1. microeconomics spring 2016 the analytics of
session eight
dynamic pricing (I)
airline pricing - introduction
………….1
airline uniform pricing
………….2
airline pricing - dynamic approach
………….4
airline myopic pricing
……..…...5
airline strategic pricing
………….9
airline pricing - key points
..……….11
retail pricing - introduction
………...12
retail uniform pricing
……..…13
retail pricing - dynamic approach
………..15
retail myopic pricing
……..….16
retail strategic pricing
………...20
retail pricing - key points
..……….24
spring 2016
microeconomics
the analytics of
constrained optimal decisions

2. microeconomics setup assumptions airline industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
setup
 We observe an increase in airline ticket price as the date of take-off approaches
 Clearly the static/uniform pricing monopoly model is too simple to capture and explain this behavior
 We have to develop a dynamic model that would capture:
- the market participants in each time interval
- the optimal behavior of the monopolist
assumptions
 The market: consists of many buyers (more than the capacity of the plane) with different willingness to buy the airline ticket, each buyer will buy at most one ticket when the price for the ticket is below buyer’s willingness to pay
 Time frame: there are two periods considered such that
- in the first period a certain fraction of the total buyers are searching for an airline ticket and these buyers are those with the lowest willingness to pay
- in the second period the remaining fraction of the buyers (that either did not get a ticket in the first period or did not participate in the first period) are searching for an airline ticket
 No arbitrage: buyers with low valuation that buy a ticket in the first period are not able to re-sell the ticket in the second period.
 2016 Kellogg School of Management
lecture 8
page | 1

3. microeconomics uniform pricing key points airline industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
uniform pricing
► Demand: P(Q) = 500 – Q
► Marginal revenue is obtained as
MR(Q) = 500 – 2Q
► Marginal cost is MC(Q) = 0
► Optimal output (A) and price (B)
Qm = 250 and Pm = 250
 With uniform pricing tickets are sold at the same price and the whole market is served at once.
 Notice that 50 seats will be empty (airplane capacity is 300).
 Total revenue (and profit since costs are assumed to be zero) is in this case:
TR = Pm∙Qm = 250∙250 = 62,500
500
demand
capacity = 300 MR
key points
(B)
250
(A)
250
300
500
 2016 Kellogg School of Management
lecture 8
page | 2

4. microeconomics uniform pricing - model airline industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
uniform pricing - model
► How would you formulate the problem of finding the optimal solution for the uniform pricing case?
► “Ingredients”:
demand P = 500 – Q
objective maximize profit  = PQ = (500 – Q)Q for 0  Q  300
► From an algebraic perspective, just take the first derivative of the objective function (with respect to Q), set that expression to zero and solve for the Q; this will give you 500 – 2Q = 0 with solution Q = 250.
► From a modelling perspective the analyst is considering a “one-shot” market even though the market may be functioning several periods. In other words, say the market is open for two days, this formulation of the problem implies that the same price is offered in both periods.
 2016 Kellogg School of Management
lecture 8
page | 3

5. microeconomics dynamic approach - introduction airline industry
lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
dynamic approach - introduction
high value
low
value
► Let’s assume that there are 250 buyers that are searching for a ticket in the first period (and these are the buyers with the lowest valuation)
► The remaining 250 are entering the market only in the second period.
► The buyers that entered the market in the first period but could not buy a ticket could potentially re-enter the market in the second period, however they will not get a ticket in the second period for sure (since they’ll have a valuation far below the valuation of all others present in the market)
500
demand
250
500
second period
market
first period
market
 2016 Kellogg School of Management
lecture 8
page | 4

6. microeconomics myopic pricing key points airline industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
myopic pricing
► First period market
► We can think of the first period market as a “stand-alone” market with a demand P1(Q) = 250 – Q and a corresponding marginal revenue
MR1(Q) = 250 – 2Q
► Optimal output (C) and price (D)
Q1= 125 and P1 = 125
 Only 125 tickets are sold in the first period which means there are another 175 seats available for the second period market.
 Of course this means that there are 125 persons (250 total trying to buy a ticket in the first period, but only 125 got a ticket) still being in the market in the second period
first period
market
500
demand
key points
250
ticket
125
(D)
move to second period
MR1
(C)
125
250
tickets sold
first period
 2016 Kellogg School of Management
lecture 8
page | 5

