Question 1 Suppose that a monopolist can sell units of a good to four consumers who demand at most one unit of the good. Producing an extra unit of the good costs $5 to the monopolist, and consumers A, B, C and D have a private valuation of $20, $16, $12 and $8, respectively. Suppose also that there is a fixed cost of $25 that is borne by the monopolist regardless of the volume of sales made (such fixed cost can be avoided if the monopolist shuts-down and chooses to produce nothing; think for example of heating or refrigeration of the building in which production takes place). Consumers A and D share the same observable features (e.g., age) and similarly for B and C:
Identify the optimal pricing strategies under Uniform Pricing, First Degree Price Discriminations, and Third Degree price Discriminations (Definitions below; use attached spreadsheet)
We consider the following ITEM PRICING problem. Consider a finite set of items owned by a single seller, who wishes to sell them to multiple prospective buyers. The seller can price each item individually, and the price of a set of items is simply the sum of the prices of the individual items in the set. The buyers arrive in a sequence, and each buyer has her own valuation function v(S), defined on every subset S of items. We assume that the valuation functions to be subadditive, which means that v(S) + v(T) ! v(S " T) for any pair of subsets S, T of items. For some results, we shall assume the valuations to be XOS, that is, they can be expressed as the maximum of several additive functions. If a buyer buys a subset S of items S, her utility is defined as her valuation v(S) of the set minus the price of the set S. Moreover, we assume the limited supply setting where a buyer can buy only yet unsold items. We assume that every buyer is selfish and rational, and thus always buy a subset of items that maximizes her utility. The strategy used by the seller in choosing the prices of the items is allowed to be randomized, and is referred to as a pricing strategy. The revenue obtained by the seller is the sum of the amounts paid by each buyer, and our goal is to design pricing strategies that maximize the expected revenue of the seller. This problem is made difficult by the fact that the seller has no knowledge of the valuation functions of the buyers, apart from the promise that they are subadditive. This is, for instance, in contrast to the Bayesian mechanism designs for revenue maximization, which assume that the valuation functions come from a known prior distribution. Optimal mechanisms, such as that given by Myerson , exist under this knowledge.
Pricing Strategies: A uniform pricing strategy is one where at any point of time, all unsold items are assigned the same price. The seller may set prices on the items initially ...