Multiproduct Price Regulation Under Asymmetric Information

MULTIPRODUCT PRICE REGULATION UNDERASYMMETRIC INFORMATION*

Mark Armstrong{ and John Vickers{

We discuss the regulation of a multiproduct monopolist when the ¢rmhas private information about cost or demand conditions. The regulatoro¡ers the ¢rm a set of prices from which to choose. When there isprivate information only about costs, the ¢rm should always have adegree of discretion over its pricing policy. When uncertainty concernsdemand, whether discretion is desirable depends on how demandelasticities vary with the scale of demands. If a positive demand shock isassociated with a reduction in the market elasticity, discretion is goodfor overall welfare; otherwise it is not.

i. introduction

How much pricing discretion should be granted to a regulated multi-product ¢rm with market power? Policy makers have given di¡erentanswers to this question in di¡erent situations. In Britain, the largesttelecommunications ¢rm, BT, is allowed some discretion over its retailtari¡ but subsidiary constraints are imposed on individual prices. It facesan average price cap on its basic basket of servicesöquarterly line rentals,local, long-distance and international calls, at peak and o¡-peak times ofdayöwhich gives it freedom to increase one price within the basket if itreduces others. Within this overall cap, though, subsidiary constraintsinclude requirements that (i) these prices be uniform across the country;(ii) prices do not fall below some £oor (roughly given by the cost involvedin supplying the service), and (iii) the quarterly line rental charge maynot rise above a speci¢ed cap (at least until recently). In addition, otherservices, most notably the interconnection charges other ¢rms pay to useits network, are controlled under a separate cap which permits no trade-o¡s with retail prices.1

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137

THE JOURNAL OF INDUSTRIAL ECONOMICS 0022-1821Volume XLVIII June 2000 No. 2

*Financial assistance from the ESRC (grant numbers L114251026 and L114251038) isgratefully acknowledged. We thank Joe Farrell, Mike Waterson and two referees, as well asJacques Cremer, David Sappington, Lars Stole, and seminar participants at LBS, LSE,Oxford and Toulouse for helpful comments.{Authors' a¤liation: Nu¤eld College, Oxford OX1 1NF, UK.

email: [email protected]{All Souls College, Oxford OX1 4AL, UK.

email: [email protected] See Armstrong et al. [1994, ch. 4] and Oftel [1996] for more details.

There are several reasons for this mixed policy, and it is useful to discussunder four broad headings the pros and cons of granting a ¢rm discretionover its prices:

`Non-Economic' Reasons: For various reasons beyond pure economice¤ciency, and perhaps to do with political concerns about `equity', ¢rmsare often restricted in the relative prices they may charge. For instance,postal service in most countries involves nationwide uniform pricing, withletters to costly rural areas being charged at the same rate as those tocheaper urban districts.

Regulatory Burden: For practical reasons it is necessary to take intoaccount the regulator's (and the ¢rm's) time and e¡ort involved in anysystem of price control. This can cut both ways as far as pricing discretionis concerned. A ¢rm such as BT o¡ers many thousands of theoreticallydistinct services, and if distinct prices were allowed for each of these thenensuring compliance with any price control formulaöwhether involvingindividual controls or an average capöwould be extremely costly. There-fore, it is natural to require certain closely related servicesöwhich mightinclude all local calls, for instanceöto be charged at the same rate, whichreduces the ¢rm's discretion. On the other hand, an average price capof some form, rather than a very disaggregated set of controls, has theadvantage that only aggregate, not individual, factors (such as rates ofreturn, scope for productivity gains, and so on) need to be estimated whensetting the cap.

Interactions with Competition: The presence of competition in one ormore markets may well have an impact on decisions over pricing policy.For instance, if the regulated ¢rm faces possible entry in one market butother markets are captive, then forms of average price regulation mightgive incentives for aggressive behaviour in response to entry. The fact thatthe incumbent can trade o¡ low prices in the competitive market for highprices in captive markets gives it a motive, in addition to that in normalmarkets, to respond aggressively, and maybe even to price services belowmarginal costösee Armstrong and Vickers [1993] for an analysis of thise¡ect. Thus this form of asymmetric competition gives a reason to restrictthe incumbent's discretion.

By contrast consider the case where the ¢rm with market power isunregulated, at least in terms of the average level of its prices. (Relativeprices might be restricted due to policy towards price discrimination, andso on.) The question of whether a monopoly should be permitted to chargedi¡erent prices for services that cost it the same has received extensiveattentionösee the survey by Varian [1989] for instance. In general it isambiguous whether a monopoly should be permitted to engage in price

138 mark armstrong and john vickers

ß Blackwell Publishers Ltd. 2000.

discrimination. However, in oligopolistic markets where competition ismoderately intense, it is natural that laissez-faire is more likely to bebetter: ¢rms attract custom by giving consumers the pattern of prices theywant, and these patterns of prices are good for overall welfare too. Thusin this case there is a presumption that full discretion over the choice ofrelative prices should be granted, even when ¢rms have some marketpowerösee Armstrong and Vickers [1999] for more details.

Asymmetric information: The ¢nal set of issues arises from the likelihoodthat the ¢rm has superior information about cost and demand conditionsin the market. The natural assumption is that this provides an argumentfor giving the ¢rm a degree of discretion over its prices. After all,most models of optimal regulation (when transfers are used) with privateinformation about the ¢rm's costöfrom Baron and Myerson [1982] toLa¡ont and Tirole [1993]ödo involve ¢rms with di¡erent costs choosingdi¡erent contracts. In the framework of this paperöwith multiproduct¢rms where no lump-sum transfers are allowedöif a ¢rm has unknownmarginal costs and is permitted some discretion over the prices it maycharge it seems likely that it will tend to choose higher prices in marketswhere costs turn out to be higher, and this will be good for overall welfaretoo. On the other hand, in a model of optimal regulation (when transfersare used) with private information about demand, Lewis and Sappington[1988a] show that it is ambiguous whether any discretion should be givento the single-product ¢rmöif the ¢rm's marginal costs decrease withoutput then it is best to force the ¢rm to set a ¢xed price regardless ofdemand conditions. In our framework it is also unclear whether a multi-product ¢rm with superior information about its demand conditions willchoose a desirable pattern of relative prices if it is given discretion.

