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Essays on production and pricing decisions Mok, Yat-Koon 1993

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ESSAYS ON PRODUCTION AND PRICING DECISIONSByYat-Koon MokM. Sc., University of AstonM. Phil., University of CambridgeA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESCOMMERCE AND BUSINESS ADMINISTRATIONWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1993© Yat-Koon Mok, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Commerce and Business AdministrationThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1Z1Date:AbstractThere has been considerable interest in finding and explaining the basic elements that candrive product quality up. In the literature this is largely done by modelling the effects ofinvesting in learning and process improvement, and of cost reduction. In the first essay,demand is modelled as a function of price and quality. With this demand function, thefirm should produce output of higher quality, the increase in quality being dependent onconsumers’ sensitivity to quality and to price, and the effect of technological improvementon product price and quality are very different from those when the demand is a functionof price alone.Some twenty states in the U.S. have passed recycling laws which mandate consumptionof old newspaper by the newsprint industry. To study the effect of regulation, a modelis used in which two firms compete under the regulatory constraint—one firm producingthe recycled product, the other the virgin product. Assuming the regulatory constraintis binding, and the demand for the recycled product is derived solely from the legislation,interesting results such as the two firms share equal profits, and consumers pay higheraverage price in competitive equilibrium than the cartel price, are obtained in the secondessay. The two firm model is generalized to include n firms which compete under thesame kind of regulatory constraint in the third essay. Results similar to the two firm caseare obtained.When the recycled product and the virgin product are partially substitutable, regulation that mandates consumption of the recycled product results in infinitely manyequilibria. A dominating equilibrium exists if the demand parameters satisfy a certaincondition, otherwise it is not clear how to select an equilibrium. On the other hand, a11suitable tax on the virgin product, or its producer, serves to induce compliance with therecycling policy and equilibrium selection. The equilibrium prices and profits of the twofirms under the schemes of production tax, excessive consumption tax and progressiveprofit tax are examined and compared in the fourth essay. It is interesting to find thatthe tax rate for excessive consumption is comparatively low and, in equilibrium, this taxscheme collects no tax payment.111Table of ContentsAbstract iiList of Figures viiAcknowledgement viii1 Effects of Technological Improvement on Quality and Price 11.1 Introduction 11.2 Quality Costs 51.3 Effects of Technological Improvement on Quality and Manufacturing Costs 71.4 Demand and Consumer Sensitivity 101.5 The Objective Function 121.6 Analysis 131.7 Effects of Acquiring an Improved Technology 141.8 Conclusion 232 The Effects of Mandatory Recycling on the Newsprint Industry 262.1 Introduction 262.2 The Model 282.3 Price Competition 302.4 The Effects of Regulation on the Equilibrium Prices and Quantities . 322.5 The Effect of Recycling Regulation on Profits 36iv2.6 Comparison of Prices, Quantities and Profits Before and After RecyclingRegulation 372.7 Conclusion 403 Competition When Market Share Is Regulated 423.1 Introduction 423.2 Regulated Market Share 433.3 Effect of Introducing a Regulated Market 474 Competition under Recycling Regulations 494.1 Introduction 494.2 The Model and Competition under Mandatory Consumption of RecycledProduct 514.3 Competition Under Taxes 574.3.1 Taxing Excessive Consumption of Virgin Product 584.3.2 Production Tax 614.3.3 Progressive Profit Tax 624.3.4 Some Remarks 654.4 Conclusion 66Appendices 68A Proofs for Chapter 1. 68B Proofs for Chapter 2. 78C Proofs for Chapter 3. 89D Proofs for Chapter 4 91VE Background to the Mandatory Recycling Legislation for Newsprint 96Bibliography 101viList of Figures1.1 The Economic Conformance Level Model 61.2 Appraisal and Prevention Cost Reduction 93.3 Cournot and Nash Equilibrium and Cartel Prices . . . 464.4 Reaction Functions of the Two Firms 534.5 Policy Lines and Reaction Functions 544.6 Reaction Functions of the Two Firms and the Policy 564.7 Reaction Functions of the Two Firms under Tax 604.8 Reaction Functions of the Two Firms under Progressive Tax 64B.9 Reaction Functions for Firms 1 and 2 88viiAcknowledgementI would like to thank my supervisors, Professors Shelby Brumelle and Ilan Vertinsky, fortheir continual support throughout the work of the thesis. My thanks are to ProfessorBill Ziemba for being a member of my committee, and to Professor Philip Loewen for hiscomments on Chapter 1 of this thesis. I would also like to thank Drs. Darcie Booth andDon Roberts for their help and hospitality at Forestry Canada, Ottawa.Financial support from the Forest Economics and Policy Analysis Research Unitat the University of British Columbia and from Hong Kong University are gratefullyacknowledged.VII’Chapter 1Effects of Technological Improvement on Quality and Price1.1 IntroductionIn recent years, the importance of product quality for a manufacturing firm to succeed inthe market place has received major attention, motivated by the observation that manyJapanese manufacturing firms can produce outputs at high quality levels with relativelylow unit costs. Evidence is given, for example, by Garvin (1983), Hayes, Wheelwrightand Clark (1988) and Fortuna (1990). The importance of product quality is undeniable.Meyer, Nakane, Miller and Ferdows (1989) report, from their yearly ‘ManufacturingFutures’ surveys on large manufacturers in Europe, North America and Japan, thatquality was consistently on the top of the list of concerns.It is generally agreed that the kind of product quality which contributes the most tothe competitiveness of Japanese manufacturing firms is conformance to design specifications, which reflects the degree of consistency and reliability of the products. This paperformally models the effects of improved technology on the costs of conformance qualitythat are associated with the quality enhancing activities , and manufacturing cost. Therole of product quality in affecting demand is also explicitly considered in the model.Numerous studies employing analytic models provide insights into various aspects ofquality. Fine (1986) introduces a model of quality-based learning in which learning isrepresented by the sum over time of the product of quality and output volume, where1Chapter 1. Effects of Technological Improvement on Quality and Price 2the quality considered is the conformance level to design. He finds that when the manufacturing cost is a function of learning, and under certain conditions of the discount rateand the planning horizon, product quality should start at a high level and continuouslydecrease until the end of the horizon, at which time the quality is at the value which minimizes cost. In case learning can reduce quality cost, and the cost reduction parameter isa function of learning, then a manufacturing system should produce at a relatively highquality level if there is no discounting; moreover, output quality increases over time. Ourmodel in this paper is developed in the same spirit of Fine (1986). Instead of pursuingfurther the issues of learning, we look at the effects of technological improvements whenconsumers’ demand is based also on product quality.Fine (1988) and Marcellus and Dada (1991) employ modified versions of stochasticmaintenance or replacement models and regard quality level as a state variable ratherthan as a decision variable. The quality level of a station as described by Fine (1988) isrepresented by its probability of being in control. Learning is assumed to occur wheneveran inspection is made and the station is found to be out of control. Each time learningoccurs, the probability that the station is out of control is assumed to decrease, and thusquality increases. The author concludes that ignoring the learning benefits of inspectionand quality control activities may lead to under-investment in quality improvement activities. The model of Marcellus and Dada (1991) is similar to that of Fine (1988), butquality level is represented by the proportion of nondefectives. A decision maker candecide whether or not to learn. Choosing to learn incurs a cost and reduces the proportion of defectives. They show that if the expected reduction in the cost of quality is lessthan the opportunity cost of the ‘learn action’, then the ‘learn action’ is not economical.Thus if a firm’s ‘hurdle rate’ is set too high, then even short-term learning is discouraged.The optimal policy for the model, in terms of expected discounted present cost, can beinterpreted to have tradeoff between the cost of failure and the cost of prevention. OtherChapter 1. Effects of Technological Improvement on Quality and Price 3stochastic studies include those of Fine and Porteus (1989), who formulate and analysethe problem of gradual process improvement as a Markov decision process in which investment in process improvement results in a probabilistic number of improvements, Leeand Tapiero (1986), who link the optimal sampling parameters in quality control withsales which have a negative binomial distribution, and Nandakumar, Datar and Akella(1993), who develop a model for delay and tardiness costs due to poor quality and amodel for costs of internal and external failure, and obtain results analogous to Fine andPorteus (1989) for multiple products.The importance of technology, to the Japanese manufacturers is summarised byOhmae (1984) saying that, “Japanese industry did not become competitive through QCcircles and company songs alone! What really counted was the wise decision to changethe whole modus operandi of Japanese manufacturing by investing heavily in new facilities and techniques.” The importance of technological improvements is also reflected bythe Japanese concern about falling behind in process technology which was ranked thirdby the large Japanese manufacturing firms in the survey by Meyer, Nakane, Miller andFerdows (1989.) The close relationship between the motivation to adopt better technology and the desire for high product quality is clear. Fortuna (1990) also suggests that toreduce the process variation (a source of poor quality) of a stable manufacturing system,improvements in the process are necessary. Examples of process improvements givenby the author include equipment modifications and enhancements. Thus, it is safe tobelieve that improved technology has the ability to improve quality. We postulate thatit directly increases the lower bound of the domain of product quality on which a firmmakes its decision. That is, without any other quality related activities, the quality ofthe output from an improved technology is higher than that from a technology on whichno improvement has been made. Indirectly, improved technology can enhance qualityrelated activities, and thus reduce the costs relating to these activities. These two effectsChapter 1. Effects of Technological Improvement on Quality and Price 4on quality are explained in more detail and modelled in section 3.Another way to model the impact on production decisions of quality concerns is toinclude quality in the demand function. Dorfman and Steiner (1954), for example, assumethat quality improvement shifts the demand curve to the right over the relevant range.On the other hand, Schmalensee (1979) assumes that higher quality shifts the inversedemand function to the left. Based on specific assumptions made about consumers’ tastefor quality, Shapiro (1983) models demand as a function of price per unit of quality. Leeand Tapiero (1986) derive a probability distribution for sales as a function of the defectrate in a production batch, and therefore relate sales to the defect rate. However, theirmodel and objectives are very different from ours. They relate optimal sampling strategyfor quality control to the parameters of the sales distribution, whereas we are interestedin the effects of technological improvements on quality, price and output given that thedemand is a function of price as well as quality. Others largely ignore the effect of productquality on demand in their studies relating to quality. We derive the demand functionwhich includes quality as a variable in section 4 by assuming that given the price andquality of a product, a perceived fair price of quality of the product is encoded by eachconsumer. Subsequently, a consumer’s purchase decision is determined by her perceivedfair price. With quality in the demand function, the optimal decision on price andquality of the product, after adoption of an improved technology, depend on how sensitiveconsumers are to price changes, and to quality changes. If consumers are indifferent toquality variations and base their decision on price alone, then a result similar to that ofFine (1986) is obtained. Indeed if we think of technological improvements broadly andconsider continuous learning over time as continuous technological improvements, thenthe model in this paper is similar to that of a discrete jump in learning. We review inthe next section the model of economic conformance quality and the costs associatedwith it. In section 3 the effects of technological improvements on the costs of quality andChapter 1. Effects of Technological Improvement on Quality and Price 5manufacturing are modelled. A demand function based on price and quality is derived,and consumers’ sensitivities are defined, in section 4. We formulate the objective functionof a manufacturing firm facing the opportunity of improving its technology in section 5.Then in section 6 the optimal quality level and price are obtained. The effects of adoptingan improved technology, a change in its adoption time and a variation of the time-pathof the investment cost are analysed in section 7. A conclusion is given in section 8.1.2 Quality CostsIn the quality control and assurance literature, two aspects of the quality of a productare often studied. One aspect is the product design. The second is the degree to whichthe product conforms to design specifications (see, e.g., Juran 1988.) Here we considerthe latter case.The model of economic conformance level (EUL) is well known and accepted in theoperations management literature (examples are Chase and Aquilano 1981, and Hill 1983)and in quality control and management literature (examples are Juran and Gryna 1970,and Sinha and Wiliborn 1985.) In the ECL model, the quality level q of a product isdefined on an interval [0, 1], i.e., q e [0, 1] where q = 1 means the firm is producing atthe perfect quality level at which there are no defective outputs. At q = 0, the productsare completely defective.Two kinds of cost are defined over the interval of quality. Failure cost,c1(q), is incurreddue to defective outputs. This is an aggregate of costs due to rework, reinspection,downtime and other losses such as returned material, after services, warranty claims,etc. Appraisal and prevention cost, c2(q), is associated with efforts to reach a higherconformance level. It is an aggregate cost resulting from activities in quality control,Chapter 1. Effects of Technological Improvement on Quality and Price 6Costs1qFigure 1.1: The Economic Conformance Level Modelquality assurance and quality planning such as inspection, test, test gear maintenance,training, reporting and quality improvement projects. A more detailed breakdown ofthese costs is given by Hagan (1986). These quality cost functions are highly dependenton the characteristics of the production system, especially its technology. The generalshapes of the cost functions for a production system are depicted in Figure 1.1, wherethe ECL is the value of q at which the total cost of quality, ci(q) +c2(q), is minimised.Chapter 1. Effects of Technological Improvement on Quality and Price 71.3 Effects of Technological Improvement on Quality and ManufacturingCostsThe technology that we consider here can be capital equipment, method of productionprocesses or the configuration of the production system of the firm, a platform for themanufacturing system on which various productive and quality related activities arebuilt. The importance of technology as a determining factor for quality of conformanceis acknowledged by Juran (1988). Schroeder (1985) postulates that the impact of bettertechnology on quality is to shift the total quality cost downwards and towards the direction of fewer defective products. Generally, technological improvement in this respectcan have various impacts on the system output. In this paper we study some of theseimpacts by making two postulates about the effects of technological improvement on thecosts of quality:1. An improved technology can increase the effectiveness of the quality control, qualityassurance and quality planning activities so that quality appraisal and preventioncan be performed in a more efficient manner. This is consistent with the remarksonce made by an executive of Corning about its manufacturing system that mostof the gains came from technology and that to get people to work smarter involvedthe application of technology (Meadows 1984.) Therefore, as a result of increasedeffectiveness, the appraisal and prevention cost, c2(q), is lowered for any givenquality level to crc2(q), where a E (0, 1). We shall call this the indirect effect onquality cost.2. An improved technology has the effect of increasing the lower bound of the quality level to, say, , so that the appraisal and prevention cost function ac2(q) isshifted downward. We call this the direct effect on quality cost. Examples of thisare employment of advanced automatic process diagnosis and production processChapter 1. Effects of Technological Improvement on Quality and Price 8adjustment equipments to reduce process