# c 2009 Je rey A. Miron 3. Examples: Linear Demand Curves and Monopoly

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1 Lecture 0: Monooly. c 009 Je rey A. Miron Outline. Introduction. Maximizing Pro ts. Examles: Linear Demand Curves and Monooly. The Ine ciency of Monooly. The Deadweight Loss of Monooly. Price Discrimination. Natural Monooly 8. What Causes Monooly? Introduction So far, we have examined rice and uantity determination in a cometitive industry, meaning one with many rice-taking rms. We next consider the oosite extreme, monooly, meaning a market with exactly one rm in the industry. This rm is unlikely to take rice as given. A monoolist cannot choose rice and uantity indeendently, since any outcome must lie on the demand curve. But the monoolist will know that its outut decision a ects the rice at which it can sell its outut. Note that we can think of the monoolist as choosing rice and letting the demand curve determine uantity; alternatively, we can think of the monoolist as choosing uantity and letting the market determine rice. The two aroaches are euivalent. It will be more natural to describe a monoolist s decision in one way or the other deending on the situation.

2 Maximizing Pro ts Let (y) be the market inverse demand curve. Let c(y) be the cost function. The revenue function is r(y) = (y)y. Then the monoolist s roblem is max fyg (y)y c(y) Intuitively, the solution must be to choose the level of outut such that MR = MC. Why? Because if this condition did not hold, the monoolist could increase ro t by changing the level of outut. For examle, if the monoolist had chosen y such that M R > M C; the extra revenue from increasing roduction a small amount would be greater than the increase in costs from increasing outut, so the initial choice of y could not have been ro t maximizing. This aroach to choosing an outut level is analogous to what a ure cometitor does, excet that for a ure cometitor, rice does not deend on outut, so MR =. We can also solve the monoolist s roblem formally. The FOC is (y) + y 0 (y) = c 0 (y) which is exactly the statement that MR = MC. Remember that MR has two comonents: the extra unit of outut roduced brings in extra revenue in the amount (y) er unit of outut; but it also drives down the rice at which the monoolist can sell all existing units because the monoolist faces a downward sloing demand curve. Another way to resent this result is what is through a concet known as marku ricing. Rearranging the rst-order condition gives the following: (y) + y0 (y) = c 0 (y) (y) Then, imose the result that since (y) is the inverse demand curve, its derivative is the recirocal of the derivative of the demand curve itself. That is, 0 (y) = y 0 ()

3 where y() = (y) is the market demand curve. 0 B + y 0 () C A = c 0 (y) y We can therefore write the FOC as or (y) + = c 0 (y) (y) where (y) is the elasticity of the market demand curve. We write it as a function of y to remind us that the elasticity otentially di ers at various oints on the demand curve. This euation is referred to as the marku ricing formula, and is often written as (y) = MC(y) + (y) In words, this says that a monoolist chooses a level of outut such that the market rice euals MC at this level of outut, adjusted by the marku exression + Note the following things based on this version of the FOC: First, ure cometition is the case where the demand elasticity is (negative) in nity. In that case, the formula reduces to the euation (y) = MC(y) Thus, erfect cometition is a secial case of monooly in which the monoolist faces an in nitely elastic demand curve.

4 Second, note that the formula gives a non-sensical answer a negative rice if < 0 Thus, the monoolist will always choose a level of outut such that market demand is elastic, i.e.,. This makes intuitive sense. Remember that if demand is inelastic, the monoolist can increase revenue by reducing outut. But in that case, the initial level of outut could not have been ro t maximizing, since reducing outut also reduces costs. So, the monoolist should kee reducing outut as long as demand is inelastic. Examles: Linear Demand Curves and Monooly Suose a monoolist faces a linear demand curve (y) = a by Then the revenue function is r(y) = (y)y = ay by and marginal revenue is MR(y) = a by Note that MR has same vertical intercet and twice the sloe as the demand curve. This is always the case with linear demand. Assume the cost function is c(y) = y + Then the FOC is a by = y