This paper analyses the ¢nal topic in more detail, and we consider theoptimal design of the price capöthe set of permitted price vectorsöfor aregulated multiproduct monopoly. Thus we abstract from questions ofequity, regulatory burden and competition. In addition, we ignore the pos-sibility that the ¢rm can a¡ect either its costs or its demand functions byexpending costly e¡ort or investment. We consider prices and the degreeof pricing discretion as the sole instrument of policy. This assumption isrestrictive in two ways: (i) no use is made of lump-sum transfers, eventhough within the terms of the models discussed such transfers are welfareenhancing, and (ii) no use is made of the realised demands of consumers.However, the case where transfers are allowed is by now fairly wellunderstood, at least in the case of cost uncertainty, and in Section III(iv)we discuss this analysis in more detail. Without demand uncertainty, ofcourse, there is no additional bene¢t in controlling quantities as well asprices, but with unknown demands this restriction is potentially important.

multiproduct price regulation 139


(For instance, it rules out price cap regulation of the form whereby theaverage price o¡ered by the ¢rm must be no greater than some constant,where the `weights' in the de¢nition of the average are proportional torealised outputs.) Like the analysis of monopoly regulation with unknowndemands by Lewis and Sappington [1988a, 1988b], we assume that actualdemands are not observable by the regulator, although it can be veri¢edthat the ¢rm is serving all demand at its prices.

How to design optimal price regulation turns out to be a di¤cultproblem, and in this paper we present only a partial analysis. The reason itis di¤cult is because it combines two non-trivial extensions of the standardregulation models: the realistic assumption that transfers are not usedmakes the problem harder not easier, and the multiproduct nature of theproblem means that it is natural to model the private information as beingmulti-dimensional and the easier scalar analysis cannot be used.2 Becauseof these di¤culties we have not been able to solve the problem for generalspeci¢cations of private information about either cost or demand con-ditions. Instead, we discuss the issues in two, restrictive, ways.In Section II we discuss whether or not any degree of pricing discretion

is desirable. (Naturally, if we were able to solve the problem in full gener-ality this section would be redundant: we could then just characterise theconditions under which zero discretion was optimal.) The main pointsfollowing from this analysis are:

. When the regulator's uncertainty concerns costs we show that pricingdiscretion is always desirable. The reason for this is straightforward:since demands are known, the regulator has an accurate measure ofconsumer welfare for a given set of prices (a measure that does notdepend on the ¢rm's superior information), and so he can o¡er the ¢rmany set of prices that leaves consumers no worse o¡ compared to a ¢xedprice vector, and the ¢rm will usually do better with this discretion.Therefore, both parties bene¢t compared to a regime where the ¢rm isforced to o¡er ¢xed prices.

. When demand is uncertain we argue that the desirability of discretionis less clear cut: when there are `multiplicative shocks' to demand it isoptimal to grant the ¢rm zero discretion over pricing policy, whereaswith `additive shocks' some discretion is optimal. The intuition for whythe multiplicative shocks formulation means that no discretion shouldbe granted is quite simple: with this functional form for the ¢rm's

2 For instance, see La¡ont and Tirole [1993, Section II(vii)] for an analysis of the no-transfer case of their model of regulation. Also, the combination of no transfers and multi-dimensional information implies that there will usually be `pooling', and di¡erent types of¢rm will end up choosing the same price vector under any scheme; this pooling complicatesthe analysis considerably.



private information the market elasticities, which are all that is neededto implement Ramsey prices, are known, and therefore Ramsey pricescan be implemented without decentralizing decision-making to the ¢rm.Since it seems hard econometrically to distinguish `additive' from `multi-plicative' shocks to demand, this last result is perhaps discouraging forpolicy since it shows the delicacy of the regulator's problem: small changesin the environment should apparently lead to major shifts in policy.

In Sections III and IV we analyze the optimal degree of pricingdiscretion in a simple discrete model. In Section III we discuss symmetriccases, and we characterize the optimal set of allowed prices when there isprivate information about costs (Section III(i)) or demands (SectionIII(ii)). In the former case, as discussed in Section II, it is always optimalto allow discretion over pricing policy. When there is private informationabout demand, however, whether it is desirable to allow discretion dependson how demand elasticities vary with the scale of demandöif `larger'markets have less elastic demand then discretion is good, whereas if theyare more elastic it is optimal to impose ¢xed prices on the ¢rm. The reasonis again straightforward:

. In most cases the ¢rm will react to a positive demand shock in a market,if given the freedom to do so, by raising its price there. If a positivedemand shock is associated with a negative shock to the elasticity, thenstandard Ramsey principles imply that this higher price is also good foroverall welfare; otherwise it is undesirable. Therefore, in the latter casethe ¢rm should not be given the freedom to trade-o¡ high prices in onemarket with low prices elsewhere.

In all cases where discretion is optimal, the symmetry assumptionimplies that prices are always `Ramsey-like' in the sense that theymaximize welfare subject to a pro¢t constraint. In Section IV we show thatthis result is particular to the symmetric case, and, in the case withunknown costs, it is optimal to distort prices away from Ramsey pricesexcept for low cost ¢rms. Thus, just as in Baron and Myerson [1982],prices are distorted away from optimal levels except for the most e¤cient¢rm. However, a crucial di¡erence with Baron and Myerson is that wedo not allow transfers, and hence there is `no distortion' compared to(second-best) Ramsey prices rather than (¢rst-best) marginal cost prices.

ii. discretion vs. fixed prices

A pair of simple examples shows that the desirability of discretion canvary considerably depending, among other things, on the nature ofuncertainty:



Example 1. Known marginal costs and no ¢xed costs, unknowndemand.

Suppose the ¢rm has constant, known marginal cost ci for product i

and has no ¢xed costs. Then regardless of the demand functions of con-sumers, it is optimal with standard welfare measures to require the ¢rm too¡er prices equal to its marginal costs. (Since there is no ¢xed cost, thispolicy does not result in the ¢rm running at a loss.) This policy results inthe ¢rst-best outcome for all demand realizations, and any discretion givento the ¢rm can only cause welfare losses.3

Example 2. Known demand, unknown costs.4

Since consumer demands are known, let v�p� the appropriate measure ofaggregate consumer welfare when the vector of prices is p � �p1; . . . ; pn�.Let C�q� denote the ¢rm's cost function for producing outputs q ��q1; . . . ; qn�, this function not being known by the regulator. Then it cannever be optimal to grant the ¢rm zero discretion over pricing policy. For letp� be any ¢xed price vector, and let P denote the set of prices which leaveconsumers in aggregate no worse o¡ compared to p�, i.e.

