Price bundling: Difference between revisions

Revision as of 17:37, 29 October 2016

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

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 $Y$	$min {p_{A, X}, p_{B, X}} + min {p_{A, Y}, p_{B, Y}}$	$min {p_{A, X}, p_{B, X}} + min {p_{A, Y}, p_{B, Y}}$	$2 (min {p_{A, X}, p_{B, X}} + min {p_{A, Y}, p_{B, Y}})$
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 product	$p_{A, X} + p_{A, Y}$	$p_{B, X} + p_{B, Y}$	$p_{A, X} + p_{A, Y} + p_{B, X} + p_{B, Y}$

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)	Condition (geometric) for 4-tuples plotted in $R^{4}$	Codimension and dimension
Pure price bundling offers no advantage relative to no price discrimination or price bundling	The higher-valuing buyer (among $A$ and $B$ ) for $X$ is the same as the higher-valuing buyer for $Y$ , or one or both of $X$ and $Y$ is equally valued by both buyers.	$p_{A, X} - p_{B, X}$ and $p_{A, Y} - p_{B, Y}$ have the same sign, or one or both of them equals zero.	Two diametrically opposite "quarter-spaces" in $R^{4}$ . Explicitly, consider the hyperplanes defined by $p_{A, X} = p_{B, X}$ and $p_{A, Y} = p_{B, Y}$ . Together, they divide the space into four quarters. Our region of interest is one diametrically opposite pair of quarters, along with the bounding hyperplanes. We can further intersect this with the nonnegative orthant since we are interested only in positive willingness to pay values.	Codimension 0, dimension 4.
Pure price bundling offers all the advantage of perfect price discrimination	The bundle is equally valued by both buyers.	$p_{A, X} + p_{A, Y} = p_{B, X} + p_{B, Y}$	Hyperplane $p_{A, X} + p_{A, Y} = p_{B, X} + p_{B, Y}$ . We can restrict to the intersection of the hyperplane with the nonnegative orthant.	Codimension 1, dimension 3.
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.	$p_{A, X} = p_{B, X}$ and $p_{A, Y} = p_{B, Y}$	Intersection of the hyperplanes $p_{A, X} = p_{B, X}$ and $p_{A, Y} = p_{B, Y}$ . Note that this is obtained by intersecting the two subsets discussed in the two preceding rows. The subsets overlap less than what you'd expect by random chance. Specifically, the $p_{A, X} + p_{A, Y} = p_{B, X} + p_{B, Y}$ hyperplane "avoids" the interior of the two quarters and just intersects the intersection of the bounding hyperplanes. We can further restrict to the nonnegative orthant in $R^{4}$ .	Codimension 2, dimension 2.

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 $m$ 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:

$(\begin{matrix} p_{A, X} & p_{A, Y} \\ p_{B, X} & p_{B, Y} \end{matrix})$

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 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.

For simplicity, we will drop the above the cost of production qualifier when discussing prices from this section onward. In other words, the prices we describe in the rest of this section are not actual prices but rather prices over and above the cost of production. We can think of this in terms of subtracting the cost of production from all prices, so that we are effectively dealing only with the profit component of revenue.

Type of situation	Seller's pricing strategy	Seller's total profit
No price discrimination or price bundling	For $X$ : If $max {p_{A, X}, p_{B, X}} \geq 2 min {p_{A, X}, p_{B, X}}$ , 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 $max {p_{A, X} + p_{A, Y}, p_{B, X} + p_{B, Y}} \geq 2 min {p_{A, X} + p_{A, Y}, p_{B, X} + p_{B, Y}}$ , 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 product	$p_{A, X} + p_{A, Y} + p_{B, X} + p_{B, Y}$

Full case analysis

We will take some help from geometry for the case analysis. We are interesting in looking at how the various cases partition the region $R^{4}$ with coordinates $p_{A, X}, p_{B, X}, p_{A, Y}, p_{B, Y}$ .

