I ' I I I , TICKET PRICING IN THE ALPINE SKI INDUSTRY by Andrew Robert Gerl_ach A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Applied Economics MONTANA STATE UNIVERSITY Bozeman, Montana September 1990 ii APPROVAL of a thesis submitted by Andrew Robert Gerlach This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate studies. Date Chairperson, Graduate Committee Approved for the Major Department Date Chairperson, Graduate Committee ~pproved for the College of Graduate Studies Date Graduate Dean I I iii STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfillment of the requirements for a master's degree at, Montana State University, I agree that the Library shall make it available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgement of source is made. Permission for extensive quotation from or reproduction of this thesis may be granted by my major professor, or in his absence, by the Dean of Libraries when, in the opinion of either, the proposed use of the material is for scholarly purposes. Any copying or use of the material in this thesis for financial gain shall not be allowed without my written permission·. Signature Date iv ACKNOWLEDGEMENTS I would like to thank the members of my thesis committee for their patience, time and direction: Dr. Ronald Johnson, Terry Anderson, and Mike Kehoe of my committee for their patience, insight, and guidance. In particular, I wish to thank, my ch~irman, Dr. Johnson, for the overall direction and commentary which were essential to the completion of this paper. I also would like encouraged and aided me. to thank my parents who always Most of all I must express my appreciation to Teresa Brock who continually supported and aided me throughout the process of writing this thesis. v TABLE OF CONTENTS Page APPROVAL. . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . i i STATEMENT OF PERMISSION TO USE..................... iii ACKNOWLEDGEMENTS . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . i v TABLE OF CONTENTS. . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . v LIST OF TABLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES . ..•.................••....... ~ . . . . . vii ABSTRACT • ••••••••••••••••••••••••••••••••••••••••.. CHAPTER: 1. INTRODUCTION TO SKI-LIFT PRICING .......... . 2. SKI-LIFT PRICING, LITERATURE REVIEW ....... . 3. BARRO & ROMER TESTED AND QUESTIONED ....... . 4. MONOPOLISTIC SKI-LIFT PRICING •............. 5. MARKET POWER and SKI-LIFT PRICING PRACTICES REFERENCES •.••••••••••••••••••••••••••••...•...... APPENDICES . ........•••....•........................ A: Data Set ................................ . B: Computer program •••••••••.••.•••.••.••..• viii 1 5 25 41 57 68 72 73 76 vi LIST OF TABLES Table Page 1. Variables; Means and Standard Deviations ..... . 31 2. 2SLS Regression Results •..••......•........... 32 3. Price Discriminatory Ride and Lift Ticket Revenue . ••....................•...•........... 55 4. Airfare and Lodging Packages ...•...•.......•.. 63 5. Data, Regions 1-5 ............................ . 74 I I I vii LIST OF FIGURES Figure Page 1. Demand Increase . ............................ . 2. Supply Increase . ............................ . 3. Ride-Ticket verses Lift-Ticket Revenue ...... . 46 4. Non-Linear Demands •.......•••.•.••........... 51 J i viii ABSTRACT Alpine ski areas worldwide use daily lift-ticket pr1c1ng rather than individual ride-ticket pricing. Robert Barra and Paul Romer argue that the ski ride industry is a competitive market and that identical equilibriums and revenues are reached with either pricing method. They also argue that sticky lift ticket prices and lift-line queues are efficient. Lift-ticket pricing dominates because of lower monitoring costs. Tests of their model's predictions, however, do not support their model. A monopolistic ski-lift pricing model is developed. The monopoly model predicts that lift-ticket pricing would 'dominate the market due to its revenue generating advantages over ride-ticket pricing. Overall the monopoly model predicts the pricing policies that exist in the ski ride market better th~:m the competitive model of Barra and Romer. It is argued that lift-ticket pricing is an indication of the market power most ski areas possess. I i 1 CHAPTER 1 INTRODUCTION TO SKI-LIFT PRICING For the recreational pleasure of skiing down a mountain, skiers travel to ski resorts and purchase lift tickets which provide them a day's access to lift rides. Why do ski areas rely almost exclusively on lift-ticket p~icing rather than ride-ticket pricing when each ski lift ride is actually a separate item? Is this simply the most efficient method of ride allocation in a competitive market, or rather an indication of market power? How can it be efficient to have lift lines, and if market power is present what are its implications? These are a few of the questions this thesis hopes to answer. While skiing is now a recreational activity, it was initially developed as a utilitarian form of transportation during the stone-age and remained primarily as such until the later part of the 19th century (Flower, 1976, p. 22). To facilitate alpine skiing's recreational aspect, uphill transportation was developed. Such uphill transportation first appeared in the 1870s in the form of mountain trains, but was not commonplace until the late 1930s, after the invention of the drag-lift (Flower, 1976, p.l14). In 1934 the first rope­ tow in America was set up at Woods toke, Vermont, and the 2 following year the first American resort specifically developed for skiers opened in sun Valley, Idaho. Over the years the demand for uphill lift transportation has continued to grow. In America alone there are now more than 20 million skiers who account for over 53 million annual skier visits at the more than 600 American ski resorts (Ski Industries America, 1984; Symonds, 1988) . In 1989 these ski resorts offered a combined lift capacity of over 19,600,000 rides per day at an average cost of $21.00 for a day lift ticket (Enzel, 1989; Goeldner, 1989). Lift-ticket pricing as opposed to ride-ticket pricing dominates the ski ride market worldwide. The former can be regarded as a two-part tariff with a lump sum fee (the lift ticket) required for the right to ski, and an explicit price per ride of zero. Ride-ticket pricing is a single-part tariff and requires a payment for each ride consumed. Barre and Romer (1987) developed a model of a competitive ski industry (reviewed in Chapter 2) and explained the dominance of lift-ticket pricing and other aspects of the ski ride industry. While most economists, such as Yoram Barzel (1974), would label long and persistent queues for a commodity as an inefficient allocative mechanism that dissipation rents, Barre and Romer argue that this is not the case with ski ride lift lines. Instead they argue that lift lines are not due to inefficiency in the pricing mechanism, but are a necessary feature which allows lift-ticket pricing to be an efficient 3 ride allocation scheme. While most models of supply and demand assume that efficient pricing dictates that prices fluctuate with demand or supply, Barro and Romer argue that due to the lift line ride allocation mechanism and a unitary demand elasticity, lift ticket prices may remain fixed. In their competitive model, ride-ticket pricing and lift-ticket pricing results in identical total revenues and equilibrium conditions. Barro and Romer argue that due to lower monitoring and set-up costs, lift-ticket pricing schemes dominate the market. While their model has many interesting features, tests of its predictive capabilities, which are reported in Chapter 3, yield ambiguous results, at best. It will be argued that the model's predictive faults appear due to inaccurate assumptions about the underlying market structure. To better explain ski ride pricing practices, a new model of the American alpine ski ride industry is developed in Chapter 4. For this model, it is assumed that each ski area possesses some degree of market power and can price as a monopolist. It is shown that under the assumption of monopolistic pricing, a ski area can generally generate greater revenues with lift-ticket pricing rather than with ride-ticket pricing, but not under all circumstances. The latter result contrasts with the outcome obtained by Oi (1974), whereby a two-part tariff (lift-ticket pricing) would always dominate. Nevertheless, this result is an anomaly, and as far as ski ride pricing studies are concerned, this aspect 4 is of minor importance since the conditions necessary for it to occur are uncommon. When compared to actual ski ride pricing practices, the monopoly model's revenue maximization principle provides a stronger explanation for the dominance of lift-ticket pricing policies than does Barre and Romer's cost minimization contentions. Further analysis of the monopoly model and its predictive advantages over Barre and Romer's competitive model are presented in Chapter 5. It is shown that another economic anomaly may occur in the alpine ski ride industry. This anomaly is that local skiers rather than tourists may be the ones discriminated against in terms of higher (lift ticket) prices. Overall, the monopoly model's predictive and explanatory capabilities are shown to be superior to those of the competitive model. It is argued that the dominance of lift-ticket pricing schemes is due to the market power that most ski areas possess. 