Bryan Caplan  

A One-Penny Proof

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I recently tweeted:
In social science, the best arguments prove more than the best studies. Hands down.
Here's one homely example of what I have in mind.

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?
 

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COMMENTS (45 to date)
Jon Riegel writes:

Doesn't this argument assume that the cost to a business of physically instituting a price change is very low? While this may be the case for amazon, it is false for many firms.

Alex J. writes:

When I see a penny on the ground, I pick it up.

Vacslav writes:

Surely the one-penny argument is wrong: when a meaningful sale is announced it's always 10%-30%-50% sale, not a one-penny less sale. Nobody rushes to buy more jeans if its price is suddenly $49.98.

Any economic theory that deals with definite prices for which infinitesimal changes are meaningful is counter-realistic. Everybody knows that supply-demand curves do not exist, and prices are fuzzy. Many prices exist at any given time - that's why people and enterprises "shop" for a better price.

Kevin Kyaw writes:

Most people don't change their behaviour on single penny differences. What they do is remember what the price was some time in the past, and when the difference between the price then and the price now becomes signifcant then they change their behaviour.

For example, something is priced at $1.00.

Next day, the price will be increased to $1.01. People won't make a different choice.

Next day, the price will be increased to $1.02. People won't make a different choice.

So on until..

Next day, the price will be increased to $1.05. Now some people reduce consumption as they remember it used to be $1.00 and the $0.05 is significant enough to change their choice.

Your "proof" is wrong because there is a hidden assumption that people only think in terms of now and the moment before now. Their decision making is more complex than that.

Kevin Kyaw writes:

The other thing to consider is, if one-penny decision making were true, the prices of products would be so much more precise than they are. There would be products sold at $23.32, some sold at $173.82, some sold at $12.41 and so on. That doesn't happen in the real world. People only respond on significant changes in price and so firms set prices accordingly. What is considered significant is subjective of course but it is certainly not one penny.

George writes:

There are threshold effects caused by the way people perceive numbers. That's why there is a big effect at xx.99 to xx+1.00. However, there is a miniscule effect, if any at xx.98 to xx.99.

The effect, then, is price dependent and has step discontinuities in it, instead of being a smooth continuous function.

Your basic argument is based upon the idea that if you see differences of behavior with big differences in price, then you can logically regress the effects to single penny steps. This does not always follow, but occasionally does.

I ran a retail business for 14 years and had plenty of opportunity to observe the effects with tens of thousands of items.

In short, there are non-linear effects caused by the way we read and think about numbers. Your basic argument holds often, but is not a theorem, or incontrovertible.

Robinson writes:

I agree with the spirit and conclusion of the argument but I don't find it completely convincing.

Replace the price of a good with the number of grains of salt on a meal. Is there one grain of salt that would transition the meal from "not too salty" to "too salty?"

batman writes:

@Vacslav
'that's why people and enterprises "shop" for a better price'

you just proved the authors point. Yea for a penny less no one will buy the jeans because of that penny. Say electricity is 20 cents per watt and they changed it to 9 cents and after a while you notice the drop in price you use more electricity or stop caring about conserving it. Now look at those jeans if they went down 1000 pennies people might decide to buy them. But some companies deliberately keep output below equilibrium because they in effect have a monopoly on their product - i.e. tommy hilfiger, or gucci bags. It's a brand name. Keeping it below the intersection of S & D means they get to sell it for a higher price and still make more profit as opposed to lowering the price on every single tommy hilfiger jean to increase sales.

batman writes:

In the previous post i meant 10 cents per watt dropping to 9 cents.

And with regards to shopping around, people shop around for deals but that doesn't mean a 1 cent drop on $50 jeans is worth the gas or time or search cost to even notice or care about. 1000 cents might make a difference for sure. a 1 cent drop in watts of energy doesn't require shopping around but the consumer will notice it and eventually change his behavior to meet that shift in price.

Jim Glass writes:

Nobody rushes to buy more jeans if its price is suddenly $49.98.

Retailers everywhere don't price things at $X.99 ($X.98 at WalMart) because the penny doesn't matter.

Retailers do more empirical testing than all the science faculties in the USA combined and multiplied several times over. They know how much that penny matters.

Early in my career I worked for one of the most successful direct marketing firms in the country and I saw those test results -- believe me, that penny matters plenty.

students often object, "But surely no one ever changes his behavior over a single penny."

Monitor those students as they make buying decisions and check *their* behavior.

Anton Sherwood writes:

Congratulations, you've rediscovered the sorites paradox.

João Neto writes:

Using that line of reasoning, we could conclude that a reader may decide not to read a book if it has an extra letter (just keep adding letters until the book has enough size). And that does not make any sense.