For a single good, e.g. $X$ , we have two bounding lines in a two-dimensional plane for $p_{A, X}$ and $p_{B, X}$ : $p_{A, X} = 2 p_{B, X}$ and $p_{B, X} = 2 p_{A, X}$ . These lines dividing the nonnegative quadrant into three regions:

$p_{A, X} > 2 p_{B, X}$ : Sell $X$ only to $A$
$p_{B, X} > 2 p_{A, X}$ : Sell $X$ only to $B$
Middle region, where each is smaller than twice the other: Sell to both

Using the symmetry between $A$ and $B$ , we see that these three cases reduce to two types of cases. Specifically, the first two cases are the same type, with the roles of $A$ and $B$ interchanged.

We similarly can identify three regions for $Y$ , and for $X + Y$ . Note that for $X + Y$ , the regions we are dealing with are no longer along coordinate axes, but geometrically the situation is similar.

In total, we therefore have 3 X 3 X 3 = 27 possible regimes. However, only 17 of these occur in practice. The list of regimes that occur in practice is described below, grouped by type. The main constraining factor is that the behavior for the bundle must be a convex combination of the behaviors for each good, so it cannot be sold to just one person if neither of the goods are sold to just that one person.

For each of the 17 cases (6 types up to symmetry), a different linear inequality (linear inequality type up to symmetry) determines whether bundling is better. In some cases, the inequality is always true, always false, or always degenerate. In others, there is genuine uncertainty so the region is further divided into two subregions by the hyperplane for the corresponding linear equation.

Case description	Number of cases of this type (equals 1, 2, or 4, because of the nature of symmetry)	Explanation for number of cases. We can think of this in terms of specifying the symmetry in terms of subgroups and quotient groups of the Klein four-group	Conclusion on bundling
Sell to both people for each of $X$ , $Y$ , and the combination	1	There are no degrees of freedom	Bundling is at least as good as not bundling. This is the simple case discussed earlier on the page
Sell one of $X, Y$ to only one person, but sell the other one and the bundle to both	4	We can choose both the good that is sold to one person, and the person who gets just that one good	Indeterminate, i.e., the linear inequality divides it into two nonempty regions
Sell each good to a different single person, and sell the bundle to both	2	We need a matching between people and goods. Two matchings are possible: $A \leftrightarrow X, B \leftrightarrow Y$ , and $A \leftrightarrow Y, B \leftrightarrow X$ (Note: This is the only case where the symmetry is arising from the diagonal subgroup)	Indeterminate, i.e., the linear inequality divides it into two nonempty regions
Sell to the same single person for each of $X$ , $Y$ , and the bundle	2	We can choose the person to be either $A$ or $B$	The inequality degenerates, i.e., whether or not to bundle does not matter
Sell one of $X, Y$ and the bundle to only one person, but sell the other one to both	4	We can choose both the good that is sold to one person, and the person who gets just that one good	Bundling is inferior to not bundling, with indifference achieved at the region boundary. To see this, suppose $A$ buys $X$ and the bundle. In that case, the bundle gives $p_{A, X} + p_{A, Y}$ . But without bundling, we'd get $p_{A, X}$ from $X$ and something that is greater than $p_{A, Y}$ from $Y$ , since our strategy for $Y$ is to sell to both rather than just to one person.
Sell each good to a different single person, and sell the bundle to one of them	4	We can choose the person who gets the combination, and the single good that person gets.	Bundling is strictly worse than not bundling. Suppose $A$ wins for $X$ and overall, and $B$ wins for $Y$ but loses overall. Then, with bundling, we make $p_{A, X} + p_{A, Y}$ . Without bundling, we'd make $p_{A, X} + p_{B, Y}$ , and we know that $p_{B, Y} \geq 2 p_{A, Y}$ , so we are strictly better off not bundling.
Total (6 types)	17	--

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 interest in both goods.

Example class where mixed bundling > pure bundling > selling separately

$(\begin{matrix} p_{A, X} & p_{A, Y} \\ p_{B, X} & p_{B, Y} \end{matrix}) = (\begin{matrix} 20 & 1 \\ 12 & 12 \end{matrix})$

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

With a no-bundling strategy, $X$ should be sold to both $A$ and $B$ for a profit of 24 (12 times 2) units, and $Y$ should be sold only to $B$ 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 $X$ in isolation for 20 units.
Sell the bundle for 24 units.

