Price-cap regulation and market definition

Journal of Regulatory Economics; 5:337-347 (1993) �9 Kluwer Academic Publishers

Price-Cap Regulation and Market Definition

IAN BRADLEY University of Leicester

Department of Economics, Leicester, United Kingdom LE1 7RH

Abstract One of the merits claimed for certain types of price-cap regulation is the possible long-run convergence of the prices of multi-product firms to Ramsey prices. Typically such regulated firms define commodities by such devices as dividing the day into discrete periods, customers into age-groups, distances into ranges, and so on. Allowing that such division is endogenous throws doubt on the ability of Laspeyre quantity-based price-caps to encourage an efficient market definition and hence to generate an efficient price structure.

Introduction

Some types of basket price-cap regulation allow the regulated multi-product firm to change the definition of goods within the basket, transferring particular commodities from one category to another. For instance the regulation of British Telecom does not preclude the transfer of particular telecommunication routes from one price band to another (and under regulation a number of such transfers has taken place) or a change in the time of day when rates change. This regulation, known as RPI-X since the firm is constrained in changing prices so that last year's sales evaluated at this year's prices must be no more than the inflation rate minus X% above last year's revenue, can be interpreted as a modification of the mechanism of Vogelsang and Finsinger (V-F) (1979). Their mechanism allows the firm to set any prices it likes, provided the value of last year's sales at these prices is no greater than last year's costs. Thus, both RPI-X and V-F have price-caps defined as a quantity weighted Laspeyre chain index. However, in examining the welfare properties, and in particular the pricing policies of multi-product firms that such regulatory mechanisms induce, it has been common practice to take the definition of commodities and pricing structure as given. That these mechanisms under certain assumptions have the property of convergence to an efficient price vector (see for instance Vogelsang and Finsinger (1979), Brennan (1989), Bradley and Price (1988)) has been seen as a very desirable characteristic. This paper concentrates on this aspect of price regulation ignoring the extremely important issues of cost reducing incentives, strategic manipulation of the regulation, and the effects on product quality. (For a summary of the main questions, see Acton and Vogelsang (1989)). The best chance of achieving efficient price structures in the long run, under price-caps

338 IAN BRADLEY

defined as a Laspeyres chain index, is under unchanging demand and cost conditions with a myopic profit-maximizing firm ( Vogelsang 1988; Brennan 1989). We keep these strong assumptions but allow market definition or division to be endogenous. Then tariff basket regulation will not in general lead to an efficient division of the market for pricing purposes.

A General Model of Market Division

Market division, here, means the usual separation of markets and price bands associated with the standard literature on price discrimination. Thus, examples of division include time of day or season, distance, weight, age, division points on multi-part tariffs, and so on. We suppose we can represent such divisions for a monopoly firm by a vector T.

Assuming quasilinear utility, which is equivalent to assuming that we can use consumers' surplus as a measure of welfare (Varian 1985), enables us to treat demand as if it were derived from a consumer with income y and an indirect utility function

V ( p , T , y ) = v ( p , T ) + y ,

where p is a vector of the firm's prices associated with each market division. For instance, British Telecom divides the day into three price regions: peak (8 a.m. to 1 p.m.), standard (lp.m. to 6 p.m.), and cheap (6 p.m. to 8 a.m.). Defining T in such a way that the higher is Tj the more sales are defined as being in categoryj with pricepj; this could be described by

P = [Pl, P2, P3 ] = [peak rate, standard rate, cheap rate]

T = [T 1 , T 2 , T 3 ] = [1300, 1800, 0800].

We assume 1 that such characteristics are defined so that they can be varied in a continuous fashion, but the conclusions are unaffected if, for instance, it was unthinkable to have time periods starting or ending on anything but a "quarter hour."

The profits of the firm are represented by u (p,T). For Ramsey pricing, we need to maximize v (p,T). subject to u (p,T). being at least some specified level. First-order conditions are

- -+~, 0 (11

and Ov

+ = o. O R (2)

On the other hand an unconstrained firm simply divides the market so that

- - ~ 0 ~ ~ ~pi (3)

Equations (1) and (2) have a straightforward equi-marginai interpretation. At an efficient market division and set of prices, allowing an extra dollar to be made in profits should cause the same loss of consumer surplus whichever price or market division is changed to give the extra profit. Defining T (as in the examples in note 1) in such a way that the higher is Tj the

PRICE-CAP REGULATION AND MARKET DEFINITION 339

more sales are defined as being in categoryj with price pj, and the less in the neighboring categoryj + 1, then equation (2) implies that

. sign ~ j j = mgn (p j -P j+I ) (4)

When profits are constrained and market division is efficient, a firm would earn more profits by extending any category of market division into its neighboring cheaper category. There is a symmetry here between market division and prices. Raising marginally any of a set of Ramsey prices increases profits. Increasing the size ofarelatively expensive market division is the same as increasing a price.

