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Garage and curbside parking competition with search congestion Inci, Eren; Lindsey, C. Robin Jul 7, 2015

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Garage and Curbside Parking Competition withSearch CongestionEren InciySabanci UniversityC. Robin LindseyzUniversity of British Columbia07 July 2015AbstractWe consider a parking market with privately operated parking garages, publiclyoperated curbside parking, and drivers who di¤er in parking duration. In equilibrium,long-term and short-term parkers can allocate themselves to garages and curbside park-ing in various ways. The equilibrium is generally ine¢ cient because garage operatorsexercise market power, and drivers have to search for curbside parking which createsa search cost externality. A city planner can achieve the social optimum without reg-ulating garage prices if it can charge di¤erentiated curbside parking fees. Many citiesstill have parking meters that can only levy uniform fees and it is very costly to replacethem. However, numerical results show that the e¢ ciency loss due to uniform pricingis modest in most cases.JEL Classi…cation: D62; L13; R41; R48Keywords: endogenous outside option; parking; price discrimination; search costs;spatial competitionWe would like to thank Richard Arnott, Jos van Ommeren, the editor, Yves Zenou, and two anonymousreferees, and session participants at the 54th European Regional Science Association Congress (SpecialSession Organized by the Urban Economics Association), the Kuhmo Nectar Conference on TransportationEconomics 2013 at Northwestern University, the 1st Meeting on Transport Economics and Infrastructure atthe University of Barcelona, and 2014 All-Istanbul Economics Workshop at Bilgi University for very helpfulcomments. Inci would like to acknowledge …nancial support from the Scienti…c and Technological ResearchCouncil of Turkey (TUBITAK Career Grant 111K051). All errors are our responsibility.yTel.: 90-216-483-9340; fax : 90-216-483-9250. Address: Sabanci University, Faculty of Arts and SocialSciences, Orhanli / Tuzla 34956 Istanbul TURKEY. E-mail address: ereninci@sabanciuniv.edu.zTel.: 1-604-822-3323. Address: University of British Columbia, Sauder School of Business, 2053 MainMall, Vancouver, BC, CANADA V6T 1Z2. E-mail address: robin.lindsey@sauder.ubc.ca.11 IntroductionDowntown parking markets can be rather complex. Parking is often available both on ando¤ the street. Parking garages provide bulk capacity at discrete locations, and can extendover multiple storeys above or below ground. The friction of space gives them market power.Curbside parking, on the other hand, is located more widely but it is frequently in shortsupply and di¢ cult to …nd. According to some estimates, cruising for parking accounts forroughly thirty percent of tra¢ c at certain times of day (Shoup, 2005, 2006). Time spentsearching for parking increases the full price or generalized cost of curbside parking, andlimits the degree to which curbside parking constrains garages’market power.Garage and curbside parking di¤er in how they are priced. Garages generally cater todrivers who park for di¤erent lengths of time, and they usually charge hourly parking feesthat vary with parking duration. Curbside parking is typically priced at a uniform hourly ratein North American cities where conventional parking meters are used. However, nonlinearpricing is sometimes practiced in cities where labor is relatively cheap. For example, inIstanbul employees collect parking fees using hand terminal technology and the hourly feesvary by parking duration. Finally, administration arrangements vary. Curbside parking ispublicly operated in most cities whereas garage parking can be publicly or privately operatedand/or regulated. For example, some Dutch cities such as Maastricht and Almere regulategarage parking fees while garages in London and Boston are free to choose their prices.In this paper we study downtown parking markets in which spatial competition betweengarage and curbside parking, nonlinear pricing, and curbside parking search congestion aresimultaneously at play. To facilitate analysis, the model is kept simple by treating totalparking demand as …xed, ignoring through tra¢ c congestion, and considering only two typesof individuals that di¤er in the amount of time they wish to park. Nevertheless, curbsideparking search congestion creates an interdependence between parking submarkets and non-convexities in garages’pro…ts, and the derivation of market equilibria in this setting is newto the spatial competition literature. In such an environment, we attempt to answer somequestions about downtown parking markets: How does competition between parking garagesplay out when curbside parking is available as a substitute? How can garage parking feeschedules be explained? How should curbside parking fees be set to control cruising conges-tion and parking garage market power? Is a uniform hourly fee optimal, or should hourlyfees be varied with parking duration? Is regulation of garage parking necessary to achieve asocial optimum, or can curbside parking fees do the job?Several strands of literature cover part of the ground required to address these questions2(see Arnott, 2011 and Inci, 2015 for comprehensive literature reviews). Some studies considercurbside parking in isolation (Arnott and Inci, 2006, 2010). Others incorporate garageparking but omit heterogeneity with respect to parking durations (Arnott and Rowse, 2009;Arnott, Inci and Rowse, 2015). Parking duration is considered by Glazer and Niskanen(1992) and Calthrop and Proost (2006), and parking time limits by Arnott and Rowse(2013). Spatial competition between parking garages has been studied (Arnott and Rowse,1999, 2009; Anderson and de Palma, 2004, 2007; Calthrop and Proost, 2006) but withoutincluding price discrimination. Yet other studies analyze parking and pricing of parkingwhen tra¢ c congestion occurs at a bottleneck (Arnott, de Palma and Lindsey, 1991; Zhanget al., 2005; Fosgerau and de Palma, 2013). Shoup (2005, 2006), Arnott and Inci (2006,2010), and Arnott, Inci and Rowse (2015) emphasize the importance of curbside parkingsearch congestion externalities in downtown districts, and van Ommeren, Wentink, andDekkers (2011) and van Ommeren, Wentink, and Rietveld (2012) estimate the externalitiesempirically. Perhaps the most thorough analysis of parking market competition to date is byArnott (2006) who analyzes spatial competition between parking garages in his core model,which he later enriches to include curbside parking. However, he assumes that drivers areidentical and he does not consider price discrimination.Understanding the e¤ects of driver heterogeneity and price discrimination on downtownparking markets is important. It highlights the market power of parking garages that shouldbe considered in designing optimal parking policy. For example, a parking garage may in-crease the parking fee it charges to short-term parkers, some of who will then park on thecurb and increase curbside parking search congestion. This in turn increases demand fromlong-term parkers for parking in the garage, which allows the garage to charge a higher fee tothem. Such e¤ects cannot be studied in models with identical drivers and no price discrim-ination. Another important insight is that if parking garages price discriminate accordingto parking duration, it is welfare-enhancing to di¤erentiate curbside parking fees as well.1Indeed, in our model the social optimum can be achieved without any regulation of garageparking fees by di¤erentiating curbside parking fees.Spatial competition and price discrimination have been extensively studied in the indus-trial organization literature (see Gabszewicz and Thisse (1986), Varian (1989), and Stole(2007) for literature reviews). Spatial competition models such as Salop’s (1979) allow forthe possibility that some potential customers choose not to buy a product or service from any1Although parking meters in many North American cities can only charge uniform fees, the practice of notcharging for parking in the evening introduces an element of second-degree price discrimination. Time limitson parking are common in other countries. However, varying curbside parking rates in hourly incrementsaccording to duration of stay is very rare.3…rm, but select an outside option instead. These models can be adapted to the downtownparking market by treating parking garages as …rms o¤ering services that di¤er by location,and curbside parking as an outside option that is ubiquitous. The standard models typicallyassume that utility from the outside option is exogenous. However, in the parking marketexpected utility from curbside parking decreases with the number of individuals who use itbecause of search congestion. Our setting is unique in incorporating such an endogenousoutside option into a Salop-type model.A few empirical studies of competition in parking markets have recently appeared. Kobus,Gutierrez-i-Puigarnau, Rietveld and van Ommeren (2012) examine the e¤ects of parkingfees on drivers’choices between curbside and garage parking. Froeb, Tschantz, and Crooke(2003), Choné and Linnemer (2012), and De Nijs (2012) focus on the e¤ects of mergers in theparking industry. Lin and Wang (2014) examine the relationship between competition andprice discrimination. Several general lessons emerge from these studies which inspired thegeneral structure of our model. First, hourly garage parking fees generally decline steeplywith parking duration. Put another way, total payment or outlay is an increasing but steeplycurved concave function of parking duration. Second, the degree of curvature in the outlaycurve declines with increased competition. Third, the (short-run) marginal supply cost ofparking is close to zero for garages. Fourth, drivers are reluctant to walk more than a fewblocks from a parking garage to their destination.The paper is organized as follows. Section 2 describes the model. Section 3 characterizesvarious possible equilibrium allocations of driver types between garage and curbside parking.Section 4 derives the socially optimal allocation of driver types between garage and curbsideparking space, and shows how the allocation can be decentralized using di¤erentiated hourlycurbside parking fees. Section 5 uses a numerical example to illustrate how the welfare gainsfrom implementing optimal di¤erentiated fees depend on such parameters as the distancebetween parking garages, parking search costs, and walking time costs. Section 5 also assessesthe relative e¢ ciency of setting optimal uniform curbside fees. Section 6 discusses extensions,and Section 7 concludes. An (online) appendix provides derivations of the various equilibria,and technical and computational details.2 The ModelConsider a …xed set of individuals (henceforth drivers) who travel to a downtown area bycar. Drivers di¤er in their destinations and lengths of stay. A long-term parker (denoted by4L) requires parking for lL hours while a short-term parker (denoted by S) requires parkingfor lS hours, where lS < lL.2 A type i driver, i = L; S, receives a bene…t of Bi from a trip.The Bis are large enough that all potential trips are made, and total parking demand istherefore price inelastic.Each driver has a given trip destination. Destinations are uniformly distributed arounda circle with densities dL for long-term parkers and dS for short-term parkers. Demand forparking by type i drivers in hours per unit distance is hi  dili. Parking is available atparking garages and on the curb.3 Curbside parking is operated publicly and distributedcontinuously around the circle. Parking garages have …xed locations a distance D apart.Each garage is operated by a separate private …rm. Garage parking space is lumpy becauseof scale economies in garage capacity (Arnott, 2006).4Curbside parking in many cities is priced at a constant fee per