Price Discrimination

• The Effects of Third-Degree Price Discrimination

• Add-on Pricing

• Bundling

• Nonlinear Pricing

Third Degree

• Distinguish from First-Degree or Perfect Price Discrimation

• Large literature from Pigou (1920) and Robinson (1933) toAER today (Aguirre, Cowan and Vickers, Sept 2010 AER,

”Monopoly Price Discrimination and Demand Curvature”)

• Large question relates to the first welfare theorem. Withuniform price monopoly we don’t get efficiency. Perfectly

discriminating monopoly we get efficiency. What happens in

between, closer to efficiency?

• Illustate Pigou’s point with liner demand on the board.

• Get output issue in general as well as a distortion: goods notallocated to consumers with highest willingness to pay.

• Is this empirically important?

Stole Handbook version of Holmes (1989)

• Two markets = 1 2, duopolists = .

• ( ) demand of firm in market given prices (assumesyjmetric)

• Market demand () = ( ) = ( ) and market elas-ticity

() = −

()0()

• Firm 0 elasticity of demand in market ,

(

) = −

(

)

( )

which at symmetric prices = = is

() = −

()0() +

()

( )

= () +

()

where () 0 is the cross-price elasticity of demand at

symmeric prices

• Monopoly pricing rule: −

=1

()

• Bertrand duopoly pricing: −

=1

() + ()

• Can see two reasons for price discrimination in oligopoly.

Output and Welfare Effects

• Do determine effects, take two markets where 1 2 withdiscrimation and compare what happens with constraint =

1 = 2.

• Consider following procedure, assume firms have constraint2 = 1 + (so price difference is limited by arbitrage. So

symmetric FONC given is

1(1) + (1 − )1(1 1)

+2(1 + ) + (1 + − )2(2 + 2 + )

= 0

•

() = 1(∗1()) + 2(

∗1() +

• = 0 is uniform pricing. So () increasing in impliesaggregate output increases from price discrimination. The

condition that 0() 0 can be shown to be equivalent tothe condition

Figure 1:

• Comments...

Add-On Pricing (Ellison 2005 QJE)

• Two firms, ∈ {1 2}

• sell vertically differentiated goods and at prices and

• marginal cost same for both goods

• Consumers different in two dimensions is marginalutility of income

• vary in ˜ [0 1] taste match

— if buy 1 from 1 then utility is

= − − 1

— if buy 1 from 1 then utility is

= − − − 1

— if buy from 2 then replace with (1− )

Figure 2:

Timing

Lal-Matutes Benchmark: close to

• .If = clear what happens in add-on pricing game.

• Results on profits

• Results on cheapskate externality.

Figure 3:

Figure 4:

Figure 5:

Bundling (Chu, Leslie, Sorensen)

• Suppose products, draw (1 2 ) for consumer, sayzero marginal cost

• Pricing Strategies

— Mixed Bundling. (MB) Set a price for each vector (1 0 ),

(1 1 0 0 ), etc, so 2 − 1 prices

— Component Pricing (CP), price for each separate good

— Bundle Size Pricing. (BSP) Set a price for one good (of

any type), two goods of any type, 3 goods of any type.

So different prices

Example

• 2 goods, zero marginal cost

• willingness to pay 1˜ [0 ], 2˜ [0 1] and no correlation

• Pricing Strategies

— Component Pricing: 2 = 5, if = 17, then 1 = 85

— BSP, one good price equal to .9, two goods price equal to

1.1. BSP has 5.6 more profit than CP.

— Mixed Bundling.

∗ (1,0) has price of 1.13

∗ (0,1) has price of .67

∗ (1,1) has price of 1.18

Figure 1: Separation of consumers under CP and BSP

V2

V1

1.7 0.9 0.85

0.5

1.0

0

E

A

B

B

D

C

D

C

0.9

59

Figure 1 (Show) Four Points

• BSP more focused on getting consumers to purchase multiplegoods

• Negative correlation in 1 and 2 increase relative profitabilityof BSP

• Diminishing marginal utility something a big counterintuitivehere

• Complexity of BSP pricing problem

Numerical Examples

• Utility 0 − ,

— vector of valuations for profits,

— binary indicators,

— price of bundle

• drawn from multivariate distribution

• Include marginal cost (should favor CP over PB)

• show some tables from the paper

• conclusion:

Figure 6:

Figure 7:

• Next estimate the preference distribution for an example oftheater tickets

• = max ( + 0) where

— = ̄, probability

— 0, probability 1-

— ˜(Σ) (in base set = 0)

• Method of simulated moments fit

— share of consumers picking all 8 plays (1 moment)

— share choosing specific combinations of 5 plays (56 mo-

ments)

— share choosing pre-set bundle of 3 plays

— overall market shares of each play

• Impose in the model optimal price of each play and the priceof all plays.

• Use estimates to simulate a comparison of BSP and CP.

Figure 2: Distributions of profits for each pricing strategy, relative to BSP,for different values of K

UPCPPBMB

.4 .5 .6 .7 .8 .9 1 1.1 1.2

K = 2

UPCPPBMB

.4 .5 .6 .7 .8 .9 1 1.1 1.2

K = 3

UPCPPBMB

.4 .5 .6 .7 .8 .9 1 1.1 1.2

K = 4

UPCPPBMB

.4 .5 .6 .7 .8 .9 1 1.1 1.2

K = 5

Each box-plot depicts the 1st, 25th, 50th, 75th and 99th percentile of the distribution of profit relative to the

profit from MB.

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Figure 4: BSP profits relative to CP profits, as a function of correlationin consumers’ tastes across products

.8.9

11.

11.

