On Competitive Pricing and Strategic Announcement of
Future Products
William Caylor∗
November 29, 2016
Abstract
In some industries, firms reveal forthcoming improved products through (credible) announcements. In other industries, future product improvements are not announced. In
a durable goods market where sellers have private information about their future products, I show that competition with rival firms can create strategic incentives for sellers
to reveal improved future products through announcements, rather than reveal the same
information by signaling through prices. In particular, future products are announced if
the improvement is major and price competition is not too severe. Otherwise, the market
outcome is one where information about future products is signaled through prices; the
signaling distortion may lead to higher or lower prices and therefore decrease or increase
welfare, depending on the degree of product differentiation. Asymmetric information
augments the sellers’ willingness to invest in product improvement, compared to full information. (JEL D82, D83, L13, L15)
∗
Department of Economics, Southern Methodist University PO Box 750496 Dallas, TX 75275, USA.
email: [email protected]
I am indebted to Santanu Roy for many helpful conversations, and Maarten Janssen for a very fruitful suggestion.
I thank Arthur Zillante and session participants at the Southern Economic Association for helpful comments.
1
1
Introduction
In durable goods markets, buyers are likely to take into account future product improvements in making their current purchases. Often, a seller has information about the product
it will be marketing in the future, though buyers or rival firms do not generally directly
observe the future product’s characteristics. Sellers in many industries credibly reveal
improved versions of their products that they will make available in the near future. Car
and fashion shows are largely concerned with the display of such products, as are events
where electronics makers reveal forthcoming versions and make them available for testing
and rating. However, there are many durable goods markets where future products are
not directly disclosed before their arrival in the market though, arguably, similar means
of credible disclosure are available. This paper is an attempt to explain sellers’ strategic
choice to disclose future products to their buyers and rivals.
In particular, this paper will analyze the incentives to disclose for sellers with established and innovative rivals. Researchers and regulators have investigated whether product
announcements might be aimed at deterring a rival seller from entering the market or similarly improving its own product.1 While this motive might explain some announcements,
we also observe announcements by established and innovative rivals. Many markets have
several potentially innovative sellers, e.g. Intel and AMD in microprocessors, or Apple,
Samsung and Microsoft in tablet computers.
The key arguments developed in this paper take into account three important aspects
of these markets. First, even if a seller does not directly disclose future product information through announcements, such information may also be revealed to buyers indirectly
through signals such as prices. The market outcomes can differ significantly depending on
the channel of communication chosen. Second, there are often multiple established firms
in the market that, at the date of any announcement, already have a new product or products already in a development “pipeline.” Finally, these markets are often characterized
by competition between sellers whose products are significantly horizontally differentiated
(as evidenced, for instance, by customer loyalty to brands in these markets).
In particular, I analyze a two-period duopoly where sellers produce a horizontallydifferentiated durable good. Each seller is privately informed about whether it can market
an improved product in the second period.2 Firms with an improved future product have
the option to reveal that product through announcement, before setting prices for their
1
For instance, Boeing may have attempted, and failed, to pre-empt development of Airbus’ A380, by announcing plans to develop a new variant of the 747 which would match its capacity; see (Esty and Ghemawat,
2002, http://ssrn.com/abstract_id=302452). Or, announcements might help coordinate consumers and suppliers on a standard when there are network externalities or questions of compatibility; see, for instance, Dranove
and Gandal (2003).
2
See Appendix B.1 for a treatment of similar industry with no horizontal differentiation; private information
will not be communicated in that case, because private information about future products will have no effect
on current purchase decisions.
2
current output. I will assume that announcements are a credible means of disclosing
a future improved product.3 While a seller without an improved version would like to
convey its true state to buyers in order to entice them to buy now, it is difficult to find
a credible means to disclose that one will not have a better product in a future period.4
After sellers with improved products make (or do not make) their announcements, all
sellers engage in price competition; firms may be able to reveal their private information
indirectly, through the price of their current product.
The main contribution of the paper is to find a strategic motive for announcements by
sellers with established, potentially innovative rivals. Announcing allows the announcer
to avoid an outcome where distortions in price due to signaling lower market power for all
sellers, including the announcer. In this paper, I show that sellers with improved future
products will choose to announce them when the improvement in the product is major,
and the products are significantly differentiated. However, if the improvement in the good
is minor or the products are insufficiently differentiated, firms never disclose and instead
signal their private information through prices.
This analysis indicates that competition with rival sellers plays a critical role in creating
the incentives to announce a future product. A recent paper, Caylor (2016), studied the
same information problem in a monopoly setting. That paper showed that a single seller
will not announce that it will market an improved version in the future; by doing so, it
would only cannibalize its own current sales. Rather, such a seller would like to imitate
a seller without an improved future product in order to prevent buyers from waiting for
a new version. In the current paper, sellers with improved future products will also wish
to imitate those without, because consumers are willing to pay more for the current good
if they think no new version is forthcoming.5 So, a seller with a quality improvement
who imitates one without can enjoy a high price for its current output. This incentive for
imitation will induce a signaling game: a seller without a new version can avoid imitation6
by setting a price that consumers know his alter-ego would never wish to imitate.
The incentive to announce an improved future product in this paper arises from the
interdependence of sellers’ pricing decisions and the desire to soften price competition. In
an outcome where information is conveyed only through prices, a seller that will not market
an improved version in the future has to distort its current price to separate itself from
the type that has an improved version. When products are significantly differentiated,
3
For instance, an announcement might take the form of displaying a prototype, and making it available for
testing and review by third parties, or to self-selected consumers for “beta testing.”
4
Supposing that the seller can arrange for third parties verify that the seller is not currently pursuing a product improvement, internally or externally–itself not a trivial undertaking–this would also require a commitment
on the part of the seller not to improve quality soon after the verification.
5
In that case, they will enjoy the services of the unit for multiple periods, and thus have a less-elastic demand
for the current product.
6
This is beneficial, as consumers will not delay purchase, or lower their willingness to pay, if they know no
new version is coming.
3
this distortion can make the type without improved version charge a very low price; due
to competition and strategic complementarity, this distortion reduces market power for
all types and firms. In this situation, if a seller with an improved product observes a
product announcement by its rival, it will have less incentive to pretend to be a firm
without an improved product. This reduces the distortion required for a seller without
an improved good to signal its type, and allows all sellers to charge higher prices. This
creates a strategic incentive for a type with an improved future product to reveal its
private information through announcement. This effect is reinforced if the rival seller also
follows such an announcement strategy. Announcement allows the announcer to raise his
rival’s price, and thus to set a higher price in turn.
It is worth noting that while the signaling distortion is initially caused by the type of
a seller that does not have an improved product, it is announcement by the other type of
that seller that removes this distortion and increases market power for all. Indeed, in my
framework, a type that does not have an improved version and suffers directly from the
threat of being imitated has no credible means to reveal its type besides price distortion.
When the extent of future product improvement is minor, private information is revealed only through price signals. Whether, in a signaling equilibrium, prices are high or
low relative to a full-information outcome varies with the degree of horizontal differentiation. In particular, when the goods are moderately differentiated, the signaling outcome is
characterized by high prices. Though buyers are fully informed after observing prices, the
price distortion used to inform them causes a loss in welfare. But, signaling can enhance
buyer welfare for highly-differentiated goods, and even when signaling hurts welfare, it
will also augment the incentive to invest in innovation.
This paper contributes to a large literature on product pre-announcements, in which
announcements are explained as a means to deter a rival from entering, or from investing in
developing a competing product.7 The current paper investigates what motives firms have
to make an announcement when there is already a rival seller in the market. Some of these
papers, like Choi et al. (2010), try to develop a microfoundation for how announcements
can be credible in a dynamic context. In order to focus on the choice between signaling
and disclosure, we abstract from such issues and simply assume that announcement is a
credible means of disclosure.
An alternative explanation is that pre-announcement is a kind of “forward advertising,”
in which sellers contest for saliency8 and attention, in a market where consumers must
search for information about prices and characteristics. For that matter, firms might
misrepresent9 characteristics, if buyers cannot observe them or act on their experiences.
7
Bayus et al. (2001) and Haan (2003) show false announcements, or “vaporware,” might be an equilibrium
behavior. Intentionally false or optimistic announcements were at issue in United States v. IBM and one of the
behaviors initially scrutinized in United States v. Microsoft.
8
As in Haan and Moraga-Gonzalez (2011), or (Zhang et al., 2016, in press).
9
As in Rhodes and Wilson (2015).
4
The current paper is concerned with markets where – while information about future
products is private – current prices are readily known, and current products’ characteristics
readily observed. An important feature of these markets is that, even where advertising for
a new product is present, it often begins with a “launch” event, where the new product is –
often physically – presented, and its characteristics enumerated. This informs buyers and
rival sellers of the same. This paper is concerned with the incentives to thus inform them,
rather than the effects of continuing to inform or persuade them through advertising.
The existing literature on durable goods markets with product improvements has
largely ignored the implications of asymmetric information about future products. Prior
work has shown that buyer expectations of future quality improvements feed back into a
seller’s choice to improve quality,10 but have not allowed sellers to make use of private information about future products.11 One exception is Haan (2003), where a durable-goods
monopolist can use a signal to deter entry;12 differently, this paper examines a similar
information problem in a durable goods oligopoly.13 In the literature on durable goods
oligopoly, researchers have modeled the dynamics of pricing and quality improvement, but
do not attend to sellers’ private information.14
It is well established that price may function as a signal of current quality. For instance,
Daughety and Reinganum (2007) show that sellers in a differentiated duopoly can signal
high quality with a high price, and Janssen and Roy (2010) show the same in a market
for a homogeneous good. In these papers, high-quality sellers distort their price.15 In
this paper, there is no informational asymmetry about current product quality. Instead,
sellers without an improved future good distort their current prices to signal the quality
of their future products.
This paper is related to a large literature on voluntary disclosure where firms use
a credible mechanism to communicate product attributes directly. That literature also
primarily focuses on information about current products. The research question in this
paper is related to that in Daughety and Reinganum (2008) and Janssen and Roy (2014)
who study the trade-off between disclosure and signaling of current product quality. These
papers show that product disclosure by high-quality firms can be a way to avoid the
constraint of having to distinguish oneself from a low-quality type and, in Janssen and
10
See, for instance Fishman and Rob (2000), Nahm (2004), and especially Waldman (1996) on monopoly.
Deneckere and De Palma (1998) study this in an oligopoly framework.
11
Caylor (2016) examines the effect of private information in a monopoly setting.
12
Haan finds that the incumbent’s conflicting desires to deter entry and to prevent consumers from delaying
their purchases will interfere and leave only pooling outcomes, as with prior work on multi-audience signaling.
13
This paper also examines a less-drastic improvement in the good, in that the old product provides utility
even after innovation, and allows consumers to decide whether to upgrade to any new product.
14
See Goettler and Gordon (2011) on innovation in microprocessors, and Gowrisankaran and Rysman (2012)
on quality improvements in camcorders.
15
Researchers have also explored the possibility of signaling the quality of a currently available durable good
in the context of a monopoly. Utaka (2014) studies a price signal of current quality, while Gunay (2014)’s seller
delays making sales to convince consumers of the quality of its output.
5
Roy (2014), also a means to aggressively undercut rival firms without being punished by
buyers’ beliefs. In my framework, disclosure is an option only for the type that also has
an incentive to imitate, and disclosure occurs because it relaxes the incentive constraint
of the rival seller, which in turn increases market power.
In the next section I develop the basic model, where only prices are used to convey
information, and in section three find the signaling outcomes that follow under incomplete
information. Section four augments this with the option to make a credible announcement
of an improved product. The fifth section analyses the effects of asymmetric information,
and the channel of its communication, on welfare and the incentive to invest in quality
improvement. The final section discusses these results. All proofs are in the appendix.
2
The basic model
In this section I will present the basic model without an option to announce future products. As a result, sellers can only use price to convey their private information. First we
will examine a full-information version of the model, then find the incomplete-information
outcome in section 3. Later, in section 4, we will augment this model with an earlier announcement stage. The full-information outcomes found here will also be those in the
augmented model when sellers announce.
A unit mass of consumers all have the same marginal value for quality, though they
differ in their taste for the output of different sellers. There are two periods, and buyers
and sellers have a common discount factor δ between zero and one. There are two sellers,
A and B, that each sell a product of quality one in the first period. In the second period,
they may be able to sell an improved version with quality v greater than one. Both
qualities will be produced at constant marginal cost normalized to zero. Consumers are
distributed uniformly on a unit Hotelling line between the two firms, with firm A located
at zero, and B at one; we will assume that transport costs are linear.16 Then, in one
period, the utility of a buyer located at x when purchasing, at a price p, a unit of quality
s from seller A is s − p − tx, and s − p − t(1 − x) if buying from seller B. The parameter t
is the measure of horizontal differentiation in this model; if t is zero, the firms will engage
in Bertrand competition, while for very high levels of t,17 they will not interact, but sell
as local monopolists.
It is common knowledge that sellers A and B succeed at developing this new version
with respective probabilities α and β, and fail to do so with probabilities (1 − α) and
(1 − β). We will assume that the firms’ success or failure is independent. There are
four possible states of the world: F F , where A and B fail, SF , where A succeeds and
16
We have found qualitatively similar results in the case of quadratic transport costs, but present the linear
case for ease of exposition.
17
This will be made precise below in Lemma 2.
6
B fails, F S and SS. Each seller is privately informed of its own success or failure in
improving second-period quality before it engages in price competition. The firms will
set their prices for their first-period products simultaneously. Consumers observe these
prices, form beliefs about each seller’s ability to market an improved product in the
second period, and choose whether to buy the current product, and from which firm. In
the second period, each seller brings its output to market, the success or failure of each
to improve quality is publicly observed, and the firms simultaneously set a second-period
price.18 Then consumers observe these prices and choose whether to buy or not in the
second period.
This paper will focus on cases where there is moderate differentiation. In particular,
we will assume that t is not very low, so that the firms will not cover the market in the
first period, but not very high, so that the market will be fully covered in the second
period.19 The market will be fully covered when there is very low differentiation, and
t is below a critical level t. If the market is fully covered by first-period sales, there
will be no communication of private information; see Appendix B.1. When the products
are thus moderately differentiated, the firms enjoy monopoly power in the first period,
since they sell to no buyer in common with their rival. But, this monopoly power is
limited by consumer anticipation of second-period sales: they face, like Coase’s durablegoods monopolist, a problem of allocating demand between current and future periods.
This Coasian problem is mitigated by the relatively soft competition in the second-period
market.20
2.1
The full-information benchmark
We will begin with a full-information version of the model, in which the state of the world
is publicly known before first-period prices are set. In addition to a benchmark for the
signaling results found below in subsection 3, the prices and profits found here will be
those that follow on the (credible) announcement of any future improved products.
18
We will assume that firms lack the power to commit to second-period prices. If the seller had commitment
power, a similar signaling game might still ensue. One possible means of credibly disclosing a future version
is to offer a rebate to current purchasers, which could also effect a price commitment. The choice to offer a
rebate would take place against the background possibility of signaling. For instance, the seller could instead
not announce a rebate, from such consumers would infer the firm’s failure, so the seller would gain a high price
for its current good, and still induce repurchases by means of a trade-in or buyback program, instead of a rebate
(assuming consumers are at least semi-anonymous, in the sense of Fudenberg and Tirole (1998)). Then in any
equilibrium where a successful seller offers a rebate, a failed seller would need to set a price the successful seller
would not imitate.