hour. However, to allowfor price discrimination and the use of curbside parking fees to enhance market e¢ ciency,it is assumed that curbside parking fees can di¤er, with type i drivers paying an hourlyfee of pci, i = L; S. Short-term parkers therefore pay pcSlS to park for lS hours, and long-term parkers pay pcLlL to park for lL hours. Depending on how parking fees are levied andenforced, incentive compatibility constraints may apply. If pcS < pcL, a long-term parkermight be able to save money by interrupting his visit, returning to his car, and either movingit to another parking spot or feeding the meter. But doing so would be inconvenient, andwe rule it out. If, alternatively, pcS > pcL, it is possible that pcSlS > pcLlL. A short-termparker could then stay at the destination an extra lL lS hours and save pcSlSpcLlL on theparking bill. The driver might also be able to pay the long-term charge and then leave afterlS hours. To admit this possibility we will entertain the incentive compatibility constraintpcSlS  pcLlL: (1)2One interpretation is that long-term parkers are commuters and short-term parkers are making businesstrips. Another is that long-term parkers are multipurpose or comparison shoppers who shop for an extendedperiod of time, whereas short-term parkers are one-stop shoppers who need to park for a shorter time.3Parking is sometimes also available at surface lots. Surface lots are typically built as transitory usesof land after buildings are torn down, and therefore o¤er only temporary additional space to park. Surfacelots are similar to garages in that they do not contribute appreciably to search congestion. In our modelsetting, lots can be treated as equivalent to garages. There are also parking spaces supplied by businessesto employees (van Ommeren and Wentink, 2012) and customers (Hasker and Inci, 2014), which we ignore.Their presence lowers garages’local market power.4We abstract from safety issues in di¤erent parking forms. In some countries, a lot of crime is committedin parking garages. According to the Bureau of Justice Statistics, more than 1 in 10 property crimes occurredin parking lots or garages between 2004 and 2008 in the US. In Istanbul, cars parked on the curb have ahigher risk of burglary or damage due to accidents than cars parked in garages.5This constraint is not imposed in the analysis of Sections 3 and 4, but it is addressed inthe numerical analysis of Section 5. To concentrate on the behavior of garage operators,curbside parking fees pcL and pcS are treated as exogenous until Section 4.The time required to …nd a curbside parking space is assumed to be proportional to thetotal number of hours of curbside parking that is occupied between adjacent garages, T . Ifall drivers of both types park on the curb, T = (hL + hS)D. A type i driver incurs a searchcost of kiT , where ki > 0 is the type-speci…c unit search cost.5 The generalized cost ofcurbside parking for a type i driver is therefore pcili+ kiT , and the net bene…t from a trip isBi  pcili  kiT; i = L; S: (2)Parking garages charge an hourly fee of pgL for parking lL hours, and an hourly fee ofpgS for parking lS hours. The time required to locate a garage and park a vehicle there isassumed to be negligible. However, garage customers have to walk from the garage to theirdestination and back. A type i driver incurs a round-trip walking time cost of wix, where xis the distance between the parking garage and the destination, and wi is the type-speci…cwalking time cost per unit distance in both directions combined. Similar to the case with5One interpretation is that drivers require time to …nd a vacant spot, but can secure one close enoughto their destination that walking time between the parking space and the destination is negligible. Analternative interpretation is that on-street parking is available not only on the circle, but also on intersectingroads and/or on roads that run parallel to the circle one or more blocks away. As nearby spots …ll up, drivershave to travel further from the circle in order to …nd parking. Time is spent driving to a vacant spot, walkingto the circle, and later walking back to the vehicle. Since occupancy of space is proportional to length ofstay, in both interpretations the time cost depends on the total number of hours that curbside parking isused, T ; i.e. the number of vehicles that use curbside parking multiplied by average parking duration. Thefunctional dependence of the search cost function on T depends on a number of factors (Brooke et al., 2014):where curbside parking is located relative to destinations on the circle, use of curbside parking for otherpurposes (e.g., residential parking, employee parking, retail deliveries), parking turnover rate, prevalence ofillegal parking and rule enforcement, driver familiarity with parking supply and preferences (e.g., for walkingdistance, security and shade), and drivers’choice of search strategies. One assumption is that search cost isinversely proportional to the fraction of parking spots that are vacant (Axhausen et al., 1994; Anderson andde Palma, 2004). If the number of parking spots is …xed, search cost increases to in…nity as the occupancyrate of parking approaches 1. However, in the model here destinations and curbside parking space areboth dispersed over space. As more drivers seek curbside parking, the range of locations over which theysearch expands. If curbside parking is equally prevalent on and o¤ the circle, drivers might search withina diamond-shaped area centred about their destination. Since the number of spaces available within thearea would rise with the square of its linear dimension, expected search costs might increase with the squareroot of the number of searchers. However, this would not be the case if parking availability decreases withdistance from the circle. A further complication is that search costs tend to be higher for individuals withdestinations midway between garages since parking search will be most intense there. The linear functionalform adopted in equation (2) is an intermediate case between concave and convex functional dependenciesthat has the advantage of analytical tractability.6curbside parking, an incentive compatibility constraint may apply:pgSlS  pgLlL: (3)As discussed later, it is also possible for the short-term garage parking fee to exceed thelong-term fee (i.e., pgS > pgL).A type i driver who parks at a garage a distance x from his destination incurs a generalizedcost of pgili + wix and gains a net bene…t ofBi  pgili  wix; i = L; S: (4)This speci…cation embodies the assumption that a driver has a …xed parking duration whichis equal to the sum of walking time to/from the destination and the visit duration. Visitduration therefore decreases with distance from a parking garage. This assumption simpli…esthe analysis. It also precludes the possibility for garage operators to price discriminatebetween drivers on the basis of their walking distance.A parking garage incurs a cost of  for each hour that a car is parked.6 Thus, it earns apro…t of (pgi  )li from a type i driver. To assure that parking garages can earn positivepro…ts in equilibrium, it is assumed that the generalized cost of curbside parking when alldrivers of both types park on the curb exceeds the supply cost of garage parking during theirvisit:pcili + ki (hL + hS)D > li; i = L; S: (5)Following common practice in the literature on spatial competition, attention is focusedon symmetric equilibria in which all garages employ the same parking fee schedules. Considerone garage called the “home garage.”Given the simple, deterministic nature of demand,competition on the circle is localized and the home garage only competes directly either withcurbside parking or with the nearest garages on either side.7 In any candidate symmetric6This cost includes any costs related to transactions, security, vehicle damage, and wear and tear on thegarage. Since the cost of curbside parking is set to zero in the model, it is assumed that garage parking ismore costly to provide than curbside parking. Garage parking usually entails construction above or belowground, while curbside parking requires only allocation of existing road space. We neglect the opportunitycost of using space for curbside parking rather than moving tra¢ c. Arnott, Inci, and Rowse (2015) accountfor this trade-o¤.7If the home garage sets parking fees much lower than its neighbors’fees, it would gain the business notonly of all customers located between it and its nearest neighbors but also some customers located on thefar side of the neighbors. The literature has sometimes adopted a ‘no-mill-price-undercutting’assumptionto preclude hyper-competitive pricing and customer ‘leapfrogging’of this sort. The potential pro…tability ofundercutting is muted in the model here because utility from the outside alternative— curbside parking—7equilibrium, the nearest neighbors adopt the same parking fee schedules as the home garage.The home garage therefore attracts type i drivers from a distance xi on either side. It earnsa pro…t of 2(pgi  )hixi from type i drivers, and a total pro…t of (pgL; pgS) = 2 (pgL  )hLxL + 2 (pgS  )hSxS: (6)The home garage chooses pgL and pgS to maximize this pro…t expression.3 Market EquilibriumIn any symmetric equilibrium, the home garage competes for each type of driver eitherwith curbside parking or with its nearest garage neighbors. There are nine possible marketcon…gurations in all. Two are illustrated in Figure 1 by showing a section of the circle betweenthe home garage and one of its neighbors. In panel (a), garage parking fees are so low thatno one parks on the curb. The market boundary for type i drivers between the home garageand its neighbor is located at a distance xi from the home garage and a distance x0i = Dxifrom the neighbor. A type i driver with a destination on the market boundary is indi¤erentbetween parking at the home garage and parking at the neighbor, while preferring bothto parking on the curb. We call this driver the “marginal type i driver”. The boundariescan di¤er for long-term and short-term parkers although in a symmetric equilibrium bothboundaries are located mid-way between the garages (i.e., xL = xS = D=2). Because nodrivers park on the curb, T = 0 in this con…guration. We call this market con…gurationregime Lg + Sg, meaning that both driver types use only parking garages.Another con…guration in which some drivers park on the curb is shown in panel (b) ofFigure 1. Type i drivers located within a distance xi of the home garage park there, thoselocated within a distance x0i of the neighbor park there, and those located in the centralregion with a span of D  xi  x0i park on the curb. Thus, the marginal type i driver whois located a distance xi from the home garage is indi¤erent between parking at the homegarage and on the curb. Similarly, the marginal type i driver located distance x0i from theneighbor is indi¤erent between parking at the neighbor and curbside parking. Hence, thetotal number of hours spent parking on the curb isT = hL (D  xL  x0L) + hS (D  xS  x0S) : (7)increases as the price of garage parking drops. Undercutting is not pro…table in the symmetric equilibria wederive.8Figure 1: Alternative market con…gurationsWe call this market con…guration regime Int, meaning that each driver type uses both garageand curbside parking, and the market boundaries are in the interior of the market segmentbetween the two garages (i.e., 0 < xL < D=2 and 0 < xS < D=2).The full set of nine market con…gurations is depicted in Figure 2. The location of thehome garage’s market boundary for long-term parkers is plotted on the horizontal axis, andthe location of the boundary for short-term parkers is plotted on the vertical axis. In regimeLg + Sg, shown in panel (a) of Figure 1, both driver types park exclusively at garages. Inregime Int, shown in panel (b) of Figure 1, both types split between garages and the curb.In regime