Proof of Lemma 1.

From eqs. (5) and (6), consumer m will stockpile if (a) S^im=UimS−UjmS≥0 and (b) S~im=UimS−UmNS≥0. Suppose condition (a) holds, with UimS≥UjmS which implies (εim−pi1)−(εjm−pj1)≥0. It then follows from eq. (4) that UmNS=uim(pi1)+max{uAm(pA2e),uBm(pB2e)}−2κ, such that condition (b) can be rewritten as S~im=(εim−pi1)−max{εim−pi2e,εjm−pj2e}+κ≥0. Therefore, S^im (or S~im) is then strictly (or weakly) increasing in consumer m’s relative brand preference for firm i, (εim−εjm)∈[−μ,μ].    □

Proof of Lemma 2.

If ψAs=ψBs, all consumers stockpile. Here, using eq. (2), any given consumer will buy two units from i rather than j in period 1 if 2(εi−pi1)−κ≥2(εj−pj1)−κ. This comparison reduces down to that in the benchmark, εi−pi1≥εj−pj1. Hence, firm i’s total period 1 demand equals Q^i1(.)=2Qi1(.), where Qi1(pi1,pj1) coincides with the benchmark demand, eq. (1).

If, instead, ψAs>ψBs, some consumers only buy one unit in period 1. As in the benchmark, such consumers will buy one unit from firm i rather than j in period 1 if uim(pi1)−κ≥ujm(pj1)−κ. Hence, as coinciding with the benchmark demand in eq. (1), a total of Qi1(pi1,pj1) consumers buy from firm i in period 1. Of these, Qi1(pi1,pj1)−Xi(ψis) consumers buy one unit and Xi(ψis) consumers buy two units, such that total demand equals Q^i1(.)=Qi1(pi1,pj1)+Xi(ψis).   □

Proof of Lemma 3.

If ψAs=ψBs, all consumers stockpile and so Q^i2(.)=0. If instead, ψAs>ψBs, then consumers with ψm∈(ψBS,ψAS) did not stockpile and so remain active in period 2. As in the benchmark, any such consumer will then buy one unit from firm i rather than j in period 2 if uim(pi2)−κ≥ujm(pj2)−κ, and one can define ψ2 as the value of ψm=pA2−pB2 at which such a consumer would be indifferent. Around any symmetric equilibrium, ψ2∈(ψBs,ψAs). Hence, there is a positive measure of consumers with ψm∈(ψBS,ψ2) that strictly prefer to buy from firm B and a positive measure of consumers with ψm∈(ψ2,ψAS) that strictly prefer to buy from firm A. As in eq. (8), this implies that firm i^ʹs total period 2 demand is equal to the benchmark demand, Qi2(pi2,pj2) from eq. (1), minus those consumers that stockpiled from firm i in period 1, Xi(ψis).   □

Proof of Lemma 4.

Suppose ψAs>ψBs. Then in any symmetric equilibrium, it must be the case that ψ2∈(ψBs,ψAs) such that both firms have positive demand. After defining firm i’s period 2 profit function as πi2(.)=pi2Q^i2(.) and substituting in eq. (8), one can derive firm i’s period 2 best response for given stockpiling levels, pi2(pj2)=pj22+μ2(12−Xi(ψis)) for i, j ≠ i ∈ {A, B}. After repeating for firm j and solving simultaneously one obtains the unique period 2 equilibrium prices, pi2∗(.)=μ6[3−4Xi(ψis)−2Xj(ψjs)]. Substituting these back into eq. (8) gives Q^i2∗(.)=16[3−4Xi(ψis)−2Xj(ψjs)]. All such prices and demands are positive if 4Xi(ψis)+2Xj(ψjs)<3 for all i,j≠i∈A,B, which ensures ψ2∈(ψBs,ψAs) as claimed.   □

Proof of Proposition 1.

Having derived period 2 equilibrium prices, we first consider consumers’ stockpiling decisions, before deriving the equilibrium levels of stockpiling demand as a function of period 1 prices, in eq. (10).

