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How do producers charge different consumers different prices? September 29, 2015

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Price discrimination is the act of charging different consumers different prices based on how much they’re willing to pay. There are a few different forms of price discrimination, and it can be achieved different ways depending on how much information a seller has.

Haggling, or negotiating to find an exact willingness to pay, is an effective form of price discrimination for large purchases. For example, a car salesman can often start with a high price, and when the customer refuses, he can incrementally lower the price (or otherwise adjust the offer) until he find a deal that the buyer is just barely willing to accept. This has one huge advantage – it gets the most that the customer is willing to give up (or, in other words, it extracts the customer’s maximum willingness to pay). It is, however, very costly for a salesman. Just imagine if the salesman were to spend a whole day negotiating only to realize the buyer wasn’t willing to pay enough to cover the cost of the car. Then, the salesman loses the chance to make a sale at all that day. Since it’s costly, this method is most useful for high-priced items like cars and houses.

If you ask a consumer what he’s willing to pay, he’ll lowball you; haggling helps force the price (and the profit) up.

Direct segmentation allows a market to be broken up based on some visible characteristic. In the previous post, I discussed my father-in-law’s senior citizen discount on haircuts and how he pays less than I do for the same cut. He does this by asking for a senior citizen discount, which I’m not eligible for.

Direct segmentation involves breaking the market up into different groups and intentionally charging different prices to those different kinds of people. It works best when one group has a higher willingness to pay – so, since I’m not a college student, and not a senior citizen, my (relatively) high income means I don’t ask for a discount. Similarly, I pay a lower price to have my blazers dry-cleaned than my department chair does, even though her blazers are made up of a smaller amount of fabric. Dry-cleaners just automatically charge a higher price for a woman’s garment than a man’s, even if the garment is similar.

This sometimes leads to unpleasant outcomes. NPR did a story on a 12-year-old girl who had to pay a premium to play Temple Run as a female character; non-white-male characters were all in-app purchases that cost money.

Indirect segmentation is like direct segmentation, but requires the consumer to do some work to get the lower price. A good example of this would be a volume discount. I have a strong preference for Crayola An Du Septic dustless chalk. (I like its weight and erasability.) When I purchase chalk to use in the classroom, my buying options include paying about $3.50 for a single box or about $12 for a 12-box package. No sane person who isn’t a teacher has any use for 12 boxes of blackboard chalk, so I signal my price sensitivity by buying a larger amount at once.

Another way people reveal their types is by clipping coupons. A coupon is like a little badge that says “I’m cheap! Give me a lower price!” By doing a bit of extra work to signal my cheapness, I qualify myself for a lower price just as much as if I’d haggled with the guy behind the counter.

Price Discrimination September 28, 2015

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I wear my hair short. Like, really short – it’s buzzed on the sides and scissor-cut on top, so that it’s low-maintenance, and I trim my own beard using a storebought clipper. My father-in-law does the same thing – except he pays a little bit less than I do, because he tells the barber he’s old and cheap. Why does that work?

A business, we assume, wants to make money. As such, it wants to sell its good at the highest price possible to each consumer. Consumers, though, want to spend as little as possible, to maximize the difference between their willingness to pay and the actual price they pay. (Economists call that difference consumer surplus.) Most of the time, it’s difficult to charge people different prices based on their willingness to pay. To do so requires three big elements.

First, the market has to be segmented. This means that consumers have to have different willingnesses to pay. Think about  a price-sensitive consumer like my father-in-law – he’s getting ready to retire. His wife is already retired. He needs to adjust to spending less money than he’s used to. A lot of his fellow senior citizens feel his pain. Meanwhile, I’m a young(ish) guy. I teach at a community college, I have no kids, and I have a long time before I retire (so my money has a lot of time to grow). I’m willing to pay a little bit more for a haircut than he is. In addition to senior citizens, college students are often given discounts just for being students.

