yeah? Then the logic of your argument means that if the restaurant charges you $45.28, and then $45.29, and then $45.30, and then $45.31, and so on, heck they could even charge you $500,045.28 and it shouldn’t matter you! Thus, your claim that you don’t care about a single penny is wrong, and you DO make choices based on one penny. You’re just too stupid to see what I know you’re thinking.”

    Um, sorry Caplan, but that is changing the context. The context is a one penny increase, not a $500,000 increase.

    ————————————–

    What I think is going on here is Caplan is jealously guarding his neoclassical cardinal mathematical interpretations of utility, as a means to tacitly jab at the alleged incompleteness of Rothbardian interpretations of utility. Remember “Why I am not an Austrian”? One of the reasons given was Rothbard’s alleged “mistake” of inferring that neoclassical economists actually believe that utility is cardinal. Caplan said no, that’s not true. They call recognize that utility is ordinal, they just use mathematical calculus to “represent” ordinal utility. As long as the transition is monotonic, it’s fine.

    Well Caplan, this is a great example of why Rothbard’s interpretation of utility is SUPERIOR to the neoclassical mathematical calculus “representation” of ordinal utility. For I myself interpret utility along Rothbardian terms, and I suspect that the reason why you just casually added penny after penny, is that this is where mathematical calculus reasoning takes you. You are mentally forced into insisting, beyond all reason, that humans value and make choices at margins of infinitesimal changes. That is why you believe people MUST be taking into account penny changes. For if we “plug one penny increase” into the mathematical utility equation, then because it’s monotonic, the output should yield a utility that represents actual human utility.

    HUMANS ARE NOT MATHEMATICAL AUTOMATONS.

    Rothbard and other Austrians use ordinal scale utility frameworks like

    1. $50.01

    2. Dinner for two at Denny’s

    3. $45.26

    is because Austrians base utility on the choices that humans actually make, not what mathematical formulas say they OUGHT to make.

    In the neoclassical cardinal interpretation, we are forced to conclude that people make choices based off of one penny changes. That is why Caplan had to switch the context to two, three, four, 500 million penny increases at a time to “make his point.”

    To paraphrase Caplan, this is (one reason among many) why I am not a neoclassical, and why I am an Austrian.

    The way Austrians think is much closer to how people actually behave in the real world, rather than some nice mathematical representation where applying it to the real world compels us to insisting regardless of the evidence, that humans make choices based on one penny differences.

    ——————–

    One caveat to the above. I do not want to convey the impression that humans NEVER make decisions based on one penny price changes. After all, there is a good reason why we see so many goods priced at $XX.99, rather than $XX.99 + $0.01. The one penny increase from $XX.99 to $XX.99 + $0.01 does affect people’s behavior. But no worries, for Rothbardian ordinal utility has no problems taking that into account. Rothbardian utility is not compelled to insist that people don’t make choices based on every one penny change. They may make choices on one penny, or they may not. It depends on the CONTEXT.

  10. Bob Murphy says:

    I agree with RPLong and MajorFreedom on this one. They are expressing exactly my POV. For Austrians in particular, it should be clear that the relevant unit of analysis is defined subjectively by each individual. For some people and some transactions, the unit is a penny, for others it is even less than a penny (stock market transactions maybe). But Bryan’s “proof” doesn’t work at all. You can’t prove that people care about a penny, by showing that at some point they will care about massive multiples of pennies. At best, this is a dubious philosophical conundrum; it’s not the obvious “aha!” that can be invoked to solve an economics problem without even acknowledging how dubious it is.

  11. Blackadder says:

    Bob,

    I disagree that Caplan is assuming his conclusion here. The question is not whether producers care about an extra penny, but whether they care about making an extra penny per transaction. Given the large number of transactions involved, we’re talking about a lot of money, and it makes sense to ask whether businesses would really just leave all that money on the table.

    • Bob Murphy says:

      BA in some markets, yes. You honestly think if I go down to a car dealership, the sticker price is what it is, because if they charged one penny more, their sales would go down by at least one unit? (And because it’s “too expensive” to the person, not because of psychological reasons for preferring round numbers in the price?)

      • Blackadder says:

        Bob,

        The car example is complicated because there is haggling involved.

      • Blackadder says:

        In any event, my point is that Caplan’s argument wasn’t question begging. Caring about a penny per transaction isn’t the same as caring about a single penny.

  12. Blackadder says:

    the producers of infomercial products charge $19.99 because they think that if they charge $20.00 their sales will go down. But, if they charged $19.98 their sales would go down too!

    Why do you think sales would go down if they reduced the price from $19.99 to $19.98?

    • Bob Murphy says:

      Because it looks funny.

      • Blackadder says:

        Bob,

        Is this an effect you have evidence for, or are you just assuming that’s the case?

        If products are priced at $19.99 instead of $19.98 I would think it would be because this maximizes profits not because it maximizes sales.

    • Major_Freedom says:

      I think it’s an illustration of neoclassical price determination that sellers charge X to maximize profits (marginal revenues=marginal costs, etc), where one penny more or less would lead to lower revenues and hence lower profits.

      For me, I didn’t get from that quote that it is Murphy’s own view. I think it is clear from his emphasis on prices being subjectively determined, and not according to rigid “rational” utility formulas, that he is actually criticizing the neoclassical view due to where it takes us.

  13. Kevin F. says:

    The world of direct response marketing is a GOLDMINE of information on real world pricing strategy and consumer behavior. These folks test, test and test some more to find ways to increase the response to an ad campaign by just a fraction of one percent. That can add up to big bucks when you’re mailing a million pieces every week. I would highly recommend reading anything by Dan S. Kennedy, who is a genuine expert in this field, having worked in direct response as a consultant, copywriter and producer of infomercials for decades. This book is a good start:

    http://www.amazon.com/B-S-Price-Strategy-Prisoners-Prosperity/dp/1599184001/ref=sr_1_8?ie=UTF8&qid=1344008481&sr=8-8&keywords=Dan+S.+Kennedy

  14. Silas Barta says:

    I stopped reading here because of the major oversight:

    Given the plausible assumption that firms want to make as much money as possible, we can infer that every firm expects that raising any price by a penny would lead to lower profits. This is only possible if every firm expects that raising any price by a penny would change some customers’ behavior.

    This doesn’t follow at all, simply because of menu costs. Even if the penny price increase would change *no one*’s behavior, it may cost more than the increase in revenue to change that one price. Less trivial examples might include a one-off business deal where one party has pre-committed that it would sell at X. Even a mere increase to X + $0.01 would have severe costs, even if (prior to the deal at least), it would have been worth it to pay that much.

    • Major_Freedom says:

      I can’t wait for digital menus.

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