Subsequently, multi-product monopolist; substitute and complement. Emphasized this meant each product could not be priced in isolation of the other. Now the puzzle. Why would a seller introduce a substitute to itself? Recalling discussion of durables good monopolist, this seems like lunacy. A bright spark suggested that the substitute product might appeal to a segment that one is not currently selling to. Yes, but wouldn’t that cannibalize sales from existing product? Time for a model! Before getting to model, formally introduced price discrimination.

Day 7, talked briefly about homework and role of mathematics in economic analysis. Recalled the question of regulating the monopolist. Lowering price benefits consumers but harms seller. Do the benefits of customers exceed harm done to seller? Blah, blah cannot settle the issue. Need a model and have to analyze it to come to a conclusion. While we represent the world (or at least a part of it) mathematically, it does not follow that every mathematical object corresponds to something in reality. Made this point by pointing them to the homework question with demand curve having a constant elasticity of 1. Profit maximizing price is infinity, which is clearly silly. Differentiating and setting to zero is not a substitute for thinking.

Went on to focus on versioning and bundling. Versioning provides natural setting to talk about cannibalization and catering to new segment. Went through a model to show how the competing forces play out. Then to bundling.

Discussion of reasons to bundle that do not involve price discrimination. Then a model and its analysis. Motivated it by asking whether they would prefer to have ala carte programming from cable providers. In the model, unbundling results in higher prices which surprises them and was a good note to end on.

]]>Thus far class had been limited to a uniform price monopolist. Pivoted to discussing a multi-product monopolist by way of a small example of a durables good monopolist selling over two periods. Had the class act out out the role of buyers and me the seller cutting price over time. It provided an opportunity to discuss the role of commitment and tie it back to the ultimatum game played Day 1. On day 6 will revisit this with a discussion of JC Penny, which will allow one to get to next item on the agenda: price discrimination.

]]>Double marginalization was followed by computing total consumer surplus by integrating the inverse demand function. Ended on optimal regulation of monopolist, showing that pricing at marginal cost maximizes producer plus consumer surplus. Brief discussion of incentives to be a monopolist if such regulation was in place. Then, asked the class to consider regulating a monopsonist and whether a minimum wage would be a good idea.

]]>To motivate the notion of marginal cost, I ask how they would define cost per unit to convey the idea that this is an ambiguous concept. A possible candidate is average cost but ist not helpful maing decisions about whether to increase of decrease output. For this, what we want is marginal cost. Define marginal cost, and onto constant, decreasing and increasing returns to scale and discussion of technologies that would satisfy each of these. Solving quadratics is a good example. The time to solve each is the marginal cost. If you have decreasing returns to scale in solving quadratics, a wit suggested, correctly, that one should give up mathematics.

Next, where do cost functions come from? Opportunity to introduce capital and labor and production function. Cost function is minimum cost way of combining K and L to produce a target quantity. Numerical example with Cobb-Douglas. Without explicitly mentioning isoquants and level curves, solved problem graphically (draw feasible region, move objective function hyperplane) as well as algebraically. Discussed impact of price change of inputs on mix used to produce target volume. Marginal productivity of labor, capital and marginal rate of technical substitution. Eyes glazing over. Why am I wasting time with this stuff? This is reading aloud. Never again.

Onto marginal revenue. By this time they should have realized the word marginal means derivative. Thankfully, they don’t ask why a new word is needed to describe something that already has a label: derivative. Marginal revenue should get their goat. Its a derivative of revenue, but with respect to what? Price or quantity? The term gives no clue. Furthermore, marginal revenue sounds like price. The result? Some students set price equal to marginal cost to maximize profit because thats what the slogan marginal revenue = marginal cost means. To compound matters, we then say the area under the marginal revenue curve is revenue. If marginal revenue is the derivative wrt quantity then integrating it should return the revenue. Does this really deserve comment? Perhaps watching paint dry would be more exciting. Wish I had the courage to dispense with the word `marginal’ altogether. Perhaps next year. Imagine the shock of my colleagues when the phrase `marginal blank’ is greeted with puzzled looks.

