International Journal of Business and Econoics, 2006, Vol. 5, No. 1, 1-16 A Siple Model of Reliability, Warranties, and Price-Capping Donald A. R. George Manageent School & Econoics, University of Edinburgh, UK Abstract The paper presents a odel of reliability which captures the process by which reliability is actually deterined ore accurately than the conventional analysis. In contrast to that conventional analysis, which is based on the characteristics approach, the odel of this paper defines reliability as the objective probability of product failure, not as a characteristic of individual goods. Reliability, thus defined, is treated as a choice variable of the fir. The resulting odel is applied to a onopolist subject to a price cap. The onopolist can vary reliability and the ters of a warranty or copensation deal in response to price-capping. The onopoly outcoe, price-capped onopoly outcoe, and Pareto-efficient outcoe are copared. The odel provides a theoretical explanation of soe epirical results in the literature on electricity regulation. Key words: reliability; warranties; price-capping JEL classification: L15; L51; M21 1. Introduction Monopolies are often subject to soe for of price-capping. In the UK, for exaple, the utilities (telecos, electricity, gas, and water) all have regulators who frequently exercise their powers to cap prices. One would expect that the rational, profit-axiising onopolist would respond to a binding price cap by changing other variables, in an attept to shift the deand and cost curves in his favour. Obvious candidates for this role are (a) reliability of the product or service and (b) the contractual ters on which the product or service is offered to the arket; for exaple, the ters of any warranty or copensation deal which the onopolist ay provide along with its output. Throughout the paper the generic ter output will be used instead of product or service. There is soe relevant epirical evidence Received Noveber 7, 2005, revised May 24, 2006, accepted June 16, 2006. Correspondence to: Manageent School & Econoics, University of Edinburgh, George Sq., Edinburgh EH8 9JY, UK. Eail: D.George@ed.ac.uk. Precursors of this paper have been presented in seinars at the University of Edinburgh, University of Melbourne (Australia), University of Waikato (New Zealand), University of Canterbury (New Zealand), Hailton College/Colgate University joint seinar (USA), and Queen s University (Canada). I a grateful for the invitations to those seinars and for the constructive coents ade. I a also indebted to Les Oxley, Ahed Anwar, Sion Clark, and to the Editor and an anonyous Co-Editor of this journal.
2 International Journal of Business and Econoics on this atter. For exaple Fraser (1994a, b) considers the supply of electricity by a private onopoly, showing that a pure price cap induces the fir to reduce reliability of supply. In order to analyse this proble theoretically it is necessary to develop a odel which allows the fir to deterine reliability and the ters of any warranty or copensation deal as part of its overall profit-axiising decision. Unfortunately, the existing theoretical literature does not provide uch of a guide on these atters, partly because the conventional analysis of output quality is based on the characteristics approach to odelling production. This paper draws a sharp distinction between quality and reliability. It defines the latter as the objective probability (relative frequency) of output failure, a probability which the fir can choose, for exaple via the quality control process, the choice of technique, or the design of output. The paper argues that this approach coes closer to describing the process by which reliability is actually deterined than does the conventional output quality approach, which is ore relevant to aspects of production other than reliability. For exaple, the conventional analysis explains well why soe coputers have fast processors or large hard drives, but not why they break down so often. The paper develops this analysis of reliability and incorporates it into a siple odel of onopoly, which is then applied to the price-capping proble. Section 2 develops the odel of reliability. In Section 3 the deand side is developed, assuing risk averse consuers, with a reservation price varying across the arket. Section 4 focuses on the supply side, incorporating reliability costs and warranty costs into the onopolist s profit function. In Section 5 a Pareto-efficient allocation of resources is characterised. Sections 6 and 7 characterise onopoly equilibriu without and with a price cap, respectively. Coparing these two types of equilibriu with each other and with the Pareto-efficient allocation allows an analysis of the effects of price-capping. The atheatical details are relegated to an Appendix. Section 8 concludes. 