First, consider consumers’ stockpiling decisions and initially suppose that each firm has positive period 2 demand with ψ2∈(ψBs,ψAs). This implies that a consumer at ψis makes her stockpiling decision by comparing (i) the net marginal benefits of stockpiling from firm i, with (ii) the net marginal benefits of waiting to buy from firm i, rather than firm j, in period 2. From eq. (6), this implies S~im(ψis)=(εim−pi1)−(εim−pi2∗(.)−κ). By construction, the consumer at ψis is indifferent between stockpiling, such that S~im(ψis)=0. Hence, this indifference requires pi1=pi2∗(Xi(.),Xj(.))+κ.

We are now in a position to derive the equilibrium levels of stockpiling demand as a function of period 1 prices. First, consider the top line of eq. (10). Here, pA1>pA2∗(0,0)+κ and pB1>pB2∗(0,0)+κ such that no consumer finds it optimal to stockpile, XA∗(.)=XB∗(.)=0. From eq. (9), pi2∗(0,0)=μ2 for both i = {A, B}, and so this case occurs when pA1>μ2+κ and pB1>μ2+κ.

Second, consider the bottom line of eq. (10). Here, pA1≤pA2∗(12,12)+κ and pB1≤pB2∗(12,12)+κ such that all consumers find it optimal to stockpile, XA∗(.)=XB∗(.)=12. Period 2 prices are unspecified as period 2 is inactive. However, if the marginal consumer at ψAs=ψBs=0 were to deviate from stockpiling, we know from eq. (9) that she should rationally expect zero period 2 prices, limXi→0.5pi2∗(Xi,12)=0. Hence, this case occurs when pA1≤κ and pB1≤κ.

Third, consider the middle line of eq. (10). Here, there exists a unique level of equilibrium stockpiling, Xi∗(.)∈(0,12) and Xj∗(.)∈(0,12), such that pi1=pi2∗(Xi(.),Xj(.))+κ holds for each firm. To find such Xi∗(.) and Xj∗(.), one can insert pi2∗(Xi(.),Xj(.)) from eq. (9) to obtain

After deriving a similar equation for Xj∗(.) and solving simultaneously, one finds a unique level of Xi∗(.)=12−(2pi1−pj1−κμ) for i, j ≠ i ∈ {A, B}. For Xi∗(.)∈(0,12), we require pi1∈(κ+pj12,μ2+κ+pj12] for each firm.

Finally, note that the levels of stockpiling and associated conditions in eq. (10) are continuous as (i) 12−(2pi1−pj1−κμ)=12 when pi1=pj1=κ, (ii) 12−(2pi1−pj1−κμ)=0 when pi1=pj1=μ2+κ, (iii) κ+pj12=κ when pi1=κ, and iv) 12[μ2+κ+pi1]=μ2+κ when pi1=μ2+κ.   □

Proof of Proposition 2.

From eq. (10), XA∗(.)=XB∗(.)=0 necessarily requires pA1>μ2+κ and pB1>μ2+κ. However, we know from the benchmark analysis in Section 2.2 that XA∗(.)=XB∗()=0 is consistent with p1=p2=pB=μ2. This then leads to a contradiction as p1<μ2+κ for all κ > 0.   □

Proof of Proposition 3.

First suppose that period 2 is active with ψAs>ψBs such that XA∗(.)+XB∗(.)<1. In any symmetric equilibrium each firm has positive period 2 demand with ψ2∈(ψBs,ψAs). Using eqs. (8) and (9), firm i’s profit function from eq. (11) can then be rewritten as:

where Qi1(.)=12+pj1−pi1μ from eq. (1), and where Xi∗(.) and Xj∗(.) are given in eq. (10). To maximise eq. (15) with respect to pi1 note that ∂πi1∂pi1 equals

where ∂Qi1(.)∂pi1=−1μ, ∂Xi∗(.)∂pi1=−2μ and ∂Xj∗(.)∂pi1=∂Xi∗(.)∂pj1=1μ. After expanding, enforcing symmetry with pi1=pj1=p1∗, and setting equal to zero, one obtains a unique value for p1∗=μ−κ2. There are no profitable local deviations as the associated second-order condition ensures local concavity, ∂2πi(.)∂pi12=−4μ<0. Then substituting p1∗ into eqs. (10), (9), (7) and (8) provides the unique values for X∗, p2∗, Q^1∗ and Q^2∗ as claimed. For period 2 to be active as assumed, we require X∗=3κ2μ<0.5. This implies κ<μ/3, which further ensures that the equilibrium is well-defined with non-negative prices. Hence, together with κ > 0, this case requires κ∈(0,μ3).