�1� P � fp j v�p� � v�p��g:(See Figure 1 for an illustration of this choice set.)

Then allowing the ¢rm to choose any price from P must increasestandard measures of welfare compared to the case where the ¢rm wasrequired to o¡er p�. (The ¢rm can never be worse o¡, for some costrealizations it will be strictly better o¡, and by construction, consumers asa whole are never worse o¡.) This form of price control has the propertythat the chosen prices are always Ramsey-like, in that consumer surplus ismaximised subject to a pro¢t constraint. We deduce that a degree of dis-cretion must be optimal if consumer demands are accurately known butcosts are not. Note, though, that we do not claim that a set of prices of theform P in Equation (1) is optimal, only that zero discretion is not optimal.We will see below that the optimal set of price vectors does not generallytake the form Equation (1).

Example 2, which is static, is closely related to Vogelsang's [1989]dynamic mechanism for achieving Ramsey-like prices when both cost and

3 If marginal costs varied with outputs then these costs depend on the private informationof the ¢rm, and the ¢rm cannot simply be instructed to price at cost. Similarly, if transfers areused and there is a social bene¢t to taxing the ¢rm's pro¢ts (i.e. there is a social cost of publicfunds a© la La¡ont and Tirole [1993]) then this result does not hold: demand-side informationis needed to calculate the optimal prices.

4 The following example is taken from Armstrong, Cowan and Vickers [1994, SectionIII(iii.ii)].



demand functions are unknown but stationary over time. He assumes thatrealized demands can be observed at the end of each period. If ptÿ1 wasthe previous period's price vector then the mechanism allows the ¢rm inthe current period t to choose any price in the set

Pt � p jXn

i�1piqi�ptÿ1� �

Xn

i�1ptÿ1

i qi�ptÿ1�( )

:

Using the convexity of v it follows that Pt lies inside the set P in (1) whenp� � ptÿ1, and so this system of dynamic regulation causes consumerwelfare to rise over time. Vogelsang shows that a non-myopic ¢rm willchoose Ramsey-like prices in the steady state, so that in the long run theprices maximize welfare for a given level of pro¢t. Clearly this is preferableto imposing on the ¢rm a ¢xed price p0 in each period.The next two examples show that when there is uncertain demand it

is ambiguous whether or not discretion is desirable. Unlike the rest ofthe paper, we assume here that the ¢rm does not learn its privateinformation until after the regulatory contract is signed, and so the ¢rm'sparticipation constraint has to be satis¢ed only on average (rather than

Figure 1The bene¢ts of discretion with known demands



state-by-state).5 Implicit in this formulation is an assumption that the¢rm can commit in advance to produce for all states of demand (or,equivalently, the ¢rm invests in sunk costs before it knows the actualdemand function).

Here and throughout the paper, welfare is measured by the weightedsum of (expected) consumer surplus and pro¢ts where the relative weighton pro¢ts is 0 � a � 1, i.e. welfare is E�v� ap�, where E�� denotes theregulator's expectations over the relevant private information.

Example 3. Multiplicative shocks to demand

Let h be a vector of random variables a¡ecting consumer demand. Con-sumers have quasi-linear utility, the consumer surplus function is

Pn

i�1yivi�pi�;and demand for product i is Qi�pi; yi� � yiqi�pi� where qi � ÿv0i. Therefore,there are no cross-price e¡ects in consumer demand, and the parameter yi is amultiplicative shock to demand for product i. Costs are known, andC�Q� � F� c �Q where F > 0. (The case where F � 0 is covered in Example1 above.) Pro¢t therefore is

Pn

i�1 yipi�pi� ÿ F, where pi�p� � qi�p��pÿ ci�.Let P be any set of price vectors from which the ¢rm is allowed to

choose, and for each h let p�h� maximize pro¢ts subject to p 2 P. The exante participation constraint is

�2� EXn

i�1yipi�pi�h��

" #� F;

and the regulator aims to maximize

�3� EXn

i�1yi vi�pi�h�� api�pi�h�� " #

subject to Equation (2) and the function p�h� being implementableöi.e. achoice set P can be found that induces the ¢rm to choose prices p�h�.

In general, determining which functions p�h� are implementable is a dif-¢cult task. However, if we ignore the constraint that p�h� be implementableand allow any function p, and if we ¢nd that the solution to this relaxedproblem is itself implementable, then it must be the solution to the fullyconstrained problem. So let us ¢nd the unrestricted function p��h� that

5 The main reason we do this is so that we do not have to consider the set of ¢rms thatchoose not to produce. The analysis in Armstrong [1996] shows that when there is an ex postconstraint in screening models with multidimensional, continuously distributed private in-formation we expect that the set of ¢rms which do not produce will have positive measure.This tends to make examples di¤cult to compute, and so we consider the ex ante type ofconstraint. Another method of overcoming this `exclusion' result is to consider simple discretemodels which, even with an ex post constraint, can be consistent with full production. (Thislatter method is used from Section III onwards.)



maximizes Equation (3) subject to Equation (2). Using standard argu-ments the function p��h� solves this relaxed problem when we choose amultiplier l > 0 such that

�4� p��h� maximizes EXn

i�1yi vi�pi�h�� a� l�pi�pi�h�� " #

and the resulting p��h� makes Equation (2) hold with equality. However,it is clear that for any l the function p��h� that maximizes the aboveexpression is constant: p�i is just chosen to maximize vi�pi� � �a� l�pi�pi�.We deduce that zero discretion is optimal in this case. The optimal ¢xedprice vector is the Ramsey price vector p� that maximizes

Pn

i�1 �yivi�pi�subject to

Pn

i�1 �yipi�pi� � F, where �yi is the mean of yi: Obviously there isno di¤culty implementing this (or any other) ¢xed price vector, and so thesolution to the `relaxed' problem solves the fully constrained problem.6

Example 4. Additive shocks to demand

Suppose the ¢rm has the same cost function as in Example 3. Now letconsumer surplus be v�p� �Pn

i�1��pi ÿ pi�yi, where v is a known consumersurplus function and �p is some (high) reference price vector. Therefore,consumer demands are Qi�p; h� � qi�p� � yi where qi � ÿ@[email protected] let p� be any ¢xed vector of prices that just satis¢es the ¢rm's ex

ante participation constraint, so that

�5�Xn

i�1�p�i ÿ ci��qi�p�� hi� � F;

where �yi is the mean of hi. Expected welfare is

v�p�� Xn

i�1��pi ÿ p�i ��hi � a

Xn

i�1�p�i ÿ ci��qi�p�� hi� ÿ F

" #which from (5) is

�6� W �p�� Xn

i�1��pi ÿ ci��hi ;

where W �p� � v�p� �Pn

i�1�pi ÿ ci�qi�p� ÿ F. De¢ne

�7� P � fp j W �p� � W �p�� Eg

6 As with Example 1 this is non-robust in a number of ways. For instance, we need marginalcosts to be constant for the argument to work. We also need there to be no cross-price e¡ectsin demand across markets. (The one exception to this is that the scalar formulation, wherebyconsumer surplus is yv�p� and all demands, yqi�p�, are shifted up and down by the same scalarshock y, works in the same way.) Unfortunately, this `relaxed method' seems to work in noother interesting cases.



where E is a positive number de¢ned belowösee Figure 2 for anillustration of this set. (Note that, since W is maximised at p� � c, forsmall enough E this set is non-empty provided p� 6� c, something that mustbe the case whenever F > 0 if the participation constraint is satis¢ed.)

We claim that for a particular choice of E welfare is strictly higherwhen the ¢rm is granted a choice from P compared to the case where itmust o¡er the ¢xed price vector p�. Clearly, when there is somevariability in the parameter h pro¢ts are strictly higher with the choiceset P when E � 0 in Equation (7) since the ¢rm then has greater freedomof choice. Therefore, by continuity, for E strictly positive but smallenough the ¢rm's expected pro¢ts are still higher with P than with the¢xed vector p�; and so the ex ante participation constraint is satis¢ed.Therefore, let E be the largest value of this constant which just satis¢esthe ¢rm's participation constraint when discretion is allowed. Sinceexpected pro¢ts are zero with this choice of E, from Equation (7),expected welfare with the set P is at least

W �p�� E�Xn

i�1��pi ÿ ci��hi

which is greater than Equation (6). In sum, unless marginal cost pricing

Figure 2The bene¢ts of discretion with additive shocks to demand



is feasible, with additive shocks to demand it is never optimal to have zerodiscretion.

The intuitive reason for why discretion is desirable in the case ofadditive but not multiplicative shocks is quite straightforward. What isneeded to implement Ramsey prices is knowledge of marginal costs anddemand elasticities. In both examples marginal costs are known, but withmultiplicative shocks demand elasticities are also knownöfor any givenprice the elasticity in a market does not depend on the shock yi. Therefore,in combination with the ex ante participation constraint, all relevant in-formation is known for calculating Ramsey prices (and such prices do notdepend on demand realisations). In Example 4, by contrast, elasticitiesdo vary with demand realisations, and it is optimal to decentralize decision-making to the ¢rm to some extent. We will see in Section III(ii) that ifdiscretion is to be bene¢cial it is important that elasticities vary with thescale demands in the `right' way, i.e. that a positive shock to demand in amarket causes the elasticity to fall.

iii. optimal pricing discretion: symmetric cases

These four examples have shed some light on the question of when to givepricing discretion to the ¢rmöa degree of discretion is always optimalwhen the uncertainty concerns only the ¢rm's costs, and it is ambiguouswhether discretion is desirable with demand uncertainty. However, exceptfor the simple no-discretion results in Examples 1 and 3, we have noexamples of what the optimal set of allowed prices should be. In thissection we examine this question using a stylized model where privateinformation is binary and symmetric.7

III(i). Cost Uncertainty

Suppose there are two independent markets, labelled 1 and 2, each withthe known consumer demand function q�p� and consumer surplus functionv�p�. We assume that the elasticity of demand Z�p� � ÿpq0�p�=q�p� is(weakly) increasing in price.8 The constant unit cost in each market can beeither cL or cH, where cL < cH.

9 Suppose the probability that the ¢rm has

7 See Armstrong and Rochet [1999] for an analysis of multi-dimensional screening modelswhere private information takes this binary form, but where there are transfers between theprincipal and agent.

8 This is just a regularity condition that guarantees the ¢rst-order conditions for Ramseypricing are su¤cient for calculating Ramsey prices. A su¤cient condition for this to hold isfor q�p� to be log-concave.

9 The following analysis is not a¡ected signi¢cantly by allowing for (i) cross-price e¡ectsin demand or (ii) non-constant marginal costs, and an earlier version of this paper examinedthese minor extensions.



low cost realizations in both markets is bL L , that the ¢rm has high costsin both markets is bHH, and that the probability that the ¢rm has low costin market 1 and high cost in market 2 is the same as the probability thatit has the reverse con¢guration, denoted b. Naturally, bL L � 2b� bHH � 1.There is a ¢xed cost of F. We will suppose in what follows that it isoptimal to produce both products for all cost realizations.

Write pi�p� � q�p��pÿ ci� for i � L ;H, and suppose that the two pro¢tfunctions are increasing in price over the relevant range of prices. Bysymmetry, the regulator o¡ers the ¢rm a choice of four price vectors, say

P � f�pL L ; pL L �; �pHH; pHH�; �pL ; pH�; �pH; pL �g;where �pL L ; pL L � is designed to be chosen by the type-L L ¢rm, �pHH; pHH�by the type-HH ¢rm, �pL ; pH� by the type-L H ¢rm, and �pH; pL � by thetype-HL ¢rm.

One thing that is immediate from the assumption that pro¢ts areincreasing in prices is that if two distinct vectors �pL L ; pL L � and �pHH; pHH�can be chosen, no type of ¢rm will choose the lower of the pair. In otherwords, incentive compatibility implies that pL L � pHH � p, say. Given this,there are ¢ve constraints on the three prices p; pL and pH: one relevantparticipation constraint

�8� 2pH�p� � F

and four incentive compatibility constraints

�9� 2pH�p� � pH�pL � � pH�pH��10� 2pL �p� � pL �pL � � pL �pH��11� pL �pL � � pH�pH� � pL �p� � pH�p��12� pL �pL � � pH�pH� � pL �pH� � pH�pL �:(The participation constraints for the other three types of ¢rm areautomatically satis¢ed given the incentive constraints.) Note that (12)holds if and only if pL � pH.

