Competition in Markets with Quality Uncertainty and Network Effects

Bjørn-Atle Reme

doi:10.1515/rne-2019-0061

Published by De Gruyter October 6, 2020

Competition in Markets with Quality Uncertainty and Network Effects

Bjørn-Atle Reme

From the journal Review of Network Economics

https://doi.org/10.1515/rne-2019-0061

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Abstract

We study how quality uncertainty among consumers affects price competition in the presence of network effects. Our main result is that quality uncertainty has non-monotonic effects on firms’ price setting behavior. Prices and industry profit is first falling, then increasing, in quality uncertainty. In addition we show that quality uncertainty can force a high quality provider to be aggressive to the point where its price in the first period is below that of a low quality provider. We also analyse the incentives for compatibility under quality uncertainty, and find that when quality uncertainty is sufficiently high, compatibility may be used as a means of softening price competition.

Keywords: network effects; vertical differentiation; duopoly; compatibility

JEL Classification: L13; L14; L15

Corresponding author: Bjørn-Atle Reme, Norwegian Institute of Public Health, Oslo, Norway, E-mail: Bjorn-Atle.Reme@fhi.no

A Appendix

A.1 Generalizing to Myopic Consumers

This section briefly lays out why the results presented in Sections 3, 4 and 5 in the main text also generalizes to the assumption that consumers are myopic (cf. footnote ^[25]). When consumers are myopic, the first period consumer chooses the product that appears to offer the highest net utility without forming expectations about what might happen in period 2. This corresponds to the choice made by a first period forward looking consumer, as described in the previous section.

The second period consumers make their choices only considering past choices, hence if the first period consumer chose product 1, they choose product 1 if v₁ + θ − p₁₂ > v₂ − p₂₂. Likewise, if the first period consumer chose product 2, they choose product 2 if v₂ + θ − p₂₂ > v₁ − p₁₂. Since there will be price competition à la Bertrand in the second period, and having assumed that θ > v₁ − v₂, it follows directly that the second period consumers will choose the same product as was chosen by the first period consumer.

The behavior of myopic consumers can now be summed up as follows: The first period consumer chooses the product that appears to offer the highest net utility without forming expectations about what might happen in the second period. The second period consumers choose the product that offers the highest net utility, taking into account the choice made by the first period consumer. In equilibrium the second period consumers choose the same product as the first period consumer. This consumer behavior is identical to that of forward looking consumers, hence the equilibrium in prices will also be the same.

A.2 Proof Lemma 1

We assume that θ > v₁ − v₂. This ensures that the utility value from the network effect from the choice in the first period surpasses the difference in stand-alone benefit. The implication of this assumption is that whatever product is preferred by the first period consumer also will be chosen by the second period consumers, assuming they are able coordinate on a payoff-dominant equilibrium.

Using backwards induction we start with the second period consumer choice. Second period consumers make their choices simultaneously. Since the second period consumers coordinate on the payoff-dominant equilibrium, and θ > v₁ − v₂, the same product as was chosen in the first period. There are hence two possible sub-games depending on which product was chosen in the first period.

First period consumer chose product 1: Given that they coordinate on the payoff-dominant equilibrium, they will choose firm 1, and not regret their choice as long as

p22−p12>v1−v2+θ.

In response, firm 2 will lower its price until p₂₂ = c. Hence, it follows that the unique pure strategy equilibrium in prices is

(33)p12*=c+v1−v2+θ

(34)p22*=c,

At these prices all consumers choose product 1, beliefs are confirmed in equilibrium, and no other equilibrium exists that could make the consumers better off. To see this, assume that p12>p12*. Then it can easily be verified that the second period consumers would be better off coordinating on product 2. Another way of finding this equilibrium is to assume that the N second period consumers act as if they were one consumer. The second period equilibrium profits are then

(35)π12*=N[v1−v2+θ]

(36)π22*=0.

First period consumer chose product 2: Using the same logic as above, it can be shown that the unique equilibrium prices must be

(37)p12*=c

(38)p22*=c−(v1−v2)+θ,

with corresponding beliefs – all consumers choose product 2. The second period equilibrium profits are then

(39)π22*=0

(40)π21*=N[θ−(v1−v2)].

