Price bundling

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This article describes a pricing strategy used by sellers, typically in markets that suffer from imperfect competition, significant transaction costs or imperfect information.
View other pricing strategies

Definition

Price bundling is a strategy whereby a seller bundles together many different goods/items being sold and offers the entire bundle at a single price.

There are two forms of price bundling -- pure bundling, where the seller does not offer buyers the option of buying the items separately, and mixed bundling, where the seller offers the items separately at higher individual prices. Mixed bundling is usually preferable to pure bundling, both because there are fewer legal regulations forbidding it, and because the reference price effect makes it appear even more attractive to buyers.

Motivation behind price bundling: exploit different valuations for different goods by different buyers

Toy example

Suppose there are two buyers, and , and two products, and . Suppose buyer values product at units above the cost of production, and values at units above the cost of production. Suppose buyer values at units above the cost of production, and at units above the cost of production. Here is a simple of the value the buyers place.

Value place on product above the cost of production (i.e., reservation price - cost of production) Value placed on product above the cost of production Value placed on a bundle of and above the cost of production = Sum of preceding two columns
Buyer 20 15 35
Buyer 15 20 35

The ideal thing for the seller would be to practice price discrimination: charge each buyer the maximum that buyer is willing to pay. However, this may be forbidden by law or otherwise difficult to implement.

Instead, the seller can pursue the following bundling strategy: charge slightly under units above production cost for the combination of and . Since both buyers value the combination at units above the cost of production, this deal appeals to both buyers. This allows the seller the obtain the entire social surplus as producer surplus. (It isn't true in general that bundling allows the seller to capture the entire social surplus -- that is a special feature of this situation because both buyers have similar reservation prices for the total bundle. However, bundling does allow the seller to capture more of the social surplus in many situations).

The seller can even make this a mixed bundling strategy: offer both and individually for units above the cost of production, and offer the combination for slightly less than units above the cost of production.

General setup

Simple case with two buyers, two goods, both goods being sold

Consider the same two-buyer situation as above with buyers and and goods and , but with more arbitrary numbers. Denote by the prices that buyers are willing to pay over and above the cost of production. Thus, for instance, is the maximum price that buyer is willing to pay for over and above the cost of production of . We will assume that all four numbers are positive.

Here are the three basic cases where the seller prices so as to sell both goods to both buyers:

Type of situation Seller's pricing strategy Seller's profit from buyer Seller's profit from buyer Seller's total profit
No price discrimination or price bundling Charge above cost of production for and above cost of production for
Pure price bundling Charge above cost of production for the bundle of and
Perfect price discrimination Charge each buyer the maximum the buyer is willing to pay for each product

With some basic algebra, we see that the pure price bundling strategy generates a profit intermediate between the strategy with no price discrimination or price bundling, and the strategy with perfect price discrimination. The following are the edge cases:

Conclusion Condition (verbal) Condition (algebraic)
Pure price bundling offers no advantage relative to no price discrimination or price bundling The higher-valuing buyer (among and ) for is the same as the higher-valuing buyer for , or one or both of and is equally valued by both buyers. and have the same sign, or one or both of them equals zero.
Pure price bundling offers all the advantage of perfect price discrimination The bundle is equally valued by both buyers.
All three cases coincide, i.e., no price discrimination, pure price bundling, and perfect price discrimination yield the same profit. The two buyers have identical willingness to pay. and

In other words, pure price bundling offers an advantage over no price bundling only if the buyers have opposite preference orders for the two products.

Generalization to multiple buyers and multiple goods

The price bundling problem for buyers and goods can be viewed as follows: we are given a matrix that describes the buyers' willingness to pay over and above the cost of production. Our goal is to find a way of partitioning the goods into bundles so as to maximize profits. Assume that we want to sell all inventory, i.e., we do not want to price any buyer out, and that all the entries of the matrix are positive. These are reasonable assumptions in cases that the marginal cost of production is close to zero.

