Multi-Product Duopoly With
Cross-Product Cost
Interdependencies
Gary Biglaiser
Andrei Hagiu
Working Paper
16-010
July 21, 2015
Copyright © 2015 by Gary Biglaiser and Andrei Hagiu
Working papers are in draft form. This working paper is distributed for purposes of comment and
discussion only. It may not be reproduced without permission of the copyright holder. Copies of working
papers are available from the author.
Multi-Product Duopoly With Cross-Product Cost Interdependencies
Gary Biglaiser and Andrei Hagiuy
July 21, 2015
Abstract
Many multi-product …rms incur a complexity …xed cost when o¤ering di¤erent product lines in
di¤erent quality tiers relative to the case when o¤ering all products lines in the same quality tier
(high or low). Such …xed costs create an interdependency between …rms’ choices of quality tiers
across di¤erent product lines, even when demands are independent. We investigate the e¤ects of
this interdependency on equilibrium pro…ts in a Stackelberg duopoly game. Both …rms’pro…ts are
(weakly) higher when the complexity cost is in…nite than when it is 0. The Stackelberg leader’s
pro…ts are always (weakly) higher with a positive complexity …xed cost, but its pro…ts can be nonmonotonic in the magnitude of this cost. The Stackelberg follower’s pro…ts can be lower when the
complexity …xed cost is positive than when it is equal to 0.
JEL Classi…cations: L1, L2, L8
Keywords: multi-product duopoly, vertical di¤erentiation, …xed costs.
1
Introduction
In many industries, multi-product …rms do not have full ‡exibility to choose di¤erent quality tiers
for di¤erent product categories (lines). Once committed to a certain quality tier (high or low) in
one product line, it is usually more costly to o¤er another product line in a di¤erent quality tier
instead of o¤ering it in the same tier. This is one of the main reasons for which many multi-product
…rms in various industries typically o¤er a consistent product and service quality for most of their
product categories. Examples include retailers (e.g. Neiman Marcus, Saks Fifth Avenue at the highend vs. T.J. Maxx, Wal-Mart at the low-end), e-commerce (e.g. Eastbay and Zappos selling shoes
and clothing with a high level of customer service vs. eBay relying on third-party sellers with typically
lower-quality service), furniture stores (e.g. Crate & Barrel at the mid- to high-end vs. Ikea at
y
University of North Carolina, [email protected]
Harvard Business School, Boston, MA 02163, E-mail: [email protected]
1
the low-end), airlines (full-price carriers such as American Airlines, Emirates, United vs. low-cost
carriers such as EasyJet, Ryan Air, Southwest), etc. Some companies have managed to maintain
di¤erent quality tiers for di¤erent product lines, e.g. Sears (low-end in clothing, but lower-upper end
in tools), Amazon (retailer with high level of service on some products, but eBay-like marketplace on
other products), Singapore Airlines (high-end airline, but also operates Scoot, a low-cost carrier, on
some, typically shorter, routes).1 However, other companies’attempts to do so have failed, e.g. J.C.
Penney’s unsuccessful attempt to become an upscale department store between 2011-2013,2 the failure
of Delta’s low-cost carrier Song,3 etc. These examples illustrate that in many circumstances there are
diseconomies of scope when o¤ering multiple quality tiers on di¤erent product lines or categories. Such
diseconomies can stem from a combination of brand stickiness - it is di¢ cult to cover multiple quality
tiers with the same brand or operate multiple brands in the same organization - and operational
complexity - di¤erent quality tiers have di¤erent internal processes (see Rivkin, 2006).
We analyze a Stackelberg multi-product duopoly game in which …rms incur a higher …xed cost
when o¤ering di¤erent product lines in di¤erent quality tiers (mixed mode) than when o¤ering all
product lines in the same quality tier (high or low). To the best of our knowledge, this type of interdependence has not been analyzed in the previous literature on multi-product oligopoly, despite its
real-world relevance. In our model, each …rm can o¤er any subset of a given range of products that
have identical and independent demands. Each product can be o¤ered in one of two qualities (or
modes) - high or low - and a …rm that o¤ers positive numbers of products in each mode incurs a
…xed complexity cost F
0. Firms compete product-by-product, so the only inter-dependence across
products is driven by costs, not by demand.
We …rst show that both …rms’ pro…ts are higher when the complexity cost is in…nite (i.e. each
…rm must o¤er all of its products in the same mode) than when the complexity …xed cost is 0 (i.e.
…rms can costlessly use a mixed mode). Indeed, the inability to o¤er di¤erent products in di¤erent
modes makes it easier for …rms to sustain di¤erentiation with respect to the o¤ered products, so that
each …rm ends up being a monopolist on the products it o¤ers. In contrast, when …rms can use a
mixed mode costlessly, the choices of mode (high or low) for individual products are independent of
one another, so …rms cannot avoid competing on each product as a vertical di¤erentiated duopoly.
The determination of the equilibrium is more complex when the …xed cost is positive but …nite.
The Stackelberg leader’s (…rm 1’s) pro…ts are higher for all F > 0 than for F = 0, but they can be
non-monotonic in F . The reason is that the …xed complexity cost can have two opposing e¤ects on
…rm 1’s pro…ts: i) a direct, negative e¤ect when …rm 1 chooses a mixed mode strategy (i.e. o¤ers some
products in each mode), and ii) an indirect, positive e¤ect, by lowering the pro…ts that the Stackelberg
follower (…rm 2) can achieve with a mixed mode, thereby relaxing the constraints on …rm 1. On the
other hand, the Stackelberg follower’s pro…ts can be lower when the complexity cost is positive and
1
See W. Smit and C. Dula, "Singapore Airlines’ Branding of its Low-Cost Carriers," Financial Times, March 26,
2014.
2
See T. Hsu and W. Hamilton, "J.C. Penney Ourst CEO Ron Johnson, Brings Back His Predecessor," Los Angeles
Times. April 9, 2013.
3
See M. Maynard, "Delta to Close Song, its Low-Fare Airline," New York Times, October 29, 2005.
2
…nite than when it is equal to 0. Furthermore, consumer surplus is higher when the …rms can freely
o¤er products in multiple modes, since consumer surplus is higher in a di¤erentiated duopoly than in
a single quality monopoly market.