19
See Lemma 1 below for the first-period partial-coverage condition. We treat the outcome when there is
full coverage and incomplete information in Appendix B.1. If there is partial coverage in both periods, the two
firms enjoy monopoly power over their segments of the line over the whole life of the model, and the situation
can simply be analyzed as a durable-goods monopoly.
20
This is somewhat like Gul (1987), in that competition between durable-goods oligopolists raises market
power, compared to monopoly.
7
We will find the full-information outcome in the symmetric states, and then in the
asymmetric states of SF and F S. In an asymmetric state, where one seller has an improved future good and one does not, the successful seller must choose, in the second
period, between selling the improved product at a high price to few consumers, or at a
low price to many. Which of these is best depends on the degree of differentiation, as
captured by t.
State F F
Suppose the state of the world is publicly known to be F F . Let x1A be the
location of the first-period marginal buyer from A, and likewise x1B from B. If firm A has
sold to the consumers from zero up to x1A and B to those from x1B to one, in the second
period they will compete to sell to the consumers in the uncovered segment (x1A , x1B );
see Figure 1. The buyer indifferent between buying from firms A and B will be located
at x2 = (p2B − p2A + t/2)/t. Seller A will choose p2A to maximize profit p2A (x2 − x1A )
FF
and likewise B with p2B and its profit p2B (x1B − x2 ). Let the maximized profits be π2A
FF .
and π2B
0
x1 A
x2
x1 B
1
Figure 1: Sales in state F F , first period dashed, second solid.
F F and B, p (1 − x ) + δπ F F .
Then in the first period, A has profit p1A x1A + δπ2A
1B
1B
2B
Take the consumer indifferent between buying from A and not buying, located at x1A
such that
(1)
1 − p1A − tx1A + δ(1 − tx1A ) = 0 + δ(1 − pF2AF − tx1A ),
and therefore x1A = (1 − p1A + δpF2AF )/t. A similar condition will determine the location
x1B of B’s marginal first-period buyer. Both firms will choose a first-period price to
maximize their total discounted profit.21 Let pF1 F denote this full-information price of a
failed firm, when both sellers have failed.
Note that the sellers must offer a lower price in the second period than the first to
induce the uncovered customers to buy, and all consumers anticipate this and require a
Coasian discount on the good in the first period.22 The firms’ inability to commit not to
sell to new customers in the second period, and the discipline imposed by second-period
competition, reduce the first-period prices.
21
In equilibrium, two firms of the same type will choose symmetric first-period prices, so these prices and
profits will be equal.
22
F
That is, the consumers’ maximum willingness-to-pay of 1 + δpF
2 − tx is less than the value of the product’s
services, 1 + δ − tx, because of the possibility of waiting to buy tomorrow at a low price.
8
State SS Let the publicly known state of the world be SS. In the second period, a
seller’s former customers have a homogeneous willingness-to-pay for any new version. The
price which makes a former customer indifferent to repurchasing from (say) A satisfies
v − p2 − tx = 1 − tx. Since the difference between the buyer’s ideal product and the
seller’s products are identical, either seller can induce repurchase by the full measure of
its former customers with a price less than (v − 1). The firms will again maximize secondperiod profit given their first-period sales, with the indifferent consumer having a location
x2 = (p2B − p2A + t)/2t. Then the second-period profits are p2A (x2A − x1A × Iv−1≤p2A )
and p2B (1 − (1 − x1B )Iv−1<p2A − x2 ), where Iv−1<p2j is one when v − 1 is less than p2j ,
and zero otherwise. In state SS, the two firms will split the market, and realize profits
like those in a single-shot Hotelling game; see Figure 2. The only difference will be that
the firms will price so that all former customers also repurchase. Each firm will charge a
second-period price of min{t, v − 1}, for a profit of
SS
SS
π2A
= π2B
= {t, v − 1}/2.
(2)
0
x1 A
x2
x1 B
1
Figure 2: Sales in state SS, first period dashed, second solid.
The firms and consumers know this will be the their optimal second-period behavior
and condition their first-period actions on this. Because the first-period marginal buyer
from A knows she will buy again in the second period, she values the current product only
for its services this period. Accordingly, her location x1A will satisfy
(3)
x1A = (1 − p1A )/t, and similarly,
x1B = 1 − (1 − p1B )/t.
Notice that consumers are willing to pay for only the current services of the current good,
though in state F F they were willing to pay for a portion of the second-period services
today. When consumers are not informed ex ante of the state of the world, this difference
in consumers’ willingness-to-pay for the current output of firms with and without an
improved future product will give firms with improved goods an incentive to price as
though they will not market an improved product.
Asymmetric states If instead the state of the world is asymmetric, SF or F S, the
pattern of second-period sales will depend on the level of differentiation t. To simplify
exposition, we will examine state SF , in which A can market a new version in the second
9
period, but B cannot.23 Then A can choose between selling at the margin to uncovered
consumers, or those in the rival’s segment; see Figures 3 and 4 below.
0
x1 A
x1 B
x2
1
Figure 3: Sales under regime R, first period dashed, second solid.
0
x1 A
x1 B
x2
1
Figure 4: Sales under regime U , first period dashed, second solid.
I will show that for low transport costs, the successful seller will do best by selling
at the margin to one of the rival’s former customers; we will call this regime R. In this
regime, all former customers of the successful firm repurchase, since that seller must set
a price less than (v − 1) to induce any of the rival’s former customers to buy again.24
The failed seller will make no sales in the second period. For higher transport costs, the
successful firm will do best by not selling to its own former customers, but selling to
uncovered consumers, which we will call regime U .
Regime R Under regime R, the successful seller chooses the second-period price to
maximize revenue p2A x2 , since the marginal buyer’s location satisfies v − p2A − tx2 =
1 − t(1 − x2 ). Then a successful A’s profit will be maximized by pSF
2A to give the profit
below.25
(4)
SF
FS
= π2B
= π2R =
π2A
(t + v − 1)2
8t
R
Then the total profit for the successful firm will be ΠR
A = p1A x1A (p1A ) + δπ2 . Because
the marginal buyer from A will always repurchase in the second period, the first-period
buyer’s location again satisfies equation 3, and likewise for the buyer located at x1B . The
failed seller makes no second-period sales in this case, so it chooses p1B to maximize ΠR
BF .
SF
In the first period, A will choose price p1A to maximize ΠR
AS = p1A x1A (p1A ) + δπ2A .
23
The equilibrium prices and profits here mirror those in state F S where B has succeeded and A failed.
I assume the firms cannot price discriminate against their former consumers.
25
If differentiation is higher, with t ≥ v − 1, this price will no longer induce repurchase from all consumers,
and a successful seller selling according to regime R would choose another price below (v − 1); but, the resulting
profit would be below that from regime U below, so successful firms will simply sell according to U instead.
24
10
Regime U
Under regime U in state SF , the two firms will compete over the uncovered
segment in the second period. The location x2 of the indifferent consumer satisfies v −
p2A − tx2 = 1 − p2B − t(1 − x2 ). Then x2 = (v − 1 − p2A + p2B + t)/2t and the firms
simultaneously choose prices p2A and p2B to maximize respective profits p2A (x2 − x1A )
U S and π U F .
and p2B (x1B − x2 ). Let the resulting profits be π2A
2B
US
Then A’s total profit over the two periods is ΠU
AS = p1A x1A (p1A ) + δπ2A . But in this
regime, the successful firm will set a price high enough no former customer will buy,26 so
that the indifferent buyer from A is located at
(5)
x1A = (1 + δ(p2A − (v − 1)) − p1A )/t.
Similarly, the first-period marginal buyer from B in state SF has an address x1B =
U F 27
(1 + δp2B − p1B )/t and ΠU
BF = p1B (1 − x1B ) + δπ2B .
A will choose p1A and B choose p1B to maximize their discounted profits over the two
periods. Having calculated the profits under each regime, we compare them to see when
each regime will be optimal for a successful seller in an asymmetric state. Though we
have focused on state SF where A has succeeded, the same comparison will hold for a
successful B in state F S.
2.2
The degree of differentiation and pattern of sales
See Figure 5 for a comparison of the profits of a successful seller in an asymmetric state
under regimes R and U as a function of t. Let t? be the level of differentiation at which the
profit from selling according to the two regimes is equal. For low t, selling at the margin
to a former customer of the rival, and reselling to his own former customers is best. If
instead t is above t∗ , regime U –selling at the margin to a buyer in the uncovered segment
and not to his own former consumers–is better. This is formally stated in Proposition 1.
Proposition 1. When t is less than t∗ , a successful seller in an asymmetric state under
full information will choose to sell according to regime R. If t is greater than t∗ , a
successful seller in an asymmetric state will sell according to regime U .
As differentiation increases, a successful seller selling under regime R must offer a
steeper discount to induce a rival’s former customers to switch brands and purchase the
improved good. This lowers the revenue from all of the seller’s own former customers.
Once the differentiation parameter t is high enough, this will make selling at the margin
to customers nearer the firm more attractive; the seller will trade off volume of sales for
a relatively high price it can charge on the uncovered segment; this price, and the profit
26
That is, v − p2A − tx1A ≤ 1 − tx1A ; but A chose p2A to make a buyer at x2 to the right of x1A indifferent,
so that v − p2A − tx1A > v − p2A − tx2 ≥ 0.
27
U
There will be similar expressions for ΠU
AF and ΠBS .
11
t*
Figure 5: Profits under regimes R (solid) and U (dashed), for δ = 9/10 and v = 5/2.
under regime U , are increasing in t.28
In section B.1, we derive cutoff levels of t such our assumptions about full and partial
coverage are satisfied, and comment on the outcome when t is very low. Let t be a level of
differentiation such that, for t < t, sellers under regime R will cover the whole market in
one period of sales. Similarly let t be a cutoff such that, for t > t, the sellers will not cover
the market after two periods of sales. In what follows, we will assume that t is neither
very low nor very high, i.e. t < t < t.
3
Price signaling
In this section, I will find the outcome under asymmetric information. That is, the state
of the world is not publicly observed, and firms do not have, or do not exercise, the
option to announce a future improved good. Under that assumption, I find the symmetric
separating equilibria under each of regimes R and U . The profits found here will be the
firms’ payoffs if successful firms choose not to announce in the augmented model below
in section 4. I have also found that no pooling outcome will survive refinement; see the
appendix at B.2 for details.
When psychic transport costs are not very low (t is above t), the firms will not cover the
market in the first period. When the market is not fully covered, consumers’ expectations
about the future availability of a new version will affect their first-period willingness28
The second-period competition under regime U is not unlike that in the usual Hotelling model, where high
differentiation softens competition and allows firms to set a higher price.
12
to-pay. This results in firms signaling their success or failure to develop a new version
through their price for the current version.
Consumers are willing to pay for the present and future services of the good, if they
expect no new version. Because of this, a firm with a new version would gain from
imitating the price of a firm without one. Then a seller that deceived consumers would be
paid more for its current output if its rival fails, and might also induce more repurchase
when its improved output is revealed.
The solution concept is perfect Bayesian equilibrium. As we are interested in the
revelation of future versions by present prices, we will attend to symmetric separating
equilibria, and refine them with the Intuitive Criterion of Cho and Kreps (1987).29 A
symmetric separating equilibrium will be a pair of first-period prices pS and pF , such
that the successful type of each seller sets pS and the failed sets pF ; these prices are best
responses to the expected price of the rival; and neither type has an incentive to imitate its
alter ego. Then the realized type of each seller is revealed in the first period. Consumers
observe these prices, update their beliefs and choose to buy or not accordingly.
Because the state of the world is not publicly observed, a buyer’s willingness to pay
for the good in the first period will depend on the second-period prices she expects to
face. This will depend on her beliefs about the types of each seller, and the degree of
differentiation. Consider the buyer indifferent to buying from A in the first period. This
buyer’s position will satisfy a weighted sum of the positions found above, where the weights
are the probabilities consumers assign to each state.30 Additionally, the prices she expects
to face will depend on how successful sellers will choose to sell in the second period, and
thus on the degree of differentiation. If t is high, she will anticipate the sellers will price
according to regime U , but if low, she will anticipate the second-period prices found under
regime R.
In the next subsection, we will find signaling outcomes for regime R, and in subsection
3.2 for regime U. Regime R will obtain for low t and U for high under incomplete information, as they did under full information; but because (in regime U ) a successful type
best-responds to a rival’s signaling distortion with a different price than that charged in
full information, the boundary between the two regimes will shift.
3.1
Signaling with low differentiation
Suppose that it is best for a successful seller facing an unsuccessful rival to sell at the
margin to the rival’s former customers. When this is the case, an unsuccessful seller will
29
That criterion is a two-part test, given an equilibrium and a set of out-of-equilibrium beliefs. When observing
a deviation from an equilibrium, receivers first eliminate from consideration all types of sender who could not,
even given the most favorable response to a signal, gain by so deviating; their responses will be best-responses
only to the remaining types. If, among the remaining types, there is one that will gain from deviating even if
receivers make the least-favorable such response, the equilibrium fails.
30
See equation 13 in the appendix.
13
set a higher price than a successful seller in the first period, and so signal type.
When we examined the location of the indifferent buyer under regime R above, the
anticipated future prices only affected the buyer’s willingness to pay if both firms had
failed. Under incomplete information, the same will be true, given the buyers’ assesment31
of the probability that the state of the world is F F . The indifferent first-period buyer
from A will have the following location.
x1A (p1A , α, β) = (1 − p1A + (1 − α)(1 − β)δpF2AF )/t
(6)
Similarly, x1B = 1 − (1 − p1B + (1 − α)(1 − β)δpF2BF )/t. There is extra value to buying the
good in the first period if the buyer assigns positive probability to state F F , and as long
as the premium paid is lower in expected present value than the second-period price for
the original good.
If seller A has succeeded in developing a new version, it will choose a first-period
quantity to maximize its expected profit, below.32 The second-period profits will be those
found in section 2.1, except that the first-period marginal buyers will be located at x1A
as given in equation 6, and similarly for x1B .
SS
R
ΠR
AS (p1A , α) = p1A Eβ (x1A (p1A , α)) + δβπ2A + δ(1 − β)π2
Expected first-period sales33 Eβ (x1A (p1A , α)) are βx1A (p1A , α|β̂ = 1)+(1−β)x1A (p1A , α|β̂ =
0). Likewise, if seller A will not be able to market a new version,34 it will choose firstperiod price to maximize its expected profit:
FF
ΠR
BF (p1A , α) = p1A Eβ (x1A (p1A , α)) + δ(1 − β)π2A .
We will now look for prices which will sustain separating equilibria, in which each
type of a seller sets a different price, given its expectation (in common with consumers) of
the rival’s success or failure in developing a new version of the good. Since the firms set
their first-period prices simultaneously and non-cooperatively, they will each best-respond
to the others’ expected first-period price, rather than the (interim) price each seller sets
with knowledge of its realized type. We will use the Intuitive Criterion of Cho and Kreps
(1987) to select separating equilibria with “reasonable” out-of-equilibrium beliefs.
31
We will abuse notation and not, for the moment, differentiate between prior and posterior probabilities.
We will use α for the consumers’ assessment that firm A will market an improved good, and β for the same
assessment about firm B.
32
Since the firms set prices simultaneously, expected first-period demand will be the quantities obtaining when
the rival is of each type, multiplied by the prior probabilities the rival is of each type.
33
Note that we have imposed the condition that the outcome is fully revealing, that is, β̂ is zero or one. If
instead the outcome is a pooling one, β̂ = β and expected sales are simply x1A (p1A , α|β).