Lg, long-term parkers use only garages while short-term parkers split betweengarages and curbside parking (i.e., xL = D=2 and 0 < xS < D=2). Regime Sg is de…nedanalogously. In regime Lc, long-term parkers use only curbside parking while short-termparkers split between garages and the curb (i.e., xL = 0 and 0 < xS < D=2). RegimeSc is de…ned analogously. In regime Lc + Sc, both types of drivers park only on the curb(i.e., xL = xS = 0). In regime Lg + Sc, long-term parkers use only garages while short-term parkers use only the curb (i.e., xL = D=2 and xS = 0). Regime Lc + Sg is de…nedanalogously. Equation (7) is modi…ed accordingly for each regime. Which one of the regimesprevails in equilibrium depends on parameter values including the curbside parking fees, pcL9Figure 2: Candidate equilibrium regimesand pcS. Since pcL and pcS are treated as given in this section, the equilibrium will be calledthe “current equilibrium”. Regime Int is examined in the next subsection in more detail inorder to provide a better sense of the model’s properties and the nature of equilibrium. Allother regimes are examined in the (online) appendix.3.1 Equilibrium for regime IntIn regime Int, shown in panel (b) of Figure 1, some drivers of each type park at garagesand the rest park on the curb. The home garage chooses pgL and pgS to maximize its pro…tgiven in (6). Several conditions must hold for this regime to be an equilibrium. First, themarginal driver of each type must be indi¤erent between parking at the home garage andparking on the curb:Bi  pgili  wixi = Bi  pcili  kiT; i = L; S: (8)Second, the net bene…t to a marginal type i driver of parking at the home garage must bestrictly greater than the net bene…t of parking at the neighbor:Bi  pgili  wixi > Bi  p0gili  wi (D  xi) ; i = L; S: (9)10This guarantees that some drivers of each type park on the curb. Third, if practicallyrelevant, the incentive compatibility constraint in (3) must hold.The market boundaries, xL, xS, x0L, x0S, and total time spent parking on the curb, T , canbe solved using the two conditions in (8), the two analogous conditions for the neighboringgarage, and equation (7). There are …ve linear equations in …ve unknowns. The solution forxL isxL =wLwSkL (hL + hS)D + wSkLhLlLp0gL  pgLwL (wLwS + 2kLhLwS + 2kShSwL)+(wS + 2kShS) lL (pcL  pgL) kLhSlS2pcS  pgS  p0gSwLwS + 2wSkLhL + 2wLkShS: (10)The formula for xS is obtained from (10) by interchanging the L and S subscripts. As shownin online Appendix A.1, the equilibrium is stable in the sense that a perturbation in thenumber of drivers of either type who park on the curb induces adjustments that return thesystem to equilibrium asymptotically.A notable feature of (10) is that the home garage’s market boundary depends on theparking fees set by the neighbor, p0gL and p0gS, even though the garages do not competedirectly with each other. The garage markets are interdependent because the fees chargedby one garage a¤ect the number of drivers who choose to park on the curb, the intensity ofcurbside parking search congestion, and hence the demand for parking at the neighboringgarages.The …rst-order conditions for an interior pro…t maximum for the home garage are@@pgL= 2hLxL + (pgL  ) @xL@pgL+ 2hS (pgS  ) @xS@pgL= 0; (11)@@pgS= 2hSxS + (pgS  ) @xS@pgS+ 2hL (pgL  ) @xL@pgS= 0: (12)These equations yield closed-form but very long expressions for pgL and pgS, as well as longexpressions for xL and xS when the formulas for pgL and pgS are substituted into (10) andits counterpart for xS.The comparative statics properties of equilibrium garage parking fees and market bound-aries are mostly ambiguous. Table 1 lists some properties for pgL and xL for the case wherecurbside parking fees are undi¤erentiated (i.e., pcL = pcS = pc).8 The garage market bound-8The comparative statics properties for pgS and xS are mirror images of those for pgL and xL except forthe condition lLlS , which is unchanged.11lL lS wL wS pcdL; dS;kL; kSpgL  if kL = kS + if kL = kS + if lL= lSand kL= kS + if kL= kS ?xLSame signas pc  Opposite signto pc   if lL = lSand kL = kS+ if lL= lSand kL= kS+ if lL = lSand kL= kS?Table 1: Comparative statics properties of interior equilibriumary for long-term parkers, xL, expands with parking duration, lL, if the curbside parkingfee is higher than the cost of garage parking, and decreases if the curbside parking fee islower. This is because the relative monetary costs of garage and curbside parking becomemore important compared to curbside parking access costs as parking duration increases.By contrast, an increase in parking duration of short-term parkers, lS, causes the garagemarket for long-term parkers to contract if pc >  because it induces short-term parkersto use less curbside parking. This reduces search costs for curbside parking, and attractssome long-term parkers away from garages. The market boundary xL also tends to contractif walking cost, wL, rises. But garages tend to raise the parking fee, pgL, because garageparking demand becomes less price elastic. An increase in wS has the opposite e¤ect on xL,but it has a similar e¤ect on pgL because curbside parking becomes less elastic. Finally, asexpected, both the garage market and the garage parking fee tend to increase if curbsideparking becomes more expensive.It is possible in regime Int for the hourly garage parking fee to be higher for long-termparkers than short-term parkers.9 Conversely, it is also possible for the long-term fee to be somuch lower that incentive compatibility constraint (3) is violated. For example, this occursif lL and lS are similar, and if short-term parkers have higher walking and search costs thanlong-term parkers. Section 5 includes a numerical example in which this happens.One, degenerate, case in which (11) and (12) yield simple solutions occurs if search costsfor curbside parking are zero (i.e., kL = kS = 0). In this case, garage parking fees andmarket boundaries work out topgi =+ pci2; i = L; S (13)xi =(pci  ) li2wi; i = L; S: (14)9For instance, assume for the moment that pcL = pcS and kL=kS = lL=lS . One can easily show thatpgL = pgS in this case. Then, within a neighborhood of this point there are parameter combinations forwhich pgL > pgS .12If kL = kS = 0, curbside parking serves as an outside good with exogenous utility as inSalop’s (1979) model. Garages can then attract a type i driver without losing money onlyif pci > . They set their fees for each driver type half way between the cost of providinggarage parking, , and the fee for curbside parking, pci, as per equation (13). The garagemarket for each type in (14) varies proportionally with the type’s parking duration, li, andinversely with its walking cost to and from garages, wi. The simplicity of formulas (13) and(14) shows that search costs are essential to make the model interesting and useful for policyanalysis.l w pc d kpg  + + + +x+=Same sign as pc    ++=+ if pc  +=+ if pc  Table 2: Comparative statics properties of interior equilibrium with homogeneous driversA second case with a simple solution is the case in which drivers are identical so thatthere is e¤ectively only one driver type. Garages then set only one parking fee, pg, and thereis only one market boundary, x, between garage parking and curbside parking. The homegarage’s problem then is to choose a single parking fee, pg, to maximize pro…t, 2(pg )dlx,subject to the constraints pgl+wx = pcl+ kT , p0gl+wx0 = pcl+ kT , and T = h(Dxx0).There is no incentive compatibility constraint with just one type. The symmetric equilibriumsolution is given bypg =pcw + wdkD +  (w + kh)2w + kh; (15)x =kh (w + kh)D + (pc  ) l (w + kh)(2w + kh) (w + 2kh): (16)The comparative statics properties of this simpli…ed model, listed in Table 2, follow bydi¤erentiating (15) and (16). The e¤ects of market density, d, and search time costs, k,are now de…nitive. If the density of parkers increases, and the curbside parking fee exceedsthe supply cost of garage parking, the garage parking fee increases and the garage marketexpands. This happens because the garage market operates with constant returns to scalewhereas, due to search costs, curbside parking su¤ers from diminishing returns. An increasein the size of the market therefore favors garage parking. If search costs for curbside parkingincrease, and the curbside parking fee is less than the supply cost of garage parking, thegarage parking fee increases and the garage market expands. However, if curbside parking isvery costly, garages have substantial market power and they can increase their fees so much13that the garage market area contracts.Derivations of equilibria for the remaining regimes are relegated to (online) AppendicesA.2-A.9. These derivations are novel in dealing with endogeneous outside options in Salop-type spatial competition models.4 Social OptimumSince total parking demand is …xed, the social optimum corresponds to a total cost minimum.Total costs are determined by how many drivers of each type park in garages, and how manypark on the curb. All drivers who park at a garage should patronize the nearest one, andwith garages spaced at equal intervals the social optimum can be derived by minimizing totalcosts within a distance D=2 on either side of the home garage.Total costs (TC) are the sum of garage supply costs (GC), walking time costs (WC),and curbside parking search costs (SC). The component costs in a symmetric equilibriumare tallied as follows. The number of type i drivers on either side of the home garage whopark there is 2dixi. Type i drivers park for li hours, and it costs  dollars to provide an hourof parking. Therefore, garage operating costs areGC = 2 (hLxL + hSxS) : (17)A type i driver bound for a destination x units away from a garage incurs a walking costfor the round-trip of wix. Total walking time for type i drivers is 2diR xi0xdx. Total walkingtime costs for all drivers are thereforeWC = 2dLwLZ xL0xdx+ 2dSwSZ xS0xdx = dLwLx2L + dSwSx2S: (18)There are di(D  2xi) type i drivers who each park for li hours on the curb and incur asearch cost of kiT , with T given by (7). Total curbside parking search costs are thereforeSC = [kLdL (D  2xL) + kSdS (D  2xS)] [hL (D  2xL) + hS (D  2xS)] : (19)The social optimum is derived by choosing xL and xS to minimize total costs, TC =GC + WC + SC. In principle, the optimum can fall into any one of the nine regimesdescribed in Section 3 for the current equilibrium. However, assumption (5) rules out regime14Lc + Sc in which garage parking is not used. The model assumptions also rule out regimeLg+Sg in which curbside parking is not used. The reasoning is as follows. First, the supplyof curbside parking is …xed so that providing curbside parking has no opportunity cost.Second, administration costs of curbside parking and fee collection are assumed to be zero.Third, drivers do not contribute to through tra¢ c congestion while they are searching forparking. Fourth, the marginal external cost of search approaches zero as hours of curbsideparking used, T , approach zero. Finally, garage parking involves a supply cost per vehicle-hour of  > 0, and drivers incur walking time costs when they use garages. Therefore, themarginal social cost of using a small amount of curbside parking is lower than the marginalsocial cost of using a small amount of garage parking so that some curbside parking shouldalways be used.10It is straightforward to derive the social optimum for each of the seven remaining regimes.Formulas for the optimal market boundaries are complicated and opaque, and are not pre-sented here (formulas for regime Int are given in online Appendix A.10). In general thesocial optimum di¤ers from the market equilibrium described in Section 3. In some cases theequilibrium regime di¤ers as demonstrated in the numerical example of Section 5. In othercases the regime is the same, but garage fees di¤er. The market equilibrium is generallyine¢ cient because it is distorted in two ways. First, parking garages