George writes:

I believe that the required assumptions are that the demand function be 1) monotone de(in)creasing, and 2) smooth and everywhere continuous and differentiable. The first assumption is valid, but the second one is violated. That is not to say that there are price regimes that exist where demand changes by the penny, but there are many factors in customer behavior which can make the demand function flat for a while.

This means that you cannot, in general make a logical argument which proves the point made by Dr. Kaplan. It involves assumptions which have to be shown to be everywhere true. The second assumption is not shown to be proved. The function might be stepwise differentiable, but there may be disontinuities. Also, it hasn't been proved that the function has no zero derivatives - i.e. regions where demand does not change with price.

I'm not arguing that the case does not exist, but that it is not necessarily universal.

George writes:

BTW, Dr. Kaplan would seem to be correct in securities markets where it can be plainly seen that transactions occur due to fractions of pennies. Retail sales is a different matter.

simon... writes:

You don't have to go any farther then a local gas station to find a proof that apparently 0.01 penny makes a difference.

Norman Pfyster writes:

Simon beat me to my answer...apparently, none of the objectors have ever bought gas for their car (or ever done price comparison shopping).

Curtis writes:

@João Neto

Using that line of reasoning, we could conclude that a reader may decide not to read a book if it has an extra letter (just keep adding letters until the book has enough size). And that does not make any sense.

A few points:

1) People rarely know the letter/word count of the book they are about to read (although perhaps some people can estimate the word count simply by looking at the size of the book and its format). However prices are usually known in advance, and when they're not they typically can be haggled. Such determinacy matters.

2) Adding letters, which have individual significance that varies depending on the order in which you add them, to a book is not the same as adding pennies, which are uniform and fungible, to the cost of a thing being purchased.

3) Some people do avoid books because of their size, and the point at a book is too long varies by person. It may be difficult to predict that point, but it occurs, and for those people there is a marginal value.

4) Other people would get more value out of having a longer book -- i.e., when they're done reading a particular book, they wish it would continue. Adding letters/words in a meaningful way would make such people value the book more. Simply increasing the cost of a thing, however, doesn't necessarily make people value that thing more.

Phil writes:

I've thought of this argument myself, and never used it. I coulda been famous!

My version goes something like this: how many fewer Camrys would Toyota sell if it raised its price by a penny?

Last year, they sold around 300,000. Let us assume that, if they doubled the price from $30,000 to $60,000, they would sell zero. (The argument doesn't change much if you assume they'd still sell a few.)

So, a $30,000 increase results in 300,000 fewer sales. Each penny increase, then, on average, is 0.1 fewer cars sold.

But, someone might argue, that's only the average. Not every penny is equal. It's not the FIRST penny that matters ... it only matters around, say, $38,000.

But ... that can't be right. Not just because Toyota would immediately raise their price to $38,000, but because you've got to think that the early pennies matter more than the late pennies.

In any case, you have to admit that, if you draw a graph of sales between $30K and $60K, it starts at 300,000 and drops to zero. You can posit a graph that stays flat for a while before dropping -- but it MUST drop. The average still has to be 0.1 car per penny.

If you want to insist on zero, the best you can do is argue WHERE a penny increase makes sales drop, not IF a penny makes sales drop.

egd writes:

A brief real world example:

There are two sandwich stores near my office, one we'll call "Metro" and the other "Landrace".

Metro used to have a deal where I could buy a sandwich of a specified length for $5. They have two sandwiches I like.

Landrace likewise has two sandwiches I like, they cost $5.70 and $5.30.

I could go to either, but I usually went to Metro (because it was cheaper) and decided what type of sandwich I wanted once I got there.

Eventually, the first sandwich I like at Metro went up to $5.50, the second stayed at $5.00. This meant that I had to decide before going out which type of sandwich I wanted, giving Landrace about 50% of my business (from almost 0)

Then the second sandwich went to $5.50. Landrace's $5.70 sandwich was better than either of Metro's sandwiches (although I wasn't willing to pay $5.70 for a sandwich before). It's no longer a better deal to go to Metro, so I go to Landrace almost every day.

Metro, by simply increasing their price by $0.50 (not a penny, but still fairly small), has lost nearly all of my business.

Small changes in price do lead to large changes in behavior, because it makes the competition more attractive.

Chris K. writes:

Bryan's post is spot on and the logic is sound. He draws two conclusions:

1) People can, do, and must occasionally change their behavior over a single penny.
2) Behavior that responds to a one-penny change isn't just a theoretical curiosity

These are absolutely true. Most of the disagreements in the comments above just boil down to arguments over elasticity. I.e., they are a matter of degree. Sure, there are times when demand is inelastic, but Bryan doesn't argue otherwise. Also, the fact that S-D curves are discontinuous doesn't change the fact that, at some point, a delta of one penny will create a delta in demand (unless demand is perfectly inelastic...does that ever happen?). Bryan doesn't state that every penny matters, just that pennies do matter sometimes.