In most of the literature on multi-product monopoly and duopoly (e.g. Mussa and Rosen 1978, GalOr 1983, Champsaur and Rochet 1989, Klemperer 1992, and Johnson and Myatt 2003), the …xed costs
of o¤ering an additional quality version or product line are typically 0 (marginal costs are, however,
allowed to depend on product quality). In contrast, the key feature of our model is the interdependence
of …rms’quality choices across product lines due to …xed costs. This type of interdependency has not
previously been studied, despite its real-world relevance.
Furthermore, unlike most of the literature on multi-product monopoly and duopoly, which assumes all product lines are within the same product category and therefore substitutes, we focus on
independent product lines. In other words, in our model consumer demand for each product line is independent of demand for the other product lines. Thus, we allow for two dimensions of di¤erentiation
between …rms: vertical di¤erentiation within the same product line and product line di¤erentiation
(i.e. …rms may o¤er di¤erent product lines). A prominent exception to the single product category
assumption is Klemperer (1992), which analyzes a circular Hotelling-type setting where duopolists are
restricted to o¤ering N products at 0 …xed costs, but consumers have disutility from making purchases
at multiple …rms (e.g. switching costs). The di¤erence is that in our model demands are independent
across product lines, therefore our focus is on supply-side interdependencies, not on demand drivers
as in Klemperer (1992).
The rest of the paper is organized as follows. In section 2, we present the basic model: we use a
reduced form set-up that encompasses a large set of demand functions. We treat the baseline cases with
complexity cost equal to 0 and in…nity in section 3. This allows us to establish some basic intuition
and de…ne two important strategies for the Stackelberg leader (full-spectrum and accommodation).
Section 4 contains the analysis for all …nite complexity cost. We analyze extensions in section 5 and
conclude in the …nal section 6.
2
Model set-up
There are two competing …rms, 1 and 2. Each …rm i 2 f1; 2g can choose a continuous number of
products Ni 2 [0; N ] to o¤er and one of two possible modes, L or H, for each product (in section
5.2 we allow …rms to o¤er a product in both modes). Demands are identical and independent across
products, i.e. competition occurs product by product. The two modes should be interpreted as
corresponding to two vertically di¤erentiated versions: high and low intrinsic qualities of the product
or high and low levels of service accompanying the product.
If one of the two …rms is a monopolist on any given product, its revenues net of marginal costs are
RH if it o¤ers the product in H mode and RL if it o¤ers the product in L mode. If both …rms o¤er
3
a given product in the same mode, Bertrand competition with no di¤erentiation implies that their
respective revenues net of marginal costs are 0. If …rm i o¤ers a product in H mode, whereas …rm j
o¤ers it in L mode, then …rm i’s equilibrium revenues net of marginal costs are rH , whereas …rm j’s
are rL . The …xed costs required to make one product available in H mode and, respectively, L mode
are fH and fL , where fH
fL
0. Denote then the net pro…ts corresponding to the revenues de…ned
above as follows:
H
RH
fH
L
RL
fL
H
rH
fH
L
rL
fL
We make the following two assumptions throughout.
Assumption 1
Assumption 2
H
>
L
>
L
>0
and
H
>
H
>
L
> 0.
Demand for each product is such that a monopolist never …nds it pro…table to
o¤ er the product in both modes.
The …rst assumption simply ensures consistency with the interpretation of two vertically di¤erentiated product versions: the H mode is more pro…table than the L mode, even accounting for …xed
costs. The second assumption rules out price discrimination reasons for o¤ering multiple (vertically
di¤erentiated) versions of the same product - such price discrimination motivations are studied indepth by Johnson and Myatt (2003). This assumption implies that throughout most of the analysis
neither …rm has any incentive to o¤er any given product in both modes. The only exception is when
a …rm wishes to foreclose the other …rm, a possibility we discuss in Section 5.2.
The following example satis…es both assumptions above. Consumers are indexed by v, uniformly
distributed on [0; 1]. Consumer v’s willingness-to-pay for a product of quality q is qv. The L mode
corresponds to quality qL and the H mode corresponds to quality qH > qL . Assume 0 marginal costs.
With this demand structure, pro…ts are:
H
=
H
=
qH
fH
4
2 (q
4qH
H
(4qH
and
L
qL )
qL )
=
qL
4
fH and
2
fL
L
=
qH qL (qH qL )
(4qH qL )2
fL
It is easily veri…ed that this example satis…es both assumptions for any (qH ; qL ), provided fH
not too
fL is
large.4
It is worth emphasizing that fL and fH do not play any signi…cant role in our analysis, which is
why they are subsumed in the terms
4
H,
Also, …xing qL , there exists qH > qL such that
L,
H
H
<
and
L
4
L.
for qL < qH < qH and
H
>
L
for qH > qH .
On the other hand, the key ingredient of our model will be the …xed cost associated with o¤ering
products in both the H and L modes. To keep things as simple as possible, we assume that a …rm that
o¤ers positive numbers of products in both modes incurs a "complexity …xed cost" F
0. In other
words, this cost is incurred whenever a …rm does not o¤er all of its products in the same mode.5 This
…xed cost may correspond to the risk of confusing consumers by o¤ering two vertically di¤erentiated
modes under the same brand or to the operational complexity of running two di¤erent business models.
The fundamental e¤ect of the complexity …xed cost is that it creates an interdependence between a
…rm’s choices of modes across products.
The timing of the duopoly game in all scenarios is as follows:
1. Firm 1 chooses the respective numbers of products to o¤er in each mode, N1H and N1L such that
N1H + N1L
N.
2. Firm 2 chooses the respective numbers of products to o¤er in each mode, N2H and N2L such that
N2H + N2L
N.
3. Firms simultaneously compete in prices product by product and pro…ts are realized.
Assuming the product selection stage precedes the pricing game is standard in the multi-product
duopoly literature (see for example Champsaur and Rochet, 1989) and re‡ects the fact that product
selection is typically a long-run decision, whereas prices can be modi…ed more easily and frequently.
It is then natural to also assume that …rms choose their prices simultaneously.