exploit their marketpower by setting parking fees above marginal costs and discriminating between short- andlong-term parkers. Second, parking garages do not internalize the congestion externalityassociated with search for curbside parking.In the balance of this section we o¤er some general insights about the social optimumand how it can be decentralized. A numerical example is developed in the next section.The social optimum can be achieved in several ways. The most direct approach is for thecity to regulate garage parking fees or take over control of garages altogether. The city canimplement …rst-best pricing of both garage and curbside parking. Denote this approach witha superscript o. Garage parking is priced at marginal supply cost so that pogL = pogS = .Curbside parking is priced to internalize the marginal external cost of search. Because thisexternality is proportional to parking duration, hourly curbside parking fees will be equal,(i.e., pocL = pocS). We will refer to the common value, poc, as the “…rst-best uniform curbsideparking fee”. Given total curbside parking search costs in (19), the marginal external hourlycost imposed by a type i driver is [kLdL(D  2xL) + kSdS(D  2xS)]li, and the …rst-best10This di¤ers from some other parking models such as that in Arnott, Inci, and Rowse (2015) whereallocating space to curbside parking reduces road capacity, and cruising for parking contributes to congestionfor through tra¢ c.15uniform curbside fee is therefore poc = kLdL(D  2xL) + kSdS(D  2xS).Since total parking demand for each driver type is …xed, …rst-best pricing is not theonly way to support the social optimum. Indeed, any set of garage and curbside parkingfees that maintains the appropriate di¤erential between garage and curbside parking fees foreach driver type does the job. For example, in the symmetric equilibrium for regime Int themarket boundary for long-term parkers given in (10) is a function of pcL pgL and pcS pgS,but it does not depend on the levels of the fees. If the city can regulate garage fees, it can…x curbside fees, pcL and pcS, at any level (subject to nonnegativity constraints) and thenset pgL and pgS, correspondingly. If parking garage operators can set their rates freely, thecity can still attain the optimum by setting curbside fees appropriately. The hourly curbsideparking fees for the two driver types will typically di¤er because their garage parking demandelasticities di¤er, and garages therefore impose di¤erent markups on their rates. We call thecurbside parking fees in this case “…rst-best di¤erentiated curbside parking fees”and denotethem with a caret (i.e., bpcL and bpcS).Di¤erentiating curbside fees is di¢ cult using conventional parking meter technology, andit may be opposed by interest groups or ruled out on public policy grounds. If so, hourlycurbside fees are constrained to be uniform and the social optimum cannot, in general, bereached if garage parking is privately controlled. We denote this scheme by a superscript u,and refer to the common fee, puc , as the “second-best uniform curbside parking fee”.Regardless of which equilibrium regime prevails at the optimum, the …rst-best and second-best curbside parking fees de…ned above are positive for both driver types. This follows fromtwo observations. First, curbside parking creates a negative search congestion externality.The …rst-best (Pigouvian) curbside parking fees are therefore positive. Second, when garageparking is privately controlled it is overpriced because the operators exercise market power.Because garage operators do not su¤er from congestion themselves, they do not internal-ize congestion costs in the way that private toll road operators do (Verhoef, Nijkamp, andRietveld, 1996). Indeed, garage operators bene…t from the congestion induced by curb-side parking search. Since curbside parking is a substitute for garage parking, second-bestcurbside parking fees exceed …rst-best fees.5 A Numerical ExampleThe numerical example is designed to establish a sense of which of the various candidateequilibrium regimes prevails under alternative parameter assumptions. It also compares the16…rst-best and second-best pricing schemes described in Section 4. Online Appendix A.11summarizes the numerical solution procedure that was used to derive equilibria.5.1 ParameterizationBase-case parameter values for the example are listed in Table 3. Some of the values aretaken from Arnott (2006), Arnott and Inci (2006, 2010), and Arnott and Rowse (2009).Suitable values for the di, li, wi, and ki parameters depend on such factors as trip purposeand income, and possibly also trip frequency and familiarity with parking availability. Thereis a large empirical literature on values of driving time and walking time, and how they varywith individual socioeconomic characteristics. There are very few estimates of the value oftime spent cruising for parking,11 and such estimates cannot be translated into values for theki parameters because search time is measured in terms of aggregate vehicle-hours of parkingrather than minutes. Another complication is that suitable parameter values depend on howlong-term and short-term parkers are interpreted. To be agnostic, we have chosen for thebase case identical parameter values for the two driver types except for parking duration.pcL = pcS D  lL lS wL = wS kL = kS dL = dS$1:00=hr 0:125 miles $2:50=hr 2 hr 1 hr $16=mile $0:16=vehicle-hr 100=mileTable 3: Base-case parameter valuesThe curbside parking fee for both types is $1:00 per hour; the distance between twoconsecutive parking garages is 0:125 miles; the cost of providing a parking garage space is$2:50 for each hour that a car is parked; parking duration is 2 hours for long-term parkers and1 hour for short-term parkers; walking time cost for both types is $16 per mile; unit parkingsearch cost for both types is $0:16 per vehicle-hour; and the density of trip destinations foreach type is 100 per mile. With these parameter values, the cost of walking for the round-tripbetween a garage and the mid-point between parking garages is $2:00 for each type. Forshort-term parkers, the generalized cost of curbside parking ranges from pcSlS = $1:00 if noone uses curbside parking to pcSlS+kST = pcSlS+kS(hL+hS)D = $7:00 if all drivers of bothtypes park on the curb. The corresponding minimum and maximum values for long-termparkers are $2:00 and $8:00, respectively.11From a meta-analysis of UK values of travel time Abrantes and Wardman (2011) report that the valueof time spent searching for parking is 38% higher than the value of time spent on normal driving.175.2 ResultsResults for various parameter combinations are reported in Tables 4-6. As shown in Table4, for the base case (case 1), the current equilibrium, denoted by a , falls in regime Int.Hourly garage parking fees are pgL = $2:81 and pgS = $3:74. Only 9 percent of long-termparkers use garages, but 97 percent of short-term parkers do so. Total costs are about $85per garage. This value should be interpreted as costs incurred every 2 hours: the parkingduration of long-term parkers.The social optimum falls into regime Sg (see Table 5). All short-term parkers use garageparking, and a larger fraction (33 percent) of long-term parkers uses garages than in the cur-rent equilibrium. This pattern re‡ects the underusage of garages in the current equilibrium,described earlier, that results from a combination of underpricing of curbside parking andexercise of garage market power. Total costs in the social optimum are about $81 which isroughly 5 percent lower than in the current equilibrium.The social optimum can be attained by pricing garage parking at marginal social cost(i.e., pogL = pogS = $2:50), and charging curbside users the marginal search congestion ex-ternality cost. For the base-case parameter values the optimal curbside fee is poc = $1:33=hrwhich is not much higher than the $1:00=hr assumed for the current equilibrium. Considerthe long-term parker who is indi¤erent between parking on the curb and at the home garage.To park at the home garage he would pay a total parking fee of pogLlL = 2:5  2 = $5 andincur a walking time cost of wLlL = 16 (0:1250:50:33) = $0:33 for a total cost of $5:33.If he parked on the curb instead, he would pay a parking fee of pocLlL = 1:33  2 = $2:66and incur a search time cost of kLT = kLdLlL(D  2xH) = 0:16  100  2  0:083 = $2:66for the same total cost of $5:33. All long-term parkers with destinations closer to the homegarage than this individual strictly prefer parking at the home garage. All long-term parkerswith destinations further away from the home garage than this individual strictly preferparking on the curb. Meanwhile, all short-term parkers are strictly better o¤ parking at agarage. The driver bound for a destination midway between garages pays a parking fee ofpogSlS = 2:5  1 = $2:50 and incurs a walking cost of wSlS = 16  (0:125  0:5) = $1, for atotal cost of $3:50. Other short-term parkers incur a lower cost. Any short-term parker whoparked on the curb would pay a parking fee of pocSlS = 1:33 1 = $1:33, and incur a searchtime cost of kST = $2:66 for a total cost of $3:99: higher than the cost of garage parking.As explained in Section 4, even if garages are operated privately and free of regulation, thesocial optimum can be decentralized by setting …rst-best di¤erentiated curbside parking fees.For Case 1, bpcL is $1:61, and bpcS is at least $2:83 to deter short-term parkers from parking18on the curb. Starting from the current equilibrium, garages raise the rate for short-termparkers from pgS = $3:74 to bpgS = $4:50. However, they lower the rate for long-term parkersslightly from pgL = $2:81 to bpgL = $2:78 because short-term parkers no longer park on thecurb which makes curbside parking more attractive to long-term parkers. This illustrateshow curbside parking congestion creates an interdependence between the parking marketsfor long-term and short-term parkers.The second-best uniform curbside parking fee is puc = $1:85 (see Table 6) which is inter-mediate between the …rst-best di¤erentiated fees. The uniform fee fails to support the socialoptimum. Its relative e¢ ciency can be measured by the ratioeff  TC  TCuTC  TCo 2 [0; 1] ; (20)where TC denotes total costs in the current equilibrium, TCo total costs in the …rst-bestoptimum, and TCu total costs with the uniform curbside parking fee. For the base-caseparameter values, eff = 0:77 so that the second-best uniform fee achieves more than three-quarters of the e¢ ciency gains from …rst-best di¤erentiated fees.Results for eight alternative parameter combinations are reported in Tables 4-6, withparameter variations identi…ed in bold type in Table 4.Case 2 : The hourly curbside parking fee for each driver type is raised from $1:00 to $4:00.The current equilibrium shifts to regime Lg. Because long-term parkers park for twice aslong as short-term parkers, they are hit twice as hard by the higher curbside parking fees andall of them end up parking at garages. By contrast, short-term parkers shift toward curbsideparking because of reduced search congestion. The curbside parking fee is higher than thegarage parking fees in this case, which is the practice in the Netherlands and Belgium.Similar to the base case, congestion in curbside parking creates an interdependence betweenthe parking markets. The social optimum is unchanged, of course, by the change in currentcurbside parking fees. However, the welfare gain from optimal curbside parking fees morethan quadruples compared to the base case, and the relative e¢ ciency of the second-bestuniform curbside parking fee rises to 94 percent.Case 3 : Curbside parking is made free. The current equilibrium shifts to regime Lc,and both driver types use less garage parking than in the base case. The welfare gainfrom optimal curbside parking fees is similar to Case 2, and the relative e¢ ciency of thesecond-best uniform curbside parking fee rises to 93 percent.Case 