For our purposes, we need to examine the Ramsey efficient consumption vector implied by equations (1) and (2).

By Roy's identity, equation (1) implies that the demand for good i is

~V (p,T, y)

bPi _ bv = 3+ ~ Xi = OV (p,r, y) OPi OPi (5)

A firm regulated by mechanisms such as the British RPI-X tariff basket, Vogelsang - Finsinger, or other similar price-caps defined as a Laspeyres chain index will maximize

rc (p,T) subject top.x (-1) being less than a specified amount. (x (-1) representing a vector of quantities sold in the previous period.)

In this case, myopic profit seeking will lead to

b~ lax! -I) = 0

7 p / -

while for right-hand derivatives

~ x J - l ) l + j _<0

and for left-hand derivatives

(6)

(7)

++ M'"] ++-,+,++j-v>,,l. ++ ] _>o. (8)

The use of base quantity weights necessitates this consideration of left- and right-hand

derivatives. Unless demand is perfectly inelastic, then, because Pj+~Pj+I, xJ -1) is not differentiable with respect to Tj at the base market division value. 2

Under stationary cost and demand conditions and with a price-cap determined either in the Vogelsang and Finsinger manner (that is the firm is constrained to set prices such that last year's sales would result in non-positive profits), or simply that the cap is unchanging in the sense that last year's output valued at this year's prices must be no greater than last

340 IAN BRADLEY

year's receipts, equation (6) converges to

l'lXi- OPi" (9)

This is the equivalent of equation (5) and hence the first-order conditions on price required for efficiency (equation(l)) are satisfied. However, equations (7) and (8) clearly allow pricing structures that do not satisfy (2), and hence the market can remain divided in an inefficient manner in the long run.

Some Illustrative Applications

To understand the determinants of efficiency or inefficiency, it is sufficient to consider some very simple examples. We suppose that marginal production costs are zero, demand is unchanging, and the price cap is simply that prices this year must be such that last years output valued at this years prices would not exceed last years total revenue. The forces driving towards efficiency are obvious here. The firm only changes prices if profits (revenue) increases, while any change means that consumers in aggregate could if they wished buy last years outputs for no more than last years expenditure. Every price change benefits the firm and increases consumers' surplus. But this does not mean that all price changes that could benefit both parties will be made. The condition that would guarantee such changes taking place is that equations (7) and (8) are equivalent to (2). That is that the left-hand and right-hand derivatives of xj with respect to Tj are equal and

av - 0"-~j = (PJ-PJ +1) OTj " (10)

Since the equality of right- and left-hand derivatives at a division point would imply perfectly inelastic demand, these conditions will not be satisfied at many interesting long-run equilibria. 3

While, in general, inefficient market divisions will persist into the future, this persistence will often be more marked and easier to understand when demands for categories of goods are clearly interdependent. Consider, for instance, the typical division between cheap and dear rates by time of day. British Telecom has stuck since privatization and price-cap regulation to a 6 p.m. division which goes back under the pre- privatization regime beyond most living memories. It is not unreasonable to suppose that this division causes the delay of some calls in addition to the generation of extra calls through cheaper rates. The number of 'units' used between 1800 and 1815 hours is likely to be very much larger than between 1745 and 1800 hours. Under price-capping, this will lead to stickiness of the 1800 hours division even if it is in everyone's interests to move it. A move to 1745 compensated for the film by slight price increases in both periods may well be beneficial to consumers. However, because of the current relatively low usage between 1745 and 1800, a base- weighted price cap would effectively allow no price rise at all. The situation of heavy use at this period after such a change is not allowed for in the price-cap. Equally, delaying the cheap rate until 1815 is an unlikely decision even if desirable. Because of the mass of calls at the start of the cheap period, any shortening of the length of that period would under


price-capping have to be accompanied by 'unnecessarily' large price falls. This is what the inequalities (7) and (8) indicate. They measure the range of prices for any division of the market that will result in there being no incentive to change the division.

Although the presence of demand interdependence as in the above example makes the problem apparent, it is by no means necessary for the problem to be significant. A particular example may serve to demonstrate this and the resulting difficulties for the regulator.

