Proof of Lemma 1

Assume for now that pt, dti, and dtu take on values such that eqs (8) and (9) imply θti∈(−(1−A),A) and θtu∈(−(1−A),A). Rearrange eq. (10) to obtain

and substitute into eq. (11) to give

Define X(Ft)≡dtu/(αm(A−pt)), which is independent of m, since θtu is independent of m, and since dtu=αm(A−θtu). Write

Substituting eq. (18) into (16) yields

Substituting eq. (18) into (17) and solving for X(Ft) gives eq. (12), so

If pt∈[0,A), then eqs (18), (19), and (12), together with λ<1−A and m≤1, imply 0<dtu+dti<m≤1. Looking at eqs (8) and (9), and again using λ<1−A, confirms that θti∈(−(1−A),A) and θtu∈(−(1−A),A), so demand is uniquely determined by eqs (18), (19), and (12). If instead pt≥A, then demand is uniquely determined by dti=dtu=0.   □

Proof of Lemma 2

Let f denote the pdf of F, and ft the pdf of Ft, for t∈{1,2}. Bayes’ rule implies

with P(K,p1)=∫01P(K,p1|m′)dF(m′), where P(K,p1|m) follows from the firm’s equilibrium strategy. Consumers update beliefs according to Bayes’ rule, except they use P(K,p1|m)=P(K,p1). Substituting into eq. (20) yields f1(m|K,p1)=f(m), or equivalently F1(m|K,p1)=F(m), confirming that period-1 beliefs are given by the prior.

For period 2, Bayes’ rule implies

with P(p2|K,p1,q1)=∫01P(p2|m′,K,p1,q1)dF(m′), where P(p2|m,K,p1,q1) follows from the firm’s equilibrium strategy. Consumers update beliefs according to eq. (21) except they use P(p2|m,K,p1,q1)=P(p2|K,p1,q1). Together with f1(m|K,p1)=f(m), this implies

where integrating gives the distribution function

with ∫01P(q1|m′,K,p1)dF(m′)=P(q1|K,p1). The probability P(q1|m,K,p1) follows from consumer equilibrium strategies and the firm’s choice of K and p1. Specifically, demand d1(p1,F) is given by eq. (13), with sales q1=min{d1,K}.

From eq. (13), d1 is strictly increasing in m, for any p1<A. Hence, for any p1<A and resulting d1, there is a unique market size m(d1,p1) consistent with this demand and price, which is increasing in d1. The right-hand side of eq. (13) is strictly decreasing in p1, so m(d1,p1) is strictly increasing in p1.

If d1<K, then consumers observe q1=min{d1,K}<K, and they infer d1=q1. This implies P(q1|m′,K,p1)=1 for m′=m(d1,p1) and P(q1|m′,K,p1)=0 for all m′/=m(d1,p1), where m(d1,p1)=m, the true market size. Eq. (22) then reduces to Fm, given by eq. (14), as required.

If d1≥K, then consumers observe q1=min{d1,K}=K, and they infer d1≥K. This implies P(q1|m′,K,p1)=1 for all m′≥m(K,p1) and P(q1|m′,K,p1)=0 for all m′<m(K,p1). Eq. (22) then reduces to Fm(K,p1)+, given by eq. (15), as required. The threshold m(K,p1) is strictly increasing in p1 and K, since m(d1,p1) is strictly increasing in both arguments. When K=d1, the threshold reduces to m(d1,p1)=m, the true market size.   □

Proof of Proposition 1

Suppose d1 is revealed after period 1. By the proof of Lemma 2, period-2 beliefs are then F2=Fm, which are independent of K, p1, and q1. Demand is d2(p2,Fm), which by eq. (13) is proportional to (A−p2). Hence, period-2 profits, π2=p2d2(p2,Fm), are proportional to p2(A−p2), provided that K≥d2(p2,Fm). Taking the first-order condition implies an optimal period-2 price of p2=A/2, provided that K≥d2(A/2,Fm). Period-2 profits are therefore π2=d2(A/2,Fm)A/2 if K≥d2(A/2,Fm), and π2<d2(A/2,Fm)A/2 if K<d2(A/2,Fm). They are independent of p1.

Period-1 demand is d1(p1,F), which by eq. (13) is proportional to (A−p1). Hence, period-1 profits, π1=p1d1(p1,F), are proportional to p1(A−p1), provided that K≥d1(p1,F). Taking the first-order condition implies an optimal period-1 price of p1=A/2, provided that K≥d1(A/2,F). Period-1 profits are therefore π1=d1(A/2,F)A/2 if K≥d1(A/2,F) and π1<d1(A/2,F)A/2 if K<d1(A/2,F). Hence, the firm maximizes its objective function, eq. (2), by setting p1=p2=A/2, with K≥max{d1(A/2,F),d2(A/2,Fm)}. The firm can only sell out in both periods if m=m0, where m0 is the unique value of m for which d1(p,F)=d2(p,Fm). The probability that m=m0 equals zero, since the market size is drawn from an atomless distribution.   □

Proof of Proposition 2

Let πso denote the maximum profits the firm can earn given price and capacity that yield a period-1 sellout. Let πec denote the maximum profits it can earn given price and capacity that yield excess period-1 capacity. Specifically, by eq. (2), F1=F and Lemma 2, write

where static pricing gives the additional constraint p1=p2≡p.

Demand in eq. (23) is identical to that in the baseline, so Proposition 1 implies

which the firm earns by setting p=A/2, with K≥max{d1(A/2,F),d2(A/2,Fm)}.

From eq. (24), write πso=pK+δmin{d2(p,F2),K}, where F2=Fm(K,p)+. If d1(p,F)>K and d1(p2,F2)>K, then πso=pK(1+δ). Deviating to p+ϵ, for ε > 0 but small, then yields strictly higher profits, π=(p+ϵ)K(1+δ)>πso, so d1(p,F)>K and d1(p2,F2)>K cannot be optimal.

If d1(p,F)>K and d2(p,F2)≤K, then πso=p(K+δd2(p,Fm(K,p)+)). Deviating to capacity K+ϵ, for ε > 0 but small, yields profits π=p(K+ϵ+δmin{d2(p,Fm(K+ϵ,p)+),K+ϵ}). To establish π>πso, I now show that d2(p,Fm(K+ϵ,p)+)>d2(p,Fm(K,p)+). Notice first that Lemma 2 implies m(K,p)<m(K+ϵ,p). Setting Ft=Fm(K,p)+ in eq. (12) gives

The expression in each integral is positive and increasing in m^ʹ, and eq. (15) implies Fm(K+ϵ,p)+≤Fm(K,p)+ for all m^ʹ∈[0,1], with Fm(K+ϵ,p)+<Fm(K,p)+ for all m(K,p) < m^ʹ < 1. Thus, X(Fm(K+ϵ,p)+)>X(Fm(K,p)+), so that eq. (13) implies d2(p,Fm(K+ϵ,p)+)>d2(p,Fm(K,p1)+). It follows that d1(p,F)>K and d1(p,F2)≤K cannot be optimal. Hence, to earn πso, the firm must set price and capacity such that d1(p,F)=K, where Lemma 2 then implies F2=Fm+.

Write πso=pK+δmin{d2(p,Fm+),K}, with d1(p,F)=K. The above argument, to establish d2(p,Fm(K+ϵ,p)+)>d2(p,Fm(K,p1)+), showed that d2(p,Fm(K,p)+) is increasing in m(K,p). Moreover, by eq. (15), beliefs Fm(K,p)+ are identical to the prior F if and only if m(K,p) = 0. Thus, eq. (13) implies d2(p,Fm(K,p)+)≥d1(p,F) for all m(K,p)∈[0,1]. It follows that profits can be written as πso=pd1(p,F)(1+δ). Eq. (13) shows that d1(p,F) is proportional to (A−p), so the firm maximizes profits when selling out by setting p=A/2, and K=d1(p,F), to earn πso=A2d1(A/2,F)(1+δ).

