37.2 Dynamic Pricing and Bundling
- A monopolist sells products in a bundle to homogenize consumer heterogeneity in willingness to pay for individual products. and to sell individual products to price discriminate
37.2.1 (Prasad, Venkatesh, and Mahajan 2017) Bundling
3 static bundling strategies:
Products are sold separately (pure components PC)
Products are sold only in a bundle (pure bundling PB)
Products are sold under both (mixed bundling MB) second-degree price discrimination
Selling products under the
pure-bundling in two stages is optimal when the marginal costs are low
pure components in two stages is optimal when marginal costs are high (with lots of myopic consumers)
pure bundling then pure components is optimal when marginal costs are moderate (with lots of strategic consumers)
mixed bundling in both stages when the market is mostly strategic consumers.
Scenario:
PB-PC: bundle first then component
PC-PB: component then bundle
Assumptions:
Monopoly with either high or low marginal cost
Heterogeneous consumers (regarding product preference)
2 types in the population: myopic (\(\alpha\)) (maximize at each stage independently) and strategic \((1- \alpha)\) (maximize both stages)
Marginal cost of the bundle = sum(component costs)
Interesting
- optimal bundle price can be higher than the sum of individual products
Literature:
Bundling
Inter-temporal pricing
PC-PC
Demand for each product from myopic consumers in the first stage is \((1-P) \alpha\)
\(P - \delta\) is the price for the second stage where \(\delta\) is the discount.
Demand for each product from myopic consumer in the second stage is \(\alpha \delta\)
Demand for each product from strategic consumer is \((1-\alpha)(1 - P + \delta)\)
For the seller, the first stage profit is \(2(P-c) (1-P)\alpha\)
and the second stage profit is \(2(P - \delta - c)[\alpha \delta +(1 - \alpha)(1 - P + \delta)]\)
Hence, optimal prices are
\[ P_1 = \frac{1 + c}{2} + \frac{(2 - \alpha)(1 - c)}{2(4 - \alpha)} \]
and
\[ P_2 = \frac{1 + c}{2} - \frac{\alpha (1-c)}{2 (4 - \alpha)} \]
the total profit is
\[ \frac{2(1-c)^2}{4 - \alpha} \]
37.2.2 (Diao, Harutyunyan, and Jiang 2019) Intertemporal price, consumer fairness concern
Consumer fairness concerns (unfair due to increased price) lead to lower retail prices in both periods while the wholesale prices might be higher in the first period.
Consumer fairness concerns can decrease the first and second period retail prices.
- The intuition that retailers increase in the first period and decrease in the second period only hold under a centralized system (wholesale to consumer), not under a decentralized one.
If the demand intercept does not raise much in the second period, the cost reduction incentive usually prevails over fairness-mitigation incentive.
A larger second-period demand can cause both the producer and the store to lower their pricing.
When consumers have fairness costs, the second-period equilibrium unit sales is actually still higher than when they don’t, because fairness concerns alleviate the double-marginalization problem.
To model fairness concern heterogeneity, they model 2 types of consumers (those with fairness concern and those without), where \(\alpha\) is the fraction of customer with fairness concerns. And it should affect the retailer’s second-period demand function.
37.2.3 (Dana and Williams 2022) Intertemporal price
Oligopoly model
Pricing on capacity:
- fixed capacity, firms dynamically set price.
Stage:
Choose capacity
Compete in prices
Inventory control (i.e., sales limit restriction) can help mitigate price competition.
Contributions:
Equilibrium prices are flat or uniform over time
Intetemporal price discrimination is possible when firms adopt inventory controls (if the demand is more inelastic over time).
Firm choices: capacity and 2 prices (2 periods)
37.2.4 (Kolay and Tyagi 2022) Event bundle
Two events sold as a bundle in 2 periods.
Contribution to current literature:
Consumer expectations of future prices
Durable goods pricing
Product obsolescence
Loyalty programs
Consumer uncertainty regarding valuations
Features:
Events with different popularity
Popular event in stage 1 or 2
Consumers can be uncertain about their valuation of the second event
Uncertainty in seller’s commitment to the price of a future event
Consumers valuations are independently distributed or positively correlated
Examine when it’s optimal to sell
event independently
as a bundle
or mixed
Events are different from products in that are
perishable
temporal
If the seller can choose the order of introduction, popular events should be saved for later. (“save the best for last”). Similar to that we introduce the lower quality products before the higher quality ones. (K. S. Moorthy and Png 1992)
Model
Consumer valuation of the less popular event \(L\) is \(v_L \sim U(0,1)\) and of the more popular event \(H\) is \(v_H \sim U(0, 1 +x)\) where \(x\) represents how more popular the second one is (i.e., the proportion of consumers preferring \(H\) over \(L\)).
Consumers understand the order and importance of events before the game begins.
Demand function depends on seller’s strategies
No bundling at all: consumers only buy products \(L, H\) can only be purchased separately
Pure bundling: Either buy bundle or not
Full and partial mixed bundling: can buy either \(L, H\) separately or in a bundle.