Price bundling
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
Contents
 1 Definition
 2 Motivation behind price bundling: exploit different valuations for different goods by different buyers
 3 General setup
 3.1 Simple case with two buyers, two goods, both goods being sold
 3.2 Generalization to multiple buyers and multiple goods
 3.3 Introducing more complexity: introducing monopolistic pricing of buyers out of the market
 3.4 Introducing more complexity: adding mixed bundling
 3.5 Introducing nonflat marginal cost of production
 4 External links
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
Further information: Twogood twobuyer pure price bundling analysis assuming everything is sold
Consider the same twobuyer 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 good 
Our overall conclusion is:
 From the seller's perspective, pure price bundling is at least as good as no bundling and at most as good as perfect price discrimination. Equality occurs under some conditions.
 Total economic surplus is unaffected by bundling as long as everything gets sold. The gains to surplus for the seller are cancelled by corresponding losses in surplus to buyers, and the losses are shared equally between both buyers.
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 and the buyers do not differ too significantly from each other.
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 good  The sum of all the entries of the matrix. 
Among various partitionbased 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
Further information: Twogood twobuyer pure price bundling analysis allowing for monopolistic pricing out
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 twogood twobuyer 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 willingnesstopay 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 good 
Below are a couple of example classes where bundling is not beneficial. For a fuller discussion, see twogood twobuyer pure price bundling analysis allowing for monopolistic pricing out.
Introducing more complexity: adding mixed bundling
Further information: Twogood twobuyer mixed price bunding analysis allowing for monopolistic pricing out
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 twogood twobuyer case, it offers an advantage over both pure bundling and no bundling when:
 One of the buyers has interest skewed heavily toward one good.
 The other buyer has similar levels of interest in both goods.
Explicitly, for the following mixed bundling strategy to be optimal and strictly superior to both pure bundling and no bundling:
 buys the bundle of and
 buys only
the necessary and sufficient conditions are as follows:
An example class satisfying this pair of conditions is:
An explicit example is:
Introducing nonflat marginal cost of production
In the discussion so far, we have assumed that the cost of production for two units is twice the cost of production for one unit, so that we can subtract this cost of production and only look at the willingnesstopay over and above the cost of production. However, we can imagine situations where the cost of production for two units is less than twice that for one unit (economies of scale) or where it is more (diseconomies of scale).
Denote by and the total cost of production of one and two units respectively of (over and above the cost of producing nothing, i.e., the cost of being idle), and similarly for . Also, now, instead of using to denote the willingness to pay over and above the cost of production, just use it to denote total' willingness to pay.
We assume that there are no complementarities in production of the two goods, i.e., .
Type of situation  Seller's pricing strategy  Seller's total profit 

No price discrimination or price bundling  For : Consider the three quantities below. Depending on which is maximum, follow the instruction in that line, and reap total profit equal to the value: 0: Do not sell : Charge and sell to the highervaluing buyer : Charge and sell to both buyers Similarly for 

Pure price bundling  Consider the three quantities below. Depending on which is maximum, follow the instruction in that line, and reap total profit equal to the value: 0: Do not sell : Charge and sell to the highervaluing buyer : Charge and sell to both buyers 

Perfect price discrimination  For : Consider the three quantities below. Depending on which is maximum, follow the instruction in that line, and reap total profit equal to the value: 0: Do not sell : Charge and sell to the highervaluing buyer : Charge to and to and sell to both buyers Similarly for 
+ 
External links
Simulations
 Bundling demo by Buck Shlegeris covers the twogood, twobuyer case and compares bundling against nobundling.
Weblog entries/articles
 Search for bundling on the Marginal Revolution weblog
 Search for bundling on the Library of Economics
Particular weblog entries/articles of interest:
 Why can't you choose your cable channels?  Alex Tabarrok, Marginal Revolution, April 2004
 Sorting out bundling and antitrust law from a seat at the Saturday double feature  Hal Varian, New York Times, July 26, 2001
 Bundling  Arnold Kling, Econlog, 26 Match 2004
 Bundling ii  Arnold Kling, Econlog, April 5, 2004