There are two reasons for assuming Stackelberg timing in product selection. First, the Stackelberg
timing re‡ects realistic scenarios in which …rm 1 is an incumbent or industry leader, who makes
product line choices …rst, whereas …rm 2 is an entrant or a follower. Second, from a methodological
standpoint, Stackelberg timing provides a natural way of ensuring a unique pure strategy equilibrium.
In contrast, with Nash (simultaneous) timing, there will typically be multiple equilibria and sometimes
there may only exist mixed strategy equilibria.
3
Baseline results
In this section, we derive the baseline results for two simple (and polar opposite) cases.
Suppose …rst that there is no complexity …xed cost associated with operating under both modes,
i.e. F = 0. This means that each …rm’s choices of modes are independent across products. In this
case, the equilibrium is easily obtained.
Proposition 1 When …rms’ choices of modes are independent across products, there is a unique
equilibrium in which …rm 1 o¤ ers all N products in H mode and …rm 2 o¤ ers all N products in L
mode. Equilibrium pro…ts are N
H
for …rm 1 and N
5
L
for …rm 2.
In the extensions section we also analyze an alternative formulation, in which o¤ering any positive number of products
in H mode requires a "capability" …xed cost FH 0 and o¤ering any positive number of products in L mode requires a
"capability" …xed cost FL 0. The analysis is quite similar.
5
Not surprisingly, since choices of modes across products are independent, this is the same (unique)
equilibrium that would prevail if there was only one product. Firms o¤er vertically di¤erentiated
versions of each product, with …rm 1 choosing the more pro…table H mode and …rm 2 choosing the L
mode.
At the opposite extreme, suppose now that each …rm must choose the same mode for all products it
o¤ers. This corresponds to the case when the complexity …xed cost is prohibitively high, i.e. F ! 1.
In this case, each …rm chooses one of the two modes - H or L - and the number of products to o¤er
in that mode. Thus, suppose …rm 1 has chosen the H mode in stage 1 and N1
N products. Firm 2
has two options:
Choose the L mode: clearly Firm 2 will o¤er all products in L mode, since
L
> 0. Firm 2’s
equilibrium pro…ts are then
N1
L
+ (N
N1 )
Choose the H mode: …rm 2 will only o¤er the N
L.
N1 products not o¤ered by …rm 1. Firm 2’s
equilibrium pro…ts are then
(N
N1 )
H.
Comparing, …rm 2 chooses the L mode if and only if
N1
L
+ (N
N1 )
> (N
L
N1 )
H
which can also be written as:
H
N1 > N
L
H
L
+
L
NH
Thus, if the number of products o¤ered by …rm 1 in H mode is larger than the threshold NH , …rm 2
responds by competing on all products with a vertically di¤erentiated version. If, on the other hand,
N1 is below the threshold NH then …rm 2 …nds it pro…table to not compete on the products o¤ered
by …rm 1, so that both …rms end up with H mode monopoly pro…ts on their respective products. The
threshold is increasing in
H
and decreasing in
L
and
L:
the higher the monopoly H pro…ts and the
lower L pro…ts (monopoly and duopoly), the easier it is to induce …rm 2 to be content with being an H
monopolist on N
N1 products instead of o¤ering all products in L mode (which involves competing
on the N1 products o¤ered by …rm 1).
Stepping back to stage 1, …rm 1 pro…ts are N1
H
if N1
NH and N1
that when …rm 1 chooses the H mode, its optimal pro…ts are max fN
H
H ; NH
if N1 > NH . We conclude
H g.
In other words, …rm
1 has a choice between duopoly pro…ts on all relevant products and monopoly pro…ts on a subset of
these products. Relegating the rest of the proof to the appendix, we obtain the following proposition.
Proposition 2 When each …rm must o¤ er all products in the same mode, the duopoly equilibrium is
as follows:
If
H
L
L H
H
H
then …rm 1 chooses the H mode, o¤ ers N1 =
6
N(
H
H
L)
L+ L
products and
makes pro…ts
N
pro…ts
If
H
N
H.
H
L
N(
L H
L+ L
<
L)
H
H
H
L+ L
N
. Firm 2 o¤ ers the other
H
L
L+ L
products in H mode and makes
.
L H
H
H
then …rm 1 chooses the H mode, o¤ ers N1 = N products and makes pro…ts
Firm 2 o¤ ers all products in L mode and makes pro…ts N
L.
Comparing Propositions 1 and 2 reveals that …rm 1 makes (weakly) higher pro…ts when …rms are
constrained to o¤er all products under the same mode. In particular, pro…ts are strictly higher when
H
is su¢ ciently large relative to both
L
and
H.
Furthermore, …rm 2 also makes (weakly) higher
pro…ts when F = +1. Indeed,
N
H
because
L
>
L
L
H
+
>N
L
L
L.
The intuition behind this result is that the inability to choose di¤erent modes for di¤erent products
allows the …rms to also di¤erentiate with respect to the products they o¤er. This can be a more e¤ective
form of di¤erentiation than the vertical di¤erentiation within each product, provided that monopoly H
pro…ts are su¢ ciently large relative to monopoly and duopoly L pro…ts. Indeed, if monopoly L pro…ts
are too close to monopoly H pro…ts, then …rm 1 cannot induce …rm 2 to o¤er a restricted number of
H products instead of all products in L mode. Anticipating this, …rm 1 has no choice but to resign
itself to duopoly H pro…ts on all products.
Finally, note that consumer welfare is strictly lower in the equilibrium in which …rm 1 o¤ers N1 < N
products in H mode and induces …rm 2 to o¤er the remaining products in H mode as well. In this
equilibrium, all products are o¤ered, but each product is only o¤ered by a monopolist. By contrast, in
the equilibrium in which …rm 1 o¤ers all N products in H mode, each products ends up being o¤ered
by a vertically di¤erentiated duopoly, so consumer welfare is higher.
In what follows, we will refer to …rm 1’s strategy of o¤ering all products in H mode as the "fullspectrum strategy" and to its strategy of only o¤ering a limited number of products in H mode (thereby
inducing …rm 2 to only o¤er the other products in H mode) as the "accommodation strategy."
4
Equilibrium with …nite complexity …xed cost
In this section, we study the case in which …rms are not forced to use the same mode for all the
products they o¤er, but they need to incur a …nite complexity …xed cost F > 0 if they o¤er positive
numbers of products in each mode.