variations. More examples of quality costreductions with equipment, method and process types of technology are given byTaguchi, Elsayed and Hsiang (1989).The total (direct and indirect) effect of technological improvement is that the appraisaland prevention cost function becomes cxc2(q) — 0, as shown in Figure 1.2. Note thatac2(q) — 0 < 0 for q near 0. We assume throughout this work that the total costof quality ci(q) + ac2(q) — 0 is positive for all q, and note that the optimal solutionsdescribed below never enter the region where the revised appraisal and prevention costis negative. We choose not to model the resulting cost, after technological improvement,as a nonsmooth function which is zero for q e [0, ] and crc2(q) — 0 for q E [, 1], for thisdoes not add much to the insight that can be obtained from the model developed in thispaper.Often an improved technology, apart from being better able to enhance quality activities, also has the property of reducing manufacturing cost at any given output level.Let c3 be the unit cost of manufacturing. Constant unit manufacturing cost is assumedhere to simplify exposition. Most results in this paper still hold so long as the total manufacturing cost as a function of output is convex and increasing. After the adoption ofan improved production technology the unit manufacturing cost becomes /3c, /3 E (0, 1).We can consider an additional downward shift of the manufacturing cost due to technological improvement. However this has the same effect as 0, and is thus absorbed inthis term, since total quality and manufacturing unit cost after technology acquisition isci(q) + oc2(q) + /3c — 0.We assume in this paper that ci(.) is strictly convex and decreasing, ci(l) = 0 andci(0) = +co. c2(.) is strictly convex and increasing, c2(l) = +oo and c2(0) = 0.Chapter 1. Effects of Technological Improvement on Quality and Price 9Costsac2(q)c2(q)—OqFigure 1.2: Appraisal and Prevention Cost ReductionChapter 1. Effects of Technological Improvement on Quality and Price 10We consider the case in which the production technology that improves over time isexogenous and the firm can acquire it whenever it feels appropriate to do so. Thus thecost reduction parameters a, 0 and /3 are functions of time, t. It is further assumed thata(.) is decreasing, a(0) = 1, a(T) = a, where T is some finite horizon and e (0, 1).0(.) is increasing, 0(0) = 0, 0(T) = and is finite such that ci(q) + a(t)c2(q) — 0(t) >0,Vt e [0, T] and ‘v’q E [0, 1]. /3(.) is decreasing, 3(0) = 1, /3(T)= / and $ e (0, 1).The cost involved in adopting a technology can be an increasing or decreasing functionof time. These two cases will be studied separately. The cost of adopting the technology,I(.), is assumed to be independent of the quality level of the output it produces.1.4 Demand and Consumer SensitivityA consumer’s propensity to purchase product quality is assumed to be captured by anindex of consumer characteristics, E R+, which may reflect the budget’s constraint ofthe consumers. Propensity to purchase quality may vary among consumers. A productwith quality q and price p is subject to the evaluation of each consumer and the resultof individual judgement is the formation of a function, W(,p, q), which is commonacross all consumers. Thus for a given product of quality q sold at price p, W(, p, q)represents the buyer’s perceived fair price of quality for a buyer with index . There isevidence to suggest that buyers often translate actual price into perceptions of price. Forexample, Jacoby and Olson (1977) distinguish between objective price (the monetaryprice) of a product and perceived price (the price as encoded by the consumer.) Themonetary price is frequently not the price encoded by consumers (Zeithaml 1988.) Asurvey of internal reference price theory is given by Winer (1988). Monroe (1990, p.55)also provides evidence that a change in product quality while maintaining the originalChapter 1. Effects of Technological Improvement on Quality and Price 11monetary price results in a corresponding shift in the buyers’ perceptions and purchases.A consumer with index who has a perceived fair price of quality W(, p, q) willpurchase the product at (p, q) if his perceived fair price is larger than the actual price,i.e., if W(,p, q) p. 1 In Shapiro’s (1983) model used in a study of reputation, aconsumer will make a purchase if p/q, which corresponds to W(., p, q) = q in ourmodel. We further assume consumer propensities to purchase quality do not change overtime, i.e., that is not a function of time. The production system, however, may adjustboth the price and quality at the time technology changes.In our model, all individuals rank the product in the same way but because of differences in budgets may opt to purchase different levels of quality. This diversity isrepresented by a density function of consumer type f(). So a product with price p andquality q is able to capture the expected fraction of the marketF(P,q)=f f()d, (1.1)EA(p,q)where A(p, q)= { : W(, p, q) p}. We call this expected market portion F(p, q) thedemand rate.It is reasonable to suppose that W(, p, q) is strictly increasing in q and strictlydecreasing in p. The demand F(p, q) will then have the same monotonicity properties:F(p, q) < 0 and Fq(p, q) > 0. We assume that F(.,.) is finite and jointly concave,F(p,q) > 0 for p E [0,P(q)], F(p,q) 0 for p > F(q). Let F(p,q) and Fq(p,q) bemeasures respectively for the consumers’ sensitivities to variations in price and quality,see Dorfman and Steiner (1954). We define quality-price sensitivity as1This inequality can be derived through the approach of Caplin and Nalebuff (1991) who studyexistence of price equilibrium in imperfect competition with fixed product attribute q. They assumea linear utility function across all consumers and conclude that each consumer has a reservation price,R(E, q), for the product. A consumer purchases the good if p R(., q). In this article both p and q aredecision variables and we allow the price and quality to interact.Chapter 1. Effects of Technological Improvement on Quality and Price 12s Fq(p,q)(p,q) .— —_____F(p, q)S(p, q) is a positive number obtained from the ratio of demand change due to qualityvariations and demand change due to price variations. It provides a measure of relativesensitivity of the consumers to quality against price.1.5 The Objective FunctionThe objective of the productive system is to determine the price and quality, (p, q), of itsoutput, before and after acquisition of improved technology, over the planning horizon,T, to maximize profit. There is no loss in generality in restricting the acquisition time, t1,to the interval [0, T] since we are interested in the effects of adopting improved technologyrather than the optimal number of adoptions over the horizon. On the other hand wecan view this restriction on time frame as a model of incremental technological or processupgrading in which decisions on technological improvements are made necessarily overshorter horizons (see Schonberger 1982 and Hayes and Wheelwright 1984.) The objectivefunction given below is a version of the capacity expansion model by Hinomoto (1965).(F) max {K(po,qo,pi,qi,ti)po,qo,pi,qi,titiJ et[po —c1(qo) —c2(qo) —c3]F(po,qo)dt +0J e’[p1 — ci(q1)— a(t1)c2(ql) + 0(t1) — /3(ti)c}F(pi, qi)dttl_ehI(ti)} (1.2)Chapter 1. Effects of Technological Improvement on Quality and Price 13We assume either 1(t) has taken account of the salvage value of the replaced technology or it has no salvage value, and ascribe negligible salvage value to the adoptedtechnology at the end of the horizon, since the firm may not want to sell the technologywhich may invite competition due to entry of the buyer.In the analysis below a prime will be used to denote the derivative with respect tothe argument.1.6 AnalysisDefineH°(p,q) : = [p—ci(q) —c2(q) —ca]F(p,q). (1.3)H’(p, q, t1) : = [p — ci(q) — (a(ti)c2q — 0(t1)) — /3(ti)c]F(p, q). (1.4)A solution to (F) satisfies the following set of necessary conditions:H°(po,qo)=O , Hq°(po,qo)=O. (1.5)H(pi,qi,ti) = 0 , H(pi,qi,ti) = 0. (1.6)Lemma 1 H°(.,.) and H1(., .,t1) are pseudocoricave for any t1 E [0, T].Proposition 1 The optimum quality levels of the product, qo and qi, before and afteracquisition of improved technology are higher than EGL, and they are the values at whichthe marginal quality costs, before and after acquisition, equal to S(po, qo) and S(pi, qi)respectively, where po and p1 are the optimal prices.The result in Proposition 1 is rather interesting. It suggests that a production systemshould always produce at a quaiity level higher than ECL so long as consumers preferChapter 1. Effects of Technological Improvement on Quality and Price 14higher quality at any given price. Furthermore, the optimal quality level is higher if eitherconsumers are more sensitive to quality variations or less sensitive to price variations,or both in which case the optimal quality of output is the highest. This result confirmsthe intuition that a manufacturer is better off to produce a higher quality product ifconsumers are more responsive to high quality. However, it is not so obvious that whenthe consumers are less responsive to price it is more profitable for the firm produce higherquality product. Cole (1992) and some other authors suggest that the the Japanese seethe pursuit of higher quality as a means of driving the cost curve down. (For a similarobservation see Fine (1986).) The result here suggests that the pursuit of higher qualitycan also be consumer-driven.The ECL is optimal only when Fq(p, q) = 0, i.e., when product quality has no bearingon consumer preference. Thus the ECL of quality, often targetted by the manufacturing divisions, at which the total quality cost is minimized, is suboptimal and reflects aseparation of manufacturing functions from market conditions.Proposition 2 Given arbitraryqo andqi, the optimum prices Po and p are characterisedby equating marginal revenue to marginal total cost, if a solution exists.Proposition 3 Given arbitrary po and p1, the optimum quality levels qo and q arerespectively characterised by equating marginal total cost to price, if a solution exists.1.7 Effects of Acquiring an Improved TechnologyIt is not clear how the optimal (p, q) and the value of F(p, q) change after the adoptionof an improved technology at time t1. Intuitively, one would expect that due to themanufacturing and quality cost reduction effects of the technology (i) q would increase,Chapter 1. Effects of Technological Improvement on Quality and Price 15or (ii) p would decrease, or (iii) both (i) and (ii) simultaneously. We shall study thesigns of change in the optimal (p, q) and F(p, q) with a change respectively of the costreduction parameters 0(t1), a(ti) and 9(t1). We shall suppress the argument of theseparameters which are written as 0, a and /9. The profit function after the change oftechnology is now a function of p,q, 0, a and /9,H(p, q, 8, a, /3) : [p — c1(q) — ac2(q) + 0 — /3cJF(p, q) (1.7)The optimal (p, q) satisfiesH(p, q, 8, a, /9) = 0. (1.8)Hq(p,q,0,a,/3) = 0. (1.9)By the Implicit Function Theorem, in the neighbourhood of (p, q) which satisfies theoptimality conditions (1.8) and (1.9), p and q are continuous functions of 0 with a and/3 fixed at given values. Differentiating w.r.t. 0 we haveH0 = Hpqqo + Hp9 + H0 = 0. (1.10)HqeHqqqe+Hqppo+Hq9. (1.11)where H is a shorthand for H(p,q,0,a,/3). Define 1/k := HppHqq — Hpq2 which ispositive at the optimal value of (p, q) which satifies (1.8) and (1.9), and solving the aboveequations, pe, qe and F9 are obtained as follows:P9 k(_HpeHqq + HqeHpq). (1.12)q = k(HqoHpp+HpoHqp). (1.13)F9(p,q) = Fp(p,q)pe+Fq(p,q)qo. (1.14)Similarly, to study the effect of indirect quality cost reduction property of improvedtechnology the following equations are obtained using the Implicit Function TheoremChapter 1. Effects of Technological Improvement on Quality and Price 16with 0 and ,6 fixed.= k(HpaHqq + HqcHqp). (1.15)q = k(HqaHpp + HpoHqp). (1.16)Fcr(p,q) = Fp(p,q)p+Fq(p,q)qa. (1.17)Following the same approach the effect of manufacturing cost reduction with a bettertechnology can be analysed by the following expressions which are obtained by fixing 0and a.= k(HpjHqq + HqHpq). (1.18)k(HqHpp+HpHqp). (1.19)F(p,q) = Fp(p,q)p+Fq(p,q)q. (1.20)Proposition 4 (i) If Sq(p, q) 0, then pe <0 and p > 0.(ii)(a) If S(p,q) 0, then qo 0 and q 0.(ii)(b) If S(p,q) 0 then q <0.(iii) If S(p,q) 0 and Sq(p,q) 0, then Fe(p,q) >0, Fa(p,q) <0, and Fp(p,q) <0.If the relative quality-price sensitivity of the consumers is a nonincreasing functionof quality, then the direct quality cost reduction effect, and the manufacturing costreduction effect, of adopting an improved technology render a lower optimal price. Onthe other hand, this may not be true if the quality-price sensitivity of the consumers isan increasing function of quality. Indeed, if the consumers are rather insensitive to theprice changes of a product, and the total quality cost of this product increases ratherslowly, as compared to the increases in consumers’ quality-price sensitivity due to qualityincreases, then the two cost reduction effects will each result in an increase in the priceChapter 1. Effects of Technological Improvement on Quality and Price 17of the product. Also, we cannot determine the sign of the optimal price change withrespect to the lowering of the indirect quality cost, p. It depends on how sharply thechanges in the relative quality-price sensitivity and price sensitivity of the consumers arewith quality changes, how sensitive the consumers are to price and to quality, and howsharply the total quality cost increases. It is interesting to note that the cost reductioneffects on price do not depend on how the consumers’ quality-price sensitivity changeswith price.Contrary to the common belief that cost reduction accompanies higher quality, theexpression (ii)(a) of the above proposition shows that a lower optimal quality level resultsfrom direct quality cost reduction, and from manufacturing cost reduction, when theconsumers’ quality-price sensitivity is an increasing function of price. If, in addition, theconsumers’ relative quality-price sensitivity is also a decreasing function of quality, theoptimal price decreases. This means that the manufacturing firm is better off to producethe product whose quality is more affordable. It is also interesting to note that whenthe consumers’ quality-price sensitivity does not change with price, there is no qualitychange due to the two kinds of improvement in costs above, but the optimal qualityincreases with indirect reductions of quality cost. All three types of cost improvementinduce higher optimal quality in case the consumers’ quality-price sensitivity decreaseswith price. The optimal quality may increase or decrease, as a result of indirect qualitycost reductions, when the consumers’ quality-price sensitivity increases with price. Thecost improvement effects on quality do not depend on how the consumers’ quality-pricesensitivity changes with quality.Output for the product increases with all three kinds of cost reduction due to technological improvements if the consumers’ quality-price sensitivity does not increase withprice, and does not increase with quality. This is a direct consequence of (i) and (ii) inthe above proposition. Note that the optimal price may rise as a result of the indirectChapter 1. Effects of Technological Improvement on Quality and Price 18effect of technological improvement, but the impact of the possible price rise on demandis dominated by that of the corresponding quality rise. Consequently, output quantityincreases. For the other cases, output quantity can increase, or decrease, depending alsoon the consumers’ sensitivities to price and quality, and how sharply the total qualitycost increases.It is instructive to look at the case of demand which is a function of price alone.The following proposition shows that the effects of the cost reductions due to improvedtechnology are definite, when quality is dropped out of the demand rate.Proposition 5 If the demand is a function of price alone, then(i)q<0, and qe=qp=0.(ii)pa > 0,p <0 and pp >0.