7. moved from first period
microeconomics
lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
myopic pricing
► Second period market
► We can think again of the second period market as a “stand-alone” market with a demand
P2(Q) = 500 – Q for Q ≤ 250
and a corresponding marginal revenue
MR2(Q) = 500 – 2Q for Q ≤ 250
► Optimal output (E) and price (F)
Q2 = 175 and P2 = 325
 All remaining 175 tickets are sold in the second period since the profit maximization is constrained by the available capacity.
 The buyers moved fro m first period have their willingness to pay too low to count in the second period.
second period
market
500
demand
residual capacity = 175
MR2
(F)
325
key points
(E)
125
moved from first period
175
250
375
tickets sold
second period
 2016 Kellogg School of Management
lecture 8
page | 6

8. microeconomics myopic pricing airline industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
myopic pricing
second period
market
first period
market
► Conclusion: the ticket price increases as we approach the “take-off” date, i.e. we move from period one to period two.
► The total revenue (and since costs are zero this is also the profit) is:
TR = P1∙Q1 + P2∙Q2 =
= 125∙ ∙175 = 72,500
► No surprise that the airline is doing far better than the case of uniform pricing (total revenue of 62,500)
500
ticket
residual
capacity
residual
capacity
total
capacity
MR2
325
(F)
250
ticket
(E)
125
(D)
MR1
(C)
175
250
125
250
300
tickets sold
second period
tickets sold
first period
 2016 Kellogg School of Management
lecture 8
page | 7

9. microeconomics myopic pricing - model airline industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
myopic pricing - model
► How would you formulate the problem of finding the optimal solution for the myopic pricing case?
► “Ingredients”:
demand period 1: P1 = 250 – Q1 for 0  Q1  300
period 2: P2 = 500 – Q2 for 0  Q2  300 – Q1
objective maximize per-period profit
period 1: 1 = P1Q1 = (250 – Q1)Q1
period 2: 2 = P2Q2 = (500 – Q2)Q2
► From an algebraic perspective, the solution is fairly straightforward:
period 1: (first derivative = 0) 250 – 2Q1 = 0 gives Q1 = 125 (less than 300, ok solution)
period 2: (first derivative = 0) 500 – 2Q2 = 0 gives Q2 = 250 (more than 300 – 125 = 175, constrained solution to 175)
► From a modelling perspective the analyst is considering now a “two-shot” market, i.e. charging different prices in the two periods; however, the current formulation maximizes the “per-period” profit not the sum (overall) profit.
 2016 Kellogg School of Management
lecture 8
page | 8

10. microeconomics strategic pricing - model airline industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
strategic pricing - model
► The myopic pricing does improve upon the uniform pricing as at least it charges different prices in the two periods to take advantage of the higher willingness to pay of buyers in the second period. The big short-coming is though that the profit maximization is performed “per-period” not looking at the sum of the profit over the two periods.
► This is the next step taken here: maximize the sum of the profit over the two –period.
► How would you formulate the problem of finding the optimal solution for the strategic pricing case?
► “Ingredients”:
demand period 1: P1 = 250 – Q for 0  Q1  300
period 2: P2 = 500 – Q for 0  Q2  300 – Q1
objective maximize overall profit  = P1Q1 + P2Q2 = (250 – Q1)Q1 + (500 – Q2)Q2
► From an algebraic perspective, the solution is fairly straightforward: try first a solution for which all 300 tickets are sold, then Q2 = 300 – Q1 which is plugged into the profit function:
 = P1Q1 + P2Q2 = (250 – Q1)Q1 + [500 – (300 – Q1)](300 – Q1) = Q1 – 2Q12
Taking the first derivative and set it equal to zero you’ll get 350 – 4Q1 = 0, Q1 = Use this to find Q2 = 300 – Q1 = 212.5
► From a modelling perspective the analyst is considering now a “two-shot” market and in the current formulation the analyst maximizes the overall profit.
 2016 Kellogg School of Management
lecture 8
page | 9

