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Licensed Unlicensed Requires Authentication Published by De Gruyter May 24, 2019

Transaction Costs as a Source of Consumer Stockpiling

Luke Garrod, Ruochen Li and Chris M. Wilson

Abstract

Consumers often stockpile goods to store for future consumption. The existing theoretical literature has focussed on a price-based explanation where stockpiling arises due to temporary price reductions. In contrast, this paper explores a transaction-cost-based explanation where consumers stockpile to avoid the need to incur future transaction costs. It shows how transaction costs lead to positive consumer stockpiling in an oligopoly equilibrium even when future prices are expected to fall. Relative to a no-stockpiling benchmark, such stockpiling lowers profits, but improves consumer and total welfare. Our results extend to the case of quantity discounts where stockpiling consumers pay relatively lower per-unit prices than non-stockpiling consumers, when purchasing multi-unit bundles.

JEL Classification: L13; D43; D11; M31

Acknowledgements

We are very thankful for comments from the editor (Hendrik Juerges), two anonymous referees, Tobias Wenzel, and from various conference and seminar audiences including EARIE (2017), Jornadas de Econom\’ıa Industrial (2017), Nottingham University Business School and Loughborough University. Li acknowledges the support from "The Fundamental Research Funds of Shandong University (2019HW008)".

Appendix

A

Proof of Lemma 1.

From eqs. (5) and (6), consumer m will stockpile if (a) S^im=UimSUjmS0 and (b) S~im=UimSUmNS0. Suppose condition (a) holds, with UimSUjmS which implies (εimpi1)(εjmpj1)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=(εimpi1)max{εimpi2e,εjmpj2e}+κ0. Therefore, S^im (or S~im) is then strictly (or weakly) increasing in consumer m’s relative brand preference for firm i, (εimεjm)[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(εipi1)κ2(εjpj1)κ. This comparison reduces down to that in the benchmark, εipi1εjpj1. 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)κujm(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)κujm(pj2)κ, and one can define ψ2 as the value of ψm=pA2pB2 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 iʹs 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(12Xi(ψis)) for i, ji ∈ {A, B}. After repeating for firm j and solving simultaneously one obtains the unique period 2 equilibrium prices, pi2(.)=μ6[34Xi(ψis)2Xj(ψjs)]. Substituting these back into eq. (8) gives Q^i2(.)=16[34Xi(ψis)2Xj(ψjs)]. All such prices and demands are positive if 4Xi(ψis)+2Xj(ψjs)<3 for all i,jiA,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)=(εimpi1)(εimpi2(.)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, pA1pA2(12,12)+κ and pB1pB2(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, limXi0.5pi2(Xi,12)=0. Hence, this case occurs when pA1κ and pB1κ.

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

(14)Xi(.)=34Xj(.)23(pi1κ)2μ.

After deriving a similar equation for Xj(.) and solving simultaneously, one finds a unique level of Xi(.)=12(2pi1pj1κμ) for i, ji ∈ {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(2pi1pj1κμ)=12 when pi1=pj1=κ, (ii) 12(2pi1pj1κμ)=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:

(15)πi(.)=pi1[Qi1(pi1,pj1)+Xi(.)]+μ(34Xi(.)2Xj(.)6)2

where Qi1(.)=12+pj1pi1μ from eq. (1), and where Xi(.) and Xj(.) are given in eq. (10). To maximise eq. (15) with respect to pi1 note that πi1pi1 equals

Qi1(.)+Xi(.)+pi1(Qi1(.)pi1+Xi(.)pi1)+μ3(34Xi(.)2Xj(.))(23Xi(.)pi113Xj(.)pi1),

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=μκ2. 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):

(16)πi(.)=pi1[2Qi1(pi1,pj1)]=2pi1[12+pj1pi1μ]

However, to ensure XA(.)+XB(.)=1, we know from Proposition 1 that eq. (16) must be maximised subject to pi1κ. 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 κμ3.   □

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, Di1pi1 and Dj1pj1, 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+pj1pi1μ and Qi2(.)=12+pj2pi2μ. The associated first order conditions equal πi(.)pi1=κ4pi1+2Di1+2pj1Dj1μ=0 and πi(.)Di1=4pi12pj1+2Dj14Di12κ+μ2μ=0. After expanding, enforcing symmetry with pi1=pj1=p1, and Di1=Dj1=D1, one obtains the unique values p1=μ2 and D1=μκ. 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 D1p1μ2κ back into Lemma 4 and the revised version of Proposition 1, one obtains X=2κμ and p2=μ22κ 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, Di12pi1. 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 Di12pi1=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 κμ4.

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, κμ4<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.   □

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Published Online: 2019-05-24

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