(iii) Fa(p, q) <0, Fe(p, q) > 0 and Fp(p, q) <0.When the impact of product quality on demand is not included, the optimal pricedecreases and the output quantity increases. However, the change in optimal quality as aresult of improvements is different according to the different types of cost being reduced.Direct effect on quality cost reduction, and manufacturing cost reduction effect, do notaffect optimal quality level. It is indeed the case because these two cost reduction effectsshift the total quality cost curve downward, and thus do not move the ECL away fromits position. However, an increase in optimal quality results from the effect of indirectcost reduction.There is a clear distinction in pricing and quality decisions under the two differentdemand models. The unidirectional changes in optimal price and quality, with the adoption of an improved technology, are clear and simple, when demand is a function of pricealone. When product quality is taken into account by the consumers in the purchasedecisions, these changes are no longer one-directional, but depend on the behaviour ofChapter 1. Effects of Technological Improvement on Quality and Price 19the consumers’ quality-price sensitivity on price as well as on quality.The result that indirect cost reduction causes an increase in optimal quality, is similarto that due to quality-based learning, as modelled by Fine (1986) who uses a demandrate which is not a function quality. Indeed if we look at technological improvement as adiscrete jump in learning that lowers the prevention cost, then our result can be regardedas a discrete version of Fine’s. On the other hand, in our model, technological improvement on manufacturing cost does not change the optimal quality. This is in contrastwith the result that, except at the terminal time of the horizon, optimal quality increasesif quality-based learning reduces manufacturing cost— because if increased quality enhances quality-based learning which, in turn, reduces manufacturing cost, then optimalquality should increase. With this reasoning, it is clear then if learning is volume-based,rather than quality-based, our result indicates that reduction in manufacturing cost dueto learning does not change the optimal quality, it merely reduces the optimal price ofthe product.To study how a change in adoption time t1 affects the manufacturing system’s decisionon the price and quality of its output, the Implicit Function Theorem is used again toderive the following equations.Pt1 = k(HptiHqq + HqtjHpq). (1.21)qt1 = k(HqtjHpp+HptiHqp). (1.22)F1(p,q) = F(p,q)p1+F9(p,q)qt1. (1.23)Proposition 6 (i) q,1 >0 if S(p,q) 0. (ii) F1 > 0 if S(p,q) 0 and Sq(p,q) 0.If demand is a function of price alone, then qj1 O,p1 0 and Ft,(p, q) 0.With the condition of the relative sensitivity of customers given in the proposition,the effect of the improvements over time of the technology is such that acquiring it laterChapter 1. Effects of Technological Improvement on Quality and Price 20renders higher product quality and greater demand as compared to earlier adoption. Thesign of the price change cannot be determined. Thus price may increase. But because thequality increases to such an extent that it dominates the impact of a possible price rise,and demand increases. If demand is a function price alone, then again, changes in optimalprice and optimal quality do not depend on the consumers’ quality-price sensitivity, asshown in the above proposition.When a productive system makes decision on the time to adopt a technological improvement, it does so based on an estimate of the time path of the adoption cost, 1(t), ofthe technology. A change in the time path of the investment will affect the time, t1, atwhich the technology is adopted. There is no generally agreed shape of the time-cost relationship. It can be increasing or decreasing, convex or concave. Each of these differenttime-cost patterns will be considered.If t T, the productive system simply does not acquire the improved technologyinside the time frame of planning. So attention is restricted to ti E [0, T) in which caset1 satisfies the necessary condition for (F) in the following Lemma. When ti = 0, I’(t1)is the righthand derivative.Lemma 2 An extremal value ofti for (F) satisfies the following conditionrI(ti)— I’(t1) + H°(po, qo) — H’(p1,q,t1) + exp r11 J —H’(pi, q,t1)dt = 0. (1.24), at1An interpretation of the lemma is that to adopt the technology as soon as a timeis reached at which the profit per unit time just before the adoption, H°(po, qo), plusthe foregone total benefit due to cost reductions to the end time, in case of a delay inadoption, discounted to the adoption time, which is the integral term in the lemma, equalto the profit per unit time just after the adoption, H’(pi, q, ti), plus the change in theinvestment cost in case of a delay in adoption, I’(t1), minus the relief of interest accruedChapter 1. Effects of Technological Improvement on Quality and Price 21due to the delay in adoption, rI(ti). That is, at each point in time over the planninghorizon, the decision of whether to wait or to invest is made, based on the trade-offbetween the total marginal benefit and total marginal loss due to waiting. The decisionto invest is made at the time when marginal benefit equals marginal loss. This is similarto a result of Marcellus and Dada (1991), that the policy of whether or not to invest inlearning depends on the resulting marginal benefits and costs. The difference is that ourdeterministic model allows just one opportunity to invest in technological improvementover the planning horizon, whereas their probabilistic model allows investment in learningin every period.Define a higher (lower) investment cost function, relative to 1(t), as the time-costrelationship 1(t) := (I + 77)(t), 77(t) > 0 (p7t) < O),Vt e [O,T]. It can be seen thatp7(t) which is assumed finite is the time path of the investment cost deviation (I — I)(t).A variation in I changes the extremal adoption time, t1. It is assumed that (pi, qi) isalready chosen by the producer, and is thus fixed, prior to the variation in I. Ideally,one would allow (p1, qi) to be the optimal price and quality as functions of t1. However,doing so renders the analysis intractable. Define the Gateau differential of t1 (I) fromabove (below) for any i > 0 (q < 0) and any arbitrary s E R asDti(I;7):= iimui +877) —t(I) (1.25)Moreover, it is assumed that i E c’[O, T], o and /9 are convex, and 0 concave.Proposition 7 Given (p1, qi) is fixed and t1 is the extremal value, and,(i) a relatively higher investment cost function, that is > 0, and i’ 0, and(a) if I is decreasing and convex, then Dt1(I; ) > 0;(b) if I is either decreasing and concave, increasing and convex ,or increasing and concave,then Dti(I;’q) has the same sign as w(ti) — rI’(t1)+ I”(t);(ii) a relatively lower investment cost function, that is <0, and,‘0, andChapter 1. Effects of Technological Improvement on Quality and Price 22(a) if I is decreasing and convex, then Dti(I;—i) <0;(b) ifI is either decreasing and concave, increasing and convex, or increasing and concave,then Dt1(I;—) has the opposite sign as that of w(tj) — rI’(ti) + I”(t1),where w(ti) := (e_r(T_t1) + 1)[—c2(q)a’(tl) —c3/3’(11)+ 8’(ti)jF(pi, q)+ 1/r(e_(T_t1) — 1)[—c2(qi)cx”(ti) —c3f3”(ti) + 0” (ti)]F(pi, qi).The proposition shows, for a decreasing and convex technology adoption cost function,any positive deviation from it (higher cost) ,which is nonincreasing, prolongs the adoptiontime, whereas any negative deviation (lower cost), which is nondecreasing, hastens theadoption. The object w(ti) is the increase in profit after adoption per unit time, togetherwith the foregone total discounted benefit due to cost reductions to end time, resultingfrom delay in adoption. Whether or not this sum, w(ti), and the rate of the change in theadoption cost, I”(t1), together is greater than the rate of relief in interest burden, rI’(ti),determine, for the other patterns of the adoption cost function given in the proposition,the direction of change of the adoption time. When the slopes of cost deviation aredifferent from those given in the proposition, the effect of them on the changes of theadoption time is similar to those shown in the above proposition, except that the costdeviations can only be of much smaller magnitudes, as given in the following proposition.Proposition 8 Given (pr, q) is fixed and t1 is the extremal value, and,(i) a relatively higher investment cost function, that is > 0, and i’ > 0 such that‘(t) < — p1(t), i < (r+ 1)i(t), and 5 >0, and(a) if I is decreasing and convex, then Dt1(I; ) > 0;(b) if I is either decreasing and concave, increasing and convex, or increasing and concave, then Dt1(I; i) has the same sign as w(ti) — rI’(ti) + I”(t1);(ii) a relatively lower investment cost function, that is < 0, and r’ < 0 such thati1(t) >—2 + 7(t), 2 < —(r + 1)(t), and 62 > 0, andChapter 1. Effects of Technological Improvement on Quality and Price 23(a) if I is decreasing and convex, then Dt1 (I; —ii) <0,(b) if I is either decreasing and concave, increasing and convex, or increasing and concave, then Dt1(I;—7i) has the opposite sign as that of w(ti) — rI’(ti) + I”(t1).The following proposition shows the horizon effects on the adoption time of an improved technology. If the adoption cost function is decreasing and convex, then extendingthe planning horizon delays the time of adoption. For other types of investment cost function, a longer planning horizon may shorten or prolong the adoption time.Proposition 9 Given (pr, qi) is fixed and t1 is the extremal value, if I is decreasing andconvex, then. > 0; for the other patterns of the time-path of the investment cost, •has the opposite sign as rI’(ti) — I”(t1)— w(ti).1.8 ConclusionThere has been considerable interest in finding and explaining the basic elements thatcan drive product quality up. In the literature this is largely done by modelling theeffects of investing in learning and process improvement, and of cost reduction. We lookat another aspect by modelling demand as a function also of quality, and the effects oftechnological improvement on quality cost and manufacturing cost.Our analysis shows that, indeed, when the consumers have preference for productquality the manufacturing system should produce output of higher quality than ECL. Themore sensitive the consumers are to quality variations, the higher should be the outputquality. The price sensitivity of the consumers also plays a role here in determining theproduct quality. The more insensitive the consumers to price variations the higher shouldChapter 1. Effects of Technological Improvement on Quality and Price 24be the output quality. The ECL is optimal only when the consumers are indifferent toproduct quality. This result confirms the importance of consumers’ preference for qualityin determining the product quality. What is less well known is the result that consumers’response to price change also influences the production system’s decision on quality.We model the effects of technological improvement on the production system throughcost reductions. An improved technology has the direct effect of turning out less defectiveproduct, consequently, it directly lowers the quality appraisal and prevention cost. It hasan indirect effect of providing a platform for more efficient quality control and assuranceactivities, and thus indirectly lowers the appraisal and prevention cost. Thirdly, it reducesthe manufacturing cost. Our results show that technological improvements may notalways increase the optimal product quality. This depends on consumers’ behaviour. Forexample, when the consumers’ relative quality-price sensitivity is an increasing function ofprice, and decreasing function of quality, then it is optimal for the manufacturing systemto produce at lower quality and charge a lower price, that is, it is better to produceoutput of more affordable quality. This is in contrast to the results when demand is afunction of price alone. In this case consumers’ sensitivity to price plays no role in thedetermination of quality, and an improved technology allows no decrease in quality, adecrease in price and an increase in output quantity.The timing of adopting an improvement also affects product quality and price, butit depends, again, on consumers’ behaviour. If demand is a function of price alone, thendelaying adoption does not decrease the quality, and does not increase the price. The timeto adopt is determined by the trade off of marginal benefits and marginal costs of waiting.The time-path of the investment required in adopting a technological improvement alsoaffects the timing of adoption. If the price and quality of the product are already chosenand fixed by the producer, an increase to an investment cost function which is convexand decreasing, such that this cost increment is nonincreasing over time, then this costChapter 1. Effects of Technological Improvement on Quality and Price 25increment delays the adoption time. On the other hand a decrease in cost relative to thissame investment cost function hastens adoption if the decrease in cost is nondecreasingover time.Chapter 2The Effects of Mandatory Recycling on the Newsprint Industry2.1 IntroductionThe U.S. generates about 160 million tons per year (tpy) of municipal solid waste andapproximately 84% of this waste is disposed of in landfill. During 1982-87 over 3,000landfills have been shut down. About 50 % of the landfills now in use in the U.S. willclose down over the next five years (McClay 1990.) About 75% of these landfills willbe closed by the year 2005. Since wastepaper accounts for almost half of all solid wasteand the technology for its recycling is well established, many states have passed recyclinglaws.Florida passed the first mandatory recycled newsprint usage bill in late 1988. Bylate 1989, California and Connecticut had passed laws requiring publishers to consumea certain portion of recycled newsprint. By the summer of 1991, eight states had passedlegislation setting newsprint recycling goals and timetables, and in some cases includingtaxes on virgin newsprint. These states were California, Connecticut, Florida, Arizona,Missouri, Illinois, Maryland and Wisconsin. California, for example, stipulated recyclednewsprint consumption of 25% by 1991, 30% in 1994, 35% in 1996, 40% in 1998 and50% in 2000 (Boyle 1990.) McClay (1991) reported that California, Arizona and Floridastipulated a 40% minimum-content standard as a definition of recycled-content newsprint,while all the others used a blended or aggregate target approach. Eleven states includingNew York, Pennsylvania, Maine, Massachusetts, Michigan, Iowa, Louisanna, Vermont,26Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 27South Dakota, New Hampshire and Virginia had voluntary agreements negotiated. Forexample, in the New York agreement 11% recycled newsprint is the purchasing target for1992, 23% for 1995, 31% for 1997 to 40% in 2000. The purchasing goals for New Yorkare to be based on an assessment of the state and local government’s ability to collectand process ‘old newspaper’ for recycling into newsprint and other products. An escapeclause is included to provide a waiver if recycled newsprint supply is not available at aprice and quality comparable to virgin newsprint. Also the New York agreement focuseson the aggregate use of recycled fibers rather than the recycled fiber content of each sheetof newsprint (Newsprint Reporter 1991, Boyle 1990.) More details about the recyclingregulations are given in Appendix E.Apart from the legislation on mandatory newsprint recycling to be in place on thepublishers and newsprint producers, the U.S. federal government, states and municipalities are also requiring their purchasing agents to procure newsprint and other papercontaining certain percentages of recycled fiber. As early as 1988, the federal EPA issuedthe ‘Guideline for Federal Procurement of Paper and Paper Products containing Recovered Materials’ under the Resource Conservation and Recovery Act (RCRA). One yearlater, according to section G002 of the RCRA, all of the federal government agencies responsible for procuring paper were required to give preference to recycled products, andthe state and local governments that purchase $10,000 worth or more using appropriatedfederal funds must also follow the EPA guidelines, e.g. newsprint must contain at least40% postconsumer wastepaper. By the beginning of 1990, 23 states had preferentialprocurement laws (Garcia 1990, Paper Recycler 1990.)In newsprint production, the use of pulp that includes a sizeable portion of recycledfibers derived from wastepaper was well established in Western Europe and South EastAsia; however, such pulp was rare in North America. In 1989, for example, only approximately 12% of the total North American newsprint production involved some form of useChapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 28of recycled fiber (Pulp and Paper North American Factbook 1990, p.140 and p.144.) Theintroduction of regulatory constraints on the production of newsprint is likely to affectthe nature of competition in this oligopolistic industry (see, e.g., Booth et. al. 1991.)The objective of this paper is to model competition in a market that is affected bya recycling regulation on the consumers of newsprint. We address the following modelquestions.1. What are the optimal production decisions of firms in the market?2. What are the consequences of tighter regulations in terms of the production plansof firms in the industry?In the model producers may choose to produce either recycled or virgin newsprint, whileconsumers who have preference for virgin newsprint (see e.g., McClay 1990) must nevertheless comply with regulations that require them to buy a given minimum proportion ofrecycled newsprint out of their total consumption. The total costs of producing recyclednewsprint are generally higher than the cost of producing virgin newsprint, mainly dueto the higher capital costs involved in building deinking plants (see e.g. Hatch AssociatesLtd. 