3. Study a durable good monopoly, where the point is that market power is eroded because the monopolist starts to compete with itself.

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1 14 Monooly Varian: Read the chater on Monooly. You may also want to go back and review elasticities. U to this oint we have studied rms that take all rices as given. While this may be a reasonable aroximation in highly cometitive markets (which we think of as markets where a large number of rms roduce some homogenous good), rice taking doesn t make much sense in markets where there is some obvious market ower, such as Microsoft, Terasen Gas, Coca Cola, and harmaceutical comanies roducing atented drugs. To analyze monooly ricing we will study a (artial equilibrium) model where a single rm is the only actor in the industry aart from consumers. The consumers will lay a assive role; their behavior is summarized by a demand function (which we think of as generated by otimizing consumers) for the good. We will roceed as follows: 1. Study the standard textbook monooly ricing model.. Make the observation that it is no longer obvious that the ricing scheme should be a single rice er unit sold. Some examles of various forms of rice discrimination will be given 3. Study a durable good monooly, where the oint is that market ower is eroded because the monoolist starts to comete with itself A Pro t Maximizing Monoolist Like a cometitive rm, we assume that the monoolist strives to generate as large ro ts as ossible. That is, it seeks to maximize y C(y): However, unlike a cometitive rm, a rational monoolist understands that it can only sell what the consumers are willing to buy and, since it oerates alone in the market, the rm can set both rice and the level of roduction. We let D () denote the (direct) demand 1

2 for the roduct and note that the monoolist can sell at most D() units at rice. The otimization roblem for the monoolist is thus subj to y D(): max y C(y) ;y Clearly, y = D() in a solution since otherwise the sales could be increased without lowering the rice, so we may rewrite the roblem as max D() C(D()); where we have rice as the choice variable. Alternatively we may invert the demand (view it as in the ictures where we have y as the indeendent variable and as the deendent variable). Then if (y) is the inverse demand we can write the roblem as max (y)y C(y); Inverse and Direct Demand f X x t } f 1 R t y Y Figure 1: The Inverse of a Function Mathematically seaking, if y = D() is the direct demand function, the inverse of D is

3 a function = D 1 (y) such that D 1 ( D() ) = : {z } quantity demanded at rice The concet is illustrated for a general function f in Figure 1. The oint is that if the function f takes x to a value y; then the inverse takes the value y back to the value x we started with. This must be so for each x and each f (x) ; so the inverse reverts the oeration of the original function f: As an examle, say that D() = a b Then, to nd the inverse we only need to solve out as a function of the quantity. That means that y = a b(y), (y) = a b 1 b y = A By where A = a b and B = 1 b 14. The Otimality Condition Both versions of the roblems are equivalent, so in rincile it doesn t matter which one to solve. However, it turns out that it is somewhat more convenient to work with max (y)y y C(y); The rst order condition is 0 (y)y + (y) c 0 (y) = 0, 0 (y)y + (y) = c 0 (y) To interret the condition note that (y)y is the revenue the monoolist gets if selling y units. Hence 3

4 d (y)y = dy 0 (y)y + (y) is the marginal revenue. In words, the additional revenue the monoolist gets for a small extra unit of outut C 0 (y) is just the marginal cost. Thus, the otimality condition is the rather natural condition that the additional revenue from the last small unit outweighs the cost. Observe that the condition = C 0 (y) has the exact same interretation. The di erence is that a cometitive rm treats rice as given, while the monoolist icks rice/quantity combinations on a downward sloing demand curve. Hence the trade-o is di erent for the monoolist. If the monoolist sells few units it can charge a high rice, but the more units it sells the lower rice it must charge. When it lowers the rice to get one additional unit sold it must lower the rice for all units, so the loss in revenue associated with selling one more unit will be higher the more units the monoolist sells. This logic is most easily understood in an examle with constant marginal cost and a linear demand Examle: Linear Demand Let (y) = A By c(y) = cy so the roblem is The rst order condition is max (A By) y y {z } (y) cy By + A By c = 0 4, y = A c B