34
Recall that in regime R, a seller of this type will make no second-period sales if its rival succeeds at improving
quality and therefore its second-period profit in that state is zero.
14
Without loss of generality, take seller A, and fix the levels of the prior beliefs that firms
A and B will market a new version of the good. The successful type of seller A will set a
price, pS , that is different from the failed type’s price, pF , and the types of B will act in
the same way.35 We will denote the profit of the successful type of A as ΠS (p, α(p)), and
the failed type ΠF (p, α(p)) when setting first-period price p and α(p) is the consumers’
and rival’s evaluation of the seller’s type when observing the first-period price.
pF
2.0
1.5
1.0
0.5
0.2
0.4
0.6
0.8
1.0
pS
Figure 6: The prices pe± (solid) and pb± (dashed).
The first- and second-period profits of the successful type of each firm are independent.
This means that in the first period a seller of type S is only concerned with maximizing
current revenue; it will do so with its full-information price of pS = 1/2. Then the
expected profit of a successful firm A in any separating equilibrium is the same as in full
information.36
It must be, in a separating equilibrium, that the successful type cannot gain from
imitating the failed type’s price, that is,
(7)
R
ΠR
AS (pF , α = 0) ≤ ΠAS (pS , α = 1),
35
When a type of firm A is choosing among first-period prices, it will do so under the assumption that its
rival, firm B, is choosing symmetric separating prices: pS if B has succeeded, and pF if it has failed. That is,
wherever x1B (p1B ) appears in A’s profit, A will assign the price that type of B will set in equilibrium, e.g.,
F
where we have pF
2B (x1A (p1A ), x1B (p1B )), A will evaluate x1B (p1B ) as coming from the corresponding, failed
type of B, and therefore that p1B is that type’s first-period equilibrium price, pF . Firm B will likewise anticipate
second-period prices induced by the prices that correspond to the types implied by the state.
36
Otherwise, S could deviate this price and be strictly better off, even if consumers make the least favorable
assessment of the deviation, that is comes from a seller of type S. Then any equilibrium in which S charges
another price fails.
15
â
1
0
pe−
pe+
p1A
Figure 7: Consumer beliefs α̂ that a seller charging price p will market an improved good
and likewise that the failed type of seller A cannot gain from imitating the other type’s
price:
R
ΠR
AF (pF , α = 0) ≥ ΠAF (pS , α = 1).
(8)
There will be similar equations for the two types of firm B. Inequality 7 will be satisfied
with equality by two prices, given pS . We will denote the upper of these as pe+ , and the
lower pe− , so that inequality 7 holds when pF is outside the interval (e
p− , pe+ ). Likewise,
inequality 8 will be just satisfied by p̂− and p̂+ , and strictly when p̂− < pF < p̂+ . See
Figure 6, and note that, when evaluated at pS = 1/2, p̂− = pe− and that pe+ < p̂+ .
It cannot be an equilibrium for the failed type of A to set a price in the interval
(e
p− , pe+ ),
since the successful type would gain from deviating to such a price from pS .
We will impose the following out-of-equilibrium beliefs: if consumers see a price in the
interval (e
p− , pe+ ), they will conclude it comes from the successful type of S; this is depicted
in Figure 7. These out-of-equilibrium beliefs sustain the equilibrium, because F cannot
gain by deviating to a price in (e
p− , pe+ ). We find the following.
Proposition 2. Under regime R, in any symmetric separating equilibrium satisfying the
Intuitive Criterion, the successful type of each seller will set a price of 1/2, and the failed
type of each seller will set a price of pe+ .
There is a large set of prices pF that sustain a separating equilibrium for the price pS :
{e
p− }
∪ (e
p+ , p̂+ ). By construction, prices outside (e
p− , pe+ ) are equilibrium dominated for
16
the successful type. Then a deviation to such a price will be credited to be from the failed
type. If pF is above pe+ , F can gain from lowering its price, so a price pF greater than
pe+ cannot support an intuitive equilibrium.37 Finally, in an equilibrium where F sets a
price of pe− , F can deviate to a price just above pe+ , which consumers will interpret as
coming from a seller of type F with probability near one, and the seller will gain from the
deviation. This means that only a separating equilibrium where F sets pe+ will survive
refinement by the Intuitive Criterion.
3.2
Signaling with high differentiation
When t is higher, so that the firms’ output is more differentiated, a successful firm in an
asymmetric state will sell according to regime U and ignore its own former customers in
the second period. This means that successful as well as failed firms’ second-period sales
are affected by the amount sold in the first period. This changes what kind of signal is
least costly for firms of type F , and means that successful firms’ choice of first-period
price will depend on the rival’s price. We will show that, when the probability of a rival
succeeding is low, the unique equilibrium surviving refinement is one where both types of
firm will set low prices, and when that probability is high, the unique equilibrium will be
one where both types will set high prices. For intermediate probabilities, two equilibria
will survive refinement, one where failed firms distort their price upward, and another
where their prices are distorted downward.
The marginal first-period buyer from A will have a position that makes her indifferent
between buying now and buying in the second period, given the prices she expects from
each type of each seller.38 The position of this buyer is given below.
(9)
S
UF
FF
x1A = 1 − p1A + α(1 − β)δ[pU
2A − (v − 1)] + (1 − α)βδp2A + (1 − α)(1 − β)δp2A /t
The position of the marginal first-period buyer from B will be similar.
In the second period, the success or failure of each seller is public knowledge. The
second-period prices and profits will be those found in section 2.1, with the difference that
the marginal first-period consumers will be located at x1A and x1B as defined here. Then
the interim expected profit of a successful seller A will be
SS
US
ΠU
AS (p1A , α) = p1A Eβ [x1A (p1A , α)] + δβπ2A + δ(1 − β)π2A ,
and the expected profit of an unsuccessful A will be
UF
FF
ΠU
AF (p1A , α) = p1A Eβ [x1A (p1A , α)] + δβπ2A + δ(1 − β)π2A ,
The same is true of a price below pe− ; F can gainfully deviate to a higher price.
This location satisfies a weighted sum of the positions found above under full information. See equation 13
in the appendix, and note 31 above.
37
38
17
where α(p1A ) is the rival’s and consumers’ belief that A has succeeded given the firstperiod price p1A . The expected profits for a successful and unsuccessful seller B will be
similar.
We attend to symmetric separating equilibria, where the unsuccessful types of A and
B set a different price pF from the successful price pS and are known to have failed.
The necessary conditions for this to be an equilibrium amount to a restatement, mutatis
mutandis, of inequalities 7 and 8.
(70 )
U
ΠU
AS (pF , α = 0) ≤ ΠAS (pS , α = 1)
(80 )
U
ΠU
AF (pF , α = 0) ≥ ΠAF (pS , α = 1).
There will be a pair of prices which will make a successful seller just indifferent to
imitating an unsuccessful for a given price pS . These solve equation 70 , which we will call
p− (pS ) and p+ (pS ). It cannot be an equilibrium for F to set a pF in between these prices,
because S would gain by deviating to that price. We will impose the out-of-equilibrium
belief that, for a given pS , any price in between p− and p+ must come from a seller with an
improved product with probability near one. Prices outside this interval will be believed
to come from seller of type F with probability near one. We will again use the Intuitive
Criterion to select the least-costly signaling outcome.
No price more distorted than p− or p+ will be charged in equilibrium, since, in such
an outcome, a firm of type F can deviate to a less-distorted price and gain from doing
so.39 It must also be the case that pS is the best price for a firm of type S, given that a
successful rival is also setting a price of pS and an unsuccessful chooses a price of pF .40
For a given price pF , there will be a unique best response p?S (pF ).41 In particular, p?S (pF )
is a linear, increasing function of pF ; see Figure 8. There will be two fixed points of this
system, p̌− and p̌+ , such that
p̌− = p− (p?S (p̌− )) and p̌+ = p+ (p?S (p̌+ )).
So, we have two candidate equilibria. The first is one where price is distorted down39
Suppose pF < p− , and F deviates to p0 , between pF and p− . Because p0 is outside (p− , p+ ), the deviation
will be equilibrium dominated for S, and consumers will infer the deviator is of type F . Because the deviation
is less distorted, F will be better off for deviating, and an equilibrium with a more distorted price than p− fails.
A similar argument will establish the same for pF > p+ .
40
The effect of a failed rival’s first-period price on a successful seller’s first-period price is indirect, as they
only compete in the second period. But, a low pF in the first period implies a failed rival will charge a low
price for the original good in the second period. This lowers the price a successful firm can charge for the
quality-improved good. This low future price for the improved good makes waiting to buy the new good more
attractive, and will lower consumers’ willingness to pay for the current good in the first period. The successful
seller will do best charging these consumers a lower price. Therefore, a low pF will prompt a low pS in response.
41
If a seller has succeeded and its rival has as well, the second-period profit does not depend on either’s
first-period price. If the rival has failed, the seller’s equilibrium profit will depend on the seller’s own price of
pS , and the rival’s price pF .
18
ward, in that firms of type F charge p̌− and of type S, p?S (p̌− ). The second is one where
prices are distorted upward, with pF equal to p̌+ and pS equal to p?S (p̌+ ). To see whether
these are equilibria, we must check whether F can gain by deviating in the opposite
direction, e.g., to p− (p?S (p̌+ )) in an outcome with upward-distorted prices.
pS
p- (pS )
+
p (pS )
p* S (pF )
pF
Figure 8: The minimally-distorted prices as a function of pS , and the best response p?S .
Whether F gets a higher profit from setting p− (pS ) or p+ (pS ), given the rival failed
type’s price, will depend on the probability the rival has succeeded. Suppose for ease of
exposition the priors α and β are symmetric, and equal to φ. If the rival has almost-surely
failed (φ ≈ 0) the firm loses a great deal, in terms of current sales, if it sells with a high
price of p̌+ . If instead the rival has almost-surely succeeded (φ ≈ 1), selling at a high price
and restricting current sales allows the firm to better compete against an improved good
in the second period.42 Then if φ is low enough – below a cutoff φ̌(p+ ) – in any outcome
where firms of type F set pF = p+ , a seller of type F could deviate to a price below p− ,
and gain from doing so; a similar argument will dispose of outcomes where failed firms
set p− when φ is higher than a similar cutoff φ̌(p− ).
We plot the profits from both equilibria, and the profit from deviating from each, in
Figure 9. An outcome with upward-distorted prices cannot be an equilibrium when φ is
less than φ̌(p+ ). Likewise, for an outcome where failed firms charge a low price, if it is
likely the rival has succeeded – that is, φ is greater than φ̌(p− ) – a seller of type F will
find it beneficial to set a high price instead of a low. This will be true in particular when
firms of type S are setting corresponding prices. We will define φ to be φ̌(p̌+ ) and φ to
be φ̌(p̌− ). Then we can establish the following proposition.
42
There is a large measure of consumers near the seller to whom it has not yet sold.
19
ΠF
0.7
0.6
0.5
0.4
ΠF (p-|p-)
ΠF (p+|p+)
ΠF (p-|p+)
ΠF (p+|p-)
0.3
0.2
β
β
0.1
0.0
0.2
0.4
0.6
0.8
1.0
β
Figure 9: Equilibrium and deviation profits for F as a function of the probability of the rival’s
success
Proposition 3. Under regime U , in any symmetric separating equilibrium satisfying the
intuitive criterion, firms of type F will set a price of p̌− when φ is less than φ, and a
price of p̌+ when φ is above φ, and firms of type S will set the corresponding price p?S (pF ).
When φ lies between φ and φ, it will be an equilibrium for firms of type F to set p̌− and
for S to set p?S (p̌− ), or for firms of type F to set p̌+ and S p?S (p̌+ ).
We presented this result in terms of a symmetric probability of success φ, but we
recover a general version of the proposition,43 as long as α and β are not too different.
¯ be such
Take the cutoffs α(β) and α(β), and see Figure 10. Define ∆ = |α − β|, and let ∆
that
¯ = α(α − ∆).
¯
α(α − ∆)
¯ the separating
As long as the difference between the sellers’ probabilities is less than ∆,
equilibria that survive refinement will be like those found in Proposition 3, up to the
values of α and β.
We can now say when the regimes will be optimal under incomplete information.
Proposition 1 states that full-information ΠR and ΠU are equal when t is t∗ . Recall that
under regime R a successful seller will set the same price, and gain the same profit under
incomplete information as full information. Differently, under regime U a successful seller
under incomplete information will, because of the distortion of a failed rival’s price, enjoy
43
In particular, a separating equilibrium with prices below full-information is the unique outcome is α and β
are both low, and likewise prices above full-information the unique outcome if both are high, and both outcomes
exist if the priors are intermediate.
20
1.0
ϕ
0.8
ϕ
α
0.6
α<α & β<β
α>α & β> β
0.4
0.2
ϕ
0.0
0.0
0.2
0.4
0.6
0.8
1.0
β
Figure 10: Symmetric separating equilibria for different probabilities α and β.
lower profits than under full information. Then for a degree of differentiation t∗∗ greater
than t∗ , incomplete-info ΠR and ΠU will be equal.
Corollary 3.1. When t is less than t∗∗ , a successful seller in an asymmetric state under
incomplete information will choose to sell at the margin to a former buyer of the rival
and resell to his own former customers. If t is greater than this, it will be optimal to sell
according to regime at the margin to a buyer in the uncovered segment and not to his own
former consumers.
When the rival is relatively unlikely to succeed, the least-costly signal will reduce
market power for all types of all firms, but when the rival is more likely to succeed, the
least-costly signal will be one where market power is higher than under full information.
These two outcomes will have different welfare implications, explored below in section 5,
and will also underlie sellers’ incentives to announce new versions, as we will see in the
next section.
In this section, we found the symmetric separating equilibria in prices when there is
low differentiation (t < t?? ), and high. The prices charged, and the boundary between the
low- and high-differentiation regimes, change with the information structure; see Table ??
for a summary. When differentiation is low, failed firms reveal their private information
with a price higher than under full information. When differentiation is high, failed firms
may signal their type with lower or higher prices than in full information, depending on
21
the likelihood the rival has an improved good.
4
The augmented model with announcements
In this section we will examine the choice to announce an improved future product, against
the background of signaling explored in section 3. Earlier work in this area has found two
motives for product announcements. Announcements – including false announcements of
“vaporware” – may deter a rival from developing a product. Or, an announcer might
hope to divert current demand to a subsequent period when the announcer will be the
only seller with an improved good.44 In contrast, the analysis in this paper establishes a
motive for announcements by firms in competition with current rival sellers that may also
market improved products in the future.
In particular, when competition is fierce because of low differentiation, it will not be
an equilibrium for firms who can market an improved good in the future to announce that
improved good. But, if differentiation is high and the putative improvement is major,
firms will announce their success. The option to announce will give successful firms a
chance to reduce the strategic price distortion they undertake in the high-differentiation
signaling outcome above; if their price is distorted downward they will be able to gain
from announcing, and this will break the downward-distorted signaling equilibrium we
studied above in section 3.2.
The welfare effect of seller’s choice whether to announce differs with the degree of
differentiation. For t below t? , the signaling distortion described above is hurtful to buyers,
and sellers will never exercise the option to disclose and avoid it. When differentiation
is higher, firms may disclose voluntarily, but could raise or lower prices, relative to the
signaling outcome, by doing so.