5 CHAPTER 2 SKI-LIFT PRICING, LITERATURE REVIEW Contained in this chapter is a review of an article written by Barre and Romer (1987). Their article develops a competitive model of the alpine ski industry and challenges conventional wisdom by arguing that queues are not inefficient; lump-sum lift-ticket and per unit ride-ticket pricing policies result in identical revenues and equilibrium conditions; and fluctuations in demand need not require changes in lift-ticket pricing. To demonstrate their arguments, ride-ticket pricing is considered first, followed by an analysis of lift.;..ticket pricing. A comparison of the two pricing schemes reveals that they yield identical . revenues and equilibrium conditions. Lift-ticket pricing, however, is argued to be the pricing scheme of choice for ski areas due to lower monitoring and set-up costs. A deeper analysis of the Barro-Romer model shows how prices adjust to shifts in demand, differences across ski areas, transportation costs and skier preferences. The equations and notation offered in this chapter follows that employed by Barre and Romer. Consider a ski industry where price is charged on a per ride basis. Each of n homogeneous skiers, who differ only in \ I 6 their fixed costs of going skiing, face an industry comprised of J identical competitive ski areas. Each ski area operator has a fixed lift capacity, x, · and charges the same common price per ride, P. The marginal cost of a ride is assumed to be zero for the ski area. The result is an industry whose total supply, Jx, is perfectly inelastic in the short run. Each individual skier, i, seeks to maximize utility by consuming a combination of ski ri~es, qi, and all other goods, zi, subject to the budget constraint, Yi. The budget constraint is assumed to be comprised of the cost of qi ride tickets, the cost of all other goods, zi, the price of which are set to $1.00, and the lump sum costs of going skiing, ci. The latter is comprised of the opportunity cost of going skiing and travel and boarding expenses and is independent of the number of ski rides taken per day. Hence, the budget constraint is (2. 1) The demand for ski runs is described by a downward- sloping income compensated demand curve, qi=Di(P), where Pis the price per ride. The integral of the demand curve taken from zero to qi provides a monetary measure of the willingness to pay for the qi rides, (2.2) An individual will choose to ski as long as the total cost involved in consuming the qi runs, is less than or equal 7 to the gain that the individual receives from skiing qi runs, (2. 3) As long as the "net surplus" is greater than or equal to zero the individual will choose to ski. While all individuals are assumed to have identical demand functions, they do differ in their respective lump sum costs, ci. These fixed costs vary across individuals depending on their occupations and locations. Costs can also vary throughout the year. For example, during weekends and holidays when skiers are not working, the opportunity cost of time is often less. The distribution of the ci's across individuals is described by a cumulative distribution function Fs. The subscript s is a shift parameter which describes the changes in the lump sum costs at different times. On a given day, only those individuals whose net surpluses are positive will ski. When the fixed costs are high, as during weekdays and non-vacation periods, relatively few individuals will have positive net surpluses and choose to ski. The actual number of skiers who choose to ski, N, therefore is a function of the price per ride and distribution of the lump sum costs, (2. 4) N • N(P, s) . Equilibrium, in a ride-ticket pricing system, is determined by the number of rides provided by the ski industry, Jx, and the total number of rides demanded by the 8 ski community, qiN. Therefore, the equilibrium conditions are (2. 5) Jx = D(P) * N(P, s). With lift capacity fixed in the short run, this condition will determine the equilibrium price per ride that each competitive firm takes as a given. For equilibrium to be reached the price per ride, P, must fluctuate in the same direction as demand, D(P). Over the long run, when capacity is allowed to fluctuate, this equilibrium condition also determines the price per ride. As capacity increases, ceteris paribus, the price per ride will decrease. Competition in the industry requires that the firms take this equilibrium ride ticket price, P, as a given. Barre and Romer demonstrate that these ride-ticket equilibrium conditions are identical, in terms of cost of skiing and allocation of rides, to that arrived at with a two­ part pricing system. The two-part tariff consists of a lump sum entry fee, the lift ticket, and an explicit price per ride set equal to zero. The identical equilibria are reached due to unique properties intrinsic in the lift-ticket pricing scheme's allocation of rides. Under two-part pricing, as with ride-ticket pricing, each individual will choose to ski as long as the gain from skiing is not exceeded by the sum of the lift ticket price and the costs represented by ci. While the explicit zero marginal price per ride appears to provide no constraint on the number of rides available per skier, an \ , I 9 area's lift capacity will constrain the number of rides that each skier will receive. Each skier, through the use of the lift line allocation mechanism, receives the same number of rides. In other words, though the skier faces an explicit zero marginal cost per ride he also faces a queue. The number of rides per skier per day is determined by the speed at which he skis down a run, by the number of other skiers at the hill, the rate at which they ski, and the rate at which the lifts can transport skiers up the hill. If it is assumed that all skiers ski at the same pace, each skier will receive the same quantity of rides. Under this assumption the number of runs that any one individual skier can make is solely determined by the number of skiers at the hill, n, and the lift capacity of the hill, x (in rides per day). The quantity of rides that each skier receives, qi, is simply the area's lift capacity, x, divided by the number of skiers at the lifts, n. It is assumed that facing a zero marginal price per ride each individual prefers a greater number of rides than is available. As a consequence, a lift line will arise and allocation by queuing will occur. It will be shown that this queuing form of ride allocation creates no inefficiencies due to an assumed zero opportunity cost of time for a skier at a hill. The assumption that the opportunity cost of time for an individual waiting in a lift line is zero is not intuitive. Barro and Romer do not imply that the overall opportunity cost 10 of.time for an individual is zero. They argue that when an individual is making a decision to go skiing, the individual's opportunity cost of time can be great. This opportunity cost of time is included in the fixed co~ts of going skiing, c 1, and is encompassed in the shift parameter, s. What Barre and Romer assume is that having made the decision to go skiing once the skier is at the ski resort he has a new approximately zero opportunity cost of time. They assume that the "time spent at a ski area or amusement park is inherently valuable so that the cost of time spent in the queue is approximately z~ro."(p. 876) Barre and Romer (pp. 877-878) state further that "people do not care directly about time spent waiting in lift lines . They would prefer shorter lines because they would prefer more rides; but given a fixed number of rides, they are indifferent between spending time outdoors in line or indoors in the loqge." The individual is assumed not to be contemplating other possibilities for the use of his time but rather how many runs will be obtained. With a zero opportunity cost of time, the lift line efficiently allocates a fixed number of rides equally to all skiers. The efficiency of the lift line will be demonstrated later. Consider now, the pricing decision of each individual ski firm. As with ride-ticket pricing, each individual skier will only choose to ski at a particular area as long as the total gain from skiing exceeds the total costs. In a lift-ticket pricing ski industry, an individual facing the choice between \' ' I ~ , ,. 11 two areas, j and k, will choose the area from which he will have the greater consumer surplus. Each area is initially assumed to have identical lift capacities, x, and therefore able to offer a fixed quantity of rides per individual per day. Given each area's fixed lift capacity, the quantities, qi and qk, are determined solely by the number of individuals choosing to ski at each area. Homogeneous individuals will be indifferent between the two areas as long as the net surpluses are equal. Therefore, (2. 