2BS

P pr

ofit

/ CP

prof

it

−1 −.5 0 .5 1Correlation

MC=0 MC=10

This figure plots the ratio of BSP profits to CP profits as a function of correlation. In each of the two cases shown,

the taste distribution is bivariate normal with (µ1, µ2, σ1, σ2) equal to (10,10,3,1). The difference between the

two cases is that marginal costs are equal to zero for both products in one case, and equal to 10 for both products

in the other case.

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Figure 5: Distributions of profits for each pricing strategy, relative to BSP,under different assumptions on marginal costs

UP

CP

PB

MB

.4 .5 .6 .7 .8 .9 1 1.1 1.2

Zero marginal cost

UP

CP

PB

MB

.4 .5 .6 .7 .8 .9 1 1.1 1.2

Positive and unequal marginal costs

UP

CP

PB

MB

.4 .5 .6 .7 .8 .9 1 1.1 1.2

Capacity constraints

Each box-plot depicts the 1st, 25th, 50th, 75th and 99th percentile of the distribution of profit relative to the

profit from MB.

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Figure 6: Distributions of profits for each pricing strategy, relative to BSP,for different distribution families

CPMB

.5 .6 .7 .8 .9 1 1.1 1.2 1.3 1.4

Exponential

CPMB

.5 .6 .7 .8 .9 1 1.1 1.2 1.3 1.4

Logit

CPMB

.5 .6 .7 .8 .9 1 1.1 1.2 1.3 1.4

Lognormal

CPMB

.5 .6 .7 .8 .9 1 1.1 1.2 1.3 1.4

Normal

CPMB

.5 .6 .7 .8 .9 1 1.1 1.2 1.3 1.4

Normal(v)

CPMB

.5 .6 .7 .8 .9 1 1.1 1.2 1.3 1.4

Uniform

Each box-plot depicts the 1st, 25th, 50th, 75th and 99th percentile of the distribution of profit relative to the

profit from MB.

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Table 6. Summary of ticket sales

Number of Average Ticket sales Ticket salesPlay Type Performances Attendance (subscription) (non-subscription)

A Little Night Music Musical 30 294.87 7018 1828

All My Sons Drama 33 233.85 6826 891

Bat Boy Musical 30 263.93 6782 1136

Memphis Musical 30 352.40 6999 3573

My Antonia Drama 26 312.38 7002 1120

Nickel and Dimed Drama 26 343.62 6800 2134

Proof Drama 25 319.88 6885 1112

The Fourth Wall Comedy 29 313.83 7385 1716

Total 229 302.21 55,697 13,510

Three plays (Bat Boy, All My Sons, and The Fourth Wall) were performed at the Lucie Stern Theater in Palo

Alto (capacity=428). The remaining 5 were performed at the Mountain View Center for the Performing Arts

(capacity=589).

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Table 7. Sales by purchase option

Purchase option Price per play ($) Number of consumers

Non-subscription:

1 play 40.80 8,131

2 plays 40.80 1,409

3 plays 40.80 555

4 plays 40.80 224

Subscription:

3-play bundle 36.20 205

5-play pick 37.00 2,794

8-play bundle 34.55 5,139

For non-subscription purchases, the numbers of consumers in each purchase option are computed by extrapolating

the purchase patterns of the consumers whose identities we could observe to the full sample of non-subscription

purchases. See text for an explanation. The 3-play subscription bundle was for the specific 3 plays performed at

the (smaller) Lucie Stern Theater in Palo Alto, which is why the per-play price is lower than the 5-play bundle.

Consumers purchasing the 5-play subscription could combine any 5 plays of their choice.

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Table 8a. Estimated coefficients

(1) (2) (3) (4) (5) (6) (7) (8)

Covariances (Σ)(1) 1.0000(2) 0.9357 1.2200(3) 1.2125 1.4150 1.7208(4) 0.9793 1.3381 1.4859 3.2685(5) 0.8743 1.1055 1.2207 1.7920 1.4308(6) 1.1602 1.3451 1.6211 1.9090 1.3801 1.9610(7) 0.7886 1.0600 1.2199 1.7924 1.2127 1.4517 1.5086(8) 1.1133 1.2873 1.5878 2.5509 1.5597 1.9529 2.0171 3.0732

Estimate Std. error

Price sensitivity (α) 4.5937 (0.0851)Probability of theater-lover (λ) 0.0805 (0.0060)Increment for theater-lovers (θ̄) 2.0561 (0.1665)

Market size 36055 (972)

Standard errors for Σ are in Table 12a. All parameter estimates are significant at the 1% level.

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Table 9. Counterfactual pricing

UP PB TW CP BSP MB

p1 35.60 44.55 27.79 56.41 48.25

p2 30.07 46.92 43.08

p3 38.01 34.67 41.12 40.57

p4 44.08 37.72 38.68

p5 36.68 31.46 36.80 38.11

p6 38.89 35.04 36.54

p7 33.23 34.01 35.23

p8 30.81 33.30 37.90 32.89 34.29

Revenue 66.85 63.67 67.57 67.81 68.42 69.50

CS 55.03 54.37 54.02 55.88 54.75 52.62

For UP, p1 is the optimal uniform price for a single play. For PB, p8 is the optimal per-play price for the bundle

of all 8 plays. TW is the pricing scheme currently employed by the theater company: p1 is the single-play price,

p3 is the per-play price for a specific bundle of 3 plays, p5 is the per-play price for any combination of 5 plays, and

p8 is the per-play price if you buy all 8. For CP, p1-p8 are the prices for the 8 individual plays, and for BSP, p1-p8

are the per-play prices for any bundle containing the corresponding number of plays. For MB, p1-p8 are mean

per-play prices for bundles of a given size (e.g., p1 is the mean single-play price, p2 is the mean price for all 2-play

bundles, and so forth). The revenue and consumer surplus numbers are normalized by the market size—i.e., we

report revenue per consumer.

58