11. microeconomics strategic pricing airline industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
strategic pricing
second period
market
first period
market
► Two-period market
► If the effect of period 1’s decision on period 2’s outcome is taken into account the solution is different:
Q1 = and P1 = 162.5
Q2 = and P2 = 287.5
► The total revenue (and since costs are zero this is also the profit) is:
TR = P1∙Q1 + P2∙Q2 = 75,312.5
► This is the highest profit that the airline can obtain!
► Notice that the marginal revenue is the same (and equal to 75) in both periods …
► Just a coincidence?
500
residual
capacity
residual
capacity
total
capacity
ticket
MR2
287.5
(J)
250
ticket
162.5
MR1
(H) 75 75
(I)
(G)
212.5
250
87.5
250
300
second period
first period
 2016 Kellogg School of Management
lecture 8
page | 10

12. microeconomics dynamic pricing – key points airline industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
airline industry
dynamic pricing – key points
ONE PERIOD MODEL
TWO PERIOD MODEL
UNIFORM PRICING
MYOPIC PRICING
STRATEGIC PRICING
 one demand function
 maximize one period profit
 one price for all buyers
 two demand functions
 maximize per-period profits
 per-period price discrimination
 two demand functions
 maximize sum of periods profits
 per-period price discrimination
single period
market
second
period
first
period
second
period
first
period
325
287.5
250
162.5
125
250
175
125
212.5
87.5
single
period
second period
first
period
second period
first
period
► The shaded areas represent the profit made by the airline (one-market on the far left and two-market case for the two diagrams on the right)
 2016 Kellogg School of Management
lecture 8
page | 11

13. microeconomics setup assumptions retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
setup
► We observe a decrease in price for units sold as the time passes (the all familiar and enjoyable discount sales…)
► Again, the static/uniform pricing monopoly model is too simple to capture and explain this behavior
► We have to develop a dynamic model that would capture:
- the market participants in each time interval
- the optimal behavior of the monopolist
assumptions
► The market: consists of many buyers (more than the inventory availability) with different willingness to buy
► The buyers: we assume that desire to be the first to have the “coolest” and “newest” clothing induces a buyer to acquire the “piece” immediately if the price is below his/her willingness to pay (no strategic waiting)
► Time frame: there are two periods created by the retailer as follows
- in the first period the retailer sets a price P1 and all buyers with a willingness to pay higher than P1 will buy immediately and remove themselves from the market
- in the second period is defined as the period for which the retailer changes the price to P2 and all buyers with a willingness to pay higher than P2 will buy immediately
 2016 Kellogg School of Management
lecture 8
page | 12

14. microeconomics uniform pricing key points retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
uniform pricing
► Demand: P(Q) = 500 – Q
► Marginal revenue is obtained as
MR(Q) = 500 – 2Q
► Marginal cost is MC(Q) = 0
► Optimal output (A) and price (B)
Qm = 250 and Pm = 250
 With uniform pricing units are sold at the same price and the whole market is served at once.
 Notice that 50 units will remain unsold (inventory is 300).
 Total revenue (and profit since costs are assumed to be zero) is in this case:
TR = Pm∙Qm = 250∙250 = 62,500
500
demand
capacity = 300 MR
key points
(B)
250
(A)
250
300
500
 2016 Kellogg School of Management
lecture 8
page | 13

15. microeconomics uniform pricing - model retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
uniform pricing - model
► How would you formulate the problem of finding the optimal solution for the uniform pricing case?
► “Ingredients”:
demand P = 500 – Q
objective maximize profit  = PQ = (500 – Q)Q for 0  Q  300
► From an algebraic perspective, just take the first derivative of the objective function (with respect to Q), set that expression to zero and solve for the Q; this will give you 500 – 2Q = 0 with solution Q = 250.
► From a modelling perspective the analyst is considering a “one-shot” market even though the market may be functioning several periods. In other words, say the market is open for two days, this formulation of the problem implies that the same price is offered in both periods.
 2016 Kellogg School of Management
lecture 8
page | 14