6) where,n is the lump sum lift ticket price. Once a ski area chooses its lift ticket price, the number of skiers adjusts to keep the net surpluses at each area equal.. In other words, a ski hill is a price taker with respect to the reservation value of net surplus. This reservation value of net surplus, which is the skier's gain from skiing minus the associated total costs, dictates a unique inverse relationship between the lift ticket price, n j , and the number of skiers at area j . As ski area j increases its lift ticket price, the number of skiers will decline, and the number of rides per skier increases so as to maintain the equality of net surplus across the different ski areas. Assuming that a ski area's marginal cost per ride is zero, a firm attempting to maximize profits given a fixed capacity will simply maximize total revenue (Hirshleifer, 12 1988, pp. 124-25). Total revenue is defined as the product of the lift ticket price and the number of skiers. To maximize revenue, a ski area will set its lift ticket price such that qj = xjnj = o· 1 (xjnj), the elasticity of nj with respect to11: J can be derived by implicit differentiation of the reservation value of net surplus (the left side of equation 2.6), that is, (2.7) an. -1 __ J =------- a1t j D_l ( X ) ( .2£.) n. n2 J j Multiplying both sides of this equation by -~~ - yields the n elasticity of the number of skiers with respect to the lift ticket price, (2. 8) Maximization of revenue requires the ski area to operate such that the left hand side of ( 2. 8) is equal to -1. Upon rearranging terms, (2. 9) Dividing (2.9) by qj indicates that the average price per ride is equal to the marginal willingness to pay for the q/h ride, (2.10) 1t J - D_l ( qj) . qj Since D" 1 (qj) is simply the inverse demand function, the i I 13 average price per ride in lift-ticket pricing is equal to the effective price per ride, ,.. 1t . pj = _J. qj (2.11) It follows that the fixed number of rides available per skier at area j under a lift-ticket pricing policy is equal to the quantity that would be demanded at the "effective" price per ride Pi, if each ride were sold individually, i.e., (2.12) Since each ski area is identical and since skiers have homogeneous demands, equilibrium dictates that each area offer the same lift ticket price, and serve the same quantity of skiers, each at the same effective price per ride. On any given day the number choosing to ski is a function of the effective price per ride and the fixed cost shift parameter, (2. 13) N=N(P, s) . Market equilibrium requires that the total number of rides offered, Jx, equal the total number of rides demanded, where the latter is the product of the number of rides demanded per individual, D(ft) , and the number of individuals, N(P, s) . (2 .14) Jx = D (F) *N(P, s) . This lift-ticket market equilibrium is equivalent to the ride- ticket market equilibrium described in equation (2.5). However, the price per ride, P, has been replaced with the 14 effective price per ride, P. Substitution of equation (2.12} into equation (2.11} dictates that the lift ticket price is determined by the effective price per ride. (2. 15) The lift ticket price, as equation (2.15} shows, is determined by the product of the equilibrium effective price per ride and the number of rides each skier receives. This effective price per ride, P, for q rides, would equal the equilibrium price per ride, P, reached with ride-ticket pricing. The identical effective price per ride with both ticketing schemes allows each skier to receive the same quantity of rides at the same total cost regardless of the pricing scheme used. With a lift ticket scheme, the individual rides are just packaged together. The equality of ride ticket price and effective price per ride is what Barre and Romer (p. 8 7 5) refer to as the "package-deal effect." The package-deal effect refers to the situation wherein an individual is indifferent between purchasing a group of items in a pre-determined package or on a per-unit basis. It would occur if the individual is offered each item at a price, the sum of which would equal the price of the same items offered as a package with the transaction costs of the two purchases being equal. In the current context, an individual would be indifferent between purchasing twenty ride tickets at a price per ride of $1.50, or a lift 1 ! 1. ' 15 ticket with a constraint of twenty rides for the price of $30.00. Since the same equilibrium-is reached with a ride-ticket pricing system without queues and a lift-ticket pricing system with queues, the queues must result in no efficiency loss. Skiers and ski areas make their allocative decisions, not on the explicit zero marginal price per ride but rather on the implicit marginal effective price per ride, P~D-1 (q). Each competitive firm maximizes revenue with respect to the reservation value of net surplus, resulting in an identical allocation of rides and cost per ride as that of an equilibrium in the ride-ticket pricing scheme. The lift ticket is simply a package deal whereby the consumer is offered a group of q rides, each at an effective price per ride of F. The total revenue received by the ski area and cost to the skier are the same with either pricing scheme. Since both ride-ticketing and lift-ticketing policies result in the same revenue, the method that minimizes monitoring and set-up costs is chosen. It is argued that lift-ticket pricing is chosen because ride-ticket pricing would require greater costs to set and enforce contracts. Barre and Romer believe that only with ride-ticketing would ski areas be required to continually change prices and monitor skiers. With ride-ticketing, skiers would be required to purchase a ticket for every ride up the ! '· ' i I I I ' i I, ' 16 hill or purchase a group of tickets ahead of time. The latter, however, would still require some physical action be taken upon a skier's ticket for every ride up the hill. Furthermore, Barre and Romer stress that a ride-ticketing system would require a larger menu of prices than a lift-ticketing system. In practice the demand for ski rides is not known with certainty at the beginning of the day. Throughout the day demand can change and ride-ticket pricing would need to be adjusted. For example, early in the day demand and price could be low, while later in the day demand could increase. To totally eliminate lift line queues, the ride-ticketing ski hill would have to constantly monitor the demand for ski rides and change the ride price accordingly. The lift-ticket pricing scheme with its lift line ride allocation mechanism, however, results in an automatic adjustment in the implicit price per ride through increases in the queue, thereby reducing adjustment costs. But, Barre and Romer also assume that the price elasticity of demand is approximately equal to unity. As is shown below, this automatic adjustment in the effective price per ride is such that lift ticket prices would seldom need adjustment. To demonstrate how this automatic adjustment operates, assume, as Barre and Romer do, that an increase in aggregate . demand results from a change in the shift parameter, s. The shift is caused by a decrease in the fixed costs of going skiing, which could be due to a decrease in the opportunity i I 17 cost of time associated with the weekend and holiday periods. Such a shift increases the number of people whose reservation value of net surplus exceeds the lift ticket price. As this occurs there will be more skiers per area. With a fixed capacity and an increase in demand, each skier will receive fewer rides. This results in an automatic increase in the effective price per ride. When the elasticity of demand with respect to the effective price per ride is unitary (absolute value), no lift ticket price change is necessary. A very large portion of Barro and Romer's analysis deals with what they see as the presence of a stickiness in lift ticket prices over time and across areas. During the course of a season, such as over weekends and during Christmas, the demand for skiing varies, but lift ticket prices are seen by Barro or Romer to remain fixed. Analysis of equation (2.15) reveals the conditions whereby the effective price per ride under the lift-ticket pricing scheme can vary in the same direction as demand without any adjustments to the lift-ticket price. Taking the derivative of equation (2.15) with respect to the effective price per ride yields the incremental effect of a change in the lift ticket price due to a change in the effective price per ride, (2. 16) d1t = PD1

+ D(:P} • c1P Upon rearranging terms, it can be seen that the degree and direction of change in the equilibrium lift ticket price I i i I I I 18 depends on the elasticity of demand for rides with respect to the effective price per ride, (2.17) Since all individuals have the same demand function, 1lv,P' also defines the individual's elasticity of demand for rides with respect to the effective price per ride. Equation (2.17) shows the profit maximizing change in lift ticket price due to an incremental change in the effective price per ride. When aggregate demand increases the effective price per ride will automatically increase. But, depending on the elasticity of demand, the profit maximizing lift ticket price may increase, decrease or stay the same. The pricing behavior due to an increase in demand can better be understood with the aid of Figure (1). In this figure ski areas have a total lift capacity, Jx, and face a market for ski rides which is comprised of individual demand curves such as di. The initial aggregate market demand is D0 which later increases to D1 as the number of skiers increases. The initial lift ticket price is, P"c,aq0 0, which is determined by the effective price per ride P0 , and the quantity of rides per individual q 0 • An increase in demand from D0 to D1 , ceteris paribus, would not necessitate any change in the lift ticket price if a unitary elastic demand existed. With a unitary demand the effective price per ride increases, but this increase is exactly offset by a decrease in the number of I ' I I I. G) (,) ·;:: c. I I " : Po --------r---- 1 I I I I I I I I I I I I I 19 Jx 0._----~--~--------------------~ q1 q 0 Jx Number of Rides Figure 1. Price Response to Demand Increase I ' ! 20 rides per individual, i.e., P0 aq0 0 = P1 b% 0 . If demand were elastic, the increase in demand would dictate a decrease in lif-t ticket prices because the decrease in rides would more than offset the increase in the effective price per ride. That is, P1 b%0 < P0 aq0 0 • If there existed an inelastic demand for rides with respect to the effective price per ride, lift ticket prices would increase as demand increased, i.e., Barre and Romer conclude that constant lift ticket prices result because the elasticity must remain very near unity throughout most of the year. Any potential increases in revenue from a lift ticket price change would be offset by the menu costs incurred from the price change. Barre and Romer infer that only in cases of very low demand, such as at the end of the season, does the elasticity of demand vary enough from unity to justify a change in lift'ticket prices. Their model also predicts that there should be less variation in lift ticket prices across areas and seasons, than in the number of skiers or length of lift lines. While these predictions are relative statements, Barre and Romer leave the impression that elasticity of demand must be very close to unity. The foundations of the Barre and Romer model described so far were derived upon the assumption that all ski areas are identical and that skiers differ only in their fixed cost 21 involved in going skiing. As these assumptions are relaxed to allow for differences in lift capacities, travel costs, and heterogenous individuals, the basic conclusions do not change. Barra and Romer also allow lift capacities to vary considerably across areas, but lift ticket prices remain fixed. This can be explained with the use of what they refer to as the homogeneity effect. The homogeneity effect with lift-ticket pricing refers to the case where there are identical skiers who are free to choose from among J ski areas which differ only in their respective lift capacities. The skiers. will sort themselves amongst the various areas such that, in equilibrium, the number of skiers at each area will vary one-for-one with their respective capacities and where each individual will pay an identical effective price per ride. If ski area B has twice the lift capacity as area A, area B will have twice as many skiers, and skiers at each area will receive the same number of rides. Skiers will adjust themselves between the areas until the net surplus at all the areas are equal. Thus, lift ticket prices will be identical across all ski areas. This same homogeneity effect allows for one lift ticket being good across all the lifts at one area on a given day. The runs at a hill that have the greatest demand would also . have the longest lines. In equilibrium a lift that provides a run that offers twice the distance as a neighboring run would be accompanied by a lift line that would allow for only 22 half the number of runs as the neighboring run. The net surplus in equilibrium must still be equal across every lift at a hill, as well as across all hills, otherwise an individual could gain surplus by choosing to ski at a different lift or area. Ski areas maximize revenue subject to the reservation value of net surplus, and skiers choose amongst the areas to keep the net surplus equal at all areas. Competition ensures that net surpluses are equated as equation (2.6) dictates. To further expand on this point, assume two identical areas existed at different distances from a metropolitan area but charged the same lift ticket price. The hill at the closer proximity to the metropolitan area will have a larger amount of skiers. The equilibrium condition will be reached when the difference in travel costs associated with reaching the farther hill is made up for by the gain in extra rides available at the distant hill. This results in an equilibrium condition where the net surpluses once again are equal at both of the areas. As described previously, the elasticity of demand for rides with respect to the effective price per ride determines the effect of a demand change on the optimal lift ticket price. Over the range where the elasticity is unity there need be no difference in lift price associated with the difference in travel costs. The Barro and Romer model, however, would predict that in the presence of an elastic demand efficient pricing would dictate a lift ticket price 23 that is higher the more distant the hill. An inelastic demand would dictate a higher priced ticket the closer the hill. As always, the elasticity of demand with respect to the effective price per ride determines the change in the "gain from skiing" with a change in the number of rides available and therefore determines the equilibrium lift ticket prices. As a final point, Barro and Romer consider the existence of two different types of skiers, avid skiers who demand a greater number of rides at every price than less avid skiers. The queuing mechanism intrinsic in the single lift-ticket pricing scheme allocates rides across ski slopes equally among each skier and cannot differentiate between the differences in demands of the various skiers. The solution posed by Barro and Romer is that two types of ski areas will develop such that each area will either cater to the avid or less avid skier. The avid hill will charge a higher lift ticket price such that fewer skiers will choose it. Thus, it will be able to offer a greater number of rides for each avid skier than the less expensive ski area. To summarize, Barro and Romer argue that lift-ticket pricing leads to no inefficiencies even though the explicit marginal price per ride is zero and queues exist. The lift ­ ticket equilibrium in this competitive market is simply a package of the same quantity of rides and effective price per ride that would be demanded and offered under ride-ticket pricing. The lift-ticket pricing policy is used because it can 24 be implemented with lower monitoring and set-up costs as compared to a ride-ticket pricing system. Skiers adjust themselves across areas to keep the net surplus equal at all areas. The competitive ski areas must take the skiers reservation value of net surplus as a given, but are free to set their lift ticket price. Importantly, the Barre and Romer model predicts that if the elasticity of demand is unity, ski areas will not adjust their lift ticket prices even with predictable fluctuations in demand. Barre and Romer leave the clear impression that lift ticket prices are sticky and that queuing is a prime method for allocating rides. Furthermore, in their model ski areas are price takers with respect to the effective price per ride. Each competitive area has no market power. Thus, an area has no ability to price discriminate across demands. I ! I 25 CHAPTER 3 BARRO & ROMER TESTED AND QUESTIONED Barro and Romer (1987) assert that lift ticket prices are "sticky" and that this stickiness is due to a unitary elasticity of demand. As described previously, with a unitary elasticity of demand with respect to the effective price per ride, ski areas would not need to adjust lift ticket price with changes in demand. Lift ticket prices would remain fixed over time and across areas because as demand changed skiers would simply adjust themselves until the net surpluses are once again equal. The assertion of a unitary elasticity and sticky prices can be tested in two ways. The assertion that the elasticity is unitary can be tested with statistical analysis, and the stickiness of lift ticket prices can be tested by comparing prices over time and across areas. The pricing behavior of a competitive ski area in Barro and Romer's model and its implication for testing can better be understood with the aid of Figure (2). There, ski areas have an initial market lift capacity of Jx0 which is later increased to Jx1 • The ski areas face an aggregate market demand function, Dm, which is comprised of individual demand functions such as di. At the initial lift capacity the lift ticket price is, P0aq0 0 , which is determined by the effective I ' .I (J) (.) ·;:::: c.. " Po 1 A I p1 -----------------t-- 1 I I I I I