1989.)2.2 The ModelWe assume a market with two products: virgin newsprint (i = 1) and recycled newsprint(i = 2.) We assume that two firms supply the market. Firm 1 supplies the marketwith virgin newsprint while firm 2 supplies the market with recycled newsprint. Eachfirm produces only one product. Let q1, i = 1,2 be the quantities consumed of thetwo products. Let us assume that publishers (the only consumers of these products)Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 29are required by the recycling regulation to consume at least w x 100 % of the recycledproduct on average (i.e. they must buy at least w x 100 % of the newsprint from theproducer of recycled newsprint.) We also assume that the recycled product is an inferiorgood, the demand for which is solely derived from the regulation.Indeed, recycled newsprint is considered an inferior product in North America becauseof its problems with printability, brightness, opacity, runnability consistency and linting,etc. (see e.g. Aspler 1989, Howard 1989.) Thus publishers will not consume more of therecycled product than required by regulation andq11 (2.26)where w measures the tightness of the legislation. However, it is technically acceptableto use newsprint consisting of a mix of recycled and virgin pulp (see e.g. Aspler 1989,Hee 1990, and Friday 1990.) To avoid uninteresting cases we assume the regulation willprescribe some positive amount of both virgin and recycled products, i.e. 0 < w <1. The higher is the value of w, the tighter is the legislation. The aggregate declinein newspaper readership due to some what lower quality newsprint is not likely to besignificant, and therefore the derived demand function for newsprint is not likely to beaffected by the lower industry quality standard for newsprint. Thus the demand functionQo(.) is assumed to be the same as for virgin newsprint in the absence of legislation, withthe price given as a weighted average of the prices of recycled and virgin newsprint mixthat is consumed, so thatQ0(P11+P22)=qi+q2. (2.27)q1 + ‘J2The North American newsprint market is relatively self-contained with only 10-15%of shipments exported outside of North America and total imports of between zero tofour percent of demand, and it was shown by Booth (1990) that the North AmericanChapter 2. The Effects of Mandatory Recyding on the Newsprint Industry 30newsprint demand is adequately described by a linear function Qo(p) = ao — f3op. Thepost-legislation demand functions can be derived from (2.26) and (2.27), and areQ1(pl,p2) = (1 — w)ao — (1 — w),80[(1 — w)p1 + wp2j,Q2(pl,p2) = wa0 — w,60[(1 — w)pi + wp2]. (2.28)The two products are complementary in the sense that Qi(•,•) and Q2(•,•) are bothdecreasing functions. Note that the system of linear equations in (2.28) has rank 1 and isnot invertable; thus a decision on quantities does not determine a vector of prices, ratherthere there is a half-line of prices. In our model, firms set prices first and let the marketdetermine the consumption.2.3 Price CompetitionLet the unit cost of producing product 1 (virgin) by firm 1 be c1 and the unit cost offirm 2 (recycled) be c2. Fixed costs such as the expenditures and investment cost inconverting to or acquiring the recycling technology will not be considered in this paper.The functions 7r1(.,.) and r2(.,.) defined below are the profit functions for the twofirms.iri(pi,p2)= (p1 — ci)Q(p,2),7r2(pl,p)= (p2 —c)Q(pi,p. (2.29)The reaction functions of the two producers, Ri(p2) and R2(pi) , can be obtainedby solving R1 (P2) = arg max1{iri (p1 ,p2): = ——q, Qo (P1 qi +P2 2) = q1 + q, Qo(p) =ao—I3op} and R2(pi) = argmaxp2{7r(pj,p : q. = —--q,0(PiiP2) = ql+q2, Qo(p) =a0—/3op}. The reaction functions intersect to give the following equilibrium prices:* —a0 + 2(1 — w)/oc1 — w/30c2P1— 3(1—w)/30Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 31a0 + 2w/30c— (1 — w)/30c1 (2 30)3w/30Proposition 10 Assuming (1 — w)c1 wc2, then the equilibrium prices (p,p) given in(2.30) are unique. (all proofs are given in Appendix B.)The equilibrium prices of the two products are related through their unit costs c1 andc2, and the tightness of regulation w by*< w(2—w) (1—2w)aoP1P2— w)(1 + w)c2 + (1 — w)(1 + w)o (2.31)Note that in (2.31) the coefficient of c2 decreases, and increases as w decreases. As the regulation is relaxed, the importance of the cost of producing recycledpaper in determining the price of recycled paper diminishes. Indeed when the necessaryquantity of recycled paper is minute (but positive), the producer of recycled paper willdetermine the price largely on the basis of the strategic value of the recycled product indetermining the total production. He will do so to extract maximum rents.The second term reflects the importance of the demand structure in the determinationof prices. As the quantity of the recycled product decreases, i.e., the regulation is relaxed,the objective of the recycled product producer is to capture his share of the rent bycharging higher price for the few units required of his product. He must, however,consider the elasticity of demand. As his price hikes reduce the quantity demanded theprice elasticity increases. (Note that the elasticity of the demand is infinite at the priceequal and the demand for the product vanishes.)Chapter 2. The Effects of Mandatory Recyding on the Newsprint Industry 322.4 The Effects of Regulation on the Equilibrium Prices and QuantitiesAs the regulation requires more recycling the equilibrium price of virgin newsprint increases and that of recycled newsprint decreases. This property can be verified by examining the following derivatives:a0 —Pi (w)= 3(1 — w)2fl0 > Op’(w) = <. (2.32)Note as the relative share of the producer in the supply of newsprint declines becauseof regulation, the firm will adjust the price upwards. This is because the weight ofthe price of his product in the price of the ‘bundled’ goods is lowered, thus the effectof charging a higher price for his good on the quantity of newsprint demanded will belowered.The results in (2.32) derived for a linear demand function do not hold in general for anarbitrary demand function Qo(.). The derivatives for a general demand function are givenin Lemma 3. Similar results to (2.32) can be obtained and are given in Proposition 11. InLemma 3, Q(P(w)) and Q’(F(w)) denote respectively the first and second derivativesof Qo(.) with respect to the bundle price P(w,pi,p2)evaluated at (p(w),p(w)), whereP(w,pi,p2) (1 — w)pi + wp2 is the average unit price the consumers pay for a mix ofrecycled and virgin products that satisfies the regulation. P(w, c1,c2) is the average unitcost.Lemma 3 Suppose an equilibrium exists for an arbitrary demand function Qo(.). Let(p(w),p(w)) be the equilibrium prices, then*11 — 3p(w) — 2c1 — c2 (p(w) — ci)(ci —c2)Q”(P(w))P1kw)— +* *3(1 — w) 3wG(w,p1( ),p2))— 3p(w)— 2c— c1 (p(w)—c2)(c1—c2)Q”(P(w))P2kW)—— +* * .3w 3wG(w,p( ),p))Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 33whereG(w,p(w),p(w)): =[(1 — w)(p(w)— ci) + w(p(w) —c2)]Q”(P(w)),+[P(w, p(w), j4(w))— P(w, c,2)]Q”(P(w))}.Proposition 11 Suppose an equilibrium exists for an arbitrary demand function Qo(.).Let (p(w),p(w)) be the equilibrium prices.a. If ci = c2, thenp’(w) = p(w) —ci > ,p’(w) = p(w)— c2 <0. (2.34)b. If Qo(.) is concave and c1 > c2 then p’(w) > 0.c. If Qo(.) is concave and c1 <c2 then p’(w) <0.Proposition 12 Suppose an equilibrium exists for an arbitrary demand function Qo(.).Let (p(w),p(w)) be the equilibrium prices, then1 —Wp*I(w) p(w) +p(w) — c1 — c2 (2.35)Without concavity assumption for Qo(), the sign of p’(w) and p’(w) cannot bedetermined as shown by lemma 3, there is a definite relationship between them. Oneimplication of Proposition 12 is that if relaxing the regulation causes the price of thevirgin product to increase then the price of the recycled product cannot decrease. Ifslightly relaxing the regulation causes the price of the virgin product to increase, thenthe price of the recycled product will increase by at least times as much, on the otherChapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 34hand, if the price of the recycled product decreases, then the price of the virgin productwill decrease by at least times.The way that the equilibrium bundle price P*(w) := P(w,p(w),p(w)) is affectedby the extent of the recycling requirement can also be obtained from Lemma 3 and isstated in Proposition 13.Proposition 13 Suppose an equilibrium exists for an arbitrary demand function Qo(.),Let (p(w),p(w)) be the equilibrium prices, thenp*(w) — — (ci —c2)Q(F(w)) (2 36)— wG(w,p(w),p(w))Proposition 13 shows that the unit production cost of the recycled product relativeto that of the virgin product is critical to the assessment of the effect of the regulation.If Qo(.) is concave, then G(w,p(w),p(w)) <0 and we have the following statement:*1 > <F (w)=0 c1=c2. (2.37)It is interesting to find that when the costs of the two products are identical, c1 = c2,the bundle price is not affected by the tightness of the regulation. In this case theequilibrium prices of the two products are such that they keep the bundle price constant,i.e. independent of w.The quantity Qo(P(w, P1, p2)) = Qi(P(w, Pi,p2))+Q(P(w,p1 ,p2)) is the total amountof the two products demanded in the market. Let q(w) denote Q(F(w,p,p)). A direct consequence of Proposition 13 for an arbitrary concave demand function is given in(2.38).q’(w) = Q’ (p*(w))p*I(w)>O C1C2. (2.38)Thus for and arbitrary concave demand function if the unit cost of producing therecycled product is higher than that of the virgin product the total quantity demandedChapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 35of the two products decreases when the regulation requires higher consumption of therecycled fiber. Similarly if c1 is higher than c2 then q(w) increases when the regulationprescribes a higher recycled content. When the unit costs of the two products are thesame, the total consumption of the two products is unaffected by the tightness of theregulation.Lemma 4 Suppose an equilibrium exists for an arbitrary demand function Qo(.). Let(q(w), q(w)) be the equilibrium quantities, then11 \ *11 2*11— r *f i — w1c — c21q0 w 1q1 w — —1q0w-‘- * *wG(w,p(w),p2))— *(ci —c2)q’(w)q2 (w) — q0(w)— * *. (2.39)G(w,p1(w),p))Proposition 14 Suppose an equilibrium exists for an arbitrary demand function Qo(.).Let (p(w),p(w)) be the equilibrium quantities, thena. Ifci = c2 thenq’(w) = —q(w) <0,q’(w) = q(w) > 0. (2.40)b. If Qo(.) is concave and c1 <c2, then q’(w) <0.C. If Qo(.) is concave and c1 > c2, then q’(w) > 0.Note that our analysis of changes in prices and quantities due to changes in regulationshow a symmetry of response in the following sense: as the share of one producer inthe bundle increases, his costs of production have higher impact upon quantities andprices. Similarly as his share increases (due to regulation) the impact of price increasesin terms of reduction in quantities sold increases, and he is less inclined to charge higherChapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 36prices. In the next section we will show that this symmetry results in equal profit sharingirrespective of the specific value of w.2.5 The Effect of Recycling Regulation on ProfitsIf the demand function Qo(.) is linear, the equilibrium profits of the virgin and recycledproduct producers, Tr(w) and r(w), respectively, are equal. It can be easily checkedthat the equilibrium profit for either firm:[co —/30P(w,ci,c2)]= 1,2. (2.41)9/3The following proposition shows that this phenomenon of equal equilibrium profitsbetween the two firms is true for an arbitrary demand function Qo(.).Proposition 15 If an equilibrium exists for an arbitrary demand function Qo(.), then1E(p — ci) = p — c2. Consequently, the two firms have equal profits.This result is similar to the Nash bargaining point in cooperative games; however,in contrast to the agreement point it is non-Pareto optimal. A cooperative bindingagreement will lead each of the players to charge somewhat lower prices and thus largerquantities sold than in the non-cooperative case. In the non-cooperative case each producer considers the effect of price increases only on its own revenue and not the lossesthat each price increase causes the other producer; thus, there is a tendency to chargehigher prices and produce lower quantities as shown in Proposition 16 below.Proposition 16 Let the demand function Qo(.) be log-concave. Then the non-cooperativeequilibrium bundle price is higher than the bundle price obtained by cooperation, and thetotal quantity of recycled and virgin product consumed at the non-cooperative equilibriumis lower.Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 37Proposition 17 Suppose an equilibrium exists for an arbitrary demand function Qo(.).Let (p(w),p(w)) be the equilibrium prices, then for i = 1,2ir’(w)= wG(wp(w),p(w)) {(1 - w)(p(w) - ci)Q’(P(w))2-Q(P(w))[Q’(P(w)) + (1 — w)(p(w) — ci)Q”(P(w))]}. (2.42)The relative values of the unit costs of production for the two products determine theextent of the legislation’s effect on the profits of the two firms. If the costs are equal, thenthe profits of the firms are independent of w. The first term represents Q(P*(w))F*(w)whereas the second term represents P*((w)— 4P(w, c1,c2). For a concave demandfunction Qo(.), these two terms work in the same direction. If the recycled product ismore expensive to make, the profits of both firms are depressed when the regulation istightened, i.e. 7rr(w) < 0; i = 1,2. On the other hand, if the recycled product is lessexpense to produce the profits of the firms are enhanced when the regulation is tightened,i.e. ir’(w) > O;i 1,2.Proposition 18 concerning the impact of regulation upon industry profit margin isdirectly derived from Proposition 17 as follows:Proposition 18 Suppose an equilibrium exists for an arbitrary demand function Qo(.).Let (p(w),p(w)) be the equilibrium prices, then— P(w, c, c2)) = 2(c —2)[Q’(F(w)) + (1— w)(iii(w) — cl)Q”(F(w))243dw wG(w,p1(w),p2))2.6 Comparison of Prices, Quantities and Profits Before and After RecyclingRegulationWe assume in the base case (before regulation) that the two newsprint producers produceone homogeneous product— virgin newsprint, and are engaged in a Cournot quantityChapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 38competition. There are several cases to consider. In case one, we consider different unitcosts between the two competitors before the regulation. There is no change in unit costsafter the second producer’s conversion to the production of recycled newsprint. In thesecond case we will assume that the unit costs of production for the two firms are equalbefore the regulation and that the cost of the firm converting to recycled newsprint arehigher after the regulation. The comparison of the prices before and after the regulation,for the case in which conversion to recycling does not change the unit cost of production,is given in Proposition 19 below.Proposition 19 Given a log-concave demand function Qo(.) and unit costs, c1 and c2,the equilibrium bundle price F*(w) for a mix of virgin and recycled products as requiredby the regulation is always higher than the pre-regulation Cournot equilibrium price j3*for the virgin product. Thus the equilibrium consumption before the regulation is higher.Proposition 20 Given a concave demand function Qo(.) and unit costs, c1 and c2, thend*(ci —c2)Q’(P(w))—(P (w —p ) — — —o c1—c2. (2.44dw wG(w,p(w),p(w))< >Proposition 20 shows that the price gap, P*(w) — j3*, increases as the degree of therecycling requirement is more stringent when the firm which converts to the recycledproduct has a higher unit cost. On the other hand when the firm which converts to therecycled product has a lower unit cost, the price difference decreases as the required degreeof recycling increases. When the two firms have equal unit costs, the price difference isindependent of the degree of required recycling.Proposition 21 Given a log-concave demand function Qo(.) and unit costs, c1 and c2,the equilibrium bundle price obtained by cooperation, for a mix of virgin and recycledproducts as required by the regulation, is always higher than the pre-regulation Cournotequilibrium price for the virgin product.Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 39Propositions 16, 19 and 21 together imply that the pre-regulation Cournot equilibriumprice is lower than the post-regulation bundle price obtained by cooperation and thusthe post-regulation non-cooperative equilibrium bundle price.The pre-regulation and post-regulation equilibrium profits of the firms, for a lineardemand function, when the unit costs of the two firms are equal, are compared in Proposition 22. When the unit costs are different, the changes in the profits of the firms dueto the regulation are given in Proposition 23.Proposition 22 Given a linear demand function Qo(.), if the unit costs of the two firmsare the same, then the equilibrium profits of the firms before and after the regulation areidentical, i.e. * = ir,(w); i = 1,2 where * is the Cournot equilibrium profit of firm i.Proposition 23 Given a linear demand function Qo(.).(i) Suppose c1 <c2, then(a) ir(w) <*;* <‘.*. <(b) 1r2(w)=1r2 zf2(ao —/3oci) —w/3o(c —C2).(ii) Suppose c1 > c2, then(a) 7r(w) <*,(b) r’(w)* if 2(cvo — ,6oci)(1— w)/3o(ci — C2).Note that the regulation always penalizes the efficient firm and, under certain conditions, may favour the inefficient firm. To study the effect of the unit cost of the recycledproduct on the equilibrium bundle price as compared to the pre-regulation equilibriumprice, we let c1 = c2 = c before the regulation and c2 c after the regulation.Proposition 24 Given a log-concave demand function Qo(.) and unit costs c1 = c2 = cbefore the regulation, c2 c1 = c after the regulation, the equilibrium bundle price P*(w)Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 40for a mix of virgin and recycled products as required by the regulation is always higherthan the pre-regulation Cournot equilibrium price j3* for the virgin product.In Proposition 24, P*(w) > fr even when c2 = c after the regulation. Thus the neteffect of the regulation is that it drives the price up.In the case of c2 < c = c1 after the regulation, the pre-regulation equilibrium pricemay be lower or higher than the equilibrium bundle price after the regulation, dependingon the specific value of the unit cost of the recycled product and the demand function.2.7 ConclusionSome twenty states in the U.S. have passed recycling laws which mandate consumptionof old newspaper by the newsprint industry. There are also talks of having similarlegislations on other grades of paper. To study the effect of the recycling regulation, weuse a simple model of two firms— one firm producing the virgin product and the other therecycled product, which compete under the regulatory constraint. The recycled productis assumed inferior to the virgin product, and its demand is derived solely from theregulation. The model shows that, under the regulatory constraint, firms set prices andlet the market determine the consumption. An interesting result from the model is thatthe firms set their prices such that the margins they earn, depending on the severity ofthe regulation, are in a fixed proportion to one another, consequently they earn equalprofits.In general, the effects of the tightness of the regulation depend on the unit costof the recycled product relative to that of the virgin product. Under a more severerecycling regulation, and a concave pre-regulation demand function, consumers pay ahigher (lower) average unit cost, and thus consume less (more), if the unit productionChapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 41cost of the recycled product is higher (lower) than that of the virgin product; similarly,the firms earn lower (higher) profit if the unit cost of the recycled product is higher(lower). This result lends support to the concern, of the newsprint producers in NorthAmerica, for the possible high cost in recycling ( Newsprint Reporter 1990.) If the preregulation demand function is linear, the price of the virgin product increases and thatof the recycled product decreases, when the regulation mandates more consumption ofthe recycled material.We assume the two firms produce a homogeneous