4 : Long-term parkers are assumed to be commuters who park for 9 hours. The19current equilibrium shifts to regime Sg with all short-term parkers using garages. Long-termparkers patronize mainly curbside parking although the proportion is less than in the basecase equilibrium. Total parking costs increase to nearly three times their base-case level. Theincrease is due to the higher resource costs of accommodating long-term parkers at garagesfor 9 hours as well as the greater contribution of long-term parkers to curbside congestiondue to their longer stays. The social optimum is similar to the current equilibrium, and thee¢ ciency gains from optimal curbside parking are modest. Moreover, the optimum can besupported by charging any non-negative hourly fee for short-term curbside parking so thatthe fee can be set at the optimal level for long-term parking of $1:35 and the second-bestuniform curbside parking fee achieves 100 percent e¢ ciency.Case 5 : Distance between garages is doubled. The current equilibrium remains in regimeInt, and the social optimum remains in regime Sg. But total costs increase greatly, andoptimal curbside parking fees are appreciably higher than in previous cases. Garage pro…tsare also much higher. The second-best uniform curbside parking fee of $3:34 achieves about90 percent of the welfare gains from the …rst-best di¤erentiated curbside fees. A problematicfeature of the di¤erentiated fees is that short-term parkers pay a total of pcSlS = $5:39 forparking which slightly exceeds the payment by long-term parkers of pcLlL = $5:14. Incentivecompatibility constraint (1) is therefore violated. If the incentive compatibility constraintis practically relevant, it can be accommodated by imposing it on the optimization problemwhile still treating pcL and pcS as independent control variables. Doing so would a¤ect thesolution very little in this instance because the constraint is only violated by a small margin.Case 6 : Curbside parking search costs are doubled. The current equilibrium shifts toregime Sg. It does not di¤er markedly from the social optimum so that the bene…ts fromadjusting curbside parking fees are modest.Case 7 : Curbside parking search costs are reduced by half. Both the current equilibriumand the social optimum shift to regime Lc. The social optimum is supported by rathermodest …rst-best di¤erentiated curbside fees although incentive compatibility constraint (1)is again violated.Case 8 : Walking time costs are doubled. The market power of garages increases andthey increase their hourly fees slightly. The combined e¤ect of more onerous walking andhigher garage parking fees induces short-term parkers to reduce their use of garage parking.Case 9 : Walking time costs are halved. All short-term parkers now use garage parkingin both the current equilibrium and the social optimum, and the optimum can be supportedby charging for short-term parking any fee over $1:41=hr. Since this minimum is below the20optimal fee of $1:45=hr for long-term parking, the second-best uniform curbside parking feeis $1:45=hr and supports the social optimum.Further outcomes can be realized by changing several parameters at once. For example, itis possible to create an equilibrium con…guration in which long-term and short-term parkersare fully segregated between curbside and garage parking, and to create a social optimum inwhich all long-term parkers park in garages.Overall, the numerical example illustrates a range of ways in which drivers are allocatedto parking space in the various pricing schemes. Consistent with usual practice, in most casesthe hourly fee for long-term garage parking is lower than the fee for short-term parking butthe total outlay is higher for long-term parking. In general, the current equilibrium featurestoo little garage parking because curbside parking is underpriced. (Case 2 is an exception.)The welfare gain from e¢ cient curbside parking varies strongly with parameter values. It islargest in Case 5, where the distance between garages is doubled, because costs are talliedover a large market area and because the greater friction of space endows garages with moremarket power.A second-best uniform hourly curbside parking fee supports the social optimum in somecases where the allocation of driver types is partly segregated between garages and the curb.In other cases, it falls short of the social optimum but still performs relatively well. Finally,the …rst-best uniform curbside fees are always lower than the …rst-best di¤erentiated andsecond-best uniform curbside parking fees. This is because garage pricing in the …rst-bestcase is priced at marginal social cost rather than priced at a markup by private operators.The …rst-best uniform curbside fees vary relatively little over the cases compared to the …rst-best di¤erentiated and second-best uniform curbside parking fees because curbside congestionvaries less from case to case than does garage market power.21ParametervalueCurrentequilibriump cL=p cSDl Ll SwLwSkLkSp gLp gSx Lx STCCase($/hr)($/hr)(hr)(hr)($/hr)($/hr)($/hr)($/hr)Regime($/hr)($/hr)(frac.ofD=2)($)($)11:001=82116160:160:16Int2:813:740:090:9785:415:724:001=82116160:160:16Lg3:503:671:000:7898:136:330:001=82116160:160:16Lc2:503:670:000:7896:011:341:001=89116160:160:16Sg2:554:500:221:00236:926:151:001=42116160:160:16Int3:474:880:350:88229:869:761:001=82116160:320:32Sg2:924:300:461:0096:627:371:001=82116160:080:08Lc2:503:100:000:4564:73:481:001=82132320:160:16Int2:954:000:150:6098:512:991:001=821880:160:16Sg2:583:500:191:0079:312:9Notes:1.denotescurrentequilibrium.2.Parametervaluesheldconstantare=2:5,dL=100,anddS=100.3.Parametervariationsareidenti…edinboldtype.4.Minimumvaluesareunderlined.Table4:Resultsfornumericalexample:currentequilibrium22SocialoptimumMSCpricingofcurbsideandgarageparkingDi¤erentiatedcurbsidefeesandprivategarageoperationxo Lxo STCoGainpo cpo gLpo gSbp c Lbp c Sbp g Lbp g SCaseRegime(frac.ofD 2)($)($)($/hr)($/hr)($/hr)($/hr)($/hr)($/hr)($/hr)1Sg0:331:0081:34:11:332:52:51:612:832:784:502Sg0:331:0081:316:81:332:52:51:612:832:784:503Sg0:331:0081:314:71:332:52:51:612:832:784:504Sg0:361:00231:75:21:272:52:51:350:002:584:505Sg0:611:00203:526:31:562:52:52:575:393:526:506Sg0:651:0093:03:61:412:52:51:992:683:084:507Lc0:000:8362:02:81:172:52:51:002:282:503:618Int0:380:8887:910:61:502:52:52:383:753:384:759Sg0:351:0077:81:51:292:52:51:451:412:663:50Notes:1.odenotes…rst-bestoptimumwithuniformcurbsideparkingfeesandmarginalsocialcostpricingatgarages.2.^denotes…rst-bestoptimumwithdi¤erentiatedcurbsideparkingfees.3.MSCdenotesmarginalsocialcost.4.Parametervaluesheldconstantare=2:5,dL=100,anddS=100.5.Minimumvaluesareunderlined.Maximumvaluesareoverlined.Table5:Resultsfornumericalexample:…rstbest23Uniformcurbsidefeepu cpu gLpu gSxu Lxu SeffCaseRegime($/hr)($/hr)($/hr)(frac.ofD=2)1Int1:853:013:590:390:870:772Int1:853:013:590:390:870:943Int1:853:013:590:390:870:934Sg1:352:584:500:361:001:005Int3:344:024:740:770:740:906Int2:703:373:900:780:920:647Int1:512:653:230:040:600:658Int2:713:504:160:540:600:909Sg1:452:663:500:351:001:00Notes:1.udenotessecond-bestoptimumwithuniformcurbsidefees.2.Parametervaluesheldconstantare=2:5,dL=100,anddS=100.Table6:Resultsfornumericalexample:secondbest246 ExtensionsThe analysis in this paper could be extended in various directions. One obvious priority is toextend it from two driver types to multiple types or a continuum of types. Doing so wouldnot only be more realistic, but also permit a more precise analysis of price discriminationaccording to parking duration as well as a comparison with garage pricing in practice. Atypical garage in Lin and Wang’s (2014) dataset posts …ve fees, and most garages set betweenthree and eight fees. Working with a continuum of types would be easier than with discretetypes insofar as there would be fewer equilibrium regimes to deal with. However, it would benecessary to adopt a joint frequency distribution of types in three dimensions (i.e., parkingduration, walking time cost, and search time cost) for which empirical data are lacking.It would still be necessary to contend with possible non-concave garage pro…t functions.Furthermore, basic insights would probably be more di¢ cult to obtain from a continuummodel.A second extension would be to allow parking duration to depend on the cost of parkingrather than being …xed. Third, alternative functional forms for curbside parking search costscould be entertained. Fourth, garage capacity constraints could be introduced. Parkingcapacity is an e¤ective policy tool that can be used by public authorities. Many cities andtowns regulate parking capacity by imposing minimum and maximum parking requirements.Unfortunately, adding capacity constraints to a model can introduce kinks in pro…t functionsand create problems with the existence of pure-strategy equilibria. Froeb, Tschantz, andCrooke (2003) and Arnott (2006) discuss these problems in the context of parking markets.These di¢ culties can be …nessed by assuming Cournot competition, but this approach isunsatisfactory for studying driver heterogeneity and price discrimination.Alternative market structures are yet another extension that we will brie‡y considerhere. Monopoly control of garage parking is one possibility. The polar opposite of perfectcompetition is implausible because of entry barriers. Parking garages have scale economiesand substantial construction costs, and it can take many years to recoup investments.12Opportunities to enter are limited by zoning regulations and shortage of space. Furthermore,even if entry were free, garages would still price above marginal costs in order to recoup theirentry costs.More plausible than either polar case is an intermediate market structure of oligopolisticcompetition between a mix of public and private multi-garage chains. Such markets are12For example, a build-operate-transfer contract for parking garages in Istanbul was o¤ered with anoperating period of 30 years, yet it attracted no applicants.25prevalent in many cities.13 A general spatial analysis would be challenging, but some insightscan be derived from the circle model. If all chains are privately owned one property ofequilibrium is immediate: if no chain owns garages adjacent to each other on the circle, theequilibrium is the same as if all garages were separately owned — as is the case in the modelhere.14 This is so because every garage is bounded by rival garages on either side. Rivalsset their parking fees simultaneously and independently, curbside rates are (still) exogenous,and curbside parking search zones do not extend beyond the area between adjacent garages.A private chain therefore cannot boost pro…ts at one of its garages by adjusting parkingrates at its other garages.The analysis is more complicated in a mixed oligopoly with one publicly operated chainthat has a goal of maximizing social surplus in the whole market. The public operator cantry to reduce the distortions due to garage market power and search congestion externalitiesby adjusting fees at public garages, but it cannot attain the social optimum because feesat private garages will still be nonoptimal. In this sense, public control over just part ofthe garage parking market is less e¤ective at achieving market e¢ ciency than simply pricingcurbside parking e¢ ciently.More de…nitive results can be derived in speci…c settings. One such case is a con…gurationwith public and private garages alternating around the circle, and curbside fees still setuniformly and exogenously. As shown in online Appendix A.12, when there is only onedriver type and equilibrium falls in regime Int, the maximum e¢ ciency of mixed oligopolycan be established analytically. We can de…ne the relative e¢ ciency index of mixed oligopoly,effm, by replacing TCu with TCm in equation (20), where TCm denotes total costs in themixed oligopoly equilibrium. Index effm turns out to be independent of parameters , pc,and D, and it depends on the other parameters only via the ratio v  w=(kdl). With v = 0,effm = 0. effm rises with v to a maximum value of 0:586 at v = 1:49, and then slowlydeclines. With the base-case parameter values of the numerical example, effm = 0:45 if alldrivers are of type L, and effm = 0:54 if all drivers are of type S.Mixed oligopoly is therefore most e¢ cient when the market distortion due to privategarage market power (proportional to w) is comparable in magnitude to the distortion dueto search cost externalities (proportional to kdl). Nevertheless, although the public operatorcontrols half the parking garages in the market, at best it can achieve less than 60 percent ofthe e¢ ciency gained by letting private garages price as they please and optimizing curbside13Amsterdam, Hong Kong, Istanbul, and Paris are some major cities with a mix of publicly and pri-vately operated garage parking facilities. De Nijs (2012) and Choné and Linnemer (2012) provide detaileddescriptions of the parking market structure in Paris.14Braid (1986) derives this result for the case of retail chain stores.26parking fees. The relative e¢ ciency of mixed oligopoly would be even lower if, perhaps dueto x-ine¢ ciency, public garages are operated less ine¢ ciently than private garages.7 ConclusionsIn this paper, we have developed a simple spatial parking model. Individuals drive downtownto visit destinations that are uniformly distributed around a circle. There are two typesof drivers who di¤er in how long they need to park. Privately-operated parking garagescompete for business with each other and with publicly-operated curbside parking space.Garage parking is free of tra¢ c congestion, but curbside parking is scarce and drivers incursearch costs that increase with the total number of vehicle-hours of curbside parking that isused.The model has several notable properties. First, unlike in standard models of spatialcompetition the utility derived from curbside parking (the outside good) is endogenous be-cause it decreases with the number of drivers who use it. This creates an interdependencebetween parking garages even when they do not compete for customers with each other di-rectly. If one garage changes its parking fees, it a¤ects the number of drivers who park on thecurb, and the change in parking search congestion a¤ects demand at neighboring garages.An increase in fees bene…ts rivals (and a reduction hurts them) in the same qualitativeway as when garages compete directly across a common border. Second, curbside parkingcongestion creates an interdependence between the demand for garage parking by the twodriver types. For example, if a garage raises its fee for short-term parking, some short-termparkers switch to curbside parking. This increases search congestion for long-term parkersand induces some of them to switch from curbside parking to the garage. Demand for garageparking from the two market segments is therefore interdependent, and partly substitutable,even though garages are not constrained by capacity in the number of vehicles they canaccommodate.The parking market in the model is distorted in two ways. First, search for curbsideparking creates a negative congestion externality among drivers looking for curbside park-ing. Second, parking garages have market power due to their discrete locations and thefriction of space, and they exploit it by setting parking fees above marginal costs. They alsoprice discriminate by setting di¤erent hourly fees for short- and long-term parking. Underplausible assumptions, the parking outlay is increasing and concave with parking duration,but exceptions are possible.27The degree of market failure varies with the distance between parking garages, parkingsearch costs, and (crucially) the “current” fees charged for curbside parking. In the basemodel, the public authority lacks regulatory power over garage prices, but it can set curbsideparking fees to alleviate search congestion. Moreover, since total parking demand is inelasticit can achieve the social optimum by charging di¤erentiated curbside parking fees. In somecases a uniform hourly curbside parking fee can also achieve full e¢ ciency. If garage parkingrates can be regulated, the social optimum can also be reached regardless of curbside fees byadjusting garage fees to maintain the appropriate di¤erential between garage and curbsiderates. At least in the mixed oligopoly extension of the model we consider, operating someof the parking garages publicly without also optimizing curbside fees yields only modeste¢ ciency gains. Moreover this simple extension shows that, in the case in which there areprivately owned parking garage chains, the equilibrium is the same as if all parking garageswere separately owned if no chain owns garages adjacent to each other.References[1] Abrantes, P.A.L., Wardman, M.R., 2011. Meta-analysis of UK values of travel time: Anupdate. Transportation Research Part A 45(1), 1–17.[2] Anderson, S., de Palma, A., 2004. The economics of pricing parking. Journal of UrbanEconomics 55, 1–20.[3] Anderson, S., de Palma, A., 2007. Parking in the city. Papers in Regional Science 86,621–632.[4] Arnott. R., 2006. Spatial competition between downtown parking garages and downtownparking policy. Transport Policy 13, 458–469.[5] Arnott. R., 2011. Parking economics. In: de Palma, A., Lindsey, R., Quinet, E., Vick-erman, R. (Eds.), Handbook in Transport Economics. Edward Elgar, Cheltenham, UKand Northampton, Mass, USA, 726–743.[6] Arnott, R., de Palma, A., Lindsey, R., 1991. A temporal and spatial equilibrium analysisof commuter parking. Journal of Public Economics 45, 301–335.[7] Arnott, R., Inci, E., 2006. An integrated model of downtown parking and tra¢ c con-gestion. Journal of Urban Economics 60, 418–442.28[8] Arnott, R., Inci, E., 2010. The stability of downtown parking and tra¢ c congestion.Journal of Urban Economics 68, 260–276.[9] Arnott, R., Inci, E., Rowse, J. 2015. Downtown curbside parking capacity. Journal ofUrban Economics 86, 83–97.[10] Arnott, R., Rowse, J., 1999. Modeling parking. Journal of Urban Economics 45, 97–124.[11] Arnott, R., Rowse, J., 2009. Downtown parking in auto city. Regional Science and UrbanEconomics 39, 1–14.[12] Arnott, R., Rowse, J., 2013. Curbside parking time limits. Transportation Research PartA: Policy and Practice 55, 89–110.[13] Axhausen, K.W., Polak, J.W., Boltze, M., Puzicha, J. 1994. E¤ectiveness of the parkingguidance information system in Frankfurt amMain. 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Journal of Public Economics 105, 106–115.[20] Froeb, L., Tschantz S., Crooke, P. 2003. Bertrand competition with capacity constraints:mergers among parking lots. Journal of Econometrics 113, 49–67.[21] Gabszewicz, J.J., Thisse, J. 1986. Spatial Competition and the Location of Firms. Fun-damentals of Pure and Applied Economics 5, 1–71.29[22] Glazer, A., Niskanen, E., 1992. Parking fees and congestion. Regional Science and UrbanEconomics 22, 123–132.[23] Hasker, K., Inci, E., 2014. Free parking for all in shopping malls. International EconomicReview 55, 1281–1304.[24] Inci, E. 2015. A review of the economics of parking. Economics of Transportation 4,50–63.[25] Kobus, M., Gutierrez-Puigarnau, E., Rietveld, P., van Ommeren, J. 2013. The on-streetparking premium and car drivers’choice between street and garage parking. RegionalScience and Urban Economics 43(2), 395–403.[26] Lin, H., Wang Y. 2014. Competition and price discrimination in the parking garageindustry. Journal of Industrial Economics, forthcoming.[27] Pierce, G., Shoup, D., 2013. Getting the prices right. Journal of the American PlanningAssociation 79, 67–81.[28] Salop, S., 1979. Monopolistic competition with outside goods. Bell Journal of Economics10, 141–156.[29] Shoup, D., 2005. The High Cost of Free Parking. American Planning Association,Chicago.[30] Shoup, D., 2006. Cruising for parking. Transport Policy 13, 479–486.[31] Stole, L., 2007. Price discrimination and competition. In Armstrong, M., Porter, R.(eds.) Handbook of Industrial Organization, Vol. 3, Elsevier, Chapter 34.[32] van Ommeren, J.N., Wentink, D., 2012. The (hidden) cost of employer parking policies.International Economic Review 53, 965–977.[33] van Ommeren, J., Wentink, D., Dekkers, J., 2011. The real price of parking policy.Journal of Urban Economics 70, 25–31.[34] van Ommeren, J., Wentink, D., Rietveld, P., 2012. Empirical evidence on cruising forparking. Transportation Research Part A: Policy and Practice 46, 123–130.[35] Varian, H.R., 1989. Price discrimination. In Schmalensee, R., Willig, R.D. (eds.) Hand-book of Industrial Organization, Vol. 1, Elsevier, 597–654.30[36] Verhoef, E.T., Nijkamp, P., Rietveld, P., 1996. Second-best congestion pricing: the caseof an untolled alternative. Journal of Urban Economics 40(3), 279–302.[37] Zhang, X., Yang, H., Huang, H., Zhang, H., 2005. Integrated scheduling of daily workactivities and morning-evening commutes with bottleneck congestion. TransportationResearch Part A: Policy and Practice 39(1), 41–60.31Online Appendix toGARAGE AND CURBSIDE PARKING COMPETITION WITHSEARCH CONGESTION(not intended for publication)Eren Inci and Robin Lindsey1A Appendix (not intended for publication)A.1 Stability of equilibrium in regime IntFollowing standard procedures, the stability of an interior equilibrium for given garage andcurbside parking fees can be established by constructing, for each driver type, excess demandfunctions for parking at the home garage and parking at the neighbor. Using (7), equations(8)-(9) can be written as excess demand functions:EL = pcLlL + kL (hL (D  xL  x0L) + hS (D  xS  x0S)) pgLlL  wLxL (A.1)ES = pcSlS + kS (hL (D  xL  x0L) + hS (D  xS  x0S)) pgSlS  wSxS (A.2)E 0L = pcLlL + kL (hL (D  xL  x0L) + hS (D  xS  x0S)) p0gLlL  wLx0L (A.3)E 0S = pcSlS + kS (hL (D  xL  x0L) + hS (D  xS  x0S)) p0gSlS  wSx0S: (A.4)Drivers are assumed to adjust their parking location choices when out of equilibriumfollowing the equations of motiondxLdt= LEL (A.5)dxSdt= SES (A.6)dx0Ldt= LE0L (A.7)dx0Sdt= SE0S; (A.8)where the speeds of adjustment for long and short-term parkers, L and S respectively, areboth positive and can di¤er. Let xL, xS, x0L , and x0S denote the equilibrium values of xL,xS, x0L, and x0S, and de…ne new variables yL  xL  xL, yS  xS  xS, y0L  x0L  x0L , andy0S  x0S  x0S . Given equations (A.1)-(A.8), the y variables follow the equations of motion266664dyLdtdySdtdy0Ldtdy0Sdt377775=266664L (kLhL + wL) LkLhS LkLhL LkLhSSkShL S (kShS + wS) SkShL SkShSLkLhL LkLhS L (kLhL + wL) LkLhSSkShL SkShS SkShL S (kShS + wS)377775266664yLySy0Ly0S377775(A.9)Finally, assume that the y variables evolve out of equilibrium according to the equationsyL = ZLet, yS = ZSet, y0L = Z0Let and y0S = Z0Set, where the Z variables are constants.2The characteristic equation for the system (A.9) isL (kLhL + wL) LkLhS LkLhL LkLhSSkShL S (kShS + wS) SkShL SkShSLkLhL LkLhS L (kLhL + wL) LkLhSSkShL SkShS SkShL S (kShS + wS)= 0;(A.10)which reduces to(LwL + ) (SwS + )2 +B+ C= 0; (A.11)where B  SwS + LwL + 2LkLhL + 2SkShS, and C  LSwLwS + 2LS(wSkLhL +wLkShS). The …rst two terms of the product in (A.11) yield roots 1 = LwL and 2 =SwS which are both negative. It is straightforward to show that the third, quadratic,term in (A.11) has a positive discriminant. The two roots are therefore both real and takethe form3 =B pB2  4C2; 4 =B +pB2  4C2: (A.12)Since B and C are both positive, 3 and 4 are both negative. All four roots of the charac-teristic equation are therefore negative, which proves that the system is stable.A.2 Equilibrium for regime LgIn regime Lg, all long-term parkers park at garages while short-term parkers split betweengarages and the curb. The home garage maximizes its pro…t, given by (6), subject to severalconstraints. First, the marginal long-term parker must be indi¤erent between parking at thehome garage and parking at the neighbor:BL  pgLlL  wLxL = BL  p0gLlL  wL (D  xL) : (A.13)Second, the marginal short-term parker must be indi¤erent