We will now allow a successful firm to announce, before first-period prices are posted
and simultaneously with a successful rival, that it will bring an improved product to
market in period two. This assumption about the timing of announcements is critical,
because it allows an announcement to affect the rival’s pricing decision.45 A firm without
an improved good cannot announce that fact.46 In an announcement equilibrium, sellers
of type S choose to announce. Each firm observes any announcement by its rival, and will
44
See, respectively Bayus et al. (2001) and Choi et al. (2010).
If announcements are made at the same time as the firms choose first-period prices, we will find the same
outcome as under signaling without announcements. The announcement will convey no information to a rival,
who will choose the same prices as in the signaling outcomes above. Then the announcer will also choose the
signaling prices, which are best responses to those price from the rival on the announcer’s information set.
46
We do not often observe an announcement that a firm will not have a new version. Such an announcement
would be in the interest of a firm without a quality-improved future good, in order to prevent consumers from
deferring their purchases. However, such an announcement would also be in the interest of a firm that did
have an improved version, as we saw in section 3. Further, making this sort announcement credible might be
very expensive: for instance, a third party might be invited to inspect all of the firm’s production and product
development facilities and verify that no new version can be made.
45
22
set the full-information price corresponding to the realized state of the world. Consumers
are informed of the state of the world by the announcements and buy as in full information.
4.1
Low differentiation: No Announcements
We have seen that, for low differentiation, where t is between t and t∗∗ , a successful firm
will enjoy the full-information profit whether or not consumers are informed before prices
are posted, and therefore makes no gain from making an announcement. Indeed, such
a firm will always have a gainful deviation from an outcome in which successful firms
announce their future products.
Suppose we are in such an outcome, but one successful firm deviates by not announcing,
and posts the price a failed firm would, given the rival’s price. Whether the rival has
succeeded or not, a successful firm’s full-information price is 1/2, and that this is also
a failed firm’s full-information price in an asymmetric state (SF or F S). Then if the
rival is successful and announces, it can set this price, make the full-information profit,
and is no worse off for the deviation. If instead the rival has failed, the deviator has
the same incentive to imitate the failed firm’s price as we found above in the signaling
outcome. In state F F , a failed firm’s full-information price is higher, as the firm wishes
to allocate demand between the first and second periods. The imitator will enjoy a higher
price, and make a higher profit, than by announcing and setting the equilibrium price.
Therefore, announcement will not be an equilibrium when differentiation is low. For the
same reasons, a successful firm would not gain by deviating from the signaling equilibrium
by announcing his future improved good, as he already enjoys the full-information profit.
If there is any positive cost of disclosure, a successful firm would strictly prefer not to
disclose.
Consumers would be better off under low differentiation if firms did announce their
future improved products,47 since the channel of revelation firms in fact choose makes
consumers and unsuccessful firms worse off, but not firms that are able to market improved
products.
4.2
High differentiation: Strategic Announcements
In this subsection I will show that firms will announce major improvements to the good
when differentiation is high, and that the option to announce can remove the downwarddistorted signaling outcome we explored above.
Unlike the case of low differentiation, it will also be an equilibrium for successful firms
to announce their future improved goods in the first period, if the quality improvement is
large enough. In such an equilibrium, the state of the world is revealed by the first-period
announcements, and all firms set the corresponding full-information prices and receive
47
See the formal welfare results in section 5.
23
their full-information profits. We must check whether a successful firm could gain by not
announcing.
If the firm does not announce, consumers and the rival will conclude that it will not
have a quality-improved version in period two. Then, we will check when the deviator
might gain from imitating either of his alter-ego’s full-information prices.48 We find the
expected profit from not announcing, and then imitating his alter-ego’s prices:
U
FF
βΠAS (pU
, α = 0|pF F ).
AF , α = 0|pBS ) + (1 − β)ΠAS (p
If this exceeds the (expected) full information profit gained by a truthful announcement,
then a successful A will have a gainful deviation, and the equilibrium breaks.
Imitating a failed type’s price will create high demand in the first period, but forces
the firm to sell a suboptimal amount in each period. When there is only a small quality
improvement, a successful and failed firms’ output are similar, and a successful firm’s
optimal prices are more like those of his alter ego. Then, the distortion in sales and profit
from setting a failed firm’s price if the rival has failed is small. But for larger quality
improvements, the firm’s distortions of first- and therefore second-period sales will be
larger and more painful. Let v̂ be the level of improved quality that sets the deviation
and announcement equilibrium profits equal in expectation. Then for any higher degree of
future quality, successful firms will announce their success in the first period. See Figure
11, and note that the level of quality for which this happens is increasing in the degree of
differentiation; as the firms’ output is more horizontally differentiated, price competition
with a failed rival is less sharp and the trade-off mentioned above will bind at a higher
degree of quality improvement. Then we can prove the following proposition.
Proposition 4. When t is greater than t∗ (v) and the quality of an improved good is at
least v̂(t), firms that have improved future products will announce them.
The option to announce will break the signaling outcome found above, if failed firms
distort their price downward.49 In that outcome, a seller of type S charged a price p?S (p̌− ),
an increasing function of the price p̌− charged by a failed rival. If such a seller deviated
from the equilibrium path by announcing it had an improved good, the rival would observe
the announcement. We have seen that a failed firm will in fact distort its price upward
when it knows its rival has succeeded, and this higher price will make the deviating firm
A better off in expectation. Then we have the following proposition.
48
Though the deviator might prefer some other price, note that this would be equilibrium-dominated for a
failed firm. Consumers and the rival would infer the deviation came from a successful firm. Then the deviation
would get the deviator less than his equilibrium profit for every price but the full-information price that seller
would set, given the rival’s announced success or un-announced failure.
49
The profit of a firm of type S in an upward-distorted equilibrium is above that gained from unilaterally
announcing, so there is nothing to be gained by announcing in an equilibrium where pF = p̌+ and pS = p?S (p̌+ ).
24
Proposition 5. If sellers have the option to announce, it will not be an equilibrium for
sellers of type F to charge p̌− and p?S (p̌− ).
Announcements allow firms with improved future versions to escape competing against
a very low price when the rival will not have an improved good. The distortion of pF
downward to p̌− also affects pS , as the successful type of each seller best-responds by
lowering its first-period price; profits of all types are lower because of signaling distortion.
Sellers can avoid this by announcing, and thus rasing their rival’s prices. The downwarddistorted outcome will not be an equilibrium when sellers have the option to announce a
future product, but the upward-distorted will remain: announcement changes the rival’s
price, but because the failed type of the rival is already distorting its price upward, the
announcement will alter pF enough to benefit the announcer.
This motive for announcement is different from those in the literature. Rather than
deterring a rival from entry or product development, we can see that, by announcing, a
seller aims to avoid a situation with low market power. Rather than informing buyers, the
announcer hopes to inform the rival, and alter the rival’s price path. For high enough new
quality and horizontal differentiation, this unilateral incentive is reinforced in equilibrium,
when successful types of both seller announce.
5
Implications for welfare and investment
In this section, we will evaluate the market outcomes found here, in terms of welfare and
consumer surplus, and then in light of the incentives to invest in quality improvement
that this market gives. On each dimension, we will compare the signaling outcomes to
a full-information benchmark, first because this highlights the effect of the information
structure on the market’s performance, and second because the full information outcome
will be that in an announcement equilibrium.50
5.1
Welfare Analysis
In this model, welfare in the first period is the consumer surplus of all first-period buyers,
plus the firms’ first-period revenue. This is the utility, net of psychic transport costs, of
all consumers who hold the good in the first period, as shown in equation 10.
(10)
W1 =
Z
x1A
(1 − tx) dx +
0
Z
1
(1 − t(1 − x)) dx
x1B
In describing second-period surplus, we must also account for the quality available
from each firm, and for whether or not first-period consumers of a firm with an improved
50
In the case of low differentiation, that outcome can only occur under incomplete information if disclosure
of future versions is mandated by a regulator.
25
version will repurchase. Let sj be the quality of firm j’s second-period output, v if j has
succeeded and 1 if j has failed. So, in regime R, where all former customers of a successful
firm repurchase, second-period welfare will be given by the equation below.
W2R =
(11)
Z
x2
(sA − tx) dx
0
Z
1
(sB − t(1 − x)) dx
x2
Under regime U , no first-period buyer purchases again, so we can define second-period
welfare as
(12)
W2U
= W1 +
Z
x2
(sA − tx) dx +
x1A
Z
x1B
(sB − t(1 − x)) dx.
x2
We evaluate these formulae at the equilibrium levels of sales under full and incomplete
information. For levels of differentiation t < t∗ , we will compare full-information and
incomplete information welfare under regime R, and for t > t∗∗ , incomplete and full
information under regime U . But in between t∗ and t∗∗ , we will compare full-information
welfare under regime U with incomplete information welfare under regime R. Proposition
6 summarizes our findings.
Proposition 6. For t between t and t∗ , welfare will be higher under full information than
incomplete. For transport costs in between t∗ and t∗∗ , welfare is higher under full information than incomplete information when the probability of success is less than φ∗ , and
lower otherwise. For t above t∗∗ , welfare will be higher under incomplete information than
full information in a downward-distorted signaling equilibrium, and lower in an upward.
A failed firm will set higher prices under incomplete than full information under regime
R, so first-period sales by a failed firm A or B, and therefore first-period welfare, will be
lower under incomplete information than full. In the other regime, if the price is distorted
downward, a failed firm will set a low price, and successful firms best-respond to the
possibility of competing with that low price by lowering their own first-period prices. So,
first-period sales are higher, and second period prices lower, under incomplete information
than full, as long as the probability of a rival’s success β is low enough to sustain the
downward-distorted outcome.
Then, in the region between t∗ and t∗∗ , the regime will change with the information
structure, so that we should compare incomplete information welfare under regimes U and
R. On the one hand, price distortions in regime U can be beneficial to consumers, and are
hurtful in regime R. But, all former customers of a successful firm receive the improved
good under regime R, while successful firms under U ignore their former customers in
asymmetric states.
Suppose for simplicity that both firms have the same probability φ of increasing quality,
so that α = β = φ. Then if φ is low and it is unlikely that firms succeed, welfare will be
higher under regime U than regime R, since a failed firm’s prices are distorted downward
26
in U and upward in R. The reverse is true when φ is high: under regime U , successful
firms set a high price on the current good, in order to preserve the market if they face a
failed rival.51 Therefore successful firms under regime U under-provide the original good
in a symmetric or asymmetric state, and welfare will be lower than under R. It follows
that for φ∗ between zero and one, welfare in the two regimes is equal.
5.2
Information and the incentive to innovate
We have so far neglected any investment in developing a new version. Now, suppose that
a firm must incur a fixed cost f in order to develop a new version. The firms can choose
to pay f and, with probability α for A and β for B, be able to sell an improved version
of quality v in the second period. A firm that does invest will then observe its success or
failure, before any announcements of second-period improvements or setting a price for
its first-period output. If a firm does not expend f , it will be unable to market the new
version and can sell only the original in both periods.
Then a firm will choose to invest when investing yields a higher expected profit, i.e.,
αΠS + (1 − α)ΠF > ΠF , or α (ΠS − ΠF ) > f. When a firm will invest for a higher fixed
cost under one information structure than another, we will say it has a higher incentive to
engage in product innovation. We will compare the profits realized under price signaling
to those realized in full information, which are also those resulting from any announcement
equilibrium.52 The profit when failing to innovate varies between price signaling and full
information, since it is the failed type of a firm that distorts its price to signal. Under
regime R, the profit when product development is successful, ΠS , is the same in the
separating equilibria we examined above and in full information.
Figure 12 shows that signaling profit ΠF (e
p+ , 0) is higher than the expected fullinformation profit, βΠF (1/2, 0) + (1 − β)ΠF (pF F , 0) when it is unlikely the rival will
succeed – when β is low – and lower when it is likely the rival will succeed. Let β ? be the
value of the prior probability of success for which they are equal.
Then we have the following proposition.
Proposition 7. Under regime R, a firm’s incentive to innovate will be higher under full
information than incomplete if the rival’s probability of success is less than β ? , and lower
otherwise.
When it is unlikely a rival succeeds, signaling allows a firm to set a high price despite
the likelihood of only competing with a failed rival. Signaling relaxes the firm’s Coasian
problem, as consumers’ beliefs would punish the firm for expanding current output. When
51
This statement is true of the downward-distorted equilibrium, and a fortiori of the upward-distorted.
We might also compare this to the efficient level of innovative effort; a social planner would be willing to
invest in quality improvement for a larger range of costs f than would the sellers in any situation here, because
the sellers only appropriate a portion of the surplus generated by the quality improvement.
52
27
a rival is more likely to succeed, signaling will restrict the firm’s sales, though there is a
good chance the firm will be unable to sell at all tomorrow. So, the profit when failing to
innovate will be lower than that expected under full information when the rival is likely
to innovate, and the firm’s willingness to pay for a chance to innovate therefore higher.
In regime U , a successful seller may compete with a failed rival for the uncovered
segment, and so lowers its first-period price in equilibrium. Both the successful and failed
profits will be below those in full information, but the profit when failing to innovate will
be more distorted, and the difference ΠS −ΠF will be higher under incomplete information
than full; see Figure 13.
That is, a firm would be willing to invest more to innovate under incomplete information than full, not only to gain profit by innovating, but to escape the punishment
of a low equilibrium profit when not innovating. In a downward-distorted equilibrium,
this is greater when the rival is likely to succeed at innovating, since a failed firm faces
low profit when his rival has a quality-improved good. Let β ∗∗ be the probability of the
rival improving quality that sets the willingness to innovate equal between the information
structures.53 Then we can prove the following result.
Proposition 8. Under regime U , a firm’s incentive to innovate will be higher in a
downward-distorted signaling outcome than full when the rival’s ex ante probability of
success is at least β ∗∗ , and lower otherwise.
In an upward-distorted equilibrium, the successful type’s profit is again less distorted
than the failed. The high price a failed rival will charge allows a successful firm to set a
high price and enjoy high first-period revenue. A firm is always willing to invest more in
an upward-distorted equilibrium than in full information.
Proposition 9. Under regime U , a seller’s incentive to innovate will be higher in an
upward-distorted signalling equilibrium than in full information.
In this regime, a failed firm competes in the second period with either type of rival,
so distorting the price hurts the firm’s profit at the margin in either state of the world.
What is more, not only a failed but a successful rival responds to a low price in kind,
intensifying the distortion. In fact, while under full information, a rival’s likely success
would discourage investment in quality improvement, under downward-distorted signaling,
a greater likelihood of the rival succeeding would prompt the firm to invest.54
Asymmetric information, and the channel by which it is revealed, affects the incentives
to invest in quality improvement. A firm’s incentive to invest will depend on the probability of the rival’s success if investing; we are likely to get reciprocal investment in this
setting, since the firm’s incentive to invest is large just when his rival is likely to invest
and succeed at improving quality.
Numerical comparisons show that β ∗∗ lies between the cutoffs β and β; that is, whenever both signaling
equilibria are possible, the incentive to innovate under either is stronger than under full information.
54
see Lemmas 11 and 12 in the Appendix.
