A Practical Approach to Quality-Adjusted Price Cap Regulation
By Kevin M. Currier*
Abstract
Price cap regulation is often combined with service quality regulation since price caps may
create incentives for quality degradation. A service quality adjustment factor (the Q-Factor) in
the price cap formula insures that allowed prices fall as quality declines. In this paper, we
discuss some considerations in determining the appropriate form of the Q-Factor. We first
examine the difficulties involved in exploiting the price/quality tradeoff. We then present a
quality-corrected price cap procedure –– possessing desirable properties –– that can be
implemented with reasonable informational requirements.
Key Words: Telecommunications Regulation, Price Caps, Service Quality Adjustment Factor
*Department of Economics and Legal Studies in Business, Spears School of Business, Oklahoma State University,
Stillwater, Oklahoma, 74078, e-mail kmcur@okstate.edu.
1. Introduction
Price cap regulation represents the principal innovation in regulatory policy in the last
20 years. In the U.S., price cap regulation has been applied to AT&T, the dominant long
distance carrier. In the U.K., price caps have been applied to British Telecom since
privatization. Unlike traditional rate-of-return regulation, price caps have been shown to
provide strong incentives for cost reduction and price rebalancing (Brennan, 1989; Cabral
& Riordan, 1989; Vogelsang, 2002; Cowan, 2002; Vogelsang & Finsinger, 1979).
A potential drawback to price cap regulation, however, derives from the fact that
incentives for cost reduction may lead to degradation of service quality (Sheshinski, 1976;
Sappington, 2003, 2005; Weisman, 2005). Indeed, under a price cap, the firm may be able
to increase profit by reducing costs without regard to service quality, particularly if it is
difficult for consumers to directly discern delivered service quality levels.
The empirical evidence regarding service quality degradation under price cap
regulation is mixed. Armstrong, Cowan, and Vickers (1994) find evidence of reduced
service quality for British Telecom in the mid- to late-1980s. Tardiff and Taylor (1993),
however, find no evidence of degraded service quality for the former Bell companies.
Alexander (2001) cites evidence of reduced service quality for Ameritech, and OFTA
(2004) notes evidence of telecommunications service quality problems in Hong Kong. It is
interesting that Sappington (2003) finds evidence that under price cap regulation, some
aspects of telecommunications service quality have improved while others have apparently
degraded.
Despite this mixed evidence, regulators do appear to be increasingly concerned with
the need for (additional) service quality regulations. However, the approach to service
1
quality regulation to date has been largely ad hoc. The regulator first identifies a group of
monitorable performance indicators known to be of concern to consumers. In
telecommunications in the U.S., this group consists of installation of service, operatorhandled calls, transmission and noise requirements, network call completion, customer
trouble reports, major service outages, service disconnection, billing and collection,
customer satisfaction surveys, public payphones, and 911 databases (Perez-Chavolla,
2003). The regulator may then require that the firms pay consumers rebates for
substandard delivered service quality. Alternatively, service quality targets may be
specified with bonuses paid for exceeding targets and penalties imposed for failure to meet
targets. In addition, regulators often require publication of performance pledges and
performance statistics. For example, telecommunications firms are regularly required to
submit performance statistics to regulators in Australia, the U.S., the U.K., and Canada,
among others. Finally, price cap plans may utilize a service quality adjustment factor (a
“Q-Factor”) whereby the level of the cap is reduced if quality degradation is determined to
have occurred. Thus, when a Q-Factor is employed, penalties for service quality
degradation take the form of mandated price reductions. In the U.S., Q-Factor regulation
has been employed in Rhode Island (Intven, 2000), Utah (PSCUT, 2001), New Mexico
(NMPRC, 2002), and Massachusetts (Vasington, 2003). Q-Factor regulation has also been
employed in Italy (toll motorway franchises and natural gas) as well as the U.K. (water
supply).
Ideally, a service quality adjustment factor should reflect consumers’ price-quality
tradeoff. The regulated firm should thus be “selling” higher quality to consumers through
higher prices or, alternatively, be “bribing” consumers via lower prices to accept lower
2
quality levels. However, little theoretically rigorous guidance has been offered to
regulators as to how best to exploit this tradeoff. In this paper, we examine this complex
issue and provide a practical suggestion for incorporation of service quality adjustments
into the price cap formula.
3
2. Difficulties Associated with the Q-Factor
A number of practical difficulties arise when attempting to apply a service quality
adjustment to a price cap. These difficulties can be illustrated by considering a regulated
firm that produces a single output x, which is sold at price p. Furthermore, assume that the
firm’s output is characterized by a scalar index of quality q.
Consumer surplus is V(p, q). We assume that V is twice continuously differentiable
and satisfies Roy’s Identity
We define  
V
( p , q)   x ( p , q)  0 where x(p, q) denotes market demand.
p
V
 p ,q   0 , the marginal social valuation of a change in quality at price p.
p
Note that υ is the change in the area below the demand curve and above the price, resulting
from a small change in q. In addition, it is reasonable to assume that consumer
“willingness to pay” for an increase in quality is highest when quality is low (DeFraja &
Iozzi, 2004). The isosurplus curves are therefore convex with d 2q/dp2 ≥ 0 for any fixed
surplus level. This property is formalized by assuming that V is quasiconcave. Note that
this assumption implies that
 p,q

