ESSAYS ON PRODUCTION AND PRICING DECISIONS By Yat-Koon Mok M. Sc., University of Aston M. Phil., University of Cambridge A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES COMMERCE AND BUSINESS ADMINISTRATION We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1993 © Yat-Koon Mok, 1993 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Commerce and Business Administration The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1Z1 Date: Abstract There has been considerable interest in finding and explaining the basic elements that can drive product quality up. In the literature this is largely done by modelling the effects of investing 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, the firm should produce output of higher quality, the increase in quality being dependent on consumers’ sensitivity to quality and to price, and the effect of technological improvement on product price and quality are very different from those when the demand is a function of price alone. Some twenty states in the U.S. have passed recycling laws which mandate consumption of old newspaper by the newsprint industry. To study the effect of regulation, a model is used in which two firms compete under the regulatory constraint—one firm producing the recycled product, the other the virgin product. Assuming the regulatory constraint is 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 higher average price in competitive equilibrium than the cartel price, are obtained in the second essay. The two firm model is generalized to include n firms which compete under the same kind of regulatory constraint in the third essay. Results similar to the two firm case are obtained. When the recycled product and the virgin product are partially substitutable, reg ulation that mandates consumption of the recycled product results in infinitely many equilibria. A dominating equilibrium exists if the demand parameters satisfy a certain condition, otherwise it is not clear how to select an equilibrium. On the other hand, a 11 suitable tax on the virgin product, or its producer, serves to induce compliance with the recycling policy and equilibrium selection. The equilibrium prices and profits of the two firms under the schemes of production tax, excessive consumption tax and progressive profit tax are examined and compared in the fourth essay. It is interesting to find that the tax rate for excessive consumption is comparatively low and, in equilibrium, this tax scheme collects no tax payment. 111 Table of Contents Abstract ii List of Figures vii Acknowledgement 1 2 viii Effects of Technological Improvement on Quality and Price 1 1.1 Introduction 1 1.2 Quality Costs 5 1.3 Effects of Technological Improvement on Quality and Manufacturing Costs 7 1.4 Demand and Consumer Sensitivity 10 1.5 The Objective Function 12 1.6 Analysis 13 1.7 Effects of Acquiring an Improved Technology 14 1.8 Conclusion 23 The Effects of Mandatory Recycling on the Newsprint Industry 26 2.1 Introduction 26 2.2 The Model 28 2.3 Price Competition 30 2.4 The Effects of Regulation on the Equilibrium Prices and Quantities 2.5 The Effect of Recycling Regulation on Profits iv . 32 36 2.6 2.7 3 4 Comparison of Prices, Quantities and Profits Before and After Recycling Regulation 37 Conclusion 40 Competition When Market Share Is Regulated 42 3.1 Introduction 42 3.2 Regulated Market Share 43 3.3 Effect of Introducing a Regulated Market 47 Competition under Recycling Regulations 4.1 Introduction 4.2 The Model and Competition under Mandatory Consumption of Recycled 4.3 4.4 49 49 Product 51 Competition Under Taxes 57 4.3.1 Taxing Excessive Consumption of Virgin Product 58 4.3.2 Production Tax 61 4.3.3 Progressive Profit Tax 62 4.3.4 Some Remarks 65 Conclusion 66 Appendices 68 A Proofs for Chapter 1. 68 B Proofs for Chapter 2. 78 C Proofs for Chapter 3. 89 D Proofs for Chapter 4 91 V E Background to the Mandatory Recycling Legislation for Newsprint Bibliography 96 101 vi List of Figures 1.1 The Economic Conformance Level Model 6 1.2 Appraisal and Prevention Cost Reduction 9 3.3 Cournot and Nash Equilibrium and Cartel Prices 4.4 Reaction Functions of the Two Firms 53 4.5 Policy Lines and Reaction Functions 54 4.6 Reaction Functions of the Two Firms and the Policy 56 4.7 Reaction Functions of the Two Firms under Tax 60 4.8 Reaction Functions of the Two Firms under Progressive Tax 64 B.9 Reaction Functions for Firms 1 and 2 vii . . . 46 88 Acknowledgement I would like to thank my supervisors, Professors Shelby Brumelle and Ilan Vertinsky, for their continual support throughout the work of the thesis. My thanks are to Professor Bill Ziemba for being a member of my committee, and to Professor Philip Loewen for his comments on Chapter 1 of this thesis. I would also like to thank Drs. Darcie Booth and Don Roberts for their help and hospitality at Forestry Canada, Ottawa. Financial support from the Forest Economics and Policy Analysis Research Unit at the University of British Columbia and from Hong Kong University are gratefully acknowledged. VII’ Chapter 1 Effects of Technological Improvement on Quality and Price 1.1 Introduction In recent years, the importance of product quality for a manufacturing firm to succeed in the market place has received major attention, motivated by the observation that many Japanese manufacturing firms can produce outputs at high quality levels with relatively low unit costs. Evidence is given, for example, by Garvin (1983), Hayes, Wheelwright and Clark (1988) and Fortuna (1990). The importance of product quality is undeniable. Meyer, Nakane, Miller and Ferdows (1989) report, from their yearly ‘Manufacturing Futures’ surveys on large manufacturers in Europe, North America and Japan, that quality was consistently on the top of the list of concerns. It is generally agreed that the kind of product quality which contributes the most to the competitiveness of Japanese manufacturing firms is conformance to design specifica tions, which reflects the degree of consistency and reliability of the products. This paper formally models the effects of improved technology on the costs of conformance quality that are associated with the quality enhancing activities , and manufacturing cost. The role of product quality in affecting demand is also explicitly considered in the model. Numerous studies employing analytic models provide insights into various aspects of quality. Fine (1986) introduces a model of quality-based learning in which learning is represented by the sum over time of the product of quality and output volume, where 1 Chapter 1. Effects of Technological Improvement on Quality and Price 2 the quality considered is the conformance level to design. He finds that when the manu facturing cost is a function of learning, and under certain conditions of the discount rate and the planning horizon, product quality should start at a high level and continuously decrease until the end of the horizon, at which time the quality is at the value which min imizes cost. In case learning can reduce quality cost, and the cost reduction parameter is a function of learning, then a manufacturing system should produce at a relatively high quality level if there is no discounting; moreover, output quality increases over time. Our model in this paper is developed in the same spirit of Fine (1986). Instead of pursuing further the issues of learning, we look at the effects of technological improvements when consumers’ demand is based also on product quality. Fine (1988) and Marcellus and Dada (1991) employ modified versions of stochastic maintenance or replacement models and regard quality level as a state variable rather than as a decision variable. The quality level of a station as described by Fine (1988) is represented by its probability of being in control. Learning is assumed to occur whenever an inspection is made and the station is found to be out of control. Each time learning occurs, the probability that the station is out of control is assumed to decrease, and thus quality increases. The author concludes that ignoring the learning benefits of inspection and quality control activities may lead to under-investment in quality improvement ac tivities. The model of Marcellus and Dada (1991) is similar to that of Fine (1988), but quality level is represented by the proportion of nondefectives. A decision maker can decide whether or not to learn. Choosing to learn incurs a cost and reduces the propor tion of defectives. They show that if the expected reduction in the cost of quality is less than 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 be interpreted to have tradeoff between the cost of failure and the cost of prevention. Other Chapter 1. Effects of Technological Improvement on Quality and Price 3 stochastic studies include those of Fine and Porteus (1989), who formulate and analyse the problem of gradual process improvement as a Markov decision process in which in vestment in process improvement results in a probabilistic number of improvements, Lee and Tapiero (1986), who link the optimal sampling parameters in quality control with sales 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 a model for costs of internal and external failure, and obtain results analogous to Fine and Porteus (1989) for multiple products. The importance of technology, to the Japanese manufacturers is summarised by Ohmae (1984) saying that, “Japanese industry did not become competitive through QC circles and company songs alone! What really counted was the wise decision to change the whole modus operandi of Japanese manufacturing by investing heavily in new facili ties and techniques.” The importance of technological improvements is also reflected by the Japanese concern about falling behind in process technology which was ranked third by the large Japanese manufacturing firms in the survey by Meyer, Nakane, Miller and Ferdows (1989.) The close relationship between the motivation to adopt better technol ogy and the desire for high product quality is clear. Fortuna (1990) also suggests that to reduce the process variation (a source of poor quality) of a stable manufacturing system, improvements in the process are necessary. Examples of process improvements given by the author include equipment modifications and enhancements. Thus, it is safe to believe that improved technology has the ability to improve quality. We postulate that it directly increases the lower bound of the domain of product quality on which a firm makes its decision. That is, without any other quality related activities, the quality of the output from an improved technology is higher than that from a technology on which no improvement has been made. Indirectly, improved technology can enhance quality related activities, and thus reduce the costs relating to these activities. These two effects Chapter 1. Effects of Technological Improvement on Quality and Price 4 on quality are explained in more detail and modelled in section 3. Another way to model the impact on production decisions of quality concerns is to include quality in the demand function. Dorfman and Steiner (1954), for example, assume that 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 inverse demand function to the left. Based on specific assumptions made about consumers’ taste for quality, Shapiro (1983) models demand as a function of price per unit of quality. Lee and Tapiero (1986) derive a probability distribution for sales as a function of the defect rate in a production batch, and therefore relate sales to the defect rate. However, their model and objectives are very different from ours. They relate optimal sampling strategy for quality control to the parameters of the sales distribution, whereas we are interested in the effects of technological improvements on quality, price and output given that the demand is a function of price as well as quality. Others largely ignore the effect of product quality on demand in their studies relating to quality. We derive the demand function which includes quality as a variable in section 4 by assuming that given the price and quality of a product, a perceived fair price of quality of the product is encoded by each consumer. Subsequently, a consumer’s purchase decision is determined by her perceived fair price. With quality in the demand function, the optimal decision on price and quality of the product, after adoption of an improved technology, depend on how sensitive consumers are to price changes, and to quality changes. If consumers are indifferent to quality variations and base their decision on price alone, then a result similar to that of Fine (1986) is obtained. Indeed if we think of technological improvements broadly and consider continuous learning over time as continuous technological improvements, then the model in this paper is similar to that of a discrete jump in learning. We review in the next section the model of economic conformance quality and the costs associated with it. In section 3 the effects of technological improvements on the costs of quality and Chapter 1. Effects of Technological Improvement on Quality and Price 5 manufacturing are modelled. A demand function based on price and quality is derived, and consumers’ sensitivities are defined, in section 4. We formulate the objective function of 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 adopting an improved technology, a change in its adoption time and a variation of the time-path of the investment cost are analysed in section 7. A conclusion is given in section 8. 1.2 Quality Costs In the quality control and assurance literature, two aspects of the quality of a product are often studied. One aspect is the product design. The second is the degree to which the product conforms to design specifications (see, e.g., Juran 1988.) Here we consider the latter case. The model of economic conformance level (EUL) is well known and accepted in the operations 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 is defined on an interval [0, 1], i.e., q e [0, 1] where q = 1 means the firm is producing at the perfect quality level at which there are no defective outputs. At q = 0, the products are completely defective. Two kinds of cost are defined over the interval of quality. Failure cost, 1 c ( q), is incurred due 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 higher conformance level. It is an aggregate cost resulting from activities in quality control, Chapter 1. Effects of Technological Improvement on Quality and Price 6 Costs q 1 Figure 1.1: The Economic Conformance Level Model quality assurance and quality planning such as inspection, test, test gear maintenance, training, reporting and quality improvement projects. A more detailed breakdown of these costs is given by Hagan (1986). These quality cost functions are highly dependent on the characteristics of the production system, especially its technology. The general shapes of the cost functions for a production system are depicted in Figure 1.1, where the ECL is the value of q at which the total cost of quality, ci(q) + c (q), is minimised. 2 Chapter 1. Effects of Technological Improvement on Quality and Price 1.3 7 Effects of Technological Improvement on Quality and Manufacturing Costs The technology that we consider here can be capital equipment, method of production processes or the configuration of the production system of the firm, a platform for the manufacturing system on which various productive and quality related activities are built. The importance of technology as a determining factor for quality of conformance is acknowledged by Juran (1988). Schroeder (1985) postulates that the impact of better technology on quality is to shift the total quality cost downwards and towards the di rection of fewer defective products. Generally, technological improvement in this respect can have various impacts on the system output. In this paper we study some of these impacts by making two postulates about the effects of technological improvement on the costs of quality: 1. An improved technology can increase the effectiveness of the quality control, quality assurance and quality planning activities so that quality appraisal and prevention can be performed in a more efficient manner. This is consistent with the remarks once made by an executive of Corning about its manufacturing system that most of the gains came from technology and that to get people to work smarter involved the application of technology (Meadows 1984.) Therefore, as a result of increased effectiveness, the appraisal and prevention cost, 2 c ( q), is lowered for any given quality level to crc ( 2 q), where a E (0, 1). We shall call this the indirect effect on quality cost. 2. An improved technology has the effect of increasing the lower bound of the qual ity level to, say, , so that the appraisal and prevention cost function 2 ac ( q) is shifted downward. We call this the direct effect on quality cost. Examples of this are employment of advanced automatic process diagnosis and production process Chapter 1. Effects of Technological Improvement on Quality and Price 8 adjustment equipments to reduce process variations. More examples of quality cost reductions with equipment, method and process types of technology are given by Taguchi, Elsayed and Hsiang (1989). The total (direct and indirect) effect of technological improvement is that the appraisal and prevention cost function becomes 2 cxc ( q) ac ( 2 q) — 0 < 0 for q near 0. of quality ci(q) + ac (q) 2 — — 0, as shown in Figure 1.2. Note that We assume throughout this work that the total cost 0 is positive for all q, and note that the optimal solutions described below never enter the region where the revised appraisal and prevention cost is negative. We choose not to model the resulting cost, after technological improvement, as a nonsmooth function which is zero for q e (q) 2 [0, ] and crc — 0 for q E [, 1], for this does not add much to the insight that can be obtained from the model developed in this paper. Often an improved technology, apart from being better able to enhance quality ac tivities, also has the property of reducing manufacturing cost at any given output level. Let c 3 be the unit cost of manufacturing. Constant unit manufacturing cost is assumed here to simplify exposition. Most results in this paper still hold so long as the total man ufacturing cost as a function of output is convex and increasing. After the adoption of an improved production technology the unit manufacturing cost becomes /3c , /3 E (0, 1). 