I I I I I I I I 26 Om di 0~----------~--~---------r----~------------~ Jxo Number of Rides Figure 2. Price Response to Capacity Increase 27 price per ride ~' and the quantity of rides per individual, q 0 • As lift capacity increases to Jx1 the effective price per ride would decrease from ~ to P1 and the number of rides per individual would increase from q0 to q 1 • An increase in capacity from Jx0 to Jx1 would not necessitate any change in lift ticket price if a unitary elastic demand with respect to the effective price per ride existed. With a unitary demand, the capacity increase would have no effect on lift ticket price because the decrease in the effective price per ride is exactly offset by an increase in the number of rides per individual; P0 aq0 0 - P1 b%0. If the capacity increase resulted in a lift ticket price increase it would suggest the presence of an elastic demand because the increase in rides would more than offset the decrease in the effective price per ride. That is P~b%0 > P0 aq0 0. If the lift ticket price is inversely affected by an increase in capacity, an inelastic demand would be suggested. In such a case, P1 b% 0 < P0 a q 0 0 . Similarly if the number of skiers increase with a fixed capacity, the number of rides per individual decreases and therefore the effective price per ride increases. If the correlation between the number of skiers and the lift ticket price is zero, a unitary elasticity would be implied. Thus, skier visits should have a positive effect on lift price if the demand is inelastic or a negative effect if demand is elastic, holding capacity constant. These relationships between lift ticket price, capacity 28 and skier visits suggests a simple empirical specification to test whether the effective price elasticity of demand is equal to unity as Barra and Romer contend. This specification can be described as ( 3 . 1) 1t- a + b 1 V+ b 2 C+ 1J. t:. Here,1t is the lift ticket price. The number of skier visits is represented by v, while c represents ski area lift capacity. Equation (3.1} can be estimated with a regression of lift ticket price on skier visits and capacity. This regression estimates the coefficients of visits and capacity and their significance on price. From these results Barra and Romer's assertion of a unitary elasticity can be tested as was described with Figures (1} and (2). Estimation of (3.1} with ordinary least squares, however, would result in biased estimates due to the simultaneous interaction of the two endogenous variables, visits and price (Rao and Miller, 1971} . Since price is dependent on visits, and visits is dependent on-price, visits are correlated with the error term, ut, thereby causing a violation of one of the basic assumptions of the classical linear regression model. Therefore, a two-stage least squares regression technique, which produces consistent unbiased estimates of equation (3.1}, is used. The two-stage least squares technique replaces visits with a computed instrumental value that is intended to eliminate the stochastic elements that are correlated with the error term. 1 ! ! 29 The effectiveness of this technique is dependent on the ability to develop an instrumental variable which is highly correlated to visits but not correlated with price. The variables other than lift ticket price which can have a significant impact on the yearly number of skier visits are: weather, population, personal income, and the skiing region. Weather was included to capture effects on skier visits that are due to weather conditions. The population variable was included to capture any fluctuations in skier visits due to demographic changes. Since personal income can affect the opportunity cost of going skiing it also was included. Eastern ski areas have many lifts with small vertical displacement while western ski areas have fewer lifts with large vertical displacement. Since the capacity data (VTF) used in this analysis is the product of the vertical displacement and number of rides, regional dummy variables were included to capture effects due to these regional variations. Data on lift ticket prices were obtained for the skiing seasons 1978-79 thru 1988-89 (Goeldner, 1978-88). These figures were only available as yearly average regional lift ticket prices including all discounts offered. The five regions covered are: the Northwest, East, Midwest, Rockies and the West. The nominal lift ticket prices were deflated by the Consumer Price Index using 1972 as the base year. Data on total yearly alpine area skier visits were obtained which correspond to the regions and time series lift ticket price 30 data (Kottke, 1978-88}. The capacity data were available from the magazine Ski Area Management (1978-88}. These figures are tabulated by the magazine as total vertical transfer feet per hour, VTF, and compiled to coincide with the above regions. Vertical Transfer Feet per Hour, or VTF, is the product of a chairlift's rated capacity in skiers per hour, and the vertical displacement of the lift ride. The data used to estimate the instrumental variable were gathered from a number of sources. The average number of operating days per season, which was used as the proxy for weather conditions, was available from the annual Economic Analysis of North American Ski Areas (Goeldner, 197 8-88) Regional population figures in the 14-44 year old age group were taken from the Statistical Abstracts of the U.S. (U.S. Department of Commerce, 1978-88}. This population dataset was chosen because according to a pair of studies approximately 80 percent of all skiers are in the 14-44 year old age group and 76.8 % of all skiers ski within 500 miles of their home (Ski Industries America, 1984). The regional average yearly per capita Personal Income was developed from the Statistical Abstracts of the u.s (U.S. Department of Commerce, 1978-88). It was deflated with the Consumer Price Index with 1972 as the base year. Annual data for 11 seasons beginning in 1978-79 and five regions yielded a time series database of 55 usable data points per variable. Table (1} presents the means and standard -I I I \I } I i I ! I ' I I ! } i 1 I I I I I I 31 deviations for all the variables. Table 1. Variables; Means and Standard Deviations. VARIABLE MEAN STANDARD DEVIATION Price 4.40 0.66 (1972 dollars) Visits 16.685 15.563 (Millions) Capacity 5.283 5.077 (Thousand VTF) Weather 121.24 18.502 (Days of operation) Population 33.499 35.041 (Millions) Personal Income 4354.9 484.00 (1972 Dollars) The results of the two-stage least square regression of equation (3.1) with the data just described are reported in table (2). The coefficient on the capacity variable is significant and positive at the 5 9.,-0 level, while the coefficient on skier visits is insignificant, providing inconclusive results. As described with Figure (2) the insignificance of visits on price supports the assertion of unitary elasticity of demand. On the other hand, the positive and significant coefficient on capacity suggests an underlining elastic demand for rides. These results do not provide consistent support for the assertion of a unitary elasticity of demand. 1 l 32 Table 2. 2SLS Regression Results. Variables Coefficient Visits 0.26E-3 Capacity 0.44E-7 Dummy 1 0.40E-2 Dummy 2 0.19E-2 Dummy 3 -0.25E-2 Dummy 4 0.27E-2 * R-Squared Adjusted = 0.8782 t-ratio 0.728 7.112 2.813 1. 022 -1.456 -3.105 In addition to the above test, there are a number of instances where Barre and Romer's assertion of unitary demand elasticity resulting are at odds with real world lift-ticket pricing practices. For example, Barre and Romer (p. 881) argue that a substantial interval of time should exist "where lift ticket prices would show little or no variation with demand." While their statement is in relative terms many ski areas have multiple prices and change lift ticket prices frequently. For example, Breckenridge ski area in Colorado had 78 different lift ticket prices during the 1988-89 season (Russell, 1989). Breckenridge is not the only hill with such pricing policies. The American Ski Association, an organization that offers lift ticket discounts to club members, offers lift tickets from many areas which change 10 times or more during the season (American Ski Association, 1990). These price fluctuations exist at every ski area. Most areas at least lower the face value of their lift tickets or offer discounts off the face value for weekday periods (Berry, 1990). I r I I , I I I , I I I I 33 Barro and Romer (p. 884) also assert that there is a "cross-sectional stickiness" in lift ticket prices. However, the 1987-88 Economic Analysis of North American Ski Areas shows lift ticket prices that are anything but cross­ sectionally sticky (Goeldner, 1988). The face value weekend ski lift ticket prices vary from $10.00 to over $35.00 and the average revenue per lift ticket ranged from $11.41 in the midwest to $20.91 in California and Nevada. Lift-ticket revenue per skier visit also varies greatly with respect to area capacity. Areas of small capacity receive an average revenue per lift ticket of $10.83 whereas areas with the largest capacities receive revenues of over $23.00 per