The matrix for the previous case would be:

Here are the same three situations as in the previous subsection, but described in this general case:

Type of situation Seller's pricing strategy Seller's total revenue
No price discrimination or price bundling For each good, charge the minimum among the entries for the corresponding column in the matrix. times the sum, over all columns, of the minimum value in that column.
Pure price bundling with a single bundle for all goods For the whole bundle, charge the minimum among the row sums of the matrix. times the minimum of the row sums.
Price bundling based on a partitioning of the goods into bundles (the previous two rows are special cases) For each bundle, charge the minimum among the row sums for the submatrix obtained by restricting columns to that bundle. times the sum of the prices chosen for each bundle.
Perfect price discrimination Charge each buyer the maximum the buyer is willing to pay for each product The sum of all the entries of the matrix.

Among various partition-based price bundling strategies, the no price bundling strategy performs worst and the single bundle performs best. The single bundle may still fall short of perfect price discrimination.

Introducing more complexity: introducing monopolistic pricing of buyers out of the market

The preceding analyses would suggest that price bundling always makes sense for a monopolist. However, this is driven by the assumption that we are only looking at situations where the goal is to sell every good to every buyer. In cases where buyers' willingness to pay differs significantly, price bundling may be inferior to monopolistic pricing strategies where some buyers are priced out of the market. Relatedly, the assumption that every buyer's willingness to pay exceeds the cost of production may also be unrealistic.

Here are the two-good two-buyer cases again, but relaxing the assumption that everything must be sold. We assume a flat marginal cost curve for production, so that the profit made from selling a good is not dependent on how many other units were sold. We also continue to assume that all the willingness-to-pay values are above the cost of production.

Type of situation Seller's pricing strategy Seller's total profit
No price discrimination or price bundling For : If , charge the maximum. Otherwise charge the minimum. Similarly for
Pure price bundling If , charge the max. Otherwise charge the min.
Perfect price discrimination Charge each buyer the maximum the buyer is willing to pay for each product

Perfect price discrimination continues to be the best. However, pure price bundling is no longer necessarily a superior strategy to no price bundling from the seller's perspective. Qualitatively, some observations from earlier continue to hold, though the algebra gets complicated. In particular:

  • In case both goods are being sold, price bundling is advantageous. We need to be in a regime where buyers are being priced out in order to make bundling bad.
  • Price bundling makes a difference at all (for better or worse) only if the buyers have opposite preference orderings for the two goods.
  • The more similar the buyers' total willingness to pay, the closer price bundling is to perfect price discrimination.

Therefore, to come up with an example where price bundling makes the situation worse, we need these three features:

  • For each good, it makes sense to sell at the maximum of the buyers' price to price out the other buyer (so the max, rather than the min, is operational).
  • The higher-valuing buyer is different for the two goods (so that bundling actually has some effect).
  • The two buyers differ significantly in their total value to the seller (this combined with the preceding points forces bundling to be inefficient).

Here is an example:

In this case, the no-bundling strategy would sell to at 16 units above cost, and to at 4 units above cost, for a total of 20 units of profit. On the other hand, the maximum that can be made from a price bundling strategy is 17 units.

Arithmetically, if we set:

then it can be verified that the above example works as long as . Note that this is stricter than the condition necessary for the markets for the individual goods to have only one buyer.

Introducing more complexity: adding mixed bundling

A mixed bundling strategy is one where the bundle is sold, and one or more of the goods in the bundle is also sold separately. In the two-good two-buyer case, it makes sense when:

  • One of the buyers has interest skewed heavily toward one good.
  • The other buyer has similar levels of interest in both goods.

The mixed bundling targets the single good at the buyer with skewed interest and the bundle at the buyer with similar level of interest in both goods.

Here is an example:

In this case, we can calculate profits based on the formulas in the previous subsection:

  • With a no-bundling strategy, should be sold to both and for a profit of 24 (12 times 2) units, and should be sold only to for a profit of 12 units. The total profit is 36 units.
  • With a bundling strategy, it makes sense to sell a bundle for 21 units to both buyers, for a profit of 42 units.

However, the following mixed bundling strategy is superior:

  • Sell in isolation for 20 units.
  • Sell the bundle for 24 units.

will buy , and will buy the bundle, for a total profit of 44 units.

External links

Simulations

  • Bundling demo by Buck Shlegeris covers the two-good, two-buyer case and compares bundling against no-bundling.

Weblog entries/articles

Particular weblog entries/articles of interest: