MEMO. 1 Single-Product Monopolist. 1.1 Inverse Elasticity Rule. Date: October Subject: Notes from Tirole on Monopoly Ch.1: pp

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1 MEMO To: From: File FM Date: October 2018 Subject: Notes from Tirole on Monopoly Ch.1: pp Competitive options. Consider 3 mutually excluive choices for competitive market: as follows: (i) choice 1: shape of LACs, U-shaped or horizontal;(ii) Choice 2: LACs alike or different; (iii) Choice 3: pecurniary externalities present or absent. Monoply: Assume that (i) the monopolist s products are given and that their existence and quality is known prior to purchase (Ch. 2 relaxes this), and (ii) the monopolist can charge only a single price to the market (Ch. 3 relaxes this). (We make these assumptions as the absence of these conditions forms part of an existing research literature.) We proceed to look at: a single-product monopolist a multiproduct monopolist a durable-good monopolist 1 Single-Product Monopolist 1.1 Inverse Elasticity Rule Let = ( ) be the demand for the good produced by a monopolist.. The inverse demand may then be written as = ( ). Let total costs of production be given by = () where, as usual, is a vector of factor prices. Everything is appropriately differentiable. the conventional profit max price is given by FOC is given by (all should know this) =argmax[( ) ( ( )) (1) 0 ( ( )) = ( ) 0 ( )

2 or 0 = 1 where ( 0 ) is sometimes called the Lerner Index (LI ) of monopoly power and is the demand elasticity at the monopoly price. The FOC above shows that the LI is inversely proportional to the demand elasticity. The inefficiency from the monopolist who charges a single price is that the monopolist produces at a price that is too high. If = ( ), then the real problem is that the monopolist produces too little output as in an attempt to maximize profits the monopolist charges to high a price is too far up the demand curve. Note that the monopoly always produces in the elastic region of the demand curve. Relate this to the cellophane fallacy, namely that with market power for pricing, the firm prices in the elastic portion of the demand curve. Thus substitutes which may not have been disciplinary at competitive prices could enter the market at monopoly prices. The caveat is that the finding of substitute products is not definitive to answering the question of whether the prevailing price is a monopolistic price or a competitive price. Property:The monopoly price is an non-decreasing function of marginal cost. Demonstrate 2 ways to show this: Way 1: Consider LI equation and differentiate it assuming a constant demand elasticity and we get = where so 0 0 Way 2: Consider two alternative cost functions such that 2( ) 0 1( ) 0 0. Let and where =1 2 denotes the respective monopoly output for each cost function. From the definition of each of these price and quantity variables we know that and ( 1 ) ( 2 ) ( 2 ) ( 1 ) so if we add up these two equations we obtain [ 2 ( 1 ) 2 ( 2 )] [ 1 ( 1 ) 1 ( 2 )] 0 or Z 1 [ 2( 0 ) 1( 0 )] 0 2 Because 0 2() 0 1() for all, the above equation implies that 1 2 or the monopoly price is a non-decreasing function of marginal cost.

3 1.2 Deadweight Loss From a partial equilibrium point of view, what is the loss of surplus from monopoly pricing? Suppose that = ( ) +[ ( ) 2] 2 (where 1) and that the inverse demand function is =1 Then monopoly profits are given by =(1 ) ( ) ( ) 2 2 and the monopolist s profit maximizing output is given by 1 2 ( ) ( ) =0 or = 1 ( ) 2+ ( ) but the efficient output would be given by 1 ( ) ( ) =0 (or where mc intersects the demand curve) which means that = 1 ( ) 1+ ( ) Clear that. (Note that if we had a linear cost curve say = ( ), then =(1 ( )) 2 and =(1 ( )) so that =2.) Diagrammatically the deadweight loss is given as follows (let TS = total surplus). (See diagram at the end of these notes.) ( ) () = The comment here is that even though the price distortion does increase as the price elasticity (in absolute terms) decreases the welfare loss does not necessarily increase. The idea is that for low price elasticities there can be strong price distortions but low price elasticies do mean relatively smaller quantity effects and the deadweight loss depends on both price and quantity effects. However, low price elasticities do produce large transfers from consumers to firms with monopoly pricing. To see this, do Exercise 1.1 which works with constant elasticity demand functions. (The answer to 1.1 is at the back of Ch. 1.) As a general comment, it is useful to be able to work with both linear and constant elasticity demand relationships. Unitary Demands What if consumers have unitary demands. That is the consumer s utility function is = + s.t. + where =1, if purchase, and =0otherwise, is the

4 consumer s reservation price, is a numeraire commodity, and is money income. Notice that monopoly pricing creates no welfare loss. (This specification is very useful in some models where we want to concentrate on welfare effects of some market arrangement where there is some price-setting powers for firms but we wish to rule out these obvious and well-known deadweight loss welfare effects.) Rent-Seeking Behavior Posner and others have pointed out that these deadweight losses understate the true cost once we question the source of the monopoly power. Suppose that firms garner monopoly power by virtue of governments granting exclusive franchise to one firm. Then in the competition for the exclusive privilege, each competitor will spend up to the expected profit from the monopoly privilege to secure that privilege. So if the monopoly profit to each winner irrespective of the identity of that winner were and if there were competitors and =1 denotes the likelihood of the firm s success then the deadweight loss associated with the monopoly privilege would be augmented by P = P =. As Tirole points out (page 76) there are two main axioms: (i) rent dissipation: the total expenditure by firms to obtain the rent is equal to the rent (as stated immediately above), and (ii) socially wasteful dissipation: this expenditure has no socially valuable by-products. 2 Multiproduct Monopolist Let = ( ) and () where =( 1 ) and =( 1 ). There are 4 cases: Independent demands and separable costs; dependent demands and dependent costs (the most general); dependent demands and separable costs; and independents demands and dependent costs. Independent demands and separable costs: This is a straightforward extension of the simple monopoly problem. The pricing problem can be decomposed into subsidiary pricing problems similar to little monopolies. As before, those goods with lower price elasticities will have higher markups. Dependent demands and dependent costs: = X ( ) ( 1 ( ) ( )) FOC: ( + + X 6= = X Tirole proceeds by considering two cases as per the last two listed above. Dependent demands and separable costs: In this case the cost functions becomes: () =

5 P ( ) and the corresponding FOC in mark-up form becomes 0 = 1 X 6= ( 0 ) where 0 is own price elasticity, ( ( )( )) is cross price elasticity and is the gross revenue from good. So now we can say the following: (I defined these cross price elasticities and formulae using a different sign convention from the Tirole this one is easier to see.) If goods are substitutes, then 0, andli exceeds inverse of own elasticity of demand. If such a firm were to be divided into independent divisions each of which sought to maximize its division profits, then what is the effect? Ans: Each division charges too low a price andoughttoraiseitspriceasthedivisionsdefactocompetewitheachother. (pecuniary externality) That is, overall firm profits are higher with higher prices. If goods are complements, then 0, andli falls short of inverse of own elasticity of demand. Now if the firm were run as independent price-setting units, each division charges too high a price and overall profits would be increased if each lowered its price. It may be that optimally profits would be enhanced if some complementary products were sold below marginal cost or even given away e.g. free mobile phones with sign-up with a network for a period of time with the price of calls positive. In both cases, there is an issue of internalizing an externality across the divisions of a firm. Comment: Now use this to consider what happens when a merger occurs. If two producers of substitute products were to merge, we would expect them to attempt to increase their prices. This would be subject to the demand conditions that they faced. If two firms producing complementary products were to merge, then we would expect them to lower their prices. So mergers of complementary products appear to have positive welfare effects. This lead William Baxter, former Asst, Attorney General in the US to propose that firms selling complements should be allowed to enter into contracts imposing price and non-price restrictions. If the firms sell substitutes, need to be suspicious (or absent any other effects should be disallowed). Example: Intertemporal Pricing and Goodwill (p. 71 in Tirole). A rather nice application of this dependent demand story. Consider a product sold in two consecutive periods, =1 2. At =1,demandis 1 = 1 ( 1 ). The production cost in general is = ( ( )). At =2, there is a good will effectsothatmoreofthegood purchased in period 1 yields an increase in the demand in period 2, cet. par.. Could think of this as a demonstration effect folks try the product and spread the word. We write the reduced-form demand in period 2 as 2 = 2 ( 1 2 ) where The monopolist s PV profit is then = 1 1 ( 1 ) 1 ( 1 ( 1 ))+ [ 2 2 ( 2 1 ) 2 ( 2 ( 2 1 ))] where is the discount factor.

6 It is obvious that if we simply define 2 2 and 2 2 and substitute these into the profit function above, we have a multiproduct monopolist with interdependent (but not symmetric) demands. Inspection reveals that conditional on this intertemporal spillover from the first to he second period, the monopolist sets the conventional price in period 2. In period 1, however, the monopolist sets a price below that of the conventional single-period demand as there is an additional term 2 1 which is negative in the FOC. So starting from the conventional single period monopoly price the additional term tells us to lower the price. [Illustrate this for the general principle.] 1 = 1 ( 1 ) Ã! =0 Evaluate this expression at the conventional single period monopoly price defined by 1 and we obtain à = 1 2! 2 2 =(+)( ) So PV of profits increase if we lower 1. Independent Demands and Dependent Costs: Tirole considers first a peak pricing problem in two periods. (Exercise 1.6) Here is how to deal with this. Consider the following cost function (suppress the factor price term): = + + where denotes capacity measured in terms of units of output. Suppose that is fixed at. If, then cost becomes = +. Otherwise, costs are = +( + ). Define peak inverse demand function as 2 = 2 ( 2 ) and off-peak as 1 = 1 ( 1 ) such that 1 ( 1 )= 2 ( 2 ) where 1 now write respective profits as Now calculate FOC and answer question. = 1 : 1 = 1 1 ( 1 ) 1 ( 1 ) = 2 : 2 = 2 2 ( 2 ) ( + ) 2 ( 2 ) Second example concerns learning by doing. This again reflects an intertemporal problem. Again specify a reduced-form cost function. Suppose that there are 2 periods and that costs are lower in period 2 the greater is the output in period 1. Similar to our demonstration effect problem above only now it is on the cost side. Costs in period =2are given by 2 = 2 ( 2 ( 2 ) 1 ( 1 ) (2) Now write down the PV profit function with a simple one-period discount factor to obtain: 1 1 ( 1 ) 1 ( 1 ( 1 ))+ [ 2 2 ( 2 ) 2 ( 2 ( 2 ) 1 ( 1 ))]

7 And the corresponding FOC of interest is for 1 andthisisgivenby: 1 1 = where, as before, 2 = 2. Let s examine signs to see the direction of the effect (although the intuition is easy): First 2 1 0; second, So together this means that the combined sign is 0. Thatis,themonopolistwillreduceitspricebelowthat it would conventionally charge and the reason is that a lower price stimulates demand and output with a second-period saving in costs. If this second-period effect is sufficiently strong, it is even possible that Exercise 1.7 is a formal continuous time version of this effect. 3 Durable-Good Monopolist So far we looked at a demand interdependence that said that when a customer buys today, provided the customer is not disappointed, then the customer is more likely to purchase tomorrow. This is an intertemporal demand dependence. Now look at another kind of intertemporal relationship that comes through a good that is durable beyond the basic period. In contrast to the above, anyone who purchases a durable good today is unlikely to be a repeat customer tomorrow. Another way to put the contrast is that with an intertemporal goodwill interdependence the goods in the two periods are complements whereas with an intertemporal durable good interdependence the goods in the two periods are substitutes. (Coase conjecture) In introducing his quite well-known paper on durable goods, Bulow says the following: This paper explains the special type of monopoly power held by a firm that is a monopolist in the production and sale of a durable good. This power can be substantial but is notably less than the power held by a monopolist who produces a durable good which is rented rather than sold. The distinction of the rented and the seller is what makes durable-goods monopolists interesting. when such a monopolist can rent, he can achieve all the standard results of the nondurable monopolist. Selling is a much more difficult problem, involving an important incompleteness of contracts and an expectational problem.. Basically the problem is the inability of the seller to contract credibly not to lower the price in the future. And so future production becomes a competitive weapon against current production. Tirole illustrates the problem in the following manner: Define a discount factor by (= 1 (1 + )). Suppose that each consumer buys at most one unit of the good, the good lasts forever in the sense that once a consumer buys that consumer exits the market. Define consumer reservation prices by =1 7. Further set variable costs equal to zero. We imagine seven consumers with valuations defined by = Total demand is given by =8. If this were a simple monopoly problem, then the price would be given by ( =1)=argmax =(8 ) =4

