Economics 2010a Fall 2003 Lecture 11 Edward L. Glaeser 11. Competition and Monopoly, some preliminary discussions a. Monopoly Pricing b. Cournot and Bertrand Oligopoly c. Two Part Pricing d. Price Discrimination e. Regulation Standard monopoly result is quite straightforward Q is set to maximize P(Q)Q-C(Q) which yields: P’(Q)Q?P(Q)?C’(Q) or C ? ?Q? P?Q? ? 1 ? Q P ?P ?Q ? 1 ? 1 ? Cournot oligopoly– N firms– fixed entry– choose Q to maximize: P Q i ? ? j?i Q j Q i ? C i ?Q i ? or P ? ? j Q j Q i ? P ? j Q j ? C i ? ?Q i ? or C ? ?Q? ? P?Q??1 ? Q i Q Q P ?P ?Q ? or P?Q? ? C ? ?Q? ? ?? Q i Q that’s the markup over marginal cost Claim: we know that industry profits are lower under cournot oligopoly than under monopoly (assuming identical cost curves) Is it possible that industry output will be lower under cournot oligopoly than under monopoly? Assume identical cost curves and write: P ? ?NQ?N??Q?N? ? P?NQ?N?? ? C ? ?Q?N?? Differentiation with respect to N then yields: P ? ?NQ?Q ? ?N? ? QP ?? ?NQ??Q ? NQ ? ?N?? ? P ? ?NQ??Q ? NQ ? ?N?? ? C ?? ?Q?Q ? ?N? Solving this yields: Q ? ?N? ? QP ? ?NQ??Q 2 P ?? ?NQ? C ?? ?Q???N?1?P ? ?NQ??NQP ?? ?NQ? If 0 ? P ? ?NQ? ? QP ?? ?NQ? then the expression is negative because the numerator is negative and the denominator is positive. If P ? ?NQ? ? QP ?? ?NQ? ? 0 then the numerator is positive–If N?P ? ?NQ? ? QP ?? ?NQ?? ? C ?? ?Q? ? P ? ?NQ? and then the denominator is negative and the whole expression is again negative. Second order conditions require that C ?? ?Q?N?? ? 2P ? ?NQ? ? QP ?? ?NQ? Only if 0 ? N?P ? ?NQ? ? QP ?? ?NQ?? ? C ?? ?Q? ? P ? ?NQ? is the sign reversal possible. What’s going on there? To show that overall industry output increases with N, we just need that Q ? Q ? ?N?N ? 0 Or 1 ? ?Q ? ?N?N/Q or 1 ? ?NP ? ?NQ??NQP ?? ?NQ? C ?? ?Q???N?1?P ? ?NQ??NQP ?? ?NQ? or C ?? ?Q? ? P ? ?NQ? ? 0 And that’s a fact– so we don’t know what happens to individual output, but we know that aggregate output has to go up with the number of firms. Bertrand Competition– competition along prices yields marginal cost pricing. Edgeworth conjecture– quantity precommitment ? bertrand price competition yields cournot outcomes. Proved true (essentially) by Kreps ? Scheinkman, Rand Journal 1983. Proof requires game theory. Obviously, every producer would be better off if they could restrict output to monopoly levels. A large literature has thought about the sustainability of these cartels. One side has thought about making cheating observable– the other has thought about the ability of a cartel to punish. Assume N independent producers, and an infinite time horizon. Write profits as ? Q,Q as profits based on own production and production of other firms. Q M is monopoly production (i.e. output that maximizes joint surplus), that maximizes N??Q M ,Q M ? Q O is each firm acting independently, i.e. that maximizes ? Q O ,Q O just over the first argument. Finally, Q C maximizes ? Q C ,Q M just over the first argument. Pofits under perfect monopoly are denoted ? Q M ,Q M Repeated game literature (Abreu, Abreu Pearce and Stachetti) tells us that a monopoly outcome is not sustainable if: ? Q C ,Q M ? ? Q M ,Q M ? ? 1?? ? Q M ,Q M ? ? Q O ,Q O The amount that you lose when you revert to oligopoly has got to be more than the amount that you can gain by cheating. This gives us the comparative static type implications that cooperation can only be sustained by the patient or if the non-cooperative outcome is really bad. Stigler (1962) instead focuses on monitoring. Firms observe only overall prices– not what the other firm is producing. This means that they can only infer that another firm is cheating if the prices fall a lot. This means that there is an inference problem– it also means that you may want to have price wars that only last a limited time. One way to handle this is to have a policy of price matching– this gets customers to tell you when other firms are cheating. Firms may end up trying to cheat with various rebates. Price discrimination. Certainly among the most interesting sets of things in I.O. is price discrimination. Assume a monopoly and that the demand curve comes from heterogeneous consumers with different valuation for the