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

This page describes a simple situation of price bundling: two goods, two buyers, one seller, subject to the constraint that the seller needs to ultimately have both goods sold to both buyers. We will compare the following three scenarios:

No bundling: The seller sells each good in a separate market, at a single price to both buyers.
Pure price bundling: The seller practices pure price bundling, selling both goods at a bundle at a single price to both buyers.
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.

Our overall conclusion will be:

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.
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.

Complications we assume away

This is a toy case. We ignore many complications such as:

The possibility of using mixed bundling
The possibility of monopolistic pricing to exclude some buyers (in order to get more from other buyers)
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

A simplifying assumption we will make for all our numbers is that we will subtract the cost of production from the prices. Alternatively, you can think of it as assuming zero cost of production, but note that that assumption does not lose us any 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.

Qualitatively, what really matters for our conclusions are the differences $p_{A,X} - p_{B,X}$ and $p_{A,Y} - p_{B,Y}$ . In other words, if we subtract a constant $c_X$ from both $p_{A,X}$ and $p_{B,X}$ , the qualitative conclusions won't change. Similarly for $Y$ .

Our conclusions are also invariant under scaling all four numbers by the same positive value.

Conclusions

The three strategies

Type of situation	Seller's pricing strategy	Seller's profit from buyer $A$	Seller's profit from buyer $B$	Seller's total profit
No 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	$p_{A,X} + p_{A,Y}$	$p_{B,X} + p_{B,Y}$	$p_{A,X} + p_{A,Y} + p_{B,X} + p_{B,Y}$

Comparison of the strategies

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.

The full case analysis below will shed more light on the above breakdown.

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 two cases that arise for one good at a time
Discuss how this leads to eight possible interior cases for $X$ , $Y$ , and the bundle, and the 6 of these 8 cases (2 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 $X$ , the seller would price it as follows to sell to both buyers:

$\min \{ p_{A,X}, p_{B,X} \}$

This is a piecewise linear function on the plane, and the boundary between its two piece definitions is the line $p_{A,X} = p_{B,X}$ . On one side of the line, $p_{A,X}$ is smaller, so the seller uses that as the price, and on the other side, $p_{B,X}$ is smaller, so the seller uses that as the price.

The two goods and the bundle together

Now let's look at everything together. We now have three dividing hyperplanes:

$p_{A,X} = p_{B,X}$ separates the space into two parts. The choice of part corresponds to what price would be used ( $p_{A,X}$ or $p_{B,X}$ ) if selling $X$ alone.
$p_{A,Y} = p_{B,Y}$ separates the space into two parts. The choice of part corresponds to what price would be used ( $p_{A,Y}$ or $p_{B,Y}$ if selling $Y$ alone.
$p_{A,X} + p_{A,Y} = p_{B,X} + p_{B,Y}$ separates the space into two parts. The choice of part corresponds to what price would be used if selling the bundle.

In total, we can imagine $2 \times 2 \times 2 = 8$ interior cases, based on what gets selected for each of the three bullet points above. However, only 6 of these can occur in practice. The two cases that are impossible are the ones where the same buyer determines the price on both $X$ and $Y$ , but a different buyer determines the price on $X + Y$ .

In addition to these interior cases, we have corner cases (where exact equality holds for one or more of the equations above). We will turn to corner cases after looking at interior cases.

The role of symmetry

Note that we can interchange the roles of $A$ and $B$ . We can also interchange the roles of $X$ and $Y$ . Doing so could change the specifics but will give qualitatively similar cases. By grouping qualitatively similar cases together, we can identify the different types. Each of the groupings (called equivalence classes in mathematics) can have size 1, 2, or 4.