V  p , q V is convex, i.e., upper contour sets of V are
convex. Figure 1 provides an illustration.
Under price cap regulation, the firm’s period t prices are required to satisfy
pt ≤ pt-1 (1 + RPI – X + Q)
where RPI is the percentage growth in retail prices, X is a general productivity factor
reflecting productivity improvements derived from technological change and cost saving,
and Q is a service quality adjustment factor (a Q-Factor) that insures that the required rate
of decline in prices increases as the firm’s delivered service quality decreases. In this
4
paper, our focus is on the appropriate form of Q; we thus assume for simplicity that RPI =
X, implying that the price cap adjustment formula is
pt ≤ pt-1 (1 + Q).
(1)
q
Direction of increase in V
p
Figure 1. Isosurplus Curves
The value of the Q-Factor should accurately reflect the rate at which quality can be
traded off against price. Suppose now that V  p,q  V . Since Vp = –x < 0, it follows that
along the isosurplus curve for V  p , q   V ,
dp 
 . For non-infinitesimal changes in price
dq x
and quality, there are two possible ways of evaluating this tradeoff: using lagged values
where








  p t 1 , q t 1
 t 1
  pt , qt
t




,
and
using
current
values
where
.
x
x
x p t 1 , q t 1
x t 1
x pt , qt
xt
5
In view of this, two possible forms for the period t Q-Factor may be suggested:
QL 
1
p t 1

 t 1 t
q  q t 1
t 1
x

(2)
and
QC 
1
p t 1

t t
q  q t 1
t
x

(3)
where the subscripts L and C denote the terms “lagged” and “current,” respectively.
In either case, the firm is assumed to select  p t , q t  to maximize period t profit πt
subject to (1). It should be noted that if the firm selects qt = qt-1, the price cap formula
permits the firm to select pt = pt-1. Thus, under either QL or Q C , application of (1) can
never decrease profit. Hence, πt ≥ πt-1.
As is customary, we shall define social welfare to be the unweighted sum of consumer
surplus and profit. Thus, period t welfare is W t = V t + πt. The following Proposition
provides the main result of this section.
Proposition: Application of Q L will not guarantee a welfare increase, but application of
QC will.
Proof: See Appendix.
The Proposition demonstrates that contemporaneous values of the quality indicator,
demand, and consumers’ quality valuation must be employed in the Q-Factor to guarantee a
welfare increase. The following example provides a simple illustration.
6
Suppose that V  p , q  
60q
30
30q
with Vq    2 and  V p  x  3 . Suppose further
2
p
p
p
that  p t 1 , q t 1   3, 3 , in which case V t -1 = 10,  t 1  30 9 , and x t 1  20 3 . Using Q C ,



q t  3 . Since t x t  p t 2q t , the
the period t price constraint is p t  3 1 
t
3 x


constraint may be rewritten as p t  3 
p , q   2, 3 2 .
t
t
V t  45
4
1 t


pt t
q  3 . Suppose now that the firm selects
2q t
This price-quality vector satisfies the constraint and yields
 10 . Since V t > V t -1 and πt ≥ πt-1, welfare must increase.