good. Assume further that the marginal cost of the good is c. If the density of consumers is denoted f(v) then markup will equal: P ? c f?P?P 1?F?P? f?P?P 1?F?P? ?1 ? c ? ??1 Obviously the monopolist would like to get some of the consumer surplus which is equal to ? V?P ?P ? c?dF?v? The best outcome for the monopolist is to just charge some consumers more than others. For example– discounts for the elderly or for other forms of consumers. If consumers are separable along some obvious dimension then it makes sense to set P i ? ? i ? i ?1 c– bigger markups on less elastic customers. This might explain why lower wage groups (students, the elderly, tend to have lower prices. This also helps to explain large differences across airline flights or time of day entries. Probably we should be more puzzled by the fact that prices vary little as opposed to the fact that prices vary a lot. Another way to address this is a tie in. Assume that there are two goods and 1/2 of the people value each good highly (with value v) and 1/2 of the people value each good much less (with value v) The correlation term is denoted ?, so there are .5? high-high types and .5? low-low types. If each good is owned by a separate monopoly, the monopolist will each charge v and each v?c or charge v and earn .5?v ? c? If the goods are sold jointly, there is no gain if you want to charge 2v in which case the profits are the same as in the separate case–as everyone will buy. If you charge 2v then you will only sell .5? units and be worse off than in the case that you sell these goods independently. The only advantage comes if you sell them for v ? v In that case profits are ?1 ?.5???v ? v ? 2c? This will be more than 2 ? 0.5?v ? c? if ?1 ?.5???v ? c? ?.5??v ? c? or ?1 ?.5??v ?.5?v ? ?1 ? ??c This will be more than 2?v ? c? If ?1 ?.5???v ? c? ? ?1 ?.5???v ? c? or ?1 ?.5??v ? ?c ? ?1 ?.5??v When c?0, these two conditions are 2 ? ? 1 ? v v ? 1 ? ? 1?.5? – this impossible if ? ?.7 (or so) but otherwise it’s doable. If the values are negatively correlated enough– then this can be attractive for almost all ranges. The key point is that you can get alot of extra buyers by tying the two goods together. Two part pricing (Disneyland) also gets at price discrimination. Here we have to assume both an intensive and an extensive margin (think the coke example in the workshop). Assume quasi-linear utility that is also separable in the good (x) y ? px ? v?x? ? U?z? where y is income, z is the vector of all other commodities– we are just going to ignore this vector, since the price of x doesn’t change consumption. Assume that the firm can set a price schedule p 0 ? p 1 x so that consumers pay both a per unit fee and a flat fee. Let x?p 1 ? satisfy: v ? ?x?p 1 ?? ? p 1 The flat fee must satisfy y ? p 0 ? p 1 x?p 1 ? ? v?x?p 1 ?? ? U?z? ? y ? v?0? ? U?z? or p 0 ? ?p 1 x?p 1 ? ? v?x?p 1 ?? ? v?0? Assume a constant marginal cost of c, the firm then maximizes: p 0 ? x?p 1 ??p 1 ? c? or substituting in v?x?p 1 ?? ? x?p 1 ?p 1 ? v?0? ? x?p 1 ??p 1 ? c? ? v?x?p 1 ?? ? v?0? ? cx?p 1 ? which leads to f.o.c. v ? ?x? ? c The firm prices at marginal cost and then uses the fixed fee to extract the surplus. Why can’t firms always do this? Legal Barriers certainly are one possibility A second possibility is the chance for resale Third there is the case of heterogeneous consumers (which is the norm). Assume that there are two types of consumers with different v-functions. The fixed fee must be low enough to scoop in both types of consumers (unless you only want one type). So set group "A" to be the low consumer surplus group and assume that the proportion of group A types in the population is ? The firm still maximizes: p 0 ? ?x A ?p 1 ??p 1 ? c? ? ?1 ? ??x B ?p 1 ??p 1 ? c? We know that p 0 ? v A ?x A ?p 1 ?? ? p 1 x A ?p 1 ? ? v?0? So the problem becomes: v A ?x A ?p 1 ?? ? p 1 x A ?p 1 ? ? v?0? ? ?x A ?p 1 ??p 1 ? c? ??1 ? ??x B ?p 1 ??p 1 ? c? Maximizing this over p 1 yields: v A ? ?x A ?x A ? ?p 1 ? ? p 1 x A ? ?p 1 ? ? x A ?p 1 ? ? ?x A ? ?p 1 ??p 1 ? c? ? ?1 ? ??x B ? ?p 1 ??p 1 ? c? ? ?x A ?p 1 ? ? ?1 ? ??x B ?p 1 ? ? ?x A ? ?p 1 ??p 1 ? c? ? ?1 ? ??x B ? ?p 1 ??p 1 ? c? ? ?1 ? ???x A ?p 1 ? ? x B ?p 1 ?? ? 