Second, there needs to be some element of monopoly power. My barber isn’t a monopolist, because pure monopoly is rare, but I do go out of my way to go to a place where I have a good rapport with the barber. I have a guy who cuts my hair the way I like it, and I like the atmosphere at his shop. Plus, even though I could probably shop around to find a cheaper price if I went somewhere else, I couldn’t find a price that much cheaper. Haircuts have pretty standard prices around here. That’s what the monopoly power condition is intended to enforce – if I get angsty about not getting a cheap haircut, I don’t really have other options.

Finally, the good needs to be difficult to resell. If we were talking about an oil change on my car, I might send my father-in-law into the mechanic’s shop with my car to get the senior citizen discount on an oil change. When we have a family event planned, he buys the bagels because the local place gives him a deal just for being older. It’s impossible, though, to resell a haircut, so I can’t use his senior citizen discount to my advantage here. Baseball and hockey tickets often offer student rush specials where you have to (theoretically) show a student ID to get the discount. Enforcing that would ensure that people with high willingness to pay didn’t buy the cheap tickets in the nosebleed section, but the open secret is that the Mets don’t really care if you buy cheap tickets, as long as you buy tickets.

If those three conditions exist, then it’s possible for a seller to charge different people different prices. Economists call that price discrimination. It’s not necessarily a bad thing, though – it means if you’re cheap, you can get a pretty good deal on some goods.

Diminishing Marginal Returns September 9, 2015

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Close your eyes.

Well, finish reading the next paragraph first, and then close your eyes.

I am going to offer you unlimited access to something good, something useful, something tasty. That’s right – I’m going to let you have as big a bottle as you’d like of Sriracha. As long as you can carry it away, you can put as much Sriracha as you’d like on your plate of pad thai and I won’t look askance at you. No, I might even respect you. How much are you going to take?

Open your eyes.

The funny thing about that thought experiment is that everyone can picture how much they’d put on a plate of noodles. Some people might put none at all;1 others might put a little dab on the side, while still others, possibly economics professors who operate multiple blogs with self-deprecating titles, might put a truly ridiculous amount and allow the streets to run red with the blood of the non-rooster-sauces. Almost no one would ever take an unlimited amount of sauce.

That’s because, like most goods, Sriracha demonstrates diminishing marginal returns. That means that if Sriracha is meant to create tastiness, then for every extra drop of Sriracha, the tastiness increases, but the increase gets smaller. Mathematically, that means the slope is positive, but decreasing; that’s the same as saying the function has a positive first derivative and a negative second derivative. One common function used to model diminishing marginal returns is the natural log function, y = ln(x). If we assume that tastiness is logarithmic in grams of Sriracha, the graph might look like this:

SrirachaJust about any good demonstrates diminishing marginal returns, and at some point you’ll have enough of a good that its marginal benefit no longer exceeds its marginal cost.


1 Those people are called wimps.

Elasticity and Demand March 18, 2015

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The price-elasticity of demand measures how sensitive consumers are to changes in price. There are two primary formulas for that. Most commonly, introductory courses will use \frac{\% \Delta Q^D}{\% \Delta P}, where \% \Delta means the percentage change. This means the numerator is \frac{\Delta Q^D}{Q_0}, where Q_0 is the original quantity demanded and \Delta Q^D is the change in quantity demanded. The denominator is \frac{\Delta P}{P_o}. Through the properties of fractions, that ratio is equal to \frac{\Delta Q^D}{\Delta P} \times \frac{P_0}{Q_o}, and a lot of students find that formula much easier to use.

A graph of demand and price-elasticity of demand

A graph of demand and price-elasticity of demand

Take note of the shape of that formula, and keep in mind the Law of Demand, which states that as price increases, quantity demanded decreases. At high prices, quantities are relatively low, meaning that a small change in price yields a relatively big change in quantity demanded. If the percentage change in quantity demanded is bigger than the percentage change in price, then demand is elastic and consumers are price-sensitive. On the other hand, at low prices, quantities are relatively high, meaning that a small change in price yields an even smaller change in quantity demanded. That means demand is inelastic.