5 and the solution for the ro t maximizing level of outut is y = a b + Grahically, we have

6 Grah: Monooly with Linear Demand and Marginal Cost Curves y = x y = x y = x * MC D MR * Another standard case to consider is linear demand combined with constant marginal costs. The algebra is similar; the grah looks like this:

7 Grah: Monooly with Linear Demand and Constant Marginal Costs * MC MR D * Still a third secial case often used in alications and illustrations is the constant elasticity demand curve. For examle, assume (y) = y = along with MC(y) = c. max y + cy The FOC is + y = c and the solution is 0 y = Grahically, c + Then the monoolist s roblem is C A

8 Grah: Monooly with a Constant Elasticity Demand Curve y = = x = y = (=) = x = * MC = c * Note that this last examle seems to rovide a sensible solution without imosing the restriction that demand is elastic. How can you reconcile this aarent result with the discussion above that the monoolist always oerates in the elastic ortion of the demand curve? (Hint: think about the SOC.) The Ine ciency of Monooly Now that we know how a ro t-maximizing monoolist behaves, we can think about the economic imlications of monooly. The results above show that a ro t-maximizing monoolist chooses an outut level at which the market rice exceeds the MC of roducing an additional unit of outut. This has two imlications. 8

9 The rst is that rice is higher and outut is lower under monooly than under ure cometition. The result is aarent from the grahical analysis; since the MR curve lies everywhere to the left of the demand curve, the intersection of MR and MC must be to the left of the intersection of the demand curve and MC. This latter oint is where MC =, the outut level that would revail under ure cometition. The second imlication, which follows from the rst, is that monooly is ine - cient. At the monooly level of outut, rice exceeds marginal cost. Price is what consumers would be willing to ay for an extra unit of outut. Marginal cost tells us the resources that would be necessary to roduce this extra outut. If the former exceeds the latter, it would increase overall economic welfare for at least a bit more outut to get roduced and consumed. More recisely, at some rice between the euilibrium rice and MC, it would be Pareto-imroving for the monoolist to roduce at least one more unit of outut and sell it at this rice. The consumer and the monoolist would both be better o, assuming the monoolist still received the same revenue on all the revious units. 9

10 Grah: The Ine ciency of Monooly * inefficiency to due to monooly: > MC c MC MR D * The Deadweight Loss of Monooly We can now examine the economic loss from monooly. grah: Consider the following 0

11 Grah: Deadweight Loss from Monooly 0 9 S * 8 A B D F E C D The comarison is between consumer and roducer surlus at the cometitive outcome versus the monooly outcome. Consumer surlus at the cometitive outcome is the area A+B+C. surlus at the monooly outcome is only the area A. Consumer Producer surlus at the cometitive outcome is the area D + E + F while roducer surlus at the monooly outcome is area B + D + F Therefore total surlus at the cometitive outcome is A + B + C + D + E + F. Total surlus at the monooly outcome is A + B + D + F.

12 So, the change in consumer surlus is (B + C); but B of this goes to roducers. The change in roducer surlus is B E. The net change in surlus the deadweight loss is therefore (B E) (B + C) = (C + E). A crucial assumtion that underlies this analysis is the absence of rice discrimination by the monoolist. This is a reasonable assumtion in many circumstances, for either legal or technological reasons. But in some markets, rice discrimination is not only ossible, but likely if legally ermitted. This can have signi cant imlications for the welfare e ects of monooly. We will examine this shortly. The conclusion that monooly is ine cient is not identical to the conclusion that olicy should intervene to revent or reduce monooly, for a variety of reasons. Aroriate olicy towards monooly deends on several auxiliary issues that we will discuss below. For now, kee in mind that the resence of an ine ciency does not automatically imly that olicy should attemt to reduce or eliminate the ine ciency. Price Discrimination The basic welfare story about monooly that the monoolist kees rice too high and uantity too low deends on one key assumtion: that the monoolist charges the same rice to all customers. This assumtion is demonstrably false in many settings; indeed, it is trivial to think of examles where rms charge di erent customers di erent rices for what is essentially the same roduct: airline seats; theater tickets; bundled versus unbundled software; umbrellas on sale in the rain; ketchu in bottles of di erent sizes; comuters sold to academic institutions versus students versus businesses; and so on. In some cases, the di erent rices might reresent di erences in the cost of roduction - ackaged versus loose mushrooms in the suermarket, for examle. But