53
28
6
Discussion
I have shown that firms have an incentive to announce a future product even when competing with an established, potentially innovative rival. A monopoly seller has no incentive
to announce an improved good, but firms with improved goods announce them if the improvement is large and their output is horizontally differentiated. Whether they announce
also depends on the degree of horizontal differentiation between the firms. If there is sufficient differentiation, firms will announce future improved products. If there is only low
differentiation, firms will use price signals to reveal whether they will market an improved
version in the future. In many durable-goods markets, we observe oligopolists with brand
power making announcements of future goods: electronics makers, software and video
game developers and car manufacturers regularly announce their future products well in
advance. In other markets, there is less brand loyalty: for instance, data storage devices
or home appliances from different manufacturers are good substitutes. In these markets,
we do not observe future product announcements. The results in this paper suggest this
is because of the greater substitutability of these latter manufacturers’ output.55
The welfare resulting from announcement or signaling also depends on the degree of
differentiation. If differentiation is low, firms without improved goods will convey their
true state by distorting price upward, and firms with improved goods will not voluntarily
disclose. In this case, asymmetric information is harmful to buyers.56 If differentiation
is higher, firms will voluntarily announce a large enough improvement; but the signaling
outcome in the absence of an option to announce could benefit consumers, like the signaling
in Caylor (2016). In highly-differentiated markets, a seller’s ability to make credible
announcements could be welfare-reducing.
In both cases, the sellers’ willingness to invest in improving quality will be higher under
signaling than full information, if the rival is likely to succeed. This means that, though
asymmetric information may harm buyers in a static sense, there is a trade-off between
current losses in consumer welfare and the dynamic gains from more innovation.
In this paper, I have imposed a simple product development process. I used this to
investigate the effect of different channels of revelation on firms’ incentives to invest in
quality improvement. This earlier stage of the game might be enriched in several ways.
We have examined a discrete quality improvement which is the same across firms. Instead,
firms might also realize a partial success in raising quality, or might realize a range of new
qualities with some probability. This more general framework would require stronger conditions on consumer beliefs, but might also capture another feature of uncertain product
55
It is difficult to say whether we also observe price signals, as we generally cannot observe firms’ private
information or consumer beliefs, or disentangle any distortions in price from differences in cost or current
quality.
56
One might suggest mandatory disclosure of future products to remove the asymmetry; it is difficult to
imagine such a policy being implementable, given both the risky nature of product development, and firm’s
desire to conceal information from competitors.
29
development processes. The degree of improvement could be made asymmetric, or could
become a strategy variable for the firms. A more general model might also endogenize the
credibility of announcements, perhaps by introducing a third-party product reviewer as
a strategic agent. Finally, we might allow consumers’ marginal value for quality to vary,
opening up the possibility of used-good markets, or the possibility of sellers using rebate
or trade-in policies.
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A
Appendix
Proofs of all propositions are in this appendix, as is the full form of the position of the
indifferent first-period buyer from firm A in the case of incomplete information, given
here. The left-hand side is expected utility from purchasing form A in the first period,
31
and the right-hand side that from delaying purchase. We will abuse notation and write α
for consumers’ and firm B’s assessment that A has succeeded, and likewise β for B.
SS
(13) 1 − p1A − tx + αβδ max{v − pSS
2A − tx, 1 − tx, v − p2B − t(1 − x)}
SF
+ α(1 − β)δ max{v − pSF
2A − tx, 1 − tx, 1 − p2B − t(1 − x)}
+ (1 − α)βδ max{1 − pF2AS − tx, 1 − tx, v − pF2BS − t(1 − x)}
+ (1 − α)(1 − β)δ max{1 − pF2AF − tx, 1 − tx, 1 − pF2BF − t(1 − x)}
=
SS
0 + αβδ max{v − pSS
2A − t(1 − x), 0, v − p2B − t(1 − x)}
SF
+ α(1 − β)δ max{v − pSF
2A − tx, 0, 1 − p2B − t(1 − x)}
+ (1 − α)βδ max{1 − pF2AS − tx, 0, v − pF2BS − t(1 − x)}
+ (1 − α)(1 − β)δ max{1 − pF2AF − tx, 0, 1 − pF2BF − t(1 − x)}
A.1
Cutoff levels of differentiation
In this section, we derive cutoff levels of t such our assumptions about full and partial
coverage are satisfied. Let t be a level of differentiation such that, for t < t, sellers under
regime R will cover the whole market in one period of sales. Similarly let t be a cutoff
such that, for t > t, the sellers will not cover the market after two periods of sales.
As we are interested in information transmission by firms that face a rival, the main
text is focused on differentiation that is neither very low, nor very high, i.e. t < t < t.
We show below in section B.1 that, if t < t, there will be no communication of private
information. When t > t, there may be communication through signaling, as in Caylor
(2016), but the firms will be monopolists of disjoint segments of our Hotelling city. We
will refer, hereafter, to levels of t such that regime R is best as low differentiation, and
those for which a firm will prefer regime U as high differentiation.
When t is below t? , sellers sell according to regime R. It is sufficient for full coverage
that the consumer A would choose to make indifferent under partial coverage, located at
x1A , be identical to same place B’s indifferent consumer, located at x1B . Denote by t the
level of t which makes this true. The market will be fully covered when t is less than t,
given below.
Lemma 1. Let t be one when at least one firm succeeds at improving quality, and 3/(3+2δ)
if none can market an improved quality. The market will be fully covered after first-period
sales when t is less than or equal to t, and partially covered otherwise.
Proof. When t is below t? , sellers sell according to regime R. When regime R is optimal
for a successful seller in a asymmetric state, the optimized first-period quantities can be
32
given as
(1 − α)(1 − β)δt(2x1B + 1) + 3
,
2t(4(α − 1)(β − 1)δ + 3)
(3α + 1)(β − 1)δt(2x1B + 1) + 9
=
,
2t(4(3α − 1)(β − 1)δ + 9)
t((1 − α)(1 − β)δ(2x1A + 5) + 6) − 3
=
, and
2t(4(1 − α)(1 − β)δ + 3)
t((α − 1)δ(15β + (6β + 2)x1A − 11) + 18) − 9
=
.
2t(4(α − 1)(3β − 1)δ + 9)
x1AS =
x1AF
x1BS
x1BF
We defined t as the level of t which satisfies x1A = x1B . There are four cases, corresponding
to the four states. A bit of manipulation will show that in each one, x1A = x1B when
t=
3
.
2αβδ − 2αδ − 2βδ + 2δ + 3
In particular, if at least one firm has succeeded and is known to have done so, either α̂ or
β̂ will be one, and t will be one. If instead both have failed and in equilibrium are known
to have failed, α̂ = β̂ = 0 and t will be 3/(3 + 2δ).
As regime U will be optimal for a successful firm for high t, we can also establish an
upper bound t in each state such that for t less than t, the market will be fully covered
after second period sales in that state, but will be only partially covered when t is higher.
Let sA and sB denote each firm’s second-period quality; e.g. in state F S, sA is one and
sB is v.
Lemma 2. The market will be fully covered after second-period sales when t is less than
or equal to t = (sA + sB )/(1 − x1A + 1 − (1 − x1B )), and partially covered otherwise.
Proof. We know that when transport cost t is high, the firms will sell according to regime
U ; suppose that full coverage fails in the second period. Let the highest quality firm
j can market in the second period be sj . If in this regime full coverage fails in the
second period, the marginal second period buyer from A will have location x2A such that
sA − p2A − tx2A = 0 and therefore x2A = (sA − p2A )/t. Firm A’s second-period profit will
be π2A = p2A (x2A − x1A ), which will be maximized when p2A is 1/2(sA − tx1A ). Then
A will maximize its profit when selling to a consumer located at x2A = (sA + tx1A )/(2t).
Similarly, B’s profit-maximizing marginal buyer will be at x2B = (−sB + t + tx1B )/(2t).
It will be sufficient for full coverage that the firms sell to at least one common consumer.
This means that x1A ≥ x1B , or, (sA + tx1A )/(2t) ≥ (−sB + t + tx1B)/(2t). This is true
when t ≤
sA +sB
1−x1A +x1B .
The market will be fully covered if both succeed when t is less
than or equal to 2v/(1 − x1A + x1B ), fully-covered when exactly one firm succeeds if
t ≤ (v + 1)/(1 − x1A + x1B ), and fully covered in the second period in state F F when t is
no more than 2/(1 − x1A + x1B ), where in each case x1A and x1B take their equilibrium
33
values in that state.
A.2
Proof of Proposition 1
When t is less than t∗ , a successful seller in an asymmetric state under full information
will choose to sell according to regime R. If t is greater than t∗ , a successful seller in an
asymmetric state will sell according to regime U .
Proof. We can easily check that a successful firm’s profit under regime R is decreasing in
t. Its derivative will be
−
(v − 1 − t)(v − 1 + t)
1
−δ
2
4t
8t2
which is always negative when t < v − 1; if instead t is higher, we will be under regime U
instead.
The full-information profit under regime U is increasing in t, as long as the discount
factor δ is not very low. Its derivative is
χ2 t 2 + χ1 t + χ0
4t(3 + 2δ)2 (9 + 26δ + 23δ 2 + 6δ 3 )2
(14)
where
χ2 = δ(9 + 17δ + 6δ 2 )2 (18 + 65δ + 84δ 2 + 44δ 3 + 8δ 4 )
χ1 = 2δ(81+405δ+782δ 2 +716δ 3 +304δ 4 +48δ 5 )(−15−33δ−21δ 2 −4δ 3 +v(6+16δ+15δ 2 +4δ 3 ))
χ0 = 729 + 162(45 − 14v + v 2 )δ + 9(3423 − 1906v + 224v 2 )δ 2 + 6(12119 − 9331v + 1555v 2 )δ 3
+(105949−102714v+22397v 2 )δ 4 +2(49471−57754v+15671v 2 )δ 5 +4(14819−20310v+6631v 2 )δ 6
+ 8(2751 − 4346v + 1667v 2 )δ 7 + 192(24 − 43v + 19v 2 )δ 8 + 416(−1 + v)2 δ 9 .
Since the denominator of 14 is positive, and the leading coefficient of the numerator is as
well, this will be an increasing function when t is above the roots of χ2 v 2 + χ1 v + χ0 . It is
straightforward to check that the roots of this polynomial are negative when 0 < δ ≤ 1 < v
as we have assumed. Then successful firm’s profit in an asymmetric state will be increasing
in the transport cost t.
Let t∗ be the value of t that sets them equal. Then for higher values of t, the profit
under regime U will be strictly better, and for lower selling according to regime R will be
better.
A.3
Proof of Proposition 2
Under regime R, in any symmetric separating equilibrium satisfying the Intuitive Criterion, the successful type of each firm will set a price of 1/2, and the failed type of each
34
firm will set a price of pe+
F.
Proof. The derivative of ΠAF (p, α̂ = 0) with respect to p is
2
− (β − 1)δt(4x1A (p1 ) − 2x1B (p1 ) − 1)
9
2dx1A dx1B
−
dp1
dp1
+ x1A (p1 ) + p1
dx1A
.
dp1
This will be positive for p less than ΠAF ’s maximizer p1AF , and negative otherwise. In
particular, it is decreasing for pe+ and any higher price. In any separating equilibrium
where F charges p greater than pe+ , F could deviate to a lower price p0 just above pe+ .
Since this price is equilibrium dominated for S, consumers will believe this deviation comes
from a firm of type S with probability near zero, and F will have a profit of ΠF (p0 , 0) which
is near ΠF (e
p+ , 0), itself above ΠF (p, 0); since F has a gainful deviation, this equilibrium
fails the Intuitive Criterion.
p− , 0) is less than
Finally, we use Lemma 3 to show that, for t in between t and t∗ , ΠF (e
ΠF (e
p+ , 0). In a separating equilibrium where F sets a price of pe− , F can deviate to a price
just above pe+ . Since this is equilibrium dominated for S, F ’s deviation will be perceived
as coming from a firm of type F , and F will gain. Since F has a beneficial deviation, this
equilibrium fails the Intuitive Criterion. Then in any equilibrium satisfying the intuitive
criterion, the failed type of each firm must set a price of pe+ . Lemma 4 assures us that
this equilibrium price is also incentive-compatible for failed firms.
In any equilibrium where the firms of type S set a price p besides 1/2, a successful
firm could deviate to a price of 1/2. Since, per Lemma 4, a firm of type F cannot gain
from setting this price, consumers will believe this comes from an innovative firm. When
it is believed to be of type S, a successful firm has a profit that is uniquely maximized by
pS = 1/2, so the deviation is beneficial. Only an equilibrium where successful firms set
pS = 1/2 will survive refinement by the intuitive criterion
Lemma 3. When t is between t and t∗ , F ’s profit from setting a price of pe− is less than
that from setting pe+ .
Proof. We can compare the profits from setting each of these prices; we will write A + B
for pe+ , and A − B for pe− . The difference, ΠF (e
p− , 0) − ΠF (e
p+ , 0) will be given by
(15)
2B 2A − 2(β − 1)2 δ 2 t + (β − 1)δ(3t − 2) − 1
,
t(2(1 − β)δ − 1)2
where the rival succeeds with probability β and A and B satisfy the following.
A = 1/2(1 + tδ − tβδ),
B=
1p
(β − 1)δ ((β − 1)δt2 − 2t + 2)
2
We can see that the denominator of 15 is positive, and therefore we must show that the
numerator is negative. As B is also positive, we examine the rest of the numerator, which
we can write as −2(1 − β)δ(t((1 − β)δ + 1) − 1), a negative quantity.
35
Lemma 4. The equilibrium profit for F , ΠF (e
p+ , 0), is greater than the profit F can have
when imitating S.
Proof. Recall that we defined p̂− and p̂+ as the prices which, if a firm of type F set
them in equilibrium, F would be indifferent to deviating to S’s price of 1/2. For prices in
between p̂− and p̂+ , ΠF (p, 0) ≥ ΠF (pS , 1). Since in equilibrium F will set pe+ below p̂+ ,
ΠF (e
p+ , 0) > ΠF (pS , 1).
A.4
Proof of Proposition 3
Under regime U , in any symmetric separating equilibrium satisfying the intuitive criterion,
firms of type F will set a price of p̌− when β is less than β, and a price of p̌+ when β is
above β, and firms of type S will set the corresponding price p?S (pF ). When β lies between
β and β, it will be an equilibrium for firms of type F to set p̌− and for S to set p?S (p̌− ),
or for firms of type F to set p̌+ and S p?S (p̌+ ).
Proof. Fix an equilibrium price for the successful types pS , and recall that any price above
p+ (pS ) or below p− (pS ) will prevent the successful type of a firm from imitating the failed.
Note that a failed firm’s profit given an equilibrium price pF , ΠF (p, 0|pF ), is increasing
in price below its maximizer p0F and decreasing in price above p0F , which lies between
p− (pS ) and p+ (pS ). In any separating equilibrium in which the failed types set a price p
below p− (pS ), a failed firm can deviate beneficially to a price p0 = p− (pS ) − ε, where ε is
positive and smaller than p− (pS ) − p. Since this is below p− (pS ), this price is equilibrium
dominated for a successful firm, and consumers will believe it comes from a failed firm
with probability near one. Since p0 is greater than p, this deviation will be beneficial.
Then such an equilibrium fails the intuitive criterion. A similar argument will dispose of
equilibria where the failed types set prices above p+ (pS ).
We use Lemma 5 to show that, in an equilibrium where the failed types set p̌+ (pS ),
the failed types have a beneficial deviation if β is below β̌(p+ ), and also if β is higher than
β̌(p− ) and they set a price of p̌− (pS ). We have eliminated upward-distorted signaling
outcomes when β is low, and downward-distorted when β is high.
We must show that when β is intermediate, both of these exist. So, if β is below β̌(p− ),
there is no beneficial deviation for the failed types when they set a price of p− (pS ): prices
more extreme than p− and p+ will make the firm worse off, and by construction so will p+ .