$A$ will buy $X$ , and $B$ will buy the bundle, for a total profit of 44 units.

The more general example class for this is:

$(\begin{matrix} p_{A, X} & p_{A, Y} \\ p_{B, X} & p_{B, Y} \end{matrix}) = (\begin{matrix} 2 u & 1 \\ u + 2 & u + 2 \end{matrix})$

where $u > 4$ .

With a no-bundling strategy, $X$ should be sold to both $A$ and $B$ for a profit of $2 (u + 2)$ units, and $Y$ should be sold only to $B$ for a profit of $2 (u + 2)$ units. The total profit is $3 (u + 2) = 3 u + 6$ units.
With a bundling strategy, it makes sense to sell to both for $2 u + 1$ units, for a profit of $2 (2 u + 1) = 4 u + 2$ units.

Since $u > 4$ , we obtain that $4 u + 2 > 3 u + 6$ , so that pure bundling is superior to no bundling.

On the other hand, the following mixed bundling strategy is superior:

Sell $X$ in isolation for $2 u$ units.
Sell the bundle for $2 (u + 2)$ units.

$A$ will buy $X$ , and $B$ will buy the bundle, for a total profit of $4 u + 4$ units. This strategy beats the profit of $4 u + 2$ from pure bundling by 2 units.

Example where mixed bundling > selling separately > pure bundling

Consider the example:

$(\begin{matrix} p_{A, X} & p_{A, Y} \\ p_{B, X} & p_{B, Y} \end{matrix}) = (\begin{matrix} 20 & 1 \\ 11 & 21 \end{matrix})$

With a no-bundling strategy, $X$ should be sold to both $A$ and $B$ for a profit of 22 (11 times 2) units, and $Y$ should be sold only to $B$ for a profit of 21 units. The total profit is 43 units.
With bundling, the bundle should be sold for 21 units to both, for a profit of 42 units.

The following mixed bundling strategy is superior:

Charge 20 units for X
Charge 32 units for the combination of X and Y

This yields a profit of 52 units.

$(\begin{matrix} p_{A, X} & p_{A, Y} \\ p_{B, X} & p_{B, Y} \end{matrix}) = (\begin{matrix} 2 u & 1 \\ u + 1 & 2 u + 1 \end{matrix})$

where $u > 1$ .

With a no-bundling strategy, $X$ should be sold to both $A$ and $B$ for a profit of $2 (u + 1)$ units, and $Y$ should be sold only to $B$ for a profit of $2 u + 1$ units. The total profit is $4 u + 3$ units.
With bundling, the bundle should be sold for $2 u + 1$ units, for a profit of $4 u + 2$ units.

We therefore see that no bundling is better than pure bundling.

The following mixed bundling strategy is superior:

Charge $2 u$ units for X.
Charge $3 u + 2$ units for the bundle of X and Y.

This gives a total profit of $5 u + 2$ units. Since $u > 1$ , this exceeds $4 u + 3$ , the profit from selling separately.

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 $T C (X, 1)$ and $T C (X, 2)$ 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., $T C (X + Y, i) = T C (X, i) + T C (Y, i)$ .