Comparing πso=A2d1(A/2,F)(1+δ) with πec=A2(d1(A/2,F)+δd2(A/2,Fm) yields πso≥πec if and only if d1(A/2,F)≥d2(A/2,Fm). Recall that m0∈(0,1) is defined as the value of m satisfying X(Fm)=X(F), and that X(Fm)=1/(1−λm) is strictly increasing in m. Thus, from eq. (13), it follows that πso≥πec if and only if m≤m0.   □

Proof of Proposition 3

Consider eqs (23) and (24) from the proof of Proposition 2, where once again, demand in eq. (23) is identical to that in the baseline. This implies that πec is given by eq. (25), which the firm earns by setting p1=p2=A/2, with K≥max{d1(A/2,F),d2(A/2,Fm)}.

Write πso=Kp1+δmin{d2(p2,F2),K}p2, where F2=Fm(K,p1)+. If d2(p2,F2)>K, then πso=Kp1+δKp2. Deviating to p2+ϵ, for ε > 0 but small, then yields strictly higher profits, π=K1p1+δK(p2+ϵ)>πso. It follows that the firm sets d2(p2,F2)≤K. If d2(p2,F2)≤K and d1(p1,F)>K, then πso=Kp1+δd2(p2,Fm(K,p1)+)p2. Deviating to p1+ϵ, for ε > 0 but small, yields π=K(p1+ϵ)+δmin{d2(p2,Fm(K,p1+ϵ)+),K}p2. To establish π>πso, it is sufficient to show that d2(p2,Fm(K,p1+ϵ)+)>d2(p2,Fm(K,p1)+). Notice that Lemma 2 implies m(K,p1)<m(K,p1+ϵ). Thus, the same argument as in the proof of Proposition 2 that showed X(Fm(K+ϵ,p)+)>X(Fm(K,p)+), also implies X(Fm(K,p1+ϵ)+)>X(Fm(K,p1)+). Combined with eq. (13), this yields d2(p2,Fm(K,p1+ϵ)+)>d2(p2,Fm(K,p1)+). It follows that the firm sets d1(p1,F)=K.

Hence, profits from selling out are πso=d1(p1,F)p1+δd2(p2,Fm+)p2, where d1(p1,F)=K and d2(p2,Fm+)≤K. I now show that d2(p2,Fm+)=K. Suppose instead that d2(p2,Fm+)<K. Then by eq. (13), the firm must set p2=A/2, to maximize period-2 profits, d2(p2,Fm+)p2. The inequality d1(p1,F)>d2(A/2,Fm+) then implies p1<A/2. This is because d(p,F)<d(p,Fm+) for all m > 0, by F=F0+ from eq. (15), and by ∂∂md(p,Fm+). Moreover, p1<A/2 implies ∂∂p1d1(p1,F)p1>0, by eq. (13). Write πso=d1(p1,F)p1+δd2(A/2,Fm+)(A2), where K=d1(p1,F). Now suppose the firm deviates to period-1 price p1+ϵ and capacity d1(p1+ϵ,F)>d2(A/2,Fm+), for ε > 0 but small. This deviation yields π=d1(p1+ϵ,F)(p1+ϵ)+δd2(A/2,Fm+)(A2)>πso, by ∂∂p1d1(p1,F)p1>0. It follows that the firm will set d1(p1,F)=d2(p2,Fm+)=K, so that

evaluated at the optimal K. To solve for this optimal capacity, and by extension the corresponding prices, define

using eq. (13), which are independent of price. Substituting eqs (27) and (28) into eq. (26) gives

where differentiating with respect to K yields optimal capacity

Profits πso are therefore defined by eqs (26) and (29). Differentiating eq. (29) with respect to δ and simplifying yields

which is strictly positive, since X(Fm+)>X(F) implies C(Fm+)>C(F). Hence, eq. (29) is strictly increasing in the discount factor, tending to C(F)(A/2) as δ approaches zero and to C(Fm+)(A/2) as δ approaches infinity. Equations (27) and (28) therefore imply d1(A/2,F)<K<d2(A/2,Fm+) for all δ > 0. This is equivalent to p1<A/2<p2, by d1(p1,F)=d2(p2,Fm+)=K.