5 Note that (A By)y is the revenue, so A By is the marginal revenue. The roblem and its solution can thus be deicted in a grah as in a 6 Tc TT cc c T cc sloe TT b c Pro t c cc T TT c sloe b cc T c T MR cc TT Demand c T c - y y Figure : A Pro t Maximizing Monoolist Figure, where the demand curve as well as the marginal revenue is drawn. To some of you it may be geometrically obvious that the solution is to set the quantity halfway in between 0 and the quantity where the demand curve intersects the horizontal line at height c: Solving c = A By we get y = A c; so B y is indeed this middle oint. To understand geometrically that this is the solution to the ro t maximization roblem, observe that the monoolist aims to make the rectangle reresenting ro ts as large as it can. If the quantity sold is small, the ro t is a high thin rectangle and there are few units to loose revenues from if the quantity is increased/rice decreased. If the quantity is close to A By; then ro ts is a low fat rectangle and there are many units to gain revenues from if quantity is decreased/rice increased It may be visually clear that the ro t is the higher with y at the midoint between 0 and A By and one can also show it by ure algebra (no derivatives involved) or elementary (=retty hard) geometric reasoning. Intuitively: 5

6 Additional revenue from extra unit due to increased sales is aroximately (y + 1) = A B(y + 1) A By Loss in revenue in terms of lower rice on all units is ((y + 1) (y)) y = (A B(y + 1) A + By) y = By so the change in revenue is aroximately a By: 14.4 Elasticity The basic trade-o for a monoolist is that: If the rice is high, the ro t er unit is high but it sells only a few units. If the rice is low, the ro t er unit is low, but it sells many units. The outcome of this trade-o is as we saw a comromise between quantity sold and ro t er unit. Clearly, it is imortant for this trade-o how resonsive demand is to changes in the rice. The most obvious measure of resonsiveness to rice is simle D 0 (); the derivative of the demand function. However, for several reasons (that I will not detail) economists usually measure resonsiveness of demand in terms of elasticities. The elasticity of a demand function D() is given by e() = D0 () D() = dy d dy y = y : The reason why this is convenient is that the elasticity is a good aroximation of the ercentage change in quantity demanded over the ercentage change in rice. This can be seen by observing that for a small change in the rice d D( 0 ) D() + D 0 ()( 0 ) 6

7 and utting this into the de nition of the demand elasticity we get e() (D(0 ) D()) D()( 0 ) = y z } { D( 0 ) D() D() {z } ( 0 ) = y y The main usefulness with the elasticity comared to the sloe of the demand function is that it is a unit-free measurement that makes it easy for economists to comare emirical ndings for di erent markets with each other and sometimes even get a sense for whether rms have monooly ower or not The elasticity of a linear demand function. Let D() = a b ) D 0 () = b the elasticity is thus e() = D0 () D() = Now, since elasticities are negative and it is easier to think about ositive numbers we tyically convert elasticities to absolute values. a b b 1. If je()j = 1 ) b a b = 1, b = a b, = a b. je()j > 1 ) b a b > 1, b > a b, > a b 3. je()j < 1 ) b a b < 1, b < a b, < a b b a b! 0 as! 0 b a b! 1 as! a b since b! a and a b! 0 as! a b : DRAW! 7

8 14.6 Elasticities and The Monoolist Problem For this it is convenient to use the version of the monooly roblem with the direct demand, i.e., we solve max D() C(D()) The rst order condition is D() + D 0 () C 0 (D())D 0 () = 0: The interretation of this the same as in the revious version of the rst order condition, but it is a little harder to see: D() + D 0 () is the change in revenue from a slight change in the rice C 0 (D())D 0 () is the change in cost from a slight change in the rice Hence the condition has the same interretation in terms of marginal revenue being equalized to marginal cost (it is the same roblem, so it should). The rst order conditions can be reorganized as Now this tells us that D() + D 0 () = C 0 (D())D 0 (), /divide by D 0 ()/ : D() D 0 () + = C0 (D()), D() D 0 () + 1 = C 0 (D()), 1 e() + 1 = C 0 (D()) 1 e() + 1 > 0 D() D 0 () = 1 e() since otherwise the marginal revenue is negative. Translating the negative elasticity to an absolute value this means that 1 je()j + 1 > 0 8

