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

• 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

The widget example

• Price 8.75; demand 3000; total contribution $11250

• What should be done, if only 2000 widgets can be manufactured?

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

Supply constraints• Service providers: a hotel only has a number of rooms; a gas

pipeline is constrained by the capacity of its pipes; a barbershop by the number of seats

• Manufacturers: physical constraints on the production amount – 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. fashion goods are only ordered once, so are the electronic goods at the end of their life cycle

• Intrinsically scarce items: beachfront property, flawless blue diamonds, van Gogh paintings, Stradivarius violins – marginal cost is either meaningless (paintings) or extremely low relative to the scracity rent (diamonds)

Hard and soft constraints

• Hard cannot be violated at any price – hotels and gas pipelines

• Soft – freight carriers can lease space on other carriers in case of excess capacity

• Timing factor – an airline learning of a very high demand two months in advance will be able to assign a new aircraft, but not when this becomes evident a week before the fact

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)

The runout price, also a Solver problem• d(p|)=10000-800p|=2000->p|=$10;contrib=$10000• Choose maximum of p* and p|• And remember, for the future, if constraint was 2001:• d(p|)=10000-800p|=2001->p|~$10;contrib=$10002.5

.

• 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

Total and marginal opportunity cost

• Total opportunity cost is measured as the loss in optimal contribution – e.g. at 2000 units it is $11250-$10000=$1250

Marginal opportunity cost

• At 2000, the optimal contribution is $10000; at 2001, it is $10002.50

• Thus, at that level the marginal opportunity cost (the potential increase in contribution margin) is $2.50

• As it is the increase in profit, it is the rent that the producer would be willing to pay; or purchase a unit for up to $2.50+$5.00(cost)=$7.50 from elsewhere

• Marginal opportunity cost is zero when the capacity constraint is non binding

Market segmentation and supply constraints

• The football game between Stanford and the University of California at Berkeley (a.k.a. „the Big Game“) has 60000 seats to sell

• We could (but lets instead do it analytically):

Single price to all?: analytical solution, see rm12.xlsx, sheet6*** for Solver

• MarginalRevenue=MarginalCost; in case of the tickets of a sporting event MC=0; thus MR=0

• Total demand: d(p)=140000-4250p• Revenue=p*d(p)=140000p-4250p^2• MR=0->(140000p-4250p^2)’=0->140000-8500p=0• p=16.47• We also find that at that price, all the students will be priced out• ds(p)=0 at 20000-1250p=0->p=16• Also, the demand would exceed 60000• 140000-4250*16.47=70000• Thus, we find a price, at which the demand for general public is 60000• 120000-3000p|=60000• p|=20 – this is the optimal single price

Two prices, analytical solution, see rm12.xlsx sheet7 for Solver

• Prices should be set so that the marginal revenues from both segments are equal with each other (here also with zero, which is MC)

• Also, the supply constraint should be satisfied• Let’s find MRs, silmplify and equate them: 120000-6000pg

and 20000-2500ps->20-pg=8-ps->pg=ps+12• Now lets take the demand constraint and simplify:

(120000-3000pg)+(20000-1250ps)=60000->3pg+1.25ps=80• Solving these two equations gives: ps=$10.35 and

pg=$22.35; general public gets 52941 tickets and students get 7059 tickets… with total revenue of $1256471, which is 4.7% over the single ticket price case

• As the ticket prices show, students win, while the general public loses; the price is increased for the less price sensitive general public and lowered for the price sensitive students

• This assumes that the „fence“ between students and the general public was perfect

• Also, that the marginal costs, and ancillary revenues, were the same for the two groups

Variable pricing• Telephone companies used to charge different

prices for daytime and evening calls• The San Francisco Opera charges a lower price

for weeknights than for weekends, the most expensive box seats cost $175 and $195 respecitvely on Wednesdays and Saturdays and the least expensive balcony side tickets cost $25 and $28

• The Colorado Rockies baseball team uses a four-tier

pricing system

Variable pricing

• The Gulf Power Company serves 370000 retail customers in northwestern Florida; the majority pay $0.057 per kilowatt hour; Residential Select Variable Pricing program with off-peak ($0.035=1030PM-6AMSummer, 1000PM-530AMWinter), on-peak ($0.093=1130AM-8.30PMSummer, 6AM-noonWinter) and shoulder ($0.046) periods

• The company can declare up to 88 hours of critical period annually (cost $0.290)

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

A theme park can serve up to 1000 customers, marginal cost is zero

Single price – wide variation in utilization and a large number of turndowns; solve independently for each day for variable pricing – are the demands

independent?

• Number of customers served rises 29%, total revenue rises 30%, utilization rises from 68.5% to 89%

• Average price rises only from $25.00 to $25.03

Variable pricing with diversion (demand shifting), Robert Cross (1997)

• A barbershop that turns away customers on Saturdays, while Tuesdays are slow

• Some of the Saturday customers are working people, but some are also retirees and schoolchildren, who could come on any day

• Prices were risen 20% on Saturday and reduced 20% on Tuesday – turnaways decreased, Saturday service improved and total revenue increased by almost 20%

• It is a two-edged sword, though

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

Easy to describe, difficult to model

• If we knew wtp across the population• Describes well, but modeling requires a lot of

information for deriving a multidimensional demand function

• It would require estimating seven own-price elasticities and 42 cross price elasticities

• It is unlikely, that a theme part or most other companies would have enough data available to estimate a credible model with this many parameters

A simpler model – eight customers will shift fom one day to another for $1

• If prices are $20 and $22 for Monday and Tuesday – we calculate the demand independently and then shift 16 people; and do this for the entire week