2. Modelling Reliability In the odel presented here reliability will be defined as the probability of an event disliked by consuers not occurring. Exaples of such an event are: A: A consuer durable (e.g., coputers, cars, washing achines, or televisions) breaking down within soe given tie period. B: An interediate good such as a silicon chip (or other coponent) failing to function correctly (here the consuer is another fir). C: A train or plane not arriving within soe predeterined tie period of its scheduled arrival tie. D: Electricity, gas, water supplies, or telephone services being interrupted for ore than soe predeterined period. Note that coercial contracts for these services ay allow soe interruption of service without penalty (e.g., the coercial supply of gas). Matsukawa and Fujii (1994) study Japanese electricity
Donald A. R. George 3 consuers and show, aong other things, that they face a trade-off between price and reliability of electricity supply. The probability referred to here is an objective relative frequency. For exaple, if a coputer anufacturer produces 100,000 coputers each year and 93,000 do not break down within a given tie period (say one year), the reliability of these coputers is 0.93. Thus the coputer fir knows for sure that 7,000 of its coputers will break down within the year, but it neither knows nor cares which 7,000 they will be. Now suppose the fir offers a one-year warranty with its coputers, proising copensation in the event of a breakdown. In this odel the fir faces no uncertainty concerning its profits: it knows its revenue and production costs, it knows that there will be 7,000 clais under the warranty (though not which custoers will ake the), and it knows how uch it will have to pay out per clai (that can either be treated as endogenous or iposed by a regulator). There is therefore no uncertainty about its profits. It will be assued that consuers (who, in the case of interediate goods entioned in point B above, will be other firs) have no knowledge about individual products or services but do know the reliability of each fir s output (in the sense defined here). Consuers read Which? agazine or Consuer Reports or obtain inforation on reliability fro other sources. For exaple, in the UK, the Strategic Rail Authority publishes inforation on the average punctuality of the different rail operators. Supplying firs will be assued to have the sae inforation. This is a plausible assuption because it is usually ipossible or extreely costly for firs to obtain inforation on each exaple of its output before it is sold. Firs will be assued to vary reliability (as defined here), for exaple via product design, choice of technique, and quality control procedures. It will be assued that higher reliability entails higher production costs. Thus, the coputer anufacturer will be able to increase or decrease the nuber of breakdowns in a given tie period without knowing (or caring) which coputers will break down and which will not. It will therefore be assued to know the reliability of its output (as defined here), without knowing which exaples of its output will break down. In this odel the fir faces no uncertainty, though this is not true of consuers, who are assued to be risk averse. This assuption is readily justified, for exaple, in the consuer durables arket, where each consuer typically owns one exaple of the good and is thus extreely concerned at the prospect of its breaking down. The fir, by contrast, supplies any exaples of the good and ay well find it profitable to operate a risk-pooling warranty schee. Under these assuptions there arises a deand, on the part of consuers, for insurance. This ight, as entioned above, be provided in the for of a product warranty offered by the fir or an insurance policy provided jointly with the product. In the case of interediate goods, warranties ay be thought of as copensation clauses built into standard supply contracts. A siilar interpretation applies to services such as electricity and gas. In the case of transport services, it is clearly possible for suppliers to offer copensation to dissatisfied passengers. Throughout the paper attention will be confined to voluntarily offered
4 International Journal of Business and Econoics warranties or copensation, though the odel is readily odified to include legally copelled copensation. It could also be odified to cover ore than one undesired event (e.g., different degrees of product breakdown) or to cover product hazard and safety issues. Warranties, whether voluntary or legally copelled, have an iportant bearing on decisions affecting reliability because the higher the reliability of a fir s arketed output, the lower the warranty costs experienced by the fir ceteris paribus. In the odel developed in this paper, warranties play the role of allocating risk and providing an incentive to supply reliable output, in contrast to uch of the existing literature on warranties where they have a signalling role. As noted in Section 1, the odel developed in this paper differs sharply fro that presented in the literature on product quality. That literature deals either with search goods or with experience goods. In the forer case both supplier and consuer know all relevant characteristics of each good or service before sale takes place (e.g., the coputer has a 3.0GHz processor and an 80Gb hard drive, or the train has a good restaurant). See for exaple Mussa and Rosen (1978) and Matthews and Moore (1987) for odels of this type. In the case of experience goods, there is an asyetry of inforation. Nature dictates all relevant characteristics of each good or service to the supplier before sale, but these are unknown to the consuer at that stage (e.g., the coputer s hard drive will fail in the first year, or the train will be 2 hours late). The supplier s proble is thus one of signalling. Perhaps by eans of advertising, or offering a warranty or copensation deal, the supplier of high quality output seeks to signal his high quality to consuers in a credible way. See for exaple Grossan (1981), Milgro and Roberts (1982, 1986), Kreps and Wilson (1982), Klein and Leffler (1981), Shapiro (1983), and McClure and Spector (1991) for odels of this type. Neither of these approaches is of uch use in analysing reliability. Search good odels assue too uch inforation on both sides of the arket, while experience good odels assue too uch inforation on the supply side and not enough on the deand side. Moreover, the latter kind of odel is based on an exogenously given quality level, while reliability (as defined in this paper) will be deterined endogenously by the supplier s decisions, as discussed above. A standard proble, often assued away in the literature, is that of oral hazard on the part of consuers. If consuers can theselves influence the probability or size of a clai under the warranty, for exaple by failing to take proper care of the good during consuption, then the econoic role of warranties ay be reduced. See for exaple McKean (1970), Oi (1973), Priest (1981), and Goering (1997), who discusses the proble of oral hazard facing a durable goods onopolist. For siplicity oral hazard will be assued away in this paper. It should be noted that the odel presented here focuses on reliability and warranties, deliberately suppressing soe other aspects of the arkets discussed above. It is essentially a static odel, and is not intended to deal with the issue of dynaic consistency in durable goods arkets. Moreover, it is a odel of syetric inforation. In such a odel nothing can be gained by aditting the possibility of repeat purchasing since neither side of the arket can learn anything useful about the other.
Donald A. R. George 5 3. The Deand Side In our odel, consuers preferences have three distinct aspects: 1. The consuer s preference for the good in its working state. This varies across consuers and is exogenous. 2. The consuer s degree of risk aversion. For convenience this is assued constant across consuers and is exogenous. 3. The probability of the good not breaking down within soe given tie period (i.e., the reliability of the good). Subjective and objective probabilities are, by definition, identical in this odel. It is an essential feature of the odel that this probability is endogenous (deterined by firs decisions) and the sae for all consuers. The details of consuers utility functions are developed below. The deand side of the arket will be assued to consist of z consuers, each consuing a single unit of the output. Each consuer has a different reservation price, and hence the arket deand curve is downward sloping. For siplicity we take z to be a strictly positive real variable. Each consuer has a oney budget M available and pays a price p for the output. As discussed in Section 2, two states of the world are assued: either the undesired event occurs or it does not. In the latter case, the z th consuer receives a strea of services which she values at f( z ) (perhaps generated by a durable good). Note that z > 0 and f ( z) < 0. In the forer case the consuer values the strea of services at zero, but the fir akes a voluntary warranty (or copensation) payent of β to her. Costs of writing and enforcing the warranty (or copensation) contract are ignored. Thus the z th consuer receives incoe strea: x = M p + f (z) (1) if the undesired event does not occur, and y = M p + β (2) if it does. The reliability of a product will be defined as in Section 2, as the objective probability R of the undesired event not occurring. Consuers are assued to be risk-averse axiisers of expected utility. As discussed in Section 2, it will be assued that consuers are fully infored about reliability, so that the subjective probability of the undesired event not occurring is equal to the objective probability. Of course R is deterined endogenously by the anageent decisions of the onopolist. The z th consuer axiises expected utility: ( M p + f ( z) ) U ( M + β ) V = RU p. (3)
6 International Journal of Business and Econoics Clearly U () > 0, and, to ensure risk aversion, it is assued that U () < 0 (i.e., the function U () is assued strictly concave). Note that the z th consuer is indifferent between consuing and not consuing when: ( M p + f ( z) ) U ( M p + ) U ( M ) V = RU β = (4) since U (M ) is the expected utility she would get by not consuing the output (she will be referred to as the arginal consuer ). Equation (4) generates, for given values of R and β, a relationship between p and z, naely the arket deand curve. Each consuer has a different reservation price, and thus the arket deand curve slopes downwards (see Figure 1). Note that R and β are deterined by the decisions of the onopolist, so that consuers can be thought of as consuing a bundle consisting of a strea of services (perhaps provided by a durable good), its reliability, and the warranty deal. They are not able to unbundle these three parts. If the onopolist raises R or β, the deand curve will shift upwards, except that, when a full oney back warranty is offered ( β = p ), the arginal consuer will be indifferent as to whether the undesired event occurs or not (since x = y when β = p ). In this case, as R changes, the deand curve will rotate about the equilibriu, which will itself be iune to variations in R. Note also that the f ( z ) curve ust be steeper than the deand curve (see Figure 1) because it is the relationship between p and z which would hold if β were continually kept equal to p (this is clear fro equation (4)). Of course the deand curve is defined ceteris paribus (i.e., holding everything constant except p and z ). Figure 1. The Distribution of Tastes ( f ( z ) ) and the Market Deand Curve ( DzRβ (,, )). p f (z) D( z, R, β ) z 4. The Supply Side Firs costs will depend on the reliability of their output for a nuber of different reasons:
Donald A. R. George 7 1. Reliability can be designed into the output. Higher reliability designs will, in general, be ore costly to produce than lower reliability ones. 2. Techniques of production can be adopted which generate higher reliability. Techniques generating higher reliability output will, in general, be ore costly to operate than those generating lower reliability output. 3. The stringency of quality control can be varied. Stricter quality control will, in general, raise reliability, but will also raise scrap or rework costs. 