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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.

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.

Don’t Discount the Importance of Patience April 9, 2013

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Uncertainty is one explanation for why interest rates vary. Tolerance for uncertainty is called risk aversion, and it can be pretty complicated. (We’ll talk about it a little bit later on.) Another big concept is patience. Willingness to wait is also pretty complicated, but that’s our topic for today.

It’s easy to imagine some reasons that people would have different levels of patience. For one, you’d expect a healthy thirty-year-old (named Jim) to be more patient than a ninety-year-old (named Methuselah). What if someone (named Peter) offered us a choice between $100 today or a larger amount of money a year from now? How much would it take for Jim and Methuselah to take the delayed payoff? Would they take $100 a year from now? A lot can change in a year:

  • There could be a whole bunch of inflation, and the $100 will be worth less next year than it is now. Boom, we’ve lost.
  • We could put the money in the bank and earn a few basis points of interest. Boom, we’ve lost.
  • We could die and not be able to pick up the money. Boom, we’ve lost.
  • Peter could die and we wouldn’t be able to collect. Boom, we’ve lost.

Based on these, we’ll want a little bit more money next year than this year in order to be willing to take the money later instead of the money now. Statistically, though, Jim is more likely than Methuselah to be there to pick up the money.

Neither would take any less than $100 next year, but that’s just a lower limit. According to Bankrate.com, Discover Bank is paying 0.8% APY, which means that the $100 would be worth .8% more next year – just by putting the money in the bank, we can trade the risk that the bank goes bust (really unlikely) for the risk of Peter dying. That’s an improvement in risk and an improvement in payoff, so there’s no reason to take any less than $100.80. Again, though, this is a lower bound. Peter still has to pay for making them wait. That’s where the third point comes into play.

Methuselah is probably not going to live another year. It’s much more likely that he’ll get to spend the $100 than whatever he gets in a year; in order to make it worth the wait, the payoff would have to be huge. Methuselah views money later as worth a lot less than money today. He might need $200 to make it worth the wait. Jim, on the other hand, might only need $125. He has more time, so he’s much more patient.

This level of patience is called a discount rate and is usually called β. You can do this sort of experiment to figure out someone’s patience level. You’d then be able to set up an equation like this, where the benefit is the $100 and the cost is what you give up later:

100 = \beta \times Payoff

Methuselah, then, would have

100 = \beta \times 200

so β = 1/2.

Jim would have the following equation:

100 = \beta \times 125

so β = 4/5.

Based on this, we can say that Methuselah values money one year from now at 50% of its current value, but Jim values money one year from now at 80% of its current value. Everyone’s discount rate is going to be a little bit different, and different discount rates can lead people to make different choices. If Peter offers $100 today or $150 tomorrow, Jim will wait patiently for $150. Methuselah will jump at the $100 today. Both of them are rational even though their choices are different.

Evaluating Different Market Structures December 13, 2012

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Market structures, like perfect competition, monopoly, and Cournot competition have different implications for the consumer and the firm. Measuring the differences can be very informative, but first we have to understand how to do it.

Measuring the firm’s welfare is fairly simple. Most of the time we’re thinking about firms, what we’re thinking about will be their profit. A business’s profit function is always of the form

Profit = Total Revenue – Total Costs

Total revenue is the total money a firm takes in. In a simple one-good market, this is just the number of goods sold (the quantity) times the amount charged for each good (the price). Marginal revenue represents how much extra money will be taken in for producing another unit. Total costs need to take into account two pieces: the fixed cost, which represents things the firm cannot avoid paying in the short term (like rent and bills that are already due) and the variable cost, which is the cost of producing each unit. If a firm has a constant variable cost then the cost of producing the third item is the same as the cost of producing the 1000th; in other words, constant variable costs imply a constant marginal cost as well. If marginal cost is falling, then there’s efficiency in producing more goods; if it’s rising, then each unit is more expensive than the last. The marginal cost is the derivative of the variable cost, but it can also be figured out by looking at the change in cost from one unit to the next.