Alternatively, if QL is applied, the period t constraint is p t  p t 1 1 

or p t  3 


1 t 1 t
q  3 
t 1
t 1
p
x




1 t
q  3 . If the firm again selects q t  3 2 , p t  9 4 now satisfies the
2


constraint but with  p t , q t   9 4 , 3 2 , we have V t  80 9  10 . Thus a welfare increase
cannot be insured.
While QL has the desirable feature of employing lagged values of demands and quality
valuations, it will not in general lead to a welfare increase. DeFraja and Iozzi (2004) have
provided a generalization of the Vogelsang-Finsinger (1979) procedure, which leads to
efficient pricing and quality provision (in the long run) using lagged values of demands and
quality valuations. However, implementation of the procedure requires that the firm and
the regulator perform complex calculations involving “distance” constraints, global
properties of demand functions, and various “scaling factors.”
7
These calculations are needed to insure that the firm’s constraint sets are non-empty,
the process doesn’t lead to regulatory “cycles,” and is monotonically increasing in welfare.
These additional constraints, however, are likely to be an impediment to practical
implementation. Thus, within this context, QC provides the most direct approach to setting
the Q-Factor.
Unfortunately, there are several reasons why the regulator is likely to find application
of QC administratively impractical as well. First, the fact that contemporaneous values of
demands and quality valuations are used necessitates forecasting. Period t demands, for
example, are not realized until the period t price-quality vector is actually selected by the
firm since x t  x  p t , q t  . Thus, application of QC requires that the regulator be able to
forecast xt before period t begins. The likelihood of forecasting error and expensive, timeconsuming arbitration between the firm and the regulator regarding the demand estimates
renders application of QC potentially problematical (Brennan, 1989).
However, even if accurate demand estimates can be obtained by the regulator,
determination of the period t quality valuation t    p t , q t  is likely to be a more severe
impediment to application of QC. Structured survey techniques such as the Contingent
Valuation Method and the Contingent Choice Method have been employed to (directly or
indirectly) infer customers’ willingness to pay for such things as air quality improvements,
food safety, and decreased risk of illness, etc. (Bjornstad and Kahn, 1996; Hanemann and
Kanninen, 1998). Application of such techniques within the context of determining QC and
consumers’ price/quality tradeoff will require that quality valuations be known at the
beginning of the regulatory period, again involving forecasting of these values or a priori
8
determination of consumers’ quality valuations over a broad range of price-quality
combinations. Although such exercises have been attempted, the expense of implementing
such a procedure at the beginning of every regulatory procedure is likely to render this
approach prohibitively costly.1
Additional difficulties arise from the fact that in general, the regulated firm will
typically produce several outputs, each with multiple service quality dimensions. In such a
case, generalizations of the “willingness to pay” assumption and the result contained in the
Proposition are not possible without imposing additional structure on consumer
preferences. Moreover, if there are n-regulated outputs and m-quality dimensions, there are
m × n price/quality tradeoffs to be determined and exploited.
The problems discussed above clearly reveal the need for a practical method for
implementing quality-adjusted price cap regulation in a general (multiple output, multiple
quality dimension) context. In the following section, we present such a procedure.2
9
3. A Practical Approach to the Service Quality Adjustment Factor
In this section, we present an approach to quality-adjusted price cap regulation that can
be applied in a straightforward manner with reasonable informational requirements. The
process is based on the premise that a reduction in quality is a hidden form of a price
increase and that consumers’ utility is a function of “quality-adjusted” consumption levels.
Specifically, changes in quality levels are assumed to augment the “services” that the goods
provide to consumers (Fisher and Shell, 1972, 1998).
Suppose, for example, that the “service” yielded by a telephone call is the
“transmission of information.” Then for a phone call of a given duration, a higher value of
voice transmission quality may imply that a greater amount of information is transmitted.
Or consider the response time for operator services. A reduction in operator response time
may increase the amount of information transmitted in any given amount of time (including
the time spent with the operator).
Consider now a regulated firm producing output vector x = (x1, …, xn) with price
vector p = (p1, …, pn). There are m overall quality of service measures for the firm:
Q = (Q1, …, Qm). In addition, each output has a quality of service index qi = f i (Q1, …, Qm)
where f i satisfies f i Q j  0 , i = 1, …, n and 1, …, m. Thus the firm’s overall quality of
service indicators imply a specific level of service quality for each regulated output.
Furthermore, f i Q j measures the effect of a small change in Qj on the service quality
index for good i. If the overall service quality index Qj has no impact on the quality index
for output i, then f i Q j  0 . For example, suppose that voice quality transmission
statistics are compiled separately for local calls, domestic long distance, and international
10
calls. Then the quality index of local calls is not likely to depend on voice quality
transmission of international calls.
Consumer utility is U  y1 , , y n  , which is assumed strictly concave. Furthermore,
y i  hi q i  xi , i = 1, …, n where hi  0 and hi  0 for each i. Thus, quality levels augment
the “services” yi generated by output xi. Since hi  0 , a ceteris paribus increase in the
product-specific quality index qi increases “services” generated yi at a decreasing rate.
Consumers maximize consumer surplus U  y1 , , y n    pi xi or equivalently
n
i 1
U  y1 ,  , y n    p̂i y
n
(4)
i 1
where p̂i 
pi
p̂
, i = 1, …, n. Maximization of (4) implies that MRS yr , ys  r for any
hi qi 
p̂ s
(r, s) pair. Thus, consumers equate the marginal rate of substitution between any two
 pr
 ps
“services” to the corresponding ratio of the quality-adjusted prices 
 p
  hs q s  
 