A.3 Lemma 2

Let πiI denote firm i’s profit in the interior equilibrium and πiCS its profit in the corner solution. The interior equilibrium dominates the corner solution when πiI>πiCS for i = 1, 2. Since firm 2 has zero profit in the corner equilibrium and non-zero profit in the interior equilibrium, we need only focus on firm 1. We want to identify the noise level where the equilibrium switches from the corner solution to the interior solution. In addition we want to show that this switching point is unique. Using (8) and (16), we have that

(41)π1I>π1CS

(42)118(3ε‾+(2N+1)(v1−v2))2ε‾>(v1−v2)(2N+1)−ε‾.

Solving the above expression using equality, we find that π1I=π1CS when ε‾=(1+2N)3(v1−v2).^[26] This is the noise level where W^* = 1, given by (18), and hence where the equilibrium switches. It can easily be verified that this switching point is unique: First we have that ∂π1CS/∂ε‾<0 ∀ ε‾>0. In addition it is shown below that ∂π1I/∂ε‾>0∀ε‾>(1+2N)3(v1−v2):

(43)∂π1I∂ε‾=−118ε‾2(3ε‾+(v1−v2)(1+2N))(−3ε‾+(v1−v2)(1+2N)).

Assuming that ε‾>(1+2N)3(v1−v2)

(44)∂π1I∂ε‾=−118ε‾2⏟<0(3ε‾+(v1−v2)(1+2N))⏟>0(−3ε‾+(v1−v2)(1+2N))⏟<0>0.

Now define ε˜≡(1+2N)3(v1−v2). It follows from the above calculations that ε‾=ε˜ must be the unique switching point between the corner solution (below ε˜) and interior solution (above ε˜).

A.4 Proof of Proposition 1

In the corner solution firm 1 wins the market with certainty (W = 1). Firm 1 will prefer this as long as its price is at least the same as in the interior solution, i.e., v1−v2+ε‾+p21 (cf. Eq. (6)) must be weakly larger than the right hand side of (12), which implies that

(45)p21≥c+3ε‾−(v1−v2)(N+1)−Nθ

Moreover, for any feasible p₂₁, (2) holds if

(46)c−[θ−(v1−v2)]N>c+3ε‾−(v1−v2)(N+1)−Nθ

which solved for ε‾ yields

(47)ε‾<(1+2N)3(v1−v2)

From Eqs. (8) and (16) we have that firm 1’s profits in the corner and interior solutions are

(48)π1CS=(v1−v2)(2N+1)−ε‾

and

(49)π1I=118(3ε‾+(2N+1)(v1−v2))2ε‾,

respectively. It follows directly that ∂π1CS/∂ε‾<0 ∀ ε‾>0.

For the interior equilibrium we have that for firm 1

(50)∂π1I∂ε‾=−118ε‾2(3ε‾+(v1−v2)(1+2N))(−3ε‾+(v1−v2)(1+2N)),

with a threshold for when the equilibrium switches equal to ε‾=(1+2N).3(v1−v2). Inserting for any ε‾ above this threshold, it can easily be verified that:

(51)∂π1I∂ε‾=−118ε‾2⏟<0(3ε‾+(v1−v2)(1+2N))⏟>0(−3ε‾+(v1−v2)(1+2N))⏟<0>0.

For firm 2, profit is equal to zero in the corner equilibrium, while in the interior equilibrium it is

(52)π2=118(3ε‾−(v1−v2)(2N+1))2ε‾.

Analogously as for firm 1, by differentiating with respect to ε‾, and inserting for any ε‾>(1+2N).3(v1−v2), it can easily be verified that ∂π2I/∂ε‾>0.

That firm 1’s prices are first decreasing, then increasing in ε‾ can be seen directly from considering the effect on its prices from a marginal increase in ε‾ in the corner solution and interior solution, given by (7) and (14):

(53)p11CS=c+(v1−v2)(N+1)−θN−ε‾

and

(54)p11I=c+ε‾−Nθ−13(v1−v2)(N−1).

