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) to AER today (Aguirre, Cowan and Vickers, Sept 2010 AER, Monopoly Price Discrimination and Demand Curvature ) Large question relates to the first welfare theorem. With uniform 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 not allocated to consumers with highest willingness to pay. Is this empirically important?
Stole Handbook version of Holmes (1989) Two markets =1 2, duopolists =. ( )demandoffirm in market given prices (assume syjmetric) Market demand ( ) = ( ) = ( ) and market elasticity ( ) = ( ) 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 with discrimation and compare what happens with constraint = 1 = 2. Consider following procedure, assume firms have constraint 2 = 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 ( ) increasingin implies aggregate output increases from price discrimination. The condition that 0 ( ) 0canbeshowntobeequivalentto the condition
Comments... Figure 1:
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 marginal utility of income vary in [0 1] taste match
if buy 1 from 1 then utility is = 1 if buy 1 from1thenutilityis = 1 if buy from 2 then replace with (1 )
Timing Figure 2:
Lal-Matutes Benchmark: close to.if = clear what happens in add-on pricing game.
Results on profits Results on cheapskate externality.
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Bundling (Chu, Leslie, Sorensen) Suppose products, draw ( 1 2 ) for consumer, say zero 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 =1 7, 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 1.0 0.9 C V 2 D A B 0.5 E D B C 0 0.85 0.9 1.7 V 1 59
Figure 1 (Show) Four Points BSP more focused on getting consumers to purchase multiple goods Negative correlation in 1 and 2 increase relative profitability of BSP Diminishing marginal utility something a big counterintuitive here 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:
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Next estimate the preference distribution for an example of theater 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 moments)
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 price of 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 K = 2 UP CP PB MB.4.5.6.7.8.9 1 1.1 1.2 K = 3 UP CP PB MB.4.5.6.7.8.9 1 1.1 1.2 K = 4 UP CP PB MB.4.5.6.7.8.9 1 1.1 1.2 K = 5 UP CP PB MB.4.5.6.7.8.9 1 1.1 1.2 Each box-plot depicts the 1st, 25th, 50th, 75th and 99th percentile of the distribution of profit relative to the profit from MB. 60
Figure 4: BSP profits relative to CP profits, as a function of correlation in consumers tastes across products BSP profit / CP profit.8.9 1 1.1 1.2 1.5 0.5 1 Correlation 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 (µ 1, µ 2, σ 1, σ 2) 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. 62
Figure 5: Distributions of profits for each pricing strategy, relative to BSP, under different assumptions on marginal costs 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 UP CP PB MB.4.5.6.7.8.9 1 1.1 1.2 Each box-plot depicts the 1st, 25th, 50th, 75th and 99th percentile of the distribution of profit relative to the profit from MB. 63
Figure 6: Distributions of profits for each pricing strategy, relative to BSP, for different distribution families Exponential CP MB.5.6.7.8.9 1 1.1 1.2 1.3 1.4 Logit CP MB.5.6.7.8.9 1 1.1 1.2 1.3 1.4 Lognormal CP MB.5.6.7.8.9 1 1.1 1.2 1.3 1.4 Normal CP MB.5.6.7.8.9 1 1.1 1.2 1.3 1.4 Normal(v) CP MB.5.6.7.8.9 1 1.1 1.2 1.3 1.4 Uniform CP MB.5.6.7.8.9 1 1.1 1.2 1.3 1.4 Each box-plot depicts the 1st, 25th, 50th, 75th and 99th percentile of the distribution of profit relative to the profit from MB. 64
Table 6. Summary of ticket sales Number of Average Ticket sales Ticket sales Play 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). 54
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. 55
Table 8a. Estimated coefficients (1) (2) (3) (4) (5) (6) (7) (8) 56 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 ( θ) 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.
Table 9. Counterfactual pricing UP PB TW CP BSP MB p 1 35.60 44.55 27.79 56.41 48.25 p 2 30.07 46.92 43.08 p 3 38.01 34.67 41.12 40.57 p 4 44.08 37.72 38.68 p 5 36.68 31.46 36.80 38.11 p 6 38.89 35.04 36.54 p 7 33.23 34.01 35.23 p 8 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, p 1 is the optimal uniform price for a single play. For PB, p 8 is the optimal per-play price for the bundle of all 8 plays. TW is the pricing scheme currently employed by the theater company: p 1 is the single-play price, p 3 is the per-play price for a specific bundle of 3 plays, p 5 is the per-play price for any combination of 5 plays, and p 8 is the per-play price if you buy all 8. For CP, p 1-p 8 are the prices for the 8 individual plays, and for BSP, p 1-p 8 are the per-play prices for any bundle containing the corresponding number of plays. For MB, p 1-p 8 are mean per-play prices for bundles of a given size (e.g., p 1 is the mean single-play price, p 2 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