Price discrimination and bundling, economics, Microeconomics

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We consider two regions A and B. Each market has the same size (i.e. number
of consumers) but differs in the willingness to pay for one unit of the good proposed
by the firm. On market i a consumer has a unit-demand for the good and her willingness
to pay is equal to Bi with i = A,B and with BA > BB. The firm incurs no cost.

1. The monopoly has perfect and verifiable information on consumer characteristics
(location and willingness to pay) and thus is able to price discriminate. Find the
optimal prices set by the monopoly in both regions. Is this pricing policy robust to
arbitrage if there is no transport cost between both regions?

2. What is the optimal price without price discrimination?
Assume now that BA < 2BB. Moreover, the firm may propose to consumers a
service in addition to the good. The valuation for that service is equal to s in both
regions. The transport cost of the service is infinite.

3. If the monopoly decides to price discriminate, determine the price for each
product in both regions. Is that pricing policy robust to arbitrage?
The monopoly introduces tie-in sales so that each consumer is now constrained to
buy the bundle "good plus service".

4. Determine the price of each bundle if the monopoly price discriminate. Show
that the discriminatory pricing policy is robust to arbitrage if and only if s < BA-BB.
Explain this result.

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