# 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.
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## 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, $A$ and $B$, and two products, $X$ and $Y$. Suppose buyer $A$ values product $X$ at $20$ units above the cost of production, and values $Y$ at $15$ units above the cost of production. Suppose buyer $B$ values $Y$ at $20$ units above the cost of production, and $X$ at $15$ units above the cost of production. Here is a simple $2 \times 2$ of the value the buyers place.

Value place on product $X$ above the cost of production (i.e., reservation price - cost of production) Value placed on product $Y$ above the cost of production Value placed on a bundle of $X$ and $Y$ above the cost of production = Sum of preceding two columns
Buyer $A$ 20 15 35
Buyer $B$ 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 $35$ units above production cost for the combination of $X$ and $Y$. Since both buyers value the combination at $35$ 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 $X$ and $Y$ individually for $20$ units above the cost of production, and offer the combination for slightly less than $35$ units above the cost of production.

## General setup

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

Further information: Two-good two-buyer pure price bundling analysis assuming everything is sold

Consider the same two-buyer situation as above with buyers $A$ and $B$ and goods $X$ and $Y$, but with more arbitrary numbers. Denote by $p_{A,X}, p_{A,Y}, p_{B,X}, p_{B,Y}$ the prices that buyers are willing to pay over and above the cost of production. Thus, for instance, $p_{A,X}$ is the maximum price that buyer $A$ is willing to pay for $X$ over and above the cost of production of $X$. 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 $A$ Seller's profit from buyer $B$ Seller's total profit
No price discrimination or price bundling Charge $\min \{ p_{A,X}, p_{B,X} \}$ above cost of production for $X$ and $\min \{ p_{A,Y}, p_{B,Y} \}$ above cost of production for $Y$ $\min \{ p_{A,X}, p_{B,X} \} + \min \{ p_{A,Y}, p_{B,Y} \}$ $\min \{ p_{A,X}, p_{B,X} \} + \min \{ p_{A,Y}, p_{B,Y} \}$ $2\left(\min \{ p_{A,X}, p_{B,X} \} + \min \{ p_{A,Y}, p_{B,Y} \}\right)$
Pure price bundling Charge $\min \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \}$ above cost of production for the bundle of $X$ and $Y$ $\min \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \}$ $\min \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \}$ $2\min \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \}$
Perfect price discrimination Charge each buyer the maximum the buyer is willing to pay for each good $p_{A,X} + p_{A,Y}$ $p_{B,X} + p_{B,Y}$ $p_{A,X} + p_{A,Y} + p_{B,X} + p_{B,Y}$

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 $m$ buyers and $n$ goods can be viewed as follows: we are given a $m \times n$ 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:

$\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix}$

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. $m$ 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. $m$ 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. $m$ 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 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 $X$: If $\max \{ p_{A,X}, p_{B,X} \} \ge 2 \min \{ p_{A,X}, p_{B,X} \}$, charge the maximum. Otherwise charge the minimum. Similarly for $Y$ $\max \{ \max \{ p_{A,X}, p_{B,X} \}, 2\min \{ p_{A,X}, p_{B,X} \} \} + \max \{ \max \{ p_{A,Y}, p_{B,Y} \}, 2\min \{ p_{A,Y}, p_{B,Y} \} \}$
Pure price bundling If $\max \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \} \ge 2 \min \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \}$, charge the max. Otherwise charge the min. $\max \{ \max \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \}, 2 \min \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \} \}$
Perfect price discrimination Charge each buyer the maximum the buyer is willing to pay for each good $p_{A,X} + p_{A,Y} + p_{B,X} + p_{B,Y}$

Below are a couple of example classes where bundling is not beneficial. For a fuller discussion, see two-good two-buyer pure price bundling analysis allowing for monopolistic pricing out.

$\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix} = \begin{pmatrix} u & 1 \\ 1 & u + v \\\end{pmatrix}, u \ge v > 2$

$\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix} = \begin{pmatrix} u + 1 & 2u - 1 \\ 1 & 2u \\\end{pmatrix}, u > 3$

### 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 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:

• $A$ buys the bundle of $X$ and $Y$
• $B$ buys only $Y$

the necessary and sufficient conditions are as follows:

$p_{A,X} + p_{A,Y} > 2p_{B,X} + p_{B,Y}, p_{A,Y} < p_{B,Y}$

An example class satisfying this pair of conditions is:

$\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix} = \begin{pmatrix} u + 2 & u + 2 \\ 1 & 2u \\\end{pmatrix}, u > 2$

An explicit example is:

$\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix} = \begin{pmatrix} 5 & 5 \\ 1 & 6 \\\end{pmatrix}$

### Introducing non-flat 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 willingness-to-pay 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 $TC(X,1)$ and $TC(X,2)$ the total cost of production of one and two units respectively of $X$ (over and above the cost of producing nothing, i.e., the cost of being idle), and similarly for $Y$. Also, now, instead of using $p_{A,X}$ 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., $TC(X + Y, i) = TC(X,i) + TC(Y, i)$.

Type of situation Seller's pricing strategy Seller's total profit
No price discrimination or price bundling For $X$: 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
$\max \{ p_{A,X}, p_{B,X} \} - TC(X,1)$: Charge $\max \{ p_{A,X}, p_{B,X} \}$ and sell to the higher-valuing buyer
$2 \min \{ p_{A,X}, p_{B,X} \} - TC(X,2)$: Charge $\min \{ p_{A,X}, p_{B,X} \}$ and sell to both buyers
Similarly for $Y$
$\max \{ 0, \max \{ p_{A,X}, p_{B,X} \} - TC(X, 1), 2\min \{ p_{A,X}, p_{B,X} \} - TC(X, 2)\} \} + \max \{ 0, \max \{ p_{A,Y}, p_{B,Y} \} - TC(Y, 1), 2\min \{ p_{A,Y}, p_{B,Y} \} - TC(Y, 2)\} \}$
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
$\max \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \} - (TC(X,1) + TC(X,2))$: Charge $\max \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \}$ and sell to the higher-valuing buyer
$2 \min \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \} - TC(X,2)$: Charge $\min \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \}$ and sell to both buyers
$\max \{ 0, \max \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \} - TC(X, 1) - TC(Y,1), 2\min \{ p_{A,X} + p_{A,Y}, p_{B,X} + p_{B,Y} \} - TC(X, 2) - TC(Y,2)\} \}$
Perfect price discrimination For $X$: 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
$\max \{ p_{A,X}, p_{B,X} \} - TC(X,1)$: Charge $\max \{ p_{A,X}, p_{B,X} \}$ and sell to the higher-valuing buyer
$p_{A,X} + p_{B,X} - TC(X,2)$: Charge $p_{A,X}$ to $A$ and $p_{B,X}$ to $B$ and sell to both buyers
Similarly for $Y$
$\max \{ 0, \max \{ p_{A,X}, p_{B,X} \} - TC(X,1), p_{A,X} + p_{B,X} - TC(X,2) \}$ + $\max \{ 0, \max \{ p_{A,Y}, p_{B,Y} \} - TC(Y,1), p_{A,Y} + p_{B,Y} - TC(Y,2) \}$