lift ticket. These pricing differences would not be observed if the homogeneity effect suggested by Barro and Romer were prevalent. The cross-sectional variation in lift ticket prices is especially evident in the Wasatch Mountains east of Salt Lake city, Utah where 7 major ski resorts are within a few miles of one another (Ski Utah, 1989). Snowbird, the area closest to Salt Lake City, charges $32.00 per lift ticket. A mile further up Little Cottonwood canyon at Alta, a day's lift ticket sells for $19.00. In the neighboring Big Cottonwood Canyon, Brighton sel1s weekday lift tickets for $14.00 and weekend or holiday tickets for $18.00. Across the canyon from Brighton at Solitude, weekday lift tickets are $17.00 and holiday and weekend tickets are $21.00. A single day's lift ticket good I i i l I I I I I I I . i I i i i I i I 34 for both Solitude and Brighton can be purchased for $28.00. The ski resorts of Park City, Deer Valley and Park West, which operate on the other side of the ridge from Brighton, have substantially different lift ticket prices. These three areas have lift tickets selling for $40.00, $35. oo and $24. oo dollars, respectively. The pricing practices observed in the Wasatch Range are clearly not cross-sectionally sticky. Further, the higher prices for the interconnect lift-tickets good for both Brighton and Solitude would not exist under Barre and Romer's assumptions. There are a number of additional assumptions in the Barre and Romer model which seem at odds with actual behavior. For example, Barre and Romer assume that the willingness to pay per ski ride is independent of the number of skiers at a hill. Given this assumption, the only determining factor in the willingness to pay for a lift ticket is the number of rides available per individual. There may, however, be an additional congestion effect which is not accounted for in the Barre and Romer model. This congestion effect is caused by a disutility that each additional skier's presence imposes on every other skier's overall skiing experience. As more people ski, the queues are assumed to absorb each additional skier, while the number of skiers on each run remains the same. A more realistic assumption, borne out by observation, is that more skiers create more crowded runs. Assuming that skiers prefer to ski on uncrowded slopes, the value of each ride would be I I I 35 inversely related to the number of skiers at an area. The inclusion of a congestion effect could negate Barra and Romer's homogeneity effect. Furthermore, this congestion effect can explain why ski areas often limit the number of skiers during peak periods. As Uel Gardner, manager of Wintergreen ski resort in Virginia, puts it, "Another policy is to limit the number of skiers to minimize overcrowding on the trails and to reduce the lift line wait." (Ayers, 1989) Ski area operators, of course, limit the number of skiers by raising price. Another problem with the Barra and Romer model is that lift lines may not always fluctuate with demand so as to automatically adjust the effective price per ride. If the number of skiers is restrained or if lift lines do not exist, the automatic adjustment cannot take place. Only during the holiday periods do most ski areas approach 100% utilization. Utilization often drops to as low as 25% for destination resorts and 50% for day areas during non-holiday periods (McKinsey and Co., 1989). During the 1989-90 ski season, Bridger Bowl near Bozeman, Montana, averaged 1208 skiers per day (Travel Montana, 1989). With a capacity of 45,600 rides per day, the average skier could have 38 rides per day and face no.lift lines. In the Barra and Romer model a requirement for equilibrium is that the total number of rides demanded always equal the number supplied. A lift line must always exist and be allowed to fluctuate with demand to automatically I ' I ' i l 36 adjust the effective price per ride with demand changes. They argue that lift-tick~ts are chosen over ride tickets because the automatic price adjustment lessens monitoring and set-up costs. Since this automatic adjustment often cannot operate, the cost advantages of lift-ticketing over ride-ticketing are reduced or even negated. There is another assumption that raises doubt that lift~ ticketing is simply chosen over ride-ticket pricing because of lower monitoring costs. This assumption is that the opportunity cost of time is negligible. Barre and Romer (p. 880) state that "if the typical skier's fixed cost, c, for getting to the ski area is large, and if waiting in line is preferred to spending time on other available activities" then ·a positive opportunity cost of time would be relatively unimportant. Furthermore, they state that "skiers receive the same number of rides at the same cost in each case. The same people end up participating, and each ski area receives the same revenue." (Barre and Romer, p. 879) What the Barre and Romer model does not explain is, if the same number of people are participating and receiving the same number of rides with either pricing scheme, where will the people who stand in the lift lines go under a ride-ticket pricing scheme? There are, for example, restaurants at most ski areas. Thus with the alternative activities to standing in lift lines that are available at a ski resort, then the greater the opportunity cost of time the less efficient the lift-line ride allocation I I 37 system would be. A positive opportunity cost of time decreases any advantage of lift-ticketing over ride-ticketing policy because people would be willing to pay a price to avoid waiting in line. Consider now, that Barre and Romer's formulation of the function assumes only the possibility of a rotating market demand curve. Barre and Romer state (p. 877) "Overall, we can write the number of persons N who choose to ski as a function, of N = N(P,s) ." Here, (s) defines the shift parameter which describes the changes in the distribution of the fixed costs, ci, involved in going skiing. When the ci's are low, as during weekends and holidays, many people will choose to ski. When the fixed costs are higher, as during weekdays and non-holiday periods, fewer people will choose to ski. The market demand curve rotates outward (as shown in Figure (1)) as the fixed costs decrease, and rotates inward as the fixed costs increase. An alternative assumption would be to allow· rotating as well as shifting demands. Early in the season when people have not skied in a long time, their demand curves may be relatively high. At the end of the season, after skiing all winter, demands may shift inward. such behavior could explain the large discounts on lift tickets at the end of the season (Russell, 1990). But, if individual demands are shifting over time, then Barre and Romer's·simple relationship between the elasticity of demand and the lift-ticket price no longer holds, and the empirical test offered in Table (2) does not 38 provide a test of whether the underlying elasticity of demand is equal to unity. Perhaps the most important assumption that shapes the foundation of the Barre and Romer model is that individual ski operators are price takers. Competition, which forces ski areas to be price takers with respect to the effective price per ride, is essential to the Barre and Romer model. There is evidence that this assumption may not accurately describe the alpine ski industry. In the Barre and Romer model (p. 877), each consumer faces a market comprised of J firms, each of which "have a negligible impact on aggregate quantities." While over 600 ski areas exist in the u.s., they are scattered and the travel costs required to reach each can vary substantially (Enzel, 1989). The market that each skier faces is limited to those areas where the total costs, including travel costs, are similar, and exceeded by the consumer surplus received from skiing at the hill. With scattered ski areas and positive travel costs, there are likely to be many instances where a skier's net surplus is negative, except at a few nearby areas. This suggests that each ski area will possess some degree of market power, and the price taking behavior that Barre and Romer model is unlikely. Furthermore, the presence of more than one ski area operating in the same location is not enough to guarantee a competitive marketplace, at least according to the supreme 39 Court. It was ruled that Aspen Skiing Co. monopolized the market for downhill skiing services in Aspen, Colorado (Aspen Ski Co. v. Aspen Highlands Skiing Corp., 1985). While there were two firms operating ski areas in the same general location, the Court nevertheless found that Aspen Skiing Co. possessed monopoly power. The presence of market power can be further witnessed by the price discriminating practices used throughout the alpine ski ride industry. Phlips (1983, p. 6) defines price discrimination as "two varieties of a commodity are sold to two buyers at different net prices." Bridger Bowl near Bozeman, Montana for example, offers coupons, for $ 5.00 off of lift tickets, available only to skiers staying at local hotels (Lippke, 1989). They also offer lift tickets for $13.00, a 35 % savings, but only if the skier is willing to purchase an item at Burger King. Most ski areas offer discounts which are only available to skiers who join a national ski club, such as the American Ski Association (1990). As men~ioned earlier, Breckenridge ski a~ea had 78 different lift tickets during one season (Russell, 1989). It is quite evident that lift tickets were commonly sold to different customers for different prices on the same day. On any given day, there are many different lift-ticket prices offered to different individuals, for entry onto the same hill. This is just one more instance where it has been shown I I I 40 that the alpine ski market envisioned and modeled by Barro and Romer differs substantially from the one that actually exists. Regression analysis does not confirm the presence of a unitary elasticity of demand, and analysis of pricing shows lift ticket prices that are not sticky. Furthermore, lift lines often do not exist so the automatic adjustment in the effective price per ride cannot take place. Importantly, the assumption of a competitive ski ride market appears tenuous. 