And, for firm 2, the equilibrium price in the corner solution is equal to marginal cost (cf. (14)), while in the interior solution it is

(55)p21I=c+ε‾−Nθ+13(v1−v2)(N−1),

which is also clearly increasing in ε‾.

A.5 Proof of Proposition 2

A.5.1 First Period

Consider the first period equilibrium prices in the corner solution:

(56)p11=c+(v1−v2)(N+1)−θN−ε‾

and

(57)p21=c−N(θ−(v1−v2)).

Comparing them we see that

v1−v2>ε‾→p11CS>p21CS∧v1−v2<ε‾→p11CS<p21CS.

This implies that if the equilibrium switches from the corner solution to the interior equilibrium at

(58)ε‾>v1−v2,

The equilibrium prices of firm 1 and firm 2 must cross once inside the corner solution. Consider the equilibrium prices in the interior solution, given by (14) and (15):

(59)p11I=c+ε‾−(N−1)3(v1−v2)−Nθ

(60)p21I=c+ε‾+(N−1)3(v1−v2)−Nθ

From these we see that

(61)p11I<p21I∀N>1∧p11I=p21I when N=1

In the above section we identified that the noise level at which the equilibrium switches (ε˜) is equal to 2N+13(v1−v2). By assuming N > 1, we have that (58) is satisfied and that (61) implies p11I<p21I. Hence, we can conclude that when N > 1 we have the following price pattern: (i) p11*>p21* when ε‾<v1−v2, (ii) p11*<p21* when ε‾>v1−v2 and (iii) p11*=p21* when ε‾=v1−v2. Note that the noise level where p11*=p21* always belongs to the corner solution.

A.5.2 Second Period Prices

In the corner solution firm 1 always wins and charges p12*=c+θ+v1−v2. Firm 2 charges p22*=c. This is the unique Nash equilibrium in prices.

In the interior solution firm 1 also charges p12*=c+θ+v1−v2 if it wins the first period, and firm 2 chooses a price p22*=c. If firm 2 wins however, we have p12*=c+θ−(v1−v2) and p11*=c.

Note that since firm 1 chooses a lower first period price than firm 2 in the interior solution, it also has a higher probability of winning the market. It follows that the expected second period price is higher for firm 1.

A.6 Proof of Proposition 3

A.6.1 Switching from Corner Solution to Interior Solution Under Compatibility

Assuming incompatibility we found in Section 7.3 that the noise level at which the equilibrium switches is ε‾=(1+2N)3(v1−v2).

As shown in Section 4, the equilibrium prices under compatibility are different than the equilibrium prices under incompatibility. This implies that the noise level at which the equilibrium switches may also be different (since we have a covered market, this will depend on whether the equilibrium price difference changes).

To find the switching point under compatibility, we solve W(p11*,p12*)=1 for ε‾ using the equilibrium prices given by (27) and (28). We find this to be ε‾=13(v1−v2). As a robustness check, we look at whether this holds when comparing the equilibrium profit of the corner- and interior solutions.

Let ε˜=ε‾ such that ε‾=13(v1−v2) and let π1CS and π1I be firm 1’s equilibrium profit under compatibility in the corner- and interior solution, respectively. If ε‾=13(v1−v2) is the point where the equilibrium switches, then the following cannot be true:

(62)π1CS(ε‾=ε˜)>π1I(ε‾=ε˜)

(63)(v1−v2)(N+1)−ε˜>118(3ε˜+v1−v2)2ε˜+(v1−v2)N.

We insert for ε˜=ε‾ and find

(64)23(v1−v2)>23(v1−v2),

which cannot be true. This leads to the conclusion that when products are compatible, the equilibrium switches at ε‾=13(v1−v2). Since the equilibrium switches from the corner solution to the interior solution at different noise levels depending on whether products are compatible or not, we divide our proof into three regions of ε‾.

A.6.2 Corner Solution (0<ε‾<13(v1−v2))

We first look at period 2. Comparing (33) and (19), it is easy to see that prices in the second period are higher under incompatibility.