Denote:
NH (F )
min
N(
H
L)
H
L
+
L
;
F
;
L
The number NH (F ) 2 [0; N ] represents the maximum number of products that …rm 1 can o¤er
in H mode such that …rm 2’s best response is to only o¤er the remaining N
NH (F ) products in
H mode. The …rst term in the min fg operator (equal to NH (+1)) is the same as the threshold in
Proposition 2. This term is the maximum number of products that …rm 1 can o¤er in H mode such
7
that …rm 2 prefers o¤ering the remaining products in H mode as a monopolist to o¤ering all products
in L mode. The second term is new and represents the maximum number NH of products that …rm
1 can o¤er in H mode such that …rm 2 is content to be a monopolist in mode H on the remaining
N
NH products and does not want to also compete in L mode on the NH products o¤ered by …rm
1 (this would yield additional pro…ts
L NH
at cost F ).
With this notation, the maximum pro…ts that …rm 1 can achieve with the accommodation strategy
are NH (F )
H.
As before, these pro…ts need to be compared with N
H,
the pro…ts resulting from
the full-spectrum strategy, but now …rm 1 can also consider a mixed mode strategy, i.e. o¤ering
some products in H mode and others in L mode. The following proposition (proven in the appendix),
provides the full characterization of …rm 1’s optimal strategy.
Proposition 3 When …rms incur a …xed complexity cost F > 0 for using a mixed mode, …rm 1 may
o¤ er products in H mode only or in both modes, …rm 2 only o¤ ers products in a single mode and
if
F
N
<
L H
L+ H
then …rm 1’s optimal pro…ts are
max N
if
F
N
>
L H
L+ H
H ; NH
(F )
H; N L
H
+
2 F; N
H
L
+
L
2 F
(1)
H
then …rm 1’s optimal pro…ts are
(
max N
H ; NH
(F )
+
L+
H H
H;
2
L
N
+
L+
L L
F;
H
2
H
N
F
H
)
Thus, when F is su¢ ciently small, …rm 1’s optimal pro…ts (from (1)) are simply N
same as in Proposition 1 (F = 0). In this case, …rm 2’s pro…ts are N
L.
(2)
H,
i.e. the
Meanwhile, when F becomes
large enough, the expression of …rm 1’s optimal pro…ts (from (2)) becomes
max N
H;
N(
L)
H
H
L
+
H
,
L
i.e. the same as in Proposition 2 (F = +1). The …rst two terms for both regions of …xed costs
in Proposition 3 (expressions (1) and (2)) are simply the full-spectrum and accommodation strategy
pro…ts. The third and fourth terms for both regions correspond to mixed mode strategies for …rm 1.
To understand the di¤erence between the two cases in the proposition, it is …rst useful to note that
in any equilibrium …rm 1 precludes …rm 2 from using a mixed mode strategy. Indeed, if …rm 2 used a
mixed mode strategy in equilibrium, then it would necessarily cover the entire product range, since it
is already incurring the …xed cost F and adding an additional product yields at least
L
> 0. Denoting
by N2 the number of products o¤ered in H mode by …rm 2, we have 0 < N2 < N . In this equilibrium,
…rm 1’s pro…ts would be N2
are dominated by N
H,
L
or (N
N2 )
H
or N2
L + (N
N2 )
H
F . But each of these pro…ts
the pro…t that …rm 1 can guarantee by choosing the full-spectrum strategy
(all products in H mode).
8
As discussed before the proposition, the necessity of preventing …rm 2 from using a mixed mode
strategy explains the new expression of NH (F ) in …rm 1’s accommodation strategy. When …rm 1 uses
the accommodation strategy, …rm 2’s resulting pro…ts are (N
NH (F ))
H.
Focusing now on equilibria in which …rm 1 chooses a mixed mode strategy, the di¤erence between
the two cases is whether or not the constraint of having to prevent …rm 2 from using a mixed mode is
F
binding. In the …rst case ( N
<
L H
L+ H
), the complexity …xed cost is su¢ ciently small, so …rm 1 needs
to take into account that …rm 2 can pro…tably o¤er products in both modes. To prevent …rm 2 from
doing so, the number N1 of products that …rm 1 o¤ers in H mode must either be small enough so that
…rm 2 does not want to o¤er those same products in L mode (N1
L
< F ), or large enough so that …rm
2 does not want to o¤er in H mode the same products that …rm 1 o¤ers in L mode (
H (N
N1 )
F ).
These two possibilities lead to the third and fourth terms in (1). The corresponding …rm 2 pro…ts are
F
N
H
L
F
when …rm 1 chooses the third option in (1) and N
H
L
when …rm 1 chooses the
fourth option in (1). In particular, note that when …rm 1 chooses the fourth option, …rm 2’s pro…ts
are strictly lower than N
In the second case,
F
(N
L,
>
the pro…ts it makes when F = 0.
L H
L+ H
), the …xed cost incurred when adopting a mixed mode is su¢ ciently
high that …rm 2 never chooses the mixed mode option in response to …rm 1 choosing the mixed mode
strategy. Firm 1 still has two options with a mixed mode strategy:
Induce …rm 2 to only o¤er H products, speci…cally the N
N1 products that …rm 1 o¤ers in L
mode. This leaves …rm 1 as an H-mode monopolist on N1 products and an L-mode duopolist
on N
than N
N1 products (third term in (2)). Firm 2’s pro…ts in this case are
L,
N
H L
L+ H
, strictly lower
the pro…ts it makes when F = 0.
Induce …rm 2 to only o¤er L products, speci…cally the N1 products that 1 o¤ers in H mode. This
leaves …rm 1 as an H-mode duopolist on N1 products and an L-mode monopolist on N
products (fourth term in (2)). Firm 2’s pro…ts in this case are once again
N
L H
L+ H
N1
.
It is interesting to discuss the e¤ect of the …xed cost F on …rm 1’s equilibrium pro…ts.
Corollary 1 If
F
N
<
L H
L+ H
then …rm 1’s equilibrium pro…ts are weakly increasing in F ; if
F
N
>
L H
L+ H
then …rm 1’s equilibrium pro…ts can be either increasing or decreasing in F .