3 We can consider an additional downward shift of the manufacturing cost due to tech nological improvement. However this has the same effect as 0, and is thus absorbed in this term, since total quality and manufacturing unit cost after technology acquisition is ci(q) + 2 oc ( q) + /3c 3 — 0. We assume in this paper that ci(.) is strictly convex and decreasing, ci(l) ci(0) = (.) is strictly convex and increasing, c 2 (l) 2 +co. c = (0) 2 +oo and c = 0. = 0 and Chapter 1. Effects of Technological Improvement on Quality and Price 9 Costs (q) 2 ac (q)—O 2 c q Figure 1.2: Appraisal and Prevention Cost Reduction Chapter 1. Effects of Technological Improvement on Quality and Price 10 We consider the case in which the production technology that improves over time is exogenous and the firm can acquire it whenever it feels appropriate to do so. Thus the cost reduction parameters a, 0 and /3 are functions of time, t. It is further assumed that a(.) is decreasing, a(0) 0(.) is increasing, 0(0) 0,Vt e = = 1, a(T) 0, 0(T) = = a, e where T is some finite horizon and and is finite such that ci(q) + a(t)c (q) 2 [0, T] and ‘v’q E [0, 1]. /3(.) is decreasing, 3(0) = 1, /3(T) = / and $e — (0, 1). 0(t) > (0, 1). The cost involved in adopting a technology can be an increasing or decreasing function of 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 Sensitivity A consumer’s propensity to purchase product quality is assumed to be captured by an index of consumer characteristics, E R+, which may reflect the budget’s constraint of the consumers. Propensity to purchase quality may vary among consumers. A product with quality q and price p is subject to the evaluation of each consumer and the result of individual judgement is the formation of a function, W(,p, q), which is common across 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 is evidence to suggest that buyers often translate actual price into perceptions of price. For example, Jacoby and Olson (1977) distinguish between objective price (the monetary price) of a product and perceived price (the price as encoded by the consumer.) The monetary price is frequently not the price encoded by consumers (Zeithaml 1988.) A survey 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 original Chapter 1. Effects of Technological Improvement on Quality and Price 11 monetary 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) will purchase 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, a consumer will make a purchase if p/q, which corresponds to W(., p, q) = q in our model. We further assume consumer propensities to purchase quality do not change over time, i.e., that is not a function of time. The production system, however, may adjust both the price and quality at the time technology changes. In our model, all individuals rank the product in the same way but because of dif ferences in budgets may opt to purchase different levels of quality. This diversity is represented by a density function of consumer type f(). So a product with price p and quality q is able to capture the expected fraction of the market F(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) the demand rate. It is reasonable to suppose that W(, p, q) is strictly increasing in q and strictly decreasing 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) be measures respectively for the consumers’ sensitivities to variations in price and quality, see Dorfman and Steiner (1954). We define quality-price sensitivity as This inequality can be derived through the approach of Caplin and Nalebuff (1991) who study 1 existence of price equilibrium in imperfect competition with fixed product attribute q. They assume a 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 are decision variables and we allow the price and quality to interact. Chapter 1. Effects of Technological Improvement on Quality and Price s (p,q) .— 12 Fq(p,q) F(p, q) —_____ S(p, q) is a positive number obtained from the ratio of demand change due to quality variations and demand change due to price variations. It provides a measure of relative sensitivity of the consumers to quality against price. 1.5 The Objective Function The objective of the productive system is to determine the price and quality, (p, q), of its output, 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, t , 1 to the interval [0, T] since we are interested in the effects of adopting improved technology rather than the optimal number of adoptions over the horizon. On the other hand we can view this restriction on time frame as a model of incremental technological or process upgrading in which decisions on technological improvements are made necessarily over shorter horizons (see Schonberger 1982 and Hayes and Wheelwright 1984.) The objective function given below is a version of the capacity expansion model by Hinomoto (1965). (F) max po,qo,pi,qi,ti {K(po,qo,pi,qi,ti) ti J J et[po — (qo) 1 c — (qo) 2 c — ]F(po,qo)dt + 3 c 0 1 e’[p — ) 1 ci(q — )c2(ql) 1 a(t ) 1 + 0(t — /3(ti)c } 3 F(pi, qi)dt tl hI(ti)} t _e (1.2) Chapter 1. Effects of Technological Improvement on Quality and Price 13 We assume either 1(t ) has taken account of the salvage value of the replaced tech 1 nology or it has no salvage value, and ascribe negligible salvage value to the adopted technology at the end of the horizon, since the firm may not want to sell the technology which may invite competition due to entry of the buyer. In the analysis below a prime will be used to denote the derivative with respect to the argument. 1.6 Analysis Define H°(p,q) : = [p—ci(q) 2 —c ( q) —ca]F(p,q). H’(p, q, t ) : 1 = [p — ci(q) — (a(ti)c ( 2 q) — (1.3) 0(t ) 1 ) — /3(ti)c ] 3 F(p, q). (1.4) A solution to (F) satisfies the following set of necessary conditions: H°(po,qo)=O H(pi,qi,ti) = 0 , Hq°(po,qo)=O. , H(pi,qi,ti) = (1.5) 0. (1.6) Lemma 1 H°(.,.) and 1 H ( ., .,t ) are pseudocoricave for any 1 1 t E [0, T]. Proposition 1 The optimum quality levels of the product, qo and qi, before and after acquisition of improved technology are higher than EGL, and they are the values at which the 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 system should always produce at a quaiity level higher than ECL so long as consumers prefer Chapter 1. Effects of Technological Improvement on Quality and Price 14 higher quality at any given price. Furthermore, the optimal quality level is higher if either consumers 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 confirms the intuition that a manufacturer is better off to produce a higher quality product if consumers are more responsive to high quality. However, it is not so obvious that when the consumers are less responsive to price it is more profitable for the firm produce higher quality product. Cole (1992) and some other authors suggest that the the Japanese see the pursuit of higher quality as a means of driving the cost curve down. (For a similar observation see Fine (1986).) The result here suggests that the pursuit of higher quality can also be consumer-driven. The ECL is optimal only when Fq(p, q) = 0, i.e., when product quality has no bearing on consumer preference. Thus the ECL of quality, often targetted by the manufactur ing divisions, at which the total quality cost is minimized, is suboptimal and reflects a separation of manufacturing functions from market conditions. Proposition 2 Given arbitraryqo andqi, the optimum prices Po and p are characterised by 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 are respectively characterised by equating marginal total cost to price, if a solution exists. 1.7 Effects of Acquiring an Improved Technology It is not clear how the optimal (p, q) and the value of F(p, q) change after the adoption of an improved technology at time t . Intuitively, one would expect that due to the 1 manufacturing and quality cost reduction effects of the technology (i) q would increase, Chapter 1. Effects of Technological Improvement on Quality and Price 15 or (ii) p would decrease, or (iii) both (i) and (ii) simultaneously. We shall study the signs of change in the optimal (p, q) and F(p, q) with a change respectively of the cost reduction parameters 0(t ), a(ti) and 9(t 1 ). We shall suppress the argument of these 1 /9. The and /9, parameters which are written as 0, a and technology is now a function of p,q, 0, a H(p, q, 8, a, /3) : [p — (q) 1 c — profit function after the change of (q) + 0 2 ac — JF(p, q) 3 /3c (1.7) The optimal (p, q) satisfies H(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 the optimality 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 have 0 H Hpqqo + 9 Hp + H 0 0. (1.10) 0. 9 HqeHqqqe+Hqppo+Hq (1.11) = where H is a shorthand for H(p,q,0,a,/3). = Define 1/k := HppHqq — 2 which is Hpq positive at the optimal value of (p, q) which satifies (1.8) and (1.9), and solving the above equations, pe, qe and F 9 are obtained as follows: P9 k(_HpeHqq + HqeHpq). (1.12) q = k(HqoHpp+HpoHqp). (1.13) (p,q) 9 F = Fp(p,q)pe+Fq(p,q)qo. (1.14) Similarly, to study the effect of indirect quality cost reduction property of improved technology the following equations are obtained using the Implicit Function Theorem Chapter 1. Effects of Technological Improvement on Quality and Price 16 with 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 better technology can be analysed by the following expressions which are obtained by fixing 0 and a. = F(p,q) = k(HpjHqq + HqHpq). (1.18) k(HqHpp+HpHqp). (1.19) 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 (ii)(b) If S(p,q) 0 then q <0. (iii) If S(p,q) 0 and Sq(p,q) 0. 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 function of quality, then the direct quality cost reduction effect, and the manufacturing cost reduction effect, of adopting an improved technology render a lower optimal price. On the other hand, this may not be true if the quality-price sensitivity of the consumers is an increasing function of quality. Indeed, if the consumers are rather insensitive to the price changes of a product, and the total quality cost of this product increases rather slowly, as compared to the increases in consumers’ quality-price sensitivity due to quality increases, then the two cost reduction effects will each result in an increase in the price Chapter 1. Effects of Technological Improvement on Quality and Price 17 of the product. Also, we cannot determine the sign of the optimal price change with respect to the lowering of the indirect quality cost, p. It depends on how sharply the changes in the relative quality-price sensitivity and price sensitivity of the consumers are with quality changes, how sensitive the consumers are to price and to quality, and how sharply the total quality cost increases. It is interesting to note that the cost reduction effects on price do not depend on how the consumers’ quality-price sensitivity changes with price. Contrary to the common belief that cost reduction accompanies higher quality, the expression (ii)(a) of the above proposition shows that a lower optimal quality level results from direct quality cost reduction, and from manufacturing cost reduction, when the consumers’ quality-price sensitivity is an increasing function of price. If, in addition, the consumers’ relative quality-price sensitivity is also a decreasing function of quality, the optimal price decreases. This means that the manufacturing firm is better off to produce the product whose quality is more affordable. It is also interesting to note that when the consumers’ quality-price sensitivity does not change with price, there is no quality change due to the two kinds of improvement in costs above, but the optimal quality increases with indirect reductions of quality cost. All three types of cost improvement induce higher optimal quality in case the consumers’ quality-price sensitivity decreases with price. The optimal quality may increase or decrease, as a result of indirect quality cost reductions, when the consumers’ quality-price sensitivity increases with price. The cost improvement effects on quality do not depend on how the consumers’ quality-price sensitivity changes with quality. Output for the product increases with all three kinds of cost reduction due to tech nological improvements if the consumers’ quality-price sensitivity does not increase with price, and does not increase with quality. This is a direct consequence of (i) and (ii) in the above proposition. Note that the optimal price may rise as a result of the indirect Chapter 1. Effects of Technological Improvement on Quality and Price 18 effect of technological improvement, but the impact of the possible price rise on demand is dominated by that of the corresponding quality rise. Consequently, output quantity increases. For the other cases, output quantity can increase, or decrease, depending also on the consumers’ sensitivities to price and quality, and how sharply the total quality cost 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 improved technology 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 price decreases and the output quantity increases. However, the change in optimal quality as a result 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 not affect optimal quality level. It is indeed the case because these two cost reduction effects shift the total quality cost curve downward, and thus do not move the ECL away from its position. However, an increase in optimal quality results from the effect of indirect cost reduction. There is a clear distinction in pricing and quality decisions under the two different demand models. The unidirectional changes in optimal price and quality, with the adop tion of an improved technology, are clear and simple, when demand is a function of price alone. When product quality is taken into account by the consumers in the purchase decisions, these changes are no longer one-directional, but depend on the behaviour of Chapter 1. Effects of Technological Improvement on Quality and Price 19 the consumers’ quality-price sensitivity on price as well as on quality. The result that indirect cost reduction causes an increase in optimal quality, is similar to that due to quality-based learning, as modelled by Fine (1986) who uses a demand rate which is not a function quality. Indeed if we look at technological improvement as a discrete jump in learning that lowers the prevention cost, then our result can be regarded as a discrete version of Fine’s. On the other hand, in our model, technological improve ment on manufacturing cost does not change the optimal quality. This is in contrast with the result that, except at the terminal time of the horizon, optimal quality increases if quality-based learning reduces manufacturing cost— because if increased quality en hances quality-based learning which, in turn, reduces manufacturing cost, then optimal quality 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 due to learning does not change the optimal quality, it merely reduces the optimal price of the product. To study how a change in adoption time t 1 affects the manufacturing system’s decision on the price and quality of its output, the Implicit Function Theorem is used again to derive the following equations. 1 Pt = k(HptiHqq + HqtjHpq). (1.21) 1 qt = k(HqtjHpp+HptiHqp). (1.22) F ( 1 p,q) = 1 +1 F(p,q)p (p,q)qt 9 F . (1.23) Proposition 6 (i) q, 1 >0 if S(p,q) 0. (ii) F 1 > 0 if S(p,q) If demand is a function of price alone, then qj 1 1 O,p 0 and Sq(p,q) 0 and Ft,(p, q) 0. 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 later Chapter 1. Effects of Technological Improvement on Quality and Price 20 renders higher product quality and greater demand as compared to earlier adoption. The sign of the price change cannot be determined. Thus price may increase. But because the quality 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 optimal price and optimal quality do not depend on the consumers’ quality-price sensitivity, as shown in the above proposition. When a productive system makes decision on the time to adopt a technological im provement, it does so based on an estimate of the time path of the adoption cost, 1(t), of the technology. A change in the time path of the investment will affect the time, t , at 1 which the technology is adopted. There is no generally agreed shape of the time-cost re lationship. It can be increasing or decreasing, convex or concave. Each of these different time-cost patterns will be considered. If t T, the productive system simply does not acquire the improved technology inside the time frame of planning. So attention is restricted to ti E [0, T) in which case 1 satisfies the necessary condition for (F) in the following Lemma. When t ti = ) 1 0, I’(t —H’(pi, q, t )dt 1 = 0. (1.24) is the righthand derivative. Lemma 2 An extremal value ofti for (F) satisfies the following condition rI(ti) — ) + H°(po, qo) 1 I’(t — , q, t 1 H’(p ) + exp r1 1 1 J , 1 at An interpretation of the lemma is that to adopt the technology as soon as a time is reached at which the profit per unit time just before the adoption, H°(po, qo), plus the foregone total benefit due to cost reductions to the end time, in case of a delay in adoption, discounted to the adoption time, which is the integral term in the lemma, equal to the profit per unit time just after the adoption, H’(pi, q, ti), plus the change in the investment cost in case of a delay in adoption, 1 I’(t ) , minus the relief of interest accrued Chapter 1. Effects of Technological Improvement on Quality and Price 21 due to the delay in adoption, rI(ti). That is, at each point in time over the planning horizon, the decision of whether to wait or to invest is made, based on the trade-off between the total marginal benefit and total marginal loss due to waiting. The decision to invest is made at the time when marginal benefit equals marginal loss. This is similar to a result of Marcellus and Dada (1991), that the policy of whether or not to invest in learning depends on the resulting marginal benefits and costs. The difference is that our deterministic model allows just one opportunity to invest in technological improvement over the planning horizon, whereas their probabilistic model allows investment in learning in every period. Define a higher (lower) investment cost function, relative to 1(t), as the time-cost relationship 1(t) := (I + 77 )(t), 77 (t) > 0 (p (t) < O),Vt 7 e [O,T]. It can be seen that p ( 7 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, t . It is assumed that (pi, qi) is 1 already 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 t . However, 1 doing so renders the analysis intractable. Define the Gateau differential of t 1 (I) from above (below) for any i > 0 (q < 0) and any arbitrary s E R as ) 7 Dti(I; Moreover, it is assumed that i := iimui +877) —t(I) (1.25) E c’[O, T], o and /9 are convex, and 0 concave. Proposition 7 Given (p1, qi) is fixed and t 1 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 Dt (I; ) > 0; 1 (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) — ) + I”(t); 1 rI’(t (ii) a relatively lower investment cost function, that is <0, and ,‘ 0, and Chapter 1. Effects of Technological Improvement on Quality and Price 22 (a) if I is decreasing and convex, then Dti(I; —i) <0; (b) if I is either decreasing and concave, increasing and convex, or increasing and concave, then 1 Dt ( I; —) has the opposite sign as that of w(tj) where w(ti) + := (e_r(T_t1) 1/r(e_(T_t1) )a’(tl) 1 + 1)[—c2(q — — rI’(ti) + I”(t ), 1 /1 3 c ) 3’(1 + 8’(ti)jF(pi, q) — 1)[—c2(qi)cx”(ti) — c f3”(ti) + 0” (ti)]F(pi, qi). 