To complete the proof, it remains to show that there exists m1 and m2, with m0<m1<m2<1, such that πso>πec for all m∈[0,m1], and πec>πso for all m∈(m2,1]. Recall m0 is defined by X(Fm0)=X(F), from eqs (12) and (14), or equivalently by d1(p,F)=d2(p,Fm0).

Suppose m≤m0. In this case, d1(p,F)≥d2(p,Fm), so that eq. (25) implies πec≤d1(A/2,F)A/2+δd1(A/2,F)A/2. Now say the firm sets p1=p2=A/2 and K=d1(A/2,F). Then it earns d1(A/2,F)A/2+δd1(A/2,F)A/2, which is strictly less than πso, since the optimal prices when selling out satisfy p1<A/2<p2. It follows that πso>πec. Moreover, both πso and πec are continuous in m, so there exists m1>m0 such that πso>πec for all m∈[0,m1].

Now suppose m = 1. In this case, πec=d1(A/2,F)A/2+δd2(A/2,Fm=1)A/2 and πso=d1(p1,F)p1+δd2(p2,F(m=1)+)p2, with p1<A/2<p2. Equations (14) and (15) imply Fm=1=F(m=1)+. Moreover, eq. (13) shows that d1(p1,F)p1 and d2(p2,Fm=1)p2 have a unique maximum at p1=A/2 and p2=A/2. It follows that πec>πso when m = 1, by p1<A/2<p2. Both πso and πec are continuous in m, so there exists m2<1 such that πec>πso for all m∈(m2,1].   □

Proof of Proposition 4

Suppose m∈[0,m1), so that πso>πec. Proposition 3 shows that the equilibrium prices, under dynamic pricing, satisfy d1(p1,F)=d2(p2,Fm+)=K, given eqs (27), (28), and (29). I now vary various parameters and demonstrate the impact on these equilibrium prices. In particular, we have ∂pt∂δ=∂pt∂K∂K∂δ for t∈{1,2}. Clearly, ∂pt∂K<0 for t∈{1,2}, since d1(p,F) and d2(p,Fm+) are both decreasing in p. Since ∂K∂δ>0 by eq. (30), it follows that ∂p1∂δ<0 and ∂p2∂δ<0.

Write p1=A−KC(F) and p2=A−KC(Fm+), by eqs (27) and (28). Using eq. (29) to substitute for K in these expressions gives

which immediately yields Sign(∂p1∂x)=Sign(∂∂xC(F)C(Fm+))=−Sign(∂p2∂x), for x∈{m,λ,α}.

Suppose α = 1. In this case, eqs (27) and (28) imply C(F) = mX(F) and C(Fm+)=mX(Fm+), so that

Setting α = 1 in eq. (12) gives X(F)=1/(1−λE(m|F)) and X(Fm+)=1/(1−λE(m|Fm+)), where I have written the integral ∫01m′dFt(m′) as E(m|Ft). If x = m, then ∂X(F)∂x=0 and ∂X(Fm+)∂x>0, so that eq. (31) is strictly negative. This implies ∂p1∂m<0 and ∂p2∂m>0 when evaluated at α = 1. If x = λ, then the numerator of eq. (31) becomes

which is strictly negative by E(m|Fm+)>E(m|F). It follows that ∂p1∂λ<0 and ∂p2∂λ>0 when evaluated at α = 1. All expressions are continuous in α, so there exists αˉ∈(0,1) such that ∂p1∂m<0, ∂p1∂λ<0, ∂p2∂m>0, and ∂p2∂λ>0 for all α∈(αˉ,1].

