Automatic Markdowns

Mariano Tappata Sauder School of Business University of British Columbia

IIOC 2010 Vancouver, May 2010

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What are Automatic Markdowns?

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The Pioneer

“According to the system, every article is marked with a tag showing the price and the date the article was first put on sale. Twelve days later, if it has not been sold, it is reduced by 25 percent. Six selling days later, it is cut by 50 percent and after an additional six days, it is offered at 75 percent off the original price. After six more days — or a total of 30 — if it is not sold, it is given to charity.” (Boston’s Favorite Bargain Store, NYT, April 18, 1982)

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Why do we observe descending prices?

Price discrimination Learning and production cost Learning valuations

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Why do we observe descending prices?

Price discrimination Learning and production cost Learning valuations Environment matters: Unique good Perishable goods Network goods Heterogeneous valuations

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Lazzear (1986) and Automatic Markdowns

Lazzear (1986) Active seller Pasive consumers

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Lazzear (1986) and Automatic Markdowns

Lazzear (1986) Active seller Pasive consumers

This paper: Strategic consumers + pasive seller When is it optimal to buy? When is it best to use Automatic Markdowns?

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Example

𝑡 = {1, 2} 𝑛=2 𝑣𝑖 ∼ 𝑈 [0, 1] 𝑝1 = 𝛿1 , 𝑝2 = 𝛿2 ; 1 ≥ 𝛿1 >≥ 𝛿2 ≥ 0.

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Lazear (1986)

Lazear 1986. Pricing with uniform valuation, 2 periods and arrivals in every period. 

$  1 

2/3 

1/3 

1 

2

t

 

The problem with strategic consumers is that as long as you receive the visit by a buyer in t=1, Automatic Markdowns 7 / 30 anticipate prices (or the prices could be announced). In the case of P1=1, P2=P3=0, and p=2/3 

The problem with strategic consumers is that as long as you receive the visit by a buyer in t=1, anticipate prices (or the prices could be announced). In the case of P1=1, P2=P3=0, and p=2/3  delta=1/2, v’=5/6 (0.833) 

Lazear (1986)

$  1 

  2/3 

1/3 

1 

2

t

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With Automatic Markdowns, n=2, PRIVATE VALUES, Pr(tao=1)=1 Pr(tao=2)= Pr(tao=3)=0  Profits are 5.7% higher with AM  

Strategic Consumers

$  0.76  0.59  0.57  0.52 

1 

2

t

   

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With Automatic Markdowns, n=2, COMMON VALUES, Pr(tao=1)=Pr(tao=2)=1/2 , Pr(tao=3)=0  Profits are 36% higher with AM [Pr=1/3 generates very similar result, profits are 18% higher, p= delta*p=0.47, v’=0.73] 

Strategic Consumers (common values)

$ 

0.72  0.61  0.45 

1 

2

t

  With Automatic Markdowns, n=2, COMMON VALUES, Pr(tao=1)=1, Pr(tao=2)=Pr(tao=3)=0  Automatic Markdowns 10 / 30 Profits are 20% higher with AM 

A Model of Automatic Markdowns

Monopolist sells a unique good: 𝑝 (𝑡) = 𝑣 (1 − 𝛿𝑡) , 𝛿 > 1/ 𝜏 𝑛 potential consumers Random arrivals: 𝜏𝑖 ∼ 𝐹 (⋅) , 𝜏𝑖 ∈ [0, 𝜏 ] Common values: 𝑣𝑖 = 𝑣 ∼ 𝐺 (⋅) , 𝑣 ∈ [0, 𝑣] with 𝑣 ≥ 1 Returning/transaction cost 𝛼 : [𝛼𝐿 , 𝛼𝐼 , 𝛼𝐻 ]

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A Model of Automatic Markdowns

“Bidding” strategies

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A Model of Automatic Markdowns

“Bidding” strategies Should visitors return later?

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A Model of Automatic Markdowns

“Bidding” strategies Should visitors return later?

{ 𝑢𝑖 (𝑏𝑖 , 𝑏−𝑖 ; 𝜏𝑖 , 𝑣) =

𝑣 − 𝑣 (1 − 𝛿𝑏𝑖 ) , 𝑏𝑖 = 𝜏𝑖 [𝑣 − 𝑣 (1 − 𝛿𝑏𝑖 )] ⋅ 1𝑏𝑖
Early Birds: 𝜏˜ =

𝑣−𝑣 𝛿𝑣

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High Return Costs: 𝛼𝐻

_

Α>Α - Uniform Arrivals

Τ 1∆

Ž Τ HvL

LC Τ* HvL

EB

0

v0 =

∆v n

v v

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No Return Costs

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No Return Costs Early birds should always return

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No Return Costs Early birds should always return Some latecomers too...