Comparing (7) and (21) we find that prices are higher in period 1 under compatibility if

(65)c+(v1−v2)−ε‾>c+(v1−v2)(N+1)−θN−ε‾

(66)θ>v1−v2.

Which holds by assumption.

A.6.3 Mixed Solution (13(v1−v2)<ε‾<(1+2N)3(v1−v2))

Period 2

Under incompatibility we have the corner solution, hence the market price will be p12IC=c+θ+v1−v2. If products are compatible, firm 1 will win the second period with certainty and charge a price p12C=c+v1−v2. It follows immediately that second period prices are higher under incompatibility.

Period 1

Under incompatibility we now have the corner solution, hence the market price will be p11IC=c+(v1−v2)(N+1)−θN−ε‾. If products are compatible, the prices charged are given by the interior equilibrium, hence

(67)p11C=c+ε‾+v1−v23,

(68)p21C=c+ε‾−v1−v23.

The expected price in period 1 under compatibility is

(69)EC(p)=p22C(1−W(p11C,p21C))+p12CW(p11C,p21C),

(70)=c+ε‾+(v1−v2)29ε‾.

Comparing prices, we find that incompatibility has lower first period prices when

(71)c+(v1−v2)(N+1)−θN−ε‾<c+ε‾+(v1−v2)29ε‾,

(72)(v1−v2)(N+1)<2ε‾+(v1−v2)29ε‾+θN.

We see that this inequality is dependent on ε‾. We insert for ε‾=13(v1−v2). We then have:

(73)(v1−v2)(N+1)<(v1−v2)+θN.

This clearly holds by assumption for ε‾=13(v1−v2). To make sure (72) holds for all noise levels in the range ε‾∈[13(v1−v2),(1+2N)3(v1−v2)], we differentiate the right hand side of (72) with respect to ε‾. If this is positive for all ε‾∈[13(v1−v2),(1+2N)3(v1−v2)], then (72) must hold. We denote the right hand side of (72) as L. We have that

(74)∂L∂ε‾=2−(v1−v2)29ε‾2,

which implies that

(75)∂L∂ε‾>0 when ε‾>v1−v218.

Since 13(v1−v2)>118(v1−v2), (72) is true ∀ε‾∈[13(v1−v2),(1+2N)3(v1−v2)].

A.6.4 Interior Solution (ε‾>(1+2N)3(v1−v2))

Period 2

When products are compatible, firm 1 dominates the market and charges a price p12C=c+v1−v2, as given by (19). If there is incompatibility, the expected market price, E_IC(p), is given by

(76)EIC(p)=p22IC(1−W(p11IC,p21IC))+p12ICW(p11IC,p21IC),

where pi2IC is firm i’s equilibrium price in the second period if it wins the first period consumer. p11IC and p21IC are the equilibrium prices in the first period of the interior solution under incompatibility. W(p11IC,p21IC) is the probability with which firm 1 wins the market, given by (18). The second period prices are

(77)p12IC=c+θ+(v1−v2),

(78)p22IC=c+θ−(v1−v2).

Since p12IC>p22IC∧p22IC>p12C, it must hold that EIC(p)>p12C∀W∈[0,1].

Period 1

We now compare expected prices in the first period interior equilibrium between compatibility and incompatibility: EC(p)=p21C(1−W(p11C,p21C))+p11CW(p11C,p21C) and EIC(p)=p21IC(1−W(p11IC,p21IC))+p11ICW(p11IC,p21IC). There is now a shortcut that can help us out: Note that under compatibility, we have equilibrium prices:

(79)p11C=c+ε‾+v1−v23,

(80)p21C=c+ε‾−v1−v23.

While under incompatibility they are

(81)p11IC=c+ε‾−Nθ−13(v1−v2)(N−1),

(82)p21IC=c+ε‾−Nθ+13(v1−v2)(N−1).

Since θ > v₁ − v₂, the price for both firm 1 and firm 2 is higher under compatibility.

