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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.

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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 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.

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)

or

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.

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.

Note:
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

or

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

or

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.

Perfectly Competitive Markets December 10, 2012

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When solving economic problems, the type of firm you’re dealing with can lead you to use different techniques to figure out the firm’s rational choice of action. This week, I’ll set up a thumbnail sketch of how to solve different firms’ types of problems, since a common exam question in intermediate microeconomics is to set up a firm’s production function and ask a series of different questions. The important thing to remember about all types of markets is that every economic agent is optimizing something.

In a perfectly competitive market, three conditions hold:

  • All goods are identical. If the seller is selling apples, then all apples are the same – there are no MacIntosh apples, no Red Delicious apples, just apples.
  • There are lots of sellers, so sellers can’t price-fix because there will always be another seller who will undercut.
  • There are lots of buyers, so a buyer boycotting won’t make a difference.

The last two conditions sum up together to mean that no one has any market power. That means, essentially, that no action an individual buyer or seller takes can affect the price of the goods. If ANY of these conditions isn’t true, then we’re not dealing with a perfectly competitive market – it might be a monopoly or a monopsony, or it might be possible to price-discriminate, but you’ll have to do a bit more to find an equilibrium.

Speaking of that, an equilibrium in microeconomics happens when we find a price where buyers are willing to buy exactly as much as sellers are willing to sell. Mathematically, an equilibrium price is a price such that QS(P) = QD(P), where QS is the quantity supplied, QD is the quantity demanded, and the (P) means that the quantities depend on the price P. Since the quantity is the same, economists sometimes call an equilibrium quantity Q* and the equilibrium price P*.

Consumers are optimizing their utility, or happiness. This might be represented using something called a utility function, or it might be aggregated and presented as a market demand function where the quantity demanded by everyone in the world is decided as a function of the price of the good. A common demand function would look like this:

QD(P) = 100 – 2*P

That means if the price is $0, there are 100 people willing to buy one good each; at a price of $1, there are (100 – 2*1) = 98 people willing to buy one good each; and so on, until no one is willing to buy if the price is $50. Demand curves slope downward because as price goes up, demand goes down. Essentially, a demand function allows us to ignore the consumer optimization step. Demand represents the marginal buyer’s willingness to pay; price equalling willingness to pay is something to remember.

Firms optimize profit, which is defined as Total revenue, minus total costs. If we have a firm’s costs, we can figure out how much they’d need to charge to break even on each sale. Let’s say that it costs a firm $39 to produce a each good. They won’t produce at all until they’ll at least break even – or, until their marginal benefit is at least equal to their marginal cost, at which point they’ll be indifferent. Then, as the price rises above $39, charging more will lead to more profit. Even if the firm’s marginal cost changes as they produce more unity, the price of the marginal unit will need to be at least as much as the marginal cost for that unit. Otherwise, selling it wouldn’t make sense.

The first condition to remember when solving microeconomics problems is that in a perfectly competitive market, a firm will set Price equal to Marginal Cost. If you have price and a marginal cost function, you can find the equilibrium quantity. If you have supply and demand functions, set QS(P) = QD(P) and solve for the price, or simply graph the functions and figure out where they meet.