4. Higher reliability will reduce the nuber of clais under the warranty and hence, for a given warranty payent, reduce warranty costs. The odel developed here foralises these costs by assuing that production costs are increasing in the reliability of output and by incorporating warranty costs into the fir s profit-axiising decision. Average and arginal production costs, at a given reliability level, will be assued constant. Note that z is the onopolist s output. Adopting the assuptions set out above, a suitable production cost function is: zc( R ), (5) where C ( > 0 and C ( > 0 for 0< R < 1. The nuber of ties that the undesired event occurs is clearly z( 1, and thus warranty or copensation costs are given by: β z( 1. (6) Thus the onopolist axiises the profit function: Φ = pz zc( β z(1. (7) 5. Pareto-Efficiency We now characterise a Pareto-efficient allocation of resources. Since there are no non-convexities, externalities, or public goods in the odel, a Pareto-efficient allocation of resources would be brought about by the operation of a perfectly copetitive arket. A Pareto-efficient allocation is defined as a 4-tuple ( pzrβ,,, ) that axiises each consuer s expected utility subject to the constraint that profits are non-negative. This proble is easily solved by taking a ultiplier ( λ ) for the profit constraint and foring the Lagrangian: L = RU ( M p + f ( z)) U ( M p + β ) + λ( pz zc( βz(1 ). (8) By considering the first-order conditions of this proble, two iportant results can be obtained. The proofs are relegated to the Appendix. Note that an asterisk denotes the efficient level of a variable. First, the profit-constraint is binding; that is, Pareto-efficiency requires zero (supernoral) profits and consuers extract all the surplus (see Appendix,
8 International Journal of Business and Econoics Proposition 1). In the absence of onopoly this would be achieved by free entry. Note that price discriination by the onopolist is assued away, so one would not expect onopoly equilibriu to be Pareto-efficient. Second, Pareto-efficiency requires that the arginal consuer is fully insured (Appendix, Proposition 2). Thus a full oney back warranty ust be offered (i.e., β = p so that x = y and the arginal consuer is indifferent as to whether the undesired event occurs or not). 6. Monopoly Equilibriu In onopoly equilibriu there is a single supplier axiising her profits subject to the voluntary participation constraint. This is the constraint that each consuer obtains at least as uch expected utility fro purchasing the output as fro not doing so. Matheatically it is siply: RU ( M p + f ( z)) U ( M p + β ) U ( M ). (9) In equilibriu z is deterined at a level which akes this constraint bind (i.e., the z th consuer is the arginal consuer, who is just on the point of leaving the arket and z is the onopolist s total output). A onopoly equilibriu is easily characterised by taking a Lagrange ultiplier ( μ ) for the constraint (9) (noting equation (7), which specifies the onopolist s profits) and foring the Lagrangian: M = pz zc( β z(1 + μ( RU ( x) U ( M )). (10) Fro the first-order conditions for this proble the following results can be obtained (proofs in Appendix). First, the voluntary participation constraint is binding (Appendix, Proposition 3). This allows the onopoly output ( z ) to be deterined. Note that the superscript denotes the value of a variable in onopoly equilibriu without price-capping and the superscript c denotes the value of a variable in onopoly equilibriu with price-capping. Second, in onopoly equilibriu without price-capping, risk is efficiently allocated (i.e., the arginal consuer is fully insured, so that p = β ; see Appendix, Proposition 4). The onopoly equilibriu is illustrated in Figure 2. Manipulating the first-order conditions allows a coparison between the onopoly levels of the relevant variables (without a price cap) with their efficient levels. In particular: R > R ; β > β ; p > p ; z < z. (11) See Appendix, Proposition 5 and Corollaries 2, 3, and 4. The onopolist raises reliability above its efficient level but lowers output and raises price in the usual way. As rearked above, it allocates risk efficiently.
Donald A. R. George 9 Figure 2. M Is the Uncapped Monopoly Equilibriu, C Is the Capped Monopoly Equilibriu, and E Is the Pareto-Efficient Outcoe. p f (z) p M D( z, R, β ) c p p C c c D( z, R, β ) E z z z z 7. Monopoly Equilibriu with Price-Capping The onopoly odel of Section 5 illustrates the standard sources of inefficiency under onopoly, naely a restricted output and raised price (relative to the efficient levels). In addition to this, inefficiency arises because the onopolist supplies a reliability level above the efficient level. However, the onopolist does allocate risk efficiently by offering a full oney back warranty. In this section, we consider the c consequences of a binding