10 Therefore, the problem is to choose p; pL

and pH in order to maximize expected total welfare subject to the aboveconstraints. Suppose the prices that solve this problem are p�; p�L and p�H.We will solve the problem by ignoring the constraints (9), (10) and

pL � pH, and then check ex post that these constraints are indeed satis¢edat the optimum. But (8) binding implies that p� is given by

�13� pH�p�� 12

F

10 The fourth incentive constraint can be written as �q�pL � ÿ q�pH��cH ÿ cL � � 0 which issatis¢ed if and only if pL � pH.



and there is average cost pricing for the symmetric high cost ¢rm.Secondly, in the case where the ¢rm has asymmetric cost realizationsmaximizing total welfare subject to (11) and (13) is the same problem asmaximizing v�pL � � v�pH� subject to (11) with p � p� given by Equation(13). This is just a standard Ramsey problem, and has the familiar ¢rst-order conditions involving inverse elasticities:

�14� p�L ÿ cL

p�L� l

Z�p�L �;

p�H ÿ cH

p�H� l

Z�p�H�where l > 0 is chosen to make (11) bind. However, this expression,together with the assumption that Z is weakly increasing in p, implies thatp�L < p�H and hence (12) holds. (Note that the strict inequality implies thatdiscretion is optimal.) It is also straightforward to show that the remainingincentive constraints (9) and (10) hold. Therefore the regulator shouldo¡er the ¢rm a choice from the three price vectors f�p�; p��; �p�L ; p�H�;�p�H; p�L �g. Figure 3 illustrates the three price vectors at the optimum.

We deduce that (i) subject to the condition that production is alwaysdesirable and that the probabilities of the cost realizations are symmetric,

Figure 3Cost uncertainty



the optimal set of allowed prices does not depend on the probabilitydistribution for costs; (ii) the incentive constraints are such that thebinding constraint is that asymmetric ¢rms not be tempted to choose theprices designed for the symmetric ¢rms, and not vice versa; (iii) the optimalset of price vectors does not take the form whereby the ¢rm can chooseany price vector which guarantees consumers a given level of surplus asillustrated in Figure 1 above; (iv) as in the standard Ramsey problem withfull information, the optimum does not depend on a, the weight placedon pro¢t in social welfare, and ¢nally (v) for each cost realization theprices chosen are always Ramsey-like prices, that is, prices that maximizeconsumer surplus subject to a pro¢t constraint. We will see in Section IVthat some of these properties depend on the symmetry of this example.

III(ii). Demand Uncertainty

Suppose again that there are two markets, and that the ¢rm has a knownconstant marginal cost c of serving each market. (See the discussion at theend of this section for the extension to non-constant marginal costs.) Ineach market, the demand function can be either qL �p� or qH�p�. (Again,there are no cross-price e¡ects across marketsösee the discussion at theend of this section for the extension to inter-dependent demands.) Assumethat qL � qH. Let vi�p� be the consumer surplus function corresponding todemand function qi�p�. Let Zi�p� denote the elasticity of demand in statei � L ;H, and again suppose that Z0i � 0. Write pi�p� � qi�p��pÿ c� fori � L ;H, and suppose that these pro¢t functions are increasing over therelevant range, and that

�15� pH�p� ÿ pL �p� is increasing in p

(provided p > c).11 This mild assumption implies that when demands areasymmetric the ¢rm prefers to set the higher price in the higher demandmarket.

As in the previous section, by symmetry the regulator o¡ers the ¢rm achoice of three price vectors

P � f�p; p�; �pL ; pH�; �pH; pL �g;and there are ¢ve relevant constraints on the three prices: one relevantparticipation constraint

�16� pL �p�� 12

F

(which implies there is average cost pricing for the symmetric low demand

11Note that this condition is automatically satis¢ed in the previous case of unknowncosts.



¢rm) and the four incentive compatibility constraints (9)^(12) above.Condition (15) implies, as with cost uncertainty, that (12) is satis¢ed if andonly if pL � pH. Therefore, the problem is to choose p; pL and pH in orderto maximize expected total welfare subject to the above constraints.Suppose the prices that solve this problem are p�; p�L and p�H.

There are two cases of interest:

Case 1: ZH�p� � ZL �p�.This is the case where the high demand realization is less elastic than

low demand, as in the case of additive shocks to demand in example 4above. (The case of multiplicative shocks is the knife-edge case that dividesCases 1 and 2 here.) We again solve the problem by ignoring the threeconstraints (9), (10) and pL � pH, and checking ex post that theseconstraints are satis¢ed at the optimum.

When the ¢rm has asymmetric demand realizations maximizing totalwelfare subject to Equations (11) and (16) is the same problem asmaximizing vL �pL � � vH�pH� subject to (11) where p � p� is given by (16).This is just a Ramsey problem with ¢rst-order conditions

�17� p�L ÿ c

p�L� l

ZL �p�L �;

p�H ÿ c

p�H� l

ZH�p�H�where l > 0 is chosen to make (11) bind. However, this expression,together with the assumption that each Zi is increasing in p and ZH � ZL ,implies that p�L � p�H and so constraint (12) holds. Discretion is optimalprovided that p�L < p�H which is the case when ZH < ZL . Again, it isstraightforward to show that the remaining incentive constraints (9) and(10) also hold. Figure 4 illustrates the three price vectors at the optimum.

Case 2: ZH�p� � ZL �p�.This is the case where the high demand realization implies that demand

is more elastic (but this e¡ect is not so severe that condition (15) isviolated). Here, our strategy is to maximize expected welfare subject to(11), but now to introduce the necessary incentive constraint pL � pH. Asbefore, p� is given by Equation (16).