3 The proposition shows, for a decreasing and convex technology adoption cost function, any positive deviation from it (higher cost) ,which is nonincreasing, prolongs the adoption time, whereas any negative deviation (lower cost), which is nondecreasing, hastens the adoption. The object w(ti) is the increase in profit after adoption per unit time, together with the foregone total discounted benefit due to cost reductions to end time, resulting from delay in adoption. Whether or not this sum, w(ti), and the rate of the change in the adoption cost, 1 I”(t ) , 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 are different from those given in the proposition, the effect of them on the changes of the adoption time is similar to those shown in the above proposition, except that the cost deviations can only be of much smaller magnitudes, as given in the following proposition. Proposition 8 Given (pr, q) is fixed and t 1 is the extremal value, and, (i) a relatively higher investment cost function, that is ‘(t) < — p1(t), i < (r+ > 0, and i’ > 0 such that 1)i(t), and 5 >0, and (a) if I is decreasing and convex, then 1 Dt ( I; ) > 0; (b) if I is either decreasing and concave, increasing and convex, or increasing and con cave, then 1 Dt ( I; i) has the same sign as w(ti) — rI’(ti) + I”(t ); 1 (ii) a relatively lower investment cost function, that is i ( 1 t) > —2 (t), +7 2 < —(r + 1)(t), and 62 > 0, and < 0, and r’ < 0 such that Chapter 1. Effects of Technological Improvement on Quality and Price (a) if I is decreasing and convex, then Dt 1 (I; —ii) 23 <0, (b) if I is either decreasing and concave, increasing and convex, or increasing and con cave, then 1 Dt ( I; —7i) has the opposite sign as that of w(ti) — rI’(ti) + I”(t ). 1 The following proposition shows the horizon effects on the adoption time of an im proved technology. If the adoption cost function is decreasing and convex, then extending the planning horizon delays the time of adoption. For other types of investment cost func tion, a longer planning horizon may shorten or prolong the adoption time. Proposition 9 Given convex, then . > (pr, qi) 0; for the other patterns of the time-path of the investment cost, has the opposite sign as rI’(ti) 1.8 is fixed and t 1 is the extremal value, if I is decreasing and — ) 1 I”(t — • w(ti). Conclusion There has been considerable interest in finding and explaining the basic elements that can drive product quality up. In the literature this is largely done by modelling the effects of investing in learning and process improvement, and of cost reduction. We look at another aspect by modelling demand as a function also of quality, and the effects of technological improvement on quality cost and manufacturing cost. Our analysis shows that, indeed, when the consumers have preference for product quality the manufacturing system should produce output of higher quality than ECL. The more sensitive the consumers are to quality variations, the higher should be the output quality. The price sensitivity of the consumers also plays a role here in determining the product quality. The more insensitive the consumers to price variations the higher should Chapter 1. Effects of Technological Improvement on Quality and Price 24 be the output quality. The ECL is optimal only when the consumers are indifferent to product quality. This result confirms the importance of consumers’ preference for quality in 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 through cost reductions. An improved technology has the direct effect of turning out less defective product, consequently, it directly lowers the quality appraisal and prevention cost. It has an indirect effect of providing a platform for more efficient quality control and assurance activities, and thus indirectly lowers the appraisal and prevention cost. Thirdly, it reduces the manufacturing cost. Our results show that technological improvements may not always increase the optimal product quality. This depends on consumers’ behaviour. For example, when the consumers’ relative quality-price sensitivity is an increasing function of price, and decreasing function of quality, then it is optimal for the manufacturing system to produce at lower quality and charge a lower price, that is, it is better to produce output of more affordable quality. This is in contrast to the results when demand is a function of price alone. In this case consumers’ sensitivity to price plays no role in the determination of quality, and an improved technology allows no decrease in quality, a decrease in price and an increase in output quantity. The timing of adopting an improvement also affects product quality and price, but it depends, again, on consumers’ behaviour. If demand is a function of price alone, then delaying adoption does not decrease the quality, and does not increase the price. The time to 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 also affects the timing of adoption. If the price and quality of the product are already chosen and fixed by the producer, an increase to an investment cost function which is convex and decreasing, such that this cost increment is nonincreasing over time, then this cost Chapter 1. Effects of Technological Improvement on Quality and Price 25 increment delays the adoption time. On the other hand a decrease in cost relative to this same investment cost function hastens adoption if the decrease in cost is nondecreasing over time. Chapter 2 The Effects of Mandatory Recycling on the Newsprint Industry Introduction 2.1 The U.S. generates about 160 million tons per year (tpy) of municipal solid waste and approximately 84% of this waste is disposed of in landfill. During 1982-87 over 3,000 landfills have been shut down. About 50 % of the landfills now in use in the U.S. will close down over the next five years (McClay 1990.) About 75% of these landfills will be closed by the year 2005. Since wastepaper accounts for almost half of all solid waste and the technology for its recycling is well established, many states have passed recycling laws. Florida passed the first mandatory recycled newsprint usage bill in late 1988. By late 1989, California and Connecticut had passed laws requiring publishers to consume a certain portion of recycled newsprint. By the summer of 1991, eight states had passed legislation setting newsprint recycling goals and timetables, and in some cases including taxes on virgin newsprint. These states were California, Connecticut, Florida, Arizona, Missouri, Illinois, Maryland and Wisconsin. California, for example, stipulated recycled newsprint consumption of 25% by 1991, 30% in 1994, 35% in 1996, 40% in 1998 and 50% in 2000 (Boyle 1990.) McClay (1991) reported that California, Arizona and Florida stipulated 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 including New York, Pennsylvania, Maine, Massachusetts, Michigan, Iowa, Louisanna, Vermont, 26 Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 27 South Dakota, New Hampshire and Virginia had voluntary agreements negotiated. For example, in the New York agreement 11% recycled newsprint 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 and local government’s ability to collect and process ‘old newspaper’ for recycling into newsprint and other products. An escape clause is included to provide a waiver if recycled newsprint supply is not available at a price and quality comparable to virgin newsprint. Also the New York agreement focuses on the aggregate use of recycled fibers rather than the recycled fiber content of each sheet of newsprint (Newsprint Reporter 1991, Boyle 1990.) More details about the recycling regulations are given in Appendix E. Apart from the legislation on mandatory newsprint recycling to be in place on the publishers and newsprint producers, the U.S. federal government, states and municipal ities are also requiring their purchasing agents to procure newsprint and other paper containing certain percentages of recycled fiber. As early as 1988, the federal EPA issued the ‘Guideline for Federal Procurement of Paper and Paper Products containing Recov ered Materials’ under the Resource Conservation and Recovery Act (RCRA). One year later, according to section G002 of the RCRA, all of the federal government agencies re sponsible for procuring paper were required to give preference to recycled products, and the state and local governments that purchase $10,000 worth or more using appropriated federal funds must also follow the EPA guidelines, e.g. newsprint must contain at least 40% postconsumer wastepaper. By the beginning of 1990, 23 states had preferential procurement laws (Garcia 1990, Paper Recycler 1990.) In newsprint production, the use of pulp that includes a sizeable portion of recycled fibers derived from wastepaper was well established in Western Europe and South East Asia; however, such pulp was rare in North America. In 1989, for example, only approxi mately 12% of the total North American newsprint production involved some form of use Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 28 of recycled fiber (Pulp and Paper North American Factbook 1990, p. 140 and p.144.) The introduction of regulatory constraints on the production of newsprint is likely to affect the 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 by a recycling regulation on the consumers of newsprint. We address the following model questions. 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 plans of firms in the industry? In the model producers may choose to produce either recycled or virgin newsprint, while consumers who have preference for virgin newsprint (see e.g., McClay 1990) must never theless comply with regulations that require them to buy a given minimum proportion of recycled newsprint out of their total consumption. The total costs of producing recycled newsprint are generally higher than the cost of producing virgin newsprint, mainly due to the higher capital costs involved in building deinking plants (see e.g. Hatch Associates Ltd. 1989.) 2.2 The Model We assume a market with two products: virgin newsprint (i (i = = 1) and recycled newsprint 2.) We assume that two firms supply the market. Firm 1 supplies the market with virgin newsprint while firm 2 supplies the market with recycled newsprint. Each firm produces only one product. Let q ,i 1 = 1,2 be the quantities consumed of the two products. Let us assume that publishers (the only consumers of these products) Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 29 are required by the recycling regulation to consume at least w x 100 % of the recycled product on average (i.e. they must buy at least w x 100 % of the newsprint from the producer of recycled newsprint.) We also assume that the recycled product is an inferior good, the demand for which is solely derived from the regulation. Indeed, recycled newsprint is considered an inferior product in North America because of 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 the recycled product than required by regulation and 1 1 q (2.26) where w measures the tightness of the legislation. However, it is technically acceptable to 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 will prescribe 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 decline in newspaper readership due to some what lower quality newsprint is not likely to be significant, and therefore the derived demand function for newsprint is not likely to be affected by the lower industry quality standard for newsprint. Thus the demand function Qo(.) is assumed to be the same as for virgin newsprint in the absence of legislation, with the price given as a weighted average of the prices of recycled and virgin newsprint mix that is consumed, so that Q ( 0 P11+P22) 1 q + =qi+q2. (2.27) ‘J2 The 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 to four percent of demand, and it was shown by Booth (1990) that the North American Chapter 2. The Effects of Mandatory Recyding on the Newsprint Industry newsprint demand is adequately described by a linear function Qo(p) = ao 30 — op. The 3 f post-legislation demand functions can be derived from (2.26) and (2.27), and are Q1(pl,p2) = (1 Q2(pl,p2) = 0 wa w)ao — — — (1 [(1 0 w,6 — — [(1 0 w),8 — 1 + wp w)p j, 2 w)pi + wp ]. 2 (2.28) The two products are complementary in the sense that Qi(•,•) and Q2(•,•) are both decreasing functions. Note that the system of linear equations in (2.28) has rank 1 and is not invertable; thus a decision on quantities does not determine a vector of prices, rather there there is a half-line of prices. In our model, firms set prices first and let the market determine the consumption. Price Competition 2.3 Let the unit cost of producing product 1 (virgin) by firm 1 be c 1 and the unit cost of firm 2 (recycled) be c . Fixed costs such as the expenditures and investment cost in 2 converting to or acquiring the recycling technology will not be considered in this paper. The functions 7r (.,.) and r 1 (.,.) defined below are the profit functions for the two 2 firms. ) = (p1 2 iri(pi,p — ) = (p2 2 7r2(pl,p 2 ( ci)Q , 1 ) p p , — ) c ( 2 ) pi,p Q . (2.29) The reaction functions of the two producers, Ri(p ) and R 2 (pi) 2 by solving R 1 (P2) = (pi) 2 op} and R 3 ao—I 0 a — op}. 3 / arg max 1 {iri (p1 ,p2): = ——q, Qo (P1 qi +P2 2) = 2 { argmaxp ( ) pj,p 7r : q. = can be obtained , = (PiiP2) 0 —--q, Q 1 + q, Qo(p) q = ql+q2, Qo(p) = The reaction functions intersect to give the following equilibrium prices: * — 1 P — 0 a 1 + 2(1 w)/oc 0 3(1—w)/3 — — 2 c 0 w/3 = Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 0+0 a 2w/3 (1 2 c 0 3w/3 — Proposition 10 Assuming (1 — 1 w)c 1 c 0 w)/3 — 31 (2 30) , then the equilibrium prices (p,p) given in 2 wc (2.30) are unique. (all proofs are given in Appendix B.) The equilibrium prices of the two products are related through their unit costs c 1 and , and the tightness of regulation w by 2 c * P1P2 < w(2—w) (1—2w)ao w)c2 + w)(1 + (1 w)(1 + w)o — (2.31) — Note that in (2.31) the coefficient of c 2 decreases, and increases as w de creases. As the regulation is relaxed, the importance of the cost of producing recycled paper in determining the price of recycled paper diminishes. Indeed when the necessary quantity of recycled paper is minute (but positive), the producer of recycled paper will determine the price largely on the basis of the strategic value of the recycled product in determining the total production. He will do so to extract maximum rents. The second term reflects the importance of the demand structure in the determination of 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 by charging 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 the price elasticity increases. (Note that the elasticity of the demand is infinite at the price equal and the demand for the product vanishes.) Chapter 2. The Effects of Mandatory Recyding on the Newsprint Industry 2.4 32 The Effects of Regulation on the Equilibrium Prices and Quantities As the regulation requires more recycling the equilibrium price of virgin newsprint in creases and that of recycled newsprint decreases. This property can be verified by exam ining the following derivatives: 0 a Pi (w) = p’(w) 3(1 — — 0 f 2 w) l > O (2.32) <. = Note as the relative share of the producer in the supply of newsprint declines because of regulation, the firm will adjust the price upwards. This is because the weight of the price of his product in the price of the ‘bundled’ goods is lowered, thus the effect of charging a higher price for his good on the quantity of newsprint demanded will be lowered. The results in (2.32) derived for a linear demand function do not hold in general for an arbitrary demand function Qo(.). The derivatives for a general demand function are given in Lemma 3. Similar results to (2.32) can be obtained and are given in Proposition 11. In Lemma 3, Q(P(w)) and Q’(F(w)) denote respectively the first and second derivatives of Qo(.) with respect to the bundle price P(w,pi,p ) evaluated at (p(w),p(w)), where 2 (1 ) 2 P(w,pi,p — w)pi + wp 2 is the average unit price the consumers pay for a mix of recycled and virgin products that satisfies the regulation. P(w, c ,c 1 ) is the average unit 2 cost. Lemma 3 Suppose an equilibrium exists for an arbitrary demand function Qo(.). Let (p(w),p(w)) be the equilibrium prices, then *11 — P1kw)— 3p(w) 1 2c 2 (p(w) ci)(ci c c )Q”(P(w)) 2 + * * 3(1 w) (w),p 1 3wG(w,p ( 2 w)) 3p(w) 2c 2 c (p(w) 1 1 )(c c 2 c )Q”(P(w)) 2 + * * . 3w (w),p 1 3wG(w,p ( 2 w)) — — — — — — kW) 2 P — — — — — — Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 33 where G(w,p(w),p(w)): = [(1 — w)(p(w) — ci) + w(p(w) — )]Q”(P(w)), 2 c + [P(w, p(w), j4(w)) — )]Q”(P(w))}. 