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 $max {p_{A, X}, p_{B, X}} - T C (X, 1)$ : Charge $max {p_{A, X}, p_{B, X}}$ and sell to the higher-valuing buyer $2 min {p_{A, X}, p_{B, X}} - T C (X, 2)$ : Charge $min {p_{A, X}, p_{B, X}}$ and sell to both buyers Similarly for $Y$	$max {0, max {p_{A, X}, p_{B, X}} - T C (X, 1), 2 min {p_{A, X}, p_{B, X}} - T C (X, 2)}} + max {0, max {p_{A, Y}, p_{B, Y}} - T C (Y, 1), 2 min {p_{A, Y}, p_{B, Y}} - T C (Y, 2)}}$
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 $max {p_{A, X} + p_{A, Y}, p_{B, X} + p_{B, Y}} - (T C (X, 1) + T C (X, 2))$ : Charge $max {p_{A, X} + p_{A, Y}, p_{B, X} + p_{B, Y}}$ and sell to the higher-valuing buyer $2 min {p_{A, X} + p_{A, Y}, p_{B, X} + p_{B, Y}} - T C (X, 2)$ : Charge $min {p_{A, X} + p_{A, Y}, p_{B, X} + p_{B, Y}}$ and sell to both buyers	$max {0, max {p_{A, X} + p_{A, Y}, p_{B, X} + p_{B, Y}} - T C (X, 1) - T C (Y, 1), 2 min {p_{A, X} + p_{A, Y}, p_{B, X} + p_{B, Y}} - T C (X, 2) - T C (Y, 2)}}$
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 $max {p_{A, X}, p_{B, X}} - T C (X, 1)$ : Charge $max {p_{A, X}, p_{B, X}}$ and sell to the higher-valuing buyer $p_{A, X} + p_{B, X} - T C (X, 2)$ : Charge $p_{A, X}$ to $A$ and $p_{B, X}$ to $B$ and sell to both buyers Similarly for $Y$	$max {0, max {p_{A, X}, p_{B, X}} - T C (X, 1), p_{A, X} + p_{B, X} - T C (X, 2)}$ + $max {0, max {p_{A, Y}, p_{B, Y}} - T C (Y, 1), p_{A, Y} + p_{B, Y} - T C (Y, 2)}$

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:

@@ Line 102: / Line 102: @@
 | Perfect price discrimination || Charge each buyer the maximum the buyer is willing to pay for each product || <math>p_{A,X} + p_{A,Y} + p_{B,X} + p_{B,Y}</math>
 |}
-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 at least one 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.
-* The two buyers differ significantly in their total value to the seller (this combined with the preceding points forces bundling to be inefficient).
-==== Example class 1 ====
-Here is an example:
-<math>\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix} = \begin{pmatrix} 6 & 1 \\ 1 & 3 \\\end{pmatrix}</math>
-In this case, the bundles are worth 7 and 4 units respectively, so that with bundling, the best strategy is to price at 4 units and make 8 units. On the other hand, selling separately, you can make 6 units from X and 3 units from Y, for a total of 9 units.
-More generally, the following example class works:
-<math>\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix} = \begin{pmatrix} u + v & 1 \\ 1 & u \\\end{pmatrix}</math>
-where <math>u > 2</math> and <math>v > 2</math>.
-In this general, case, the bundles are worth <math>x + v</math> and <math>x</math> units respectively. With bundling, you'll make <math>\max \{ 2(u + 1), u + v + 1\}</math>. Without bundling, you'd make a total of <math>u + v</math> from <math>X</math> and <math>v</math> from <math>Y</math>, giving a total of <math>2u + v</math>.
-Since <math>v > 2</math>, we get:
-<math>2u + v > 2(u + 1)</math>
-Since <math>u > 2</math>, we get:
-<math>u + (u + v) > 2 + (u + v) > u + v + 1</math>
-Combining, we get that:
-Profit from selling separately = <math>2u + v > \max \{ 2(u + 1), u + v + 1\}</math> = Profit from bundling
-====Example class 2 ====
-Here is an example:
-<math>\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix} = \begin{pmatrix} 16 & 1 \\ 4 & 4 \\\end{pmatrix}</math>
-In this case, the no-bundling strategy would sell <math>X</math> to <math>A</math> at 16 units above cost, and <math>Y</math> to <math>B</math> 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:
-<math>\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix} = \begin{pmatrix} x^2 & 1 \\ x & x \\\end{pmatrix}</math>
-then it can be verified that the above example works as long as <math>x > 3</math>. Note that this is stricter than the <math>x > 2</math> condition necessary for the markets for the individual goods to have only one buyer.
 ==== Full case analysis ====