Lemma 1. If the prices pL and pH are such that pL � pH and

pL �pL � � pH�pH� � pL �p�� pH�p��then

vL �pL � � vH�pH� � vL �p�� vH�p��:

Proof. First note that



�18� p0H�pH�p0L �pL �

<v0H�pH�v0L �pL �

whenever pH � pL �� c�. For

p0i�pi� � qi�pi� 1ÿ Zi�pi�pi ÿ c

pi

� �and so Equation (18) holds if and only if

ZL �pL �pL ÿ c

pL

< ZH�pH�pH ÿ c

pH

which holds if pH � pL �� c� given that ZL �p� � ZH�p� and Z0i � 0.But (18) implies that, in the region pL � pH, the `iso-pro¢t' curve

through the point �p�; p�� lies inside the `iso-consumer surplus' curvethrough the same pointösee Figure 5.

In particular, any price vector in the region pL � pH that the ¢rm prefersto �p�; p�� makes consumers worse o¡.

This lemma implies that the solution to the problem of maximizing

Figure 4Demand uncertainty: Case 1



welfare subject to the two constraints (11) and pL � pH is given byp � pL � pH � p�, where p� is given by Equation (16). In particular, nodiscretion should be given to the ¢rm.12

This analysis of pricing discretion with unknown demand functionscontrasts in important respects with the case of unknown cost functions.As is clear from Figure 1 above, and corroborated with the more detailedanalysis in Section III(i), it is always optimal to allow the ¢rm somediscretion in its pricing policy when there is uncertainty about costs butnot about consumer demand: in this case the ¢rm's private incentiveswhen choosing its relative prices can be aligned to socially desirableoutcomes.

Matters are less clear-cut when there is uncertainty about consumerdemand. (This ambiguity has already been made clear by Examples 3 and4.) The pro¢t di¡erence pH ÿ pL being increasing in p implies, loosely

Figure 5Demand uncertainty: Case 2

12 As mentioned above, the case of multiplicative shocks to demand (Example 3 above) isthe functional form that divides Cases 1 and 2. Thus the reason that zero discretion is optimalin that case is not so much that the ¢rm's incentives go strictly the `wrong way', but ratherthat the regulator does not wish to o¡er non-uniform prices to a ¢rm with asymmetricdemand.



speaking, that the ¢rm with one low and one high demand market wouldlike to set a higher price in the larger market. In Case 1, when largermarkets have lower elasticities, the interests of social welfare are alsoserved if there is a higher price in the larger market. (This is just thestandard Ramsey inverse elasticity result.) Therefore, both the ¢rm's andsociety's interests are roughly aligned, which implies that some discretioncan bene¢cially be given to the ¢rm. In Case 2, by contrast, these interestsdiverge: because the larger market now has a greater elasticity, socialwelfare is enhanced by setting a lower price in the larger market. Since the¢rm cannot be trusted to act in the public interest, it is no longer optimalto grant any discretion to the ¢rm.13

This analysis is closely related to that of Lewis and Sappington [1988a]who examine optimal regulation (using lump-sum transfers) of a single-product ¢rm with a known cost function but private information aboutconsumer demand. They show that the solution to the problem dependscrucially on whether marginal cost is decreasing or increasing in output. Inthe latter case they show that the ¢rst best outcome, in which price alwaysequals marginal cost can be obtained, whereas when marginal costs aredecreasing it is optimal to give the ¢rm zero discretion over pricing policy.In more detail, they write the demand function as Q�p; y�; where Qy > 0;and the pro¢t function as p�p; y� � pQ�p; y� ÿ C�Q�p; y��. They assumethat py�p; y� is increasing in p which is the continuous version of (15)above. This single-crossing condition implies that under any regulatoryscheme the ¢rm's price p�y� must be (weakly) increasing in y. By contrast,if C is concave then the optimal price (with full information) is decreasingin y, in which case no discretion should be given to the ¢rm. As Lewisand Sappington (p. 993) put it: `when the regulated price must rise asdemand increases to ensure incentive compatibility, price is proceeding ina direction opposite to that of the ¢rst-best price. Thus, allowing the ¢rmto use its private knowledge of demand to select the market price becomesvery costly . . . so costly, in fact, that no delegation of pricing authorityis allowed.'

This suggests that an extra e¡ect is introduced if we allow marginalcosts to vary. So suppose now that the cost function in each market isC�q� rather than cq. In order to be able to use only the ¢rst-orderconditions for Ramsey prices, suppose that the Lerner indexpÿ C0�qi�p�� =p is increasing in price (so that marginal cost does not

13 The principles underlying this analysis are the same as those in the duopoly model ofprice discrimination in Armstrong and Vickers [1999, Section IV(i)], where two ¢rms competeagainst each other in two separate markets. In that paper we show that when the `morecompetitive' market (i.e. the market with the lower equilibrium price) is also more elastic then¢rms should usually be permitted to set di¡erent prices in the two markets. However, if themore competitive market is less elastic, then Ramsey principles imply that welfare is higher ifprice discrimination is banned.



decrease too fast). Then the above analysis can be extended to show thatCase 1 occurs provided that

ZL �p�pÿ C0�qL �p��

p� ZH�p�

pÿ C0�qH�p��p

in which case discretion is again optimal. If this inequality does not holdthen Case 2 holds and zero discretion is optimal. Thus if marginal cost C0

is increasing then we are more likely to want discretion, and the converseis the case if marginal costs decrease.

III(iii). Cost and Demand Uncertainty

Having examined cost and demand uncertainty individually, we nowdiscuss implications of their combination for the desirability of giving the¢rm some pricing discretion.14 Thus in each market there are fourpossibilities: the ¢rm could have high or low cost, and high or lowdemand.

With Case 1 demand uncertainty (i.e. lower elasticity in the largermarket), the addition of cost uncertainty naturally reinforces the case forhaving some discretion: private and social incentives are aligned withrespect to both kinds of uncertainty. However, with Case 2 demand un-certainty (i.e. higher elasticity in the larger market), it is ambiguous whetherthe addition of cost uncertainty makes some discretion desirable. With`enough' cost uncertainty, zero discretion ceases to be optimal. To see thissuppose by contrast that zero discretion was optimal. If production isalways desirable this ¢xed price vector is given by �p�; p�� where p� is givenby

�p� ÿ cH�qL �p�� 12

F:

(This price vector ensures non-negative pro¢t even in the worst state oflow demand and high cost in both markets.) Now consider allowing the¢rm to choose any price vector that consumers weakly prefer to �p�; p�� inall states, so that

�19� P � fp j vi�p1� � vj�p2� � vi�p�� vj�p�� for all i; j � L ;Hg:Figure 6 illustrates.