8 Consider what happens next as there are two periods. Suppose that the firm were to charge =4in the first period. If all consumers with valuations 4 purchased, then demand in the second period would be =4 andthesecondperiodpricewouldbe ( =2)=2. Consider the consumer defined by 4 =4. If the consumer purchases in the first period, then this consumer receives a net surplus of 4 =4 4=0. But if this consumer waits one period, then the consumer would receive in =1terms (4 2) 0, sothereisno equilibrium here. In fact the equilibrium is a sequence of prices and consumer expectations such that expectations are rational given the firm s behavior and the firm s behavior is optimal given consumers expectations. The problem is that the producer cannot commit nottolowerthepriceoncetheinitialperiodshavepassed. To illustrate the solution and the idea of breaking the intertemporal commitment problem through leasing, we follow the Tirole s example (pp of the textbook). Let demand ineachofthetwoperiodsbegivenby ( ) =1. Supposethatthemonopolistcould lease or rent the good in each of the two periods or could sell it in each of the two periods. There is no depreciation in the good so that the good produced and used in period 1 may be re-used in period 2 without any loss. Variable costs are zero. Let s set up the two options. Lease: This is simple. Let choices be in quantities for simplicity and take the inverse demand function. In each period, the monopolist sets =argmax (1 ) = 5 and the corresponding = 5 so that profits are = = 25(1 + ). Sell: There is now an intertemporal link. A buyer in period 1 is prepared to lease the good to another consumer in period 2 when the market price in period 2 exceeds the value placed on the good by the first-period buyer. Suppose that the firm sells 1 in period 1. Now the monopolist offers 2 in the second period. The price in the second period 2 must clear the market,thatisitmustbesuchthatboth 1 and 2 would clear the market as the quantity 1 from the first period is still out there and constitutes a secondary source of supply. Not all whoboughtinperiod1arepreparedtosellorleasebuttheyareinthemarketassecondary suppliers. This constitutes the intertemporal link. Therefore the (inverse) demand in the second period is given by 2 =1 1 2.Inthisperiod,thechoiceisgivenby 2 =argmax 2 (1 1 2 )= 5(1 1 ) and the corresponding price is 2 =1 1 2 = 5(1 1 ).Theprofit is 2 = 25(1 1 ) 2. Now go to the first period.the price that first-period buyers are willing to pay depends on their expectations of second-period prices. We want them to have rational expectations and so we say that the price that they expect is the price that is charged. That is, first-period buyers can work out this problem. Their correct expectations places pricing discipline of the monopolist. Define the expected second-period price as 2. Therefore consumers in the first period are willing to pay 1 = That is they could lease the good in the second period and reduce their first-period price by 2. Now rat ex means that 2 = 2.

9 Now substitute so that 1 = (1 1 )=(1 1 )(1 + 5 ) So the price in period 1 is now lower than the price under leasing when the monopolist could commit not to produce in period 2. The PV maximization problem is now given by and the solution is or max 1 = 1 (1 1 )(1 + 5 )+ 25 (1 1 ) 2 (1 1 1)(1 + 5 ) 5 (1 1)= =0 or 1 1 = 2+ 5 and correspondingly (1 + 5 )2 1 = 2+ 5 and if we define the corresponding PV profit as The idea is that first-period buyers will be unwilling to pay a very high price for a good when they know that sellers can floodthemarketinperiod2. Leasingisonearoundthis problem for the monopolist seller. But leasing is not always possible, for example, steel used in rail lines. On page 325 of Bulow s article he has a nice discussion of a Coase problem in macro when the government cannot commit not to print large amounts of money in the future. On page 329, Bulow also discusses other commitment issues. For example, a monopolist may use intertemporal price guarantees, such as most-favored nation promising to remit to past buyers any price differential below their purchase price. Consider a service contract on a durable. Bulow notes that service contracts are nondurable and so a monopolist may try to collect monopoly rents through the service contract, perhaps tying it to the purchase of the machine. (This is interesting in light of the Kodak decision.). Also the monopolist seller may try to reduce the liquidity of the second-hand market by agreeing to guarantee the performance of the durable only so long as it is held by the original purchaser.

10 $ CS = welfare and deadweight loss C E, G F P(q) MR q