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No Return Costs Early birds should always return Some latecomers too... Equilibrium properties 

Symmetric equilibrium:     1/  

   

   

0,

̂ 

         

̃ 

 

̂ 

1/

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1 Return Costs No ̃ 1 1

Pr

̃

 

   

0 

𝐸𝑈 𝛽 (𝑏𝑖 ; 𝜏𝑖 , 𝑣)

=

( ) ̃  [𝑣 − 𝑣 (1 − 𝛿𝑏𝑖 )] Pr min 𝛽 (𝜏𝑗 ) > 𝑏𝑖 ∣ min 𝛽 (𝜏𝑗 ) > 𝜏 𝑖 𝑗∕=𝑖

 

𝑗∕=𝑖

  1 1

Pr

 

  0 

 

̃ 

 

 

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No Return Costs

Α=0 - Uniform Arrivals

Τ 1∆

LCB Ž Τ HvL

EB

0

v1 =v

` Τ HvL

Τ* HvL

LCA

∆-1 n-1

v v

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0 < 𝛼 < 𝛼𝐻

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0 < 𝛼 < 𝛼𝐻 Returning might not be profitable now

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0 < 𝛼 < 𝛼𝐻 Returning might not be profitable now Two cases: high and low “relative” return cost

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̃ 

 

0 < 𝛼 < 𝛼𝐻

̂ 

1/

 

 

Returning might not be profitable now  

Two cases: high and low “relative” return cost    

Case 1:  

         

 

Case 2:

1/  

 

1/

  0,

̂   ̃  ̃

     

̃ 

̂  

1/

̂ ̃

 

1/

 

   

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1 1

Pr

 

Intermediate Costs 𝛼𝐼

  0 

 

̃ 

 

 

   

𝐸𝑈𝛾 (𝑏𝑖 ; 𝜏𝑖 ≤ 𝜏ˆ𝛼 , 𝑣) = [𝑣 − 𝑣 (1 − 𝛿𝑏)] Pr (𝑊 ) − 𝛼

  1

Pr

̂

1 ̂

1

 

  0 

 

  1

̃ 

̂  

̃

 

 

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Intermediate Return Costs

Τ

0<Α<Α - Uniform Arrivals

1∆

A.3 B.3

A.2 Ž Τ HvL

B.2

Τ* HvL

` Τ Α HvL

A.1

B.1 0

v1 =v

∆-1 n-1

v'

v v

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Intermediate Return Costs

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Expected Revenues

Fixed Posted Prices: Π𝐹 (𝑝) = 𝑝 [1 − 𝐺 (𝑝)] 𝐹𝜏(1) (𝑇 ) 𝑝∗ =

[1 − 𝐺 (𝑝∗ )] 𝑔 (𝑝∗ )

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Expected Revenues

Automatic Markdowns: Π𝐴 𝛼𝐿

∫𝑣 ∫1/𝛿 𝑣 [1 − 𝛿𝜂𝛼𝐿 (𝜏, 𝑣)]𝑛 [1 − 𝐹 (𝜏 )]𝑛−1 𝑑𝐹 𝑑𝐺 = | {z } 0

0

price

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Expected Revenues

Automatic Markdowns: Π𝐴 𝛼𝐿

∫𝑣 ∫1/𝛿 𝑣 [1 − 𝛿𝜂𝛼𝐿 (𝜏, 𝑣)]𝑛 [1 − 𝐹 (𝜏 )]𝑛−1 𝑑𝐹 𝑑𝐺 = | {z } 0

Π𝐴 𝛼𝐻 =

0

∫𝑣 ∫1/𝛿 0 𝜏˜(𝑣)

price

𝜏 (𝑣))]𝑛−1 𝑑𝐹 𝑑𝐺 𝑣 (1 − 𝛿𝜏 )𝑛 [1 − 𝐹 (𝜏 ) + 𝐹 (˜ | {z } price

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Expected Revenues

Price paid by the first visitor (𝛼 = 0): 𝑝 (𝜏 = 0, 𝑣) = 𝑣 − 𝑣𝛿 (𝜏 ∗ − 𝜏˜) [1 − 𝐹 (𝜏 ∗ )]𝑛−1

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Expected Revenues

Price paid by the first visitor (𝛼 = 0): 𝑝 (𝜏 = 0, 𝑣) = 𝑣 − 𝑣𝛿 (𝜏 ∗ − 𝜏˜) [1 − 𝐹 (𝜏 ∗ )]𝑛−1

AM and Market Thickness There exists 𝑛∗ such that AM dominates fixed posted prices for 𝑛 ≥ 𝑛∗ .

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Uniform Arrivals and Valuations

EHvL PΑAL PΑAH

PF

∆=2

0

2

10

n

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Uniform Arrivals and Valuations

EHvL

PΑAL PΑAH PF ∆=7

0

2

10

n

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Storage costs Expected Price -Expected Storage Cost Ep-ESc HEp-EScLΑL

HEp-EScLΑH

n=5,sc=0.6

0

1

2

n

∆

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Storage costs Expected Price -Expected Storage Cost Ep-ESc

HEp-EScLΑL HEp-EScLΑH

n=15

0

1

2

∆

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Conclusions

AM as an alternative to fixed prices or sales Strategic consumers and equilibrium Revenues and market thickness Revenues and transaction costs Storage costs and 𝛿

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Back Automatic Markdowns 30 / 30