The taxonomy of interior cases

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	Algebraic formulation (one line per case)	Conclusion on bundling
The same buyer has a lower willingness to pay for both goods	2	We can choose which of the buyers has a lower willingness to pay	$p_{A,X} < p_{B,X}$ and $p_{A,Y} < p_{B,Y}$ $p_{B,X} < p_{A,X}$ and $p_{B,Y} < p_{A,Y}$	Bundling is irrelevant. It neither helps nor hurts.
Different buyers have lower willingness to pay for the two goods	4	We can choose which of the buyers has a lower willingness to pay for the bundle, and we can then choose which of the goods that buyer has a lower willingness to pay for	$p_{A,X} + p_{A,Y} < p_{B,X} + p_{B,Y}$ , $p_{A,X} < p_{B,X}$ , and $p_{A,Y} > p_{B,Y}$ $p_{A,X} + p_{A,Y} < p_{B,X} + p_{B,Y}$ , $p_{A,X} > p_{B,X}$ , and $p_{A,Y} < p_{B,Y}$ $p_{A,X} + p_{A,Y} > p_{B,X} + p_{B,Y}$ , $p_{A,X} < p_{B,X}$ , and $p_{A,Y} > p_{B,Y}$ $p_{A,X} + p_{A,Y} > p_{B,X} + p_{B,Y}$ , $p_{A,X} > p_{B,X}$ , and $p_{A,Y} < p_{B,Y}$	Bundling helps. The seller's gain from bundling, in the four respective cases, is: $2(p_{A,Y} - p_{B,Y})$ $2(p_{A,X} - p_{B,X})$ $2(p_{B,X} - p_{A,X})$ $2(p_{B,Y} - p_{A,Y})$
Total (2 types)	6	--	--	--

Full social surplus analysis: case where bundling helps seller

We consider the case:

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

Item	Value for $X$	Value for $Y$	Total value with no bundling	Value with pure price bundling	Increase from bundling
Seller's surplus	$2p_{A,X}$	$2p_{B,Y}$	$2(p_{A,X} + p_{B,Y})$	$2(p_{A,X} + p_{A,Y})$	$2(p_{A,Y} - p_{B,Y})$ (positive)
Buyer $A$ 's surplus	0	$p_{A,Y} - p_{B,Y}$	$p_{A,Y} - p_{B,Y}$	0	$p_{B,Y} - p_{A,Y}$ (negative)
Buyer $B$ 's surplus	$p_{B,X} - p_{A,X}$	0	$p_{B,X} - p_{A,X}$	$(p_{B,X} + p_{B,Y}) - (p_{A,X} + p_{A,Y})$	$p_{B,Y} - p_{A,Y}$ (negative)
Total	0	0	0	0	0

We see that the gains in surplus to the seller come at the expense of both buyers, to an equal extent (though the breakdown per good is a little different).

The taxonomy of corner cases

We now consider all cases including cases where the point in question lies on one or more of the hyperplanes $p_{A,X} = p_{B,X}$ , $p_{A,Y} = p_{B,Y}$ and $p_{A,X} + p_{A,Y} = p_{B,X} + p_{B,Y}$ . In principle, we get $3 \times 3 \times 3 = 27$ possible combinations, of which 8 possible interior cases have been considered already. That leaves 19 cases. Of these 19, it turns out that only 7 can occur in practice.

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	Algebraic formulation (one line per case)	Codimension and dimension	Conclusion on bundling	Conclusion on price discrimination
One good is equally valued by both buyers, the other is not	4	We can choose which good is not equally valued, and what buyer values it less	$p_{A,X} = p_{B,X}, p_{A,Y} > p_{B,Y}$ $p_{A,X} = p_{B,X}, p_{B,Y} > p_{A,Y}$ $p_{A,Y} = p_{B,Y}, p_{A,X} > p_{B,X}$ $p_{A,Y} = p_{B,Y}, p_{B,X} > p_{A,X}$	Codimension 1, dimension 3	Bundling is irrelevant	Perfect price discrimination helps on the unequally valued good and on the bundle but not on the equally valued good
The bundle is equally valued by both buyers, but the individual goods are not	2	We can choose a matching of buyers to goods based on what buyer has lower willingness to pay for a good	$p_{A,X} + p_{A,Y} = p_{B,X} + p_{B,Y}, p_{A,X} > p_{B,X}$ $p_{A,X} + p_{A,Y} = p_{B,X} + p_{B,Y}, p_{B,X} > p_{A,X}$	Codimension 1, dimension 3	Bundling helps	Perfect price discrimination on the bundle yields no additional gains relative to bundling
Each good is equally valued by both buyers	1	No degrees of freedom	$p_{A,X} = p_{B,X}, p_{A,Y} = p_{B,Y}$	Codimension 2, dimension 2	Bundling is irrelevant	Perfect price discrimination is unnecessary
Total (3 types)	7	--	--	--	--	--