To establish ∂∂δP(q1=q2=K)>0, I need to show that dπsodδ>dπecdδ when evaluated at m such that πso=πec, with πec and πso given by eqs (25), (26) and (27). Such an m exists since πso>πec for m∈[0,m1) and πso<πec for m∈(m2,1], and since both πso and πec are continuous in m. Moreover, P(q1=q2=K)≥F(m0) follows immediately from m1>m0.

From eq. (25), write

From eq. (26), write

The envelope theorem implies ∂πso∂K=0 at the optimal capacity, eq. (29), so that

Comparing eqs (32) with (33) shows dπsodδ−dπecdδ=Kp2−A2d2(A/2,Fm). Rewrite πso−πec=0 as

where K=d1(p1,F)=d2(p2,Fm+). The first expression in large brackets is strictly negative, since p1<A/2, and since pd1(p,F) has a unique maximum at p=A/2. It then follows from δ > 0 that Kp2−A2d2(A/2,Fm)>0, and hence dπsodδ>dπecdδ, as required.   □

Proof of Proposition 5

Expected profits from Proposition 3 are ∫01max{πso,πec}dF, defined by eqs (25), (26), and (29). Expected profits in the baseline are ∫01πecdF. Proposition 3 shows that P(πso>πec)>F(m0)>0, which implies ∫01max{πso,πec}dF>∫01πecdF.

To establish the results for consumer payoffs, it is sufficient to consider values of m for which πso>πec, since otherwise the firm sets the baseline price and capacity. First consider period-1 consumers. Proposition 3 shows that p1<A/2, with d1(p1,F)=q1(p1,F)=K. By eq. (1), a type θ consumer earns θ+λd1(p1,F)−p1 from buying. She buys when θ+λd1(p1,F)−p1≥0 if informed, and when θ+λ∫01d1(p1,F)dF−p1≥0 if uninformed. In the baseline, a type θ consumer earns θ+λd1(A/2,F)−A/2 from buying. She buys when θ+λd1(A/2,F)−A/2≥0 if informed, and when θ+λ∫01d1(A/2,F)dF−A/2≥0 if uninformed. Hence, p1<A/2 and ∂d1(p1,F)∂p<0 imply that all consumers who buy in the baseline also buy according to Proposition 3, and earn a strictly higher payoff. Informed consumers who do not buy in the baseline cannot earn less according to Proposition 3, as they will only buy if their payoff exceeds zero.

Now consider period-2 consumers. Proposition 3 shows that p2>A/2, with d2(p2,Fm+)=q2(p2,Fm+)=K. An uninformed type θ consumer earns θ+λd2(p2,Fm+)−p2 from buying, and she buys if θ+λ∫01d2(p2,Fm+)dFm+−p2≥0. Define θ^ʹ as the supremum of the set θ<p2−λ∫01d2(p2,Fm+)dFm+, taken over all m such that πso>πec. For any ϵ∈(0,1−θ′], there exists a market size m such that (i) πso>πec holds, and (ii) an uninformed consumer of type θ′+ϵ buys, i.e. θ′+ϵ+λ∫01d2(p2,Fm+)dFm+−p2≥0. Now take a sequence ϵl→0. It must be that θ′+ϵl+λ∫01d2(p2,Fm+)dFm+−p2→0, for all m such that πso>πec, by the definition of θ^ʹ. Recall that d2(p2,Fm+)<∫01d2(p2,Fm+)dFm+. Thus, for l sufficiently large, type θ+ϵl will only buy when θ′+ϵl+λd2(p2,Fm+)−p2<0. She therefore earns a negative expected payoff, taken over all m such that πso>πec, whereas her payoff in the baseline is bounded below by zero.