16. microeconomics dynamic approach retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
dynamic approach
high value
low
value
► Let’s assume that the retailer sets a price P1 for a new line of clothing – all buyers with willingness to pay greater than P1 will buy at this price
► The retailer sells Q1 units.
► The retailer basically created two markets: the starting market at price P1 and quantity Q1 and then for the remaining buyers there will be a new price P2
500
demand P1 Q1
500
first period
market
second period
market
 2016 Kellogg School of Management
lecture 8
page | 15

17. microeconomics myopic pricing key points retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
myopic pricing
first period
market
► First period market
► Since at the start of the “game” the retailer serves all the market the first period demand is actually
P1(Q) = 500 – Q
and corresponding marginal revenue
MR1(Q) = 500 – 2Q
► Optimal output (C) and price (D)
Q1= 250 and P1 = 250
 Only 250 units are sold in the first period which means there are another 50 units available for the second period market.
500
demand
capacity = 300
MR1
(D)
250
key points
(C)
250
300
500
first period
market
residual
capacity
 2016 Kellogg School of Management
lecture 8
page | 16

18. microeconomics myopic pricing key points retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
myopic pricing
second period
market
► Second period market
► The remaining buyers represent now the second period market with demand
P2(Q) = 250 – Q
and corresponding marginal revenue
MR1(Q) = 250 – 2Q
► Optimal output (E) and price (F)
Q2= 50 and P2 = 200
 All remaining 50 units are sold in the second period; the profit maximization is constrained by the available capacity.
500
demand
residual capacity = 50
MR1
250
key points
(F)
200
(E)
MR2 50
250
first period
market
second
period market
 2016 Kellogg School of Management
lecture 8
page | 17

19. microeconomics myopic pricing retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
myopic pricing
first period
market
second period
market
► Two-period market
► Conclusion: the per unit price decreases once the high-willing to pay buyers are making their purchase and exit the market.
► The total revenue (and since costs are zero this is also the profit) is:
TR = P1∙Q1 + P2∙Q2 =
= 250∙ ∙50 = 72,500
► No surprise that the airline is doing far better than the case of uniform pricing (total revenue of 62,500)
500
total
capacity
residual
capacity
buys
MR1
residual capacity
250
250
(D)
buys
200
(F)
(E)
MR2
(C)
250
300 50
250
units sold first period
units sold second period
 2016 Kellogg School of Management
lecture 8
page | 18

20. microeconomics myopic pricing - model retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
myopic pricing - model
► How would you formulate the problem of finding the optimal solution for the myopic pricing case?
► “Ingredients”:
demand period 1: P1 = 500 – Q for 0  Q1  300
period 2: P2 = (500 – Q1) – Q for 0  Q2  300 – Q1
objective maximize per-period profit
period 1: 1 = P1Q1 = (500 – Q1)Q1
period 2: 2 = P2Q2 = (500 – Q1 – Q2)Q2
► From an algebraic perspective, the solution is fairly straightforward:
period 1: (first derivative = 0) 500 – 2Q1 = 0 gives Q1 = 250 (less than 300, ok solution)
period 2: (first derivative = 0) (500 – Q1) – 2Q2 = 0 implies Q2 = 125 (constrained to 50)
► From a modelling perspective the analyst is considering now a “two-shot” market, i.e. charging different prices in the two periods; however, the current formulation maximizes the “per-period” profit not the sum (overall) profit.
 2016 Kellogg School of Management
lecture 8
page | 19

21. microeconomics strategic pricing - model retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
strategic pricing - model
► Two-period market
► Let’s analyze the pricing strategy. The choice of prices in the two steps:
- the price is chosen first for period 1 such that it maximizes the profit in period 1
- the remaining units (residual capacity/inventory) is then available for period 2
► Clearly this is an improvement over the uniform pricing but can the airline do better? How can the retailer improve its pricing strategy, i.e. to get an even higher profit?
► Notice that the retailer chooses the quantity in period 1 without factoring in the decision in period 2, i.e. this is a period-by-period maximization: myopic optimization.
► The profit obtained in period 2 clearly depends on the decision taken in period 1 – this has to be taken into account when the retailer sets the price and quantity in period 1…
► This is again an issue addressed through strategic optimization, or backward induction …
 2016 Kellogg School of Management
lecture 8
page | 20