9 or, je()j > 1: We can then rearrange the rst order condition once more as = je()j je()j 1 C 0 (D()); which is often referred to as a marku ricing formula. The oint with the exression is that we see that rice must exceed the cost. However, since the monoolist icks the elasticity when icking a oint on the demand curve we should kee in mind that the elasticity in general varies (recall the linear examle) The Ine ciency of Monooly Pricing Consumer(s) value the last unit roduced at the monooly rice m : However, the monoolist only gives u the marginal cost C 0 (y m ) if roducing an extra unit, which the consumer would value at m : Due to this wedge between the marginal cost and the willingness to ay for the last unit all agents could be made haier if, in addition to the units traded at monooly rice, the monoolist and the consumer(s) could trade some additional units at a lower rice. You may note that the roblem is the di erence between the marginal cost and the rice the consumers ay, so a tax on a good sold by a cometitive market (which in the constant returns case would imly that the equilibrium rice must be c + t) creates an ine ciency for exactly the same reason Justifying Deadweight Loss Tringles Often times the distortion is quanti ed in a grah using the deadweight loss, which is illustrated for the case of a constant marginal cost in Figure 3. The intuitive idea is that the demand curve gives the willingness to ay for the marginal unit, the that the distance between the demand and the marginal cost curve gives the dollars lost for that unit not being roduced. To really make sense of this we d have to do more careful welfare analysis. However if utility is quasi-linear then it is rather easy to demonstrate that this is a valid way to roceed. Suose U(x; y) = x + v(y) and that the rice of good x is normalized to one. 9

10 a 6 Tc TT cc T c cc sloe b c T Pro t TT T sloe T TT c cc c b cc y MR T T c cc Demand c c - y Figure 3: The Deadweight Loss of a Monooly The roblem for a utility maximizing consumer is then s.t x + y m max x + v(y) Plugging in the constraint and otimizing we get the rst order condition v 0 (y) = This condition de nes the inverse demand function for y (as long as m is large enough so as to guarantee that the solution is interior), that is (y) = v 0 (y) Observe that v(y) v(0) = so since Z y v 0 (y)dy = Z y 0 0 (y)dy = Area under inverse demand; x + v(y) is the utility of consuming (x; y) m + v(0) is the utility of consuming x = m and y = 0 10

11 we have that the consumer is haier if consuming (x; y) than (m; 0) if and only if x + v(y) m + v(0), Area under inverse demand = v(y) v(0) m x Thus, the area under the inverse demand tells you how many units of the other good the consumer would be willing to give u for y units of good y if facing an all-or-nothing choice Price Discrimination The basic reason why a monooly (charging a uniform rice) is ine cient is that even though it would be bene cial for society, the monoolist doesn t want to increase outut since this would reduce the ro ts earned for all the units sold at the monooly rice. However, if: 1. The monoolist can make the rice contingent on quantity, and. Individualize rice/quantity o ers (in case of more than one consumer), then this trade-o disaears. This is illustrated in Figure 4 where the horizontal line indicates the (constant) marginal cost of roduction. If the market would be cometitive, then we know that the equilibrium rice would equal; the marginal cost and that there would be no distortion. The associated demands are given marginal cost ricing are given by y1; y; ::::; yn in the gure. The standard monooly analysis would be to add u all the demand curves and then set a rice so that the ro ts are maximized (given that the rice is the same for all customers and the same no matter how much each customer buys. However, let A i be the area under the demand curve in between 0 and yi for agent i and suose gives the following o er to each agent: Consumer 1 can ay A 1 and get y 1 or ay nothing and get nothing (and this is a non-negotiable o er) 11

12 y1 Consumer 1 y y Consumer y yn Consumer n - y Figure 4: Perfect Price Discrimination Consumer can ay A and get y or ay nothing and get nothing (and this is a non-negotiable o er) and so on for all consumers in the economy. All consumers would be willing to buy (strictly so if the monoolist sweetens the deal with a enny o )! Note, This is fully e cient! It is imossible to increase consumer surlus without reducing the ro t for the monoolist (which resumably has some owner or owners who would have to decrease the consumtion if the ro ts would be reduced). We say that the monoolist fully extracts the surlus from the consumers by using the individualized take it or leave it-o ers. The form of rice discrimination considered in this examle is referred to as erfect rice discrimination Is Perfect Price Discrimination Reasonable? Most economists would robably say that erfect rice-discrimination is unrealistic in most circumstances. There are several reasons for this. For examle: 1. The monoolist is assumed to know the demand function (that is know references) for each and every individual in the economy. While rms robably have a rather good idea about the aggregate demand, this seems rather unreasonable. 1