Let time of day (a) or age of demander or any other market quality range from 0 to 1, and let us suppose demand is linear and given by

qa = 1 - a - p . (11)

We retain, for simplicity, the assumption that marginal costs are zero and suppose that the market is divided into just two parts, so that for a < T price is Pl and for a > T price is P2. Figure 1 indicates the pricing policy of an unregulated profit maximizer for different market divisions. Points U and V (at T = 0.4) are the prices for the market definition that allows highest profits. There is akink in the unregulatedpl schedule at T= 2/3 andpl = 1/3, because it never pays the firm to lower Pl below 1/3 ( the profit-maximizing uniform price for an undivided market).

P

0.5

0.4

0.2

= ,

~ g u l a t e d Pl

v

I I I I

0.4 0.5 1.0 T

Figure 1 Routine application of conditions (1), (2), and (3) shows that, for Ramsey pricing with

the most efficient single division of the market, we require

(1 - 27 ) (2 - 7) (12) Pl - (2 - 3T)

342 IAN BRADLEY

and P2 = 1 - 2T. (13)

What this means is that the efficient division increases from T = 0.4 to T = 0.5 as prices decrease with a lower allowable level of profits or higher fixed costs. This is illuslrated by the lines UR and VR in figure 2 representing equations (12) and (13) respectively. In this case, the length of the "dear period" increases, and consequently the length of the "cheap period" shortens to retain efficiency in the face of more severe price control. This is not a general result.

P

0.5

0.4

0.2

U

0 0.4 0.5 1.o T

Figure 2

The firm that has profits squeezed out of it in the Vogelsang-Finsinger manner, or is simply price regulated so that last year 's output valued at this year 's prices does not exceed last year 's receipts will alter prices until (see equations (6),(7), and (8))

= T T (14) T(1- 2Pl - ~) = IXT(1- Pl - 2 ) = IXX1

[(1 - T-P2 )2 ] Ix [(1 - T-P2 )2] (15) 3p2 - 2 P2 (1 - T - P 2 ) = 2 = laX2

Pl (1 - T - P l ) -P2 ( 1 - T - P 2 ) - I x (p l -P2) ( 1 - T -P2 )<O (16)

i.e., 1 - T - p 1 -P2 - Ix(1 - T - p 2 ) < 0

Pl (1 - T - P l ) -P2 (1 - T - P 2 ) - Ix (Pl - P 2 ) (1 - T - P l ) > 0 (17)


i.e.,1 - T - p ] -P2 - Ix( 1 - T - p l ) > O.

Conditions (14) and (15), taken with (16), imply that if

Pl <P2 + T/Z (18)

(and hence P2 -< T/2), then for all T less than 0.4 , although an increase in T is socially desirable, there is no incentive for the firm to lengthen the expensive 'period' or equivalently re-categorize the market.

Using (14) and (15), with (17), we can likewise obtain conditions which will lead to an equilibrium but 'too short' a cheap 'period'. Any Ramsey prices for fixed T (> 0.4) that satisfy

( r + ~ ) P l < l 2 (19)

will leave such a situation. In figure 3, the lines UO and VO show the boundaries for condition (18), and the curves US and VS show condition (19). While this is a particular example, one suspects that it shares certain of its characteristics with many other demand and cost structures. The likelihood of being stuck with the 'wrong' market definition is higher the lower are prices (the higher are fixed costs or the more severe is the regulator's definition of profits), but, if prior to the imposition of the cap, the market division was a long way from efficiency then there may be some movement in the right direction. However, there is nothing definite that the regulator can use to say whether the firm is at or near an efficient market division. Equilibrium points such as a , b , or c in figure 4 represent very

p

0.5

0.4

0.2

0 0.4 0.5 1.0 T

Figure 3

344 IAN BRADLEY

P

0 . 5

0.4 U

a

0.2

0 O.4 0.5 1.0 T

Figure 4

different market divisions, but if these points give the firm equal profits then consumers' surplus is highest at b. If at a, the firm needs encouragement to lower prices with a compensatory reduction in the length of the 'cheap' period and vice versa if at c. Price-cap regulation does not allow such moves as a to b, (nor b to a of course) since

~_Pb qa > ~ga qa (and ~Pa qb > ~Pb qb )"