virgin product, and are engaged inCournot quantity competition, before the recycling regulation. With a log-concave preregulation demand function, consumers pay more for a unit of the product mix underthe regulation than for a unit of the virgin product before the regulation; also, underthe regulation, the average unit price that results from price competition is higher thanthat obtained from cooperation, and the average unit price obtained from cooperationis higher than the Cournot equilibrium price. If the pre-regulation demand function islinear, then the model shows that the recycling regulation penalizes the efficient firm byhampering its profit, and in certain conditions depending on the demand function, thecost structure of the industry and the tightness of the regulation, it may even favor theinefficient firm.Chapter 3Competition When Market Share Is Regulated3.1 IntroductionIn this chapter the two firm model in Chapter 2 is generalized to include n firms whichcompete under the same kind of regulation. The regulation mandates consumption ofa fixed proportion of each product out of the total consumption of all n products. Thecase can also arise if the n producers produce distinct products and consumers are characterized by a Leontieff type consumption technology (e.g., they use the goods boughtfrom the n producers as inputs to a Leontieff type production function.)The motive for a government to mandate the bundling of outputs of different producers of a commodity may be diverse. For example, environmental concerns may requirethe use of different energy types by utilities to ensure that the impacts of generating electricity will not exceed the absorptive capacities of the environment. The diversificationof sources may fulfil this objective. Alternatively, regional equity concerns may lead toregulations mandating consumption of products produced by different producers locatedin different regions.42Chapter 3. Competition When Market Share Is Regulated 433.2 Regulated Market ShareThere are n firms, each of which produces a distinct product. These products are technologically substitutable, but differ in their effects on the environment. In order to minimizethe impact of these products on the environment, the government has imposed regulations which stipulate for each product its proportion of the total consumption of all nproducts. That is, the regulation specifies a vector of market shares w = (wi, w2,.. . , w,)such that= 1 (3.45)w>Ofori=1,2,...,nThe proportion of product i which is consumed must be exactly the fraction w of thetotal of all n products. We have assumed that the regulation keeps all firms in the market(w > 0). Some of our results do not hold in the cases where some of the w are zero.However, these cases may be considered by reformulating the model to exclude the firmswhich the regulation eliminates from the market.We assume that the total demand for the products depends only on the price of aunit bundle consisting of w units of product i (i 1,2,. . . , n). Let p denote the pricecharged by firm i for product i. Then the unit price of a bundle is wp, the scalar productof the market share vector w and the price vector p= (pl,p2,. . . ,p,j. Let Q denote thedemand as a function of the unit bundle price, so that Q(wp) is the total demand if themarket share vector is w and the price vector is p. The demand for product i isqj = w:Q(wp) for i = 1,2,... ,Ti. (3.46)The demand function is, of course, nonnegative and decreasing on its domain, whichis an interval of nonnegative prices. To avoid technical problems with existence questionsChapter 3. Competition When Market Share Is Regulated 44concerning equilibria, we further assume that the percent decrease in demand per unit increase in price x, —Q’(x)/Q(x), is assumed to be continuous, increasing, and unbounded.We will sometimes write Q’/Q in different forms —= (-lnQ)’(x) = (3.47)where the elasticity at price x is E(x) —xQ’(x)/Q(x). Our assumption that —Q’(x)/Q(x)is increasing is equivalent to the demand function, Q, being log concave and implies thatelasticity is increasing at least at a linear rate.The profit for firm i is= (Pt — cjw2Q(wp). (3.48)Under the assumptions in the previous paragraph, the unique Nash equilibrium is obtained at a price vector p such thatthr.Im*’1ZU’=Ofori=1,2,...,n. (3.49).9piBy evaluating the partial derivative, this condition can be rewritten as*wp 1— c)= E(wp*) = (_lnQ)(wp*) (3.50)The profit for firm i at the equilibrium price vector p is— wp’ Q(wp*)E(wpjIf each firm prices its product according to the Nash equilibrium price, then no firmwould unilaterally change their price. Each firm will maximize their profit, given theother firms’ prices. If we consider a cooperative version of this model, the monopoly orcartel bundle price, wpm, is the unique solution tom 1wp —wc=. (3.52)(_ in Q)(wpm)The above discussion is summarized in the following proposition. It is interesting tonote that the cartel price is less than the more competitive Nash equilibrium price.Chapter 3. Competition When Market Share Is Regulated 45Proposition 25 Assume that the regulated market share vector w is such that the costof a unit bundle wc is in the interior of the domain of the demand function.(i) The unique Nash equilibrium price of a bundle satisfiesW — WC= (— in Q)(wp*)’(3.53)whereas the unique cartel price of a bundle at the Nash bargaining point satisfies display(3.52).(ii) The equilibrium price is larger than the cartel price.(iii) The combined profits for all firms at the Nash equilibrium price isn (wpj Q(wp*) ()1E(wp*)and is= wpmQ(wpm)(3.55)14E(wpm)at the cartel price. The Nash equilibrium profits are shared equally among the firms.It is interesting to consider Figure 3.3 (or displays (3.52) and (3.53)) in the casewhere each firm has the same unit cost, say c. In this case wc = c for all w, so thatthe three functions appearing in this figure do not depend on w. It follows that theNash equilibrium and cartel prices for a bundle are independent of the regulated marketshares.If the costs are not all identical, then the effect of changing the regulated marketshares is more difficult to describe without making the model more specific. However, wecan make a few observations when the market shares are perturbed so that the cost, wc,of producing a unit bundle increases. (A similar analysis can be made if the perturbationcauses the unit bundle cost to decrease.)Since wc increases, the two lines having x-intercept wc in Figure 3.3 keep the sameslope but move to the right and the decreasing function l/(— ln Q)’(x) is not affected. SoChapter 3. Competition When Market Share Is Regulated 461/(— in Q)’(x)n(x—)x — wc(x—wc)/nFigure 3.3: Cournot and Nash Equilibrium and Cartel PricesChapter 3. Competition When Market Share Is Regulated 47each of the Nash equilibrium and cartel prices for a bundle (i.e. wp and wpm) increaseand for each firm i, profit 4 decreases.Since we are considering perturbations of market shares which increase the weightedaverage cost, wc, there must be at least one firm whose market share increases, say firmk. For such a firm, price p decreases. The decrease in p is clear since from display(3.50)*_1 1Pk— wk(_lnQ)(wp*)’and each term on the right hand side decreases when w is perturbed so as to increase Wkand the price of a bundle wp.The prices charged by firms whose market share decreases can move in either direction. However, since wp’ increases, Q(wp*) decreases. So if the market share of firm idecreases, so does its production qj w1Q(wp*).3.3 Effect of Introducing a Regulated MarketSuppose that before regulation, the firms are all producing the same product at thelevels corresponding to the Cournot equilibrium and costs c. The price associated withthis equilibium will be denoted by p and the quantities by q. The inverse demandfunction is P(.)= Q’(.). The vector e is the n-vector consisting of all ones. Note thateq = q. We will also write c = ec/n to represent the average unit cost. The firstorder conditions for the Cournot equilibrium are•P(eq)—c356— —P’(eq) • ( . )Summing these equations yieldseq— (3.57)Chapter 3. Competition When Market Share Is Regulated 48Proposition 26 (i) If c is in the domain of Q, then there is a unique Cournot equilibrium.(ii) If c and wc are in the interior of the domain of Q and wc > , thenp• <wpm (3.58)Chapter 4Competition under Recycling Regulations4.1 IntroductionThe lack of suitable locations for landfill is becoming a serious problem for municipalwaste disposal in the U.S. (McClay 1990, Paper Recycler 1900.) Since waste papersconstitute a significant portion of the municipal solid waste (Paper Recycler 1990), federal and state governments are passing laws that mandate recycling of certain grades ofwaste paper (Boyle 1990, Edwards 1991.) For example, California stipulated recyclednewsprint consumption of 25% in 1991, 30% in 1994, 35% in 1996, 40% in 1998 and 50%in 2000 (Boyle 1990.) It is not surprising that the legislation will be requiring highervolumes of recycling and covering wider ranges of paper grades. Taxation policies thataim at penalizing consumption of virgin newsprint are also in place in a number of states(Boyle 1990, Newsprint Reporter 1991.) Mandatory recycling of post-consumer items ortaxing consumption, or production, of virgin products definitely changes the competitionbetween the recyclers and the users of virgin fiber. The objective of this paper is tostudy how competitive equilibria change under two approaches of recycling regulation:(i) mandatory consumption of a recycled product and, (ii) taxing a virgin product, orits producer, under a recycling policy. To this end a stylized model is used in whichtwo firms, a recycler and a producer of virgin product, compete under the recyclingregulations to maximize their own profits.49Chapter 4. Competition under Recycling Regulations 50In the enviromnental economics literature, the use of marketable permits and emission charges for environmental management have received widespread attention. Bothapproaches are meant to induce firms to find efficient ways to comply with environmentalstandards. They have been applied mainly to the control of air and water pollution (see,for example, Cropper and Oates 1992, and Hahn 1989.) Similar approaches can be usedto discourage consumption of virgin fiber. Charging production of virgin products issimilar to levying tax on the products, or their producers. Pigouvian tax, the purpose ofwhich is to correct for externalities, is also widely studied in the context of environmentalpollution (Cropper and Oates 1992.) An example of the tax to compensate for the socialcosts associated with litter and waste disposal is given by Dobbs (1991). But the taxeson the virgin product, or its producer, that we will consider are meant to induce, undercompetition, compliance with the recycling policy. Some common taxation schemes fornatural resources are franchise tax, profit tax, progressive profit tax, value tax, severancetax, property tax, and revenue tax (see, for example, Heaps 1985, Englin and Klan 1990.)Not all of these taxes can ensure that the policy objective is achieved, however. We willconsider production tax, progressive profit tax and taxing excessive consumption of thevirgin product. In this paper, we are interested in the pricing decisions, the resultingprofits for the firms, and the tax rate required under these tax schemes.In chapter 2, the recycled product is assumed to be an inferior product and its demand is assumed to be derived solely from the regulation. In this chapter, a marketof two partially substitutable products, virgin and recycled, are assumed. Two firms,one producing the virgin product, the other the recycled product, compete under therecycling regulations. The policy objective for the tax schemes, and for the regulation ofmandatory recycling, is to accomplish the target of consumption of the recycled productat a certain proportion, w%, of the total consumption of the two products. The modelChapter 4. Competition under Recycling Regulations 51and competition under a regulation that stipulates percentage consumption of the recycled product is given in the next section. We show that this regulation induces multipleequilibria. Under certain conditions, one equilibrium dominates; but in general, an equilibrium selection mechanism is needed to be added to the model. A suitable tax on thevirgin product, or its producer, serves the dual purposes of inducement of compliancewith the recycling policy and of equilibrium selection. In section 3 the effects of thevarious taxes on the firms are studied. It is found that the equilibrium profit of the recycler is unaffected by the different tax schemes, and the severity of the recycling policyfavours the recycler in the sense that its profit increases. The virgin product producerearns highest profit in the scheme of excessive consumption tax. The tax rate requiredto induce compliance with the policy is low for the scheme of excessive consumption taxas compared to the scheme of production tax and, in equilibrium, it does not collect anytax payment. Section 4 concludes the paper.4.2 The Model and Competition under Mandatory Consumption of Recycled ProductConsider two firms, indexed by i = 1,2; which compete in price. Firm 1 produces thevirgin product, firm 2 the recycled product. Firm 1 and firm 2 each have a constant unitcost of c1 and c2, respectively. The firms face a linear demand system,qi(pi,p) = i —7iipi +712P2, (4.59)q2(pl,p2) =a2—72p+ipi, (4.60)where > 0 and 722 > 0 are the own price effects, 712 > 0 and 721 > 0 are the crossprice effects. It is common in the economics literature to assume a symmetric systemof demands that greatly simplifies analyses; where a = a2, ‘y = 722 and 712 = 721.Chapter 4. Competition under Recycling Regulations 52Examples are Kiemperer and Meyer (1986) in their study of the role of uncertainty oncompetition, Roller and Tombak (1990) in the analysis of the strategic role of flexiblemanufacturing systems, and the work of Furth and Kovenock (1990) on price leadershipunder capacity constraints. However, symmetry is not assumed in this paper since it isunlikely that the demand schedule of the recycled product is the same as that of thevirgin product.In the absence of the recycling regulation, the equilibrium of competition is determined by the reaction functions of the two firms. The profit functions of the two firmsarelri(pi,p2)= (p1 — ci)qi(pi,p2), (4.61)ir2(pi,p)= (p2 — c2)q(pl,p). (4.62)The reaction functions are obtained, respectively, by Ri(p2) = arg max17r1(pj,p2) andR2(pr) = arg max2 ir2(p1, P2) which are given below:R1(p2) 01 + 712P2 + 7nCi (4.63)27a2 + 721P1 + 722CR2(pi) =. (4.64)2722The reaction functions are shown in Figure 4.4. The slope of Ri(p2) is 2711/712, andthe slope of R2(pi) is 721/(2722). For an equilibrium to exist, the reaction functionsmust intersect in the positive domain. This requires the condition that 4711722 > 7i27i•Ri(p2) is shifted to the right by an increase in c1 or a bigger value of 712, and shiftedto the left by a greater value of -y. Similarly, R2(pi) is shifted upwards by an increasein c2 or a larger value of 721, and shifted downwards with a greater value of 722• Theequilibrium prices are given below:* — 201722 + 02712 + 2y1172c+7122CP1 — , (4.65)4711722— 712721Chapter 4. Competition under Recycling Regulations 53P2 R1(p2)increase in C2, 721R2(p1)increase in 722a2+y22 C22y22‘ increase in c1, 712increase in 711/ P1a1+y1 Cl2ynFigure 4.4: Reaction Functions of the Two Firms*a1721 + 2ay11 +7112C + 2711-y2cP2 = (4.66)4711722— 712721Assume that the policy of the mandatory recycling law is to require the publishers toconsume at least w% of the recycled product, that is, qi(pi,p) (100 — w)q2(pi,p)/w.However, in what follows it is much less cumbersome to work with the policy constraintbelow:rql(pl,p2) q2(pi,p) (4.67)where r e (0, co). The regulation mandates consumption of no virgin product if r = co,Chapter 4. Competition under Recycling Regulations 54P2 Ri(p2)policy line,policy line,O <r <R2(pj)Y22policy line, r = 00a2+2:c2 / P1aj+y11C, ai..2y Y11Figure 4.5: Policy Lines and Reaction Functionsand no recycled product if r = 0. The policy line corresponding to rql(pl,p2) = q2(pl,p2)is shown in Figure 4.5. Any equilibrium in the region under the policy line in the positivedomain satisfies the policy constraint. Likewise, an equilibrium lying above the policyline violates the regulation. An increase in the recycling requirement shifts the policyline downwards. The slope of the policy line is (r711 +-y21)/(r’y12 + ‘722), which is boundbetween 721/722 (r = 0), and 711/712 (r = co). To guarantee that the policy line meetsat least one of the reaction functions, and is thus attainable, it must either meet R2(pi)when r = 00, or R1(p2)when r = 0. In both cases, the condition is 2711722 > 712721.Chapter 4. Competition under Recycling Regulations 55Suppose the policy constraint is binding; the policy loses its meaning if the unregulated equilibrium already satisfies it. As shown in Figure 4.6, the unregulated equilibrium,(p,p), does not meet the policy requirement, and the policy line intersects Ri(p2)and R2(p1), respectively, at F’ = (pt, p’2) and P2 = (p, p).1 — 2712 + 1722 + (1 + r)c,-yi,7i2p1 — (4.68)—712721 + 2711722 + 7,,712r— 2c2’y,, +a1721 +c1712 — a,711r+ ci71rP2 — , 4.69—712721 + 2711722 + 7112T2 — 272 — C272 — 27l2’ — 2a,72r—c27,2rp1 — , 4.70—721722 +7,21r— 27,,72r2 — —(1 + r)c2712 —a2711r— a,721rP2 — . 