13 . Even if the monoolist could see who has what demand, the monoolist must somehow revent consumers from trading with each other. However, the imortant insight from the examle with refect rice discrimination is that when we think about a monoolist there is no a riori reason to restrict attention to trading mechanisms where the roducer sets a rice and customers decide how much to buy. Exlicitly introducing the considerations above (the reasons why erfect rice discrimination is not ossible) into the formal model of a ro t maximizing monoolist we can allow more clever ricing schemes than a xed er unit rice equal for all customers. Deending on details of the model we can then make sense of several ricing ractices that we do observe in the real world. For examle: 1. If the monoolist can not see which consumer is high demand and which is a low demand consumer, the monoolist can still cature some extra ro ts comared to a uniform rice by non-linear ricing, which means that the ricing scheme involves quantity discounts.. We can also make sense of student and senior citizen discounts by thinking of being a student/senior citizen as an imerfect measure of willingness to ay. 3. Sometimes the quantity dimension may be relaced by a quality dimension (think of demand for airline tickets) and here the same tye of logic that rationalizes non-linear ricing can be used to rationalize why airlines wants to make economy seats really crummy in order to make business travellers to self-select into exensive but nice seats with caviar and chamagne served Justi cations of the Standard Monooly Model Otional: Which roblem that is the more aroriate descrition of monooly ricing behavior deends on what may aear as tiny details. As just mentioned various forms of rice discrimination can be rationalized as ways to increase the ro ts for the monoolist: there are monooly ricing models that can 13

14 exlain student and senior citizen discounts, bundling (of for examle browsers and oerating systems), frequent yer rograms, how a monoolist can sometimes send extra money to make goods less useful, etc. 1 Becasuse of this lethora of models, it is of some interest to ask under what circumstances the standard model described in the beginning of the section makes sense. There are essentially two ways to justify that model: 1. Arbitrage ossibilities: if consumers can trade with each other, this makes it very hard to rice discriminate. As an examle, suose that the er unit rice would be decreasing in the number of units sold. Then, a single consumer has an incentive to buy a large quantity of the good (at a low rice er unit) and resell to consumers that are facing higher rices er unit if buying directly from the monoolist. Note: whether this is ossible or not deends on the good in question. Airline tickets, cable TV services, electricity are examles of non-transferable goods, and for these goods it is either imossible or very hard/costly for the consumers to do arbitrage.. Unit demands and uncertainty about references. One can then show that, if the monoolist cannot observe the references of the consumer, then the otimal ricing mechanism is alying the standard monooly ricing formula, where the demand function is given by D () = 1 F () = 1 Pr [v ] ; that is, F is the cumulative distribution over ossible valuations for the object A Durable Good Monoolist Many goods are durable: cars, refrigerators, light bulbs, art, comuters etc. all last for some time. This durability introduces some quite interesting issues concerning the market ower (or lack thereof) for a monoolist. Intuitively, the main ideas are as follows: Suose that some customers urchase the roduct at date t: The crucial observation is that, the customers that bought the roduct tend to be those with high valuations. 1 An examle of the last ractice was when the 486 chi was introduced. A cheaer, lower quality, variant, 486S, was also avaliable. This was roduced by taking a regular 486 chi and disabling certain functions. 14

15 But then, at date t + 1 the monoolist has an incentive to lower its rice to cature some ro t also from the customers with lower valuations. But then,..., the roblem is that all customers will look forward to the future decrease in rice, which erodes the monooly ower at date t: This is referred to as a time inconsistency roblem, which is similar to Birgittas dilemma in the battle of the sexes game discussed in game theory. If Birgitta could credibly commit to always go to the Oera, then she would get a higher ayo than in the backwards induction solution. Similarly, the durable good monoolist would do better if it could commit never to descrease its rice. The main result in the literature is that, if the monoolist can change its rice often enough, then almost all customers will ay a rice close to the marginal cost. That is, the cometition against future incarnations of itself makes the monooly ower go away almost comletely. This tye of analysis is too advanced for the scoe of this course. Instead, we will show something weaker: that the monooly ro t is always less than commitment to the static monooly rice in both eriods. This we can do in a model with only two eriods by alying the backwards induction rocedure reviously discussed Model 1. There are two eriods, indexed by t = 1;. Both consumers and the monoolist discounts future utility and ro ts resectively at a common discount factor < 1 3. Monoolist sells an indivisible good 4. The (direct) demand is given by D () = 1. For interretation, it is a good idea to think of D () as the robability that the valuation for the good is less than. 5. For simlicity, we let the constant marginal cost be c = 0: 15