0 Or p 1 ? c ? ?1????x A ?p 1 ??x B ?p 1 ?? ??x A ? ?p 1 ???1???x B ? ?p 1 ? The intuition of this is that you charge a premium if the marginal group consumes less of the product and a discount if the marginal group consumes more of the product. The key is to use the the variable cost/fixed cost combination to extract rents. If marginal group is high quantity, then reducing the price for the variable cost gives you a chance to make it up big time on the fixed cost– but if the marginal group is low quantity then reducing the marginal cost gives you much less of a kick on the fixed cost. Final notes– how to write a theory paper: (1) A highbrow theory paper– go talk to Jerry or Drew– don’t listen to me. (2) A lowbrow or applied theory paper. You must start with an interesting real world puzzle. You must have a compelling idea that answers this puzzle, which needs to be both novel and at least plausibly true. You must write a parsimonious model that makes your point clearly. Ideally, you would want your model to have other testable implications that could be used to confirm your theory. You might want to discuss a little bit of the evidence yourself. A few hints on how to write a paper of this form: (1) Do not include a literature summary– weave references elegantly into the text. (2) Make the paragraph sentence of the opening paragraph– your question– if possible– it might even by an appropriate title for the paper. This will keep you focused. (3) Keep your abstract below 100 words. (4) Keep your introduction below four pages. (5) Get to your explanation of the puzzle by the third paragraph. (6) If you can’t convince your mother that you are working on an interesting problem, it is a good bet that you won’t be able to convince a journal editor either (at least in applied theory). (7) After your introduction, you can either have a two-three page section explaining the puzzle or not. This requires some facts, and you would want to use this to weave in the literature. Again, it should be heavy on facts. (8) After this, you will want to get to the model. Be very precise about setting up the model. Make sure that everyone can understand what you did. But do not put your calculations in the text, unless you are really sure that a particular first order condition gives huge amounts of insight. (9) Better to write everything in proposition/proof format. Even comparative statics, i.e. the equilibrium level of x is falling with y and rising with z if the following conditions hold .... (10) A good figure is worth alot. (11) Do not include every calculation in the paper. Just because you did it, does not mean that anyone else wants to read it. In many cases, extra work is helpful because it helps guarantee the generality of results. Often just citing the work in a footnote is fine. (12) In the model, focus on explaining variation across time and space as well as explaining your core puzzle. If you are trying to explain a behavior why do we see it in some places and not others, etc. (13) Do include a discussion section after the model. This would be a good time to argue how the model fits the available evidence well. (14) Keep your conclusion to a page or less. (15) Economists care a lot about being cited– the easiest way to make an enemy is to fail to cite someone who think that he or she has done important work in your area. A brief aside on empirical papers: My understanding is that they come in really four flavors: (1) a stylized facts paper, (2) estimating a single parameter– usually an elasticity of some x on some z, (3) structural empirical work– writing down a formal model and using the moments of the data to fit the exact model you are writing (4) a puzzle driven paper– i.e. why are poor countries poor or why has income risen, or something like that, that lists a set of hypotheses and then essentially tries to figure out a decomposition of the form: Y i ? ?? j X j,i or Y i ? Y k ? ?? j ?X j,i ? X j,k ? Essentially the goal is to have a set of factors ? measure the difference in the factors between the places, time periods, people, etc., and then multiply those differences times the estimated parameter that tells you the effect of this. Of these– stylized