between parking at the homegarage and curbside parking:BS  pgSlS  wSxS = BS  pcSlS  kST: (A.14)3This condition is the same as condition (8) for regime Int, except now T = hS(DxS x0S)because only short-term parkers use curbside parking. Third, the marginal long-term parkermust weakly prefer parking at the home garage to curbside parking:BL  pgLlL  wLxL  BL  pcLlL  kLT: (A.15)Fourth, the marginal short-term parker must strictly prefer parking at the home garage toparking at the neighbor:BS  pgSlS  wSxS > BS  p0gSlS  wS (D  xS) : (A.16)This condition assures that some short-term parkers prefer curbside parking to garage park-ing. Finally, if practically relevant, the incentive compatibility constraint given in (3) musthold. The equilibrium solution depends on whether condition (A.15) binds or not. The twocases are examined in turn.A.2.1 Long-term parkers strictly prefer garage parking to curbside parkingIn this case, constraint (A.15) does not bind. The market boundary for long-term parkers isobtained from (A.13):xL =D2+p0gL  pgL2wLlL: (A.17)The market boundary for short-term parkers is determined by solving (A.14) and its coun-terpart for the neighboring garage. The solution for xS isxS =kShSD + lSpcSwS + 2kShS+kShSp0gS  (wS + kShS) pgSwS (wS + 2kShS)lS: (A.18)Substituting (A.17) and (A.18) into (6), and di¤erentiating with respect to pgL and pgS,yields the home garage’s best response functions for garage parking fees:pgL =2+wLD2lL+p0gL2(A.19)pgS =2+kSdSwSD + wSpcS + kShSp0gS2 (wS + kShS): (A.20)4In a symmetric equilibrium, p0gL = pgL and p0gS = pgS. Then, equations (A.19) and (A.20)yield the symmetric-equilibrium parking fees:pgL = +wLDlL; (A.21)pgS =(wS + kShS)+ kSdSwSD + wSpcS2wS + kShS: (A.22)By applying symmetry in (A.17) and substituting (A.22) into (A.18), we get the marketboundaries:xL =D2; (A.23)xS =wS + kShS(wS + 2kShS) (2wS + kShS)(kShSD + lS (pcS  )) : (A.24)Unlike regime Int, in regime Lg the formulas for garage parking fees and market bound-aries for one driver type do not depend on parameters speci…c to the other type because thetwo types no longer interact. Equation (A.21) can be rewritten as (pgL)lL = wLD whichshows that the pro…t earned from a long-term parker is independent of parking duration,lL.15 The pro…t increases with the market power of a garage, which varies proportionallywith the distance between garages, D, and the walking time cost for long-term parkers, wL.The equilibrium garage parking fee for short-term parkers in (A.22) is a more complicatedfunction. It increases, but less than proportionally, with the cost of providing service, , andthe curbside parking fee, pcS.16 The fee pgS also increases with the distance between garagesbecause more drivers then park on the curb, which increases search costs and makes curbsideparking less attractive as a substitute for garage parking. Finally, pgS decreases with park-ing duration, lS, but unlike for long-term parkers, the pro…t earned from short-term parkersincreases with parking duration.Similar to regime Int, it is possible for the garage parking fee to be higher for long-termparkers than short-term parkers (i.e., pgL > pgS), and conversely also possible for the totaloutlay on garage parking to be lower for long-term parkers than short-term parkers (i.e.,pgLlL < pgSlS). That is, the incentive compatibility constraint can be active. Section 5provides an example with pgLlL < pgSlS.15This result is attributable to the linearity of the model.16If  and pcS both increase by some amount , then pgS also increases by  because total parkingdemand is price inelastic.5Two further derivatives of interest are@pgS@wSs= pcS + kSdSD  ; (A.25)@pgS@kSs= +2wSDlS pcS > 0; (A.26)where s= means “identical in sign.”Assumption (5) does not assure that the right-hand sideof (A.25) is positive, but it is likely to be satis…ed unless the supply cost of garage parkingis high. If the right-hand side is positive, the garage parking fee for short-term parkers risesas their walking time cost increases. Using (A.24) and the condition xS < D=2, it is easy toshow that the right-hand side of (A.26) is positive. Therefore, pgS rises with the search costfor curbside parking, as expected.A.2.2 Long-term parkers indi¤erent between garage and curbside parkingIf long-term parkers are indi¤erent between garage and curbside parking, constraint (A.15)binds and multiple symmetric equilibria for pgL and pgS can exist. This complication arisesbecause the garage’s pro…t function is kinked. To see this, consider the candidate symmetricequilibrium shown in Figure 3. The full cost of parking at the home garage, line AB, and thefull cost of parking at the neighbor, CD, intersect at point E where xL = x0L = D=2. Thecost of curbside parking, FG, intersects the other two curves at point E as well. If the homegarage lowers pgL, line AB shifts down and intersects line CD at point E1. All long-termparkers now strictly prefer parking at one of the garages to parking on the curb. If the homegarage instead raises pgL, line AB shifts up. The market boundary with the neighbor isbroken, and some long-term parkers now park on the curb. This raises the search cost ofcurbside parking, and line FG shifts upwards. A new market boundary between the homegarage and curbside parking forms at point E2, and the equilibrium changes from regime Lgto regime Int.One can easily show that the home garage’s market shifts more quickly when pgL increasesthan when it decreases, which suggests that the pro…t function has a “downward” kinkat the candidate equilibrium. To see this, suppose that the marginal long-term parker isindi¤erent between the home garage, the neighboring garage, and curbside parking. If thehome garage reduces pgL, the marginal long-term parker now strictly prefers garage parking.Regime Lg applies and the market boundary for long-term parkers is de…ned by (A.17):6Figure 3: Costs of parking in regime LgxL = (D=2) + ((p0gL  pgL)lL)=(2wL). The slope of the home garage’s market demand is@xL@pgLLg=  lL2wL:If the home garage instead increases pgL, regime Int prevails and the market boundaryfor long-term parkers is given by (10). The slope of the home garage’s market demand is@xL@pgLInt=hLlLkLwS + wL (wS + 2hSkS) lLwL (wLwS + 2hLkLwS + 2hSkSwL)=  lL2wL2hLkLwS + 2wL (wS + 2hSkS)2hLkLwS + wL (wS + 2hSkS)<  lL2wL: (A.27)Since@xL=@pgLjLg < @xL=@pgLInt, the market demand curve for long-term parkersis kinked. This suggests that the pro…t function has a “downward”kink at the candidateequilibrium. However, because the cost of curbside parking rises if pgL is increased, someshort-term parkers shift from curbside parking to the home garage, which increases the homegarage’s pro…ts from short-term parkers. The direction of the kink in the pro…t function isthus ambiguous.The pro…t function is also kinked for changes in pgS. If pgS rises, more short-term parkers7park on the curb, and curbside parking becomes less attractive for long-term parkers. LineFG rises and the equilibrium for long-term parkers is una¤ected. If pgS drops instead, fewershort-term parkers park on the curb, and curbside parking becomes more attractive for long-term parkers. Line FG drops, and the home garage loses some long-term parkers. The homegarage’s pro…t function therefore has a downward kink with respect to pgS at the candidateequilibrium. The kink occurs for similar reasons as in Salop’s (1979) model although utilityfrom the outside good is exogenous in his model whereas utility from curbside parking isendogenous here.The kinks explained above in the home garage’s pro…t function complicate the derivationof equilibria for regime Lg. Let (pgL; pgS) be candidate symmetric equilibrium fees, Intthe home garage’s pro…t function in regime Int, and Lg the home garage’s pro…t functionin regime Lg. If condition (A.15) binds, several local conditions must be satis…ed for acandidate symmetric equilibrium in regime Lg to prevail. First, raising pgL slightly shouldnot increase pro…t. Raising pgL slightly shifts the equilibrium from regime Lg to regime Int.An increase is unpro…table iflimpgL!p +gL@Int@pgL 0:This condition can be written as a lower bound on pgL:pgL  pMingL ; (A.28)where pMingL is a complicated expression.Second, lowering pgL slightly should not increase pro…t. A slight reduction leaves equi-librium in regime Lg. The requisite condition is thereforelimpgL!p gL@Lg@pgL 0:This condition can be written as an upper bound on pgL:pgL  pMaxgL = +wLDlL: (A.29)Third, raising pgS should not increase pro…ts. Since a slight increase in pgS leaves equi-librium in regime Lg, the condition for deviation to be unpro…table islimpgS!p +gS@Lg@pgS 0;8Figure 4: Example with multiple equilibria in Regime Lgwhich can be written as a lower bound on pgS:pgS  pMingS =(wS + kShS)+ kSdSwSD + wSpcS2wS + kShS: (A.30)Fourth, lowering pgS should be unpro…table. Since lowering pgS shifts equilibrium toregime Int, the relevant condition islimpgS!pe gS@Int@pgS 0:This can be written as an upper bound on pgS:pgS  pMaxgS ; (A.31)where pMaxgS is a complicated expression.A …nal local condition for a symmetric equilibrium is that condition (A.15) in the textholds as an equality with xL = D=2; this yieldspgS =2lL (wS + 2hSkS) pgL  2lL (wS + 2kShS) pcL + 4kLhSlSpcS4hSlSkL+(wL (wS + 2hSkS) 2hSwSkL)D4hSlSkL: (A.32)9Figure 4 illustrates conditions (A.28)-(A.32) for a numerical example with parametervalues pcL = pcS = 2:5, D = 0:125,  = 2:5, lL = 2, lS = 1, wL = 8, wS = 8, kL = 0:2, kS =0:08, dL = 100, and dS = 100. Inequality conditions (A.28)-(A.31) are all satis…ed withinthe polygonal region of (pgL; pgS) with vertices at points A, B, C and D. All points withinthis region are candidate equilibria. Condition (A.32) lies within the region between pointsE and F . All points on the line segment EF are therefore candidate symmetric equilibria.We arbitrarily choose point E with the lowest garage parking fees as the equilibrium forregime Lg.The necessary conditions for a pro…t maximum listed above are all local conditions. Tocon…rm that regime Lg is a global equilibrium it is also necessary to check that it yieldsa higher pro…t to the home garage than any other regime. Doing so entails considerationof non-local deviations by the home garage in garage parking fees that shift equilibrium toother regimes. This process is described in Appendix A.11 below. A systematic check is notpractical analytically, but it is straightforward to do numerically as is done in Section 5.A.3 Equilibrium for regime SgRegime Sg is a mirror image of regime Lg. The solution can therefore be obtained from thesolution to regime Lg described in the text by interchanging L and S subscripts.A.4 Equilibrium for regime LcIn regime Lc, all long-term parkers park on the curb. Short-term parkers split betweengarages and the curb, and the marginal short-term parker is indi¤erent between parking at agarage and parking on the curb. Thus, the market boundary for short-term parkers is solvedby using (8) with T = hLD + hS(D  xS  x0S) and the counterpart of (8) for the neighbor.In a symmetric equilibrium with p0gS = pgS, the solution isxS =kS (hS + hL)D + lS (pcS  pgS)wS + 2kShS: (A.33)The home garage’s pro…t is given by (6) with xL = 0:  = 2 (pgS  )hSxS. The localpro…t-maximizing parking fee for short-term parkers works out topgS =(wS + kShS) lS+ wSlSpcS + wSkS (hS + hL)D(2wS + kShS) lS: (A.34)10Several conditions must be satis…ed for this solution to be consistent with regime Lc aswell as with a global pro…t maximum. First, pgL must be high enough that no long-termparkers are drawn to park at a garage. The lower bound on pgL is an increasing function of pgSbecause a higher value of pgS induces more short-term parkers to use curbside parking, whichmakes curbside parking less attractive for long-term parkers, and increases their demand forgarage parking. The constraint thus takes the formpgL  pMingL (pgS) ; (A.35)where pMingL (pgS) is a complicated, increasing, linear function of pgS.A second