Similarly, if β is higher than β̌(p− ), there is none if they set a price of p̌+ (pS ): deviation
prices more extreme than p− (pS ) and p+ (pS ) are ruled out, and because β is high enough,
deviating to p− (pS ) will make the firm worse off. That is, when β is in between β(pS )
and β(pS ), there is no beneficial deviation from either and upward- or downward-distorted
signaling outcome for firms of type F . This will be true in particular for these cutoffs
evaluated at the equilibrium prices for firms of type S, that is, β defined as β̌(p̌+ ) and β,
defined as β̌(p̌− ).
36
Lemma 7 shows us a failed firm cannot gain by imitating a firm that has succeeded,
as long as transport costs are lower than t̂.
It remains to show that the successful types will set, prices of p?S (p̌− ) and p?S (p̌+ ). Our
out-of-equilibrium beliefs ensure that a firm with an improved future good cannot gain
by setting a price at which consumers would believe it will not have an improved version.
Among the prices for which this firm will be believed to have succeeded, p?S (pF ) is the
unique maximizer of a successful firm’s profit for a given price pF from the rival; the firm
cannot gain from setting some other price.
Lemma 5. For a given price pF from a failed rival, the profit from setting p− (pS ),
ΠF (p− (pS ), 0|pF ) is greater than ΠF (p+ (pS ), 0|pF ), that from setting p+ , when β is less
than β̌, and lower otherwise.
Proof. It will be convenient to write the failed types’ profit, which is quadratic in their
own price, in ‘vertex form,’ as below:
ΠF (p|pF ) = Π0F − λF0 (p0F − p)2
where Π0F is the maximum profit given that consumers evaluate the firm as having failed,
and p0F is the price which would give us this profit. The coefficient λF0 is given by
−8(β − 2)δ 5 + (128 − 68β)δ 4 − 172(β − 2)δ 3 + (400 − 167β)δ 2 + (201 − 53β)δ + 36
(δ + 2)2 (4δ 2 + 8δ + 3)2 t
.
Fix a successful type’s price pS and suppose that
ΠF (p− (pS ), 0|p− (pS )) ≤ ΠF (p+ (pS ), 0|p+ (pS )),
that is, Π0F − λF0 (p0F − p+ )2 ≤ Π0F − λF0 (p0F − p+ )2 and therefore (p0F − p− )2 ≥ (p0F − p+ )2
and, since this is a difference of squares p0F − p− + p0F − p+ (p+ − p− ) ≥ 0. Note that
p+ is greater than p− by construction, and these two prices are equidistant from p0S , the
maximizer of a successful firm’s profit when consumers believe it has failed, and given the
equilibrium prices of both types of its rival. Then (p+ − p− ) is positive, and the term in
square brackets is 2(p0F − p0S ). We know from Lemma 6 that this is negative when β is
below β̌(pS , pF ). Therefore it must be that ΠF (p− (pS ), 0|p+ (pS )) > ΠF (p+ (pS ), 0|p+ (pS ))
when β is less than β̌(pS , p+ ), and otherwise lower. Likewise, ΠF (p− (pS ), 0|p− (pS )) >
ΠF (p+ (pS ), 0|p− (pS )) when β is less than β̌(pS , p− ), and otherwise lower.
Lemma 6. The failed type’s best interim price, p0F , is less than the successful type’s best
shirking price p0S , when β is less than β̌, and higher otherwise.
Proof. Fix a price pS for successful firms, and a corresponding price pF . We find the
prices p0S and p0F from the first-order conditions of each type’s profit, given that consumers
37
believe the firm to be of type F . Without loss of generality, we will consider firm A, so
the probability of the rival being type S is β.
Then the difference p0S − p0F will be given by the equation below; the numerator will
have one root where 0 < β < 1, which we will call β̌.
γ2 β 2 + γ1 β + γ0
,
6(2δ + 3) (8δ 3 + 28δ 2 + 28δ + 9) (6βδ 2 + 5βδ + 2δ 2 + 5δ + 3)
where
γ2 = 3δ 2 48δ 4 + 256δ 3 + 480δ 2 + 376δ + 105 (2pF − 2pS + v − 1)
γ1 = −δ −12 32δ 4 + 164δ 3 + 296δ 2 + 227δ + 63 pF
+ 6 16δ 5 + 144δ 4 + 448δ 3 + 624δ 2 + 403δ + 99 pS
+ (2δ + 3) 3 48δ 5 + 288δ 4 + 624δ 3 + 600δ 2 + 253δ + 36 t + 64δ 5 (v − 1)
+ 8δ 4 (43v − 85) + 4δ 3 (163v − 463) + 2δ 2 (257v − 965) + δ(145v − 781) − 81
γ0 = δ −6 48δ 5 + 320δ 4 + 808δ 3 + 968δ 2 + 559δ + 126 pF
+ 6 32δ 5 + 208δ 4 + 504δ 3 + 568δ 2 + 304δ + 63 pS
+ (2δ + 3) 16δ 5 (3t − 4) + 16δ 4 (18t − 23) + 8δ 3 (75t − 92) + 8δ 2 (63t − 71)
+ 7δ(21t − 16) + 4 16δ 5 + 80δ 4 + 148δ 3 + 130δ 2 + 55δ + 9 v + 27
and the rival’s types are charging prices pS and pF . Seeing that the denominator and
the leading coefficient of the numerator are positive,57 we know the difference will be
positive when β is below its first root, and negative when higher. We call that first root
β̌.
It is useful for out purposes to note that this root, β̌(pS , pF ), will be decreasing in each
of pS and pF , so that β̌(pS , p+ ) will be below β̌(pS , p− ), and in particular β will be less
than β.
Lemma 7. When transport costs t are below t̂, a failed firm cannot do better by imitating
a successful firm’s price than in equilibrium.
Proof. Let ΠF (p, 1|pF , pS ) be a filed firm’s interim expected profit given the equilibrium
prices of the types of the rival, when setting p and consumers evaluate the firm to be of
type S with probability one, and let p1F (pF , pS ) be its maximizer. If this maximum shirking
profit exceeds F ’s equilibrium profit, and there is a price F can set and be believed to be
innovative, the equilibrium will fail.
Let t̂ be the level of differentiation that makes this true. This maximum shirking profit
will be the same as F ’s equilibrium profit of ΠF (p̌− , 0), that is, t̂ satisfies
ΠF (p1F , 1|p̌− , p?S (p̌− )) = ΠF (p̌− , 0|p̌− , p?S (p̌− )).
57
−
p
This is clear when pF is distorted upward, above pS . When pF is distorted downward, recall that our prices
and p+ are centered on p0S > p1S , and that pS will be p1S in equilibrium.
38
For a range of parameter values, this t̂ will be below tSF , and there will be no symmetric
separating equilibrium; when transport costs are this high, two failed firms will not cover
the whole market with their second-period sales, but instead act as local monopolists.
This monopoly power means a failed firm facing a failed rival can gain from setting a
higher price than p̌− , even though consumers’ out-of-equilibrium beliefs – that such a
deviation comes from a seller with an improved version – will mean it makes somewhat
lower first-period sales.
But, when t is lower, it is impossible for a firm of type F to gain form such an action,
and consumers’ out-of-equilibrium beliefs sustain the signaling equilibrium.
A.5
Proof of Proposition 4
When t is greater than t∗ and the quality of an improved good is at least v̂(t), firms that
have improved future products will announce them.
Proof. The alternative is for a firm with an improved future product not to announce,
and subsequently imitate one of the full-information prices of a failed firm. (See note 48
above on the possibility of imitating other prices.)
The expected profit from doing so will be
βΠS (pFF SU , α̂ = 0|pB = pFBSSU ) + (1 − β)ΠS (pF F , α̂ = 0|pB = pF F )
which we can compare the firm’s expected profit under full information,
βΠSS + (1 − β)ΠSF
We subtract them, shirking profit less full-information. The result will be given by
(16)
ξ2 v 2 + ξ1 v + ξ0
36(δ + 1)2 (2δ + 3)2 (6δ 2 + 17δ + 9)2 t
where the coefficient on v 2 is a negative quantity:
ξ2 =δ 2 +3456δ 7 + 29760δ 6 + 106024δ 5 + 203132δ 4 + 226924δ 3 + 148197δ 2 + 52470δ + 7776
−βδ 2 3456δ 7 + 29616δ 6 + 105880δ 5 + 204032δ 4 + 227860δ 3 + 147081δ 2 + 50742δ + 7209 .
The denominator of 16 is positive, and the numerator will be negative when v is greater
than its second root, which we have called v̂. Then for lower v, a firm with an improved
future version will gain from not announcing and imitating its failed alter ego, but for
higher future qualities, firms will announce their forthcoming products.
39
A.5.1
Proof of Proposition 5
Proof. Suppose we are in that outcome, and firm A has succeeded at improving the quality
of its second-period output. If A deviates by announcing this before first-period prices are
set, firm B will be informed of the type of its rival. As stated in Proposition 3, a failed
firm will in fact distort its price upward when it knows its rival has succeeded, and this
higher price will make the deviating firm A better off in expectation.
In particular, consider the incentive compatibility constraint for the successful type
of B when A is known to have succeeded, so the state of the world is SS. Accordingly,
a successful B will set a price of 1/2, the full-information price in this state, and enjoy
the full-information profit, which is better than that in a downward-distorted signaling
equilibrium.
ΠS (1/2, β = 1) ≥ ΠS (pF , β = 0)
This will be satisfied with equality for two prices above and below 1/2, but it will be least
costly for a failed B to set a high price.58 Knowing this, firm A will charge a higher price
than in equilibrium as a best response, and will benefit from this smaller distortion by a
failed rival.
A.6
Proof of Proposition 6
For t between t and t∗ , welfare will be higher under full information than incomplete.
For transport costs in between t∗ and t∗∗ , welfare is higher under full information than
incomplete information when the probability of success is less than φ∗ , and lower otherwise.
For t above t∗∗ , welfare will be higher under incomplete information than full information
in a downward-distorted signaling equilibrium, and lower in an upward.
Proof. To see that welfare is higher under full than incomplete information for t ∈ (t, t∗ ),
recall that the equilibrium price for F will be pe+ , above that in full information, and that
in full or incomplete information, S will charge pS = 1/2. Formally, W1 will be decreasing
in pF , and pF is higher under incomplete information than full. Then in every state but
SS, consumers under incomplete information face at least one first-period price that is
higher than in full information, and fewer buy.
For intermediate t, the comparison is between full-information welfare under R and
incomplete information welfare under U . See Lemma 8 below.
When t higher than t∗∗ , we can compare the prices consumers face in each state,
recalling that p̌− and p?S (p̌− ) are lower than what a firm of that type would charge under
full information, and p̌+ and p?S (p̌+ ) are higher.
58
Recall that a successful B has no need to preserve the market in case he faces a failed rival, and a failed B
would like to preserve some demand for second-period competition.
40
We will first treat the case of a downward-distorted signaling outcome. In state SS,
consumers will see prices of p?S (p̌− ) and buy more in the first period than they would in
full information, and face lower second-period prices for the improved good as well. In
state F F , they will likewise buy more in the first period at a low price of p̌− , and face
lower second-period prices for the original good. In states SF (with a similar expression
for F S) welfare as a function of the type’s prices pS and pF will be
1
δ(t + v − 1)x2 (pS , pF ) − δtx22 (pS , pF ) − tx21A (pS , pF ) − δvx1A (pS , pF )
2
1 2 2
+ δx1A (pS , pF ) + x1A (pS , pF ) − δ tx1B (pS , pF ) + δ 2 (t − 1)x1B (pS , PF )
2
δ 2 t δt
+ δ2 + δ −
− ,
2
2
which is decreasing in pS and in pF on an interval including our incomplete- and fullinformation prices. Consumers face prices of p?S (p̌− ) and p̌− , which are above each above
the full-information prices for each type. Recall, too that p?S (pF ) is a linear, increasing
function of pF . Then, in an asymmetric state,
tween the full-information pF and
p̌− ,
dW
dpF
=
?
∂W dpS
∂p?S dpF
+
∂W
∂pF
is negative in be-
and therefore welfare is higher under incomplete
information with a downward-distorted signaling equilibrium, than under full information.
A similar argument will show that welfare is lower in an upward-distorted signaling
equilibrium than in full information.
Lemma 8. For transport costs in between t∗ and t∗∗ , welfare is higher under full information in regime U than incomplete information in regime R when the probability of success
is less than φ∗ , and lower otherwise.
Proof. Let the firm’s probability of success be symmetric, so that α = β = φ. We can
see with Lemma 9 that welfare under U is higher than under R when φ is low, and with
Lemma 10 that the reverse is true when φ is high. Then, for some φ∗ in between zero
and one, expected welfare under the two regimes are equal, and expected welfare under
U is higher when the probability is below φ∗ , and lower than expected welfare under R
otherwise.
Lemma 9. When the probability of each firm improving quality is zero, full information
welfare under regime U will exceed incomplete information welfare under regime R.
Proof. Incomplete-information welfare under regime R will be
p
δ 2 −5t2 + 20t + 6 − δ t2 − 11 − 4δ 3 (t − 4)t + 3 4δt − (δ(4 − t) + 1) δ (δt2 + 2t − 2)
WR (0) =
+
4(2δ + 1)2 t
4(2δ + 1)2 t
when φ is zero. Likewise, welfare under regime U will be
δ 4 −4t2 + 20t + 39 − 8δ 3 4t2 − 20t − 23 − 8δ 2 8t2 − 42t − 19 + 16δ(4t + 11) + 48
WU (0) =
16(δ + 4)(2δ + 1)2 t
41
Assume to the contrary that WR (0) > WU (0). This simplifies to the requirement that
− 4(4 + δ)(δ(4 − t) + 1)
p
δ (δt2 + 2t − 2) >
δ 39δ 3 + 160δ 2 + 12δ + 4 3δ 3 + 13δ 2 + 5δ + 4 t2 − 44(4 + δ)δ 2 t − 12 .
The left-hand side of this inequality is negative for t ≤ t < 4, but we can see that the
right hand side is positive, a contradiction.
The right hand side is quadratic in t, and is positive with no real roots as long as δ is
not very low, in particular, as long as δ is at least 0.247311.
Lemma 10. When the probability of each firm improving quality is one, incomplete information welfare under regime R will exceed full information welfare under regime U .
Proof. Expected welfare under regime R when φ is one is
WR (φ = 1) = −
−4δst + δt2 − 3
4t
and full-information welfare under regime U is
3 − δ −4st + 4s + t2 − 4
WU (φ = 1) =
.
4t
The difference WR − WU is 4δ(v − 1), a positive quantity.
A.7
Proof of Proposition 7
Under regime R, a firm’s incentive to innovate will be higher under full information than
incomplete if the rival’s probability of success is less than β ? , and lower otherwise.
Proof. Because a firm with an improved future good enjoys the same profit under either
information structure, the incentive to innovate will be stronger under signaling when a
failed firm’s signaling profit is lower than its full-information, and weaker otherwise.
We can write a failed firm’s profit in the signaling equilibrium, ΠF (e
p+ , 0) as
(17)
−4(β − 1)3 δ 3 t2 + 4(β − 1)2 δ 2 t2 − 2(β − 1)δ t2 − 2t + 3 + 1
4t(1 − 2(β − 1)δ)2
p
(β − 1)δ(βδt − δt − t + 1) (β − 1)δ (βδt2 − δt2 − 2t + 2)
.
t(2βδ − 2δ − 1)2
Notice that the signaling distortion disappears as the probability β of the rival’s success
tends to one: a failed firm’s alter ego gains nothing from imitation if he faces a rival with
an improved future good, so very little distortion is needed to signal type if the rival is
likely to succeed.