13 in many cases it is hard to think of convincing cost di erences that could exlain the di erences in rice. Airlines tickets with or without a Saturday night stayover is one examle. The ractice of charging di erence rices to di erent customers is known as rice discrimination. The label is unfortunate, since it imlies the ractice is necessarily bad. In fact, rice discrimination can lead a monoolist to roduce the e cient level of outut. Before examining this result, note that several kinds of rice discrimination occur. The standard labels for these are rst-degree, second-degree, and third-degree, but these labels are not intuitive and therefore not esecially helful. For now, we focus solely on what is known as rst-degree, or erfect, rice discrimination. The term erfect is more helful since it suggests what occurs under this kind of rice discrimination. We will discuss other varieties in the next lecture. Perfect rice discrimination means that a monoolist otentially charges a di erent rice for each di erent unit of outut, even if these are sold to the same erson. In articular, each unit of the good is sold to the individual who values it most highly, at the maximum rice that this individual is willing to ay. A given individual might ay di erent rices for the rst unit urchased, the second unit urchased, and so on, because the willingness to ay for each additional unit di ers. This is easiest to formalize if we consider a discrete good, and assume each erson buys at most one unit of the good. Each erson has a willingness to ay (W T P ) for one unit (for some eole this W T P is zero, i.e., these eole are not interested in buying the good). Under these assumtions, we can resent the situation grahically as follows:

14 Grah: Perfect Price Discrimination with a Discrete Good 0 WTP9 WTP8 WTP The height of each bar is each consumer s WTP, and we have lined u the consumers in descending order of WTP. If the number of units of the good sold is large, so each unit is small relative to the overall market, then we can aroximate this demand curve with a smooth, continuous curve. For the sake of convenience, assume the monoolist has constant marginal costs. In this case, the otential consumer surlus is the area under the demand curve and above the marginal costs curve. If the monoolist sets MR = MC; consumer surlus is less than this entire area:

15 Grah: Monoolist with Constant Marginal Costs * consumer surlus under monooly w/o rice discrimination MR MC D * If the monoolist engages in erfect rice discrimination, however, the consumer surlus all becomes roducer surlus.

16 Grah: Perfectly Discriminating Monoolist roducer surlus under monooly with erfect rice discrimination * MC MR D * Under these conditions, the monoolist roduces the e cient amount of outut. That means roducing u to the oint where the W T P euals the MC of roduction. Note that there is no consumer surlus - the roducer gets it all. Under rst-degree rice discrimination, the outcome under monooly is e cient. The roducer wants to maximize the sum of roducer lus consumer surlus. In considering whether to roduce another unit, the monoolist comares the extra revenue on that unit to the marginal cost. The monoolist will always roduce the extra unit if one consumer has a W T P that exceeds marginal cost; this is also what is necessary for an e cient outcome. The key di erence between this version of the model and the standard one is that under erfect rice discrimination, lowering the rice for a customer with a low WTP does not involve lowering the rice for all other customers. So, to reeat, a erfectly discriminating monoolist roduces the e cient level of outut.