41 CHAPTER 4 MONOPOLISTIC SKI-LIFT PRICING In this chapter a model of a ski industry is developed in which each ski area is assumed to posses market power. Both simple monopolistic and price discriminatory pricing are considered for ride-ticket and lift-ticket pricing. Comparison of ride-ticket and lift-ticket pricing in the monopoly model reveals that revenues derived using lift-ticket pricing generally exceed those of ride-ticket pricing. The monopoly model predicts that lift-ticketing would be the dominant pricing scheme due to its revenue maximizing capabilities. In contrast, Barre and Romer's competitive model shows ride and lift-ticketing to result in identical equilibria and revenues and that ski areas choose lift-ticket pricing on purely~a cost minimization basis. Skiers face a market that is limited to those areas where the total costs, including travel costs, are exceeded by the consumer surplus received from skiing at the hill. over 600 ski areas are scattered throughout the United States and the travel costs required to reach each varies substantially, depending upon a skiers location. With scattered ski areas, scattered skiers, and positive travel costs, there are likely many instances where a skier's choice set will be restricted ' I 1, } 42 to only local areas. This suggests that each ski area will posses some degree of market power and be a price setter. To capture this market structure, it will be assumed that each ski area faces a market comprised of various identifiable groups of skiers, comprised of Ni skiers. The skiers are assumed to have homogenous demand characteristics within a group, but these characteristics vary across groups. As in Barro and Romer, skiers can differ in their demand for ski runs across groups but not in the time it takes them to ski each ski run. This assumption allows every skier at a hill using a lift-ticket pricing scheme to receive the same quantity of rides. The queuing mechanism remains the method by which a ski area using lift-ticket pricing can distribute a fixed number of rides equally to all skiers. Each monopolistic ski area has the option of using ride­ ticket pricing, where each ride is sold individually at a single price, P, or lift-ticket pricing with a lump-sum entry fee and a zero price per ride. The lift-ticket pricing scheme is designed to capture the entire net consumer surplus and therefore cannot exceed the gain minus the fixed costs of skiing. For simplicity in this analysis, however, fixed costs are ignored. With a fixed capacity of K rides and a marginal cost per ride that is assumed to be zero, the ski area will .choose the ride-ticket or lift-ticket pricing scheme that maximizes revenue. This comparison of a single and two-part tariff differs ' \ 43 from that analyzed by Oi (1971) in his Disneyland ride pricing model. Ride allocation in Oi' s model allows the number of rides allocated per individual to vary across individuals with either pricing scheme. With the per individual ride allocation mechanism of Oi's model, two-part tariff revenue always exceed single-part tariff revenue. Unlike oi•s Disneyland model, a ski area can differentiate the number of rides offered to each individual only with ride-ticket pricing. With lift-ticket pricing the ski lift line allocates rides equally across all skiers. This lift-ticket ride allocation mechanism constrains the revenue generating capabilities of a lift-ticketing policy as compared to oi•s model. This constraint suggests that conditions may exist where ride ticket revenue exceeds lift-ticket revenue. To analyze these revenue generating differences, consider first a ski area that faces one group of N1 homogeneous skiers. Assume that the inverse market demand function for this group is linear and defined by (4 0 1) b1Q P -a1-]i 1 1 P1 is the individual price per ride, b 1 is each individual's demand slope, and Q is the number of rides available to the group. Under ride ticket pricing and a zero marginal cost of supplying a ride, the ski area will operate at full capacity as long as marginal revenue is greater than zero. Substituting l ''} 44 B1 for b 1/N1 and the ski area ride capacity, K for Q, yields the total ride-ticketing revenue (4. 2) TRlrt - alK- B1K2. Under a lift-ticket pricing policy the ski area can generate revenue equal to the product of each individual lift ticket price, 1t 1 , and the number of skiers N1 • Here, the revenue maximizing ski area would set the lift ticket price equal to each individual's total willingness to pay. If the ski area charges anything greater than this, the skier would choose not to ski. If anything less than this is charged, the ski area is not maximizing revenue. The total willingness to pay is equal to the area under the individual's demand curve from zero to the rides available per individual, q 1 ; that is (4. 3) 4> (q) -fa q1D-1 (q) dq. Equating willingness to pay with lift ticket price and assuming linear demands, we obtain (4. 4) 1tl =a1 q1 - bl qt 2 The number of rides available per individual, q 1 , is equal to K/N1 • The product of the lift ticket price 1t 1 , and the number of skiers, N1 , as well as substitution of K/N1 for q 1 , and B1 for b 1jN1 , results in a total lift ticket revenue, (4. 5) TRlLT .. alK- B1K2 2 . 45 Comparison of equations (4.2) and (4.5), (4. 6) alK-BlK2 :;; a K- B1K2 1 -2- I indicates that a ski area serving just one group of homogeneous skiers would maximize revenue by using a lift- ticket pricing scheme because its revenue exceeds ride ticket revenue by B1K 2/2. Notice that if the demand function is perfectly elastic, B1 is equal to zero and both ride and lift ticketing revenues would be equal. Hence the Barra and Romer assumption of competitive pricing is a special case. The revenue generating capabilities of ride-ticketing and lift-ticketing for a ski area serving a single homogeneous group are shown diagrammatically in Figure (3). If the lift capacity is K0 the total revenue under a ride ticket policy is equal to OP0bqb, since qb ride tickets would be purchased at price per ticket of P0 • With lift-ticketing, revenue equals Oabqb. Since, at a capacity of K0 the ski area would be operating at full capacity with either pricing method, each skier would receive the same K0/N rides with either ride­ ticketing or lift-ticketing. The effective price per ride, however, is larger with lift-ticketing because each individual is forced to pay a greater amount for the same number of rides. Clearly lift-ticketing revenue exceeds ride-ticketing revenue by P0ab. The figure illustrates another revenue generating advantage that lift-ticketing has over ride-ticketing. If the C1) u 't: a. ) I I 46 Ko K1 K2 a Po p1 o~--------~~--~--------~~--------~ Number of Rides Figure 3. Ride-Ticket versus Lift-Ticket Revenue 01 47 capacity is increased to K1 , ride ticket revenue would only rise to OP1cqc. The ski area would sell only qc rides because marginal revenue would be negative at full capacity. With a capacity of K1 , however, a lift-ticketing scheme would still generate positive marginal revenue at full capacity with total lift ticket revenue equal to Oad~. At this capacity the number of rides sold with lift-ticketing would exceed those sold with ride-ticketing. Only if the ski area has a capacity greater than K2 will it operate below full capacity with either scheme. At capacities greater than K2 , note that qe rides would be taken. This is exactly twice the qc rides and twice the revenue which would maximize revenue under ride ticketing. Accordingly, lift-ticketing provides an incentive for a ski area to operate at a greater capacity. Now consider the pricing options for a ski area facing not only the N1 skiers in group 1 but also a second different group of N2 skiers and assume that these two separable groups have different demand elasticities. The demand equation for group 1 remains as previously described in equation (4.1), while the demand equation for group 2 is (4.7) P2 = a2- B202 There are two possible relationships for the two demand functions. If .the two groups are