 .
  hr q r  
p

Maximization of (4) yields demand functions yi  p̂1 , , p̂ n   yi  1 , , n  ,
hn q n  
 h1 q1 
i = 1, …, n. Substitution of these demands back into (4) yields the consumer surplus
 p1
pn 
 . V is a convex function of quality-adjusted
, ,
hn q n  
 h1 q1 
function V  p̂1 , , p̂ n   V 
prices  p̂1 , , p̂ n  and satisfies Roy’s Identity
V
  yi  p̂1 , , p̂ n  , i = 1, …, n. Since V
p̂i
is convex, it is quasiconvex.3 Using an argument similar to the one used in the proof of the
Proposition in Section 2, it is straightforward to show that
11
n
 p̂ it y it  1
i 1
n

i 1

p̂ it  1
y it  1

1
implies V  p̂1t , , p̂ nt   V p̂1t 1 , , p̂ nt 1 . Since p̂i 
(5)
pi
, (5) may be expressed as
hi qi 

t  t 1
p
i
t
 xi
i  1  hi

1
n
t 1 t 1
 pi xi
n
 hit  1
 
(6)
i 1
where hit 1  hi qit 1  and hit  hi qit  . Figure 2 provides an illustration for n = 2.
p̂ 2
slope =




 y1 p̂1t 1 , p̂2t 1
y 2 p̂1t 1 , p̂2t 1
t 1
t 1
●  p̂1 , p̂2 
Direction of increase in V
isosurplus V = V t - 1
p̂1
Figure 2. The Quality-Adjusted Pricing Constraint
12


In Figure 2, period t – 1 quality-adjusted prices are p̂1t 1 , p̂ 2t 1 . By Roy’s Identity, the


slope of the line tangent to the isosurplus curve passing through p̂1t 1 , p̂ 2t 1 is

y  p̂
.