This pattern of high prices corresponding to elastic demand and low prices corresponding to inelastic demand holds for most goods. At a very high price, firms can make a small change in price to try to encourage new buyers to buy their product, whereas at a very low price, firms can jiggle the price up a little bit to try to snap up some extra revenue without dissuading most of their buyers from purchasing the product.

A slightly more accurate formula for price-elasticity of demand is \frac{dQ}{dP} \times \frac{P}{Q}, which looks surprisingly like the previous formula but doesn’t depend on choosing an original value.

The graph in this post shows market demand Q^D = 1000 - P in blue and elasticity in orange. Note the high elasticity at high prices and low elasticity at low prices.

Elasticity (SPROING~!) March 17, 2015

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When we think about elasticity in the real world, we often think about the properties of things like rubber bands or the waists of sweatpants. If a solid has high elasticity, that means it’s very sensitive to having forces applied to it – so while something like Silly Putty or latex is very elastic, other materials like steel or titanium are not. A small amount of force yields a lot of deformation for Silly Putty, but not much at all for steel.

Elasticity in economics works the same way. It measures how responsive one measurement is to a small change in some other measurement.

When economists say “elasticity” without any qualifiers, they typically mean the price-elasticity of demand, which measures how sensitive purchases are to small changes in price. Elasticity, \epsilon, is expressed as a ratio:

\epsilon = \frac{\% \Delta Q^D}{\% \Delta P}

where \% \Delta Q^D refers to the percentage change in quantity demanded1 (the actual change divided by the starting value) and \% \Delta P refers to the percentage change in price (again, the actual change in price divided by the starting value).2 This leads to three cases:

  1. \% \Delta Q^D > \% \Delta P – a small price change yields a big change in quantity demanded. This means that buyers of the good are price sensitive, and (equivalently) demand for the good is elastic. Note that in this case, \epsilon > 1.
  2. \% \Delta Q^D < \% \Delta P – a small price change yields an even smaller change in quantity demanded. This means that buyers of the good are not price sensitive, and demand for the good is inelastic. Here, \epsilon < 1.
  3. \% \Delta Q^D = \% \Delta P – a small price change yields exactly the same change in quantity demanded. The term for this type of demand is unit-elastic. When demand is unit-elastic, \epsilon = 1.

It’s tempting to treat elasticity as very complicated, when it has a really simply mathematical interpretation. It answers the question “Which change is bigger – price, or quantity?”

Also interesting is the question of why some goods are demanded elastically and some are demanded inelastically. Typically, goods with many alternatives are demanded elastically. Alternatives can come in many forms. Most commonly, they’ll show up as substitute goods, or goods which you can use instead of another good. For example, bread has many substitutes (naan, grits, cornbread, rice, tortillas, English muffins….), and so if the price of bread rose significantly, you’d see many people substituting away from using bread. However, there are other forms of alternatives, too. You may see elastic demand for goods that cost a large proportion of the buyers’ income or that can be purchased over a longer timescale. A college education is an example of both of these – a small change in the level of tuition can lead to big changes in the behavior of students, who will often take a year off to earn money.

Anything with few alternatives will typically be demanded inelastically. Salt is the classic example, because it has no alternatives – it’s necessary for flavoring food, allowing our bodies to function properly, and (in the case of iodized salt) preventing certain illnesses. However, anything that is addictive (like tobacco or heroin), necessary for many uses (like cell phone plans), or difficult to switch away from (it’s not like you can put diesel fuel in your gas-engine car!) will typically have inelastic demand.

1 Quantity demanded means the number of goods people are willing to buy at a certain price.
2 Usually one of these will be negative and the other positive, because of the Law of Demand; economists, ever economical with their notation, simply ignore this and use the absolute value.