14 See Lewis and Sappington [1988b] and Armstrong [1999] for analyses of optimalregulation (where use is made of lump-sum transfers) of a single-product ¢rm facing unknowndemand and cost functions. Again there is in general a con£ict between social and privateincentives, and given discretion the ¢rm with higher marginal costs will not always choose thehigher price. One result is that it may be optimal to make the ¢rm choose a price belowmarginal cost.



Consider the pro¢tability of a small price change to �p� � dp1;p� ÿ dp2�, where qH�p��dp1 � qL �p��dp2 > 0, so that the new price vector isin P and consumers are certainly no worse o¡ compared to �p�; p��: If costlevels in markets 1 and 2 are cH and cL respectively, then the change inpro¢t is

�20�dp � qH � �p� ÿ cH�q0H�p�� dp1 ÿ qL � �p� ÿ cL �q0L �p�� dp2

� p� ÿ cL

p�

� �ZL �p�� ÿ

p� ÿ cH

p�

� �ZH�p��

� �qH�p��dp1:

From Equation (20) and the de¢nition of p� it follows that dp > 0 if

�21� cH ÿ cL

cL

� �>

F

2cL qL

� �ZH ÿ ZL

ZL

� �:

If the relative cost di¡erence is large enough relative to the relativeelasticity di¡erence so that Equation (21) holds, then the ¢rm canpro¢tably use the discretion allowed in P. Since by construction consumersare never worse o¡ with this discretion, we deduce that some pricingdiscretion is certainly desirable.

If the relative cost di¡erence is small, however, then zero discretion

Figure 6Price set that ensures consumers gain



remains optimal in Case 2. If the cost di¡erence is small enough in relationto the elasticity di¡erence, then the ¢rm facing asymmetric demand willalways prefer to charge the higher of two prices in the market with highdemand and high elasticityöeven if cost is lower in that market. Whendemands are asymmetric, that is bad for welfare, and doubly so when costis lower in the more elastic market. If the probability of an asymmetricdemand realization is large enough, this welfare loss outweighs any gainsfrom discretion in other states (e.g. that of symmetric demands andasymmetric costs). Thus the result that zero discretion may be optimal isrobust to the introduction of at least some cost uncertainty.

III(iv). Policy with Lump-sum Transfers

In this paper we discuss pricing policy when regulation does not permitthe use of lump-sum transfers. As a result prices must be chosen that allowthe ¢rm to generate su¤cient revenue to cover its costs. It is natural, how-ever, to consider how policy is modi¢ed when transfers are permitted.

The case of uncertain costs has been analysed in Dana [1993] andArmstrong and Rochet [1999, Section III(ii)], both of which assume thesame binary distribution for costs as we do here. For our purposes themain results are: (i) it is always optimal to give the ¢rm some discretionover its prices; (ii) the optimal menu of contracts depends on theprobability distribution for costs (even in symmetric cases); (iii) theamount of distortion away from ¢rst-best marginal cost pricing dependson a, the relative weight placed on pro¢ts compared to consumer surplus,and (iv) except in extreme cases (negative correlation in costs combinedwith large asymmetries across markets) it is optimal to have marginal costpricing in any market where the ¢rm has a low cost realisation. (Thuspoint (iv) is a multiproduct generalization of Baron and Myerson's [1982]famous result that the most e¤cient ¢rm should not o¡er a distortedprice.)

This analysis shows that some of the conclusions of the model in SectionIII(i)önamely, that the optimal prices did not vary with the welfareweight a or the probability distribution on costsödepended on theassumption that transfers were not possible. In the next section we see thatthese results are also sensitive to the assumption that the two markets aresymmetric.

iv. an asymmetric case: optimal departures from ramsey pricing

Here we modify the model in Section III(i) in two ways: we allow thetwo markets to be asymmetric in terms of both costs and demands, and weconsider cost uncertainty in only one market. The case of cost uncertaintyin both markets when there is asymmetry involves a more complicated



analysis and several di¡erent cases (depending on parameter values), andfor simplicity we discuss the straightforward, one-market uncertainty case.This simple analysis is su¤cient to show that some of the conclusions ofthe symmetric model discussed above are fragile.

Therefore, suppose that in market 1 the ¢rm's unit cost could be cL

with probability b and cH > cL with probability 1ÿ b. In market 2 theunit cost is c for certain. Known demand in markets 1 and 2 is q1�p� andq2�p� respectively. Write pi�p� � q1�p��pÿ ci� for i � L ;H, and p�p� �q2�p� �pÿ c�. As before, we assume that all pro¢t functions are increasingin price over the relevant range.

Suppose the regulator o¡ers the ¢rm the choice of two price vectors:

P � f�pL1 ; p

L2 �; �pH

1 ; pH2 �g:

There is just one relevant participation constraint

�23� pH�pH1 � � p�pH

2 � � F

and two incentive compatibility constraints

�24� pH�pH1 � � p�pH

2 � � pH�pL1 � � p�pL

2 ��25� pL �pL

1 � � p�pL2 � � pL �pH

1 � � p�pH2 �:

Note that, provided Equation (25) is binding, Equation (24) is satis¢edif and only if pL

1 � pH1 : We use the usual method and try to solve the

problem subject to the two constraints (23) and (25) and check ex postthat pL

1 � pH1 holds.

De¢ne V �R� to be the maximum consumer surplus given that the lowcost ¢rm receives rent R � 0, i.e.