16 Static Benchmark (Commitment) First consider the case where the monoolist commits to never changing the rice. Since consumers discount their utilities, nobody will then buy anything at time t = : The best (common) rice the monoolist can charge is then the solution to max (1 ) This roblem has solution = 1 ; and the associated quantity sold is q = D( ) = 1 = 1 ; so the monoolist makes a ro t = q = 1 4 if it somehow can romise never to change the rice in the future (we have not shown it, but this is actually the best the monoolist can do if it is able to commit to rices in both eriods before any sales are made) Selling in Both Periods (Non-Commitment) We will solve the roblem by alying the backwards induction rocedure that we discussed in the section on game theory. To imlement this, we must gure out what haens in the second eriod after any history of lay. A history can be described as either a quantity solved in the rst eriod or a rice. One can roceed either way, but I nd it somewhat easier to work with the quantity; Consider the second eriod, and assume that in the rst eriod the monoolist sold q 1 units. That means that customers with valuations above q 1 have left the market already, so the second eriod inverse demand (as a function of the arbitrary q 1 ) is D (; q 1 ) = 1 q 1 {z } New intercet q Second eriod roblem is thus max q q (1 q 1 q ) ; which has a solution (which deends on q 1 ) given by q (q 1 ) = (1 q 1) (q 1 ) = (1 q 1) : 16

17 The second eriod ro t is thus (q 1 ) = (1 q 1) Now, in the rst eriod a consumer with willingness to ay v will buy the roduct if 4 v 1 0 and v 1 (v ) The rice in the second eriod will be below the rice in the rst eriod, so we can ignore the rst inequality. Now, if q 1 is the quantity sold, it will be bought by the q 1 agents with the highest valuation, so the agents with valuation 1 q 1 v 1 are the ones that buy in the rst eriod, and (1 q 1 ) 1 = ((1 q 1 ) ) ; since the agent with valuation v = 1 q 1 must be indi erent between buying in eriod 1 and buying in eriod. But we know that when the monoolist re-otimizes in the second eriod, then so, the indi erence condition becomes (q 1 ) = (1 q 1) ; (1 q 1 ) 1 = ((1 q 1 ) (q 1 )) = (1 q 1 ) 1 q 1 1 = = (1 q 1) ) 1 (1 q 1 ): The rst eriod ro t maximization roblem for the monoolist is thus max q 1 (1 q 1 ) 1 q 1 + (1 q 1) ; 4 17

18 which gives rst order condition (1 q 1 ) 1 q 1 (1 q 1) = 0, = 1 = 1 q1 (1 ) = 4 3 Now, when we have the solution in terms of the rst eriod quantity we notice: 1. Previously, we exressed the second eriod rice, quantity sold and ro t as q (q 1 ) = 1 q 1 (q 1 ) = 1 q 1 1 q1 (q 1 ) = : By evaluating these exressions at q 1 we get the values that will be chosen after the best rst eriod choice as q = q (q 1) = 1 q 1 = 1 (1 ) 4 3 = (q1) = 1 q 1 = q = (q1) = 1 = 1 4 = (1 ) 4 3 = Next, we observe that the indi erence condition for buying in rst versus second eriod was 1 = so, the rice charged at eriod 1 is 1 = = 1 (1 q 1 ) (1 ) 1 (1 4 3 ) 4 3 (1 ) = ( ) (4 3) :

19 3. Hence, the discounted value of the monoolists ro t is (I write it as a function of as that is a key arameter) 4. Observe that that is: () = 1 + = 1q1 + q ( ) (1 ) = + 1 (4 3) 4 3 {z } {z } {z } q 1 1 = 1 [4 (1 ) + ] (0) = 1 4 = (1) = 1 1 [4 (1 1) + 1] = 1 1 [1] = ; when = 0; nobody cares about the next eriod so the roblem becomes like the static roblem when = 1; the monoolist charges ( 1) (4 3) = 1 (or higher) in the rst eriod, so nobody actually buys anything in the rst eriod. The second eriod is then just like the only eriod in the static model. For intermediate values of we have that 0 < < ; imlying that [4 (1 ) + ] = 4 3 ( ) 4 3 = < = 1 Hence () = 1 4 [4 (1 ) + ] < : That is, the otion to sell to those customers who don t buy at time 1 unambiguously hurts the monoolist. 19

20 Discussion The conclusion that not selling in the second eriod is better for the monoolist is often considered counter intuitive: exibility, which is often times considered valuable, is bad. The oint is that consumers are not going to be fooled. If the monoolist will do what is otimal in the second eriod, otential buyers will understand this, so the monoolist is better o if it can somehow commit not to change the rice in the second eriod even though the second eriod incentive obviously is to change the rice to sell more units. 0