Two-good two-buyer pure price bundling analysis allowing for monopolistic pricing out

This page describes a simple situation of price bundling that is a logical next step in the analysis of price bundling after two-good two-buyer pure price bundling analysis assuming everything is sold. We compare three situations:

• No bundling: The seller sells each good in a separate market, at a single price to both buyers. It is possible that one of the buyers is unable to afford the good at that price.
• Pure price bundling: The seller practices pure price bundling, selling both goods at a bundle at a single price to both buyers. It is possible that one of the buyers is unable to afford the bundle at that price.
• Perfect price discrimination: The seller practices perfect price discrimination, selling each good to each buyer at a price that could depend on that good and that buyer.

For both the individual goods in the no bundling scenario and for the price bundle in the pure price bundling scenario, we additionally have the option of either pricing so that both buyers can buy the good, or pricing high enough that only one buyer can buy the good. This additional feature distinguishes it from the previous, simpler analysis of two-good two-buyer pure price bundling analysis assuming everything is sold.

Complications we will assume away

• The possibility of using mixed bundling
• Nonconstant marginal costs of production
• Complementarity or substitution in the production costs of the two goods
• Complementarity or substitution in the consumption of the two goods

Subtracting the cost of production

We operate under the assumption of constant marginal cost of production, i.e., the cost of producing two units is twice the cost of producing one unit.

To simplify matters without loss of generality, we will subtract the unit cost of production from the prices. This is similar to assuming zero marginal cost of production, but does not actually assume that (i.e., there is no loss of generality).

With this convention, we see that the profit the seller gets from a particular buyer equals the price of the sale, and the total profit equals total revenue.

Nomenclature and invariance

We will call our two buyers $A$ and $B$ and our two goods $X$ and $Y$.

We will denote by $p_{A,X}$ the maximum price that buyer $A$ is willing to pay for $X$ over and above the cost of production. Similarly, define $p_{B,X}$, $p_{A,Y}$, and $p_{B,Y}$. When referencing these values verbally, we will call them the "willingness to pay".

The four numbers can be conveniently put in a matrix:

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

We will assume that all four numbers are positive.

Unlike the simple case where we assume everything is sold, we care about the actual values $p_{A,X}, p_{B,X}$ and not just the difference $p_{A,X} - p_{B,X}$. Therefore, if we subtract a constant from all entries in a column the qualitative behavior could change.

All our results depend qualitatively only on the numbers up to scaling. In other words, if all four entries of the matrix are multiplied by a positive number, the qualitative conclusions do not change.

Conclusions

The three strategies

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	$p_{A,X} + p_{A,Y} + p_{B,X} + p_{B,Y}$

Broad conclusions

• If there is no pricing out of any buyer (i.e., every buyer buys everything) then the analysis reduces to that for two-good two-buyer pure price bundling analysis assuming everything is sold. In particular, under this condition, pure price bundling is always at least as good for the seller as no bundling, and is strictly better if the rank order of buyers' willingness to pay is opposite for the two goods.
• If the relative order of the two buyers is the same for the two goods (i.e., the buyer with lower willingness to pay for $X$ is the same as the buyer with lower willingness to pay for $Y$) then pure price bundling is at most as good for the seller as no bundling.
• If, for the pure price bundle, it makes more sense to sell to one buyer than to both (i.e., it makes sense to price out) then pure price bundling is at most as good for the seller as no bundling.
• There are a number of other cases where pure price bundling is better, as well as where it is worse, than no bundling.
• Bundling can improve social surplus in cases where the no bundling scenario would price buyers out.

Linear programming interpretation

We now discuss in a little more detail the interpretation of the above in terms of linear algebra. Our scenario is as follows. To describe a given set of prices, we need to look in four-dimensional space $\R^4$ with coordinates for $p_{A,X}, p_{B,X}, p_{A,Y}, p_{B,Y}$. Every point in the positive orthant of this space (all coordinates positive) corresponds to a possible set of prices that the seller faces.

We will proceed as follows:

• Discuss the four cases that arise for one good at a time
• Discuss how this leads to 64 possible interior cases for $X$, $Y$, and the bundle, and the 36 of these 64 cases (in 10 types) that are actually possible
• Group the cases into types (math jargon: equivalence classes) and discuss the qualitative conclusions for each type

One good at a time

Projecting down to one good $X$, we obtain a plane with coordinates $p_{A,X}$ and $p_{B,X}$, and we are interested in the positive quadrant of this plane.

If the seller were selling just $X$, there are four regions of interest, with three bounding lines.

$p_{A,X} = 2p_{B,X}, p_{A,X} = p_{B,X}, p_{B,X} = 2p_{A,X}$

The four interior cases are as follows:

Region	Price chosen by seller	Who buys	Seller revenue
$p_{B,X} > 2p_{A,X}$	$p_{B,X}$	Only $B$	$p_{B,X}$
$2p_{A,X} > p_{B,X} > p_{A,X}$	$p_{A,X}$	$A$ and $B$	$2p_{A,X}$
$2p_{B,X} > p_{A,X} > p_{B,X}$	$p_{B,X}$	$A$ and $B$	$2p_{B,X}$
$p_{A,X} > 2p_{B,X}$	$p_{A,X}$	Only $A$	$p_{A,X}$

The two goods and the bundle together

Now let's look at everything together. We have four interior regions for $X$, four for $Y$, and four for the bundle. Combining, we could get as many as $4 \times 4 \times 4 = 64$ regions. However, in practice, we won't see all these regions appear, because the region allowed for the bundle has to be in the convex hull of the regions for $X$ and $Y$. It turns out that only 36 of the 64 possible interior cases can actually occur, and that they fall in 10 types.