price cap. Suppose a binding price cap ( p ) is iposed on c the onopolist such that p > p p. A reduction in reliability will clearly lower costs and ay therefore increase the fir s profits. With a oney back warranty in place, the arginal consuer is indifferent as to whether the undesired event occurs or not, and consequently the arket equilibriu will be iune to variations in reliability. In fact, in the odel set out above, the effect of price-capping is rather ore coplex than this because it induces under-insurance and an inefficient allocation of risk. Because of the under-insurance, the arket deand curve now shifts in response to changes in R (rather than siply rotating about the equilibriu). In particular, lowering R shifts the deand curve downwards (see Figure 2). The analysis of price-capping is best approached by odifying the Lagrangian of Section 5 (equation (10)), by adding a constraint on the price: c p p. (12) We take a ultiplier ν for this constraint, odifying equation (10) as follows: ( p c p) M = pz zc( β z(1 + μ( RU ( x) U ( M )) + ν. (13)
10 International Journal of Business and Econoics The ultiplier ν represents the onopolist s arginal valuation of the price cap, i.e., the aount by which she could increase her profits if the price cap were relaxed by one arginal unit (or the axiu she would be willing to pay as a bribe in order to get a arginal, one-unit increase in the price cap). We are interested, in this paper, in effective price-capping and will therefore assue that, under pricecapping, ν > 0. Note that equation (13) includes the uncapped case (i.e., when ν = 0 ). In the Appendix, the first-order conditions of the Lagrangian (equation (13)) are derived, thus incorporating the price-capped and uncapped cases into the sae atheatical proble. Using these first-order conditions the onopoly equilibriu with price-capping ( ν > 0 ) can be characterised. Coparing this equilibriu with the uncapped case ( ν = 0 ), the effects of price-capping can be deduced. In particular, onopoly output with price-capping is below the efficient level (Appendix, Proposition 6). Moreover, price-capping induces the onopolist to provide a less-than-oney-back warranty (Appendix, Proposition 4 and Corollary 1). Thus, the arginal consuer is not fully insured and risk is not efficiently allocated. The arket deand curve now shifts in response to changes in reliability. Price-capping also induces the onopolist to reduce reliability (Appendix, Proposition 7), which, as noted above, was inefficiently high initially. The cost savings associated with the reliability reduction and the under-insurance are partly passed to consuers via the lower price and partly taken by the onopolist as (supernoral) profits. 8. Conclusions This paper has introduced a new definition of reliability based on the objective probability of output failing in the hands of consuers. This definition has been incorporated into a siple odel of onopoly in which the fir can vary output reliability and has an incentive voluntarily to offer a warranty (or copensation deal) to the arket. The odel suggests that, even though the iposition of an effective price cap on the onopolist has the standard effect of increasing quantity and decreasing price, it will also induce the onopolist to reduce reliability. This is consistent, for exaple, with the findings of Fraser (1994a, b), who shows that a pure price cap iposed on a private onopoly supplier of electricity induced the fir to reduce the reliability of supply. The odel also iplies that an uncapped onopolist will offer a full oney back warranty, thus allocating risk efficiently. An effective price cap induces the onopolist to worsen the warranty (or copensation) ters, generating underinsurance and an inefficient allocation of risk. The cost savings arising fro reliability reduction and under-insurance are partly passed on to consuers via the lower price and partly taken by the onopolist as (supernoral) profits. Appendix This Appendix contains the proofs of the results discussed in the ain text. Part A characterises a Pareto-efficient allocation of resources. Part B characterises