22. microeconomics strategic pricing - model retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
strategic pricing - model
► The myopic pricing does improve upon the uniform pricing as at least it charges different prices in the two periods to take advantage of the higher willingness to pay of buyers in the second period. The big short-coming is though that the profit maximization is performed “per-period” not looking at the sum of the profit over the two periods.
► This is the next step taken here: maximize the sum of the profit over the two –period.
► How would you formulate the problem of finding the optimal solution for the strategic pricing case?
► “Ingredients”:
demand period 1: P1 = 500 – Q for 0  Q1  300
period 2: P2 = 500 – Q1 – Q for 0  Q2  300 – Q1
objective maximize overall profit  = P1Q1 + P2Q2 = (500 – Q1)Q1 + (500 – Q1 – Q2)Q2
► First notice that there is really no period market before period 1 market is closed (we need Q1 sold in period 1 to figure the demand in period 2). The fact that the choice of Q1 affects the demand function of period 2 means that when we calculate the marginal revenue relative to period 1, i.e. increase Q1 by one unit, we have to take into account how this extra unit affects the revenue in period 2.
► A (small) change in Q1 will bring a change in total revenue, i.e. marginal revenue MR1 equal to
MR1 = [500 – 2Q1] + [–Q2] = 500 – 2Q1 – Q2
► A (small) change in Q2 will bring a change in total revenue, i.e. marginal revenue MR2 equal to
MR2 = 500 – Q1 – 2Q2
period 1
period 2
 2016 Kellogg School of Management
lecture 8
page | 21

23. microeconomics strategic pricing - model retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
strategic pricing - model
► With the two marginal revenues available, MR1 giving the change in total revenue when an extra unit is added to Q1 and MR2 giving the change in total revenue when an extra unit is added to Q2, at the optimum
MR1 = MR2
that is
500 – 2Q1 – Q2 = 500 – Q1 – 2Q2
with solution
Q1 = Q2
► The second condition is that
Q1 + Q2 = 300
► The optimal allocation is thus
Q1 = Q2 = 150
► The prices are
period 1: P1 = 500 – Q1 = 350
period 2: P2 = 500 – Q1 – Q2 = 200
 2016 Kellogg School of Management
lecture 8
page | 22

24. microeconomics strategic pricing retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
strategic pricing
first period
market
second period
market
► Two-period market
► If the effect of period 1’s decision on period 2’s outcome is taken into account the solution is different:
Q1 = 150 and P1 = 350
Q2 = 150 and P2 = 200
► The total revenue (and since costs are zero this is also the profit) is:
TR = P1∙Q1 + P2∙Q2 = 82,500
► This is the highest profit that the retailer can obtain!
► The dotted line in the diagram on the left represents the marginal revenue when Q1 changes without taking into account the effect of this change on period 1.
500
buys
total
capacity
residual
capacity
350
(H)
350
buys
residual capacity
Q2 = 150
200
(J)
MR1
MR2
(G) 50 50
(I)
150
175
250
300
150
175
units sold
first period
units sold
second period
 2016 Kellogg School of Management
lecture 8
page | 23

25. microeconomics dynamic pricing – key points retail industry lecture 8
dynamic pricing (I)
the analytics of constrained optimal decisions
retail industry
dynamic pricing – key points
ONE PERIOD MODEL
TWO PERIOD MODEL
UNIFORM PRICING
MYOPIC PRICING
STRATEGIC PRICING
 one demand function
 maximize one period profit
 one price for all buyers
 two demand functions
 maximize per-period profits
 per-period price discrimination
 two demand functions
 maximize sum of periods profits
 per-period price discrimination
single period
market
first
period
second
period
first
period
second
period
350
250
250
200
200
250
250 50
150
150
single
period
first
period
second
period
first
period
second
period
► The shaded areas represent the profit made by the airline (one-market on the far left and two-market case for the two diagrams on the right)
 2016 Kellogg School of Management
lecture 8
page | 24