A particular example might illustrate that this is rather more than a possible perverse curiosity. Let us suppose that prior to regulation the firm has been efficiently maximizing profits with [Pl, P2, T] = [0.4, 0.2, 0.4] and then straightforward Vogelsang- Finsinger is applied. The prices and market division in the first period of regulation must satisfy the specific equivalents of (14) ,(15 ),(16),and(17), namely

r (1 - 2p 1 - 0.573 - ~ t ( o . s r - 0 . 5 ~ - 0.08) = 0 (20)

0.5 - 2 p 2 + 1 . 5 p ~ - r ( 1 - 2p 2 - 0.5~3 - ~ t (0 .32 - 0 . 8 r + 0 .5 /2 ) =- 0 (21)

1 - T - g ( 0 . 8 - T) - P l - P2 < 0 ( 2 2 )

1 - T - Ix(0.6 - T) -Pl -P2 > 0 (23)

and the regulatory constraint

Pl (0.8T- 0.5I 2 - 0.08) +P2 (0.32 - 0.8T+ 0.5T 2 ) - F < 0 . (24)

This conslraint merely says that the value of last period's outputs valued at the new prices


should not exceed fixed costs, F. (We assumed earlier there were only fixed costs.) Our earlier analysis tells us that at the same time as prices decrease under the regulation it would be welfare improving for T to increase.

Routine work shows that T will always remain at 0.4 for the first regulatory round. (An easy way to see this is to put T=0.4 in (18) giving Ix = 2 - 5pl and hence showing that (20) and (21) will be strict inequalities at T=0.4 if pl < 0.2 +P2 andp2 < 0.2. This will always be so as long as (22) is binding-- that is that the unregulated firm had been making profit.) Numerical values of the new prices can be calculated from

2 . 8 - ~ 7 . 3 6 - 60F F P2= 6 and Pl = 0.16 - 0"5P2"

Thus, we haveforexample F Pl P2 .07 .3523 .1704 .06 .3033 .1435 .05 .2531 .1187

And for instance with F=0.05, the next round of the regulatory exercise would again leave T at 0.4 and drive p 1 and P2 down to 0.18483 and 0.08258 respectively, with sales of 0.2461 in market 1 and 0.1339 in market 2. Here revenue is 0.05645 compared to the fixed costs of .05 so the regulation still has more work to do. However, the socially desirable increase in T will never be made, and nor will it be made for any value for fixed costs. The first regulatory round drives pl andp2 closer together (Pl - P 2 < 0.2) and inequality (18) holds. For a given and immoveable T of 0.4, the Ramsey prices that the mechanism approaches are such that Pl = (1.6p2)/(0.6 +P2) with both prices lower the lower are fLxed costs. We already know from the tent-like shape of figure 4, that since the Ramsey prices are such that Pl < 0.4 and P2 < 0.2, then once at them there would be no incentive to increase T. This example merely reinforces what was suggested earlier. In some sense, since our starting point (the profit-maximizing point ) is itself a Ramsey efficient point (with fixed costs of .08) both for relative prices and market division, then we are already relatively close to efficiency and product definition will not change. It is only if, prior to regulation, either relative prices or market division were a long way from Ramsey optimality that any improving changes in definition will be made.

Conc lus ions

For ease, we have just been supposing there is one market division in this example. Allowing more than one makes things no easier for the regulator. If there are costs to the firm in actually making exlra divisions, then we may presume that a regulated finn may make too few divisions, but this does not help to tell us where divisions should be. It is likely that history will determine the definitions, rather than the incentives, provided by regulation.

The shortcoming in Vogelsang- Finsinger mechanisms that allows inefficient market divisions to be maintained is that such mechanisms only search out social improvements that involve consumers being able now to buy that which they used to at less expenditure than before. This, of course, does not exhaust all possible price changes of benefit to both

3 ~ IANBRADLEY

the firm and the consumer. In general, it seems that such mechanisms have as far as market definition is concerned a built in conservative element. Getting market divisions wrong in the first place may result in them staying wrong.

As was mentioned above, demand interdependence will increase the conservative nature of the regulation, but again its presence provides few clues to the regulator as to how to overcome the problem. In our illustrative example of the use of telephones around 6 p.m., the large difference in units used just before and just after the boundary (which difference we presume will continue round any new boundary point) discourages a change to either an earlier or a later time. The consequence that regulated finns may choose wrong divisions is perhaps not surprising, but the casual analysis above indicates that the extent of the wrongness may be considerable. That the direction of the inefficiency is ambiguous is unfortunate for price-cap regulators.