They’ve been very patient. Before class ends there should be a payoff. Show that marginal revenue = marginal cost is a necessary condition for profit maximization and is sufficient when we have decreasing returns to scale. This seems like small beer. What happens when we have increasing returns to scale? Why does this break down? Some pictures, of why the slogan is no longer sufficient and a discussion of how this relates to pricing for firms with increasing returns like a producer of an app who must rent server space and gets a quantity discount.

]]>Over to you, Walter.

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So let me offer a distinction between `repugnant exchanges’, where the good itself is fine but buying and selling it is repugnant and `repugnant good/services’ where the good or service are what is repugnant, even if for whatever reason what we actually outlaw is the transaction.

Most of the examples that Roth gives fall into the `repugnant good/service’ category rather than `repugnant exchange’. Such is the case of buying and selling recreational drugs, endangered species, imported cultural property. Are there any examples for `repugnant exchanges’ in addition to selling human organs ? Well, there is `renting’ organs, as in surrogate motherhood. Anything else ? An interesting example is lending money with interest, which used to be repugnant in the West (we got over it already): The very idea of lending money was never considered repugnant. What was repugnant is doing it for payment in terms of interest.

Finally, there is prostitution, which is illegal in the US. Repugnant service or repugnant exchange ? depends on your reasoning. Anti-prostitution laws have an unlikely coalition of supporters. There are the religious moralists, for whom the service (extra-marriage sexual intercourse) is what makes the transaction repugnant. They go for prostitution just because that’s what they can outlaw in the US. (They go further in Iran.) But there are also feminists and liberals who view prostitution as exploitation, as I view selling human organs. They find the exchange repugnant even if they have no problem with the service itself.

Note that the example of prostitution shows the difficulty in the distinction I make between `repugnant good/service’ and `repugnant exchange': It relies on unobservable reasoning. Just by knowing the laws and customs of a society you don’t know to which category a forbidden transaction belongs. Moreover, since different people may have different reasoning, the category is sometimes not uniquely defined. But I still think it’s a useful distinction.

]]>Over the summer Thanh Nguyen and myself completed a paper about stable matchings. The abstract is reproduced below.

The National Resident Matching program strives for a stable matching of medical students to teaching hospitals. With the presence of couples, stable matchings need not exist. For any student preferences, we show that each instance of a stable matching problem has a `nearby’ instance with a stable matching. The nearby instance is obtained by perturbing the capacities of the hospitals. Specifically, given a reported capacity for each hospital , we find a redistribution of the slot capacities satisfying for all hospitals and , such that a stable matching exists with respect to . Our approach is general and applies to other type of complementarities, as well as matchings with side constraints and contracts.

In other words, with the addition of at most 9 additional slots, one can guarantee the existence of a stable matchings. This is independent of the size of the market or doctors preferences (it does assume responsive preferences on the part of hospitals). The key tool is Scarf’s lemma which is a wonderful device for converting results about cardinal matching problems into results about ordinal matching problems. For more on this, consult the paper by Kiralyi and Pap, who should be credited with a formulation of Scarf’s lemma that makes its usefulness evident.

]]>Before he was a Nobel Laureate Roth was a Nancy L. Schwartz Memorial Lecturer. His talk was about kidney exchanges — these are exchanges between several pairs of donor+recipient involving no money but only kidneys — and he started with a survey of the audience: who is in favor of allowing selling and buying of kidneys in the free market ? (I am glad I didn’t raise my hand. The next question was about selling and buying of living hearts.) I remember noticing that there was a correlation between raised hands and seniority: For whatever reason, seniors were more likely to be in favor of the free market than juniors.

In the dinner after the talk I ended up in a table of juniors & spouses and we got to discuss our objection to the idea of letting Bob sell his Kidney to Alice, so that Bob can afford to send his daughter to college, and in doing so save Alice’s small child from orphanhood. Turned out we agreed on the policy but for different reasons. I don’t remember which was my reason. I still find both of them convincing, though less so simultaneously.