4.71721722 + 7,21r— 2-y,,72rThe above expressions show that p’ and p’ are functions of c1 only, p, and p arefunctions of c2.Suppose the firms comply with the recycling policy. Then the regulated reactionfunction for firm 1 is the segment of the R,(p2) below P’ and the segment of the policyline above F’. This is so because ir1 (p1, P2) is strictly concave in p, and for any P2 > pthe best price for firm 1 is the P1 on the policy line that corresponds to P2. Similarly, theregulated reaction function for firm 2 is the segment of R2(pa) above P2 and the segmentof the policy line below F2. Since r2(p’, P2) is strictly concave in P2, then for any p <pthe most that firm 2 can earn is by setting P2 on the policy line that corresponds top,. The regulated reaction functions of the two firms meet along the segment P12.Thus any point on this segment is a Nash equilibrium. This result is summarised in thefollowing proposition. It will be shown in the proof of the proposition that, indeed, atany point on P12,each firm has no incentive to unilaterally deviate from it.Chapter 4. Competition under Recycling Regulations 56‘(P1)P1P2 Ri(p2)policy linep2Figure 4.6: Reaction Functions of the Two Firms and the PolicyChapter 4. Competition under Recycling Regulations 57Proposition 27 Suppose the two firms comply with the recycling policy under the regulation which mandates consumption of the recycled product, then each point on the segmentof the policy line between P’ and F2 is a Nash equilibrium.Proofs of propositions are given in Appendix D. Since there are infinitely many possible equilibria that satisfy the regulation, it is natural for the firms to identify an equilibrium, if it exists, from the set of equilibria which maximizes the profits of the two firms.If-y12721 711722, then the profits of the firms on the line segment P’P2 increase towardsF2, and P2 dominates the rest of the equilibria. This is summarised in Proposition 28.Proposition 28 Suppose the two firms comply with the recycling policy under the regulation which mandates consumption of the recycled product, if 712721 711722, then P2is the dominating equilibrium.However, when the condition given in the above proposition for P2 to be the dominating equilibrium is not satisfied, it not clear how to select a workable equilibrium. Asthe two firms are not allowed to collude, a chaotic market for the products may resultfrom competition under the regulation. This leads to the use of tax as an alternativeapproach to induce compliance with the recycling policy.4.3 Competition Under TaxesThe analysis in the previous section assumed that the firms, competing to maximizetheir own profits, voluntarily observe the regulation that mandates consumption of therecycled product. The price competition results in multiple equilibria on the policy line.Moreover, we cannot be sure that the firms will voluntarily comply with the regulation.Thus a suitable tax system plays two roles of ensuring that the firms do not violate theChapter 4. Competition under Recycling Regulations 58recycling policy (4.67) and, at the same time, selecting an equilibrium out of the manypossible ones. We assume that only the virgin product, or its producer, is taxed. Theunregulated reaction function of the virgin product producer is thus modified by the tax,so that the taxed reaction function, that meets with the reaction function of the recycler,yields an equilibrium which belongs to the set of equilibria on the policy line as given inProposition 27. Not all taxes can modify the behaviour of a firm, however. For example,profit tax does not change the unregulated reaction function and, therefore, does notserve the purpose of policy compliance and equilibrium selection.We will consider a number of different tax schemes. One scheme is to tax the producerfor the amount of its product consumed in excess of that required by the policy. Thisturns out to be a good tax system in the sense that a producer who observes the policydoes not pay any tax. Another approach is to tax production which modifies the costof the firm being taxed. Production tax is equivalent to sales tax which acts to changethe demand of the product being taxed. Under progressive profit tax the cost of thefirm as well as the demand for the taxed product are affected, and, in place of a reactionfunction, there are two best responses to the price set by the opponent. These will belooked into in more detail in the following.4.3.1 Taxing Excessive Consumption of Virgin ProductUnder this tax with tax rate, 11, the profit function and the reaction function of firm 1are given below:ir(p1,p2t)= (p1 — ci)qi(pi,p2—t1(qp,2)— _q2(p1,)f, (4.72)R(p2,t1) = r(ai + ciYll + 712P2) + (721 + 711r)t (4.73)2711rChapter 4. Competition under Recycling Regulations 59where (x)+ is the nonnegative value of x. When t’ = 0, the taxed reaction function,R(p2,0), is just the reaction function, R1(p2). An increase in the tax rate, t, shifts thetaxed reaction function of firm 1 to the right, as shown in Figure 4.7, until t’ = t1’’ atwhich the taxed reaction function of firm 1 meets with the reaction function of firm 2 aton the policy line. Then F2 is the equilibrium satisfying the recycling policy, whichis induced by the tax. The equilibrium prices as a function of the tax rate arep1(t)= +211(721722 +71122r) (474)r(4711’y22— 712721)p2(t1)= +11(72 +7112r) (4.75)r(4711722— 712721)Substituting the above set of equations for the equilibrium prices into the equationfor the policy line, the tax rate that selects the regulated equilibrium is obtained.11*= [a1722(271r— 721) +a2711(-yir— 2722)+C17n(712721r — 271172r— 721722) + c2722(7ii-yl2r— 712721 + 2711722)]r. (4.76)(711r+721)(—’y12r+ 271172r+ 721722)In this tax scheme, no tax is collected from firm 1, so long as the equilibrium satisfiesthe recycling policy. Thus it is a kind of tax without taxing. The regulated equilibriumprofit for firm 2 is r(p,p) which is given below together with that of firm 1.— 72[a7i1 + Li721 +c2(712’y1— 711722)]— /i71272T—7122T— 721722)x[a2(712r 722) + 2cv172r+c(7122r+ 72)+ci(712721r— 271172r— 721722)], (4.77)I — 722{a1i+ aI72l +c2(71272l —71i722)]r— / ( )7122ir— L71172r— 721722)Chapter 4. Competition under Recycling Regulations 60lineR(p2,tc*),P2 R1(p2) R(p2,tl*)P1Figure 4.7: Reaction Functions of the Two Firms under TaxChapter 4. Competition under Recycling Regulations 614.3.2 Production TaxLet the tax rate in this system be t. The profit function and the reaction function offirm 1 under production tax are given below:ir(pi,p2,tc)= (P1 — c1 — tc)(ai— 7nPi + 712P2), (4.79)R(p2,tc) = a1 + C1711 + 712P2 + (4.80)2711With zero tax rate, the taxed reaction function for firm 1 is the reaction function, R1(p2).The taxed reaction function shifts to the right with an increase in t. In Figure 4.7, theproduction tax rate, t, at which the taxed reaction function of firm 1 intersects withthe reaction function of firm 2 and the policy line, yields the equilibrium P2. The taxedreaction function of firm 1 and the reaction function of firm 2 together give the followingset of equilibrium prices as functions of the tax rate t.p(tc)= p+2711722tc, (4.81)4Y11Y22 — 712721p(tC)= p +711721(4.82)4711722— 712721Substituting the above prices into the expression for the policy line, t is obtained.trn” = [a1-y22(271r— 721) +cv2-y11(72r— 2722)+c171(722r— 271172r— 721722) +c272(711-yr— 712721 + 2711722)]x1. (4.83)711(—722ir+ 271172r+ 721722)Comparing t-’ and t1, we have the following proposition.Proposition 29 The production tax rate, t, is higher than the tax rate for excessiveconsumption of virgin product, t’, for the regulated equilibrium, P2.The equilibrium profit for firm 2 under the production tax, ir, is the same as the profitobtained at F2, and thus equals ir in expression (4.78). The equilibrium profit underChapter 4. Competition under Recycling Regulations 62the production tax for firm 1, the virgin product producer, is given below:— 722[a1i+a1721 + c2Qy12yi— 711722)12—. 4.711(712721r— 27172r— 721722)2Proposition 30 The equilibrium profit of the virgin product producer which satisfies therecycling policy under the production tax decreases with a higher recycling requirement.Proposition 31 Under the recycling policy and the production tax, the equilibrium profitof the virgin product producer is722/(-yiir) times that of the recycler.Prop osition 32 Given the recycling policy, the virgin product producer earns lower equilibrium profit under the production tax scheme than that obtained under the scheme oftaxing excessive consumption of virgin product.4.3.3 Progressive Profit TaxLet ti-1(p1, P2) be the tax rate in the progressive profit tax scheme. The profit of firm 1under the tax scheme is given below:ir(p1,p2t8)= (p1 —ci)(ai —7iipi +712p2)[1 — 18(pi — ci)(c1—+ 712p2)]. (4.85)The first order condition of the profit function is:81 43\uliP1,P2, ) s= (ai +c1y1 —27iiPi +712p2)G(pi,p,t )(Ii= 0, (4.86)where G(pi,p2,t8) = 1 — 2t(pi—ci)(cvi— YiiPi + 712p2). If t8 = 0, then G(pi,p2,0) = 1,and the expressions in (4.86) give the unregulated reaction function for firm 1. Whent > 0, the best responses, which is not unique in this case, for the p2 set by firm 2,Chapter 4. Competition under Recycling Regulations 63is contained in G(p1,p2t) = 0. The taxed reaction curve from G(pi,p2,t3)= 0 in thepositive domain is shown in Figure 4.8. It intersects with the reaction function of firm2 at two points. It is not difficult to select an equilibrium from these two equilibria,however, because the government can simply choose the value of t8 that correponds tothe desired equilibrium on the policy line. The reaction curve slides down R1(p2) asincreases. On the reaction curve, the profit of firm 1, ir, is constant as shown by thefollowing proposition.Proposition 33 The tax payment of the virgin product producer, tir(pi,p2)and itsprofit, Tr(pi,p2,t8), both equal to l/(4ts) on the taxed reaction curve G(pi,p2,t8)= 0.Proposition 34 Given the recycling policy, the virgin product producer earns less profitunder the progressive profit tax scheme than the scheme of taxing excessive virgin productconsumption.Finding the equilibrium prices, as functions of t, for the two firms becomes very messyindeed by solving G(pi,p2,t8)= 0 and the reaction function of firm 2. Thus to find thevalue of t’ which selects the equilibrium F2, substitute (p,p) into G(pi,p2,t) = 0.Note that G(p,p2,t’) is linear in t8. So given (pl,p2), t is uniquely determined.ts* — (—721722 +712’y2ir — 2-yll-y22r) 4 87- H ‘ (.)where H =2722(c2-y11 + a-y21 +c2712-y1 —c27lly2)(—a 722 —c1-y212 +c27+a2’-y12r+c11221r+ 2a1’ynr — 2c1’-y12r+c2-y127r).The profit of firm 2, is the same as It follows from Proposition (33) that theprofit of firm 1 isH (4.88)4(—721722 +71221r— 2y1172r)Chapter 4. Competition under Recycling RegulationsP2policy line(P1)64Reaction CurvePiR1 (P2)p2p1Figure 4.8: Reaction Functions of the Two Firms under Progressive TaxChapter 4. Competition under Recycling Regulations 65Proposition 35 If the recycling policy is severe enough, then the virgin product producerearns less profit under production tax than progressive profit tax.4.3.4 Some Remarks(a) The equilibrium regulated prices of the two firms, p and p, under the taxes considered, increase as the policy requires more recycling. This can be seen from Figure 4.5,since as r increases the policy line shifts to the right.(b) The cost of the recycled product, c2, determines the equilibrium prices of the twoproducts under the taxes, while the cost of the virgin product, c1, has no effect on theprices. This is shown in expressions (4.70) and (4.71). An inspection of Figure 4.4 showsthat an increase in the cost of the recycled product results in higher prices of both products. On the other hand, the more efficient the virgin product producer, that is, the loweris c1, the further to the left is its reaction function. Then a higher tax rate is required toattain the regulated equilibrium (p,p) for the production tax and for the tax schemefor excessive consumption of the virgin product. In the case of progressive tax, Figure4.8 suggests that a bigger value of the marginal tax rate, I?”, is required.(c) The price effects, 721 and 722, on the recycled product affect the equilibrium pricesas Figure 4.4 exhibits; those on the virgin product do not. A decrease in the own priceeffect, 722, an increase in the cross price effect, 721, raises the equilibrium prices. On theother hand, the reaction function of the virgin product producer is shifted to the leftby an increase in the own price effect, y, and a decrease in the cross price effect, 712.Consequently, the tax rates t” and t, and the marginal tax rate t’ increase.Chapter 4. Competition under Recycling Regulations 66(d) The profit of the recycler does not change under the various tax schemes considered, because it is not taxed by these schemes. The following proposition shows that therecycling policy is to the advantage of the recycler.Proposition 36 In the tax schemes considered, the more severe is the policy, the higheris the profit of the recycled product producer.4.4 ConclusionWhen the recycling regulation mandates consumption of the recycled product as a certainproportion of the total consumption, in a market where the recycled and virgin productsare partially substitutable, there are infinitely many possible equilibria of price competition between the two producers that satisfy the recycling requirement. 1f712721 yii22,an equilibrium exists which maximizes the profits of the firms over the set of equilibria.The firms, thus, select this equilibrium. If the above criterion is not satisfied, the competition between the firms may become chaotic, unless an equilibrium selection mechanismis introduced.A suitable tax scheme chosen by the regulator can play the dual roles of complianceinducement for the recycling policy and equilibrium selection. The same equilibrium,through taxing the virgin product or its producer, under the different schemes of production tax, progressive profit tax and taxing consumption of the virgin product, is obtained.The equilibrium profit of the recycler is unaffected by the different tax schemes, and theseverity of the recycling policy favors the recycler in the sense that its profit increases.While the profit of the virgin product producer decreases with the severity of the recycling policy under production tax. The virgin product producer earns highest profit inthe scheme of taxing excessive consumption of the virgin product, and when the policyChapter 4. Competition under Recycling Regulations 67is severe enough, its profit is higher under progressive profit tax than under productiontax. An interesting feature of the progressive profit tax is that the profit of the firmbeing taxed equals its tax payment which is a quarter of the inverse of the marginal taxrate. The equilibrium regulated prices of both products increase with (i) the recyclingrequirement of the policy, (ii) the cost of the recycled product, (iii) the effect of the priceof the virgin product on the recycled product, and (iv) a lower own price effect of therecycled product.The tax rate required to induce compliance with the policy is higher for the productiontax scheme than the scheme of taxing excessive consumption of the virgin product. Inthe progressive profit tax scheme, 18(lri) = tslri can be looked at as the variable tax rate,and this tax rate always equals half. Higher tax rates, for the different tax schemes, arerequired to attain the regulated equilibrium if (i) the virgin product producer is moreefficient, (ii) the more sensitive is the demand for the virgin product to its own price, and,(iii) the less sensitive is the demand for the virgin product to the price of the recycledproduct. It is interesting to find that in equilibrium, taxing excessive consumption ofthe virgin product does not collect any tax payment, and the tax rate required is lowerthan that of the production tax. Thus from the stand point of solely inducing recycling,rather than increasing tax income, this is a good scheme to follow.Appendix AProofs for Chapter 1.Proof of Lemma 1. Consider the domain on which the following profit function is nonnegative.H1(p, q, t1) := (—/3c3 + p + 0 — c1 (q) — c(q))F(p, q).The first factor on the right hand side is concave in (p, q). The second factor is concave byassumption. Thus H1(p, q, t1) is pseudoconcave in (p, q) (Mangasarian 1970.) It followsthat H°(p, q) is also pseudoconcave. 0Proof of Propostion 1.q, t1) = F(p, q) + (p — c3 /3(t1) — c1 (q) — cr(tj )c(q) + 0(t1 ))F(p, q)= 0. (A.89)H(p,q,ti) = F(p,q)(—cj’(q)—(tj)c2’(q) +(p—c813(tj) — c1 (q)—o(t’ )c(q) + 0(t1 ))Fq(p, q)= 0. (A.90)(A.89)xFq(p,q)- (A.90)xF(p,q) givesF(p,q)F(p,q) + (F(p,q)ci’(q) + cr(ti)F(p,q)c21(q))Fp(p,q)Fq(p,q) +(cj’(q)+c(tj)c’(q))F(p,q) = 0.68Appendix A. Proofs for Chapter 1. 69Simplifying, we havecj’(q) + a(ti)c’(q)= -:S(p,q)> 0. D (A.91)Define the revenue function, with q fixed, as R(p, q) pF(p, q). Then differentiatingand re-arranging terms to obtain the marginal revenueMR(p,q) = +p. (A.92)Proof of Proposition 2. From (A.89) above, given q we haveF(p,q)+p= cg(tj)+c1(q)+a(t)c2q)—0(t1).p(p, q)The left hand side is marginal revenue, and the right hand side is marginal total cost. 0Proof of Proposition 3. From (A.90) above, given p, we haveF(p,q) (c1’(q)+a(t1c2(q))c(t + c1(q) +a(ti)c(q) —0(t1 =p.q(p,q)The left hand side is marginal total cost. 0To prove Proposition 4 we need the following two lemmas.Lemma 5 S(p,q)O if Fp(p,q)Fpq(p,q)— Fq(p,q)Fpp(p,q)O.Proof: differentiate S(p, q) with respect to p to get the following expression.— —F(p, q)Fpq(p, q) + Fq(p, q)F(p, q)p(p,q)— P1 \2 ( . )i p, q1The result follows. 0Appendix A. Proofs for Chapter 1. 70Lemma 6 Sq(p, q)0 if F(p, q)Fqq(p, q) — Fq(p, q)Fpq(p, q)0.Proof: differentiate S(p, q) with respect to p to get the following expression.— —Fp(p,q)Fqq(p,q) + Fq(p,q)Fpq(p,q)Sq(p,q)_ 2F(p, q)The result follows. 0Proof of Proposition 4.(A.94)H(p,q,O,a,/3)H(p,q,8,a,/3)H(p, q, 0, a, /3)Hpq(p, q, 0, a, /3)H9(p, q, 8, a, /3)H0(p, q, 0, a, /3)q, 0, a, /3)Hq(p, q, 0, a, /3)Hqq(p, q, 0, a, /3)Hqs(p, q, 8, a, /3)Hqfl(p, q, 0, a, /3)H(p, q, 0, a, i3)= (—/3c3 + p + 0 — ci (q) —ac2(q))F(p, q).= F(p,q)+ (—/3c3 +p+O— c1(q) —ac2(q))F(p,q).= 2F(p,q)+(—f3cs +p+O— cj(q) —ac(q))F(p,q).= Fq(p, q) — (ci ‘(q) + ac’(q))F(p, q) +(—/3c.g + p + 0 — c1 (q) — ac2(q))Fpq(p, q).= F(p,q).= —csF(p,q).= —c(q)F2(p, .= F(p,q)(—ci’(q)— ac’(q)) +(—/3c3 + p + 0 — ci (q) — ac2(q))Fq(p, q).= F(p,q)(—cj”(q)—ac”(q))+2(—ci’(q)—ac’(q))Fq(p,q)+(—/3c+ p + 8 — ci (q) — cxc2(q))Fqq(p, q).= Fq(p,q).= —csFq(p,q).= —F(p,q)c2’(q — cr(q)Fq(p,q).To sign the following expressions, we use the following properties: (a) the quality costsAppendix A. Proofs for Chapter 1. 