17 Given this, why are economists (and the ublic) so negative about monooly? There are two reasons. First, erfect rice discrimination is not esecially common in ractice. The reason is that identifying customer W T P is often hard or imossible. Customers have an obvious incentive to ortray themselves as having low W T P so that they can buy at low rices. Indeed, they have this incentive even if they have no interest in the good at all, since they might be able to re-sell the good at a higher rice. So, erfect rice discrimination is rare in ractice. Second, even if erfect rice discrimination were ossible, many eole do not like the distributional imlications, seci cally, the fact that roducers get all the surlus. Whether this is something olicy should worry about is a more comlicated issue, but without uestion, it is one reason for oosition to monooly. The fact that erfect rice discrimination is di cult to carry out does not mean the rice discrimination result is irrelevant. As we will discuss, many other kinds of rice discrimination are ossible and common. These do not necessarily lead to a Pareto-e cient outcome, but, in many instances, they move the outcome in that direction. So, the degree to which we should be concerned about monooly is a somewhat subtle issue. We will see examles in the next lecture. Natural Monooly The fact that monooly is ine cient, at least as we have modeled it so far, is imortant. Before drawing strong conclusions about aroriate olicy resonses, however, it is imortant to think about why monooly exists in the rst lace. Deending on the reason, the aroriate olicy resonses can be radically di erent. To see this, consider one ossible olicy resonse to monooly: mandating that the monoolist set outut at the level where = MC, rather than at the oint where MR = MC. Grahically, this would amount to the following:

18 Grah: Fixing Monooly by Mandating = MC MC * MR D This aroach looks romising in rincile, but it is roblematic for a number of reasons. Consider the following gure: 8

19 Grah: Natural Monooly MC AC AC =MC D 0 The key feature of this market is that average cost is declining over the relevant range. This is a natural assumtion if roduction of a articular good involves large xed costs but small or zero marginal costs over a broad range of outut levels. This situation is known as a natural monooly because with ever-declining average costs, it is natural to want one roducer who can take advantage of high volume to achieve lower costs. Unfortunately, this means having one sulier and therefore the likelihood of monooly ricing. As an examle, the cost curve might be c(y) = ay + F Then the average cost function is AC(y) = F=y + a This curve starts out at an arbitrarily high level, but then asymtotes to a. In this articular case, average cost never turns back u. This condition is not necessary to have a natural monooly; the only reuirement is that over a broad range, the AC curve is declining or at. 9

20 Under this condition in articular, when the oint of increasing cost occurs at an outut level that is large relative to demand one rm can suly all of the market at an average cost that is lower than would occur with more than one rm. So, for a market like that ictured, consider what would haen if the government mandated that the rm set rice eual to marginal cost. The monoolist would lose money and would not want to stay in business. So, this aroach is infeasible. What are the alternatives?. Regulation that forces the monoolist to set rice eual to average cost, i.e., setting the outut level such that demand intersects the average cost curve. This seems like a reasonable comromise. But it is still ine cient, and it is messy in ractice because the regulator needs to know the rm s cost structure.. Government ownershi and oeration. The government entity could set rice eual to marginal cost and receive a lum-sum subsidy from taxayer funds. This is also messy in ractice, and government entities have a tendency to oerate ine ciently for searate reasons.. Let the monoolist rice discriminate, as we will discuss next time.. Do nothing and let the monooly ro ts rovide an incentive for innovation that will gradually erode the monooly. 8 What Causes Monooly? A declining long-run average cost curve is one ossible reason for monooly. useful to note brie y some other ossible reasons. It is A closely related condition occurs when the minimum e cient scale of oeration is large relative to the market: 0

21 Grah: Minimum E cient Scale and Monooly D 0 0 D 0 0 Note that, in this case, regulation is at least feasible, since marginal cost ricing allows the monoolist to make some ro t. But determining costs is still messy in ractice.

22 A di erent ossible situation is that of cartels, meaning an agreement between rms to restrict outut and set a high rice. We will examine this in more detail later. Another ossible reason is rst-mover advantage or reutation. this under oligooly. We will study The nal reason for monooly is government olicy. A broad range of government interventions create monooly or at least restrict cometition. A rime examle is atents. Other examles include ublic schools; the ost o ce; restrictions on cell towers or service via other channels; rofessional licenses; ermits and fees; trade restrictions; and economic regulation generally. We will discuss these kinds of monooly more in later lectures.

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