�26� V �R� � max : fv1�p1� � v2�p2� j pL �p1� � p�p2� � F� Rg:Naturally, V �R� and V �R� � R are decreasing in R. Moreover, given ourincreasing elasticity assumption it follows that the Ramsey prices thatsolve Equation (26) are each increasing in R. Given that the twoconstraints (23) and (25) bind it follows that the rent of the low cost ¢rm isR�pH

1 � � �cH ÿ cL �q1�pH1 � > 0, and so the rent of the low cost ¢rm is

decreasing in the high cost ¢rm's price in market 1. Therefore, the problembecomes one of choosing �pH

1 ; pH2 � to maximize

�27� �1ÿ b� v1�pH1 � � v2�pH

2 �� b V �R�pH

1 �� aR�pH1 �

� �subject to Equation (23). (Recall that a � 1 is the weight placed on pro¢tsin the welfare function.) Notice that since V �R� � R is decreasing in R forR, V �R� � aR is decreasing in R; and hence the second bracketed term in(27) is increasing in pH

1 .Let �pR

1 ; pR2 � denote the Ramsey prices for the high cost ¢rm under full-

information, i.e. these prices maximize consumer surplus v1�pH1 � � v2�pH

2 �



subject to (23). Then since the second bracketed term in (27) is increasingin pH

1 , it follows that the solution to (27) involves distortions of the form

�28� pH1 > pR

1 ; pH2 < pR

2 :

All that remains to do is to check that pL1 � pH

1 holds:

Lemma 2. If �pH1 ; p

H2 � maximizes (27) and �pL

1 ; pL2 � is the solution to

problem (26) with R � �cH ÿ cL �q1�pH1 �, then pL

1 � pH1 and hence this pair of

price vectors maximizes welfare.

Proof. As a ¢rst step, let �p�1; p�2� maximize consumer surplus subject to

�29� pL �p1� � p�p2� � pL �pR1 � � p�pR

2 �;where �pR

1 ; pR2 � are the Ramsey prices for the high cost ¢rm. (In other

words, �p�1; p�2� is the solution to (26) with R � �cH ÿ cL �q1�pR1 �.) Then it

must be that p�1 � pR1 . For if not, so that p�1 > pR

1 , then pro¢ts for the highcost ¢rm are strictly higher with prices �p�1; p�2� than with �pR

1 ; pR2 �, which in

turn implies that v1�p�1� � v2�p�2� < v1�pR1 � � v2�pR

2 � (since �pR1 ; p

R2 � maximizes

welfare subject to the budget constraint). However, since �pR1 ; p

R2 � is a

feasible choice of prices when maximizing welfare subject to Equation(29), it follows that v1�p�1� � v2�p�2� � v1�pR

1 � � v2�pR2 � which is a contra-

diction. We deduce that p�1 � pR1 .

However, Equation (28) tells us that pR1 � pH

1 . In particular, the rent R

given to the low cost ¢rm is lower than that corresponding to prices�pR

1 ; pR2 � and since a tightened budget constraint lowers all Ramsey prices,

it follows that p�1 � pL1 and the result is proved.

The prices designed for the high cost ¢rm are distorted away from full-information Ramsey prices, and in particular the price for the market withuncertain cost is distorted upwards in order to reduce the rent paid to thelow cost ¢rm. It is straightforward to show that this distortion is greaterwhen (i) the probability b of the ¢rm having low costs is high, and (ii) theweight a placed on pro¢ts is low. In particular, aspects of the symmetricmodel in Section III(i) do not carry over to this case, including the featuresthat the optimum there did not depend on the probability weights nor onthe welfare weight placed on pro¢ts compared to consumer surplus, andthat prices in all cost realizations were Ramsey-like. In this asymmetricsetting, then, we obtain a `second-best' version of Baron and Myerson's[1982] result that the most e¤cient ¢rm should not o¡er distorted pricesbut less e¤cient ¢rms should restrict output. However, the crucial dif-ference with Baron and Myerson is that here we do not allow transfers,and hence there is `no distortion' for e¤cient ¢rms when compared toRamsey prices rather than marginal cost prices.



v. conclusions

The desirability of pricing discretion depends on whether private andsocial interests concerning its exercise are aligned or opposed. Whenuncertainty is about costs they are aligned: it is good for both pro¢ts andwelfare to have higher relative prices in markets where relative costs arehigher. With demand uncertainty, however, private and social interestsmay or may not be aligned. It is socially desirable (for Ramsey reasons) tohave higher prices where demand elasticities are lower, but it is generallypro¢table to have higher prices where the scale of demand is greater. If theelasticity and scale of demand are negatively related, discretion is thereforedesirable. But if they are positively related (and there is no costuncertainty), there might be no way to exploit the ¢rm's privateinformation that is both incentive compatible and socially bene¢cial; insuch cases zero discretion is best.

ACCEPTED APRIL 1999

references

Armstrong, M., 1996, `Multiproduct Nonlinear Pricing', Econometrica, 64,pp. 51^75.

Armstrong, M., 1999, `Optimal Regulation with Unknown Demand and CostFunctions', Journal of Economic Theory, 84, pp. 196^215.

Armstrong, M., Cowan, S. and Vickers, J., 1994, Regulatory Reform: EconomicAnalysis and British Experience, MIT Press, Cambridge, MA.

Armstrong, M. and Rochet, J. C., 1999, `Multi-Dimensional Screening: A User'sGuide', European Economic Review, 43, pp. 959^979.

Armstrong, M. and Vickers, J., 1993, `Price Discrimination, Competition andRegulation', Journal of Industrial Economics, 41, pp. 335^360.

Armstrong, M. and Vickers, J., 1999, `Competitive Price Discrimination', mimeo.Baron, D. and Myerson, R., 1982, `Regulating a Monopolist with UnknownCosts', Econometrica, 50, pp. 911^930.

Dana, J., 1993, `The Organization and Scope of Agents', Journal of EconomicTheory, 59, pp. 288^310.

La¡ont, J. J. and Tirole, J., 1993, A Theory of Incentives in Procurement andRegulation, MIT Press, Cambridge, MA.

Lewis, T. and Sappington, D., 1988a, `Regulating a Monopolist with UnknownDemand', American Economic Review, 78, pp. 986^998.

Lewis, T. and Sappington, D., 1988b, `Regulating a Monopolist with UnknownDemand and Cost Functions', Rand Journal of Economics, 19, pp. 438^457.

Oftel, 1996, Pricing of Telecommunications Services from 1997: A Statement (Oftel,London).

Varian, H., 1989, `Price Discrimination', in Schmalensee, R. and Willig, R. (eds.),Handbook of Industrial Organization (North Holland, Amsterdam).

Vogelsang, I., 1989, `Price cap regulation of telecommunications services: a long-run approach', in Crew, M. (ed.), Price-Cap Regulation and Incentive Regulation(Kluwer, Amsterdam).



Multiproduct Price Regulation Under Asymmetric Information

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