F
N
. The term NH (F ) H is weakly increasing in F and the only way
in which one of the mixed mode strategy pro…ts in (1) dominates the baseline pro…t N H is if H
2>0
Proof. Consider the case
<
L H
L+ H
L
or
H
L
2 > 0, which means those pro…ts are also increasing in F . If
F
N
>
L H
L+ H
, then if …rm 1 uses the
accommodation strategy pro…ts are increasing in F , while it if uses either mixed mode strategy it is
falling in F .
When F is not too large, it a¤ects …rm 1’s pro…ts both directly and indirectly. The direct e¤ect is
always negative: it reduces the pro…ts that …rm 1 can obtain with a mixed-mode strategy. In contrast,
the indirect e¤ect is positive: a larger F reduces …rm 2’s pro…ts from using a mixed mode, which in
turn relaxes the constraints faced by …rm 1 and therefore enhances (weakly) its pro…ts. The negative
9
direct e¤ect only comes into play when …rm 1 uses a mixed-mode strategy, but for such a strategy
to be dominant, it is necessary that the positive indirect e¤ect of F on …rm 1’s pro…ts outweighs the
negative direct e¤ect. Thus, …rm 1’s pro…ts are weakly increasing in F in this parameter range.
Consider now the case when F is above the threshold
N
L H
L+ H
. If …rm 1 chooses the accommodation
strategy then its pro…ts are weakly increasing in F because F only a¤ects …rm 1’s pro…ts indirectly:
a larger F relaxes the constraint of having to prevent …rm 2 from using a mixed mode. On the other
hand, if …rm 1 chooses either one of the two mixed-mode strategies then F only has the direct negative
e¤ect on its equilibrium pro…ts since …rm 2 would never consider choosing a mixed mode in this case.
This is why the overall e¤ect of F on …rm 1’s pro…ts is ambiguous on this parameter range.
Figures 1 and 2 illustrate the two possible cases emerging from this discussion. Both use the linear
vertical di¤erentiation example presented in Section 2 with fL = fH = 0, but with di¤erent values of
(qL ; qH ). In …gure 1 (qL = 5 and qH = 9), …rm 1 pro…ts are everywhere weakly increasing in F . In
…gure 2 (qL = 5 and qH = 5:25), …rm 1 pro…ts are non-monotonic in F .
Figure 1
Figure 2
10
5
Extensions
5.1
Equilibrium with capability …xed costs
An alternative way to introduce interdependence between a …rm’s choices of modes across products
lines through …xed costs is to assume that each mode requires a …xed "capability" investment, instead
of the complexity …xed cost studied above. Speci…cally, suppose that o¤ering any positive number of
products in H mode requires a …xed cost FH
mode requires a …xed cost FL
0 and o¤ering any positive number of products in L
0. We still maintain the assumption that the …rm can only o¤er one
mode per product. Consistent with the interpretation of vertical di¤erentiation (H corresponds to the
higher quality mode), we assume the …xed cost required for the H capability is larger than the …xed
cost required for the L capability:
FH > FL .
Furthermore, we also assume that …xed costs are not too high, such that both modes are viable
for a monopolist as a well as for a duopolist:
N
L
> FL and N
H
> FH
Denote:
NH
NL
N(
min
(
L)
H
H
(FH
+
L
L
0
FH FL N (
H ( H
L)
H
L)
FL ) FL
;
L
if FH
FL
N(
H
L)
0
if FH
FL
N(
H
L)
>0
The expression of NH is very similar to NH (F ) from section 4 and has the same meaning: it is
the maximum number of products that …rm 1 can o¤er in H mode such that …rm 2’s best response
is to only o¤er the remaining N
NH products in H mode (accommodation strategy). Similarly, NL
represents the maximum number of products that …rm 1 can o¤er in L mode such that …rm 2’s best
response is to only o¤er the remaining N
NL products in L mode. Note that NL
0 in all cases.
In the appendix, we derive the full equilibrium summarized in the following proposition.
Proposition 4 Suppose the …rms incur …xed costs FH and FL for operating in H mode and L mode
respectively.
If
FL
L
+
FH
H
< N then …rm 1’s optimal pro…ts are
max
8
>
>
<
>
>
: N
L
+
H
L
N
H
FH ; N
NH
H
FH ; NL
L
FH ; N
+
2 FL
11
FL ;
L
H
FL ;
L
H
2 FH
FL
9
>
>
=
>
>
;
(3)
If
FL
+
L
FH
H
> N then …rm 1’s optimal pro…ts are
max
8
>
<
>
:
N
N
H
FH ; N
NH
H
FH ; NL
L
+(
L)
H
FL ;
L
L
FL
L
FL ;
FH
FL
9
>
=
>
;
(4)
The …rst two terms for both cases are the full-spectrum strategy pro…ts. The di¤erence with respect
to Proposition 3 is that now …rm 1 may …nd it pro…table to o¤er the full spectrum of products under
either the H mode or the L mode, depending on which option yields higher revenues net of …xed costs
(in Proposition 3, the full-spectrum strategy in H mode always dominated the full spectrum strategy
in L mode). The third and fourth terms in both (3) and (4) are the accommodation strategy pro…ts.
Once again, the di¤erence with respect to Proposition 3 is that either the H or the L mode may be
optimal for the accommodation strategy.
When
FL
L
H
+ FH
< N , the …fth and sixth terms in (3) correspond to mixed mode strategies for …rm
1 and they are almost identical to the corresponding terms in Proposition 3 (save for the di¤erence
in the …xed cost structure). Their interpretation is the same: …rm 1 needs to take into account that
…rm 2 can pro…tably enter with a mixed strategy. To prevent this, …rm 1 must o¤er a number N1
of products in H mode that is either small enough so that …rm 2 does not want to o¤er those same
products in L mode (N1
L
< FL ) or large enough so that …rm 2 does not want to o¤er in H mode the
products that …rm 1 o¤ers in L mode ((N
When
FL
L
+
FH
H
N1 )
H
< FH ).