A.7 Proof of Proposition 4

There are three different regions of ε‾ that we need to check: (i) the corner solution under both regimes, (ii) the interior solution under compatibility and corner solution under incompatibility and (iii) the interior solution under both regimes. In the proof of proposition 3 it is shown that (i) is where 0≤ε‾<13(v1−v2), (ii) is 13(v1−v2)<ε‾<(1+2N)3(v1−v2) and (iii) is ε‾>(1+2N)3(v1−v2).

A.7.1 Corner Solution (0<ε‾<13(v1−v2))

For firm 1 the equilibrium profit under incompatibility is given by

(83)π1*=p11*+p12*N,

where p11* and p12* are given by (21) and (19), respectively. Hence π1*=(v1−v2)(N+1)−ε‾. Comparing this with the profit under incompatibility for the same region (5), it is immediately clear that incompatibility dominates compatibility for firm 1 in this region. Firm 2 is indifferent since its profit is zero in either case.

A.7.2 Mixed Solutions (13(v1−v2)<ε‾<(1+2N)3(v1−v2))

Firm 2 clearly prefers compatibility, since incompatibility gives zero profit (corner solution), while compatibility gives non-zero profit. For firm 1 there is a need for more analysis. We compare its equilibrium profit in the corner solution under incompatibility with the equilibrium profit in the interior solution under compatibility.

Let ε˜⊂ε‾ where ε‾∈[13(v1−v2),(1+2N)3(v1−v2)] and let πICCS be firm 1’s equilibrium profit under incompatibility in the corner solution. πCI is firm 1’s profit under compatibility in the interior solution. If incompatibility is to be preferred in this region, then the following must hold πICCS>πCI∀ε‾∈ε˜:

(v1−v2)(2N+1)−ε‾>118(3ε‾+v1−v2)2ε‾+(v1−v2)N

Insert for ε‾=(1+2N)3(v1−v2). The above expression is now true when

2(v1−v2)(2N+1)3≥318((v1−v2)(2N+2))2(v1−v2)(2N+1)+(v1−v2)N4(v1−v2)(2N+1)≥(2N+2)2(v1−v2)(2N+1)+6(v1−v2)N4(2N+1)≥(2N+2)2(2N+1)+6N2N+4≥(2N+2)2(2N+1)(2N+4)(2N+1)≥(2N+2)24N2+2N+8N+4≥4N2+8N+42N≥0

This reveals that incompatibility is preferred by firm 1 at the switching point of the incompatibility equilibrium. Remember that this switching point is higher than for the compatibility equilibrium. If we now evaluate ∂πICCS∂ε‾ and ∂πCI∂ε‾ at ε‾∈[13(v1−v2),(1+2N)3(v1−v2)], it can easily be shown that ∂πICCS∂ε‾<0 and ∂πCI∂ε‾>0, implying that as we consider ε‾<(1+2N)3(v1−v2), the difference πICCS−πCI would only increase. This concludes the proof for this region.

A.7.3 Interior Solution (ε‾>(1+2N)3(v1−v2))

Firm 2 prefers compatibility whenever its equilibrium profit under compatibility exceeds the profit under incompatibility (given by (17)). The equilibrium profit under compatibility is derived by inserting for the equilibrium prices (27) and (28) into (18) and then inserting into (24). Doing this, we find that compatibility is preferred whenever

118(3ε‾−(v1−v2))2ε‾>118(3ε‾−(v1−v2)(2N+1))2ε‾(3ε‾−(v1−v2))2>(3ε‾−(v1−v2)(2N+1))2

We now analyze the interior solution, i.e. ε‾>(1+2N)3(v1−v2). The expressions within the above parentheses must therefore be positive on both the left and the right hand side. It then follows that inequality is always satisfied for the relevant range of ε‾.

Compatibility is preferred by firm 1 whenever (27) > (16):

118(3ε‾+v1−v2)2ε‾+(v1−v2)N>118(3ε‾+(v1−v2)(2N+1))2ε‾(3ε‾+v1−v2)2+18ε‾(v1−v2)N>(3ε‾+(v1−v2)(2N+1))26ε‾(v1−v2)+(v1−v2)2+18ε‾(v1−v2)N>6ε‾(v1−v2)(2N+1)+(v1−v2)2(2N+1)26ε‾+(v1−v2)+18ε‾N>6ε‾(2N+1)+(v1−v2)(2N+1)26ε‾N>(v1−v2)(2N+1)2−(v1−v2)ε‾>(v1−v2)(2N+1)2−(v1−v2)6Nε‾>2(v1−v2)(N+1)3.

