This section examines the equilibrium quality levels obtaining in the absence of any policy intervention. We are particularly interested in how the fraction of informed customersαaffects the equilibrium quality investment decisions.
We first derive the quality choice that maximizes firm L’s profit (1−α)RL(qˆL, ˆqH) +
αRL(qL, ˆqH)−C(qL)when the beliefs are(qˆL, ˆqH). As mentioned above, revenues from
uninformed customers do not depend on the actual qualityqL. Therefore, the restricted
best response bL(qˆH,α) ≡ arg max0≤q≤qˆH αR
L(q, ˆq
H)−C(q) of the low quality firm
depends on the belief ˆqH but not on the belief ˆqL. This observation will be important for
our analysis below. Since any belief ˆqL that differs frombL(qˆH,α) is inconsistent with
the firm’s desire to maximize its profit, we will also refer to bL(qˆH,α) as the quality
which firmLmaycrediblyproduce. Note thatbLis only a restricted best response since firm L is constrained to provide a quality below ˆqH.24 The properties of RL and C
ensure thatbL(qˆH,α) >0 and that the associated optimality condition
αRLqL(qL, ˆqH) = C
0
(qL) (4.2)
uniquely characterizes the best response wheneverbL(qˆH,α) < qˆH. Observe that the
best response bL approaches 0 as α → 0 since RqLL is bounded above and C 0
(0) = 0. 23These formulae obtain from the ones already presented in the limit forq
L→0.
24We will later assure that the low quality firm has no incentives to provide a higher quality than its
If bL(qH,α) < qH, then differentiating equation (4.2) and using the homogeneity of degree 0 ofRLqL yields25 bLqH(qH,α) = b L(q H,α) qH 1− C00(bL(qH,α)) αRLqL,qL(bL(qH,α),qH) < bL(qH,α) qH .
Similarly, let bH(qˆL,α) ≡ arg maxq≥qˆLαR
H(qˆ
L,q)−C(q) denote the firms’ restricted
best response when having to provide a quality which is at least as high as the belief ˆ
qL about the competitor’s quality. Since RHqH decreases in qH and C is convex, b
H is
uniquely defined by the optimality condition
αRHqH(qˆL,qH) = C
0
(qH) (4.3)
wheneverbH(qˆL,α) >qˆL.26 In this case, after differentiating (4.3) we obtain
bqHL(qL,α) = b H(q L,α) qL 1− C00(bH(qL,α)) αRHqH,qH(qL,bH(qL,α)) < bH(qL,α) qL .
Since the marginal revenue is downward sloping in the own quality level, condi- tions (4.2) and (4.3) imply that the restricted best response increases in the fraction of informed customers when holding the rival’s quality constant. Hence,bαL(qH,α) ≥0
andbαH(qL,α) ≥0 with a strict inequality ifbL(qH,α) <qH andbH(qL,α) >qL, respec-
tively.
We can apply a similar reasoning as Ronnen (1991), who has examined the special case withα =1, to establish that at most one equilibrium exists.27
Proposition 4.1. For any α ∈ (0, 1], there is a unique pair of quality levels (q∗L,q∗H) that
satisfies conditions(4.2)and(4.3). When consumers correctly anticipate that the firms produce at these quality levels, the profits areΠH(q∗L,q∗H) >ΠL(q∗L,q∗H) >0. These quality levels are an equilibrium if C000 ≥0.
25This property impliesRi
qL,qL+R
i
qL,qHr=0.
26In the proof of Proposition 4.1 we show thatbH(qˆ
L,α)always exists and is bounded above.
27It seems there is a minor mistake in the proof of Ronnen (1991), Theorem 1 which has been corrected
Proof. See Appendix A4.1.
Proposition 4.1 first stipulates that there is a unique candidate equilibrium where each firm’s quality maximizes its profit when one firm is restricted to offer a quality below and the other to choose a quality above that of the rival. This result relies on the ob- servation that the best responses of both firms are increasing and that their slope is bounded and satisfiesbqLHbqHL <1 on the relevant range. Proposition 4.1 further shows that the high quality firm earns higher profits than the low quality provider. This re- sult relies on the functional form of the firms’ revenue functions and on the convexity of the costs and is not intuitive. In contrast, after rewriting the equilibrium profit as ΠL∗ =´q∗L
0 RLqL(q,q ∗
H)−C
0
(q)dq, it is easy to see that the low quality firm earns a pos- itive profit. The optimality condition (4.2) implies that the marginal profit with respect to qL, RLqL(qL,qH)−C
0
(qL), must be non-negative at (qL∗,q∗H). The profit of firm Lis
thus positive, because each firm’s marginal profit decreases in the own quality. Since both firms earn positive profits, we may conclude that it is optimal for them to enter the market in the first place. Moreover, the proposition verifies that firm H has no incentive to deviate to a quality level that lies below that of its rival. Conversely, as- suming that the cost of quality exhibits non-decreasing convexity (C000 ≥ 0 ) is grossly sufficient to guarantee that it is unprofitable for the low quality firm to deviate to a higher quality level than its competitor.