The Do-Nothing Alternative March 16, 2015

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Consider the following situation: You are at a casino. You have a crisp new $100 bill in your pocket and an hour before your friend arrives. There are several options available: blackjack, poker, and slot machines. Each has its advantages and disadvantages. Blackjack offers a 45% probability that you will double your money over the next hour, but a 55% probability you will lose it all. Based on your understanding of statistics, you know this means you should expect to have about $90 at the end of the hour. Poker is a better proposition – since it is a game of skill, you have a 60% chance of earning an extra $50 (for a total of $150), but a 40% chance of losing all of your money. That means you can expect to have about $90 in your pocket at the end of the hour. Slot machines, to go to the other extreme, are a highly negative expected-value proposition. You stand a 1% chance of winning $1000, but a 99% chance of losing all of your money. As a result, you could expect to have about $1 in your pocket at the end of the hour.

Thinking like an economist, you quickly winnow your options down to blackjack or poker, since you cannot abide such a risky proposition. Then, however, you’re stuck – the expected values are the same. Which game is it rational to play?

Similarly, consider this problem raised in a freshman course on ethics: You are on your way out of a coffee shop carrying a double shot of espresso and a $1 bill you received as change. Two homeless people, one man and one woman, each step toward you and simultaneously ask you for the dollar. Since you don’t have any coins, you cannot split the value between the two people. Who should you give your dollar to?1

What do these two situations have in common? In each of them, you are attempting to choose between two options that result in negative consequences for you. In the gambling scenario, you have two options, each with the expectation of losing $10. In the coffee shop scenario, you have two people each asking for $1. In neither case is there a compelling reason to choose one option over the other. The underlying assumption, though, is that we must choose an option at all.

The do-nothing alternative is often (but not always) a hidden option when making choices. For example, in the gambling scenario, you have the option to literally do nothing for an hour until your friend arrives. This leaves you $100 with certainty. In the coffee shop scenario, you have the option to politely refuse each person’s request, leaving you free to keep your dollar. Not every situation allows a do-nothing option; for example, a baseball manager faced with the option of starting his worst starting pitcher or a pitcher who is usually used only in long relief cannot opt to simply start no pitcher. However, a voter who is disgusted with all available candidates may bemoan his “forced” vote for the lesser of two evils without acknowledging that he has the option simply not to vote at all. The do-nothing option is often low-cost but has low returns as well, making it a great way to avoid choosing the best of a bad lot, but a lousy choice for a firm seeking growth.


1 This was met with considerable debate about the probability that the homeless woman had children.

The Good and Bad of Goods and Bads January 25, 2015

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When students first hear the word “goods” pertaining to economic goods, they sometimes find it a little funny. When they hear some sorts of goods called “bads,” they usually find it ridiculous. Let’s talk a little about what those words mean and how they pertain to preferences.

Goods are called that because, well, they’re good. Typically, a person who doesn’t have a good would, if given the choice, want it. Examples of goods might be cars, TVs, iPads, or colored chalk. Since people want this good if they don’t have it, they’d be willing to pay for it. Consequently, goods have positive prices.

That doesn’t mean that everyone wants as much of any good as they could possibly have. When purchasing, people consider the price of a good – that is, how much money they would have to spend to obtain that good. However, that’s not because money has any particular value. It’s because money can be exchanged for goods and services, but you can only spend money once, meaning that buying one good means giving up the chance to buy a different one.

Bads are called that because they’re not good. A bad is something you might be willing to pay someone to get rid of for you, like a ton of pollution, a load of trash, a punch in the face, or Taylor Swift. Because you would pay not to have the bad, bads can be modeled as goods with negative prices.

Typically, a demand curve slopes downward because of the negative relationship between price and quantity. This is true for goods – as price increases, people face an increasing opportunity cost to consume one more of a good. If goods are being given away for free, people will consume a lot of them, but as the price rises the tradeoff increases as well. Bads, on the other hand, act a bit different. If free disposal of trash is an option, most people will not keep much trash at all in their apartments. However, as the cost of trash disposal (the “negative price”) rises, people will hold on to trash longer and longer to avoid paying the cost. Consider how often you’d take your trash to the curb if you had to pay $50 for every trip! You might also look to substitutes for disposal, like reusing glass bottles or newspaper in different ways, to lower the overall amount of trash you had to pay to dispose of.