Given our assumption that compatibility is a bilateral decision, compatibility is the equilibrium outcome when ε‾>2(v1−v2)(N+1)3.

A.8 Proof of Proposition 5

No proof is needed for the firms, since they gain from compatibility by assumption. Since we are interested in compatibility as an equilibrium outcome, we need only focus on the effect on consumer surplus for ε‾>2(v1−v2)(N+1)3, hence compare interior solutions.

A.8.1 Consumer Surplus Under Compatibility

Since we have compatibility, all N + 1 consumers enjoy the network effects: θN. We then find that total consumer surplus is

CSc=θN(N+1)+(v1−p11c)Wc+(v2−p21c)(1−Wc)+N(v1−p12c).

Using (27), (19) and (28) to insert for p1,1c, p1,2c and p2,1c, we get

CSc=θN(N+1)+N(v2−c)+(v1−(c+ε‾+v1−v23))(12+v1−v26ε‾)+(v2−(c+ε‾−v1−v23))(12−v1−v26ε‾).

Which may be reduced to

CSc=118ε‾(v12−18ε‾2−18cε‾+v22+9ε‾v1+9ε‾v2−2v1v2−18Ncε‾+18Nθε‾+18Nε‾v2+18N2θ ε‾)

A.8.2 Consumer Surplus Under Incompatibility

Also here all consumers enjoy the full network benefits, since all consumers will follow the first period consumer in equilibrium. The consumer surplus is therefore

CSnc=θN(N+1)+[(v1−p11nc)+N(v1−p12nc)]Wnc+[(v2−p21nc)+N(v2−p22nc)](1−Wnc)

We use (14), (33) and (15) to insert for p11nc, p12nc and p21nc. In addition we know that p22nc=θ−(v1−v2).

CSnc=((N+1)v1−(c+ε‾−(N−1)3(v1−v2)−Nθ)−N(c+θ+v1−v2))(12+16(2N+1)v1−v2ε‾)+θN(N+1)((N+1)v2−(c+ε‾+(N−1)3(v1−v2)−Nθ)−N(c+θ−v1+v2))(12−16(2N+1)v1−v2ε‾).

which can be reduced to

CSnc=118ε‾(v12−18ε‾2−2N2v12−2N2v22−18cε‾+v22+9ε‾v1+9ε‾v2−2v1v2+Nv12+Nv22+4N2v1v2−18Ncε‾+18Nθε‾+9Nε‾v1+9Nε‾v2−2Nv1v2+18N2θε‾)

We now have that consumers are better off under compatibility whenever

CSc>CSnc,

which can be reduced to

(84)−118Nε‾(v1−v2)(9ε‾+v1−v2−2Nv1+2Nv2)>0

(85)9ε‾+v1−v2−2Nv1+2Nv2<0

(86)ε‾<(2N−1)(v1−v2)9.

We know that ε‾>(1+2N)3(v1−v2) in the interior equilibrium, hence the consumers are worse off under compatibility whenever compatibility is an equilibrium outcome.

A.9 Proof of Proposition 6

A.9.1 Total Profit Increase

Firm 1’s gain from compatibility:

△π1=π1c−π1nc△π1=118(3ε‾+v1−v2+2Nv1−2Nv2)2ε‾−118(3ε‾+v1−v2)2ε‾−N(v1−v2)△π1=19Nε‾(v1−v2)(3ε−2v1+2v2−2Nv1+2Nv2).

Firm 2’s gain from compatibility:

△π2=π2c−π2nc△π2=118(3ε‾−v1+v2)2ε‾−118(−3ε+v1−v2+2Nv1−2Nv2)2ε‾△π2=29Nε‾(v1−v2)(3ε‾−v1+v2−Nv1+Nv2).