2 P(w, c, c Proposition 11 Suppose an equilibrium exists for an arbitrary demand function Qo(.). Let (p(w),p(w)) be the equilibrium prices. a. If ci = , then 2 c p’(w) p’(w) = p(w) —ci = > p(w)— c 2 , <0. (2.34) b. If Qo(.) is concave and 1 c > c 2 then p’(w) > 0. c. If Qo(.) is concave and 1 c 2 <c 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, then 1 —W p*I(w) p(w) +p(w) — 1 c Without concavity assumption for Qo(), the sign of — 2 c p’(w) (2.35) and p’(w) cannot be determined as shown by lemma 3, there is a definite relationship between them. One implication of Proposition 12 is that if relaxing the regulation causes the price of the virgin product to increase then the price of the recycled product cannot decrease. If slightly relaxing the regulation causes the price of the virgin product to increase, then the price of the recycled product will increase by at least times as much, on the other Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 34 hand, if the price of the recycled product decreases, then the price of the virgin product will decrease by at least times. The way that the equilibrium bundle price P*(w) := P(w,p(w),p(w)) is affected by the extent of the recycling requirement can also be obtained from Lemma 3 and is stated in Proposition 13. Proposition 13 Suppose an equilibrium exists for an arbitrary demand function Qo(.), Let (p(w),p(w)) be the equilibrium prices, then p*(w) — — — )Q(F(w)) 2 (ci c wG(w,p(w),p(w)) — (2 36) Proposition 13 shows that the unit production cost of the recycled product relative to 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: > F *1 (w)=0 < 2 = 1 c . c (2.37) It is interesting to find that when the costs of the two products are identical, c 1 = , 2 c the bundle price is not affected by the tightness of the regulation. In this case the equilibrium 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, 2 )P(w, p ( )+Q p1 ,p2)) is the total amount of the two products demanded in the market. Let q(w) denote Q(F(w,p,p)). A di rect 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 the recycled product is higher than that of the virgin product the total quantity demanded Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 35 of the two products decreases when the regulation requires higher consumption of the recycled fiber. Similarly if c 1 is higher than c 2 then q(w) increases when the regulation prescribes a higher recycled content. When the unit costs of the two products are the same, the total consumption of the two products is unaffected by the tightness of the regulation. Lemma 4 Suppose an equilibrium exists for an arbitrary demand function Qo(.). Let (q(w), q(w)) be the equilibrium quantities, then 1 q*11w 1 2 (w) q — — — — r *f — w 0 q 1 * q ( 0 w) — 11 — i c 1 w — \ *11 2 q 2 c 0 1 w 1 1 -‘- * * 2 ( 1 wG(w,p ( w),p w)) (ci — 2 )q’(w) c * * (w),p 1 p ( w)) G(w, 2 . (2.39) Proposition 14 Suppose an equilibrium exists for an arbitrary demand function Qo(.). Let (p(w),p(w)) be the equilibrium quantities, then a. Ifci = 2 then c q’(w) = —q(w) <0, q’(w) = q(w) > 0. (2.40) b. If Qo(.) is concave and c 1 <c , then q’(w) <0. 2 C. If Qo(.) is concave and c 1 > c , then q’(w) > 0. 2 Note that our analysis of changes in prices and quantities due to changes in regulation show a symmetry of response in the following sense: as the share of one producer in the bundle increases, his costs of production have higher impact upon quantities and prices. Similarly as his share increases (due to regulation) the impact of price increases in terms of reduction in quantities sold increases, and he is less inclined to charge higher Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 36 prices. In the next section we will show that this symmetry results in equal profit sharing irrespective of the specific value of w. 2.5 The Effect of Recycling Regulation on Profits If the demand function Qo(.) is linear, the equilibrium profits of the virgin and recycled product producers, Tr(w) and r(w), respectively, are equal. It can be easily checked that the equilibrium profit for either firm: [co — ] P(w,ci,c 0 /3 ) 2 9/3 = 1,2. (2.41) The following proposition shows that this phenomenon of equal equilibrium profits between the two firms is true for an arbitrary demand function Qo(.). Proposition 15 If an equilibrium exists for an arbitrary demand function Qo(.), then E(p 1 — ci) = p — . Consequently, the two firms have equal profits. 2 c 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 binding agreement will lead each of the players to charge somewhat lower prices and thus larger quantities sold than in the non-cooperative case. In the non-cooperative case each pro ducer considers the effect of price increases only on its own revenue and not the losses that each price increase causes the other producer; thus, there is a tendency to charge higher prices and produce lower quantities as shown in Proposition 16 below. Proposition 16 Let the demand function Qo(.) be log-concave. Then the non-cooperative equilibrium bundle price is higher than the bundle price obtained by cooperation, and the total quantity of recycled and virgin product consumed at the non-cooperative equilibrium is lower. Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 37 Proposition 17 Suppose an equilibrium exists for an arbitrary demand function Qo(.). Let (p(w),p(w)) be the equilibrium prices, then for i ir’(w) = wG(wp(w),p(w)) {(1 1,2 w)(p(w) - Q(P(w))[Q’(P(w)) + (1 = — - w)(p(w) 2 ci)Q’(P(w)) — - ci)Q”(P(w))]}. (2.42) The relative values of the unit costs of production for the two products determine the extent of the legislation’s effect on the profits of the two firms. If the costs are equal, then the profits of the firms are independent of w. The first term represents Q(P*(w))F*(w) whereas the second term represents P*((w) — ). 2 ,c 1 4P(w, c For a concave demand function Qo(.), these two terms work in the same direction. If the recycled product is more expensive to make, the profits of both firms are depressed when the regulation is tightened, i.e. 7rr(w) < 0; i = 1,2. On the other hand, if the recycled product is less expense 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 is directly 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 dw 2.6 — P(w, c, c2)) = 2(c — )c 2 [Q’(F(w)) + (1— w)(iii(w) (w),p 1 wG(w,p ( 2 w)) — ) 243 cl)Q”(F(w)) Comparison of Prices, Quantities and Profits Before and After Recycling Regulation We assume in the base case (before regulation) that the two newsprint producers produce one homogeneous product— virgin newsprint, and are engaged in a Cournot quantity Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 38 competition. There are several cases to consider. In case one, we consider different unit costs between the two competitors before the regulation. There is no change in unit costs after the second producer’s conversion to the production of recycled newsprint. In the second case we will assume that the unit costs of production for the two firms are equal before the regulation and that the cost of the firm converting to recycled newsprint are higher 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, c , 2 1 and c the equilibrium bundle price F*(w) for a mix of virgin and recycled products as required by 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, c 1 and d —(P * (w —p dw ) (ci c )Q’(P(w)) 2 —o wG(w,p(w),p(w))< c2, then — — — Proposition 20 shows that the price gap, P*(w) — 2 — 1 c . c > (2.44 j3*, increases as the degree of the recycling requirement is more stringent when the firm which converts to the recycled product has a higher unit cost. On the other hand when the firm which converts to the recycled product has a lower unit cost, the price difference decreases as the required degree of recycling increases. When the two firms have equal unit costs, the price difference is independent of the degree of required recycling. Proposition 21 Given a log-concave demand function Qo(.) and unit costs, c , 2 1 and c the equilibrium bundle price obtained by cooperation, for a mix of virgin and recycled products as required by the regulation, is always higher than the pre-regulation Cournot equilibrium price for the virgin product. Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 39 Propositions 16, 19 and 21 together imply that the pre-regulation Cournot equilibrium price is lower than the post-regulation bundle price obtained by cooperation and thus the post-regulation non-cooperative equilibrium bundle price. The pre-regulation and post-regulation equilibrium profits of the firms, for a linear demand function, when the unit costs of the two firms are equal, are compared in Propo sition 22. When the unit costs are different, the changes in the profits of the firms due to the regulation are given in Proposition 23. Proposition 22 Given a linear demand function Qo(.), if the unit costs of the two firms are the same, then the equilibrium profits of the firms before and after the regulation are identical, 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 c 1 <c , then 2 (a) ir(w) <*; (b) * <‘.*. 1r2(w)=1r2 zf2(ao —/3oci) < —w/3o(c —C2). (ii) Suppose 1 c > c , then 2 (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 condi tions, may favour the inefficient firm. To study the effect of the unit cost of the recycled product on the equilibrium bundle price as compared to the pre-regulation equilibrium price, we let c 1 = 2 c = c before the regulation and c 2 c after the regulation. Proposition 24 Given a log-concave demand function Qo(.) and unit costs c 1 before the regulation, c 2 1 c = = 2 c = c c after the regulation, the equilibrium bundle price P*(w) Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 40 for a mix of virgin and recycled products as required by the regulation is always higher than the pre-regulation Cournot equilibrium price j3* for the virgin product. In Proposition 24, P*(w) > fr even when c 2 = c after the regulation. Thus the net effect of the regulation is that it drives the price up. 2 < c In the case of c = 1 after the regulation, the pre-regulation equilibrium price c may be lower or higher than the equilibrium bundle price after the regulation, depending on the specific value of the unit cost of the recycled product and the demand function. 2.7 Conclusion Some twenty states in the U.S. have passed recycling laws which mandate consumption of old newspaper by the newsprint industry. There are also talks of having similar legislations on other grades of paper. To study the effect of the recycling regulation, we use a simple model of two firms— one firm producing the virgin product and the other the recycled product, which compete under the regulatory constraint. The recycled product is assumed inferior to the virgin product, and its demand is derived solely from the regulation. The model shows that, under the regulatory constraint, firms set prices and let the market determine the consumption. An interesting result from the model is that the firms set their prices such that the margins they earn, depending on the severity of the regulation, are in a fixed proportion to one another, consequently they earn equal profits. In general, the effects of the tightness of the regulation depend on the unit cost of the recycled product relative to that of the virgin product. Under a more severe recycling regulation, and a concave pre-regulation demand function, consumers pay a higher (lower) average unit cost, and thus consume less (more), if the unit production Chapter 2. The Effects of Mandatory Recycling on the Newsprint Industry 41 cost 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 North America, for the possible high cost in recycling ( Newsprint Reporter 1990.) If the pre regulation demand function is linear, the price of the virgin product increases and that of the recycled product decreases, when the regulation mandates more consumption of the recycled material. We assume the two firms produce a homogeneous virgin product, and are engaged in Cournot quantity competition, before the recycling regulation. With a log-concave pre regulation demand function, consumers pay more for a unit of the product mix under the regulation than for a unit of the virgin product before the regulation; also, under the regulation, the average unit price that results from price competition is higher than that obtained from cooperation, and the average unit price obtained from cooperation is higher than the Cournot equilibrium price. If the pre-regulation demand function is linear, then the model shows that the recycling regulation penalizes the efficient firm by hampering its profit, and in certain conditions depending on the demand function, the cost structure of the industry and the tightness of the regulation, it may even favor the inefficient firm. Chapter 3 Competition When Market Share Is Regulated 3.1 Introduction In this chapter the two firm model in Chapter 2 is generalized to include n firms which compete under the same kind of regulation. The regulation mandates consumption of a fixed proportion of each product out of the total consumption of all n products. The case can also arise if the n producers produce distinct products and consumers are char acterized by a Leontieff type consumption technology (e.g., they use the goods bought from the n producers as inputs to a Leontieff type production function.) The motive for a government to mandate the bundling of outputs of different produc ers of a commodity may be diverse. For example, environmental concerns may require the use of different energy types by utilities to ensure that the impacts of generating elec tricity will not exceed the absorptive capacities of the environment. The diversification of sources may fulfil this objective. Alternatively, regional equity concerns may lead to regulations mandating consumption of products produced by different producers located in different regions. 42 Chapter 3. Competition When Market Share Is Regulated 3.2 43 Regulated Market Share There are n firms, each of which produces a distinct product. These products are techno logically substitutable, but differ in their effects on the environment. In order to minimize the impact of these products on the environment, the government has imposed regula tions which stipulate for each product its proportion of the total consumption of all n products. That is, the regulation specifies a vector of market shares w = ,.. 2 (wi, w . , w,) such that = 1 (3.45) w>Ofori=1,2,...,n The proportion of product i which is consumed must be exactly the fraction w of the total 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 firms which the regulation eliminates from the market. We assume that the total demand for the products depends only on the price of a unit bundle consisting of w units of product i (i 1,2,. . . , n). Let p denote the price charged by firm i for product i. Then the unit price of a bundle is wp, the scalar product of the market share vector w and the price vector p = (pl,p2,. . . ,p,j. Let Q denote the demand as a function of the unit bundle price, so that Q(wp) is the total demand if the market share vector is w and the price vector is p. The demand for product i is qj = w:Q(wp) for i = 1,2,... ,Ti. (3.46) The demand function is, of course, nonnegative and decreasing on its domain, which is an interval of nonnegative prices. To avoid technical problems with existence questions Chapter 3. Competition When Market Share Is Regulated 44 concerning equilibria, we further assume that the percent decrease in demand per unit in crease 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) where the elasticity at price x is E(x) (3.47) = —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 that elasticity is increasing at least at a linear rate. The profit for firm i is = (Pt — cjw Q 2 (wp). (3.48) Under the assumptions in the previous paragraph, the unique Nash equilibrium is ob tained at a price vector p such that thr.Im*’1 ZU’ .i p 9 =Ofori=1,2,...,n. (3.49) By evaluating the partial derivative, this condition can be rewritten as * — c) = wp E(wp*) = 1 (_lnQ)(wp*) (3.50) The profit for firm i at the equilibrium price vector p is — wp’ Q(wp*) E(wpj If each firm prices its product according to the Nash equilibrium price, then no firm would unilaterally change their price. Each firm will maximize their profit, given the other firms’ prices. If we consider a cooperative version of this model, the monopoly or cartel bundle price, wpm, is the unique solution to wp m —wc= 1 Q)(wpm) (_ in . (3.52) The above discussion is summarized in the following proposition. It is interesting to note that the cartel price is less than the more competitive Nash equilibrium price. Chapter 3. Competition When Market Share Is Regulated 45 Proposition 25 Assume that the regulated market share vector w is such that the cost of a unit bundle wc is in the interior of the domain of the demand function. (i) The unique Nash equilibrium price of a bundle satisfies — WC W = (— (3.53) in Q)(wp*)’ 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 is 1 n (wpj Q(wp*) E(wp*) () wpmQ(wpm) 4E(wpm) (3.55) and is = 1 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 case where each firm has the same unit cost, say c. In this case wc = c for all w, so that the three functions appearing in this figure do not depend on w. It follows that the Nash equilibrium and cartel prices for a bundle are independent of the regulated market shares. If the costs are not all identical, then the effect of changing the regulated market shares is more difficult to describe without making the model more specific. However, we can 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 perturbation causes the unit bundle cost to decrease.) Since wc increases, the two lines having x-intercept wc in Figure 3.3 keep the same slope but move to the right and the decreasing function l/(— ln Q)’(x) is not affected. So Chapter 3. Competition When Market Share Is Regulated 46 1/(— in Q)’(x) n(x—) x — wc (x—wc)/n Figure 3.3: Cournot and Nash Equilibrium and Cartel Prices Chapter 3. Competition When Market Share Is Regulated 47 each of the Nash equilibrium and cartel prices for a bundle (i.e. wp and wpm) increase and for each firm i, profit 4 decreases. Since we are considering perturbations of market shares which increase the weighted average cost, wc, there must be at least one firm whose market share increases, say firm k. For such a firm, price p decreases. The decrease in p is clear since from display (3.50) _1 1 wk(_lnQ)(wp*)’ * Pk — and each term on the right hand side decreases when w is perturbed so as to increase Wk and the price of a bundle wp. The prices charged by firms whose market share decreases can move in either direc tion. However, since wp’ increases, Q(wp*) decreases. So if the market share of firm i w Q 1 (wp*). decreases, so does its production qj 3.3 Effect of Introducing a Regulated Market Suppose that before regulation, the firms are all producing the same product at the levels corresponding to the Cournot equilibrium and costs c. The price associated with this equilibium will be denoted by p and the quantities by q. The inverse demand function is P(.) eq = = Q’(.). The vector e is the n-vector consisting of all ones. Note that q. We will also write c = ec/n to represent the average unit cost. The first order conditions for the Cournot equilibrium are •P(eq)—c —P’(eq) — • ( 356 ) . Summing these equations yields eq — (3.57) Chapter 3. Competition When Market Share Is Regulated 48 Proposition 26 (i) If c is in the domain of Q, then there is a unique Cournot equilib rium. (ii) If c and wc are in the interior of the domain of Q and wc > m p• <wp , then (3.58) Chapter 4 Competition under Recycling Regulations 4.1 Introduction The lack of suitable locations for landfill is becoming a serious problem for municipal waste disposal in the U.S. (McClay 1990, Paper Recycler 1900.) Since waste papers constitute a significant portion of the municipal solid waste (Paper Recycler 1990), fed eral and state governments are passing laws that mandate recycling of certain grades of waste paper (Boyle 1990, Edwards 1991.) For example, California stipulated recycled newsprint 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 higher volumes of recycling and covering wider ranges of paper grades. Taxation policies that aim 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 or taxing consumption, or production, of virgin products definitely changes the competition between the recyclers and the users of virgin fiber. The objective of this paper is to study how competitive equilibria change under two approaches of recycling regulation: (i) mandatory consumption of a recycled product and, (ii) taxing a virgin product, or its producer, under a recycling policy. To this end a stylized model is used in which two firms, a recycler and a producer of virgin product, compete under the recycling regulations to maximize their own profits. 49 Chapter 4. Competition under Recycling Regulations 50 In the enviromnental economics literature, the use of marketable permits and emis sion charges for environmental management have received widespread attention. Both approaches are meant to induce firms to find efficient ways to comply with environmental standards. 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 used to discourage consumption of virgin fiber. Charging production of virgin products is similar to levying tax on the products, or their producers. Pigouvian tax, the purpose of which is to correct for externalities, is also widely studied in the context of environmental pollution (Cropper and Oates 1992.) An example of the tax to compensate for the social costs associated with litter and waste disposal is given by Dobbs (1991). But the taxes on the virgin product, or its producer, that we will consider are meant to induce, under competition, compliance with the recycling policy. Some common taxation schemes for natural resources are franchise tax, profit tax, progressive profit tax, value tax, severance tax, 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 will consider production tax, progressive profit tax and taxing excessive consumption of the virgin product. In this paper, we are interested in the pricing decisions, the resulting profits 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 de mand is assumed to be derived solely from the regulation. In this chapter, a market of two partially substitutable products, virgin and recycled, are assumed. Two firms, one producing the virgin product, the other the recycled product, compete under the recycling regulations. The policy objective for the tax schemes, and for the regulation of mandatory recycling, is to accomplish the target of consumption of the recycled product at a certain proportion, w%, of the total consumption of the two products. The model Chapter 4. Competition under Recycling Regulations 51 and competition under a regulation that stipulates percentage consumption of the recy cled product is given in the next section. We show that this regulation induces multiple equilibria. Under certain conditions, one equilibrium dominates; but in general, an equi librium selection mechanism is needed to be added to the model. A suitable tax on the virgin product, or its producer, serves the dual purposes of inducement of compliance with the recycling policy and of equilibrium selection. In section 3 the effects of the various taxes on the firms are studied. It is found that the equilibrium profit of the re cycler is unaffected by the different tax schemes, and the severity of the recycling policy favours the recycler in the sense that its profit increases. The virgin product producer earns highest profit in the scheme of excessive consumption tax. The tax rate required to induce compliance with the policy is low for the scheme of excessive consumption tax as compared to the scheme of production tax and, in equilibrium, it does not collect any tax payment. Section 4 concludes the paper. 4.2 The Model and Competition under Mandatory Consumption of Recy cled Product Consider two firms, indexed by i 1,2; which compete in price. Firm 1 produces the = virgin product, firm 2 the recycled product. Firm 1 and firm 2 each have a constant unit 1 and c cost of c , respectively. The firms face a linear demand system, 2 where > 0 and price effects. It of 722 > qi(pi,p) = i +712P2, (4.59) q2(pl,p2) = — a + 2 p 22 i pi, 7 7 (4.60) 0 are the is common in —7iipi own price effects, 712 > 0 and 721 > 0 are the cross the economics literature to assume a symmetric system demands that greatly simplifies analyses; where a a ‘y , = 2 = 722 and 712 = 721. Chapter 4. Competition under Recycling Regulations 52 Examples are Kiemperer and Meyer (1986) in their study of the role of uncertainty on competition, Roller and Tombak (1990) in the analysis of the strategic role of flexible manufacturing systems, and the work of Furth and Kovenock (1990) on price leadership under capacity constraints. However, symmetry is not assumed in this paper since it is unlikely that the demand schedule of the recycled product is the same as that of the virgin product. In the absence of the recycling regulation, the equilibrium of competition is deter mined by the reaction functions of the two firms. The profit functions of the two firms are ) 2 lri(pi,p = (p1 (pi,p ir ) 2 = (p2 — — ci)qi(pi,p ) 2 , (4.61) (pl,p 2 c2)q ) . (4.62) The reaction functions are obtained, respectively, by Ri(p ) 2 2 (pr) R = = 1 arg max 2 ( 1 7r ) pj,p , and arg max 22 ir (p1, P2) which are given below: 01 2 ( 1 R ) p (pi) 2 R + 712P2 + 7nCi + 22 7 2 C (4.63) 27 2+ a 721P1 = 2722 (4.64) . The reaction functions are shown in Figure 4.4. The slope of Ri(p ) is 2711/712, and 2 the slope of R (pi) is 721/(2722). For an equilibrium to exist, the reaction functions 2 must intersect in the positive domain. This requires the condition that ) is shifted to the right by an increase in c 2 Ri(p 1 or a bigger value of 4711722 > 7i27i• 712, and shifted to the left by a greater value of -y. Similarly, R (pi) is shifted upwards by an increase 2 in c 2 or a larger value of 721, and shifted downwards with a greater value of 722• The equilibrium prices are given below: * 1 P — 201722 + 02712 22 + 7 11 2y 1 +c C 1 2 7 2 2 , — 4711722 — 712721 (4.65) Chapter 4. Competition under Recycling Regulations 53 2 ( 1 R ) p P2 increase in , 721 2 C 2 (p1) R increase in 722 a2+y22 C2 2y22 ‘ increase in c , 1 712 increase in 711 / P1 11 + 1 a y Cl 2yn Figure 4.4: Reaction Functions of the Two Firms a 2 7 1 1 + * P2 2a 1 y 2 1 = 11 + 27 2 7 1 2 c 22 -y 11 +C 4711722 — (4.66) 712721 Assume that the policy of the mandatory recycling law is to require the publishers to consume at least w% of the recycled product, that is, qi(pi,p) (100 — (pi,p 2 w)q ) /w. However, in what follows it is much less cumbersome to work with the policy constraint below: rql(pl,p2) where r e (pi,p 2 q ) (0, co). The regulation mandates consumption of no virgin product if r (4.67) = co, Chapter 4. Competition under Recycling Regulations 54 ) 2 Ri(p P2 policy line, policy line,O <r < R ( 2 pj) Y22 policy line, r a2+2:c2 / aj+y11C, ai.. 2y Y11 = 00 P1 Figure 4.5: Policy Lines and Reaction Functions and 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 positive domain satisfies the policy constraint. Likewise, an equilibrium lying above the policy line violates the regulation. An increase in the recycling requirement shifts the policy line downwards. The slope of the policy line is 11 (r7 + 21 -y + 12 )/(r’y between 721/722 (r = 0), and 711/712 (r = ‘722), which is bound co). To guarantee that the policy line meets at least one of the reaction functions, and is thus attainable, it must either meet 2 R ( pi) when r = 00, or ) 2 ( 1 R p when r = 0. In both cases, the condition is 2711722 > 712721. Chapter 4. Competition under Recycling Regulations 55 Suppose the policy constraint is binding; the policy loses its meaning if the unregu lated 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(p ) 2 and R 2 (p1), respectively, at F’ 1 1 p — — — P2 2 1 p — — — = 2 1 7 2+ 2 ) and P 2 (pt, p’ = (p, p). 2 + (1 + r)c,-yi,7i r 12 —712721 + 2711722 + 7,,7 21 + 7 7 1 11 a,7 1 c 2 r + ci7 11 2c2’y,, + a r 1 , 2T 7 1 —712721 + 2711722 + 71 22 C 7 2 22 7 2 r r 22 2a,7 7 c 2 7 2 2 , 27l2’ , 21 27,,7 7 2 7, r 22 —721722 + r —(1 + 7 2 r)c 2 1 2 11 7 2 a r r 21 a,7 21 2-y,,72 7 2 7, r 2 721722 + r 1722 (4.68) — — — — 4.69 — 4.70 — 2 P2 — — — — 4.71 . — The above expressions show that p’ and p’ are functions of c 1 only, p, and p are functions of c . 2 Suppose the firms comply with the recycling policy. Then the regulated reaction function for firm 1 is the segment of the R,(p ) below P’ and the segment of the policy 2 line above F’. This is so because 1 ir (p1, P2) is strictly concave in p, and for any the best price for firm 1 is the P1 on the policy line that corresponds to P2. P2 > p Similarly, the regulated reaction function for firm 2 is the segment of R 2 (pa) above P 2 and the segment of the policy line below F . Since 2 2 r (p’, P2) is strictly concave in the most that firm 2 can earn is by setting P2 P2, then for any p <p on the policy line that corresponds to . 2 1 p,. The regulated reaction functions of the two firms meet along the segment P Thus any point on this segment is a Nash equilibrium. This result is summarised in the following proposition. It will be shown in the proof of the proposition that, indeed, at any point on P , each firm has no incentive to unilaterally deviate from it. 2 1 Chapter 4. Competition under Recycling Regulations P2 56 ) 2 Ri(p policy line ‘(P1) 2 p P1 Figure 4.6: Reaction Functions of the Two Firms and the Policy Chapter 4. Competition under Recycling Regulations 57 Proposition 27 Suppose the two firms comply with the recycling policy under the regula tion which mandates consumption of the recycled product, then each point on the segment of the policy line between P’ and F 2 is a Nash equilibrium. Proofs of propositions are given in Appendix D. Since there are infinitely many possi ble equilibria that satisfy the regulation, it is natural for the firms to identify an equilib rium, if it exists, from the set of equilibria which maximizes the profits of the two firms. 12721 If-y 711722, then the profits of the firms on the line segment P’P 2 increase towards , and P 2 F 2 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 regu lation which mandates consumption of the recycled product, if 712721 711722, then P 2 is the dominating equilibrium. However, when the condition given in the above proposition for P 2 to be the domi nating equilibrium is not satisfied, it not clear how to select a workable equilibrium. As the two firms are not allowed to collude, a chaotic market for the products may result from competition under the regulation. This leads to the use of tax as an alternative approach to induce compliance with the recycling policy. 4.3 Competition Under Taxes The analysis in the previous section assumed that the firms, competing to maximize their own profits, voluntarily observe the regulation that mandates consumption of the recycled 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 the Chapter 4. Competition under Recycling Regulations 58 recycling policy (4.67) and, at the same time, selecting an equilibrium out of the many possible ones. We assume that only the virgin product, or its producer, is taxed. The unregulated 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 in Proposition 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 not serve the purpose of policy compliance and equilibrium selection. We will consider a number of different tax schemes. One scheme is to tax the producer for the amount of its product consumed in excess of that required by the policy. This turns out to be a good tax system in the sense that a producer who observes the policy does not pay any tax. Another approach is to tax production which modifies the cost of the firm being taxed. Production tax is equivalent to sales tax which acts to change the demand of the product being taxed. Under progressive profit tax the cost of the firm as well as the demand for the taxed product are affected, and, in place of a reaction function, there are two best responses to the price set by the opponent. These will be looked into in more detail in the following. 4.3.1 Taxing Excessive Consumption of Virgin Product Under this tax with tax rate, 11, the profit function and the reaction function of firm 1 are given below: ,p 1 ir(p , 2 ) t ) 1 ,t 2 R(p ) 2 ci)qi(pi,p = (p1 = r(ai + — ciYll + — 2 t ( , 1 ) p q p 712P2) r 11 27 + (721 (p 2 _q , 1 ) f, p — 1 r)t + 711 (4.72) (4.73) Chapter 4. Competition under Recycling Regulations where (x)+ is the nonnegative value of x. When t’ = 59 0, the taxed reaction function, , 0), is just the reaction function, ) 2 R(p 2 ( 1 R p . An increase in the tax rate, t , shifts the taxed reaction function of firm 1 to the right, as shown in Figure 4.7, until t’ = ’t’ at 1 which the taxed reaction function of firm 1 meets with the reaction function of firm 2 at on the policy line. Then F 2 is the equilibrium satisfying the recycling policy, which is induced by the tax. The equilibrium prices as a function of the tax rate are 12 2 7 1 + r) 211(721722 (t 1 p ) + = 2 ’y2 11 r(47 + = (474) 712721) 11 2 7 + r) 11(72 1 ( 2 p ) t — r(47 2 7 11 2 (4.75) 712721) — Substituting the above set of equations for the equilibrium prices into the equation for the policy line, the tax rate that selects the regulated equilibrium is obtained. 11* = (27 2 7 1 [a r 11 2 — +C17n(712721r 7 a ( 1 r -yi 1 +2 721) — r 2 7 11 27 2 — — 2722) 721722) + r r+r 11 (7 21 + 27 7 12 )(—’y r 2 7 11 2+ c2722(7ii-yl2r — 712721 . + 2711722)] (4.76) 721722) In this tax scheme, no tax is collected from firm 1, so long as the equilibrium satisfies the recycling policy. Thus it is a kind of tax without taxing. The regulated equilibrium profit for firm 2 is r(p,p) which is given below together with that of firm 1. — — 7i 2 [a 72 1 + 12 ( 2 x[a r 7 — — {a 2 i 1 7 2 2 — + — 12 ( 2 c 21 ’y 7 l 2 aI7 — — — 711722)] 721722) 22 + r 7 1 2cv (7 2 c 2 7 12 2+ +r r 2 7 11 27 2 / 1r 7 i 2 2 + T 1 2 7 2 722) r 2 7 12 +ci(7 1 I 21 Li7 / iT 2 7 12 i7 + — 721722)], c ( 2 71272l L71 2 2 7 1 r — — 22 i 1 7 r 2 )] 7 721722) 72) (4.77) ( ) Chapter 4. Competition under Recycling Regulations 60 R(p tc*), , 2 P2 2 ( 1 R ) p 2 R(p , tl*) line P1 Figure 4.7: Reaction Functions of the Two Firms under Tax Chapter 4. 4.3.2 Competition under Recycling Regulations 61 Production Tax Let the tax rate in this system be t. The profit function and the reaction function of firm 1 under production tax are given below: ir(pi,p , 2 tc) R(p , 2 tc) = (P1 = 1 + a — 1 c — tc)(ai — C1711 + 712P2 2711 7nPi + + 712P2), (4.79) (4.80) With zero tax rate, the taxed reaction function for firm 1 is the reaction function, ) 2 ( 1 R p . The taxed reaction function shifts to the right with an increase in t. In Figure 4.7, the production tax rate, t, at which the taxed reaction function of firm 1 intersects with the reaction function of firm 2 and the policy line, yields the equilibrium P . The taxed 2 reaction function of firm 1 and the reaction function of firm 2 together give the following set of equilibrium prices as functions of the tax rate t. p(tc) 711722 2 tc = p+ (4.81) , 4 Y 11Y22 712721 711721 — p(tC) = p+ (4.82) 4711722 — 712721 Substituting the above prices into the expression for the policy line, t is obtained. trn” = 22 1 [a r 11 (27 y — r 1 1 +c 2 7 12 (7 1 x 721) — -y 2 cv r 12 (7 + 11 r 2 7 11 27 2 — 1 1r i 2 7 12 (—7 1+r 22 + 7 11 27 — 2722) 721722) 22 7 2 c r 12 -y 11 + (7 — 712721 + 2711722)] (4.83) . 721722) Comparing t-’ and t , we have the following proposition. 1 Proposition 29 The production tax rate, t, is higher than the tax rate for excessive consumption of virgin product, t’, for the regulated equilibrium, P . 2 The equilibrium profit for firm 2 under the production tax, obtained at F , 2 ir, is the same as the profit and thus equals ir in expression (4.78). The equilibrium profit under Chapter 4. Competition under Recycling Regulations 62 the production tax for firm 1, the virgin product producer, is given below: — 21 + c2Qy 7 1 i 711722)12 2 y 12 +a 721722)2 21 2 7 12 (7 172 1 27 r 711 r [a 2 i 1 7 2 2 — — 4. . — — Proposition 30 The equilibrium profit of the virgin product producer which satisfies the recycling policy under the production tax decreases with a higher recycling requirement. Proposition 31 Under the recycling policy and the production tax, the equilibrium profit of the virgin product producer is 2 /(-yiir 2 7 ) 2 times that of the recycler. Prop osition 32 Given the recycling policy, the virgin product producer earns lower equi librium profit under the production tax scheme than that obtained under the scheme of taxing excessive consumption of virgin product. 