Importantly, when some consumers cannot discern the actual qualities, each firm would improve its profits if it could commit to producing higher quality goods. In equilibrium, the beliefs are correct, and the optimality conditions (4.2) and (4.3) im- ply that Πiqi(q∗L,q∗H) = (1−α)Riqi(q∗L,q∗H) > 0. Thus, if a firm increased slightly its
quality and consumers adapted their beliefs accordingly, the firm’s profit would in- crease. However, the proportion 1−α of consumers does not react to changes of the
actualquality of a good, so that each firm has insufficient incentives to invest in qual- ity. Therefore, it installs a production technology that maximizes αRi(qL,qH)−C(qi)
instead of the whole profitRi(qL,qH)−C(qi).28
For later reference, we note that a single monopolist sets the quality level q∗M so as to maximize αRM(qM)−C(qM). From limqL→0R
H(q
L,qM) = RM(qM)and RHqH,qL >0
follows that for anyα >0, the equilibrium quality level of the high quality firm exceeds
the equilibrium quality level of a monopolist.
Our next result relies on the observations that each firm’s best response increases inα
and that the qualities are strategic complements.
Lemma 4.2. The quality levels q∗L and q∗H increase in the fraction of informed customersα.
Proof. In the proof of Proposition 4.1 we argue thatq∗L necessarily satisfiesB(q∗L,α) =0
with B(q,α) ≡ bL bH(q,α),α−q and Bq(q,α) < 0 on the relevant range. We have
Bα =
bαL+bLqHbαH > 0, where the arguments are omitted for brevity. Therefore, q∗L increases inαby the implicit function theorem. Sinceq∗H =bH(q∗L,α)and bqHL >0, also
q∗H increases inα.
It is useful to relate the equilibrium qualities to those that maximize social welfare in order to assess the scope for governmental interventions. Our interest is cen- tered on instruments that affect the quality choice, but do not directly intervene in the price stage.29 Accordingly, a pair of quality levels is second-best, if the ensuing price equilibrium maximizes welfare: (qSBL ,qSBH ) ≡ arg maxqL,qHW(qL,qH). Differen-
tiating equation (4.1) and comparing to the marginal profits yields WqL > Π
L
qL and
WqH >Π
H qH.
30 The firms’ first order conditions (4.2) and (4.3) imply thatΠi qi(q
∗
L,q∗H) = (1−α)Riqi(q∗L,q∗H) >0 fori ∈ {L,H}. Therefore, we may conclude thatlocallyincreas-
ing each firm’s quality raises welfare. The following Lemma asserts that indeed both second-best quality levels lie above the equilibrium values.
Lemma 4.3. For any fraction of informed customersα ∈ (0, 1], both firms’ equilibrium quality
levels are socially too low: qSBL >q∗L and qSBH >q∗H. Proof. See Appendix A4.1.
According to Lemma 4.3, a social planner that controls the quality choices but not the ensuing prices would implement a higher quality level than each firm does in equilib- rium. The equilibrium quality levels do not correspond to those that maximize social 29Clearly, if the government could also determine the firms’ prices, it would be optimal that a single
firm serves the whole market at a price of zero. The first best quality level satisfiesC0(qFB) =´01θdθ= 12.
30Formally,W qL(qL,qH) = r2(20r−17) 2(4r−1)3 −C 0 (qL)> r 2(4r−7) (4r−1)3 −C 0 (qL) =ΠqLL(qL,qH)andWqH(qL,qH) = 24r3−18r2+5r+1 (4r−1)3 −C 0 (qH)> 4r(2−3r+4r 2) (4r−1)3 −C 0 (qH) =ΠHqL(qL,qH).
welfare for various reasons. First, we have already pointed out that the equilibrium qualities plummet as the proportion of informed customers shrinks. This is in con- trast to the second best qualities which are not affected by the information level of customers. Second, firms usually chose the quality to cater the valuation of marginal customers, while a social planner cares for themean value of quality. In addition, the quality choice usually affects how much of the total surplus can be optimally extracted by a firm as pointed out by Spence (1975) in a monopoly setup. In our duopoly setup, a third effect is present because of the strategic interaction between competitors. As dis- cussed by Ronnen (1991), the total industry profit increases in the amount of product differentiation.
Importantly, according to Lemma 4.3, the socially optimal qualities even exceed the equilibrium levels (qFIL ,qFIH) when customers are fully-informed (α = 1). Put differ-
ently, our setup exhibits a general tendency that firms offer products of too-low quality that is even reinforced when there are further informational problems.