71are stricUy convex, (b) F(p,q) < 0 and Fq(p,q) > 0, and (c) from (A.91), Fq(p,q) +(c(q) + ac(q))F(p, q) = 0.pg = k{F(p,q)(c11(q + cc”(q))F(p,q) +Fq(p,q)[Fq(p,q) + (cj’(q) + cc’(q))F(p,q)] +(—/3c8 +p + 0— c1(q) — cxc(q))(—Fqq(p,q)Fp(p,q) + Fq(p,q)Fpq(p,q))}.qe = k{—Fp(p,q)[Fq(p,q)+ (cj’(q)+ac’(q))F(p,q)] +(—/c3 +p + 0— c1(q) — ac(q))(Fp(p,q)Fpq(p,q) — Fq(p,q)Fpp(p,q))}.Fg(p,q) = k{F(p,q)(c11(q +cc2”(q))F(p,q) + (—3cs +p+O — ci(q) —cc2(q))[F(p, q)(—Fqq(p, q)F(p, q) + Fq(p, q)Fpq(p, q)) +Fq(p,q)(Fp(p,q)Fpq(p,q)— Fq(p,q)Fpp(p,q))]}.From expression (A.94) of Lemma 6, if Sq(p, q) 0, then pg < 0; if S(p, q)0, thenqgO; and if S(p,q) 0 and Sq(p,q) 0 then Fg(p,q) >0.p = k{—c2(q)F(p,q)(ci”(q) +ac211(q))F(p,q) —c2(q)Fq(p, q)[Fq(p, q) + (cj’(q) + cc’(q))F(p, q)] +F(p,q)c1(q)[—F (p,q) + (c1’(q) +ac2’(q))F(p,q) —(—/3c3 +p + 0— c1(q) — crc2(q))Fpq(p,q)] +c2(q)(—I3c3+p + 0— cj(q) — crc(q))(Fqq(p,q)Fp(p,q) — Fq(p,q)Fpq(p,q))}.q = k{c(q)Fp(p,q)[Fq(p,q)+ (c1q)+ac2’(q))F(p,q)] +F(p, q)c’(q) [2F(p, q) + (—/3c3 + p + 0 — c1 (q) — ac(q))F(p, q)] —c2(q)(—/3,g +p+ 0— c1(q) — oc2(q))(Fp(p,q)Fpq(p,q) —9(p,q)F(p,q))}.Fa(p,q) = k{—(c(q)F(p,q)(cj”(q) +ac”(q))F(p,q)2+F(p,q)c’(q)Fp(p,q)[Fq(p,q) + (ci’(q)+cvc1q))F(p,q)] +(—/3c3 + p + 0—ej (q) — crc2(q))Appendix A. Proofs for Chapter 1. 72[c (q)F(p, q)(—Fq(p, q)Fpq(p, q) + Fqq(p, q)F(p, q)) —Fq(p, q)(F(p, q)Fpq(p, q) — Fq(p, q)F(p, q)) +F(p,q)c2’(q)(—F(p, q)Fpq(p, q) + Fq(p, q)F(p, q))J}.IfS(p,q) <0, then q, <0; and if S(p,q) 0 and Sq(p,q) 0, then Fa(p,q) <0. Thesign of p0, cannot be determined without making further assumptions.= k{—c3Fq(p,q)[Fq(p,q) + (c(q) + ac(q))F(p,q)1 —c3F(p, q)(c(q) + cxc’(q))F(p, q) +c3(p— ci(q) — aci(q) + 0—/3c)(Fp(p,q)Fqq(p,q) — Fq(p,q)Fpq(p,q))}.q = k{cFp(p,q)[Fq(p,q) + (c(q)+ac(q)]F(p,q))+c3(p— ci(q) — cxc2(q) + 0 —3c)(—F(p, q)Fpq(p, q) + Fq(p, q)F(p, q))}.F,3(p,q) = k{c3(p—ci(q) —cc(q)+0—/3c3[Fq(p,q)(Fq(p,q)Fpp(p,q)— Fp(p,q)Fpq(p,q)) +F(p, q)(F(p, q)Fqq(p, q) — Fq(p, q)Fpq(p, q))] —F(p,q)2caF(p, q)(c(q) + ac(q))}.From expression (A.94) of Lemma 6, if Sq(p, q) 0, then p13 > 0; if S(p, q)0, thenq1O; and if S(p, q) 0 and Sq(p, q) <0 then Fs(p, q) <0.Proof of Proposition 5. Just consider the following condition in the results of Proposition 5 to obtain the results in this proposition: Fq(p, q) 0. 0Proof of Proposition 6.Ht1(p,q,ti) = (—c(q)cx’(tj)—c318’(tj) + 0’(tj))F(p,q).Hqti (p, q, t1) = —F(p, q)a’(tj )c21(q) + (—c2q)cV(ti) —c3/3’(t1)+ 0’(t1 ))Fq(p, q).Appendix A. Proofs for Chapter 1. 73pj1 = k{—F(p, q)a’(ti)c’2q)[Fq(p, q) — (c(q) + a(ti)c(q))F(p, q) +(p — c1(q) — cx(ti)c2(q + 6(t1) — /3 (tl)c3)Fpq(p, q)] +(c2q)cx’(ti) +c3’(ti) — 0’ (ti))[—F(p, q)F(p, q)(c’1(q) + a(ti)c’2’ q)) —Fq(p, q)(Fq(p, q) + (c(q) + a(ti)c’(q))F2p,q)) +(p — c1(q) — cx(ti)c2(q + 6(t1) — /3(t1)c(F(p, q)Fqq(p, q) — Fq(p, q)Fpq(p, q))]}.qt1 = k{F(p,q)a’(ti}4(q)[2F(p,q) +F(p, q)(p — ci(q) — a(ti)c2(q + 6(t1) — /3(t1)c]+(p — c1(q) — a(ti)c2(q + 0(t1) — /3(ti)c) c2a’(ti) +c3/3’(ti) — 6’(t1))(—F(p, q)Fpq(p, q) + Fq(p, q)F(p, q))}.Ft1(p,q) = k{—F(p,q)F(p,q)2(c’’(q)+ a(ti)c’’(q))(c2(q r’(ti) +c3/3’(ti) — 0’(t1)) +(p — ci(q) — o(ti)c2(q + 6(t1) — /3(ti)c) c(q)cr’(ti) +c3/3’(ti) — 6’(t1))[F(p, q)(F(p, q)Fqq(p, q) — Fq(p, q)Fpq(p, q)) +Fq(Fq(p, q)F(p, q) — F(p, q)Fpq(p, q))]—F(p, q)ü’(ti)c’2q)(p — c1(q) — a(ti)c2(q + 6(t1) — /3(ti)c)(F(p, q)Fpq(p, q) — Fq(p,q)Fpp(p,q))}.Again using the conditions that, (a) the quality costs are strictly convex, (b) F(p, q) <0and Fq(p,q) > 0, and (c) from (A.91), Fq(p,q) + (c(q) +crc2(q))F(p,q) = 0, and expression (A.93), that (i) if S(p, q) < 0, then qt, > 0; (ii) in addition, from (A.94), ifSq(p, q) 0, then F1(p, q) > 0; and (iii) the sign of pj1 cannot be determined without further assumptions. Again, if the demand is a function of price alone, then usingFq(p, q) 0 in the above expresions, Pt1 0, q,1 0, and F1 (p, q) 0 are obtained. 0Appendix A. Proofs for Chapter 1. 74Proof of Lemma 2. Straight forward differentiation of the objective function (1.2) withrespect to t1 and setting to zero gives the required result. DProof of Proposition 7. A higher investment cost function relative to 1(t) is 1(t) =(I + j) (t), ij (t) > 0, Vt E [0, T], a lower investment cost function relative to 1(t) is similarly defined with (t) < 0, Vt E [0, T]. We use the extremal condition (1.24) for thetechnology adoption time, t1, in Lemma 2, and consider a variation, in the direction of(t), to the investment cost, 1(t). We assume the extremal (p, q), obtained for 1(t), doesnot change with the variation, because, otherwise, the problem would become intractable.The extremal condition for (F) with respect to t1 can be written in the following form.rI(ti) — I’(t1)+ H°(po,qo) — K(11) = 0, (A.95)1 — e7’(tl_T)where K(t1) = H’(pi, q, t1) + (a’(t1)c2q— O’(t1)+16’(ti)ca)F(p, q)rDefine w(ti) as in the following,dK(t1)w(1).= dt1= (e_rTt1 + 1)F(pi,qi)(—c2(qi)a’(ti) —c3fl’(ti) + O’(ti)) +eTtl —rT— 1rF(pi,qi)(—c(q )” ti) —c3f3”(ti) + O”(t1)) (A.96)>0.Now allowing a variation in I in the direction of in equation (A.95), and note thats is a positive real number.0 = rI(ti + dt1) — I’(t1 + dt1) + H°(po, qo) — K(t1 + dt1),= r(I + si7)(ti + dt1) — (I’ + s’)(t1 + dt1) + H°(pi, qi) — K(t1 + dt1),= r[(I + .s)(ti) + (I’ + sq’)(ti)dtij — (I’ + 871’)(ti) + (I” + si”)(ti)dt1+Appendix A. Proofs for Chapter 1. 75H°(po, qo) — K(t1 + dt1) + (dt1), (f(dti) —+ 0 as dt1 —* 0);= rI(ti) + rsr(ti) + rI’(ti)dti — I’(t1) — sij’(ti) — I”(ti)dt1+H°(po, qo) — K(t1 + dt1) + (dt1), ((dt1) —* 0 as dt1 —÷ 0).Substract the above expression by equation (A.95) and divide the resulting expressionwith s to obtain the following,(w(ti) — rI’(ti) + I”(t1))= r7)(ti) — 7)’(t1) +t1)(A.97)Letting s J 0,(w(ti) — rI’(t1)+I”(t1))D(I)= r7)(ti) — 7)’(t1), > 0; arid (A.98)(w(ti) — rI’(tj) + I”(t1))D(I)= r7)(ti) —i71(t), < 0. (A.99)(i) If 77> 0 and r’ <0, then the right hand side of (A.98) is positive, and,(a) if I is convex and decreasing, then the left hand side is also positive, and Dt1(I) > 0;(b) for the other cases of I, Dt1(I) has the same sign as the left hand side of (A.98).(ii) If <0 and ‘> 0, the right hand side of (A.99) is negative, and,(a) if I is decreasing and convex, then the right hand side is positive, and Dt1(I) < 0;(b) for the other cases of I, Dt1(I) has the opposite sign of the left hand side of (A.99).0Proof of Proposition 8. The preceding proposition does not hold in general for 77 > 0 andii’> 0, or i < 0 and iii’ <0. We need to restrict i to some neighbourhoods of I. DefineU(I, 6) as a weak upper 8-neighbourhood of I, 6 R, as the set of functions I whichdo not cross I from above, i.e. 1(t) > 1(t), Vt e [0,T], such thatII’ — Ill + lI’ — I’ll < 8, (A.l00)Appendix A. Proofs for Chapter 1. 76where IL II is the sup-norm; and U(I,S) as a weak lower 6-neighbourhood of I, SE R,which do not cross I from below, i.e. 1(t) < I(t),Vt e [O,T], which satisfy (A.100) above.It follows from (A.lOO) that,II1II + ?7’ <S. (A.lOl)(i) Suppose I E U+(I,81), > 0 and q’ > 0. From expression (A.101), we have‘(t) < 6 — (t),Vt E [0,T], which impliesr(t)— ‘(t) > (r + l),(t) — Si> 0, if 6 < (r+ 1)i(t).Therefore the right hand side of (A.98) is positive. Then the results in (a) and (b) followthe same proof of the preceding proposition.(ii) Suppose I U(I,62) < 0 and ‘ < 0. From expression (A.101), we have‘(t) > —6 + ii(t), Vt E [0, Tj, which impliesr(t)—7(t) < (r+l)(t)+62< 0, if 62 < —(r + l)i(t).Therefore the right hand side of (A.99) is negative. Then, again, the results in (a) and(b) follow the same proof of the preceding proposition. 0Proof of Proposition 9. Using the Implicit Function Theorem, differentiate the extremalt with respect to T in (A.95) to obtain the following,, dt1 dt1 OK(t1,T) dt1 äK(t1,T)0 = rI (t1)—(t1)—________—________(A.102)Appendix A. Proofs for Chapter 1. 77where K(t1,T) is given in (A.95) with T made explicit. Then simplifying the aboveexpression and rearranging terms we haveer(u1_T)(c2(qi)a(ti) +c3/3’(ti) — O’(t1)) = (rI’(ti) — I”(t) — w(ti)). (A.103)The left hand side of the above equation is negative, and w(ti) from (A.96) is positive.If I is decreasing and convex, then 4&# is positive. For the other patterns of I, hasthe opposite sign of rI’(ti)—I”(t1)— w(ti)). 0Appendix BIn the following proofs the equilibrium prices will be given with no asterisks, as it willbe clear from the context.cr0 — w30p2+ (1 — w)3ocipi(Th)= 2(1 — w)/30a0 — (1 — w)/30p1+ w/30c2 (B.104)p2(p1)= 2w/30Figure B.9 shows the graphical representation of the reaction functions, R1 and R2, offirm 1 and firm 2. The intercepts are:— a0 + (1 — w)fJoci —‘ — a0 + (1 — w)/30c1P1(O) — (O)__2(1 — w)f30 wf30a0 + w/0c2 1 ao + w/30c2P2(O)= 2w/% P2 (0) (1 — w)/30Assuming (1 — w)cj 4 wc2, otherwise the two reaction functions overlap. There are fourcases to consider:(i) p’(0) >p1(O),p’(O) > P2(O) implies pi > O,P2 > 0.(ii) P1(O) > p’(O),p2(0) > p(0) implies P1 <O,P2 <0.(iii) p(O) > pl(O),p2(O) > pj’(O) implies P1 <°,P2 > 0.(iv) p1(O) > p’(O),pj1( ) > P2(O) implies P’ > °,P2 <0.Thus case (i) is the only feasible equilibrium and uniqueness follows.Proofs for Chapter 2.Proof of Proposition 10. The reaction functions of the two firms with the pre-legislationlinear demand function, Qo(p) = a0 —D78Appendix B. Proofs for Chapter 2. 79Proof of Lemma 3 and Proposition 13. From (2.26), (2.27) and (2.29) we have7r1(pi,p2)= (p1 —c1)Q(pi,p2= (pi — ci)(1 — w)Qo(P(w,pi,p2).ir2(pi,p)= (p2 —c2)Q(pi,p = (p2 — )wQo(P(w,pi,p).where F(w, P1, p2) = (1 — w)p + wp2. Differentiating the above expressions to obtainÔQ0(F(w,p1,p2))Qo(F(w,pi,p2))+ (p1 — ci)Up1= Qo(P(w,p1,p2))+ (1 — w)(pi — ci)Q’(P(w,pi,p2)= 0. (B.105)ÔQ0(P(w,Pi, P2))Qo(P(w,pi,p2))+ (p2 — c2)up2= Qo(P(w,pi,p2))+ w(p2 —c)Q(F(w,pi,p)= 0. (B.106)The equilibrium prices, Pi and P2, that solve the above equations can be thought ofas a function of w, the parameter that represents the tightness of the regulation. Toreduce the burden of symbols, we will suppress the argument w of the equilibrium pricesthroughout the proofs. Differentiate pi and P2 in the expressions with respect to w toobtain the following,Q(P(w, P1, p2))dP(w,pi, p2)+ (1 — w)Q(P(w, P1,dw dw+(1 — w)(p1 —c1)0 ,P1,p2))— (p1 — ci)Q(P(w,p,p2)= O(B.107)Pi,))dP(WPl p2)+ wQ(P(w, p1,p2))+w(p2 — + (p2 —c)Q’0(P(w,pi,p)= 0. (B.108)Substitute (B.109) into (B.107) and (B.108) to obtain (B.110) and (B.111) below.dP(w,pi,p2)=—Pi +P2 + (1 — + dp2 (B.109)Appendix B. Proofs for Chapter 2. 80Then put (B.115)dp1dwdp2dw= 0. (B.110)(B.111)(B.112)(B.113)(2p— P2 — ci)Q’(P(w,pi,p)— 2(1 — w)Q(P(w,pi,p)dp2 dQ(P(w,pi,p2))—wQ0(P(w,pi,p))—— — (1 — w)(pi — ci)dw(p1 — 2p —c)Q’(P(w,pi,p)— (1 — w)Q(F(w,p,p)dp2 dQ(P(w,pi,p2))—2wQ0(P(w,pi,p))-—— — w(p2 — c2)dw = O•Solving the above set of equations, we havedp1 (3pi — 2c1 — c2) —— (p1 — ci) dQ’0(P(w,pi,p2))dw — 3(1 — w) 3Q’0(P(w,p1,2)) dw42— (3p2 — 2c — ci)+(p2 — c2) dQ(P(w,pi,p2))dw — 3w 3Q’0(F(w,pi,p2)) dwSubstituting (B.112) and (B.113) back into (B.109) to get (B.114), and thus Proposition13, as follows:dP(w,pi,p2) — (c1—c2)—dw— 3(1 — w)(pi — ci) + w(p2 — c2) Q’(P(w,pi,p))dF(w,pi,p114)3 ’0(P(w,pi,p)) dw___________—Q(P(w,pi,p2))(c c2)3Q(P(w,pi,p2))+ [(1 — w)(p1 — ci) + w(p2 — )]Q’’(P(w,pi,p)= Q(P(w,pi,p)) ci — c2) (B 115)wG(w,pi,p2into (B.112) and (B.113) to obtain Lemma 3 in the following:— 2c1— c2+(p’ — ci)(ci —2)Qg(P(w,pi,p)3(1 — w) 3wG(w,pi,p23P2 — 2c — c1+(p2 —c2)(ci —2)Qg(P(w,pi,p)3w 3wG(w,pi,pTherefore,dP(w,pi,p2) —dwProof of Proposition 11. The results follow from Lemma 3, since G(w,pi,p2)is negativeAppendix B. Proofs for Chapter 2.81with Qo(.) concave.Proof of Proposition 12. Multiply ‘- in Lemma 3 by (1, subtract in Lemmadw ‘ w3. Then use equations (B.105) and (B.106) to get the required result. DProof of Proposition 13. See Proof of Lemma 3.Proof of Lemma 4. Using (B.115) we first obtain (B.116).dQo(P(w, p, pa)) — dP(w, Pi, p2) Q(P(w, P1, p2))dw — dwQ(P(w,pj,p2))(ci—c2) (B.116)wG(w,pi,p2But Qi(P(w,pi,p2))= (1—w)Qo(P(w,pi,p2))andQ(P(w,pi,p))= wQo(P(w,pi,p2))together with (B.116), give the following:dQ1(P(w,p,p2)) dQo(P(w,pi,p2))—Qo(P(w,pi,p2))+ (1 — w)dwdw= —[Qo(P(w,pi,p))+(1 — w)(ci —wG(w,pi,p2dQ2(P(w,pi,p)) dQo(P(w,pi,p2))= Qo(P(w,pj,p2))+ wdw ciw— c2)= Qo(P(w,pi,p))— . DwG(w,pi,p2)Proof of Proposition 14. The results follow fromLemma 4. D—1—w — 1—wProof of Proposition 15. Define j3 — —--p’, c1 ——--c, and P(w,13,p2)= wQ31 + p2).The objective functions of the two firms are as follows,= — êi)Qo(P(w,13,p2)). (B.117)ir2(p1,) = w(p2 —c)Qo(P(w,,p). (B.118)Appendix B. Proofs for Chapter 2.82To obtain the reactionfunctions for the two firms, differentiate ir1 (13i ,P2)and ir2 (P1,P2)respectively with Pi and P2 as shown in thefollowing expressions.OiriQ3i,p2) —OQo(P(w,i,p2))+ Qo(P(w,231,p))}op1 ——0.(BJ19)thr2(pi,p)OQo(P(w,i,p2))+ Qo(P(w,i,p2))jw[(p2 — c2) 0P2‘9P2 —= 0.(B.120)The above system of equations imply the following relationship.c9Qo(P(w,j31,p2))= (p2—C1)Q((PP (B.121)(y3i — ci) 0P2Perform the followingdifferentiation,= wQ’(P(w,j3i,p2)).(8.122)Oj3OQ0(P(w, P1,p2)) = wQ,(P(w,j5i,p2)).(B .123)8P2Equations (B.121), (B.122)and (B.123) together imply1—w (pi — c) = P2 — c2.(B.124)1LIn particular, if (B.119) and(B.120) admit an equilibrium, then the equilibriunapricessatisfy (B.124). It follows from (B.117) and (B.118) that the two firms have equal profits.DProof of Proposition 16. For the non-cooperative case, the equilibrium price satisfyequations (B.105) and (B.106). Adding these two equations and rearrangingtrins wehave 2Q0(F(w, Pi, P2))P(w,pi,p2)— P(w,cj,c2)= — Q(P(w,pi,p)).(B.125)Appendix B. Proofs for Chapter 2.83Now consider the cooperative case in which the two firms collude to act as a monopolist.The profit function of a monopolist is given below.ir(p,w) = (p—P(w,ci,c2))Qo(p). (B.126)The monopolist bundle price ptm satisfy the following first order condition derived from(B.126).(mmOj’p —P(w,ci,c2)=—,9,1m\ (B.127)‘o’p jWe next compare (B.125) and (B.127), and show that the monopolist bundle price,ptm, cannot be higher than the competitive equilibrium bundle price, P(w, Pi, p2). First(B.125) and (B.127) show that ptm P(w,pi,p2). Secondly, suppose ptm > P(w,pi,p2).Log-concavity of Q(.) implies that_Q0(p)/Q(pm) —Qo(P(w, P’,2))/Q(P(w, Pi ,p2)).Then (B.125) and (B.127) contradict the supposition. Therefore ptm <P(w,pi,p). DProof of Proposition 17. From Proposition 15, 7r1(w) = 7r2(w),Vw E (0, 1).Then fori = 1,2,d’r(w) 1 ddw=—P(w, c1,2))Qo(F(w,pi, p2))]= [(P(w,pi,p2)— P(w, c1,2))Q’0(P(w,pi ,p2))dF(w,pi, p2)dP(w,pi, p2) — dP(w,ci, c2) )Qo(P(w,pi ,p2))] (B.128)(P(w,pi,p2)— P(w,ci,c))Q(P(w,pi,p(c — c2)2wG(w,pi ,p2)Qo(P(w,pi,p2))1Q(P(w,p,p)) c— c2)——(c1 —c2)] (B.129)2 wG(w,pi,p2—(1 —w)(p —ci)(ci—c)Q(P(w,pi,p)—wG(w,pi,p2Appendix B. Proofs for Chapter 2. 84+WG(WP1P2)(cl —c)Qo(P(w,p1,p)[Q(F(w,pi,)+(1 — w)(p1— ci)Q’0’(F(w,pi,p2))J. 0Proof of Proposition 18. Compare the second term of (B.128) and the second term ofthe last expression in the preceding proof, the result follows. 0To prove Proposition 19, we will use the following lemma.Lemma 7 Define P(.)= Q1(.) and let Qo(•) be continuously differentiable. If Qo(.) isdecreasing, then P’(q) =Proof: From the identity QQo(p) = p we have-P(Qo(p)) = P’(Qo(p))Q(p) = 1.P’(Qo(p)) = P’(q)1= Q(p)•Proof of Proposition 19. Let the inverse demand function before regulation be p =where F(.)= Q(.) and q is the industry output. In Cournot competition, firms competein terms of output (see, for example, Friedman 1983, chapter 2, for a theory of Cournotcompetition.) Let qi and q be the outputs of firm 1 and firm 2. The profit functions ofthe two firms are1(qi,q2) = (P(qi+q2)—ci)qi.r2(q1,) = (P(ql+q2)—c).Appendix B. Proofs for Chapter 2. 85The first order conditions, and thus the reaction functions, for the two firms are givenbelow.thr1(qi,q2)= P(qi + q) — 1 + P’(q1 + q)qi = 0.8*2(q,q)= P(qi +q2) — c2 + P’(qi +q2)q2 0.Let 3 and (i’, ) be the Cournot equilibrium price and quantities. Adding the abovetwo first order conditions, and using Lemma 7, the following expression is obtained.2j3 = + c1 + c2. (B.130)From equation (B.125), the equilibrium bundle price under recycling regulation satisfiesthe following relationship.P(w,pi,p2)= +P(w,ci,c2). (B.131)We il1 show that j3 cannot be higher than P(w,pi,p2). First suppose j3= P(w,pi,p2)then (B.130) and (B.131) implies2 1= + c2) — -P(w, c1,c2).Without loss of generality, assume c1 > c2, then1p—c1 = (2c—ci—P(w,ci,c)<0.This suggests that the firm with higher unit cost earns, in Cournot equilibrium, negativeprofit. Thus J3 P(w,pi,p2). Next, suppose j3 > P(w,pi,p2). Log-concavity of Qo(•)implies—Qo(j3)/QQ3) —Qo(P(w,p1,))/ (F().This gives1j3 < + c2) — P(w,c1,c2)}.Appendix B. Proofs for Chapter 2. 86Applying the same argument above will show that the firm with higher unit cost earnsnegative Cournot equilibrium profit. Therefore J3 < P(w, p1, p2). DProof of Proposition 20. (P(w,p1,p)— = P(w,pj,p2) and the result followsfrom Proposition 13. 0Proof of Proposition 21. The bundle price, ptm, obtained by cooperation under regulationsatisfies equation (B.127). The Cournot equilibrium price, j3, satisfies equation (B.130).Suppose J3 = pm, and firm 1 has higher unit cost, then j3 — c1 = c2— P(w, c1,c2) < 0 isobtained from (B.127) and (B.130). Thus the firm with higher unit cost earns negativeCournot equilibrium profit. So 1 ptm. Now suppose j3 > ptm, and firm 1 has higher unitcost. Log-concavity of Qo(•) implies —Qo(j3)/QQ3) $ Q0(pm)/Q(pm). From (B.127)and (B.130), it can be shown that J3 — c1 < c2 — P(w, ci, c2) < 0. Again the firm whichhas higher unit cost earns negative profit. Therefore, 5 <pm. 1iProof of Proposition 22. The inverse demand function for Qo(p) = a0— /3op is P(q)(ao— q)/f3o. The Cournot equilibrium profits (see, for example, Friedman 1983, chapter2) for the two firms are:(ao + 0c2 — 2/30c1)K19/3(ao + /30c1 — 2/30c)K29/3The equilibrium profits for the two firms under regulation, iri(w) and K2(w), are given in(2.41).(ao— (1 — w)floci — w/30c2) (ao + f30c2 — 2/30c1)iri(w)—K1 =—________________________= (1 +W)(Ci -c2)[2(ao-/3ocl) -(1 -w)/3(c C2)] (B.132)Appendix B. Proofs for Chapter 2. 87(ao — (1 — w)/0c1 — w130c2) (ao + /30c1 — 2/30c)ir2(w)—ir =—__________________= _(2 — w)(ci—c)[2(ao — oc2)+ wo(ci—c2)] (B.133)Ifci=c2,thenI=7r(w);i=1,2. DProof of Proposition 23. (i) Suppose c1 < c2, (a) from expression (B.132) above,iri(w) < i; (b)if 2(ao — .9oc1)— wfio(ci — c2), then the right hand side of (B.133)and ir2(w);ir.Proof of (ii) is similar to part (i) above. 