4.3.3 Progressive Profit Tax Let ti1 (p1, P2) be the tax rate in the progressive profit tax scheme. The profit of firm 1 under the tax scheme is given below: 8 , 1 ir(p , 2 ) p t = (p1 [1 —ci)(ai —7iipi — 1 ( 8 pi — 1 ci)(c +712p2) — (4.85) + 712p2)]. The first order condition of the profit function is: 81 43\ u l 1 iP1,P2, ) (Ii where G(pi,p ) 8 ,t 2 = 1 — = (ai = 0, 2t(pi — 11 y 1 +c ci)(cvi — 1 7 ) p , G(pi,p t 2 2 7 iiPi + 2 s ) (4.86) — 8 YiiPi + 712p2). If t = , 0) 2 0, then G(pi,p = 1, and the expressions in (4.86) give the unregulated reaction function for firm 1. When t > 0, the best responses, which is not unique in this case, for the p2 set by firm 2, Chapter 4. Competition under Recycling Regulations 63 is contained in , 2 , 1 G(p t) p = 0. The taxed reaction curve from 3 ,t = 0 in the 2 G(pi,p ) positive domain is shown in Figure 4.8. It intersects with the reaction function of firm 2 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 t 8 that correponds to the desired equilibrium on the policy line. The reaction curve slides down 2 (p as 1 R ) increases. On the reaction curve, the profit of firm 1, ir, is constant as shown by the following proposition. Proposition 33 The tax payment of the virgin product producer, , ) 2 tir(pi,p and its ,t8), both equal to l/(4ts) on the taxed reaction curve ) 2 profit, Tr(pi,p 8 , 2 G(pi,p t = 0. Proposition 34 Given the recycling policy, the virgin product producer earns less profit under the progressive profit tax scheme than the scheme of taxing excessive virgin product consumption. Finding the equilibrium prices, as functions of t, for the two firms becomes very messy indeed by solving ) 8 , 2 G(pi,p t = 0 and the reaction function of firm 2. Thus to find the value of t’ which selects the equilibrium F , substitute (p,p) into G(pi,p 2 ,t) = 0. 2 ,t’) is linear in t 2 Note that G(p,p . So given (pl,p2), t is uniquely determined. 8 ts* — (—721722 - ’y 1 7 i 2 r +2 H where H = 2 21 + -y 12 7 2 c 21 722(c2-y11 + a-y 11221 + 2a c r ’ynr 1 — — — 2 2-yll-y22r) ‘ 22 7 2 c )(—a2722 lly — 4 87 (.) -y + c 1 c 22 -y 21 ’-y + 2 a r 7 + 12 2 ’-y + r). 1 2c r 22 ’-y 11 -y 2 c 2 7 12 2 The profit of firm 2, is the same as It follows from Proposition (33) that the profit of firm 1 is 4(—721722 H 11 2 7 2 +r — 22 7 11 2y 2 r) (4.88) Chapter 4. Competition under Recycling Regulations P2 64 policy line 1 (P2) R (P1) Reaction Curve 2 p 1 p Pi Figure 4.8: Reaction Functions of the Two Firms under Progressive Tax Chapter 4. Competition under Recycling Regulations 65 Proposition 35 If the recycling policy is severe enough, then the virgin product producer earns 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 consid ered, 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, c , determines the equilibrium prices of the two 2 products under the taxes, while the cost of the virgin product, c , has no effect on the 1 prices. This is shown in expressions (4.70) and (4.71). An inspection of Figure 4.4 shows that an increase in the cost of the recycled product results in higher prices of both prod ucts. On the other hand, the more efficient the virgin product producer, that is, the lower is c , the further to the left is its reaction function. Then a higher tax rate is required to 1 attain the regulated equilibrium (p,p) for the production tax and for the tax scheme for excessive consumption of the virgin product. In the case of progressive tax, Figure 4.8 suggests that a bigger value of the marginal tax rate, (c) The price effects, 721 and 722, I?”, is required. on the recycled product affect the equilibrium prices as Figure 4.4 exhibits; those on the virgin product do not. A decrease in the own price effect, 722, an increase in the cross price effect, 721, raises the equilibrium prices. On the other hand, the reaction function of the virgin product producer is shifted to the left by an increase in the own price effect, y, and a decrease in the cross price effect, Consequently, the tax rates t” and t, and the marginal tax rate t’ increase. 712. Chapter 4. Competition under Recycling Regulations 66 (d) The profit of the recycler does not change under the various tax schemes consid ered, because it is not taxed by these schemes. The following proposition shows that the recycling policy is to the advantage of the recycler. Proposition 36 In the tax schemes considered, the more severe is the policy, the higher is the profit of the recycled product producer. 4.4 Conclusion When the recycling regulation mandates consumption of the recycled product as a certain proportion of the total consumption, in a market where the recycled and virgin products are partially substitutable, there are infinitely many possible equilibria of price competi tion between the two producers that satisfy the recycling requirement. 1f 712721 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 compe tition between the firms may become chaotic, unless an equilibrium selection mechanism is introduced. A suitable tax scheme chosen by the regulator can play the dual roles of compliance inducement for the recycling policy and equilibrium selection. The same equilibrium, through taxing the virgin product or its producer, under the different schemes of produc tion 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 the severity 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 recy cling policy under production tax. The virgin product producer earns highest profit in the scheme of taxing excessive consumption of the virgin product, and when the policy Chapter 4. Competition under Recycling Regulations 67 is severe enough, its profit is higher under progressive profit tax than under production tax. An interesting feature of the progressive profit tax is that the profit of the firm being taxed equals its tax payment which is a quarter of the inverse of the marginal tax rate. The equilibrium regulated prices of both products increase with (i) the recycling requirement of the policy, (ii) the cost of the recycled product, (iii) the effect of the price of the virgin product on the recycled product, and (iv) a lower own price effect of the recycled product. The tax rate required to induce compliance with the policy is higher for the production tax scheme than the scheme of taxing excessive consumption of the virgin product. In the progressive profit tax scheme, 8(lri) 1 = 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, are required to attain the regulated equilibrium if (i) the virgin product producer is more efficient, (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 recycled product. It is interesting to find that in equilibrium, taxing excessive consumption of the virgin product does not collect any tax payment, and the tax rate required is lower than 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 A Proofs for Chapter 1. Proof of Lemma 1. Consider the domain on which the following profit function is non negative. H ( 1 p, q, t ) := 3 1 (—/3c + p + 0 — 1 (q) c — c(q))F(p, q). The first factor on the right hand side is concave in (p, q). The second factor is concave by assumption. Thus 1 ) is pseudoconcave in (p, q) (Mangasarian 1970.) It follows 1 H ( p, q, t that H°(p, q) is also pseudoconcave. 0 Proof of Propostion 1. ) 1 q, t H(p,q,ti) = F(p, q) + (p = 0. = 2 F(p,q)(—cj’( ’ (q))+ q)—(tj)c (p = (A.89)xFq(p,q) - — 3 /3(t c ) 1 — 1 (q) c — cr(tj )c(q) + 0(t 1 ))F(p, q) (A.89) — c 1 8 3(tj) — 1 (q) c — o(t’ )c(q) + 0(t 1 ))Fq(p, q) 0. (A.90) (A.90)xF(p,q) gives F(p,q)F(p,q) + (F(p,q)ci’(q) + cr(ti)F(p,q)c (q))Fp(p,q)Fq(p,q) + 1 2 (cj’(q)+c(tj)c’(q))F(p,q) = 0. 68 Appendix A. Proofs for Chapter 1. 69 Simplifying, we have cj’(q) + a(ti)c’(q) = S(p,q) -: > Define the revenue function, with q fixed, as R(p, q) 0. D (A.91) pF(p, q). Then differentiating and re-arranging terms to obtain the marginal revenue MR(p,q) +p. = (A.92) Proof of Proposition 2. From (A.89) above, given q we have F(p,q) ). 1 (q)+a(t c ) 1 ( q) c —0(t +p= cg(tj)+ 2 p(p, q) The left hand side is marginal revenue, and the right hand side is marginal total cost. 0 Proof of Proposition 3. From (A.90) above, given p, we have F(p,q) ’(q) 1 1 (c ) +a(t ( t 2 ) (q) +a(ti)c(q) —0(t 1 q))c(t ) =p. 1 c+ c q(p,q) The left hand side is marginal total cost. 0 To 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. — p(p,q)— The result follows. 0 —F(p, q)Fpq(p, q) + Fq(p, q)F(p, q) P1 1\2 i p, q ( . ) Appendix A. Proofs for Chapter 1. 70 Lemma 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. — Sq(p,q)_ The result follows. —Fp(p,q)Fqq(p,q) + Fq(p,q)Fpq(p,q) F(p, q) 2 (A.94) 0 Proof of Proposition 4. H(p,q,O,a,/3) = 3 +p+0 (—/3c H(p,q,8,a,/3) = 3 F(p,q)+ (—/3c H(p, q, 0, a, /3) = 2F(p,q)+(—f3cs +p+O— cj(q) —ac (q))F(p,q). 2 Hpq(p, q, 0, a, /3) = Fq(p, q) — ac ( 2 q))F(p, q). (q) —ac 1 (q))F(p,q). 2 +p+O— c — H ( 9 p, q, 8, a, /3) = F(p,q). H ( 0 p, q, 0, a, /3) = —csF(p,q). q, 0, a, /3) = —c(q)F ( 2 p,q). Hq(p, q, 0, a, /3) = F(p,q)(—ci’(q) 3 +p+0 (—/3c = — (ci ‘(q) + ac’(q))F(p, q) + (—/ c 3 .g + p + 0 Hqq(p, q, 0, a, /3) ci (q) — — — 1 (q) c — ac2(q))Fpq(p, q). ac’(q)) + ci (q) — ac2(q))Fq(p, q). F(p,q)(—cj”(q)—ac”(q))+2(—ci’(q)—ac’(q))Fq(p,q)+ c +p+8 3 (—/ Hqs(p, q, 8, a, /3) = Fq(p,q). Hqfl(p, q, 0, a, /3) = —csFq(p,q). H(p, q, 0, a, i3) = —F(p,q)c ’ 2 (q) — — ci (q) — cxc2(q))Fqq(p, q). cr(q)Fq(p,q). To sign the following expressions, we use the following properties: (a) the quality costs Appendix A. Proofs for Chapter 1. 71 are 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) pg = = 0. (q) 1 1 k{F(p,q)(c 1 + cc”(q))F(p,q) + Fq(p,q)[Fq(p,q) + (cj’(q) + cc’(q))F(p,q)] + 8 +p + 0— c (—/3c (q) 1 qe = = cxc(q))(—Fqq(p,q)Fp(p,q) + Fq(p,q)Fpq(p,q))}. k{—Fp(p,q)[Fq(p,q)+ (cj’(q)+ac’(q))F(p,q)] + 3 +p + 0— c (—/c (q) 1 Fg(p,q) — — ac(q))(Fp(p,q)Fpq(p,q) (q) 1 1 k{F(p,q)(c 1 2 ”(q))F(p,q) + (—3cs +cc — Fq(p,q)Fpp(p,q))}. +p+O — ci(q) —cc ( 2 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) qgO; and if S(p,q) p = 0 and Sq(p,q) 0, then pg < 0; if S(p, q)0, then 0 then Fg(p,q) >0. (q)F(p,q)(ci”(q) + (q))F(p,q) 2 k{—c 11 2 ac — c2(q)Fq(p, q)[Fq(p, q) + (cj’(q) + cc’(q))F(p, q)] + (q)[—Fq(p,q) + (c 1 2 F(p,q)c ’(q))F(p,q) 2 ’(q) + ac 1 3 +p + 0— c (—/3c (q) 1 — crc2(q))Fpq(p,q)] + (q)(—I3c +p + 0— cj(q) 2 c 3 q = — crc(q))(Fqq(p,q)Fp(p,q) 3 ( 2 c c q)(—/ ,g +p+ 0— c (q) 1 = — Fq(p,q)Fpq(p,q))}. (q)Fp(p,q)[Fq(p,q)+ 1 2 k{c (c ( q) +ac ’(q))F(p,q)] + 2 F(p, q)c’(q) [2F(p, q) + 3 (—/3c + p + 0 Fa(p,q) — — — 1 (q) c — ac(q))F(p, q)] oc2(q))(Fp(p,q)Fpq(p,q) — (p,q)F(p,q))}. 9 F (q)F(p,q)(cj”(q) +ac”(q))F(p,q) 2 k{—(c )+ 2 F(p,q)c’(q)Fp(p,q)[Fq(p,q) + (ci’(q)+cvc (q))F(p,q)] + 1 2 3 +p+0 (—/3c — ej (q) — (q)) 2 crc — 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)c ’(q)(—F(p, q)Fpq(p, q) + Fq(p, q)F(p, q))J}. 2 IfS(p,q) <0, then q, <0; and if S(p,q) 0 and Sq(p,q) 0, then Fa(p,q) <0. The sign of p , cannot be determined without making further assumptions. 0 = Fq(p,q)[Fq(p,q) + (c(q) + ac(q))F(p,q)1 3 k{—c — F(p, q)(c(q) + cxc’(q))F(p, q) + 3 c (p 3 c q = = ci(q) — aci(q) + 0— /3c )(Fp(p,q)Fqq(p,q) 3 — Fq(p,q)Fpq(p,q))}. Fp(p,q)[Fq(p,q) + (c(q)+ac(q)]F(p,q))+ 3 k{c c3(p (p,q) 3 F, — — ci(q) — (p—ci(q) 3 k{c cxc ( 2 q) + 0 — )(—F(p, q)Fpq(p, q) + Fq(p, q)F(p, q))}. 3 3c (q)+0—/3c 2 —cc ) 3 [Fq(p,q)(Fq(p,q)Fpp(p,q) F(p, q)(F(p, q)Fqq(p, q) — — Fp(p,q)Fpq(p,q)) + Fq(p, q)Fpq(p, q))] — caF(p, q)(c(q) + ac(q))}. 2 F(p, q) From expression (A.94) of Lemma 6, if Sq(p, q) q O 1 ; and if S(p, q) 0, then p 3 > 0; if S(p, q)0, then 1 0 and Sq(p, q) <0 then Fs(p, q) <0. Proof of Proposition 5. Just consider the following condition in the results of Propo sition 5 to obtain the results in this proposition: Fq(p, q) 0. 0 Proof of Proposition 6. (p,q,ti) 1 Ht = (—c(q)cx’(tj) ) 1 Hqti (p, q, t = —F(p, q)a’(tj 2 )c ( 1 q) + (—c (q)cV(ti) 2 — 8’(tj) 3 c 1 + 0’(tj))F(p,q). — /3’(t + 0’(t 3 c ) 1 1 ))Fq(p, q). Appendix A. Proofs for Chapter 1. 1 pj = 73 k{—F(p, q)a’ (ti)c’ (q)[Fq(p, q) 2 (p — c ( 1 q) — cx(ti)c ( 2 q) + 6(t ) 1 (q)cx’(ti) + 3 2 (c c 3 ’(ti) — — — (c(q) + a(ti)c(q))F(p, q) + /3 (tl)c3)Fpq(p, q)] + 0’ (ti))[—F(p, q)F(p, q)(c’ (q) + 2 1 a(ti)c’ ’ (q)) — Fq(p, q)(Fq(p, q) + (c(q) + 2 a(ti)c’(q))F ( p, q)) + (p — c ( 1 q) — cx(ti)c ( 2 q) + 6(t ) 1 (F(p, q)Fqq(p, q) 1 qt = — — 3 ) 1 /3(t ) c Fq(p, q)Fpq(p, q))]}. k{F(p,q)a’(ti}4(q)[2F(p,q) + F(p, q)(p (p — c ( 1 q) — — ci(q) — a(ti)c ( 2 q) + 6(t ) 1 a(ti)c ( 2 q) + 0(t ) 1 — — 3 ) 1 /3(t ) c ]+ 2 ) 3 /3(ti)c a /3’(ti) 3 ’(ti) (c + c — 6’(t ) 1 ) (—F(p, q)Fpq(p, q) + Fq(p, q)F(p, q))}. Ft ( 1 p,q) = 2 k{—F(p,q)F ( c’’(q) (p,q) + a(ti)c’’(q))(c (q)cr’(ti) + c 2 /3’(ti) 3 (p — ci(q) — o(ti)c ( 2 q) + 6(t ) 1 [F(p, q)(F(p, q)Fqq(p, q) Fq(Fq(p, q)F(p, q) (q) (p 2 F(p, q)ü’(ti)c’ (F(p, q)Fpq(p, q) — — — — — 2 ) 3 /3(ti)c ( q)cr’(ti) (c + c /3’(ti) 3 0’(t ) 1 )+ — 6’(t ) 1 ) Fq(p, q)Fpq(p, q)) + F(p, q)Fpq(p, q))] c ( 1 q) — — — a(ti)c ( 2 q) + 6(t ) 1 — ) 3 /3(ti)c Fq(p,q)Fpp(p,q))}. Again using the conditions that, (a) the quality costs are strictly convex, (b) F(p, q) <0 and Fq(p,q) > 0, and (c) from (A.91), Fq(p,q) + (c(q) + crc (q))F(p,q) 2 = 0, and ex pression (A.93), that (i) if S(p, q) < 0, then qt, > 0; (ii) in addition, from (A.94), if Sq(p, q) 0, then 1 F ( p, q) > 0; and (iii) the sign of pj 1 cannot be determined with out further assumptions. Again, if the demand is a function of price alone, then using Fq(p, q) 0 in the above expresions, Pt 1 0, q, 1 0, and 1 F (p, q) 0 are obtained. 0 Appendix A. Proofs for Chapter 1. 74 Proof of Lemma 2. Straight forward differentiation of the objective function (1.2) with respect to t 1 and setting to zero gives the required result. D Proof of Proposition 7. A higher investment cost function relative to 1(t) is (I + j) (t), ij (t) > 1(t) = 0, Vt E [0, T], a lower investment cost function relative to 1(t) is sim ilarly defined with (t) 0, Vt E [0, T]. We use the extremal condition (1.24) for the < technology adoption time, t , in Lemma 2, and consider a variation, in the direction of 1 (t), to the investment cost, 1(t). We assume the extremal (p, q), obtained for 1(t), does not change with the variation, because, otherwise, the problem would become intractable. The extremal condition for (F) with respect to t 1 can be written in the following form. rI(ti) — ) + H°(po,qo) 1 I’(t where K(t ) 1 = — ) 1 K(1 = 0, (A.95) H’(pi, q, t ) + 1 1 )q (a’(t ( 2 ) c — )+6 1 O’(t 1 ’(ti)ca)F(p, q) 1 — e7’(tl_T) r Define w(ti) as in the following, = dK(t ) 1 1 dt = (e_rTt1 w( ) 1 . eTtl —rT — 1)F(pi,qi)(— ( qi)a’(ti) c +2 1 F(pi,qi)(—c ( 2 qi)”(ti) r — — fc 3 l’(ti) + O’(ti)) + fc 3 3”(ti) + O”(t )) 1 (A.96) >0. Now allowing a variation in I in the direction of in equation (A.95), and note that s is a positive real number. 0 = rI(ti + dt ) 1 = r(I + 7 si ) (ti + dt ) 1 = r[(I + .s)(ti) + (I’ + sq’)(ti)dtij — 1 + dt I’(t ) + H°(po, qo) 1 — — ), 1 1 + dt K(t 1 + dt (I’ + s’)(t ) + H°(pi, qi) 1 — — ), 1 1 + dt K(t (I’ + 1 87 ’ )(ti) + (I” + 1 si”)(ti)dt + ) Appendix A. Proofs for Chapter 1. H°(po, qo) — 75 1+1 K(t dt + (dt ) ), 1 rI(ti) + rsr(ti) + rI’(ti)dti = H°(po, qo) — — (f(dti) ) 1 I’(t sij’(ti) — 1+1 K(t dt + (dt ) ), 1 —+ — ) 1 ((dt 1 0 as dt —* 0); 1+ I”(ti)dt —* 1 0 as dt —÷ 0). Substract the above expression by equation (A.95) and divide the resulting expression with s to obtain the following, (w(ti) — rI’(ti) + I”(t )) 1 = r7)(ti) — )+ 1 ’(t 7) ) 1 t (A.97) Letting s J 0, (w(ti) (w(ti) (i) If 77> — — )+1 1 rI’(t ))Dt I”(t ( I) = r7)(ti) — ), 1 ’(t 7) > 0; arid (A.98) rI’(tj) + 1 ))Dt I”(t ( I) = r7)(ti) — (t 7 i ) 1 , < 0. (A.99) 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 Dt (I) 1 0; > (b) for the other cases of I, Dt (I) has the same sign as the left hand side of (A.98). 1 (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 Dt (I) 1 < 0; (b) for the other cases of I, Dt (I) has the opposite sign of the left hand side of (A.99). 1 0 Proof of Proposition 8. The preceding proposition does not hold in general for 0, or 0 and <0. We need to restrict 77 > 0 and i to some neighbourhoods of I. Define U(I, 6) as a weak upper 8-neighbourhood of I, 6 R, as the set of functions I which ii’> i < iii’ do not cross I from above, i.e. 1(t) > 1(t), Vt e II’ Ill + lI’ I’ll — — [0,T], such that < 8, (A.l00) Appendix A. Proofs for Chapter 1. where IL I 76 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 + ), 1 (i) Suppose I E U+(I,8 ‘(t) < 6 — > ?7’ <S. (A.lOl) 0 and q’ > 0. From expression (A.101), we have (t),Vt E [0,T], which implies r(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) follow the same proof of the preceding proposition. (ii) Suppose I ‘(t) > —6 + U(I,6 ) 2 , ii(t), < 0 and ‘ < 0. From expression (A.101), we have Vt E [0, Tj, which implies r(t)—7(t) < 2 (r+l)(t)+6 < 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. 0 Proof of Proposition 9. Using the Implicit Function Theorem, differentiate the extremal t with respect to T in (A.95) to obtain the following, 0 = , 1 dt rI 1 (t ) — 1 dt (t ) 1 , T) dt 1 OK(t 1 — , T) 1 äK(t — (A.102) Appendix A. Proofs for Chapter 1. 77 where K(t , T) is given in (A.95) with T made explicit. Then simplifying the above 1 expression and rearranging terms we have er(u1_T)(c ( 2 qi)a(ti) /3’(ti) 3 +c — O’(t ) 1 ) = (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, the opposite sign of rI’(ti) — ) 1 I”(t — w(ti)). 0 has Appendix B Proofs for Chapter 2. In the following proofs the equilibrium prices will be given with no asterisks, as it will be clear from the context. Proof of Proposition 10. The reaction functions of the two firms with the pre-legislation linear demand function, Qo(p) pi(Th) = 0 a — 0 cr = w3 + (1 w)3oci 2 p 0 0 2(1 w)/3 (1 w)/3 1 + w/3 p 0 2 c 0 0 2w/3 — — 0 a p2(p1) — = — — (B.104) Figure B.9 shows the graphical representation of the reaction functions, R 1 and R , of 2 firm 1 and firm 2. The intercepts are: — P1(O) P2(O) Assuming (1 — — = 0 + (1 w)fJoci a 2(1 — w)f3 0 0 + w/ a 2 c 0 2w/% —‘ — P2 1 0 + (1 a — (O)__ (0) — 1 c 0 w)/3 0 3 wf w/3 2 c 0 ao + (1 — w)/3 0 w)cj 4 wc , otherwise the two reaction functions overlap. There are four 2 cases to consider: (i) p’(0) > 1 (p 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) 1 p’(O),pj ( O) > P2(O) implies P’ > °,P2 <0. p ( O) > 1 Thus case (i) is the only feasible equilibrium and uniqueness follows. 78 D Appendix B. Proofs for Chapter 2. 