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Pricing with constrained supply

Pricing with constrained supply. Pricing with constrained supply. While considering basic PRO we learned that MC=MR and also that at optimal price contribution margin ratio equals 1 over elastisity

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Pricing with constrained supply

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  1. Pricingwithconstrainedsupply

  2. Pricing with constrained supply • While considering basic PRO we learned that MC=MR and also that at optimal price contribution margin ratio equals 1 over elastisity • However, in reality sellers have freedom to adjust prices over many periods; are unsure, how customers will respond to different prices; and satisfy constraints in their ability to satisfy demand • Sellers need to set prices in a dynamic, uncertain and constrained world • In this part we focus on the different constraints

  3. Thewidgetexample • Price 8.75; demand 3000; totalcontribution $11250 • Whatshouldbedone, ifonly 2000 widgetscanbemanufactured?

  4. Pricing with constrained supply • Most retailers replenish their stock at fixed intervals, in between they are limited to selling their current inventory • A drugstore will typically have enough toothpaste, shaving cream and aspirin in stock • It will not sell out, except in cases of an extraordinary run on a particular item – like bottled water prior to a hurricane

  5. Supplyconstraints • Service providers: a hotelonlyhas a number ofrooms; a gaspipelineisconstrainedbythecapacityofitspipes; a barbershopbythe number ofseats • Manufacturers: physicalconstraints on theproductionamount – e.g. Ford Motor Company can produce 475000 vehicles per month in North America; and sell that plus the inivitial inventory • Retailers and wholesalers: e.g. fashiongoods are onlyorderedonce, so are theelectronicgoods at the end oftheirlifecycle • Intrinsicallyscarceitems: beachfrontproperty, flawlessbluediamonds, van Goghpaintings, Stradivariusviolins – marginalcostiseithermeaningless (paintings) orextremelylowrelativetothescracity rent (diamonds)

  6. Hard and softconstraints • Hardcannotbeviolated at anyprice – hotels and gaspipelines • Soft – freightcarrierscanleasespace on othercarriersincaseofexcesscapacity • Timingfactor – anairlinelearningof a veryhighdemandtwomonthsinadvancewillbeabletoassign a newaircraft, butnotwhenthisbecomesevident a weekbeforethefact

  7. What if you are a reseller with a fixed monthly quota that you cannot exceed? • Do nothing – sell on a first come first served basis and run out of the product • Allocate the limited supply to favored customers • Raise the price until demand falls to meet supply • Combinations of second and third option – if he has segmented the market effectively, he could raise his average price by allocating most or all of the limited supply to higher-paying customers (this is the basic idea of revenue management in chapter 6)

  8. The runout price, also a Solver problem • d(p|)=10000-800p|=2000->p|=$10;contrib=$10000 • Choosemaximumof p* and p| • And remember, forthefuture, ifconstraintwas 2001: • d(p|)=10000-800p|=2001->p|~$10;contrib=$10002.5 .

  9. p|=d^-1(b) = (10000-b)/800 • As the profit-maximizing price under a supply constraint is equal to the maximum of the runout price and the unconstrained profit maximizing price, it is always greater than or equal to the unconstrained profit maximizing price • If an auto manufacturer has a strike take out 25% of its capacity, for a portion of a month, it will likely see lower profits – thus the term opportunity cost • Furthermore, a 200 room hotel that takes 50 rooms out of service for two month to be refurbished is likely to give up some potential revenue

  10. Total and marginalopportunitycost • Totalopportunitycostismeasuredasthe loss inoptimalcontribution – e.g. at 2000 unitsitis $11250-$10000=$1250

  11. Marginalopportunitycost • At 2000, theoptimalcontributionis $10000; at 2001, itis $10002.50 • Thus, at thatlevelthemarginalopportunitycost (thepotentialincreaseincontributionmargin) is $2.50 • Asitistheincreaseinprofit, itisthe rent thattheproducerwouldbewillingtopay; orpurchase a unitforupto $2.50+$5.00(cost)=$7.50 fromelsewhere • Marginalopportunitycostiszerowhenthecapacityconstraintisnonbinding

  12. Market segmentation and supplyconstraints • ThefootballgamebetweenStanford and theUniversityof California at Berkeley (a.k.a. „theBigGame“) has 60000 seatsto sell • Wecould (butletsinsteaddoitanalytically):

  13. Singlepriceto all?: analyticalsolution, see rm12.xlsx, sheet6*** for Solver • MarginalRevenue=MarginalCost; incaseoftheticketsof a sportingevent MC=0; thus MR=0 • Totaldemand: d(p)=140000-4250p • Revenue=p*d(p)=140000p-4250p^2 • MR=0->(140000p-4250p^2)’=0->140000-8500p=0 • p=16.47 • Wealsofindthat at thatprice, all thestudentswillbepricedout • ds(p)=0 at 20000-1250p=0->p=16 • Also, thedemandwouldexceed 60000 • 140000-4250*16.47=70000 • Thus, wefind a price, at whichthedemandforgeneralpublicis 60000 • 120000-3000p|=60000 • p|=20 – thisistheoptimalsingleprice