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 both $X$, $Y$, and the bundle to both buyers. The buyers have opposite relative ordering for the goods.	4	We need a matching between people and goods based on who values what less. Two matchings are possible: $A \leftrightarrow X, B \leftrightarrow Y$, and $A \leftrightarrow Y, B \leftrightarrow X$. We have an additional symmetry based on which of $A$ and $B$ determines the price for the bundle	Bundling is strictly better than not bundling
Sell both $X$, $Y$, and the bundle to both buyers. Both buyers have the same relative ordering for the goods.	2	We can choose which of $A$ and $B$ values both less.	Bundling is degenerate, i.e., it does not affect the outcome
Sell one of $X,Y$ to only one person, but sell the other one and the bundle to both. The buyers have the same relative ordering for the goods.	4	We can choose both the good that is sold to one person, and the person who gets just that one good	Bundling is strictly worse than not bundling
Sell one of $X,Y$ to only one person, but sell the other one and the bundle to both. The buyers have opposite relative ordering for the goods. The buyer with the lower price for the bundle is the one who received the single good	4	We can choose both the good that is sold to one person, and the person who gets just that one good	Bundling is strictly better than not bundling (explanation in next section)
Sell one of $X,Y$ to only one person, but sell the other one and the bundle to both. The buyers have opposite relative ordering for the goods. The buyer with the lower price for the bundle is the one who did not receive the single good	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 nonempty regions (explanation in #First indeterminate case)
Sell each good to a different single person, and sell the bundle to both	4	We need a matching between people and goods. Two matchings are possible: $A \leftrightarrow X, B \leftrightarrow Y$, and $A \leftrightarrow Y, B \leftrightarrow X$. We also need to choose which buyer has the lower price for the bundle.	Indeterminate, i.e., the linear inequality divides it into nonempty regions (explanation in #Second indeterminate case)
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. Both buyers have the same relative ordering for the two goods.	4	We can choose both the good that is sold to one person, and the person who gets just that one good	Bundling is worse than 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 one of $X,Y$ and the bundle to only one person, but sell the other one to both. Both buyers have opposite relative ordering for the two goods.	4	We can choose both the good that is sold to one person, and the person who gets just that one good	Bundling is worse than 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, even at the boundary. 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 $p_{B,Y} \ge 2p_{A,Y}$, so we are strictly better off not bundling.
Total (10 types)	36	--	--

Nontrivial determinate case

We are looking at the case: "Sell one of $X,Y$ to only one person, but sell the other one and the bundle to both. The buyers have opposite relative ordering for the goods. The buyer with the lower price for the bundle is the one who received the single good."

We will be more specific: $A$ is the person to whom the single good is sold, and the single good sold is $X$. We are then in the following region:

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

We then obtain:

• The seller revenue from no bundling is $p_{A,X} + 2p_{A,Y}$
• The seller revenue from bundling is $2(p_{A,X} + p_{A,Y})$
• Therefore, the seller's gain from bundling is $p_{A,X}$, which is positive

Here is an example that satisfies these conditions:

$\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 1 & 7 \\\end{pmatrix}$

First indeterminate case

We are looking at the case: "Sell one of $X,Y$ to only one person, but sell the other one and the bundle to both. The buyers have opposite relative ordering for the goods. The buyer with the lower price for the bundle is the one who did not receive the single good."

We will be more specific: $A$ is the person to whom the single good is sold, and the single good sold is $X$. We are then in the following region:

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

We then obtain:

• The seller's revenue from no bundling is $p_{A,X} + 2p_{A,Y}$
• The seller's revenue from bundling is $2(p_{B,X} + p_{B,Y})$
• Therefore, bundling is advantageous for the seller if and only if:

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

We therefore obtain a further partition of the region into two subregions. Below are examples to show that both subregions are feasible.

Example class where bundling is beneficial:

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

Example class where bundling is not beneficial:

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

Second indeterminate case

We are looking at the case: "Sell each good to a different single person, and sell the bundle to both."

We will be more specific: $A$ gets $X$, and $B$ gets $Y$, and $A$ has the lower willingness to pay on the bundle. We are then in the following region:

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

We then obtain:

• The seller's revenue from no bundling is $p_{A,X} + p_{B,Y}$
• The seller's revenue from bundling is $2(p_{A,X} + p_{A,Y})$
• Therefore, bundling is advantageous for the seller if and only if:

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

We therefore obtain a further partition of the region into two subregions. Below are examples to show that both subregions are feasible.

Example class where bundling is beneficial:

$\begin{pmatrix} p_{A,X} & p_{A,Y} \\ p_{B,X} & p_{B,Y} \\\end{pmatrix} = \begin{pmatrix} u & 1 \\ 2 & u \\\end{pmatrix}, u > 4$

Example class where bundling is not beneficial:

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