These price-cap mechanisms suffer from this shortcoming only if the assumption of myopic profit-maximization holds. Without this our results would be different. A change in market definition might be unprofitable with a time horizon of one period, but worthwhile when one looks at periods ahead of that with the new division incorporated in the price constraint. It is perhaps ironic that under these kinds of regulation the virtuous properties of cost reduction and asymptotic relative price efficiency depend upon non-strategic myopic behavior (see Sappington (1980)), while such behavior may well cause the retention of inefficient market divisions.

We might also mention that the analysis here sits a little uneasily next to the Happy Hours theorem of Varian (1989, 625). In one of the seemingly few discussions of endogenous market definition and solely in the context of an unregulated, discriminating monopolist, Varian argues that as social surplus at points like U and Vin our diagrams could be increased by leaving prices unchanged and reducing T, then this implies that cheap periods are always too short or requirements for concessionary fares are too stringent; in other Words Happy Hours are always too short. But, of course, the same argument could be used to say that both the price in the dear period and the cheap period were too high. Indeed they a re - - an unregulated monopolist does not maximize social surplus! However, the most efficient way of reducing prices may not be by extending a cheap period. In the particular numerical example above, we have seen that efficiency would demand that reductions in profit from an unregulated position can best increase social surplus by reducing both prices and increasing T. (The Happy Hour was, in this sense, too long). There is no general rule to tell a regulator how to encourage movements in market division. In this matter, the appeal of the light regulatory touch of Laspeyre price-caps is dampened by their bias towards the status-quo. Such price-caps may provide us with few clues as to whether Happy Hours are too short, too long, too few, or too many.

Notes

I would like to thank Tom Abbott and a referee for explaining to me what I was trying to say. Any remaining incomprehensibility and errors are my own work.

1. Other examples may be useful both to clarify and to indicate the prevalence of moveable market divisions. Inland letter post in the United Kingdom is charged according to weight, and here py (pence) would be the letter rate associated with a maximum weight of Ty (grams).

p = [24, 36, 45, 54] and T = 160, 100, 150, 200].


Having a multi-part weekly tariff of, say, $100 standing charge, $5 per unit for the first 200 units, $4 for the next 1000, and $3 per unit for more, could be represented by

p = [100, 5, 4, 3] and T = [ 0, 200, 1200, ,,o]. In all these examples, to be consistent with the analysis below, T is defined deliberately so that increasing Tj increases the category size of goods sold at unit price py and reduces the extent of category j+l.

2. A somewhat extreme version of what this non-differentiability means in practice can be seen by considering the age structure of young passengers on the many airlines that allow children under two years of age to travd free, while standard fare is charged from a child's second birthday.

3. However, noting this does point to a structure of demand where price-capping would divide the market efficiently. Suppose that demands are independent and take the form of a unit purchase if price is less than a reservation price, with the reservation price changing in the same direction as the observable and practical quality that defines the market division. Then, we effectively lose a decision variable for the firm. Deciding upon thejth price also determines T 1 as the point in the market with reservation pricepj. Hence, profits and consumers' surplus can be represented by functions of p alone and the usual mechanisms (Vogelsang and Finsinger 1979) work. But this is a very special case.

References

Acton, J. P., and I. Vogelsang. 1989. "Introduction; Symposium on Price-Cap Regulation." Rand Journal of Economics 20: 369-72.

Bradley, I., and C. Price. 1988. "The Economic Regulation of Private Industries by Price Constraints." Journal of lndustriaI Economics 37: 99-106.

Brennan, T. J. 1989. "Regulating by Capping Prices." Journal of Regulatory Economics 1" 133-47. Sappington, D. 1980. "Strategic Firm Behavior under a Dynamic Regulatory Adjustment Process."

The Bell Journal of Econornics 11." 360-72. Varian, H. R. 1985. "Price Discrimination and Social Welfare."American Economic Review 75: 870-

75. Varian, H. R. 1989. "Price Discrimination." In Handbook of Industrial Organization, edited by R.

Schmalansee and R. Willig. Amsterdam: North Holland. Vogelsang, I. 1988. "Price-Cap Regulation of Telecommunications Services: aLong-Run Approach."

In Deregulation and Diversification of Utilities:, edited by M.A. Crew. Dordrecht: Kluwer. Vogelsang, I., and J. Finsinger. 1979. "A Regulatory Adjustment Process for Optimal Pricing by

Multiproduct Monopoly Firms." The Bell Journal of Economics 10:157-171.

Price-cap regulation and market definition

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