Second suppose that period 2 is inactive with ψAs=ψBs such that XA∗(.)+XB∗(.)=1. Firm i’s profit function then equals πi(.)=pi1[Q^i1(pi1,pj1)] which can be rewritten as follows using eq. (7):

However, to ensure XA∗(.)+XB∗(.)=1, we know from Proposition 1 that eq. (16) must be maximised subject to pi1≤κ. After solving and enforcing symmetry, this leads to a unique local maximum with p1∗=min{κ,μ2} and X∗=12. There are no profitable local deviations because the associated second-order condition ensures local concavity, ∂2πi(.)∂pi12=−2μ<0. When κ∈(0,μ3), we know from above that any symmetric equilibrium must have X∗<0.5. As this is inconsistent with XA∗(.)+XB∗(.)=1, the current case must require κ≥μ3.   □

Proof of Proposition 4.

The proof proceeds by initially deriving the equilibrium, before then comparing the level of stockpiling to that in the main model. First suppose that period 2 is active, with XA∗(.)+XB∗(.)<1. In any symmetric equilibrium each firm has positive period 2 demand. Following the arguments in the text, pi2∗ is still given by Lemma 4, and the expressions for stockpiling demand in Proposition 1, Xi∗(.) and Xj∗(.), remain valid after replacing pi1 and pj1 with the new effective costs of stockpiling a second unit, Di1−pi1 and Dj1−pj1, respectively. Each firm i must then choose pi1 and Di1 to maximise its profit function, which is now given by eq. (13) where Qi1(.)=12+pj1−pi1μ and Qi2(.)=12+pj2−pi2μ. The associated first order conditions equal ∂πi(.)∂pi1=κ−4pi1+2Di1+2pj1−Dj1μ=0 and ∂πi(.)∂Di1=4pi1−2pj1+2Dj1−4Di1−2κ+μ2μ=0. After expanding, enforcing symmetry with pi1=pj1=p1∗, and Di1=Dj1=D1∗, one obtains the unique values p1∗=μ2 and D1∗=μ−κ. There are no profitable local deviations from these prices as the second order conditions ensure local concavity, ∂2πi(.)/∂2pi1=−4μ<0, and ∂2πi(.)/∂2Di1=−2μ<0. After substituting D1∗−p1∗≡μ2−κ back into Lemma 4 and the revised version of Proposition 1, one obtains X∗=2κμ and p2∗=μ2−2κ as claimed. For the period 2 market to be active as assumed, we then require X∗=2κμ<12. This leads to κ>μ/4, which also ensures that the equilibrium is well-defined with non-negative prices. Therefore, together with κ > 0, this case requires κ∈(0,μ4).

Second, suppose that period 2 is inactive such that XA∗(.)+XB∗(.)=1. In this case, as all consumers stockpile, the firms are indifferent in period 1 between using single-unit prices or the bundle price, Di1≡2pi1. Suppose both firms only use single-unit prices. Then, the proof from the main model applies directly, with p1∗=min{μ2,κ}. Alternatively, suppose both firms use bundle prices. One can then verify that Di1∗≡2pi1∗=min{2κ,μ}. Either way, we know from above that any symmetric equilibrium under quantity discounts must have X∗<0.5 when κ∈(0,μ4). As this is inconsistent with XA∗(.)+XB∗(.)=1, the current case must require κ≥μ4.

Finally, for any κ > 0, one can show that the level of stockpiling under quantity discounts is weakly higher than that in the main model because i) the level of stockpiling within the interior solution is relatively higher, X∗=2κμ>3κ2μ, and ii) the condition for the corner solution where all consumers stockpile is less strict, κ≥μ4<μ3.   □

Proof of Proposition 5.

Proceed by contradiction. Suppose no consumers stockpile in period 1. Building on the benchmark, the firms then receive a demand of (1+α)Qit(pit,pjt) in each period. Using the first-order condition for a symmetric equilibrium, pit∗=−[Qit(pit∗,pit∗)/Qit′(pit∗,pit∗)], each firm therefore sets p1=p2=pB=μ2. However, this cannot be an equilibrium because any consumer who can stockpile with κ > 0 would then have an incentive to stockpile in period 1 as p1<p2+κ. Thus, in any symmetric equilibrium each firm must receive a positive level of stockpiling demand.   □