Donald A. R. George 11 onopoly equilibriu with and without a price cap. The approach is to derive and utilise the first-order conditions of the Lagrangians given in the ain text. A. Pareto-Efficient Allocation Consider first the Lagrangian of equation (8). Its first-order conditions characterise a Pareto-efficient allocation: L = RU ( M p + f ( z)) U ( M p + β ) + λ( pz zc( βz(1 ). (A1) It is easy to prove the following proposition. Proposition 1. Pareto-efficiency requires that the profit constraint is binding (i.e., Φ = 0 ). Proof. Differentiating equation (A1) with respect to p and using equations (1) and (2) yields one of the first-order conditions for an interior axiu: L p = R ( x) (1 + λ z = 0 λ z = RU ( x) > 0. (A2) But z > 0, hence λ > 0, and it follows by copleentary slackness that the profit constraint is binding. We now establish that Pareto-efficiency requires that a full oney back warranty is offered. First it is necessary to establish three useful leas. Lea 1. Pareto-efficiency requires that β = f (z). Proof. Differentiating equation (A1) with respect to β and using equations (1) and (2) yields another first-order condition for an interior solution: L = ( 1 λz(1 = 0 β ( 1 R ) = (1 ( R ( x) ) U = R ( x) R = R ( x) x = y, using equation (A2), since R 1 and R 0 because we are seeking an interior solution, and (.) is invertible because U () < 0. Hence fro equations (1) and (2) we have β = f (z). Lea 2. Pareto-efficiency requires that β = C (. Proof. Differentiating equation (A1) with respect to R and using equations (1) and (2) yields another first-order condition for an interior solution: L R = U ( x) + λ ( zc ( + βz) = 0. (A3)
12 International Journal of Business and Econoics But x = y (fro Lea 1), λ > 0 (fro Proposition 1), and z > 0 (by definition), hence equation (A3) iplies that β = C ( as required. Lea 3. Pareto-efficiency requires that p = f (z). Proof. We have x = y fro Lea 2. So equation (4) yields: RU ( x) U ( x) = U ( M ) U ( x) = U ( M ) x = M, using the fact that U () is invertible because U () > 0. Hence, fro equation (1) p = f (z) as required. It is now straightforward to establish Proposition 2. Proposition 2. Pareto-efficiency requires that a full oney back warranty is offered (i.e., p = β ). Proof. Cobining Leas 1 and 3, we have We now establish the following useful lea. p = β as required. Lea 4. Pareto-efficiency requires that C ( R = C(. Proof. Fro Proposition 1, Pareto-efficiency requires that profits are zero. Thus, using Proposition 2 and equation (7) we obtain: pz zc( pz + pzr = 0 pr = C(. Cobining Proposition 2 and Lea 2, we have p = β = C (. Thus, C ( R = C( as required. B. Monopoly Equilibriu In this section we characterise onopoly equilibriu with and without pricecapping by deriving the first-order conditions of the Lagrangian specified in Section 6 of the ain text as equation (13): ( p c p) M = pz zc( β z(1 + μ( RU ( x) U ( M )) + ν. (A4) Note ν > 0 corresponds to the price-capped case and ν = 0 to the uncapped case. Differentiating (A4) yields first-order conditions for an interior solution. Differentiating with respect to p yields: ( R ( x) ( 1 ) = 0 z + μ ν. (A5)
Differentiating with respect to β yields: Donald A. R. George 13 z( 1 + μ (1 = 0. (A6) Differentiating with respect to R yields: [ U ( x) ] 0 zc ( + β z + μ =. (A7) Differentiating with respect to z yields: p C( β (1 + μru ( x) f ( z) = 0. (A8) It is now straightforward to establish the following results. Proposition 3. In onopoly equilibriu (price-capped and uncapped) the voluntary participation constraint binds. That is: U ( M ) = RU ( x) U. (A9) Proof. Equation (A6) iplies that μ 0 since we are seeking an interior solution (so that z > 0 and 1 > R > 0 ). Hence, by copleentary slackness, the voluntary participation constraint ust bind. Proposition 4. The uncapped onopolist provides the arginal consuer with full insurance, while the price-capped onopolist provides under-insurance. Proof. Equations (A5) and (A6) together iply that R ( ( x)) =ν μ. Thus, for the unregulated case ( ν = 0 ) we ust have x = y. (Note that U () is invertible because U () < 0.) Hence, in the uncapped case, the arginal consuer is fully insured. For the price-capped case we have ν > 0 and hence, by a siilar arguent, y > x and the arginal consuer is less than fully insured. Corollary 1. For the uncapped case we have: f ( z) = p = β, (A10) and for the price-capped case: f ( z) > p > β. (A11) Proof. These results follow fro (A9) and Proposition 4. We now prove the inequalities in (11) of the ain text which express coparisons between onopoly levels (with and without a price cap) and efficient levels of the relevant variables. We start by considering reliability R.