79 Proof of Lemma 3 and Proposition 13. From (2.26), (2.27) and (2.29) we have 2 ( 1 7r ) pi,p = (p1 (pi,p 2 ir ) = (p2 where F(w, P1, p2) (1 = — — — 2 ) c ( 1 ) pi,p Q = (pi — ) c ( 2 ) pi,p Q = (p2 — ci)(1 — 2 w)Qo(P(w,pi ) ). ,p )wQo(P(w,pi, 2 c ) ). p w)p + wp2. Differentiating the above expressions to obtain Qo(F(w,pi,p ) 2 ) + (p1 — ci) 2 ( 0 ÔQ , 1 ) F(w,p p ) 1 Up 2 , 1 Qo(P(w,p ) p ) + (1 w)(pi 2 ci)Q’(P(w,pi ) ) ,p (P(w, Pi, P2)) 0 ÔQ )) + (p2 c 2 Qo(P(w,pi,p ) 2 up2 = 2 Qo(P(w,pi,p ) ) + w(p 2 2 )Q(F(w,pi,p c ) ) = 0. = — — = 0. (B.105) — (B.106) — The equilibrium prices, Pi and P2, that solve the above equations can be thought of as a function of w, the parameter that represents the tightness of the regulation. To reduce the burden of symbols, we will suppress the argument w of the equilibrium prices throughout the proofs. Differentiate pi and P2 in the expressions with respect to w to obtain the following, Q(P(w, P1, p2)) +(1 — 1 w)(p Pi, 2 +w(p dP(w,pi, p2) + (1 dw — c ) 1 0 — w)Q(P(w, P1, ,P1,p2)) — ))dP(WPl p2) (p1 — dw 2 , 1 ci)Q(P(w,p ) p ) = O(B.107) + wQ(P(w, p1,p2)) + — (p2 — ) 2 c ( 0 ) P(w,pi,p Q’ ) = 0. (B.108) Substitute (B.109) into (B.107) and (B.108) to obtain (B.110) and (B.111) below. ) 2 dP(w,pi,p = —Pi +P2 + (1 — + 2 dp (B.109) Appendix B. Proofs for Chapter 2. (2p — P2 — 80 2 ci)Q’(P(w,pi ) ) ,p 2 dp 2 ( 0 —wQ ) P(w,pi )—— ,p (p1 — 2p — — (1 — — )Q’(P(w,pi,p 2 c ) ) 2 dp 2 ( 0 —2wQ ) P(w,pi )-—— ,p — 2(1 w)(pi (1 2 w(p — w)Q(P(w,pi,p ) 2 ) — — — dQ(P(w,pi,p ) 2 ) dw = 0. (B.110) 2 , 1 w)Q(F(w,p ) p ) ) 2 c — ci) dQ(P(w,pi,p ) 2 ) dw = O• (B.111) Solving the above set of equations, we have dp 1 dw 42 dw 1 c (3pi 2c ) 2 2 ( 0 dQ’ ) P(w,pi,p ) (p1 ci) 3(1 w) (P(w,p 0 3Q’ , 1 ) 2 p ) dw 2 ci) 2c )) 2 dQ(P(w,pi,p ) 2 c p2 3 ( (p2 + 3w 2 ( 0 3Q’ ) F(w,pi,p ) dw — — — — — — — — — (B.112) — — — (B.113) Substituting (B.112) and (B.113) back into (B.109) to get (B.114), and thus Proposition 13, as follows: dP(w,pi,p ) 2 dw — — 2 — 1 (c ) c 3 (1 w)(pi — — — ci) + w(p 2 3 — )2 2 c Q’(P(w,pi,p ) ) dF(w,pi,p 114) (P(w,pi,p 0 Q’ ) 2 ) dw Therefore, dP(w,pi,p ) 2 dw — —Q(P(w,pi,p ) 2 )(ci c ) 2 )) + [(1 w)(p 2 3Q(P(w,pi,p 1 ci) + w(p 2 2 )]Q’’(P(w,pi,p c ) ) ))(ci c 2 Q(P(w,pi,p ) 2 (B 115) ) 2 wG(w,pi,p — = — — — Then put (B.115) into (B.112) and (B.113) to obtain Lemma 3 in the following: dp 1 dw 2 dp dw 2c 2 1 c )Qg(P(w,pi,p c ) ) (p’ ci)(ci 2 + w) 3(1 ) 2 3wG(w,pi,p 2 2c 1 c )(ci 2 2 c )Qg(P(w,pi,p c ) ) P2 3 (p2 + 3w 3wG(w,pi,p ) 2 — — — — — — — — — Proof of Proposition 11. The results follow from Lemma 3, since G(w,pi,p ) is negative 2 81 Appendix B. Proofs for Chapter 2. with Qo(.) concave. in Lemma (1, subtract by 3 ma Lem in ‘w ly ltip Mu 12. n itio pos dw Pro of of Pro D (B.106) to get the required result. 3. Then use equations (B.105) and ‘ of Lemma 3. Proof of Proposition 13. See Proof we first obtain (B.116). Proof of Lemma 4. Using (B.115) dQo(P(w, p, pa)) — dP(w, Pi, p2) Q(P(w, P1, p2)) dw — dw ) 2 j,p —c ,pi ) (w Q(Pc ) ( 2 wG(w,pi,p ) 2 ,pi,p ) Qi(P(w ) But 2 = ,pi,p (P(w) 2 Q (P(w,pi,p and ) w)Qo) 2 (1— ) = (B.116) ,p , (w,pi ) o(P 2 wQ ) owing: together with (B.116), give the foll ,p ) Q p P(w 2 d ( , 1 ) dw ,pi,p ) o(P(w 2 dQ ) dw — ci (1 w)( w,pi,p + ) —[Qo(P( ) 2 wG(w,pi,p ) 2 ,pi,p ) dQo(P(w ) 2 w j,p + ) w,p Qo(P( ) 2 ciw ) 2 — c (w,pi,p (P— Qo) ) 2 wG(w,pi,p ) 2 (w,pi,p + (1 —Qo(P) ) 2 — w) — = ,pi,p ) Q P(w d ( 2 ) dw = = follow from Lemma 4. Proof of Proposition 14. The results Proof of Proposition 15. Define — 1—w 3 — j 1 —--p’, c 1—w — , -c —— 1 . D D , p (w 2 P , 3 1 and ) = 3 + p2). 1 wQ s are as follows, The objective functions of the two firm = r p ip ( 2 , 1 ) = p 2 w( — w, o(P(. ) p )Q 2 êi , 3 1 ) (B.117) — w, P(. ) p Qo( 2 ) c , 3 1 ) (B.118) 82 2. roofs for Chapter Appendix B. P 2 (P1,P2) 1 (13i ,P2) and ir erentiate ir ff di s, m fir o tw for the action functions To obtain the re pressions. the following ex in n ow sh as and P2 respectively with Pi (w,i,p (w, (P} p OQo(P) ) 2 Qo) , 1 3 ) +2 ,p 3i OiriQ ) 2 (BJ19) — p o1 — — 0. r i,p h tp ( ) 2 2 ‘ P 9 [(p 2 w — ) 2 c (w,i,p ,i,p (wj ) Qo(P) o(P 2 O ) 2 Q +) 2 0 P (B.120) — = 0. ip. llowing relationsh fo e th y pl im s of equation The above system P )Q((P 1 C (w,j3 ) (P p c9Qo , 1 ) 2 = (p2 2 P 0 ci) (y3i (B.121) — — ion, wing differentiat llo fo e th rm fo er P Oj3 w, P1,p2)) P(Q O ( 0 = 3i,p P(w,j. wQ’() ) 2 = 5i,p (w,j. ) Q,(P 2 w ) (8.122) (B .123) P2 8 imply (B.123) together d an ) 22 .1 (B ), Equations (B.121 1—w . 2 c (pi c) = P2 (B.124) — — 1L iuna prices then the equilibr , m iu br li ui eq 20) admit an its. (B.119) and (B.1 if , ar ul ic have equal prof rt s pa m fir In o tw e th (B.118) that from (B.117) and s w llo fo It ). 24 satisfy (B.1 D satisfy uilibrium price eq e th , se ca e iv non-cooperat we on 16. For the ti si po ro P nging trins of ra ar of re Pro d an s on ti e two equa 06). Adding thes .1 (B d an ) 05 .1 equations (B have (w,pi,p 2 P ) — (w,cj,c 2 P ) = — w, Pi, P2)) Q F( 0 2 ( ,pi,p ) Q(P(w ) 2 (B.125) . 83 Appendix B. Proofs for Chapter 2. two firms collude to act as a monopolist. Now consider the cooperative case in which the w. The profit function of a monopolist is given belo ir(p,w) = (B.126) . ci,c w,p) o( P( )Q 2 (p— ) derived from m satisfy the following first order condition The monopolist bundle price tp (B.126). p m (m Oj’ m\ ‘o’p j (B.127) , w,ci,c P(— = 9 — ) 2 1 , show that the monopolist bundle price, We next compare (B.125) and (B.127), and equilibrium bundle price, P(w, Pi, p2). First tm cannot be higher than the competitive p , ,p . P(w,pi ) tm > 2 p Secondly, suppose p . P(w,pi, ) 2 tm (B.125) and (B.127) show that p m) (p Q /Qp ))/Q(P(w, Pi ,p2)). 2 _ ( 0 ) m —Qo(P(w, P’, p Log-concavity of Q(.) implies that ,p D . <P(w,pi ) tm 2 osition. Therefore p Then (B.125) and (B.127) contradict the supp (w) 1 7r Proof of Proposition 17. From Proposition 15, i = = r E (0, 1). Then for w),Vw 2 (7 1,2, 1 d d’r(w) dw = = — (w,pi,p 2 [(P ) — dP(w,pi, p2) ))Qo(F(w,pi, p2))] 2 ,c 1 P(w, c ’ ,p2)) w,pi )Q P( ) 2 c ( ,0 1 P(w, c — ) 2 dP(w,ci, c — — — — — (B.128) )Qo(P(w,pi ,p2))] ) 2 c c i,p i i, ,p ) ,c (w c (P (w )Q P ) ) 2 ( 2wG(w,pi ,p2) ) 2 ,p c c ( (w p ) Q(P , ) 2 ,pi,p 1 ) Qo(P(w ) 2 ] —c ) 1 2 —(c pi,p (w, wG ) 2 2 p pi, c ) Q(P(w, — ) ) —w)(p —ci)(ci 2 (1 1 ,pi,p 2 (P(w ) — dF(w,pi, p2) (w,pi,p 2 wG ) (B.129) Appendix B. Proofs for Chapter 2. 84 +WG(WP ) 2 P 1 (cl (1 — 1 w)(p — — ))[Q(F(w,pi,p c , 1 ) 2 ) Qo(P(w p ) ,p + 2 ’ 0 ci)Q’ ) (F(w,pi,p )J. 0 Proof of Proposition 18. Compare the second term of (B.128) and the second term of the last expression in the preceding proof, the result follows. 0 To prove Proposition 19, we will use the following lemma. Lemma 7 Define decreasing, then P(.) P’(q) = Q ( 1 .) and let Qo(•) be continuously differentiable. If Qo(.) is = Proof: From the identity QQo(p) = p we have -P(Qo(p)) = P’(Qo(p)) = = P’(Qo(p))Q(p) = 1. 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 compete in terms of output (see, for example, Friedman 1983, chapter 2, for a theory of Cournot competition.) Let qi and q be the outputs of firm 1 and firm 2. The profit functions of the two firms are 2 ( 1 ) qi,q = (P(qi+q ) 2 —ci)qi. (q 2 r , 1 ) q = )q 2 (P(ql+q2)—c . Appendix B. Proofs for Chapter 2. 85 The first order conditions, and thus the reaction functions, for the two firms are given below. 2 ( 1 thr ) qi,q = P(qi + q) = P(qi +q2) 1 — (q,q 2 8* ) Let 3 and (i’, ) 2+ c — be the Cournot equilibrium 1+ + P’(q q)qi = P’(qi +q2)q2 price 0. 0. and quantities. Adding the above two first order conditions, and using Lemma 7, the following expression is obtained. 2j3 From c + +1 = . 2 c (B.130) equation (B.125), the equilibrium bundle price under recycling regulation satisfies the following relationship. ) 2 P(w,pi,p = +P(w,ci,c ) 2 . (B.131) We il1 show that j3 cannot be higher than 2 P(w,pi,p ) . First suppose j3= 2 P(w,pi,p ) , then (B.130) and (B.131) implies 2 ) 2 +c = — 1 -P(w, c ,c 1 ). 2 Without loss of generality, assume c 1 > c , then 2 1 p—c = 1 —ci—P(w,ci,c 2 (2c ) <0. This suggests that the firm with higher unit cost earns, in Cournot equilibrium, negative profit. Thus J 3 P(w,pi,p ) 2 . Next, suppose j3 > P(w,pi,p ). Log-concavity of Qo(•) 2 implies —Qo(j3)/QQ3) —Qo(P(w,p ) 2 , 1 ) )/Q(F(w,p p ). This gives j3 < 1 ) 2 +c — 2 , 1 P(w,c ) }. c Appendix B. Proofs for Chapter 2. 86 Applying the same argument above will show that the firm with higher unit cost earns negative Cournot equilibrium profit. Therefore J 3 < P(w, p1, p2). Proof of Proposition 20. 2 ,p 1 (P(w,p ) from Proposition 13. P(w,pj,p ) 2 , and the result follows = — D 0 Proof of Proposition 21. The bundle price, p t, m obtained by cooperation under regulation satisfies equation (B.127). The Cournot equilibrium price, j3, satisfies equation (B.130). Suppose J3 = pm, and firm 1 has higher unit cost, then j3 — 1 c = 2 c — P(w, c ,c 1 ) < 0 is 2 obtained from (B.127) and (B.130). Thus the firm with higher unit cost earns negative Cournot equilibrium profit. So 1 t. p m Now suppose j3 > p t, m and firm 1 has higher unit Q ( 0 pm)/Q(pm). From (B.127) cost. Log-concavity of Qo(•) implies —Qo(j3)/QQ3) $ and (B.130), it can be shown that J3 — 2 1 < c c — P(w, ci, c ) < 0. Again the firm which 2 has higher unit cost earns negative profit. Therefore, 5 <pm. Proof of Proposition 22. The inverse demand function for Qo(p) (ao — 1i = 0 a — / o 3 p is P(q) q)/f o 3 . The Cournot equilibrium profits (see, for example, Friedman 1983, chapter 2) for the two firms are: (ao + c 0 2 2 ) 1 c 0 2/3 9/3 — 1 K (ao + /3 1 c 0 2 K — ) 2 c 0 2/3 9/3 The equilibrium profits for the two firms under regulation, iri(w) and K (w), are given in 2 (2.41). iri(w) (ao — 1 K = = — (1 — w)floci — ) 2 c 0 w/3 (ao + f3 2 c 0 — 2 ) 1 c 0 2/3 — (1 +W)(Ci -c2)[2(ao-/3ocl) -(1 -w)/3(c C2)] (B.132) Appendix B. Proofs for Chapter 2. (w)—ir 2 ir (ao — (1 87 — 1 c 0 w)/ — ) 2 c 0 w13 = (ao 1 c 0 + /3 — ) 2 c 0 2/3 — _(2 = — w)(ci—c ) 2 [2(ao ,Ifci=c 2 thenI=7r(w);i=1,2. — ) + wo(ci—c 2 oc )] 2 (B.133) D Proof of Proposition 23. (i) Suppose c 1 < c , (a) from expression (B.132) above, 2 iri(w) < ) 1 .9oc i; (b)if 2(ao — — wfio(ci — c ) 2 , then the right hand side of (B.133) and . (w);ir 2 ir Proof of (ii) is similar to part (i) above. 0 Proof of Proposition 24. Let the unit costs of the two firms before the regulation be 1 c = 2 c = c, and after the regulation be c 2 = c. Then equations (B.130) and (B.131) obtained, respectively for Cournot equilibrium and equilibrium under regulation become p ) 2 P(w,pi,p First supposej3 = = — = Qo(j3) ,.+c. (p) 0 2Q iB.134 ). 2 + P(w,c,c )) 2 ( (w,pi,p 0 Q (B.135) / P(w,pi,p ) 2 , then (B.134) and (B.135) give3—c = (c—F(w,c,c ) 2 ) 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 (B.134) and (B.135) give j3 — c < (c — P(w, c, c )) 2 Cournot equilibrium profits. Thus j3 < P(w,pi,p ). 2 Again 0, and both firms make negative 0 Appendix B. Proofs for Chapter 2. 88 P2 (o) 1 p_ P2(O) 2 R Pi P1(O) (O) 1 p Figure B.9: Reaction Functions for Firms 1 and 2 Appendix C Proofs for Chapter 3. Proof of Proposition 25 (i) The necessary condition for the Nash equilibrium price at display (3.53) is obtained by summing equation (3.50) over all n firms. The necessity of (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 (x — wc and wc)/n exactly once (see Figure 3.3). Hence (3.52) and (3.53) have unique solutions. (ii) Since x to x — — ‘—+ 1/(— in Q)’(x) is continuous, its graph must cross the line corresponding wc strictly to the left of where it crosses the line corresponding to (x — wc)/n. Hence wp m <wp’, which establishes (ii). (iii) The value of the combined profits at the Nash equilibrium point is obtained by summing 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 — * — for each firm. wp*Q(wp*) (C 136) E(wp*) 0 Proof of proposition 26. (i) Using the facts that p 89 = P(eq) and that P’(eq’) = Appendix C. Proofs for Chapter 3. 90 1/Q’(p), equation (3.57) can be rewritten as — (— in Q• (C.137) Under our assumptions on the demand function, this equation has a unique solution provided 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 Propo sition 25(u). 0 Appendix D Proofs for Chapter 4 Proof of Proposition 27. The expression for the policy line in Figure 4.4 is —a 2 ‘Y22P2 r 1 +a p 1 7 r 1 1 + -y r 2 p 12 — 0. Given a price of firm 2, P2, — 721P1 + the price of firm 1 on the policy line is = 2+ —a 722P2 pl(p2) 721 + rQi + 11 7 +T Suppose both firms comply with the regulation. Let 712p2) pl(p2, e) (D 138) = pi(p) + e, i.e., the price firm 1 deviated from that on the policy line given the price of firm 2. Let 2 ),p be iri(pi(p ) the profit of firm 1 attained on the policy line given the price of firm 2, and 2 (pi(p e),p2) 1 7r , be the profit obtained at the deviated price. Then ),p 2 iri(pi(p ) = (721 — 2 +7iir) + 7 2 11722P2 = ,e),p 2 ri(pi(p ) [—2cr 1 7 2 1 11 7 1 a +r — — a 2 7 1 1 r 1 cl7 — 2 7 1 c — 712721P2 12 7 r 2 1 + e7 rJ 11 1 + C7 72 11 +p e [e(7ii7zi + 2 2 r)(p 1 7 12 7 1 p)(—712721 + 2711722 + r)] + 7iir) i+7 2 7 1 e(7i r) 1 (D.139) P2 > i7 1 7 r 712721 + 2711722 + 12 — (721 > 0 — — — Therefore given any P2 above the point F , firm 1 does not unilaterally deviate from the 1 corresponding Pi on the policy line. Now consider firm 2. Given a price of firm 1, p’, the price of firm 2 on the policy line is = 2+ a 721P1 — p2(p1) 722 91 r(ai _711P’) + ‘y12r (D.140) Appendix D. Proofs for Chapter 4 Let p2(pl, e) = p2(pl) 92 e, a price deviated from that on the policy line given p. Then — 2 r ( pi,p pi)) — (pi,e)) 2 ir ( pi,p e 2 22 + C 7 2 [—a 22 7 2 21 7 2 cv 721722P1 + r \2 r) 2 7i 22 ‘y 12 + -yi 2 c 2 7 22 + r 7 1 +2a r i7 + e7 27 r 1 p 22 22 7 12 e7 2 + r] ipir 7 2 e e(7 + 21 2 11 2 7 2 2 27 1 7 r)(p1 7fl722r)] p)(—72l722 + r 12 7 (722 + r) 2 + r) e(7 11 2 7 2 (D.141) + P1 = I — — — — r) 12 (722 +7 Thus given Pi below the point F , firm 2 does not unilateral deviate from the corre 2 sponding price on the policy line. Conditions (D.139) and (D.141) taken together implies that each point is a Nash equilibrium on the segment F’P 2 of the policy line. Proof of Proposition 28. 0 On the policy line, the price of the recycled product as a function of the price of the virgin product, p2(pl), is given by (D.140). The profit of the virgin product producer along the policy line and its first order condition are (i 7’1(p1,p2(pi)) — ci)(a 1 7 2 2+ 722 2 , 1 diri(p ( pi)) p 1 dp — -yi c 2 + 722 1 a + p1 2 ( — ci)Qy 2 7 12 1 — — 711722P1) 711722) r 12 722 +7 — > + 712721P1 12 7 +r O1722 = 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 along the policy line and its first order condition are / / , , 2IP1IP2),P2) — (p2 — )(cx 2 c yii + °1721 — 721 (pi(p dir ) 2 ) ,p 2 dp — [o2711 + 21 al7 p2 2 ( 0 if 712721 711722. — 22 i 1 7 ) 2 P 7 T + ‘fliT — 12 ) 2 C 2 7 (7 1 11 + 1 72 r 7 — > + + 712721P2 — 7n722)]r Appendix D. Proofs for Chapter 4 93 The profits of the two firms on the policy line increase with Pi and P2, and the equilibrium profits are maximized at P 2 which dominates the other equilibria. D Proof of Proposition 29. Divide expression (4.83) for t by expression (4.76) for t to obtain the following: r+ 1 7 721 >1. D Proof of Proposition 30. It follows from the fact that the denominator of (4.84) is an increasing function of r. D Proof of Proposition 31. Divide the profit for firm 1 under production tax in expres sion (4.84) by the profit for firm 2 under excessive consumption tax in (4.78) to give the equation below. 7r(r) 74(r) Proof of Proposition 32. 722 — — 11 7 2 r The profit functions of firm 1 under the scheme of taxing excessive consumption of virgin product and production tax are given below. 1 , 2 7r(pi,p ) t ir(pi,p , 2 tc) = (pi,p 2 1 7r ) 2 —t ( , 1 ) p q p = ) 2 7ri(pip — —q2(pl,p2)Y’, _tCq(p,p). (D.142) (D.143) At the equilibrium point F , the above expressions become 2 irp,p,tc*) = iri(p,p), (D.144) = iri(p,p) _tc*qi(p,p). (D.145) Appendix D. Proofs for Chapter 4 94 Comparing equations (D.144) and (D.145), and since >0, i4(p,p,tl*) > cQ,i f 2 22 c*) D Proof of Proposition 33. The profit function of firm 1 under progressive tax is 8 , 2 7rj(pi,p ) t = (pi,p r 3 —i ) ( , 1 ) Jir p p . [1 2 (D.146) Differentiating the profit function with respect to P’ to obtain the following first order condition: 1 — 2 , 1 2t?ri(p ) p = 0. Suppose t 5 > 0, then the value of Therefore, tion curve. = l/4ts, ir(pi,p , 2 t) 2 , 1 Tri(p ) p on the reaction curve of firm 1 is l/2t . 3 and the tax payment, ) ,p = 1/4t 1 tri(p 2 , on the reac 3 D Proof of Proposition 34. The profit function of firm 1 under progressive profit tax is given by expression (D.146). The profit function under the scheme of taxing excessive consumption of virgin product is given by (D.142). At the equilibrium point F , the 2 profit functions become = (p,p,t8*) (D.147) ri(p,p), = Comparing equations (D.147) and (D.148), and sincets* >0, (D.148) (p,p,tl*) > ppts*). 0 Proof of Proposition 35. At the equilibrium prices, (p(r),p(r)), the profit firm 1 earns under the progressive tax scheme equals the tax payment for all meaningful r. This is the result from Proposition 33. The profit function of firm 1 under production tax is (p(r) — ) 1 c ( p,p) q — tcqi(p,p). Since t increases with r, and by increasing r, some Appendix D. Proofs for Chapter 4 value of r is reached at which t = 95 — , then profit equals tax payment. Increasing 1 c r further tax payment outweighs profit. But at this value of r, profit still equals tax payment under the progressive tax scheme, thus profit is lower under production tax at this point onward. Note that at each r, the equilibrium prices and the demands for the two products are the same under the two tax schemes. D Proof of Proposition 36. Under the tax schemes considered, the equilibrium profit of firm 2 and its derivative with respect to r are given below. 