classified as either "avid" or "less avid 11 skiers, as described by Barra and Romer, individuals in group 1, avid skiers, demand a greater number of rides at all prices than skiers in group 2. such non- 48 intersecting group demand functions are not very interesting because under such cases the ski area would maximize revenue by serving just the avid group. The demand relationship which create a more complex pricing option is the case where the demand curves are intersecting, a possibility not considered by Barro and Romer. One group may have a higher willingness to pay for the first few rides but a smaller, marginal, willingness to pay for a large number of rides. What makes this case important is that with intersecting demand functions ride-ticketing may be the preferred option. Since lift-ticketing operates subject to the constraint that all skiers receive the same number of rides, while ride ticketing allows maximization with adjustments in ride allocation and price, ride-ticketing revenue may be greater than lift-ticketing revenue. A ski area would choose to serve both groups only if it increased revenue. Assume, for now, that it does maximize ride ticket revenue by serving the two groups of demanders with a simple monopolistic pricing scheme and that the capacity constraint is binding. To analyze the revenue generation under a ride-ticketing scheme, serving both groups with a single ride ticket price, it is necessary to sum the two demand curves horizontally to derive the total market demand curve, (4. 8} P - a1B2 + a2Bl Bl+B2 B1B2 K. Bl+B2 49 The total ride ticket revenue received by serving both groups is the sum of the ride ticket price, P, and number of rides, K, (4.9) TR - a1B2+a B 2 1 B +B K- 1 2 B1B2 K2 B1+B2 To determine whether ride-ticket pricing can exceed lift ­ ticket pricing under these conditions, compare equations (4.5) and ( 4. 9) , (4.10) a1K- B1K2 2 < > a1B2+a2B1 K­ B1+B2 B1B2 K2 B1+B2 If the left side of equation (4.10) is always greater than the right side, the revenue derived from serving both groups with a single ride ticket price can never exceed the revenue received from serving just group 1 with lift-ticket pricing. To see this, multiply equation (4.10) by (B1+B2); 2 K 2 < K2 B1B2 > a B K+a B1K-B1B2 -- 1 2 2 2 (4.11) a1B1K+a1B2K- B12K2 Subtracting a 1B2K from both sides and then dividing by B1 yields, (4.12) BK 2 < BK2 aK--1- > aK- - 2 1 2 - 2 2 50 Equation 4. 12 indicates that the comparison between ride­ ticket and lift-ticket pricing reduces to a simple comparison of lift-ticket pricing between the two groups. As long as the total lift-ticket revenue received from serving just group 1 exceeds the total lift-ticket revenue that would be generated from serving just group 2, lift-ticket pricing revenue exceeds ride ticket pricing revenue. Otherwise, the ski area would choose to serve just group 2 with lift tickets rather than just group 1. This reversal would still generate the result that lift-ticketing revenue is greater than ride-tick;eting revenue with a single price per ride. Of course, the ski area also has the option of serving both groups with singular priced lift tickets. A revenue maximizing ski area would choose to serve both groups with a singular priced lift ticket when such a policy would generate more revenue than a scheme of serving just one group. In this case, lift-ticketing would clearly beat simple monopolistic ride-ticketing. The analysis shows that when facing two groups with linear demands, simple lift-ticket pricing revenue exceeds simple monopolistic ride-ticket pricing. The above result, however, holds only as long as the assumption of linear demands is valid. Once this assumption is relaxed to allow for non-linear demands, the results can change. The simple illustration in Figure (4) shows that a ski area with a capacity of 30 rides could generate revenues of $60.00 by serving both skiers with a price per ride of $2.00. ., <.0' c: ro ~ z 0 :::::1 I c :::::1 CD jl) '""I 0 CD 3 jl) :::::1 a. (/) z c 3 g - "'' 0 0 - z c 3 C" ~ 0 0 Price Price I I I I I I I I I I I I I I I I I ·­ I I I I I I I I I I I I I­ I I I I I I I I I I a :c c.: ...&. (;) '3" ().::!. :21 u 8:22 8():28 1':;,18() 1 8u555 :25·::1.,:~. 54· 5'3 10 .. :25 :22 u (>~'2·:.1 8';15E:J 1'381 1 81! •353 257150 81 11.52 2~2u1!51 1002'3 1'3!-32 1 11 .. 457 :2~3'::J.::~B7 121 12. E,5 :22. &::~77 11204 :1. '383 1 '3 a ~:)23 2683·45 75 1 ~'2. 8'3 22. 6~5E, 1 :;~ :l ~'2(> 1'384 1 :l2n <)87 ~'28~:~6 :L Ci 1:25 :L31185· :2:~~ u sc;~~. 13541 :l '38~3 1 11.083 309381 10'3 :t .::!. u .::J.:2~ :~~3 u (J7 1 4'30~5 :l '385 1 12.835 32'33.::~3 11 '3 15.58 23 .. :24 15'352 1'387 1 14.745 35'3042 1 ""•t::' ..:;,....! 16.4 23. 1!:~~31 17155 1988 1 14.421 3'3'3'357 12'3 17.2'3 23n 5(>4 18531 1 '38'3 1 12.741 4100'35 121 1 '3. 0'3 23 • .:~·88 201 '31 i 1'37'3 2 3.763 1 '343'38 117 7. ·::J·'3 23.843 8248 I I 1980 2 4 ·-:•'? • .a.:..w 200515 '3'3 7,,57 15.846 '312-:L 1'381 ·';'t . .::. 4·. 172 :20219'3 108 '3. 37 16.126 1012':::) 1'382 2 5.064 2134'30 123 10.75 17.655 11252 1.'383 ~"2 4.256 22252(i 80 10.55 17. '365 12051 1'384 ~'2 5.:1.75 236780 125 12.26 18.267 13.:1·06 1'385 2 4. 3'34 25.::!.606 103 11.8 18.631 14555 1986 .-, ..::. 5 .. 2:L8 26194'3 113 13.71 18. '387 15518 :1.'387 2 5.816 274785 112 14.84 1 '3. 375 :1.6436 1'388 .-. ..::. 5.885 2'31121 1.-..... ~w 16.61 1'3.716 17626 1'38'3 2 5.424 301020 104 18.0:1. 1 '3. '302 18'35'3 1'37'3 3 9.743 114080 14'3 6.05 26.711 8178 1'380 3 8.682 121'324 128 7,.3 27.018 '3053 1 '381 3 7.688 1241'37 110 '3. 34 27.07'3 '3715 1'382 3 7 .. 846 1276'30 135 '3 n 07 27n316 10612 1983 3 6.2:L3 1318'31 '3'3 8.71 27.381 1 0'37':;.J 1 ':!84 3 6 .. '361 134236 1 :"23 '3. 16 27 A &!~•32 1 :I.G77 1'385 3 6.899 138 j, 10 117 '3. '3'3 27.5'38 12'::J58 1'386 3 7.201 141584 :1.35 1(lll 53 :2'7 Q 77";~ :l3687 1987 3 6. ·:.:;1'34 146055 124 10.:28 2"7 a '353 14·::J.26 1'388 3 6.783 151340 130 11.41 28.07'3 15227 198'3 3 7.013 153'3 :to 13'3 12. 16 28. 1 0'3 16:23'3 75 Table 5. (continued). ------------ . - Yeay F.:egio:on '·l:isits VTF Days F'·r i o: e F'o:op. F'I -~·--·--·---------·-- - ... ·------·-----------··- i '37':3 .::j. :l~:.s" 837 340032 13:3 7. '34 4.743 7.::!.:2'7 1'380 4 :1.7. :I.E, 3 b Ci .::1.':1 5 1.-,,.., .&::.~ '3. 41 4 .. '::J3t:?, s::2l:;s :1. '381 4 10.486 3'33(H,;2 ·::~5 10.63 ~5. 408 ·;:; :L.:.l (.~, 1'382 4 :1.5.337 424803' 1:26 :1 .-, 1::'"7 . ..::.. ~ / 5nb77 1 () :1.77 1'38~3 4 14.808 455823 132 12.88 5.841 :I.OE./3 1'384 4 16.80:1. 486475 :1.41 13 .. 8 5 .. •;171 ~ 1·:l58 :1. '38~5 .::j. 17. 62·:+ 511;7'30 141 14.87 (;" <)8'7 1 :~~ 1 E.::: :!. '386 .::j. 15.859 571603 127 15.24 6" L:Ci\3 :L ~"2E.85 1987 4 16.68 622734· 125 16.96 6" ~33 :L ~3 C> ·~ ·;1 1·::m8 4 16.564 641506 116 17.55 6.404 13620 1 '38'3 4 16.601 666656 122 18.67 6. 43'3 14363 1 '37'3 5 '3.56 305341 155 8.42 14. 16'3 8726 1'380 5 '3.473 324'38~'2 13'3 '3.4 14.581 ''3:777 1'381 5 8.401 3411'31 103 10.63 15.504 10770 1'382 C' ~J 11.004 35582'3 13'3 12. 2:2 15. '37 11802 1'383 5 :1.2. 061 374944 154 13 .. 83 16.308 122'30 1'384 5 '3. 606 3'38376 138 14.5'3 16.614 135~'26 1985 C' ~J 11" 35:2 418370 145 15.36 16.'328 14608 i 1'385 1.:' ...J '3.7'37 4467·42 143 15.63 17. :2:'36 15482 I 1':::J87 5 '3. 554 457742 12:2 15.54 17 .. 6'3 16217 r. ) 1988 5 10.255 486255 132 15.85 18.025 17168 1 '38'3 5 11.556 500'3'3 :1.35 1'3.81 18.31'3 18138 j I I S: XIGN:!IddV I I I I I ! 9L I i i I I I I 77 The following outlines the computer program referred to in chapter 4. Demand equations: P1 = a1 - b 1q1 • P2 = a2 - b2q2. Enter variables: Capacity in rides per day, K. Number of skiers in group 1, n1 • Number of skiers in group 2, n2. Y intersept for group 1, . a 1 • Y intercept for group 2, a2. Slope for skier in group 1, b 1 • Slope for skier in group 2, b2. A. Ride-ticket equilibrium and revenue. Steps: 1) Determine how many rides to offer: If K < ( a 1-a2 ) I ( 2b11n1 ) then it is best to serve just group 1, q1=KIN1 q2=0 Else if; Serve both groups. Set marginal revenues equal and allocate rides accordingly. If K < ( a 1n112b1 + azn212b2) . _ · q1 ~ ( a 1-a2 + ( 2b~Kin2 )) 1 2 (b11n1) + (b21n2) q2 - ( a2 a 1 + ( 2b1Kin1 ) ) 1 2 (b21n2) + (b11n1 ) Else if; Allocate rides at zero marginal revenue. q1=a112b1 q2=a212b2 2) Set price per ride. P1 = a 1 - b 1q 1 P2 = a2 - b2q2 3) Total ride-ticket revenue. Revenue = P1n1q1 + P2n2q2 78 B. Lift-ticket equilibrium and revenue. Steps: 1) Calculate revenue if serving just group 1. If K > a 12n1jb1 Then q 1 = a1jb1 Revenue, just group 1 = a 1a 1n1/2b1 Else q 1 = K/n1 Revenue, just group 1 = (b1/2) (K/n1) (K/n1 ) + (a1-b1K/n1) (K/n1) 2) Calculate revenue if serving just group 2. If K > a 2n2jb1 then q2 = az/bz Revenue, just group 2 = (a2a2) n2/2b2 Else q2 = K/nz Revenue, just group 2 = (b2/2) (K/n2) (K/n2) + ( a2-b2K/n2) (K/n2 ) 3) Calculate revenue if serving both groups. If K > a 1n1jb1 + a2n2/b2 then Else q 1 = a1jb1 % = az/bz Revenue,groups 1 and 2 = (qla1n1)/2 + (q2a 2n2)/2 q 1 = K/ ( n1 + n2) q 2 = K/ ( n1 + n2 ) Revenue, groups 1 and 2 = (b1/2) (q1 ) (q1 ) + (a1-b1q1 ) (q1) + (bz/2 ) (qz) (qz) + (az-bzqz) (qz) 4) Determine revenue maximizing lift ticket choice. c. Final Step: Compare A. and B.