The broader point, though, is that thinking at the margin does matter and it is the right way to think about things, and this provides an easy to understand example of why.

I actually wouldn't expect to find any disagreement to this from readers on an economics blog. In the comments on Yahoo!, maybe, but not here...

Andrew writes:
I actually wouldn't expect to find any disagreement to this from readers on an economics blog. In the comments on Yahoo!, maybe, but not here...


Spot on

Mike writes:

=============================

There would be products sold at $23.32, some sold at $173.82, some sold at $12.41 and so on

=============================

- Ebay
- NYSE/NASDAQ/etc.
- Gas stations

The effect of penny sized changes can be easily witnessed at any of the above 3. I suspect that the effect is less pronounced as the percentage change that the penny represents decreases.

yet another david writes:

Alex:

When I see a penny on the ground, I pick it up.

Me too.

BTW, Bryan, you're sounding a little Austrian here (methodologically speaking). A good thing.

BLM4L writes:

I think the students are wrong for a more fundamental reason: since demand is ordinal, and money is divisible indefinitely, there is always going to be some lumpiness.

Also, some people really do trade at a penny, or even a fraction of a penny. Think commodities like wheat or copper. Also, remember when stock used to be priced in sixteenths of a dollar? Now it's priced in pennies.

Costard writes:

The argument may be correct - it would depend upon the context - but the logic isn't.

The fact that between your house and mine is a contiguous set of properties, doesn't make us neighbors. The difference between a cat and dog is not a certain number and order of molecules, but the fact that one meows and one barks. The world people inhabit is one of ideas, and associations made from perception, not mathematical quantities; thus social science is distinct from physical science.

"Price too high" is a judgment made by a specific person at a specific time with regards to a particular transaction, weighed against the alternatives. Neither you nor I know the process by which different people arrive at such a decision -- but I really doubt that it involves a calculation of the marginal utility of pennies. It is unknowable - it is a given - and whatever you might say about it there is absolutely nothing you can "prove". If people were calculators they might act one way, and if they were carnivorous plants they might act in another, but these speculations do little to inform us about actual human behavior.

The point being that all tools - logical, empirical or otherwise - should be used responsibly. IMO economists are among the guiltiest in presuming to know what they can only guess, and the most irresponsible in using their "knowledge" to affect public policy. Social science imposes limitations - unknowns that begin with the individual - and these should be respected.

Vacslav writes:

@Costard writes "The point being that all tools - logical, empirical or otherwise - should be used responsibly."

Precisely. Straightforward, textbook economics requires suspension of disbelief to accept concepts like S and D and equilibrium prices and quantities. These concepts do not correspond to anything in reality where both prices and quantities are uncertain and defined non-locally both in space and in time.

Equilibrium prices and quantities did not exist in the 1800s and they even less likely to exist today. That's why the one-penny issue is moot.

Andrew writes:
The difference between a cat and dog is not a certain number and order of molecules, but the fact that one meows and one barks.

A cat is a cat and a dog is a dog BECAUSE of the order of molecules.

A dead cat is still a cat.

kyleN writes:

I think Dr Kaplan is correct because of marginalism.

Price changes occur at the margins.

We do not perceive them for many items because the cost of those items are large. But they become more noticeable on smaller items.

But just because we perceive them differently it does not mean they do not exist.

Carl writes:

Sounds a bit like Zeno's paradox of the arrow only substituting money for time and demand for position.

Sam Grove writes:

Isn't the nickel the new penny?

Used to be a penny was made of something fairly valuable.

Lots of prices are (dollar value) - (one cent)

Henry writes:
In any case, you have to admit that, if you draw a graph of sales between $30K and $60K, it starts at 300,000 and drops to zero. You can posit a graph that stays flat for a while before dropping -- but it MUST drop. The average still has to be 0.1 car per penny.

If you want to insist on zero, the best you can do is argue WHERE a penny increase makes sales drop, not IF a penny makes sales drop.

Spot on. I have used a similar argument against someone who claimed that lottery tickets can be a good buy because the cost is trivial but the rewards can be life-changing, contending that there were "steps" in most peoples' utility functions. But if that's the case, we should be able to identify approximately where these steps are, and I haven't seen any attempts to do so.

James A. Donald writes:

simon... writes:

You don't have to go any farther then a local gas station to find a proof that apparently 0.01 penny makes a difference.

What makes gas special, and ship maintenance services special, and several other areas of human activity special, is that the extra cost in choosing one provider rather than another is often very low, so the customer may well act, and very often does act, for a difference that is not merely a penny, but a small fraction of a penny. In other areas the barrier to action is larger, so the curve may well be non smooth or discontinuous, hence the number of items priced at x.98 or x.99.