condition is that the home garage does not …nd it pro…table to attract long-term parkers by setting pgL low enough. If it does attract any long-term parkers, the marketmoves into regime Int. Thus, the pro…t in regime Int should be non-decreasing at pMingL (pgS):@Int@pgLpMingL (pgS) 0:This condition can be written as an upper bound on pgL:pgL  pMaxgL (pgS) ; (A.36)where pMaxgL (pgS) is another complicated, increasing, linear function of pgS.Further conditions must be imposed because the pro…t function is not globally concave.If pgS is increased su¢ ciently above the level given in (A.34), curbside parking may becomeso crowded that long-term parkers start to use garage parking. Let pgS jxL=0 denote thethreshold value of pgS if it exists. At this point, the pro…t function is kinked upward withrespect to pgS. A necessary condition for (A.34) to be a global pro…t maximum is thereforethat the local maximum of regime Lc be lower than the threshold parking fee at which theregime switches from regime Lc to regime Int.pgS < pgSjxL=0 : (A.37)Figure 5 depicts a case where inequality (A.37) is violated. In this example, the parkingmarket switches from regime Lc to regime Int at point B. Thus, by default, the localmaximum in regime Lc, represented by point A, must be below the local maximum in regimeInt, represented by point C. This means that point A cannot be the global maximum under11Figure 5: Pro…t function in regime Lc, case (a)these conditions. If inequality (A.37) is satis…ed, pro…t may still increase enough within theinterior region that the global pro…t maximum lies within regime Int rather than regime Lc.A fourth condition is thereforeLc > Int; (A.38)where  denotes the maximum of the respective regime.In Figure 6, regime switching occurs (at point B) after the local maximum of regime Lc(point A); yet the local maximum of regime Int, point C, is higher. Thus, in this …gure,condition (A.37) is satis…ed but condition (A.38) is violated. Figures 7 and 8 show twofurther cases for which both conditions are satis…ed.Finally, multiple equilibria are a possibility. Figure 9 depicts an example with the sameparameter values as for the base case of the numerical example in Section 5 except for thecurbside parking fees: pcL = pcS = 0:7, D = 0:125,  = 2:5, lL = 2, lL = 1, wL = 16,wS = 16, kL = 0:16, kS = 0:16, dL = 100, and dS = 100. The locally optimal parking fee forshort-term parkers in regime Lc is pgS = 3:9. Inequality conditions (A.35) and (A.36) aresatis…ed on line segment AB. Conditions (A.37) and (A.38) are also satis…ed since pro…tswithin regime Int are decreasing to the right of the line @Int=@pgS = 0. The choice of pgLon segment AB is inconsequential because it does not a¤ect parking choices for either typeof driver.The equilibrium conditions listed above are necessary conditions for an equilibrium in12Figure 6: Pro…t function in regime Lc, case (b)Figure 7: Pro…t function in regime Lc, case (c)13Figure 8: Pro…t function in regime Lc, case (d)Figure 9: Example with multiple equilibria in regime Lc14regime Lc to prevail. To con…rm that regime Lc is a global equilibrium, it is also necessaryto check that it yields a higher pro…t to the home garage than any other regime.A.5 Equilibrium for regime ScRegime Sc is a mirror image of regime Lc, and the solution can be obtained from the solutionto regime Lc by interchanging L and S subscripts.A.6 Equilibrium for regime Lc+SgIn this regime, all long-term parkers park on the curb while all short-term parkers park at agarage. Thus, xL = 0 and xS = D=2 in a symmetric equilibrium. Total parking time on thecurb is T = hLD. Several conditions must be satis…ed for this regime to be an equilibrium.First, pgL must be high enough that no long-term parkers are drawn to park at a garage, acondition that works out topgL  pcLlL + kLhLDlL pMingL : (A.39)Second, the curbside parking fee must be high enough that no short-term parkers are drawnto curbside parking, which de…nes an upper bound on the garage parking fee that can becharged to short-term parkers:pgS pcSlS + kShLD  wSD2lS pMaxgS : (A.40)Third, it cannot be pro…table to decrease pgL and attract long-term parkers, which wouldcause a shift to regime Sg:@@pgLpMingL 0: (A.41)Fourth, it cannot be pro…table to increase pgS and induce some short-term parkers to shiftto the curb which would cause a shift to regime Lc:@@pgSpMaxgS 0: (A.42)As is the case with other regimes, these four conditions are necessary conditions for a localmaximum. To con…rm that regime Lc + Sg is a global equilibrium, it is also necessary to15check that it yields a higher pro…t to the home garage than any other regime.A.7 Equilibrium for regime Lg+ScThis regime is a mirror image of regime Lc + Sg. The solution can therefore be obtainedfrom the solution to regime Lc+ Sg by interchanging L and S subscripts.A.8 Equilibrium for regime Lg+SgAll drivers of both types in this regime use garage parking. Thus, xL = D=2 and xS = D=2in a symmetric equilibrium. For this regime to be an equilibrium, the following conditionsmust be satis…ed. First, pgL must be low enough that no long-term parkers are drawn topark on the curb:pgL  2pcLlL  wLD2lL pMaxgL : (A.43)Second, a similar condition applies for short-term parkers:pgS  2pcSlS  wSD2lS pMaxgS : (A.44)Third, it cannot be pro…table to increase pgL above pMaxgL which would cause a shift to regimeSg:@Sg@pgLpMaxgL 0: (A.45)Fourth, it cannot be pro…table to increase pgS above pMaxgS which would cause a shift toregime Lg:@Lg@pgSpMaxgS 0: (A.46)These four conditions are necessary conditions for a local maximum. To con…rm that regimeLg + Sg is an equilibrium regime it is also necessary to check numerically that it yields ahigher pro…t to the home garage than any other regime.A.9 Equilibrium for regime Lc+ScIn this regime, all drivers of both types park on the curb. Thus, xL = 0, xS = 0, andT = (hL + hS)D. Several conditions must be satis…ed for this regime to be an equilibrium.16First and second, garage parking fees must be high enough that all drivers prefer to park onthe curb. The requisite conditions arepgL  pcLlL + kL(hL + hS)DlL pMingL (A.47)pgS  pcSlS + kS(hL + hS)DlS pMingS : (A.48)Third, it cannot be pro…table to decrease pgL to attract long-term parkers and induce a shiftto regime Sc:@Sc@pgLpMingL 0: (A.49)Fourth, it cannot be pro…table to decrease pgS to attract short-term parkers and induce ashift to regime Lc:@Lc@pgSpMingS 0: (A.50)Again, these four conditions are necessary conditions for a local maximum. To con…rm thatregime Lc + Sc is an equilibrium regime, it is also necessary to check numerically that thehome garage cannot obtain positive pro…ts in another regime.A.10 The social optimum: regime IntBecause it is an interior solution, the social optimum in regime Int is characterized by …rst-order conditions @TC=@xL = 0 and @TC=@xS = 0 where TC is given by the the sum of (17),(18), and (19). The solution isxSOLInt= 2hS [dL(kLlS  kSlL)2D  (kL + kS)lS + kSlL]2hLkLDwS  hS(kL + kS)DwS + lLwS!4dLdS(kLlL  kSlS)2  4hSkSwL  4hLkLwS  wLwS (A.51)xSOSInt= 2hL [dS(kLlS  kSlL)2D  (kL + kS)lL + kLlS]2hSkSDwL  hL(kL + kS)DwL + lSwL!4dLdS(kLlL  kSlS)2  4hSkSwL  4hLkLwS  wLwS : (A.52)The second-order condition for a total cost minimum is (4kLhL + wL)(4kShS + wS) 4dLdS (kLlS + kSlL)2 > 0.17A.11 Testing for a global equilibriumAs noted above, to establish that a candidate symmetric equilibrium for a given regime is aglobal equilibrium, it is necessary to check that the home garage cannot earn higher pro…tsby deviating from the candidate and bringing about a change of regime. The followingprocedure was used for the numerical examples.Let (pegL; pegS) denote candidate symmetric equilibrium parking fees and set the neighbor-ing garage’s fees at these values. Then, allow the home garage to experiment with di¤erentcombinations of (pgL; pgS) that shift equilibrium into other regimes. For example, supposethat the candidate equilibrium regime is Int. Set (p0gL; p0gS) = (pIntgL ; pIntgS ). Then, considerdeviations from (pIntgL ; pIntgS ) by the home garage. One possibility is that the home garage willreduce pgL to the point where all long-term parkers park in garages. If short-term parkerscontinue to park both in garages and on the curb, the new regime becomes Lg. Assumethat regime Lg does prevail and derive pro…t-maximizing values of (pgL; pgS) for the homegarage while holding (p0gL; p0gS) …xed at (pIntgL ; pIntgS ). Several consistency conditions have to besatis…ed (the xi must be non-negative, etc.). If the new regime passes this initial test, checkwhether the home garage’s pro…ts are higher than in the candidate equilibrium. If they arehigher, the candidate equilibrium fails. Otherwise, the candidate equilibrium passes this testand the next alternative regime is examined.Because deviation by the home garage breaks symmetry, it is possible that the homegarage and the neighbor will serve di¤erent markets. Consequently, several asymmetricregimes must be tested in addition to the symmetric regimes described in the text. Theadditional regimes are as follows: Lc(): The home garage does not serve any long-term parkers, but the neighbor doesserve some of them. Both home garage and neighbor serve some short-term parkerswhile the remaining short-term parkers park on the curb. (Lc): Same as Lc() with the roles of home garage and neighbor reversed. Sc(): The home garage does not serve any short-term parkers, but the neighbor doesserve some of them. Both home garage and neighbor serve some long-term parkerswhile the remaining long-term parkers park on the curb. (Sc): Same as Sc() with the roles of home garage and neighbor reversed. Lc(Lc): Neither the home garage nor the neighbor serves long-term parkers. Thisregime is qualitatively the same as regime Lc although it is not symmetric.18 Sc(Sc): Neither the home garage nor the neighbor serves short-term parkers. Thisregime is qualitatively the same as regime Sc although it is not symmetric. Lg+Sc(): The home garage and neighbor together serve all long-term parkers. Thehome garage does not serve any short-term parkers, but the neighbor does serve someof them. Lc() +Sg: The home garage does not serve any long-term parkers, but the neighbordoes serve some. The home garage and neighbor together serve all short-term parkers. Lc(Lc) + Sg: Neither the home garage nor the neighbor serves long-term parkers.Together they serve all short-term parkers. This regime is qualitatively the same asregime Lc+ Sg although it is not symmetric. Lg + Sc(Sc): The home garage and neighbor together serve all long-term parkers.Neither serves any short-term parkers. This regime is qualitatively the same as regimeLg + Sc although it is not symmetric.Depending on which candidate symmetric equilibrium is being tested, some of theseadditional regimes cannot be reached by pro…table deviations by the home garage. But it issimpler to run through all the possibilities numerically than to determine analytically whichare potentially pro…table.A.12 Mixed oligopoly with identical driversWith a private oligopoly the equilibrium garage fee with one driver type in regime Int isgiven in equation (15). For the mixed oligopoly the equilibrium fee for private garages worksout topg =5v2 + 6pcv + 6v2 + 8v + 2+ (+ pc)v3 + dkv(1 + v)(2 + v)D2(1 + 6v + 5v2 + v3);where v  w=(kdl). The second-best fee at public garages ispPubg =(1 + 3v + 3v2)+ v (5 + 5v + v2) pc + (1 v  v2) dkvD1 + 6v + 5v2 + v3:Total costs in the private oligopoly and mixed oligopoly are given by long expressions.19Figure 10: E¢ ciency of mixed oligopoly equilibrium (one driver type)The e¢ ciency index de…ned in Section 6 works out toeffm =v(v + 4)(10 + 52v + 63v2 + 28v3 + 4v4)8(1 + 6v + 5v2 + v3)2. (A.53)Equation (A.53) is plotted in Figure 10. Relative e¢ ciency is zero at v = 0, increases steeplywith v to a maximum value of effm = 0:586 at v = 1:49, and declines very slowly for largervalues of v.20


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