Let the total industry profit increase be defined as

△πT=△π1+△π2.

Then

△πT=19Nε‾(v1−v2)(9ε‾−4v1+4v2−4Nv1+4Nv2).

A.9.2 Total Loss of Consumer Surplus

Let the loss of consumer surplus due to compatibility be defined as

△CS=CSc−CSnc

Which we know from (84) to be equal to

△CS=118Nε‾(v1−v2)(9ε‾+v1−v2−2Nv1+2Nv2)

A.9.3 Total Welfare

Total welfare increases under compatibility whenever

△πT>△CS16Nε‾(v1−v2)(3ε‾−3v1+3v2−2Nv1+2Nv2)>03ε‾−3v1+3v2−2Nv1+2Nv2>0ε‾>3+2N3(v1−v2)

A.10 Discussion: Consumer Awareness

The model analysis presented in this paper assumes that the first period consumer is unaware of his inability to correctly assess quality. We now provide a short discussion of the implications of consumer awareness.

In our model consumer awareness has three possible interpretations. The choice of interpretation implicitly determines what the consumer’s beliefs will be about: (i) The consumer knows the level and structure of the noise (ε∼U[−ε‾,ε‾]), in which case his beliefs must be about the true quality difference |v₁ − v₂| and which product has the higher quality. (ii) The consumer knows the quality difference between the products (|v₁ − v₂|) and must form beliefs about the structure of the noise (ε) and which product has the higher quality. (iii) The consumer knows both the noise structure (ε∼U[−ε‾,ε‾]) and the quality difference (|v₁ − v₂|), in which case beliefs must be formed about which product has the higher quality.

In situations where the consumer has a higher level of awareness the model becomes complicated, as both firms will strategically use prices to signal quality. The beliefs of the first period consumer, which determine how prices will be interpreted, then become a driving force in the model. The model used in the paper is not well suited for this type of analysis. The reason is the combination of a continuous type space (|v₁ − v₂|), two strategic senders of signals and a continuous noisy signal (ε). We therefore do not present a full analysis of the model assuming a higher level of consumer awareness, but rather sketch out the necessary conditions for the results in our equilibrium to hold.

Let the pricing strategies found in sections 3.1 and 3.2 be denoted as P^*. The first period consumer’s plan of action in our model can be described as follows: choose the product that appears to offer the highest net utility. We refer to this strategy as B^*. Note that since E(ε) = 0, the consumer’s signal about the true quality difference is informative. B^* is the optimal plan of action for a consumer that does not use prices to draw inferences about quality.

We know that P^* is the optimal response to B^*, but a higher level of consumer awareness could lead to beliefs where B^* is no longer the optimal response to P^*. Note that if the consumer has “simplistic beliefs”, in the sense that he does not know how to draw inferences about quality from observed prices, B^* and P^* will remain the equilibrium outcome. It is not unlikely that a consumer facing such a complex strategic environment will resort to such a heuristic. To illustrate the complexity involved: a Perfect Bayesian Equilibrium (PBE) with two strategic senders of signals is demanding in terms of both information and calculation: The consumer should have beliefs that are defined for all possible prices. These beliefs should be known and used by the firms when setting prices. Furthermore, the consumer should be able to understand the firms’ optimization problem and interpret prices according to Baye’s rule. In cases (i) and (ii) above, the consumer’s beliefs should also include an a priori distribution of |v₁ − v₂| and ε, respectively.

For the equilibrium in our main model to be a PBE we hence need that (i) P^* is the optimal response to B^* and (ii) B^* is a sequentially rational response to P^*, given that the consumer’s beliefs are updated according to Baye’s rule.^[27]

Acknowledgment

I would very much like to thank Lars Sørgard, Kurt Brekke, Hans Jarle Kind, Timothy Simcoe, Anette Boom, Morten Sæthre, the NORIO workshop in Copenhagen, and the CLEEN workshop in Mannheim for valuable comments, discussion and suggestions.

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Received: 2019-12-16

Accepted: 2020-09-15

Published Online: 2020-10-06

Published in Print: 2019-12-18