 y1 p̂1t 1 , p̂ 2t 1
2
t 1
1 ,
p̂ 2t 1
The set of points on or below this line is the set of all quality-adjusted


prices that satisfy (5). With  p̂1t , p̂ 2t  in this set, we are assured that V  p̂1t , p̂ 2t   V p̂1t 1 , p̂ 2t 1 .
To make the model operational, assume that each output-specific quality indicator qi is
selected to be a weighted average of the set of overall quality indicators (Q1, …, Qm). This
may be express as q = AQ where:
 q1 
 
q  
q 
 n
 a11

a
A   21
 

 a n1
 Q1 


Q  
Q 
 m
a12  a1m 

a 22   

 

a n 2  a nm 
with aij ≥ 0, i = 1, …, n and j = 1, …, m. In this formulation, aij is the marginal contribution
of overall service quality index j to the quality index for output i. Furthermore, assume that
hi(qi) = kiqi, ki ≥ 0, i = 1, …, n. Thus, for any fixed level of output i, the “level of services”
generated yi is a linear function of qi. In this case (6) may be expressed as

t 1

i

q
  i t p it  xit 1
q
i 1
n

n

i 1
p it  1 xit  1
 1.
(7)
Suppose now that the regulator imposes (7) as the regulated firm’s period t constraint. If
the firm selects prices and quality levels to maximize profit subject to (7) and qt = AQ t ,
then we are assured that consumer surplus, profit, and hence welfare must increase.4
Indeed, (7) insures that consumer surplus increases as demonstrated above. Moreover,
13


since (7) permits the firm to select  p1t , , p nt   p1t 1 , , pnt 1 and q1t , , q nt  q1t 1 , , q nt 1  ,
profit cannot fall when (7) is applied.
We conclude by noting three important points. First, a vector of prices and qualities is
efficient if consumer surplus is maximum, given the profit level of the firm. Suppose now
that our procedure is iterated, thereby generating an infinite sequence of price/quality
vectors. It is straightforward to show that long-run prices and quality levels are efficient
(i.e., the steady state price/quality vector is efficient). This is a generalization of Brennan
(1989), which demonstrates that when quality considerations are absent, Laspeyres-based
price cap regulation leads to efficient Ramsey pricing.
Second, observe that if qit  qit 1 for all i, (7) reduces to
n
 pit xit 1
i 1
n
 pit 1 xit 1
 1.
i 1
Thus, when quality is invariant, our procedure is equivalent to capping a Laspeyres index of
the regulated firm’s prices.
Finally, consider the special case in which there is one overall service quality indicator
m
for the firm Q    j Q j where βj ≥ 0, j = 1, …, m. In this case, (7) may be expressed as
j 1
n
 p t x t 1
 Q t 1  i 1 i i


 Q t  n t 1 t 1  1

  pi xi
i 1
or, equivalently,
14
n
 pit xit 1
i 1
n
 pit 1 xit 1
 Q t  Q t 1 
.
 1  
t 1