As the cost to eliminate bads increases, people will suffer through a higher quantity, so as the price of disposal increases, the quantity accepted will also increase.

Anecdote Alert: Do restaurant deposits depress attendance? January 1, 2015

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Last night I spent New Year’s Eve at one of my favorite restaurants, Verace in Islip, New York. I actually did New Year’s Eve there last year, too, and there were three very interesting changes. The upshot is that the restaurant, though it had a fantastic menu, was significantly less full than it was last year, and the crowd skewed slightly older.

First, the price of the dinner was $65 last year and $85 this year. That corresponds to about a 30% price hike. That might deter some people, but I’m skeptical. The price-elasticity of demand for restaurant meals is about 2.3, or very elastic. (That means that if the price of a restaurant meal changes by 1%, the quantity of restaurant meals sold would drop about 2.3%.) If that’s the correct elasticity to apply here, that would explain a 69% drop in attendance, but I’m not so sure that restaurant meals on New Year’s are as elastic as restaurant meals during the rest of the year. The well-known Valentine’s Day Effect causes price elasticity for certain goods (cut roses) to drop on Valentine’s Day, and since a meal at home isn’t a close substitute for a restaurant meal on a special occasion, I’m skeptical that this price change would explain the precipitous drop in attendance.

Second, the restaurant required a deposit this year – $50 per person, returned at the beginning of the meal as a gift card. This was my first hypothesis, but I’m not sure it’s much of an explanation. For one thing, I put down my deposit on Monday, so there was no real loss of value. $50 per person to hold a spot is well within the income for most demographics that you see at Verace most nights [more on this in a moment], especially since it operated as a credit on the bill. No dice here, really.

Third, this might be the big one – Verace is part of the Bohlsen Restaurant Group, which operates a couple of restaurants at slightly different price points. This year, BRG made a big deal of advertising different, keyed experiences at their different restaurants. Specifically, Teller’s was a much more expensive steakhouse offering, Verace was a meal only, but Monsoon – their lower-priced, Asian fusion restaurant – had a modular menu with options of $75 for a meal (a bit cheaper than Verace, but not much) and $75 for an open bar. The open bar and Monsoon’s dance floor almost surely made it more attractive to younger revelers. That also explains the shift in demographics – Verace’s younger crowd may have been cannibalized by another BRG restaurant.

In the alternative, our waiter’s hypothesis: The manager did a great job seating people. “This guy,” says he, “is a magician.” He may be, but I’m more interested in seeing Monsoon’s numbers.

Uncertainty Over Time (Lords) November 25, 2013

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The previous post introduced a problem that arose in the Doctor Who 50th Anniversary special in which uncertainty was a core element. To explore that problem in depth, we’ll need some tools to work with uncertainty. Also, that problem was a little grim, so let’s take the edge off.

Let’s say that a friend of mine, Matt, is visiting and I need to order food. Matt’s favorite food is fish custard, so it makes sense that I should order him a dish of fish custard. Simple enough. Of course, if my other friend Peter were coming over, and I knew Peter didn’t like fish custard and much preferred haggis, it wouldn’t make sense for me to order him fish custard. Obviously I’d order haggis for him. To make this work mathematically, let’s say that each friend of mine likes his preferred dish about the same, so we could say that Matt gets utility of uM(fish custard) = 1 from eating fish custard and uM(haggis) = 0 from eating haggis, and Peter gets the reverse – uP(fish custard) = 0 and uP(haggis) = 1. If I want to maximize my friend’s utility, then I should do so by buying each friend his favorite dish. But what if I don’t know who’s coming over?