2 , 2a [ 2 r i 1 7 2+a 21 + 12 7 1 Qy 711722)12 c i 2 7 ir r y 2 (7i 22 721722) 2 7 11 27 i[o 27 r 7 2 1 7 i +a 711722)12 21 + 21 7 1 2-y 1 c2(’y 21 + r 7 12 (—7 r 22 + 721722) 7 11 2-y D >0. — — — — d 2 —ir 1 r 2 dr — — — — Appendix E Background to the Mandatory Recycling Legislation for Newsprint It is a fact that there is a shortage of sanitary landfill in North America, especially in the United States. The U.S. generates about 160 millions tons per year (tpy) of municipal solidwaste and approximately 84% of this is disposed of in landfill. Wastepaper accounts for 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 in North America. Around 50% of the landfills now in use will close down over the next five years (McClay 1990.) About 75% of the landfills in the U.S. will close by the year 2005 (Paper Recycler 1990.) The remaining landfills are generally not located closed to urban centres; therefore the cost of transporting garbage has risen dramatically. By the year 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 office of Solid Waste Management Programs. In the same year the first Solid Waste Act was passed which was replaced by the Resource Recovery Act in 1970, and by the Resource Conservation and Recovery Act (RCRA) in 1976. The RCRA required each of the states to file with the federal EPA a solid waste management plan. About 43 states have filed such plans with the EPA. The major problem facing those who are responsible for the disposal of solid waste is the lack of landfill space. The lack of landfill sites did really set the legislation wheel in motion only after New Jersey’s garbage crises. New Jersey closed all but 20 of its landfills by 1984, and no new ones were sited. In 1987, a law was passed requiring all communities in New Jersey to 96 Appendix E. Background to the Mandatory Recycling Legislation for Newsprint 97 mandate recycling programs. Some 17 states followed New Jersey’s lead by setting up their own source separation and collection systems or by studying such programs. As these recycling programs began to generate more recyclables, it became apparent that demand for them was not growing proportionally. Some states in the interim, paid mills to take the excess old newspaper (ONP) rather than paying high tipping fees (dumping charges) at landfill or paying to transport the solid waste hundreds of miles (Paper Recycler 1990.) Tipping fees in the U.S., as suggested by Edwards (1991), are now approaching $100 per ton. Hatch Associates Ltd. (1989) and Paper Recycler (1990) pointed out that the over supply of ONP for recycling had resulted in the introduction of legislation to stimulate the usage of such recyclable products. In the U.S. such legislation is purely a reaction to limited municipal and state budgets by reducing pressure on collection and landfill services. This has focused mainly on getting newsprint producers to use more ONP in their fibre furnishes, though legislation on mandatory recycling of other grades of paper are now also being considered. Florida passed the first mandatory recycled newsprint usage bill in late 1988. Since then many states had proposed or adopted similar bills. By late 1989, California and Connecticut had passed laws requiring publishers to consume a certain portion of re cycled newsprint. By the summer of 1991, eight states had passed legislation setting newsprint recycling goals and timetables, and in some cases including taxes on virgin newsprint. These states were California, Connecticut, Florida, Arizona, Missouri, Illi nois, Maryland and Wisconsin. California, for example, stipulated recycled newsprint consumption 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 had voluntary agreements negotiated. For example, in the New York agreement 11% recycled Appendix E. Background to the Mandatory Recycling Legislation for Newsprint 98 newsprint 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 and local government’s ability to collect and process ONP for recycling into newsprint and other products. An escape clause is included to provide a waiver if recycled newsprint supply is not available at a price and quality comparable to virgin newsprint. Also for the New York agreement, focus is on aggregate use of recycled fibers rather than the recycled fiber content of each sheet of newsprint (Newsprint Reporter 1991, Boyle 1990.) McClay (1991) reported that except California, Arizona and Florida which stipulated a 40% minimum-content standard as a definition of recycled-content newsprint, all the others used a blended or aggregate target approach. Four other states, Georgia, Indiana, Ohio and Oregon had legislation pending. In 1988 Florida also pioneered an advance disposal 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 place on the publishers and newsprint producers, the U.S. federal government, states and municipalities are also requiring their purchasing agents to produce newsprint and other paper containing certain percentages of recycled fiber. As early as 1988, the federal EPA issued the ‘Guideline for Federal Procurement of Paper and Paper Products containing Recovered Materials’ under the RCRA. One year later, 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, and the state and local governments that purchase $10,000 worth or more using appropriated federal funds must also follow the EPA guidelines, e.g. newsprint must contain at least 40% postconsumer wastepaper. By the beginning of 1990, 23 states had preferential procurement laws (Garcia 1990, Paper Recycler 1990.) Jaakko Pöyry Oy (1990) pointed out the driving force for the use of waste paper had Appendix E. Background to the Mandatory Recycling Legislation for Newsprint 99 traditionally been economic in the other parts of the world: waste paper had enabled mills and countries without abundant forest resources to compete effectively on bulk grades with integrated producers of virgin based grades. This was particularly the case in regions such as Japan and Western Europe where it had been relatively easy to collect and process large volumes of wastepaper. In the Far East the local industries depended to a large extent on the cost economies of wastepaper. In North America, on the other hand, abundant forest resources had not encouraged the use of recycled fiber. Though there appeared to be general acceptance by the newsprint producers of the U.S. government mandated recycling requirements (Hatch Associates Ltd. 1989), the responses from the industry were less than enthusiastic. In 1989, even under the federal EPA preferential procurement law there was only approximately 12% of the total North American newsprint production capacity which used some percentages of recycled fiber as a raw material (Pulp and Paper North American Factbook 1990, p. 140 and p.144.) In 1990 Canada produced less than 2% newsprint containing recycled content (Lukins 1992.) The whole newsprint recycling issue in North America has been primarily legislation driven 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 Associates Ltd. 1989, Garcia 1990.) It was even said that the industry had never before been forced to absorb so much additional recycled fiber (Paper Recycler 1990.) A survey of the newsprint industry reported by Newsprint Reporter (1990) showed that the newsprint producers perceived the mandatory recycling issue as one of solid waste disposal. Thirty nine percent of the general public agreed with the newsprint producers. Obstacles to heavy recycling as perceived by the executives were the cost impact, an adequate supply of ONP, the paper quality and the quality of recycling machinery. Seventy five percent of the newsprint producers felt that the states had been too aggressive in trying to increase newsprint recycling, an obvious reference to the Appendix E. Background to the Mandatory Recycling Legislation for Newsprint 100 mandatory recycling bills. Pulp and Paper North American Factbook (1990) questioned whether or not any of the state goals could be achieved within the time frames mandated. For it was not certain how much secondary fiber could be reclaimed, as various factor such as contamination, improper handling and lack of consumer interest tended to limit availability. Newsprint Reporter (1990) also suggested that the community collection systems were not yet pro ducing high quality material. It was estimated 20-40% of the postconsumer newspaper would be difficult to process or could not be used because of contaminants. From the perspective of the demand side, McClay (1990) suggested that up until recently the newsprint market had given preference to virgin fiber product and shunted the recycled variety in North America because of its inferior quality both in terms of runnability and printability. Landegger (1992) was particularly vocal about the disec onomics of the mandatory recycling programs. He pointed out that virtually all chief executives of the industry questioned the recycling programs, and stories about mills exchanging rolls of unused newly produced papers which were then being cut up and put back into their paper stream so that they could claim the resulting papers produced to be ‘recycled.’ He also claimed that tonnage of dead leaves and grass cuttings that went to the landfill equaled to all the wastepaper and that it was much cheaper to turn them into compost or animal feed. Bibliography [1] Aspler, J. S., 1989, ‘Potential Printability Problems in Recycled Papers,’ in J. Aspler et al. (eds.), Recycling and the Canadian Pulp and Paper Industry, A Report to the Research Program Committee of PAPRICAN. [2] Booth, D. L., 1990, A Strategic Capacity Planning Tool for a Firm in the Newsprint Industry, unpublished Doctoral Thesis, University of British Columbia. [3] Booth, D. L., V. Kanetkar, I. Vertinsky and D. Whistler, 1991, ‘An Empirical Model of Capacity Expansion and Pricing in an Oligopoly with Barometric Price Leader ship: A Case Study of the Newsprint Industry in North America,’ The Journal of Industrial Economics, 35, 255-276. [4] Boyle, P., 1990, ‘Update on Federal Activity on Recycling Legislation,’ Symposium on Waste Paper in Newsprint and Printing and Writing Grades, Canadian Pulp and Paper Association. [5] Caplin, A. and B. Nalefuff, 1991, ‘Aggregation and Imperfect Competition: on the Existence of Equilibrium,’ Econometrica, 59, 1, 25-59. [6] Chase, R. B. and N. J. Acquilano, 1981, Production and Operations Managemant, 3rd Ed., Richard D. Irwin, Inc., Homewood, Ill. [7] Cole, R. E., 1992, ‘The Quality Revolution,’ Production and Operations Manage ment, 1, 1, 118-120. [8] Cropper, M. L. and W. E. Oates, 1992, ‘Environmental Economics: A Survey,’ Journal of Economic Literature, 30, 675-740. [9] De Meyer, A., Nakane, J., Miller, J.G. and K. Ferdows, 1989, ‘Flexibility: The Next Competitive Battle the Manufacturing Futures Survey,’ Strategic Managemant Journal, 10, 135-144. [10] Dobbs, I. M., 1991, ‘Litter and Waste Management: Disposal Taxes versus User Charges,’ Canadian Journal of Economics, 24, 1, 221-227. [11] Dorfman, R. and P. 0. Steiner, 1954, ‘Optimal Advertising and Optimal Quality,’ American Economic Review, 44, 826-836. 101 Bibliography 102 [12] Duncanson, J., 1990, Economics of Newsprint Manufacture— Recyled vs Virgin, Sanwa McCarthy Securities Limited. [13] Edwards, J. R., 1991, ‘Future Trends in the Secondary Fiber Industry,’ Progress in Paper Recycling, 1, 1, 24-30. [14] Englin, J. E. and M. S. Klan, 1990, ‘Optimal Taxation: Timber and Externalities,’ Journal of Environmental Economics and Management, 18, 263-275. [15] Fine, C. H., 1986, ‘Quality Improvement and Learning in Productive Systems,’ Man agement Science, 32, 10, 1301-1315. [16] Fine, C. H., 1988, ‘A Quality Control Model with Learning Effects,’ Operations Research, 36, 3, 437-444. [17] Fine, C. H. and E. L. Porteus, 1989, ‘Dynamic Process Improvement,’ Operations Research, 37, 4, 580-591. [18] Fortuna, R. M., 1990, ‘The Quality Imperative,’ in The Ernst &. Young Quality Improvement Consulting Group (eds.) Total Quality: an Executive s Guide for the 1990s, Dow Jones-Irwin, Homewood, Ill. [19] Friday, G. D., 1990, ‘Pressroom Experience— Commercial Printers,’ Symposium on Waste Paper in Newsprint and Printing and Writing Grades, Canadian Pulp and Paper Association. [20] Friedman, J., 1983, Oligopoly Theory, Cambridge University Press. [21] Furth, D. and D. Kovenock, 1990, ‘Price Leadership in a Duopoly with Capacity Constraints and Product Differentiation,’ Technical Report. [22] Garcia, D. A., 1990, ‘Recycling Capacity to Increase at Record Rates as Laws Pro liferate,’ Pulp and Paper, 64, 5, sl-s29. [23] Garvin, D. A., 1983, ‘Quality on the Line,’ Harvard Business Review, SeptemberOctober, 64-75. [24] Hagan, J. T., 1986, ‘Quality Costs,’ in L. Walsh, R. Wurster and R. J. Kimber (eds.), Quality Management Handbook, Marcel Dekker. [25] Hahn, R. W., 1989, ‘Economic Prescriptions for Environmental Problems: How the Patient Followed the Doctor’s Order,’ Journal of Economic Perspectives, 3, 2, 95-114. [26] Hatch Associates Ltd., 1989, Waste Paper Recycling, Final Draft Report to Forestry Canada. Bibliography 103 [27] Hayes, R. H. and S. C. Wheelwright, 1984, Restoring Our Competitive Edge, John Wiley and Sons. [28] Hayes, R. H., Wheelwright, S.C. and K. B. Clark, 1988, Dynamic Manufacturing, the Free Press, NY. [29] Heaps, T., 1985, ‘The Taxation of Nonreplenishable Natural Resources Revisited,’ Journal of Environmental Economics and Management, 12, 14-27. [30] Hee, W. J., 1990, ‘Pressroom Experience— Newspaper,’ Symposium on Waste Paper in Newsprint and Printing and Writing Grades, Canadian Pulp and Paper Associ ation. [31] Hill, T., 1983, Production and Operations Management, Prentice Hall. [32] Hinomoto, H., 1965, ‘Capacity Expansion with Facilities under Technological Im provements,’ Management Science, 11, 5, 581-592. [33] Howard, R. C., 1989, ‘Effect of Recycling on Paper Quality,’ in J. Aspler et al. (eds.), Recycling and the Canadian Pulp and Paper Industry, A Report to the Research Program Committee of PAPRICAN. [34] Jaakko Pöyry Oy, 1990, Recycled Fiber: An Underutilized Opportunity, Vol. I, Ex ecutive Report. [35] Jacoby, J. and J. C. Olson, 1977, ‘Consumer Response to Price: An Attitudinal, Information Processing Perspective,’ in Y. Wind and P. Greenberg (eds.) Moving Ahead with Attitude Research, American Marketing Association, Chicago, 73-86. [36] Juran, J., 1988, Quality Control Handbook, 4th Ed., McGraw-Hill, NY. [37] Juran, J. and F. M. Cryna, 1970, Quality Planning and Analysis, McGraw-Hill, NY. [38] Klemperer, P. and M. Meyer, 1986, ‘Price Competition vs. Quantity Competition: The Role of Uncertainty,’ Rand Journal of Economics, 17, 4, 618-638. [39] Landegger, C. C., 1992, ‘The True Cost of Recycling,’ Progress in Paper Recycling, 1, 2, 85-88. [40] Lee, H. L. and C. S. Tapiero, 1986, ‘Quality Control and the Sales Process,’ Naval Research Logistics Quarterly, 33, 569-587. [41] Lukins, J. E., 1992, ‘Recycled Paper Quality Will be Defined by Market, Technology, Fiber Sources,’ American Papermaker, April, 54-56. Bibliography 104 [42] Mangasarian, 0. L., 1970, ‘Convexity, Pseudo-convexity and Quasi-convexity of Composite Functions,’ Cahiers du Centre d’Etude de Rech. Oper., 12, 114-122. [43] Marcellus, R. L. and M. Dada, 1991, ‘Interactive Process Quality Improvement,’ Management Science, 37, 11, 1365-1376. [44] McClay, B., 1990, ‘Paper Recycling in Canada: An Established Practice on the Verge of Significant Growth,’ Paper Presented at Globe 90, Vancouver, B.C., Canada. [45] McClay, B., 1991, ‘Recycling in the Canadian Pulp and Paper Industry,’ The Forestry Chronicle, 67, 5, 532-534. [46] Meadows, E., 1984, ‘Success at Three Companies,’ in Editors of Fortune (eds.) Working Smarter, Penguin Books. [47] Monroe, K. B., 1990, Pricing: Making Profitable Decisions, McGraw-Hill. [481 Nandakumar, P., Datar, S. M. and R. Akella, 1993, ‘Models for Measuring and Acounting for Cost of Conformance Quality,’ Management Science, 39, 1, 1-16. [49] Newsprint Reporter, 1990, 34, 1. [50] Newsprint Reporter, 1991, 35, 1. [51] Ohmae, K, 1984, ‘The New Technologies: Japan’s Strategic Thrust,’ The McKinsey Quarterly, Winter, 20-35. [52] Paper Recycler, 1990, Miller Freeman Publications, Inc. [53] Pulp and Paper 1990 North American Factbook, 1990, ‘Newsprint,’ Miller Freeman Publications, Inc., 139-159. 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Essays on production and pricing decisions Mok, Yat-Koon 1993
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Title | Essays on production and pricing decisions |
Creator |
Mok, Yat-Koon |
Date Issued | 1993 |
Description | There has been considerable interest in finding and explaining the basic elements that can drive product quality up. In the literature this is largely done by modelling the effects of investing 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, the firm should produce output of higher quality, the increase in quality being dependent on consumers’ sensitivity to quality and to price, and the effect of technological improvement on product price and quality are very different from those when the demand is a function of price alone. Some twenty states in the U.S. have passed recycling laws which mandate consumption of old newspaper by the newsprint industry. To study the effect of regulation, a model is used in which two firms compete under the regulatory constraint—one firm producing the recycled product, the other the virgin product. Assuming the regulatory constraint is 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 higher average price in competitive equilibrium than the cartel price, are obtained in the second essay. The two firm model is generalized to include n firms which compete under the same kind of regulatory constraint in the third essay. Results similar to the two firm case are obtained. When the recycled product and the virgin product are partially substitutable, regulation that mandates consumption of the recycled product results in infinitely many equilibria. A dominating equilibrium exists if the demand parameters satisfy a certain condition, otherwise it is not clear how to select an equilibrium. On the other hand, a suitable tax on the virgin product, or its producer, serves to induce compliance with the recycling policy and equilibrium selection. The equilibrium prices and profits of the two firms under the schemes of production tax, excessive consumption tax and progressive profit tax are examined and compared in the fourth essay. It is interesting to find that the tax rate for excessive consumption is comparatively low and, in equilibrium, this tax scheme collects no tax payment. |
Extent | 2157301 bytes |
Subject |
Quality of products -- Mathematical models Manufactures -- Prices -- Mathematical models Newsprint -- Recycling -- Mathematical models Marketing -- Mathematical models Recycling (Waste, etc.) -- Law and legislation -- United States -- Mathematical models |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-04-08 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0088017 |
URI | http://hdl.handle.net/2429/6891 |
Degree |
Doctor of Philosophy - PhD |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Graduation Date | 1994-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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