Measuring the consumer’s welfare is a bit more difficult. We need to take all of the goods sold and meausre how much more people were willing to pay than they actually did. To do that we’ll need a consumer demand function, which represents the marginal buyer’s willingness to pay (that is, what the price would have to be to get one more person to buy the good). Let’s say the market demand is governed by the function

QD = 250 – 2P

That is, at a price of $0, 250 people will line up to buy the good. At a price of $125, no one wants the good (QD = 0). In between, quantity demanded is positive. We’ll also need to know what price is actually charged. Let’s try it with a few different prices, but we’ll always use the following format1:

Consumer Surplus = (1/2)*(pmax – pactual)*QD

where pmax is the price where 0 units would be sold and QD is the quantity demanded at the actual price. In our example, that’s 125.

Let’s say that we set a price of $125. Then, no goods are demanded, and anything times 0 is 0.

What about $120? At that price, the quantity demanded is (250 – 240) or 10; the price difference is (125 – 120) or 5; half of 5*10 is 25, so that’s the consumer surplus. That means that the people who bought those 10 units were willing to pay $25 more, in total, than they actually had to pay.2

Finally, at a price of $50, 100 units are demanded; the total consumer surplus is (1/2)(75)(100) or 1875.

Whenever the number of firms goes up, the price decreases, and quantity increases. When quantity increases or when price decreases, all else equal, consumer surplus will go up; consequently, more firms in competition are better for the consumer.

1 Does this remind you of the formula for the area of a triangle? Yes. Yes it does.
2 If you add up each person’s willingness to pay and subtract 120 from each, you’ll underestimate this slightly. That’s because it ignores the slope between points, meaning that there’s a bit of in-between willingness to pay necessary to make the curve a bit smoother. Breaking this up into 100 buyers instead of 10 would lead to a closer approximation, and 1000 instead of 100 even closer. This is known mathematically as taking limits.

Duopoly and Cournot Equilibrium December 12, 2012

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A few days ago, we discussed perfectly competitive markets; yesterday, we talked about monopolistic markets. Now, let’s expand into a case in between – a duopolistic, or two-seller, market. This is usually called a Cournot problem, after the economist who invented it.

We’ll maintain the assumption of identical goods, so that consumers won’t be loyal to one company or the other. We’ll also assume that each company has the same costs, so we’re looking at identical firms as well. Finally, assume that there are a lot of buyers, so the firms face a market demand of, let’s say, QD(P) = 500 – 2P, so P = 250 – QD(P)/2. Since the firms are producing the same goods, then QS(P) = q1(P) + q2(P).

Neither firm knows what the other is doing, but each firm knows the other is identical to it, and each firm knows the other knows this. Even though neither firm knows what’s going on behind the scenes, they’ll assume that a firm facing the same costs and revenues is rational and will optimize its own profit, sothey can make good, educated guesses about what the other firm will do. Each firm will determine the other firm’s likely course of action and compute its own best response. (That’s the one that maximizes its profit.)

Now, let’s take a look at what the firms’ profit functions will look like.

Recall that Total Profit = Total Revenue – Total Cost, and that Marginal Profit = Marginal Revenue – Marginal Cost. Companies will choose quantity to optimize their profit, so they’ll continue producing until their expected Marginal Profit is 0, and then produce no more. Firm 1’s total revenue is Pxq1 – revenue is always price times quantity. Keeping in mind that price is a function of quantity, we can rewrite this as (250 – QD(P)/2)xq1. Since QD(P) = q1 + q2, this is the same as writing (250 – (1/2)(q1 + q2))q1. Then, we need to come up with a total cost function. Let’s say it’s 25 + q12, where 25 is a fixed cost (representing, say, rent for the factory) and q12 is the variable cost of producing each good. Then, Firm 1’s profit function is:

Profit1 = (250 – (1/2)(q1 + q2))q1 – 25 – q12


Profit1 = (250 – q1 /2 – q2/2)q1 – 25 – q12


Profit1 = 250q1 – q12/2 – q1q2/2 – 25 – q12

The marginal profit is the change in the total profit function if Firm 1 produces one more unit; in this case it’s easier to just use the calculus concept of taking a derivative, which yields

Marginal Profit1 = 250 – q1 – q2/2 – 2q1 = 250 – 3q1 – q2/2

Since the firms are identical, though, firm 1 knows that firm 2 is doing the same optimization! So, q1 = q2, and we can substitute it in.

Marginal Profit1 = 250 – 3q1 – q1/2 = 250 – 5q1/2

This is 0 where 250 = 5q1/2, or where q1 = 100. Firm 2 will also produce 100 units. Total supplied quantity is then 200, and total price will be 200. We can figure out each firm’s profit simply by plugging in these numbers:

Total Revenue = Pxq12 = 200×100 = 20,000

Total cost = 25 + q12 = 25 + 100×100 = 25 + 10,000 = 10,025

Total Profit = 9,075

This was a bit heavier on the mathematics than some of the other problems we’ve talked about, but all that math is just getting to one big idea: it’s rational to produce when you expect your marginal benefit to be at least as much as your marginal cost.

Monopolistic Markets December 11, 2012

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Continuing our whistle-stop tour through market types, today’s topic is monopolies. Yesterday’s discussion was of perfectly competitive markets, where three conditions held:

  • Identical goods
  • Lots of sellers
  • Lots of buyers

Today, we’ll talk about what happens when that second condition doesn’t hold – that is, when sellers have market power. When sellers don’t have market power, they have to price according to what the market will bear. If they price too high, someone will undercut them, but if they price too low, they’ll lose money. The only thing they can do is price at their break-even point, where price is equal to marginal cost. (This is sometimes called the zero profit condition.)

When only one seller exists, he is called a monopolist, and the market is called a monopoly. A monopoly can arise for one of two reasons: either it can be because the owner has exclusive access to some important resource, called a natural monopoly, or the owner has an ordinary monopoly because of laws, barriers to entry, or some other reason.

A natural monopoly is one that arises not because of anticompetitive action by the monopolist but because of exclusive access to some resource. For example, owning a waterfall means you have unbridled access to it for hydroelectric purposes; being the first to lay cable or pipelines makes it inefficient for anyone else to access those resources; essentially, anything where there’s a high fixed cost and a zero marginal cost are good candidates for natural monopoly status.

Regardless of whether a monopoly is natural or ordinary, a monopolist isn’t subject to the same zero-profit condition as he would be in a perfectly competitive market, since there’s no one to undercut him if he prices higher than his own marginal cost. He’s free to do the absolute best he can – in other words, to maximize his profit. The monopolist doesn’t have to take the price, as a perfectly competitive market would force him to; he’ll choose the price himself by choosing the quantity he produces.

The monopolist’s profit-maximization condition is that his marginal revenue = marginal cost. This derives from the monopolist’s profit function, Profit = Total Revenue – Total Cost. The monopolist will produce as long as each unit provides positive profit – in other words, as long as marginal profit ≥ 0. In non-economic terms, he’ll continue producing as long as it’s worth it for him – as long as each extra unit he produces gives him at least a little bit of profit. Once his marginal profit is 0, there’s no point in producing any further, since every unit he produces will then cost him a little bit of profit. Because Profit = Total Revenue – Total Cost, another equation holds: Marginal Profit =Marginal Revenue – Marginal Cost. Saying that marginal profit is nonnegative means exactly that marginal revenue is at least as much as marginal cost.

Finally, note that marginal revenue is the price of the last (marginal) unit, but keep in mind that the monopolist has control over the quantity that’s produced. Thus, he has control over the price, and will choose quantity to get his optimal profit.


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