0Proof of Proposition 24. Let the unit costs of the two firms before the regulation bec1 = c2 = c, and after the regulation be c2 = c. Then equations (B.130) and (B.131)obtained, respectively for Cournot equilibrium and equilibrium under regulation becomeQo(j3) /p =— ,.+c. iB.1342Q0(p)P(w,pi,p2) = + P(w,c,c2). (B.135)Q0( (w,pi,p2))First supposej3 = P(w,pi,p2) then (B.134) and (B.135) give3—c = (c—F(w,c,c2))0, and both firms make no positive Cournot equilibrium profits. On the other hand,suppose j3> F(w, Pi, p2). Log-concavity of Qo() implies Again(B.134) and (B.135) give j3 — c < (c — P(w, c,c2)) 0, and both firms make negativeCournot equilibrium profits. Thus j3 < P(w,pi,p2). 0Appendix B. Proofs for Chapter 2. 88P2p_1(o)P2(O)R2PiP1(O) p1(O)Figure B.9: Reaction Functions for Firms 1 and 2Appendix CProofs for Chapter 3.Proof of Proposition 25 (i) The necessary condition for the Nash equilibrium price atdisplay (3.53) is obtained by summing equation (3.50) over all n firms. The necessityof (3.52) for a Nash Bargaining Point was already noted in the paragraph containing(3.52). From our assumptions on the demand function, it follows that that 1/(— in Q)’(x)decreases to zero and so must cross each of the lines corresponding to x — wc and(x — wc)/n exactly once (see Figure 3.3). Hence (3.52) and (3.53) have unique solutions.(ii) Since x ‘—+ 1/(— in Q)’(x) is continuous, its graph must cross the line correspondingto x — wc strictly to the left of where it crosses the line corresponding to (x — wc)/n.Hence wpm <wp’, which establishes (ii).(iii) The value of the combined profits at the Nash equilibrium point is obtained bysumming display (3.51) over all firms.The value of the combined profits to the cartel at the Nash bargaining point is (wpm —wc)Q(wpm). Substituting (3.52) into this expression verifies equation (3.55).Substituting (3.53) into (3.48) gives*— wp*Q(wp*)(C 136)— E(wp*)for each firm. 0Proof of proposition 26. (i) Using the facts that p = P(eq) and that P’(eq’) =89Appendix C. Proofs for Chapter 3. 901/Q’(p), equation (3.57) can be rewritten as— (— inQ• (C.137)Under our assumptions on the demand function, this equation has a unique solutionprovided that is in the domain of the demand function (see Figure 3.3).(ii) This inequality follows from Figure 3.3 by an argument similar to that in Proposition 25(u). 0Appendix DProofs for Chapter 4Proof of Proposition 27. The expression for the policy line in Figure 4.4 is —a2— 721P1 +‘Y22P2 + a1r —711pr+ -y12p2r 0. Given a price of firm 2, P2, the price of firm 1 on thepolicy line ispl(p2)= —a2 + 722P2 + rQi + 712p2) (D 138)721 + 711TSuppose both firms comply with the regulation. Let pl(p2, e) = pi(p) + e, i.e., the pricefirm 1 deviated from that on the policy line given the price of firm 2. Let iri(pi(p2), bethe profit of firm 1 attained on the policy line given the price of firm 2, and7r1(pi(p2,e),p2)be the profit obtained at the deviated price. Theniri(pi(p2), — ri(pi(p2,e),)= (721 +7iir)2[—2cr2711 — a1721 —c172— 712721P2+711722P2 +a171r— cl71r+7112p2r+ e71172 + C711rJ= e2[e(7ii7zi +71r)(p2— p)(—712721 + 2711722 +7112r)](721 + 7iir)> 0 P2 >— e(7i172i+71r) (D.139)—— 712721 + 2711722 +71i72rTherefore given any P2 above the point F1, firm 1 does not unilaterally deviate from thecorresponding Pi on the policy line.Now consider firm 2. Given a price of firm 1, p’, the price of firm 2 on the policy lineisp2(p1)= a2 + 721P1 — r(ai_711P’) (D.140)722 + ‘y12r91Appendix D. Proofs for Chapter 4 92Let p2(pl, e) = p2(pl) — e, a price deviated from that on the policy line given p. Thenr2(pi,p)) —ir2(pi,pi,e))e 2= I \2 [—a722 + C272 — 721722P1 + cv271r‘y227ir)+2a172r+c2712r+ -yi27ipir — 271i2pr+ e72 + e71272r]e2e(7 +71271r)(p1— p)(—72l722 +71221r—27fl722r)](722 +712r)P1 +e(72 +71221r) (D.141)(722 +712r)Thus given Pi below the point F2, firm 2 does not unilateral deviate from the corresponding price on the policy line.Conditions (D.139) and (D.141) taken together implies that each point is a Nashequilibrium on the segment F’P2 of the policy line. 0Proof of Proposition 28. On the policy line, the price of the recycled product as afunction of the price of the virgin product, p2(pl), is given by (D.140). The profit of thevirgin product producer along the policy line and its first order condition are(i — ci)(a2712 + O1722 + 712721P1 — 711722P1)7’1(p1,p2(pi)) =722 + 712rdiri(p,p2(p ))— c2-yi +a1722 + (2p1 — ci)Qy1221— 711722)dp1— 722 +712r> 0 if 712721 711722.Similarly, the price of the virgin product as a function of the price of the recycled product,pi(p2), on the recycling policy line is given by (D.138). The profit of the recycler alongthe policy line and its first order condition are/ / , , — (p2 —c2)(cx-yii + °1721 + 712721P2 —71i722P)T2IP1IP2),P2)—721 + ‘fliTdir(p (p),p—[o2711 + al721 + (2p2 —C2)(71271 — 7n722)]rdp2— 721+71r> 0 if 712721 711722.Appendix D. Proofs for Chapter 4 93The profits of the two firms on the policy line increase with Pi and P2, and the equilibriumprofits are maximized at P2 which dominates the other equilibria. DProof of Proposition 29. Divide expression (4.83) for t by expression (4.76) for tto obtain the following:71r+ 721>1. DProof of Proposition 30. It follows from the fact that the denominator of (4.84) is anincreasing function of r. DProof of Proposition 31. Divide the profit for firm 1 under production tax in expression (4.84) by the profit for firm 2 under excessive consumption tax in (4.78) to give theequation below.7r(r)— 72274(r)— 711r2Proof of Proposition 32. The profit functions of firm 1 under the scheme of taxingexcessive consumption of virgin product and production tax are given below.7r(pi,p2,t1)=7r1(pi,p2)—t1(qp,2)— —q2(pl,p2)Y’, (D.142)ir(pi,p2,tc)=7ri(pip2)_tCq(p,p). (D.143)At the equilibrium point F2, the above expressions become=iri(p,p), (D.144)irp,p,tc*)=iri(p,p) _tc*qi(p,p). (D.145)Appendix D. Proofs for Chapter 4 94Comparing equations (D.144) and (D.145), and since >0, i4(p,p,tl*) > cQ,i22fc*)DProof of Proposition 33. The profit function of firm 1 under progressive tax is7rj(pi,p2,t8)= [1—i3r(pi,p2)Jirp, . (D.146)Differentiating the profit function with respect to P’ to obtain the following first ordercondition: 1 — 2t?ri(p,p)= 0.Suppose t5 > 0, then the value of Tri(p1,p2)on the reaction curve of firm 1 is l/2t3.Therefore, ir(pi,p2,t) = l/4ts, and the tax payment, tri(p1,p2)= 1/4t3, on the reaction curve. DProof of Proposition 34. The profit function of firm 1 under progressive profit tax isgiven by expression (D.146). The profit function under the scheme of taxing excessiveconsumption of virgin product is given by (D.142). At the equilibrium point F2, theprofit functions become=ri(p,p), (D.147)(p,p,t8*) = (D.148)Comparing equations (D.147) and (D.148), and sincets* >0, (p,p,tl*) > ppts*).0Proof of Proposition 35. At the equilibrium prices, (p(r),p(r)), the profit firm 1 earnsunder the progressive tax scheme equals the tax payment for all meaningful r. This isthe result from Proposition 33. The profit function of firm 1 under production tax is(p(r)—c1)q(p,p) — tcqi(p,p). Since t increases with r, and by increasing r, someAppendix D. Proofs for Chapter 4 95value of r is reached at which t = — c1, then profit equals tax payment. Increasingr further tax payment outweighs profit. But at this value of r, profit still equals taxpayment under the progressive tax scheme, thus profit is lower under production tax atthis point onward. Note that at each r, the equilibrium prices and the demands for thetwo products are the same under the two tax schemes. DProof of Proposition 36. Under the tax schemes considered, the equilibrium profit offirm 2 and its derivative with respect to r are given below.2 , — 722r[a1i+a1721 +c2Qy127i— 711722)12— 2(7i2yir—271172r— 721722)d 2 — 27ir[o7i1+a1721 + c2(’y12-y1 — 711722)12—ir2r1 —dr (—712721r+ 2-y1172r+ 721722)>0. DAppendix EBackground to the Mandatory Recycling Legislation for NewsprintIt is a fact that there is a shortage of sanitary landfill in North America, especially in theUnited States. The U.S. generates about 160 millions tons per year (tpy) of municipalsolidwaste and approximately 84% of this is disposed of in landfill. Wastepaper accountsfor almost half of all solid waste (Paper Recycler 1990.)During the period 1982-87 over 3,000 landfills have been filled up and shut down inNorth America. Around 50% of the landfills now in use will close down over the nextfive years (McClay 1990.) About 75% of the landfills in the U.S. will close by the year2005 (Paper Recycler 1990.) The remaining landfills are generally not located closed tourban centres; therefore the cost of transporting garbage has risen dramatically. By theyear 2000, the U.S. will be short 56 million tpy of disposal capacity.According to Edwards (1991), as early as 1965, the EPA had established an officeof Solid Waste Management Programs. In the same year the first Solid Waste Act waspassed which was replaced by the Resource Recovery Act in 1970, and by the ResourceConservation and Recovery Act (RCRA) in 1976. The RCRA required each of the statesto file with the federal EPA a solid waste management plan. About 43 states have filedsuch plans with the EPA. The major problem facing those who are responsible for thedisposal of solid waste is the lack of landfill space.The lack of landfill sites did really set the legislation wheel in motion only after NewJersey’s garbage crises. New Jersey closed all but 20 of its landfills by 1984, and no newones were sited. In 1987, a law was passed requiring all communities in New Jersey to96Appendix E. Background to the Mandatory Recycling Legislation for Newsprint 97mandate recycling programs. Some 17 states followed New Jersey’s lead by setting uptheir own source separation and collection systems or by studying such programs. Asthese recycling programs began to generate more recyclables, it became apparent thatdemand for them was not growing proportionally. Some states in the interim, paid millsto take the excess old newspaper (ONP) rather than paying high tipping fees (dumpingcharges) at landfill or paying to transport the solid waste hundreds of miles (PaperRecycler 1990.) Tipping fees in the U.S., as suggested by Edwards (1991), are nowapproaching $100 per ton.Hatch Associates Ltd. (1989) and Paper Recycler (1990) pointed out that the oversupply of ONP for recycling had resulted in the introduction of legislation to stimulatethe usage of such recyclable products. In the U.S. such legislation is purely a reactionto limited municipal and state budgets by reducing pressure on collection and landfillservices. This has focused mainly on getting newsprint producers to use more ONP intheir fibre furnishes, though legislation on mandatory recycling of other grades of paperare now also being considered.Florida passed the first mandatory recycled newsprint usage bill in late 1988. Sincethen many states had proposed or adopted similar bills. By late 1989, California andConnecticut had passed laws requiring publishers to consume a certain portion of recycled newsprint. By the summer of 1991, eight states had passed legislation settingnewsprint recycling goals and timetables, and in some cases including taxes on virginnewsprint. These states were California, Connecticut, Florida, Arizona, Missouri, Illinois, Maryland and Wisconsin. California, for example, stipulated recycled newsprintconsumption of 25% in 1991, 30% in1994, 35% in 1996, 40% in 1998 and 50% in 2000(Boyle 1990.) Eleven states including New York, Pennsylvania, Maine, Massachusetts,Micigan, Iowa, Louisanna, Vermont, South Dakota, New Hampshire and Virginia hadvoluntary agreements negotiated. For example, in the New York agreement 11% recycledAppendix E. Background to the Mandatory Recycling Legislation for Newsprint 98newsprint is the purchasing target for 1992, 23% for 1995, 31% for 1997 to 40% in 2000.The purchasing goals for New York are to be based on an assessment of the state andlocal government’s ability to collect and process ONP for recycling into newsprint andother products. An escape clause is included to provide a waiver if recycled newsprintsupply is not available at a price and quality comparable to virgin newsprint. Also forthe New York agreement, focus is on aggregate use of recycled fibers rather than therecycled fiber content of each sheet of newsprint (Newsprint Reporter 1991, Boyle 1990.)McClay (1991) reported that except California, Arizona and Florida which stipulateda 40% minimum-content standard as a definition of recycled-content newsprint, all theothers used a blended or aggregate target approach. Four other states, Georgia, Indiana,Ohio and Oregon had legislation pending. In 1988 Florida also pioneered an advancedisposal fee for every ton of virgin newsprint used exceeding half of total consumption.This was followed by California and Pennsylvania (Newsprint Reporter 1991.)Apart from the state legislations on mandatory newsprint recycling to be in placeon the publishers and newsprint producers, the U.S. federal government, states andmunicipalities are also requiring their purchasing agents to produce newsprint and otherpaper containing certain percentages of recycled fiber. As early as 1988, the federal EPAissued the ‘Guideline for Federal Procurement of Paper and Paper Products containingRecovered Materials’ under the RCRA. One year later, according to section G002 ofthe RCRA, all of the federal government agencies responsible for procuring paper wererequired to give preference to recycled products, and the state and local governments thatpurchase $10,000 worth or more using appropriated federal funds must also follow theEPA guidelines, e.g. newsprint must contain at least 40% postconsumer wastepaper. Bythe beginning of 1990, 23 states had preferential procurement laws (Garcia 1990, PaperRecycler 1990.)Jaakko Pöyry Oy (1990) pointed out the driving force for the use of waste paper hadAppendix E. Background to the Mandatory Recycling Legislation for Newsprint 99traditionally been economic in the other parts of the world: waste paper had enabledmills and countries without abundant forest resources to compete effectively on bulkgrades with integrated producers of virgin based grades. This was particularly the casein regions such as Japan and Western Europe where it had been relatively easy to collectand process large volumes of wastepaper. In the Far East the local industries dependedto a large extent on the cost economies of wastepaper. In North America, on the otherhand, abundant forest resources had not encouraged the use of recycled fiber.Though there appeared to be general acceptance by the newsprint producers of theU.S. government mandated recycling requirements (Hatch Associates Ltd. 1989), theresponses from the industry were less than enthusiastic. In 1989, even under the federalEPA preferential procurement law there was only approximately 12% of the total NorthAmerican newsprint production capacity which used some percentages of recycled fiberas a raw material (Pulp and Paper North American Factbook 1990, p.140 and p.144.) In1990 Canada produced less than 2% newsprint containing recycled content (Lukins 1992.)The whole newsprint recycling issue in North America has been primarily legislationdriven by the need to reduce solid waste stream, not by a need for newsprint production,a shortage of fiber, or cost competitiveness (Newsprint Reporter 1990, Hatch AssociatesLtd. 1989, Garcia 1990.) It was even said that the industry had never before been forcedto absorb so much additional recycled fiber (Paper Recycler 1990.)A survey of the newsprint industry reported by Newsprint Reporter (1990) showedthat the newsprint producers perceived the mandatory recycling issue as one of solidwaste disposal. Thirty nine percent of the general public agreed with the newsprintproducers. Obstacles to heavy recycling as perceived by the executives were the costimpact, an adequate supply of ONP, the paper quality and the quality of recyclingmachinery. Seventy five percent of the newsprint producers felt that the states hadbeen too aggressive in trying to increase newsprint recycling, an obvious reference to theAppendix E. Background to the Mandatory Recycling Legislation for Newsprint 100mandatory recycling bills.Pulp and Paper North American Factbook (1990) questioned whether or not any ofthe state goals could be achieved within the time frames mandated. For it was not certainhow much secondary fiber could be reclaimed, as various factor such as contamination,improper handling and lack of consumer interest tended to limit availability. NewsprintReporter (1990) also suggested that the community collection systems were not yet producing high quality material. It was estimated 20-40% of the postconsumer newspaperwould be difficult to process or could not be used because of contaminants.From the perspective of the demand side, McClay (1990) suggested that up untilrecently the newsprint market had given preference to virgin fiber product and shuntedthe recycled variety in North America because of its inferior quality both in terms ofrunnability and printability. Landegger (1992) was particularly vocal about the diseconomics of the mandatory recycling programs. He pointed out that virtually all chiefexecutives of the industry questioned the recycling programs, and stories about millsexchanging rolls of unused newly produced papers which were then being cut up and putback into their paper stream so that they could claim the resulting papers produced tobe ‘recycled.’ He also claimed that tonnage of dead leaves and grass cuttings that wentto the landfill equaled to all the wastepaper and that it was much cheaper to turn theminto compost or animal feed.Bibliography[1] Aspler, J. S., 1989, ‘Potential Printability Problems in Recycled Papers,’ in J. Aspleret al. (eds.), Recycling and the Canadian Pulp and Paper Industry, A Report to theResearch Program Committee of PAPRICAN.[2] Booth, D. L., 1990, A Strategic Capacity Planning Tool for a Firm in the NewsprintIndustry, unpublished Doctoral Thesis, University of British Columbia.[3] Booth, D. L., V. Kanetkar, I. Vertinsky and D. 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