> N , …rm 1 can foreclose …rm 2 with a mixed mode strategy that denies …rm
2 viable scale in both modes - a novel possibility relative to Proposition 3. Speci…cally, …rm 1 can
o¤er N1 =
FL
L
products in H mode and N
positive pro…ts in either mode (
L N1
N1 <
= FL and
H
FH
H
(N
in L mode: as a result, …rm 2 cannot make
N1 ) < FH ). Furthermore, it turns out that
the pro…ts that …rm 1 can obtain with this foreclosure strategy dominate the maximum pro…ts it can
obtain with any other mixed strategy that allows …rm 2 to enter. This is why in this case the only
options to consider for …rm 1 are full-spectrum, accommodation and foreclosure (…fth term in (4)).
5.2
Foreclosure
Throughout the previous analysis, we have assumed that neither …rm can o¤er a product in both
modes. Assumption 2 guarantees that …rm 2 never has an incentive to do so. On the other hand, …rm
1 may …nd it pro…table to o¤er a product in both modes, but only if it leads to foreclosure of …rm
2. Consider the main model above with …xed complexity costs F . Allowing …rm 1 to foreclose simply
amounts to adding a …fth option to the ones described in the text of Proposition 3. Foreclosure yields
pro…ts for …rm 1 equal to
N(
H
fL )
F
Indeed, this dominates any strategy in which …rm 1 would only o¤er a subset n < N products in both
modes. To see why, note that assumption 2 implies that RH
(assumption 1) leads to
H
fL >
H.
rH + rL , which together with rL > fL
This means that the highest pro…ts …rm 1 can hope to achieve
12
by only o¤ering n < N products in both modes are
n(
fL )
H
F + (N
n)
H
<N(
fL )
H
F.
Thus, …rm 1’s optimal pro…ts from Proposition 3 become
n
8
>
max
N(
>
>
<
>
>
>
:
fL )
H
max N (
H
F; N
fL )
H ; NH (F )
F; N
H ; NH
H; N
(F )
H;
L+
(
H
2 F; N
H
L
2
H+ L
L+ H
)N
F;
(
+
L
H
2
L L+ H
)N
H
L+ H
2 F
F
o
if
F
N
<
L H
L+ H
if
F
N
>
L H
L+ H
It is clear from these expressions that foreclosure is not necessarily optimal, so most of the analysis
in section 4 remains valid. Note that foreclosure is more likely when both fL and F are smaller.
6
Conclusion
Many multi-product …rms incur additional …xed costs when attempting to o¤er di¤erent products
in di¤erent quality or service tiers. These costs are usually associated with brand confusion or the
necessity of maintaining di¤erent brands, as well as with the operational complexity (diseconomies
of scope) of running di¤erent business models under the same organization. Such complexity …xed
costs create interdependencies in …rms’choices of quality or service tiers across di¤erent product lines.
We have shown that these cost-driven interdependencies create interesting and novel e¤ects on the
equilibria that arise in a Stackelberg duopoly model. The Stackelberg leader always bene…ts from
positive complexity …xed costs (relative to the case with no complexity costs), but its pro…ts can be
non-monotonic in the magnitude of the complexity costs. In contrast, the Stackelberg follower’s pro…ts
can be lower when complexity …xed costs are positive and …nite, although they are (weakly) higher
when the complexity …xed cost is in…nite (i.e. when each …rm must choose a single quality tier for all
of the products it o¤ers).
There are several promising avenues for extending our analysis. First, one could investigate the
impact of complexity …xed costs on …rm pro…ts in the scenario when consumers incur switching costs
between …rms, so they prefer buying all products from the same …rm (in our analysis above, consumer
demands are independent across products). Second, one could introduce heterogeneity in product demands and investigate whether the Stackelberg leader would always o¤er the most pro…table products
or sometimes prefer to accommodate the follower by leaving some pro…table products out of its line-up
in order to reduce the follower’s incentives to compete with the leader.
References
[1] Casadesus-Masanell, R. and F. Zhu (2010) "Strategies to Fight Ad-Sponsored Rivals," Management
Science, 56 (9), 1484–1499.
13
[2] Champsaur, P. and J.-C. Rochet (1989) "Multiproduct Duopolists," Econometrica, 57 (3), 533-557.
[3] Gal-Or, E. (1983) "Quality and Quantity Competition," Bell Journal of Economics, 14 (2), 590600.
[4] Johnson, J. P. and D. P. Myatt (2003) "Multiproduct Quality Competition: Fighting Brands and
Product Line Pruning," American Economic Review, 93 (3), 748-774.
[5] Klemperer, P. (1992) "Competing Product Lines: Competing Head-to-Head May Be Less Competitive," American Economic Review, 82 (4), 740-755.
[6] Mussa, M. and S. Rosen (1978) "Monopoly and Product Quality," Journal of Economic Theory,
18 (2), 301-317.
[7] Rivkin, J. (2006) "Delta Air Lines (A): The Low-Cost Carrier Threat and Delta Air Lines (B):
The Launch of Song," Harvard Business School Teaching Note #5-706-430.
[8] Shaked A. and J. Sutton (1982) "Relaxing Price Competition Through Product Di¤erentiation,"
Review of Economic Studies, 49 (1), 3-13.
Appendix
Proof of Proposition 2
Suppose …rm 1 has chosen the H mode in stage 1 and o¤ers N1
N products. Firm 2 has two options:
choose the L mode, in which case it is optimal to o¤er all N products, leading to pro…ts
N1
L
+ (N
N1 )
L
choose the H mode, in which case it is optimal to only o¤er the N
N1 products not o¤ered by …rm 1,
leading to pro…ts
(N
N1 )
H
Comparing, …rm 2 chooses the L mode (…rst option) if and only if
N1
L
+ (N
N1 )
(N
L
N1 )
H
which can also be written:
N1
H
N
H
L
L
+
L
NH
Stepping back to stage 1, …rm 1 pro…ts are N1 H if N1
NH and N1 H if N1 > NH . We conclude that
when …rm 1 chooses the H mode, its optimal pro…ts are max fN H ; NH H g.
Suppose now …rm 1 chooses the L mode and o¤ers N1
14
N products. Again, …rm 2 has two options:
choose the H mode, in which case it is optimal to o¤er all N products, leading to pro…ts
N1
H
+ (N
N1 )
H
choose the L mode, in which case it is optimal to only o¤er the N
N1 products not o¤ered by …rm 1,
leading to pro…ts
(N
N1 )
L
so …rm 1’s pro…ts are N1 L . But these
pro…ts are always smaller than what …rm 1 can obtain by o¤ering all products in the H mode (at least N H ).