42
We will compare this to the expected profit under full information, βΠF (pFAFS , α̂ =
0)+(1−β)ΠF (pF F , α̂ = 0).It is easy to show these profits are decreasing in the probability
β a rival firm has succeeded. When β is one, signaling and full-information profits are
identically 1/4t. And the derivative of ΠF (e
p+ , 0) with respect to β evaluated when β is
one will be −((−1 + t)2 δ)/(2t), less negative than that of the full-information profit,
δ δ(2δ + 1)2 − 8δ 4 + 44δ 3 + 84δ 2 + 65δ + 18 t2 + 2 8δ 3 + 28δ 2 + 28δ + 9 t
.
4(δ + 1)2 (2δ + 3)2 t
Then signaling profit is below the expected full-information profit for lower β.
But there is another level of β that sets these equal and lies between zero and one,
which we will call β ∗ . While we can solve for it explicitly, the following argument will be
more instructive. For low probabilities of a rival succeeding, a failed firm’s profit in the
signaling equilibrium is higher than it would be in full information: the requirement to
signal that he will not have an improved good allows the failed type of a firm to set a
high price, despite the likelihood of only competing with a failed rival. In particular, if
the rival is surely a failed type, the signaling profit will be
p
6δ + 4δ 3 t2 + 4δ 2 t2 + 2δt2 + 4δ(δt + t − 1) δ (δt2 + 2t − 2) − 4δt + 1
,
4(2δ + 1)2 t
and the expected full-information profit will be,
8δ 5 t2 + 4δ 3 21t2 − 14t + 4 + δ 2 65t2 − 56t + 36 + 6δ 3t2 − 3t + 5 + 4δ 4 t(11t − 4) + 9
.
4(δ + 1)2 (2δ + 3)2 t
Subtracting these and rearranging, we find
κ3
p
δ(t2 δ + 2(t − 1)) + κ2 t2 + κ1 t + κ0
4t(1 + δ)2 (1 + 2δ)2 (3 + 2δ)2
where
κ3 =4δ(1 + δ)2 (3 + 2δ)2 (−1 + t + tδ)
κ2 = − δ 2 (41 + 186δ + 332δ 2 + 284δ 3 + 112δ 4 + 16δ 5 )
κ1 =2δ(−9 + 4δ + 102δ 2 + 192δ 3 + 136δ 4 + 32δ 5 ), &
κ0 =δ(18 + 25δ − 38δ 2 − 84δ 3 − 40δ 4 ).
We can see that this is positive. First, κ3 and the radical term are each positive. The rest
of the numerator is quadratic in t, and, while it will be negative for any t greater than one,
κ3 and the radical term will grow more quickly, so that the numerator is positive on net,
as long as δ is not very low. If t is one, the numerator will be 8δ 2 (δ + 1)2 4δ 2 + 4δ − 1 ,
p
positive for δ greater than 0.2071. For any higher t, κ3 δ(t2 δ + 2(t − 1)) will grow faster
43
than the quadratic decreases. So signaling profit will be higher when the rival surely fails.
Then we have shown that signaling profit is higher than full information, and the
incentive to innovate weaker, when the rival is unlikely to innovate, and that the signaling
profit is lower than full information and firm more willing to invest in quality improvement,
when the rival is likely to innovate.
A.8
Proof of Proposition 8
Under regime U , a firm’s incentive to innovate will be higher in a downward-distorted
signaling outcome than full when the rival’s ex ante probability of success is at least β ∗∗ ,
and lower otherwise.
Proof. We define β ∗∗ as the probability that satisfies the following equation for A:
U
(18) α βΠS (1/2, 1) + (1 − β)ΠS (pSF
, 1) + βΠF (pFASU , 0) + (1 − β)ΠS (pF F , 0)
A
= α(ΠS (p∗S , 1) − ΠS (p̌− , 0))
Then, we can invoke Lemmas 11 and 12 to show that, for higher β, it must be that a
seller who will subsequently play the signaling equilibrium for regime U will be willing to
pay more for a chance to improve the quality of the good.
Lemma 11. When the probability of a rival’s success is high enough, the incentive to
invest under signaling will be increasing in the probability of the rival’s success.
Proof. The most a firm under incomplete information is willing to pay for a chance α to
improve quality will be α (ΠS (p?S , 1) − ΠF (p̌− , 0)). To see that this will be increasing in
the probability β of a rival’s success when β is high enough, consider the derivatives of
ΠS (p?S , 1) and ΠF (p̌− , 0) with respect to β.
The latter, ∂ΠF (p̌− , 0)/∂β is
p̌−
−
dx1A (p̌− , 0) ∂ p̌−
dπ F S ∂ p̌−
dπ F S ∂ p̌−
− ∂ p̌
FS
FF
+
x
(p̌
)
+
δβ(π
−
π
)
+
δβ
+
δ(1
−
β)
,
1AF
2
2
dp̌−
∂β
∂β
dp̌− ∂β
dp̌− ∂β
which it will be convenient to write as
δβ(π2F S − π2F F ) +
∂ΠF (p̌− , 0) ∂ p̌−
.
∂ p̌−
∂β
The first term is negative, since the firm is at a competitive disadvantage when the rival
succeeds. We have seen that ΠF (p, 0) is increasing in p near p̌− , and that p̌− will be
decreasing in the rival’s probability of improving quality. Therefore ∂ΠF (p̌− , 0)/∂β < 0.
44
The first of these, ∂ΠS (p?S , 0)/∂β, will be positive when β is large enough, and negative
otherwise. We can write
∂ΠS (p?S , 0)/∂β = π2SS − π2SF +
∂p?S ? dx1AS (p?S )
∂π SF
?
pS
+
x
(p
)
+
(1
−
β)
1AS S
∂β
dp?S
∂p?s
We know the difference π2SS − π2SF is positive, and that
∂p?S
∂β
is negative, both because p̌−
is decreasing in β, and because, as the rival becomes more likely to succeed, a successful
firm is less concerned with setting a high price to preserve the market for competition
with a failed rival tomorrow. It remains to show when the terms in square brackets are
negative and positive. The first term is negative, as demand x1AS decreases in price, and
the second positive. Second-period profit when facing a failed rival is decreasing in current
price. For a high enough β, this last term will be a small enough negative number that
the terms in square brackets are negative and ∂ΠS (p?S , 0)/∂β positive. But, it is actually
sufficient for out purposes that ∂ΠS (p?S , 0)/∂β be not too negative, so that the difference
∂ΠS (p?S , 0)/∂β − ∂ΠF (p̌− , 0)/∂β is positive. This will be true for a smaller β still.
Lemma 12. When t is above ẗ, the incentive to invest under full information will be
declining in the probability of the rival’s success.
Proof. The full-information willingness to invest is as given above. Its derivative in the
probability of the rival’s success if investing will be
(19)
−α
ψ2 t2 + ψ1 t + ψ0
4(δ + 1)2 (2δ + 3)2 (6δ 2 + 17δ + 9)2 t
where the coefficients on t in the numerator are given by
2
ψ2 = δ 4δ 2 + 10δ + 5 6δ 2 + 17δ + 9 ,
2
ψ1 = 6δ 2 + 17δ + 9
8δ 3 + 28δ 2 + 28δ + 9 (δ(v − 1) − 1), and,
ψ0 = 448δ 8 + 3760δ 7 + 13256δ 6 + 25856δ 5 + 30822δ 4 + 23137δ 3 + 10722δ 2 + 2817δ + 324 v 2
− 2 448δ 8 + 4336δ 7 + 17816δ 6 + 40632δ 5 + 56454δ 4 + 49195δ 3 + 26391δ 2 + 8001δ
+ 1053 v.
Let ẗ be the second root of the numerator in t. Noting that the denominator of
19 is positive, and the numerator is quadratic in t with a positive leading coefficient of
ψ2 , we conclude that, if t is greater than ẗ, the second root of ψ2 t2 + ψ1 t + ψ0 , the
incentive to innovate will be declining in β, the probability of a rival’s improving quality
if investing.
A.9
Proof of Proposition 9
Under regime U , a seller’s incentive to innovate will be higher in an upward-distorted
signalling equilibrium than in full information.
45
Proof. We saw that the incentive to innovate in a downward-distorted signaling equilibrium will be increasing in the probability the rival will produce an improved version. In
an upward-distorted equilibrium, the incentive to innovate is decreasing in t for low β
(recall 11 and consider that p̌+ will be increasing in β). For a high-enough level of differentiation t, the willingness to invest under full information may pass above that in an
upward-distorted equilibrium. We can plot the regions where the incentive to innovate in
an upward-distorted equilibrium will be below that in a downward-distorted equilibrium,
and below that in full information in Figure 14. But, this value of t59 will be higher than
t, our restriction on differentiation to ensure full coverage; in particular, we can evaluate
the difference exactly when t = t from state SS, which will be v + 1 − p?S (p̌+ ), and find a
positive quantity. Therefore, for the relevant part of the parameter space, the incentive to
innovate under incomplete information in an upward-distorted signaling equilibrium will
be higher than that in full information.
We can also note that the incentive to innovate in an upward-distorted signaling equilibrium will be higher than that in a downward-distorted for low β, and lower otherwise.
Fix a price pS 60 and subtract
ΠS (pS , 1|p+ (pS )) − ΠF (p+ (pS ), 0|p+ (pS )) − ΠS (pS , 1|p− (pS )) − ΠF (p− (pS ), 0|p− (pS ))
It is easy to show that, for β equal to zero, the difference will be positive, and that
e t), the incentive to
this quantity will be negative when β is near one. Therefore for β(δ,
innovate in an upward-distorted equilibrium will pass below that in a downward-distorted.
See Figure 13, and note that this will be declining in the degree of differentiation.
59
Itself quadratic in β, as we can see in Figure 14.
When we evaluate this at p?S (p̌− ), term on the right will be exactly the incentive to innovate in a downwarddistorted equilibrium, and that on the left will be strictly less than the incentive to innovate in an upward; it is
sufficient for our purposes that this difference is positive.
60
46
B
Supplemental Appendix
The following appendices are included for reference, but not for publication.
B.1
Low differentiation: no signaling
In this section, we will see what occurs when there is little differentiation. We show
that, if oligopolists’ output is insufficiently horizontally differentiated, they will not signal
whether they have developed a new version of the good through the first-period price.
We first develop the limiting case of a homogeneous good which may be upgraded, then
extend this to the case of a Hotelling line. If the firms’ output is similar enough that
all consumers buy the original quality in the first period, there can be no equilibrium in
which quality is revealed by price in the first period.
If there is no differentiation between either the firms or in consumer tastes – that is,
t is zero – there is no equilibrium in which a firm with a future improved good will sell
the current good at a different price than one without. Because there is no differentiation
between the firms, gross surplus is simply the quality of the good held. For this subsection,
initial good is produced at marginal cost c less than one, and the new at marginal cost c2
no less than c; we will assume both that c is less than one, and c2 less than the incremental
quality, v − 1.61
It is worth comparing this game to a similar, full-information one where the state of the
world is commonly known. In that full-information case, there is Bertrand competition
in the first period, followed by the same in state SS, no sales in F F , or sales by the
successful firm at the choke price of (v − 1) in one of the asymmetric states. Note that
the incentive to innovate comes wholly from the rents in these last states.
Now, consider the version with private information. We proceed by backward induction. In state F F , if consumers bought in the first period, there are no second-period
sales, since no consumer is willing to pay more than the marginal cost of production. In
state SS, there is Bertrand competition in the second period, and all consumers buy at a
price of c2 . In the asymmetric states, SF and F S, the firm with a new version sells it at
the highest price consumers will bear: if none bought in the first period, this is v, or the
incremental utility of v − 1 if consumers have bought.
We might conjecture there will be a separating equilibrium, where in the first period
a firm which has realized S sets pS > pF . This will not work. In such an outcome, in an
asymmetric state, a successful firms is surely undercut in the first period and makes no
sales, while the firm which failed gets pF − c. In state SS, the firms split the market at
price pS .62 Consider a deviation to pS − ε; no matter their evaluation of the deviator’s
61
Without this assumption, there would be no incentive to innovate, either in the full-information or privateinformation versions of the game.
62
If pS exceeded unity, so there are no first-period sales by a successful firm, a firm of type S could gain in
47
type, consumers will only buy if this is the lower of the prices posted. If the rival is of
type F , the firm is undercut and makes no sales, which is no worse for the deviator than
equilibrium. If the rival succeeded, the rival is undercut, and the firm enjoys first-period
profit pS − ε − c, better than (pS − c)/2. There is always a deviation which in expectation,
and given consumer beliefs, is better than the equilibrium profit, so the equilibrium fails.
Then we have established the following.
Proposition 10. When there is no horizontal differentiation, there is no equilibrium in
which a firm that cannot market an improved version tomorrow will sell the current good
at a different price from a firm that will market an improved version.
As there is no separating equilibrium, we attend to pooling equilibria. There will be
no pooling equilibrium at a price above c, when t is zero. Suppose it were otherwise, and
both types of both firms set p? > c, so the first-period payoff would be (p? − c)/2. Then if
a type of a firm set a price p? − ε, it would gain a first-period profit of p? − ε − c, whether
consumers believed it would sell a new version or not tomorrow. This beneficial deviation
breaks the candidate equilibrium, so the only pooling equilibrium price is the marginal
cost, c. This incomplete-information game replicates the outcome of the analogous fullinformation game, and gives firms the same expected profit from investing in innovation.
This result is different from those in the literature: Janssen and Roy (2010) show that
prices can signal current quality differences to homogeneous consumers, and Daughety and
Reinganum (2007) show this in the case of horizontally differentiated goods. The current
paper is different in that there is no current quality difference, but may be a future one.
Here and in the next section, we show that this difference in future quality cannot be
communicated if the firms’ output is very similar.
B.1.1
Very Low differentiation in full information
Now we will return to the more general differentiated-good framework and find the full
information outcome when differentiation is low enough that all consumers buy in the
first period.
We will proceed by backward induction. Since all consumers have bought, a firm
without an improved version will make no second-period sales. So too, a firm with a new
version faces homogeneous demand for the improved good from his former customers:
they are all willing to buy the new good for a price no greater than v − 1. Consider
a buyer who weakly prefers repurchasing from A and holding the old good, for whom
v − p2A − tx ≥ 1 − tx. Since the transport costs tx are the same for both goods, this
simply requires that p2A ≤ v − 1.
If both firms succeed, they will play a second full-coverage Hotelling game, where the
indifferent buyer will be located at x2 such that v − p2A − tx2 = v − p2B − t(1 − x2 )
state SS, and be no worse in any other, by setting price equal to one and selling to all.
48
and therefore x2 = (p2B − p2A + t/2)/t. The two firms will maximize second-period
profits, subject to the restriction that their price never exceed v − 1; their prices will be
min{t, v − 1}, and they will split the market evenly.
If exactly one firm has succeeded at improving quality, it will sell as a monopolist
of that new good. Let x1 be the position of the marginal first-period buyer. Then, a
successful firm can sell to all consumers between him and x1 at a price of v − 1 for a profit
of x1 (v − 1), or sell at the margin to his rival’s former customers.63 Suppose only firm A
has succeeded. If A sells to some of B 0 s customers at the margin, that marginal buyer
will be located at x2 such that v − p2A − tx2 = 1 − t(1 − x2 ) and x2 = (v − 1 + t − p2A )/2t,
and firm A’s profit will be (v − 1 + t)2 /(8t) when t ≤ v − 1, but will be constrained to
x1 (v − 1) when t is higher.