 Q

i 1
The term
Q t  Q t 1
Q t 1
is the percentage change in the firm’s overall service quality index. If
there is no degradation in overall service quality from period t – 1 to period t, the firm faces
the conventional Laspeyres price cap. If Q t > Q t-1, the level of the cap is increased, and if
Q t < Q t-1, the level of the cap is reduced. It is noteworthy that this is the approach to QFactor regulation adopted for telecommunications by public service commissions in Utah
(PSCUT, 2001) and Rhode Island (Intven, 2000), among others. Our results then provide a
sound theoretical justification for such an approach.
15
4. Conclusions
To prevent service quality degradation under price cap regulation, a service quality
adjustment factor may be included in the price cap formula. Ideally, the regulator should
attempt to exploit the price/quality tradeoff by insuring that firms “sell” higher quality to
consumers via higher prices and “bribe” consumers to accept lower quality levels by
lowering prices. We have illustrated that, in general, the practical difficulties associated
with such an effort are likely to be formidable. Yet the need remains for a practical method
of adjusting the price cap formula to allow for quality variations.
We have modeled telecommunications demands by assuming that consumers derive
utility from the “services” generated by the consumption of the regulated outputs. Both the
actual consumption levels and product quality are assumed to contribute to utility. In this
way, quality reductions are essentially regarded as price increases and quality increases are
viewed as price decreases. We proposed a modified Laspeyres price cap that utilizes
“quality-corrected” prices, which are determined by actual prices as well as current and
lagged quality levels. If the process is iterated, long-run prices and qualities are efficient.
In reality, it is likely that the procedure would have to be terminated after a finite number of
iterations. For this reason, it is significant that the procedure is monotonically increasing in
consumer surplus, profit, and hence overall welfare. Finally, we have demonstrated that
our procedure provides a sound theoretical foundation for some contemporary approaches
to service quality regulation in telecommunications.
16
Endnotes
1.
The contingent valuation method is also known to suffer from potential biases since the
hypothetical nature of the transactions often causes consumers to overstate their true
willingness to pay (Murphy and Stevens, 2004).
2.
Any procedure that exploits the price/quality tradeoff must employ current values of
the quality indicators since consumer surplus in period t is determined by pt and qt.
Because of random events (e.g., weather-related service disruptions), some quality
levels may only be observable ex post whereas other quality measures (e.g., the
number of operators) may be observable at the beginning of the regulatory period.
Thus, exploitation of the price/quality tradeoff is most feasible when quality measures
can be observed in advance for the entire regulatory period.
3.
A function F is quasiconvex if F(y) ≤ F(x) implies that F ( x)  ( y  x)  0 . It should also
be noted that the assumption that each hi (qi) is concave hi  0  implies that, ceteris
paribus, consumer willingness to pay for an increase in qi (via an increase in pi) is
highest when qi is low.
4.
The system of linear equations qt = AQt has at least one solution if rank (A) = rank (A0)
 a11  a1m q1t 


 .
where A 0   
 a  a qt 
nm
n
 n1
17
Appendix
Proof of Proposition: A function F is quasiconcave when F(y) ≥ F(x), implies
( y  x)  F ( x)  0 , where F denotes the gradient (i.e., the vector of partial
derivatives) of F.
Applying this fact to the quasiconcave function V, with x =  p t 1 , q t 1  and y =  p t , q t  we
obtain V  p t , q t   V  p t 1 , q t 1  implies
p
t
 

 p t 1 , q t  q t 1   x t 1 , t 1  0 .
(A.1)

Rearranging terms, (A.1) may be expressed as p t  p t 1 1 


1
p
t 1

t 1 t
q  q t 1
t 1
x

p t  p t 1 1  Q L .
 or

(A.2)
Thus, V  p t , q t   V  p t 1 , q t 1  implies, but is not implied by (A.2). Since application of QL
is not sufficient to guarantee an increase in V, it is not sufficient to guarantee an increase in
W.
To demonstrate that QC will guarantee an increase in welfare, apply the above
technique with x   p t , q t  and y   p t 1 , q t 1  to obtain V  p t 1 , q t 1   V  p t , q t  implies
p
t 1
 

 p t , q t 1  q t   x t , t  0 .

1

p t 1
Rearranging terms (A.3) may be expressed as p t  p t 1 1 





(A.3)

 or
 

t t
q  q t 1
t
x

p t  p t 1 1  Q C . Hence, p t  p t 1 1  Q C implies V t  V p t , q t  V p t 1 , q t 1  V t 1
and we conclude that application of QC guarantees an increase in V. Since πt ≥ πt – 1, it
follows that W t  V t   t  V t 1   t 1  W t 1 .
18
Acknowledgements
I would like to express my sincere thanks to Doug Pitt, Niall Levine, and two
anonymous referees for numerous constructive suggestions.
19
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