Clearly, …rm 2 always prefers the …rst option (recall
H
>
L)
We can therefore conclude that …rm 1 always chooses the H mode in equilibrium, earning pro…ts max fN H ; NH H g.
Comparing, N H
NH H if and only if H L
H
L.
H
H
Proof of Proposition 3
Firm 1 has three options: o¤er only H products, o¤er only L products or o¤er products in both modes. Let us
examine each of these options in turn.
Option 1: …rm 1 o¤ers N1 products in H mode, 0 < N1
N
Firm 2 has three options in response:
O¤er H products only, in which case it is optimal to o¤er all N
H mode. This yields pro…ts (N
N1 )
H
N1 products not o¤ered by …rm 1 in
for …rm 2 and N1 H for …rm 1.
O¤er L products only, in which case it is optimal to o¤er all products in L mode. This yields pro…ts
(N
N1 )
L
+ N1
L
for …rm 2 and N1 H for …rm 1.
O¤er products in both modes, in which case it is optimal to o¤er all N
N1 products not o¤ered by …rm 1
in H mode and the N1 products o¤ered by …rm 1 in L mode. This yields pro…ts (N
N1 )
H +N1 L
F
for …rm 2 and N1 H for …rm 1.
If …rm 2 chooses one of the last two options, then this can be part of an equilibrium for …rm 1 only if
N1 = N , resulting in pro…ts N
H.
Thus, the only way in which N1 < N can be optimal for …rm 1 is
for …rm 2 to prefer the …rst option over the other two. In this case, …rm 1 chooses the largest N1 such that
(N
N1 ) (
H
L)
N1
L
and N1 L
N1 = NH (F )
F , i.e.
F
min
L
;
N(
L)
H
L+
H
L
Thus, the best pro…ts that …rm 1 can achieve with this option (products in H mode only) are
max fN
H ; NH
(F )
Hg
Option 2: …rm 1 o¤ers N1 products in L mode, 0 < N1
Firm 2 has the following three options in response:
15
N
O¤er H products only, in which case it is optimal to o¤er all N products in H mode. This yields pro…ts
(N
N1 )
H
+ N1
for …rm 2 and N1 L for …rm 1.
H
O¤er L products only, in which case it is optimal to o¤er all N
mode. This yields pro…ts (N
N1 )
L
N1 products not o¤ered by …rm 1 in L
for …rm 2.
O¤er products in both modes, in which case it is optimal to o¤er all N
N1 products not o¤ered by …rm 1
in L mode and the N1 products o¤ered by …rm 1 in H mode. This yields pro…ts (N
N1 )
L +N1 H
F
for …rm 2.
It is immediately clear that …rm 2 always prefers the …rst option, which means that the maximum pro…t
that …rm 1 can achieve in this scenario is N L . But this is dominated by the …rst option above: o¤ering all
products in H mode guarantees pro…ts N H > N L . Therefore, this option will never be chosen by …rm 1 in
equilibrium.
Option 3: …rm 1 o¤ers N1 products in H mode and N
N1 products in L mode,
0 < N1 < N .
Indeed, it would be a dominated strategy for …rm 1 to o¤er products in both modes without covering the
entire available product range.
Firm 2 has the following three options in response:
O¤er H products only, in which case it is optimal to o¤er only the N
L mode. This yields pro…ts (N
N1 )
for …rm 2 and N1 H + (N
H
N1 products that …rm 1 o¤ers in
N1 )
L
F for …rm 1.
O¤er L products only, in which case it is optimal to o¤er all N1 products that …rm 1 o¤ers in H mode.
This yields pro…ts N1 L for …rm 2 and N1 H + (N
N1 )
L
F for …rm 1.
O¤er products in both modes, in which case it is optimal to o¤er each product in the other mode relative
to …rm 1. This yields pro…ts N1 L + (N
…rm 1.
N1 )
H
F for …rm 2 and N1
H
+ (N
N1 )
L
F for
Note that if …rm 2 chooses the third option then …rm 1’s pro…ts are dominated by what it can obtain with
the …rst option above (all products in H mode, guaranteeing N H ). Thus, it can never be optimal for …rm 1
to induce …rm 2 to choose its third option. In other words, …rm 1 must choose N1 here such that N1 L
F
or (N
N1 )
F , i.e.
H
F
N1
or N1
F
N
L
H
There are two cases to consider.
Suppose …rst F
F
F
L H
, i.e. N
. In this case, …rm 1 must not choose N1 2 F ; N
L+ H
L
H
F
in order to avoid a mixed mode response by …rm 2. If N1
then …rm 2 o¤ers only H products, so …rm 1
L
N
F
H
L
makes pro…ts N1 H + (N N1 ) L
F
conditional on N1
is then
F , which are clearly increasing in N1 . The best that …rm 1 can achieve
L
N
L
+
H
L
16
2 F
(N
N1 )
at least N
>
L
H,
F
then …rm 2 o¤ers only L products, so …rm 1 makes pro…ts N1 H +
F . If L
L
H then these pro…ts are dominated by …rm 1’s option 1 (which guarantees
H ). Thus, for this scenario to have a chance of yielding higher pro…ts for …rm 1 it must be that
F
in which case the best that …rm 1 can achieve conditional on N1
N
is
On the other hand, if N1
N
H
H
N
H
L
+
2 F
H
F
Thus, …rm 1’s optimal pro…ts when N
max N
H ; NH
(F )
L H
L+ H
H; N L
are
H
+
2 F; N
H
L
+
L
2 F
H
F
F
Consider now the other case: N
> LL+ HH , i.e. N
< FL . In this case it is never optimal for …rm 2
H
to choose the mixed mode option in response to …rm 1 choosing it. In particular, it is easily seen that …rm 2
N H
L+ H
o¤ers only H products when N1
1’s pro…ts are
(
Since N1 H + (N
N1
H
+ (N
N1 )
L
F
for N1 <
N1
H
+ (N
N1 )
L
F
for N1 >
N1 )
N H
.