Chris K. writes:

[disclaimer] I don't follow the comments regularly enough on this blog to know if Vasclav is just a troll who should be ignored, so I apologize to the others for potentially feeding him....[/disclaimer]

@vasclav I am not a theoretical economist. I am a business man. A finance guy working in industry. I work in commercial aviation. And I can tell you that, practically speaking, S-D curves and equilibrium pricing are so real that once the airlines understood the concepts and incorporated them into revenue management models, the whole game changed. There are small armies of fresh undergrads working in revenue management groups in airlines across the world the live and breath S-D curves every day. They change a price (or "close a bucket") and they can literally watch the demand change. Why do you think airline ticket prices are so complicated? Because airlines have figured out how to "work" that S-D curve that you think is imaginary.

The concepts weren't invented, they were discovered, much like gravity. You can fight them all you want, but they'll exist whether you believe in them or not. Just like gravity.

Again, this isn't theoretical, this is in practice, observable, empirical, and fairly well understood. As I said in my prior post, you can have a logical discussion as to the degree (the elasticity), but to tout that these concepts "do not correspond to anything in reality" makes me wonder what kind of reality you live in. I suspect Hugo Chavez shares your reality.

I'll apologize in advance in the case that I misunderstood your meaning, but people who believe that S-D curves where invented by rich people to exploit poor people is a pet peeve of mine.

Piet le Roux writes:

Incredible. Judging by the comments there's a significant counter marginalist revolution abrew. No marginalism, however, means economic science goes back to the dark ages.

Andy writes:

Buying online is another situation where $0.01 makes a difference. For example on Amazon Marketplace it's common for book prices to cluster around some number, with one being slightly lower. Sometimes this is by only 1 cent.

RJB writes:

What a fun argument. it also proves that no market is perfectly competitive because a single trader influences price.

Also, sorites.

mark writes:

I wouldn't call this a proof. The so-called proof simply assumes an action that is opposite to the proposition it seeks to refute.

The proposition it seeks to refute is "no one ever changes their behevior over a single penny".

The "proof" consists of assuming ab initio that someone has raised prices - changed their behavior - by a single penny. That's just assuming the opposite of the proposition under review.

awp writes:

The confusion here is being caused by transaction costs.

Depending on the good and the location of their sales points I often appear to leave many many pennies on the table. As anecdote consider my recent shopping adventure for a tie.

At store A I found a suitable tie for $12. Mistakenly believing that the market rate for ties was actually $10, I then travelled to store B. At store B I found the same tie for $12.50, which I then bought. So then it turns out that my transaction costs for traveling between stores was less than $2 and more than $.50, and also thus appears that the $.50 "doesn't matter".


On the other hand with two gas stations on opposing corners, you can be sure that I am going the one that is one cent cheaper.

epignosis writes:

Stuff is not priced at $1.00, usually. It's priced at 99 cents. There is a reason why.

A car may be priced at $22,999.

So for the merchant in the supposition, the price change is not likely to be 1 cent, is it?

Moreover, the proof relies on a sequence of price changes to prove that a single price change matters so much. Certainly, the accumulated change of the sequence matters, but this provides no substantiation that a single change produces the same result? Failure of logic here!

Adam writes:

Here is why the one-penny proof is unsatisfying to me:

Let us remake Zeno's paradoxes in the context of this discussion. Let's say that someone has made the argument "No one can discern movement of one picometer with the naked eye." You now turn around and argue "if that were true, I could move one picometer at a time and pile on one picometer movements until I crossed vast distances, and you would never see any of the movement. Since all movement is subdivisible into picometer units, all movement, by your logic, must be invisible to the eye."

Your one-penny example is like taking the latter class of argument as proof that we can see one picometer movements with the naked eye, after all. Which...is absurd.

Chris writes:

The reason so may prices are $x.99 is precisely because people respond sharply to that last penny. I remember a study about pizza where demand dropped sharply when prices were raised by a penny to an incremental dollar amount.

Of course, this impact of the penny seems to be driven by psychology, not rational utility maximization. In any case, it's there.

David Jinkins writes:
In social science, the best arguments prove more than the best studies. Hands down.

Just another way of putting the old argument that all real advances in the sciences are theoretical. Empirics tests theory and motivates new theory.

RPLong writes:

Not so fast, Bryan. Ever heard of the Paradox of the Heap?

Major_Freedom writes:

[Comment removed pending confirmation of email address. Email the webmaster@econlib.org to request restoring this comment. A valid email address is required to post comments on EconLog and EconTalk.--Econlib Ed.]

Major_Freedom writes:

Here is my response to Caplan's post over at Murphy's blog:

http://consultingbyrpm.com/blog/2012/08/my-two-cents-on-caplans-proof.html#comment-42441

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