Intuitively, it makes sense – if Matt’s more likely to come over, I should order fish custard. If Peter’s probably on the way, haggis should go on the menu. If they’re equally likely, flip a coin. More mathematically, if the probability that Matt is coming is πM and the probability that Peter is coming is πP (and πM + πP = 1), the expected utility of my guest when I order fish custard is

E[u(fish custard)] = πMuM(fish custard) + πPuP(fish custard),

which reduces to

πMuM(fish custard) + πP(0).

Since uM(fish custard) = 1, then the expected utility of my guest from fish custard is just πM. For haggis, of course, the same logic applies – Matt gets 0 utility, so we can ignore him, and the expected utility ends up being πP. So, whichever friend of mine is more likely to show up should get his favorite dish ordered, and if I don’t know who’s coming, I might as well just draw names from a hat.

This gets a little bit more complicated if both of my friends like each dish. So, let’s consider what happens if uM(fish custard) = 1 and uM(haggis) = 0, but if uP(fish custard) = 0.5 and uP(haggis) = 1. Then, our calculation for the expected u(haggis) stays the same (it’s still πp) but now our E[u(fish custard) ends up being

πMuM(fish custard) + πPuP(fish custard) = πM*(1) + πP*(.5)

So now, in order to figure out what dish to order, we need to know what the probabilities are! If it’s fifty-fifty, then E[u(haggis)] is .5, but E[u(fish custard)] is .5 + .5*.5 = .75! In that case, we should order fish custard, even though the guys are equally likely to show up, since Peter likes fish custard a little bit, too. We don’t get to flip a coin (mathematically, we don’t reach our indifference point) until

πMuM(fish custard) + πPuP(fish custard) = πMuM(haggis) + πPuP(haggis)


πM(1) + πP(.5) = πM(0) + πP(1) => πM + πP(.5) = πP => πM = .5*πP

That means we don’t get to the indifference point until Matt is twice as likely to arrive as Peter! Funny how these probabilities influence the choices we’ll make. Sometimes things get a bit more complicated, but that’s a post for another time.

… lord.

SPOILERS: The Tenth and Eleventh Doctors’ Problem November 24, 2013

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Major, major spoilers here. If you haven’t yet seen the Doctor Who 50th Anniversary Special, please avoid this post.

Really, seriously, please.

During the 50th Anniversary Special episode of Doctor Who, there was an interesting scene where a group of characters and their malevolent clones were negotiating a treaty that would determine, in part, whether the Zygons (an empire of malevolent aliens) would take over the Earth. This presents some quite interesting problems that were solved by some quite interesting guys – the Tenth and Eleventh incarnations of the Doctor.

Obviously each side had opposite incentives. The Zygons needed a planet to live on, and the human population did as well. Take for granted that the populations can’t coexist and you can see that this is what’s known in economics as a zero-sum game. Since they would not agree to share the planet, one side would have to come out the winner in the negotiations, with the other losing. Similarly, since each Zygon cloned a human negotiator, it inherited at least some of its human’s memories, meaning that we can assume symmetric information. In cases like this, it’s usually up to the agent with the higher valuation of the asset being negotiated to make some sort of concession to the agent with the lower valuation in exchange for agreeing to give up any claims – in other words, pay them to go away. In other cases, the agents will agree to let an arbitrator make the decision for them, under the assumption that the arbitrator won’t be clouded by personal decisions and will make the “best” choice, for some value of “best” to be determined.

The Doctors (played by David Tennant and Matt Smith) opted for an entirely different (and clever) solution, albeit one that wouldn’t work in the real world: they wiped the humans’ and Zygons’ memories so that each would forget which side they were on, allowing for efficient resolution of the Three Incarnations Togetherproblem. The theory, which is similar to Richard Rorty’s Veil of Ignorance, is that people who don’t know whether their side stands to benefit will reach an equitable solution, if not one that they might have argued for in the first place. As such, it’s an example of hidden information. The same thinking generates the idea in law and economics of efficient breach. This might result in the solution that’s efficient in the economic sense, but it probably won’t lead to anyone being the best off they could be.


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