**Reason I**: The market price would be too low. Hungry people will compete selling their organs for a bowl of red pottage out of desperation. The slippery slope leads to poor people being harvested for their body parts.

**Reason II**: The market price would be too high. Only the 0.01 % will be able to afford it. The slippery slope leads to a small aristocracy who live forever by regenerating their bodies.

As I said, both (somewhat) convincing. And please don’t ask me what would be the fair price, that is neither too low nor too high.

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Began with the ultimatum game framed in terms of a seller making a take or leave it offer to the buyer. The game allows one to make two points at the very beginning of class.

1) The price seller chooses depends on their model of how the buyer will behave. One can draw this point out by asking sellers to explain how they came by their offers. Best offers to discuss are the really low ones (i.e. give most of the surplus to the buyer) and the offers that split the difference.

2) Under the assumption that `more money is better than less’, point out that the seller captures most of the gains from trade. Why? The ability to make a credible take or leave it offer.

This makes for a smooth transition into the model of quasi-linear preferences. Some toy examples of how buyers make choices based on surplus. Emphasize it captures idea that buyers make trade-offs (pay more if you get more; if its priced low enough its good enough). Someone will ask about budget constraints. A good question, ignore budget for now and come back to it later in the semester.

Next, point out that buyers do not share the same reservation price (RP) for a good or service. Introduce demand curve as vehicle for summarizing variation in RPs. Emphasize that demand curve tells you demand as you change your price holding other prices fixed.

Onto monopoly with constant unit costs and limited to a uniform price. Emphasize that monopoly in our context does not mean absence of competition, only that competition keeps price fixed as we change ours. Reason for such an assumption is to understand first how buyers respond to one sellers price changes.

How does monopoly choose profit maximizing price? Trade-off between margin and volume. Simple monopoly pricing exercise. Answer by itself is uninteresting. Want to know what profit maximizing depends upon.

Introduce elasticity of demand, its meaning and derivation. Then, a table of how profit and elasticity vary with price in the toy example introduce earlier. Point out how elasticity rises as price rises. Demand starts to drop off faster than margin rises. Explain why we don’t stop where elasticity is 1. Useful place to point out that here a small price increase is revenue neutral but total costs fall. So, uniform price is doing things: determining how much is captured from buyers and controlling total production costs. Table also illustrates that elasticity of demand matters for choosing price.

Segue into the markup formula. Explain why we should expect some kind of inverse relationship between markup and elasticity. Do derivation of markup formula with constant unit costs.

Now to something interesting to make the point that what has come before is very useful: author vs. publisher, who would prefer a higher price for the book? You’ll get all possible answers which is perfect. Start with how revenue is different from profit (authors get percentage revenue). This difference means their interests are not aligned. So, they should pick different prices. But which will be larger? Enter markup formula. Author wants price where elasticity is 1. Publisher wants to price where elasticity is bigger than 1. So, publisher wants higher price. Wait, what about e-books? Then, author and publisher want same price because unit costs are zero.

This is the perfect opportunity to introduce the Amazon letter to authors telling them that elasticity of demand for e-books at the current $14.99 price is about 2.4. Well above 1. Clearly, all parties should agree to lower the price of e-books. But what about traditional books? Surely lower e-book price will cause some readers to switch from the traditional to the e-book. Shouldn’t we look at the loss in profit from that as well? Capital point, but make life simple. Suppose we have only e-books. Notice, under the agency model where Amazon gets a percentage of revenue, everyone’s incentives appear to be aligned.

Is Amazon correct in its argument that dropping the e-book price will benefit me the author? As expressed in their letter, no. To say that the elasticity of demand for my book at the current price is 2.4 means that if I drop my price 1%, demand will rise 2.4% HOLDING OTHER PRICES FIXED. However, Amazon is not taking about dropping the price of my book alone. They are urging a drop in the price of ALL books. It may well be that a drop in price for all e-books will result in an increase in total revenues for the e-book category. This is good for Amazon. However, it is not at all clear that it is good for me. Rustling of papers, and creaking of seats is a sign that time is up.

Because of its simplicity, the i.i.d. example is in fact somewhat misleading for my purposes in this post. If you toss a coin then your belief about the parameter of the coin determines your belief about the outcome tomorrow: if at some point your belief about the parameter is given by some then your prediction about the outcome tomorrow will be the expectation of . But in a more general stationary environment, your prediction about the outcome tomorrow depends on your current belief about the parameter and also on what you have seen in the past. For example, if the process is Markov with an unknown transition matrix then to make a probabilistic prediction about the outcome tomorrow you first form a belief about the transition matrix and then uses it to predict the outcome tomorrow given the outcome today. The hidden markov case is even more complicated, and it gives rise to the distinction between the two notions of learning.

The formulation of the idea of `learning to make predictions’ goes through merging. The definition traces back at least to Blackwell and Dubins. It was popularized in game theory by the Ehuds, who used Blackwell and Dubins’ theorem to prove that rational players will end up playing approximate Nash Equilibrium. In this post I will not explicitly define merging. My goal is to give an example for the `weird’ things that can happen when one moves from the i.i.d. case to an arbitrary stationary environment. Even if you didn’t follow my previous posts, I hope the following example will be intriguing for its own sake.

Every day there is a probability for eruption of war (W). If no war erupts then the outcome is either bad economy (B) or good economy (G) and is a function of the number of peaceful days since the last war. The function from the number of peaceful days to outcome is an unknown parameter of the process. Thus, a parameter is a function . I am going to compare the predictions about the future made by two agents: Roxana, who knows and Ursula, who faces some uncertainty about represented by a uniform belief over the set of all parameters. Both Roxana and Ursula don’t know the future outcomes and since both of them are rational decision makeres, they both use Bayes’ rule to form beliefs about the unknown future given what they have seen in the past.

Consider first Roxana. In the terminology I introduced in previous posts, she faces no structural uncertainty. After a period of consecutive peaceful days Roxana believes that with probability the outcome tomorrow will be W and with probability the outcome tomorrow will be .

Now consider Ursula. While she does not initially know , as times goes by she learns it. What do I mean here by learning ? Well, suppose Ursula starts observing the outcomes and she sees G,B,W,B,G,…. From this information Ursula she deduces that , so that if a peaceful day follows a war then it has a bad economy. Next time a war pops up Ursula will know to make a prediction about the outcome tomorrow which is as accurate as Roxana’s prediction. Similarly Ursula can deduce that . This way Ursula gradually deduces the values of while she observes the process. However, and this is the punch line, for every there will be a time when Ursula observes consecutive peaceful day for the first time and at this day her prediction about the next outcome will be for war, for good economy and for bad economy. Thus there will always be infinitely many occasions in which Ursula’s prediction differ from Roxana.

So, Ursula does learn the parameter in the sense that she gradually deduce more and more values of . However, because at every point in time she may require a different value of — This is the difference between the stationary environment and the i.i.d. environment ! — there may happen infinitely many times in which she has not yet been able to deduce the value of the parameter which she needs in order to make a prediction about the outcome tomorrow.

You may notice that Ursula does succeed in making predictions most of the times. In fact, the situations when she fails become more and more rare, after observing longer and longer blocks of peaceful days. Indeed, Nabil and I formalize this idea and show that this is the case in every stationary environment with structural uncertainty: the observer makes predictions approximately as if he knew the parameter in almost every day. For that, we use a weak notion of merging which was suggested by Lehrer and Smorodinsky. If you are interested then this is a good time to look at our paper.

Finally, the example given above is our adaptation to an example that appeared first in a paper by Boris Yakovlevich Ryabko. Ryabko’s paper is part of a relatively large literature about non-Bayesian predictions in stationary environment. I will explain the relationship between that literature and our paper in another post.

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