14 International Journal of Business and Econoics Proposition 5. Reliability in uncapped onopoly equilibriu is above the Paretoefficient level. Proof. Equations (A6) and (A8) together iply that: zr ( x) p + f ( z) = C( + β (1. (A12) But in the uncapped case x = y, hence (A12) iplies: p + zrf ( z) = C( + β (1. (A13) Now note that, given x = y, (A7) iplies that: β = C (. (A14) Cobing (A13), (A14), and Corollary 1 yields: C ( RC ( = zrf ( z). (A15) Now define the function F( R ) as follows: F ( = C( RC (. (A16) Differentiating F (noting that C (.) > 0 ) gives: F ( = RC ( < 0. (A17) Using the function F we can now copare the onopoly level of R with the efficient level. Fro Lea 4: F ( R ) = 0, (A18) while (A15) gives: F ( R ) < 0, (A19) since f ( z) < 0. Hence (A17) iplies that Corollary 2. β > β. R > R as required. Proof. Corollary 2 follows fro Lea 2, (A14), and Proposition 5 and noting that C ( > 0. Corollary 3. p > p. Proof. Corollary 3 follows fro Proposition 2, (A10), and Corollary 2.
Donald A. R. George 15 Corollary 4. z < z. Proof. Corollary 4 follows fro Lea 3, (A11), and Corollary 3. Figure 3. The Function U (). U (x) H U (y) f (z) β y x We now turn to the price-capped onopolist. It is useful to establish the following lea. Lea 5. Under price-capped onopoly C ( < f ( z). Proof. Using (A6) to eliinate μ fro (A7) yields: U ( x) C ( = β +. (A20) The proof relies on the strict concavity of the utility function U () ; see Figure 3. Strict concavity iplies that: U ( x) < H < ( x y) = ( f ( z) β ) U ( x) < f ( z) β. (A21) Cobining (A20) and (A21) yields the result required. We can now establish the following proposition. Proposition 6. The price-capped onopoly output is below the efficient level. Proof. Cobining Lea 3 with (A11) yields:
16 International Journal of Business and Econoics c c f ( z ) > p p = f ( z ). (A22) Hence, noting that f (.) < 0, it follows that Finally we establish Proposition 7. z c < z as required. Proposition 7. Reliability is lower under price-capped onopoly than under uncapped onopoly. Proof. Cobining Lea 5, (A10), and (A14) yields: c c C ( R ) < f ( z ) < f ( z ) = C ( R ), (A23) noting that f () < 0 and C () > 0. Hence c R < R as required. References Fraser, R. W., (1994a), Price, Quality and Regulation: An Analysis of Price Capping and the Reliability of Electricity Supply, Energy Econoics, 16, 175-183. Fraser, R. W., (1994b), Privatisation, Price-Capping and Reliability, Utilities Policy, 4, 121-127. Goering, G. E., (1997), Product Durability and Moral Hazard, Review of Industrial Organisation, 12, 399-411. Grossan, S. J., (1981), The Inforational Role of Warranties and Private Disclosure about Product Quality, Journal of Law and Econoics, 24, 461-483. Klein, B and K. B. Leffler, (1981), The Role of Market Forces in Assuring Contractual Perforance, Journal of Political Econoy, 89, 615-641. Kreps, D. M. and R. Wilson, (1982), Reputation and Iperfect Inforation, Journal of Econoic Theory, 27, 253-279. Matsukawa, I. and Y. Fujii, (1994), Custoer Preferences for Reliable Power Supply: Using Data on Actual Choices of Back-Up Equipent, Review of Econoics and Statistics, 76, 434-446. Matthews, S. and J. Moore, (1987), Monopoly Provision of Quality and Warranties: An Exploration in the Theory of Multidiensional Screening, Econoetrica, 55, 441-467. McClure, J. E. and L. C. Spector, (1991), Joint Product Signals of Quality, Atlantic Econoic Journal, 19, 38-41. Milgro, P. and J. Roberts, (1986), Price and Advertising Signals of Product Quality, Journal of Political Econoy, 94, 796-821. Milgro, P. and J. Roberts, (1982), Liit Pricing and Entry under Incoplete Inforation: An Equilibriu Analysis, Econoetrica, 50, 443-459. Mussa, M. and S. Rosen, (1978), Monopoly and Product Quality, Journal of Econoic Theory, 18, 301-317. Shapiro, C., (1983), Preius for High Quality Products as Returns to Reputations, Quarterly Journal of Econoics, 98, 659-679.