facts paper are often enormously useful but hard to write well and even harder to publish. Leave this to a later stage in your career. Surely the most straightforward task is to estimate a single elasticity. The quality of the paper hinges on the degree to which we care about the elasticity and the degree to which people believe your estimation strategy– i.e. the degree to which your instruments are orthogonal to the error term. Remember exogeneity and orthogonality are not, not the same thing. One final suggestion– in all cases– it is best to have a sensible model that justifies whatever regression you are writing. In the case of a parameter estimate paper (which includes papers is competition good for schools, is segregation bad for minorities, etc.)– the structure can be quite simple: (1) introduction – again the paragraph sentence of the first paragraph should be the the question (2) "theory" or discussion section – in some particularly straightforward cases, this can be dropped– in other cases, you need 2 pages to set up a horserace – make it clear that the theoretical case is ambiguous and that you are trying to figure out what is true. In cases, where we think we know the sign (punishment deters crime) but we care about the magnitude– this section is really unnecessary. Alternatively, you can even write down a little model here. But if the section is more than 4 pages– you are screwing up. (3) Data and Discussion of the Instrument/Natural Experiment– where is your exogenous source of variation coming from. This is going to take about 3 pages– I would guess. (4) Main resuts– 4-6 pages. (5) Extensions and Robustness checks 2-4 pages– remember, not every specification you run needs to be in the paper. You can describe things that don’t appear in tables– you can put things in footnotes. Keep it short. (6) Conclusion– 1 page for you to wax philosophic on how important your results actually are. A broad question paper– this come in two flavors: discrete and continuous. A continuous paper starts with a correlation– cov(x,y) is big– Why? Examples are– why is there more crime in cities, why does fertility fall with income across countries, why does pollution first rise and then fall with income, etc. A discrete paper starts with a discrete fact– why do whites earn more money than minorities– why does the u.s. have a more generous welfare system than europe. In either case– the introduction can be as much as four pages– the opening paragraph should set up the fact. The next three paragraphs should explain between two and four major theories– (if you have more minor theories then dispatch them in footnotes). Then use the rest of the introduction to explain what you found. (2) In this section Section II needs to clearly explain each one of the three theories. In this case (as opposed to the previous paper)– the theories can be complementary– cities can have less crime both because of a greater density of victims and because of lower probabilities of arrest due to anonymity. I would say no formal modelling is needed, but you should have gone through the discipline of writing down a model for each one of the theories to make sure your own thinking is clear, i.e. what is needed to get each theory to work. (3) Then in Section III you need to set up the methodology, either the discrete formula Y i ? Y k ? ?? j ?X j,i ? X j,k ? or the continous formula, which is essentially: y ? f?z 1 ?x?,z 2 ?x?,z 3 ?x??? so dy dx ? ?f ?z 1 ?z 1 ?x ? ?f ?z 2 ?z 2 ?x ? ?f ?z 3 ?z 3 ?x or if you prefer elasticities: x y df dx ? z 2 y ?f ?z 2 x z 2 ?z 2 ?x ? z 2 y ?f ?z 2 x z 2 ?z 2 ?x ? z 3 y ?f ?z 3 x z 3 ?z 3 ?x Or statistically– assume all variables are mean zero, standard deviation one (just normalize): y ? ? 1 z 1 ? ? 2 z 2 ? ? 3 z 3 ?? z i ? ? i x ? ? i The estimated regression coefficient from a univariate regression of y on x will be ?? i ? i so then the game is to estimate the ? i and ? i (4) Now you need to estimate this stuff. The beta terms are straightforward– these are just correlations. The hard part is the alpha terms– sometimes you need to estimate these ? sometimes you can just take them from the existing literature. (5) Conclude– again no more than 1 page.