  14. Twoprices, analyticalsolution, see rm12.xlsx sheet7 for Solver • Pricesshouldbeset so thatthemarginalrevenuesfrombothsegments are equalwitheachother (herealsowithzero, whichis MC) • Also, thesupplyconstraintshouldbesatisfied • Let’sfindMRs, silmplify and equatethem: 120000-6000pg and 20000-2500ps->20-pg=8-ps->pg=ps+12 • Nowletstakethedemandconstraint and simplify: (120000-3000pg)+(20000-1250ps)=60000->3pg+1.25ps=80 • Solvingthesetwoequationsgives: ps=$10.35 and pg=$22.35; generalpublicgets 52941 tickets and studentsget 7059 tickets…withtotalrevenueof $1256471, whichis 4.7% overthesingleticketpricecase

  15. Astheticketprices show, studentswin, whilethegeneralpublicloses; thepriceisincreasedforthelesspricesensitivegeneralpublic and loweredforthepricesensitivestudents • Thisassumesthatthe „fence“ betweenstudents and thegeneralpublicwasperfect • Also, thatthemarginalcosts, and ancillaryrevenues, werethesameforthetwogroups

  16. Variablepricing • Telephonecompaniesusedtochargedifferentpricesfordaytime and eveningcalls • TheSan Francisco Operacharges a lowerpriceforweeknightsthanforweekends, themostexpensiveboxseatscost $175 and $195 respecitvely on Wednesdays and Saturdays and theleastexpensivebalcony side ticketscost $25 and $28 • The Colorado Rockiesbaseballteamuses a four-tier pricingsystem

  17. Variablepricing • TheGulfPowerCompanyserves 370000 retailcustomersinnorthwestern Florida; themajoritypay $0.057 perkilowatthour; ResidentialSelectVariablePricingprogramwithoff-peak ($0.035=1030PM-6AMSummer, 1000PM-530AMWinter), on-peak ($0.093=1130AM-8.30PMSummer, 6AM-noonWinter) and shoulder($0.046) periods • Thecompanycandeclareupto 88 hoursofcriticalperiodannually (cost $0.290)

  18. Variable pricing • Demand is variable, but follows a predictable pattern • The capacity of a seller is fixed in the short run (or is expensive to change) • Inventory is perishable or expansive to store, otherwise buyers would learn to predict the variation in prices and stockpile when the price is low • The seller has the ability to adjust prices in response to supply/demand imbalances • Customers can self-select

  19. A theme park can serve upto 1000 customers, marginalcostiszero

  20. Singleprice – widevariationinutilization and a large number ofturndowns; solve independentlyforeachdayforvariablepricing – are thedemandsindependent? • Number of customersservedrises 29%, totalrevenuerises 30%, utilizationrisesfrom 68.5% to 89% • Averagepricerisesonlyfrom $25.00 to $25.03

  21. Variablepricingwithdiversion (demandshifting), Robert Cross (1997) • A barbershopthatturnsawaycustomers on Saturdays, whileTuesdays are slow • SomeoftheSaturdaycustomers are workingpeople, butsome are alsoretirees and schoolchildren, whocouldcome on anyday • Priceswererisen 20% on Saturday and reduced 20% on Tuesday – turnawaysdecreased, Saturdayserviceimproved and totalrevenueincreasedbyalmost 20% • Itis a two-edgedsword, though

  22. Knowledge about individual wtp • Consumer wtp vector • $30, $18, $22, $19, $14, $18, $32 • Seller price vector • $33.87, $15.00, $17.50, $18.01, $19.00, $27.00, $38.33 • Consumer surplus vector – select Tuesday • −$3.87, $3.00, $4.50, $.99, −$5.00, −$9.00, −$6.33 • Comparison with uniform price – select Saturday • $5.00, −$7.00, −$3.00, −$6.00, −$11.00, −$7.00, $7.00

  23. Easytodescribe, difficulttomodel • If we knew wtp across the population • Describes well, but modeling requires a lot of information for deriving a multidimensionaldemand function • Itwouldrequireestimatingsevenown-priceelasticities and 42 crosspriceelasticities • Itisunlikely, that a theme part ormostothercompanieswouldhaveenoughdataavailabletoestimate a crediblemodelwiththismanyparameters

  24. A simplermodel – eightcustomerswillshiftfomonedaytoanotherfor $1 • Ifprices are $20 and $22 forMonday and Tuesday – wecalculatethedemandindependently and thenshift 16 people; and dothisfortheentireweek

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