L+ H
and only L products only when N1
Consequently, …rm
N H
L+ H
N H
L+ H
F is increasing in N1 , the best that …rm 1 can achieve conditional on
N ( H H + 2L )
H
is (slightly less than)
F . Conditional on N1 > N
, the best that …rm 1
N1 <
L+ H
L+ H
2
N ( L L+ H )
can achieve is either
F or N H F (depending on whether H <> L ). But N H F is
L+ H
L
N H
L+ H
dominated by …rm 1’s option 1, which guarantees at least N H .
F
Consequently, …rm 1’s optimal pro…ts when N
< L+ H are
L
(
max N
6.1
H ; NH
(F )
H;
N
+
H H
L
+
H
2
L
F;
N
H
L L
L
+
+
2
H
F
H
)
Proof of Proposition 4
Suppose …rst …rm 1 chooses the pure H mode, with N1
N products o¤ered in H mode. Then …rm 2’s best
three options in response are:
pure H mode, yielding (N
N1 )
pure L mode, yielding N1 L + (N
FH for …rm 2 and N1
H
N1 )
hybrid HL mode, yielding N1 L + (N
L
N1 )
H
FH for …rm 1
FL for …rm 2 and N1
H
FL
H
FH for …rm 1
FH for …rm 2 and N1
H
FH for …rm 1
Note that the second option yields positive pro…ts for …rm 2 (recall N L > FL ), so …rm 1 cannot foreclose
…rm 2 with a pure H strategy. For this to be an equilibrium with N1 < N , …rm 2 must prefer the …rst option
17
above, i.e. we must have
FL
N1
N(
and N1
L)
H
L
H
(FH
L+ L
FL )
,
which is equivalent to
N1
NH
N(
min
L)
H
H
FL ) FL
;
(FH
L+ L
.
L
If N1 > NH then …rm 1 is better o¤ o¤ering all N products in H mode.
Thus, the highest pro…ts that …rm 1 can obtain with a pure H strategy are max fN H ; NH H g FH .
Suppose next …rm 1 chooses the pure L mode, with N1
N products o¤ered in L mode. Then …rm 2’s
best three options in response are:
pure H mode, yielding N1 H + (N
pure L mode, yielding (N
N1 )
N1 )
FH for …rm 2 and N1
H
FL for …rm 2 and N1
L
hybrid HL mode, yielding N1 H + (N
N1 )
FL
L
FL for …rm 1
L
FL for …rm 1
L
FH for …rm 2 and N1
FL for …rm 1
L
Clearly, the third option is always dominated by the …rst option for …rm 2, so …rm 2 chooses the best among
the …rst two options. For this to be an equilibrium with N1 < N , …rm 2 must prefer the second option above,
i.e. we must have
N1 (
H
(
L ))
H
FH
FL
N(
L)
H
Given that N H > FH , it can be veri…ed that this inequality can only be satis…ed if FH
N( H
L ) > 0, in which case it is equivalent to
FH
N1
FL
N(
(
H
L)
H
L)
H
Thus, the highest pro…ts that …rm 1 can obtain with a pure L strategy are max fN L ; NL L g
NL
(
0
FH FL N (
H ( H
L)
H
L)
FL
if FH
FL
N(
H
L)
0
if FH
FL
N(
H
L)
>0
Finally, suppose …rm 1 chooses a hybrid mode, i.e. o¤ers N1 products in H mode and N
FL , where
N1 products in
L mode, where 0 < N1 < N . Firm 2’s best three response options are:
pure H, yielding (N
pure L, yielding N1 L
N1 )
H
FH for …rm 2 and N1
FL for …rm 2 and N1
hybrid HL, yielding N1 L + (N
for …rm 1
N1 )
H
H
FL
18
+ (N
H
+ (N
N1 )
N1 )
L
L
FH
FH for …rm 2 and N1
FH
FL for …rm 1
FL for …rm 1
H
+ (N
N1 )
L
FH
FL
The pro…ts obtained by …rm 1 under the third option are strictly lower than what …rm 1 can obtain with
the pure H strategy (at least N H FH ), therefore it cannot be an equilibrium for …rm 1 to choose the HL
strategy and induce …rm 2 to choose the third option above. Thus, we must have (N N1 ) H
FH or
N1 L FL , i.e.
FL
N1
or N1
FH
N
L
H
If both inequalities hold then …rm 2 cannot make positive pro…ts, so it is foreclosed. This is possible if and
only if N
FH
+
H
H
L
FH
Suppose N
N1
FL
H
+ (N
N1 )
…rm 1 is N1 =
FL
L
.
h
+ FLL . Then …rm can foreclose …rm 2 by choosing N1 2 N
FH
L
i
FL
;
, leading to pro…ts
H
L
FH
FL , which are increasing in N1 . As a result, the best foreclosure strategy for
, leading to …rm 1 foreclosure pro…ts
FL
H
+ N
L
FL
FH
L
FL
L
There are two other non-foreclosure possibilities for …rm 1:
N1 >
FL
L
FH
N
H
FL
N1 = N or N1 =
N1 < N
N1 = N
L
, which results in pro…ts N1 H + (N
.
FH
FL
H
L
FH
H
N1 )
, which results in pro…ts N1 H + (N
FH
L
N1 )
FL , maximized for either
FH
L
FL , maximized for
.
It is easily veri…ed that the foreclosure pro…ts dominate both of these possibilities, so foreclosure is the
optimal strategy for …rm 1 when it is feasible, i.e. when N
Suppose now N >
N1 <
FL
L
<N
FH
+
H
FH
H
FL
L
FH
+
FL
L
.
, which results in pro…ts N1 H +(N
H
L
2 FL
N1 )
L
FH
FL , maximized for N1 =
FL
L
FH
FL
, which results in pro…ts N1 H + (N N1 ) L FH
FL . These pro…ts are
FH
maximized either by N1 = N or by N1 = N
. In the …rst case, pro…ts are clearly dominated by
H
>
H
. Then …rm 1 has two options:
(slightly below), which yields N L +
N1 > N
FH
L
H
the pure H mode (N H
FH ), so only the second case is relevant.
Thus, when N > FH + FL , the best pro…ts that …rm 1 can achieve with the HL mode are
H
max N
L
L
+
H
2 FL
FH ; N
L
H
+
L
H
19
2 FH
FL