Then in the first period, in state F F , consumers know that they will hold the good for
two periods. The indifferent consumer will have a location satisfying 1 − p1A − tx1 + δ(1 −
tx1 ) = 1 − p1B − t(1 − x1 ) + δ(1 − t(1 − x1 )), so x1 = (p1B − p1A + t(1 + δ)/2)/t(1δ). The
firms will optimally choose p1A = p1B = t(1 + δ), so that they split the market evenly.
In state SS, consumers know they will repurchase in the second period. Then the
marginal first-period buyer’s location will satisfy 1−p1A −tx+δ(v−min{v−1, t}−tx) = 1−
p1B −t(1−x)+δ(v−min{v−1, t}−t(1−x)), so that again x1 = (p1B −p1A +t(1+δ)/2)/t(1δ),
with the same result: the firms set equal prices of t(1 + δ) in the first period.
In an asymmetric state, consumers take account of the possibility they may or may
not repurchase. Suppose that only firm A has succeeded, and therefore the indifferent
first-period buyer will have a location such that 1 − p1A − tx + δ(v − p2A − tx) = 1 − p1B −
t(1 − x) + δ(v − p2A − tx). This simplifies to x1 = (p2B + p2A + t/2)/t, since in the second
period a successful firm A will price to make every former consumer, and possibly some of
B’s consumers, repurchase. The first-period prices will then be p1A = p1B = t, and they
will split the market evenly.
B.1.2
Very Low differentiation with incomplete information
In this subsection we examine case of full coverage under incomplete information. If the
market is fully covered in the first period, the firms will not condition their first-period
prices on their private information.
Our no-signaling result in the case of homogeneous consumers extends to consumers
whose tastes are only somewhat heterogeneous. Recall that the firms are commonly known
to succeed with probability α for A, and β for B. The firms are privately informed of their
own success or failure before setting a first-period price. In the second period, each firm
brings its output to market, and the success or failure of each to develop the new version
is publicly observed. If the transport cost parameter t is low enough, the market will be
63
These will require a discount below v − 1 in compensation for a higher transport cost, and this lower price
ensures all of the successful firm’s first-period customers will buy again.
49
fully covered in the first period, and the indifferent consumer of the good will satisfy the
following condition, where pωti is firm i’s price at time t in state ω. The left-hand side
of this equation is the expected utility of buying from firm A in the first period (cf. to
equation 13), and the right-hand side is an analogous expression for the expected utility
of purchasing in from firm B in the first period.
SS
(20) 1 − p1A − tx + αβδ max{v − pSS
2A − tx, 1 − tx, v − p2B − t(1 − x)}
SF
+ α(1 − β)δ max{v − pSF
2A − tx, 1 − tx, 1 − p2B − t(1 − x)}
+ (1 − α)βδ max{1 − pF2AS − tx, 1 − tx, v − pF2BS − t(1 − x)}
+ (1 − α)(1 − β)δ max{1 − pF2AF − tx, 1 − tx, 1 − pF2BF − t(1 − x)}
=
SS
1 − p1B − t(1 − x) + αβδ max{v − pSS
2A − t(1 − x), 1 − t(1 − x), v − p2B − t(1 − x)}
SF
+ α(1 − β)δ max{v − pSF
2A − tx, 1 − t(1 − x), 1 − p2B − t(1 − x)}
+ (1 − α)βδ max{1 − pF2AS − tx, 1 − t(1 − x), v − pF2BS − t(1 − x)}
+ (1 − α)(1 − β)δ max{1 − pF2AF − tx, 1 − t(1 − x), 1 − pF2BF − t(1 − x)}
Notice that this simplifies64 to the static Hotelling indifference condition,
(200 )
1 − p1A − tx = 1 − p1B − t(1 − x)
and therefore the two firms play the canonical Hotelling pricing game in the first period,
regardless of their ability or inability to market a new version tomorrow. If the firms
tried to price differently, and even if they swayed consumer beliefs, consumers would not
buy differently than in a static Hotelling game, subject to the prices. An unsuccessful
firm will make no second-period sales.65 Since consumers’ first-period willingness to pay
is not affected by their beliefs about future products, a seller with no improved future
product cannot gain by distinguishing himself from a firm with a new one. A successful
firm tomorrow will sell in the second period as well as the first, either as a monopolist of
its improved product, or playing a second Hotelling game against a successful rival.66 In
both of these cases, its second-period profits do not depend, at the margin, on the level of
first-period sales, so a firm with an improved product loses nothing by pretending to be a
64
The market will bear no positive price for the original good in the second period under full coverage, and a
successful firm will never charge more than (v − 1), in order to ensure repurchase by her own former customers.
65
Since all consumers already possess a good of the best quality a failed firm can offer, conveying information
in the first period would not give the firm any second-period benefit.
66
Note that, as in section 2.1, the prices of the improved products will be constrained by former consumers’
willingness to pay for the incremental quality.
50
firm without a new product. If the output of the two firms is not sufficiently different, no
signaling is possible, as above in the case of homogeneous consumers. Indeed, there will
also be no information conveyed through disclosure.
Remark. There will be no announcement for any positive cost of announcing under very
low differentiation.
We have so far ignored the possibility of announcement. Under low differentiation,
we saw that there is no signaling equilibrium, and that the resulting pooling outcome
reproduces the full-information prices and profits. Therefore, it will only be an equilibrium
for successful firms to announce when disclosure is costless, since taking the option to
announce will not alter the prices or profits under low differentiation.
B.2
Pooling Equilibria
In this section, we will address pooling outcomes, where consumers are unable to infer
from first-period prices what qualities will be available in the second period because both
types of a firm set the same price. In each regime, we will show that no pooling equilibria
survive when we impose the Intuitive Criterion. In particular, we can construct a beneficial
deviation for one type from any putative pooling equilibrium, when out-of-equilibrium
beliefs are consistent with that criterion.
For any pooling outcome to be an equilibrium it must be that a firm of type S prefers
the equilibrium payoff to the payoff from setting any other price, subject to the out-ofequilibrium beliefs. At a given price, each type would prefer to be known not to have an
improved version, rather than being known to have a new version. Fix a pooling price p? ,
and let the prior probability of each firm’s success be φ.
Let the price that maximizes S’s profit when its true type is known be p1S (p? ). It must
also be the case that the best profit for S, given its true type is known, must be less than
the pooling profit, as shown below.
(21)
ΠS (p1S , 1) ≤ ΠS (p? , φ)
Similarly, the least favorable interpretation of a deviation by F will be that the deviator
is of type S with probability near one. The best deviation profit for F given that worst
interpretation must not exceed F ’s profit in the candidate equilibrium, which is expressed
in the equation below.
(22)
ΠF (p1F , 1) ≤ ΠF (p? , φ)
Otherwise, each type could simply deviate to the corresponding price of p1F or p1S , and
even at worst be better off than in the equilibrium. There will be two pooling prices which
satisfy each of equations 21 and 22 with equality. Since both must hold for any pooling
51
equilibrium, we will inspect pooling prices in the intersection of (pF , pF ) and (pS , pS ). This
intersection will be different in each regime. In regime R we can restrict our attention to
pooling prices pF < p < pF , as this interval is a subset of (pS , pS ), so both conditions 21
and 22 are satisfied. Under regime U , we will attend pS < p < pF .
For a price p? to be an equilibrium, it must be that, or any price p for which consumers
and the rival assign probability near zero to the type with an improved version, it must
be that
(23)
ΠS (p, 0) ≤ ΠS (p? , φ), and likewise,
(24)
ΠS (p, 0) ≤ ΠS (p? , φ).
It will be convenient to write the profits in vertex form, defining p0τ as the maximizer
of τ ’s profit when consumers and the rival believe it is almost-surely of type F . Then
0 ? 2
ΠS (p, 0) = ΠS (p0S , 0)
qcall the solutions to
q− λ(p − pS (p )) and similarly for F . We will
±
0
±
0
0
?
0
the first qe = pS ± (ΠS − ΠS )/λS , and to the second q̂ = pF ± (Π0F − Π?F )/λ0F , and
plot them, for regime R, in Figure 15. Then we can prove the following.
Proposition 11. No pooling outcome survives refinement by the intuitive criterion.
Proof. We have seen that, for any pooling equilibrium to survive refinement, conditions
21, 22, 23 and 24 must hold. In each case, we will show that there is a price q(p, φ) that
will be equilibrium dominated for S, but beneficial to F when consumers have ‘intuitive’
beliefs.
We can see that, on the relevant interval, q̂ + will exceed qe+ ; see Lemma 13 for this
in regime R, and Lemma 14 in regime U . Then if F deviates from a pooling equilibrium
at p? to q(p? ) = qe+ + ε, for ε > 0, consumers will infer that the deviation did not come
from a firm of type S, since even the most favorable consumer response to that signal
would leave a deviating S worse off than equilibrium. Then, a deviating F would gain
from setting q(p? ), since q(p? ) is less than q̂ + , and F strictly prefers a price below q̂ +
when consumers infer its true type.
Lemma 13. In regime R, the price q̂ + will exceed qe+ .
Proof. Suppose to the contrary that qe+ > q̂ + or more explicitly
p0S +
q
q
(Π0S − Π?S )/λ0S > p0F + (Π0F − Π?F )/λ0F .
We can show that p0S < p0F for p? ∈ (pF , pF ), so this means that (Π0S − Π?S )/λ0S >
(Π0F − Π?F )/λ0F . Now, λ0F < λ0S for our parameter restrictions, so our assumption that
qe+ > q̂ + implies that
Π0S − Π?S − (Π0F − Π?F ) ≥ 0.
52
The left-hand side of this inequality is quadratic in the equilibrium pooling price p? , and
will be negative between its roots, which include the whole interval (pF , pF ), contradicting
hypothesis. Therefore q̂ + is greater than qe+ for pooling prices in (pF , pF ).
Lemma 14. In regime U , the price q̂ + will exceed qe+ .
Proof. As in regime R, we can show that p0F is greater than p0S on the relevant interval.
We can see, in Figure 16, that p0F − p0S and the difference of the radical terms of qe+ and q̂ +
both eventually increase in the pooling price. The prices where they cross will be those
pooling prices where qe+ = q̂ + , which are outside our interval of interest. It remains to
show that in fact q̂ + > qe+ , from which it will follow that it is higher on the whole interval.
Assume to the contrary that
p0F − p0S ≤
q
q
(Π0S − Π?S )/λ0S − (Π0F − Π?F )/λ0F ,
in particular at pS . We know the left hand side is positive. The right hand side will be
positive and negative when a simpler function, the difference of the inside of the radicals,
is as well:
(25)
Π0F − Π?F
Π0S − Π?S
−
λ0S
λ0F
Evaluated at p? = pS – between the roots of 25 – this will be negative, a contradiction.
Therefore q̂ + will exceed qe+ for pS < p < pF .
B.3
The leasing benchmark
In this section we find the leasing outcome in this market as a benchmark for sales. There
is no signaling, because firms’ and consumers’ decisions across time are not coupled under
leasing. The information structure does not matter, in that consumers will make the same
consumption decisions in period one whether they are informed ex ante about the firms’
ability to market a new version in period two, or not.
B.3.1
Partial coverage leasing
Let differentiation t be high enough that the market is not fully covered by the lessors’
output. The indifferent renter from firm A in each period will have a location xA =
(sA − rA )/t, where s is the realized quality in that period, either v or one, and rA is the
rental rate charged by A in that period. The marginal renter from B will similarly be
located at 1 − (sB − rB )/t. Then, since each firm faces no competition, in each period the
firm maximizes profit r(s − r)/t with r = s/2 and enjoys a profit of s2 /4t.
The equilibrium marginal renters xA and xB will be located respectively at sA /2t and
1 − sB /2t. We can use this to find a condition on t that will allow full or partial coverage.
53
It is sufficient for partial coverage that A’s marginal renter under partial coverage fall to
the left of B’s, or sA /2t < 1 − sB /2t. That will be true when the differentiation parameter
exceeds the average realized quality, that is,
(26)
t > (sA + sB )/2.
We will make two remarks: first, this is exactly t if consumers hold nothing after the
first period; and the lessors do recover all their first-period output. Second, there will
partial coverage leasing in the first period whenever t is greater than one, which is t when
at least one firm has succeeded.
If both firms fail to market a new version and t is at least one-half, or if both succeed
and t is greater than v, we will have partial coverage for two periods. If condition 26 fails
in the first period for the initial quality, we will have a repeated, static Hotelling game
under full coverage.
The rest of the current paper is focused on sales when the market is at first partially
covered, and then fully covered after second-period sales. As we are interested in leasing as
a benchmark for sales, we will explore the parallel cases for leasing in the next subsection.
B.3.2
Full-coverage leasing
There are two cases in which we might find partial coverage in period one and full coverage
in period two: either both firms succeed, or only one. Suppose both firms have succeeded,
and v > t so that the market will be covered in the second period. Then the consumer
just indifferent between leasing from A or B will be located at x2 such that v −r2A −tx2 =
v − r2B − t(1 − x2 ), and thus x2 = (rB − rA + t)/2t. The two firms will choose the rental
rates to maximize their second-period profits, which in equilibrium will each be t/2.
If instead only one firm has succeeded and the average quality (v + 1)/2 exceeds t, we
will have asymmetric competition. Let A have succeeded and B failed. Then A will have
second-period demand x2 = (v − 1 + r2B − r2A + t)/2t and B demand of 1 − x2 . They will
r = (3t + v − 1)2 /18t and
simultaneously choose their rental rates and enjoy profits of π2A
r = (−3t + v − 1)2 /18t.
π2B
54
3.0
v
2.8
2.6
2.4
2.2
2.0
1.8
1.6
t
1.0
1.5
2.0
2.5
3.0
Figure 11: New quality v̂, above which firms announce future products; the region where t ≤ t∗
at left
Π
0.40
0.35
0.30
0.25
0.0
0.2
0.4
0.6
0.8
1.0
β
Figure 12: Profits for type F under signaling (solid) and full information (dashed) in regime R
55
ΠS-ΠF
0.6
0.5
-(Π-S -ΠF )
0.4
++
(Π++
S -ΠF )
(ΠSfull info -ΠFfull info )
0.3
0.2
0.1
0.2
0.4
0.6
0.8
1.0
β
Figure 13: The incentives to innovate under regime U , in full information and upward- and
downward-distorted equilibria, as a function of the rival’s probability of innovating.
2.4
2.2
2.0
++
--(Π ++
S -Π F )<(Π S -Π F )
1.8
++
full info -Π full info )
(Π ++
S -Π F )<(Π S
F
t>t
1.6
1.4
1.2
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 14: The regions where the incentive to innovate in an upward-distorted equilibrium is
less than that in a downward and in full information.
56
q(p* )
pF
˜+
q
pF
1.25
+
q
1.00
q
0.75
˜q
0.50
p*
0.25
0.4
0.6
0.8
1.0
1.2
Figure 15: Deviations from a pooling equilibrium at p? , under regime R
0.10
Π0S -Π*S 
λS
-
pS
Π0F -Π*F 
λF
pS
0.05
p*
p0F -p0S
0.2
0.4
0.6
0.8
1.0
1.2
Figure 16: Deviations from a pooling equilibrium at p? , under regime U
57