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Countervailing Power with Large and Small Retailers

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Frontiers of Dynamic Games


When concentration in the retail market increases, retailers gain more market power towards the suppliers and they hence can achieve better wholesale prices. In the 1950s, Galbraith introduced the concept of countervailing power claiming that lower wholesale prices will pass on to consumer as lower retail prices. Consequently higher concentration may turn out to be beneficial for consumers. In this model where a monopolistic supplier sells an intermediate good to M large retailers who are Cournot competitors and a competitive fringe consisting of N retailers, we show that higher concentration does not decrease retail prices and results solely to a reallocation of profits between the supplier and large retailers, thus invalidating Galbraith’s conjecture. The same result carries on when the exogenously given level of bargaining power of large retailers increases.

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  1. 1.

    See for instance [9] and [10] and the references therein.

  2. 2.

    We use capital letters M and N to denote the set, the last element of the set or the cardinality of the set, depending on the context. We use small letters m and n to denote a typical element or an index of the set M and N respectively. So we adopt the convention M = {1, …, m, …, M} and N = {1, …, n, …, N}.


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Konstantinos Papadopoulos gratefully acknowledges Research Grant no 87937 from the Aristotle University of Thessaloniki Research Committee.

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Correspondence to Konstantinos G. Papadopoulos .

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1.1.1 Derivation of the Reaction Function of the Large Retailer q m(Q m)

Using (1.4), the profit function of a Cournot retailer as defined in (1.5) can be written as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \pi _{m} &\displaystyle =&\displaystyle [a-b(q_{m}+Q_{-m}+Q_{N}(q_{m}))-c_{m}-w_{m}]q_{m}-F_{m} \\ &\displaystyle =&\displaystyle \left[ a-b\left( q_{m}+Q_{-m}+\frac{N[a-b(Q_{-m}+q_{m})-w_{n}]}{k+bN} \right) -c_{m}-w_{m}\right] q_{m}-F_{m} \end{array} \end{aligned} $$

which we differentiate with respect to q m to obtain

$$\displaystyle \begin{aligned} \ q_{m}(Q_{-m})=\frac{ak-(c_{m}+w_{m})(k+bN)-bkQ_{-m}+bNw_{n}}{2bk}. \end{aligned}$$

1.1.2 Derivation of Bargaining Outcome (1.11) and (1.12)

Let \(\tilde {\pi }_{s}=Mw_{m}q_{m}+N(F_{n}+w_{n}q_{n})\) so that from (1.9) we can write \(\pi _{s}(F_{m},w_{m})=\tilde {\pi }_{s}+MF_{m}\). Let \( \tilde {\pi }_{m}=[p-c_{m}-w_{m}]q_{m}\) so that from (1.10) we can write \(\pi _{m}(F_{m,}w_{m})=\tilde {\pi }_{m}-F_{m}\). Cournot retailers are symmetric so \(\sum _{m-1}^{M}\gamma _{m}=M\gamma _{m}\) and given that \(\bar {\pi }_{m}=0\), (1.8) reduces to

$$\displaystyle \begin{aligned} \max_{\left( F_{m},w_{m}\right) }\left[ \tilde{\pi}_{s}+MF_{m}-\bar{\pi}_{s} \right] ^{\left( 1-M\gamma _{m}\right) }\left[ \tilde{\pi}_{m}-F_{m}\right] ^{M\gamma _{m}}. {} \end{aligned} $$

The first order condition with respect to F m is

$$\displaystyle \begin{aligned} \begin{array}{rcl} 0= &\displaystyle &\displaystyle M\left( 1-M\gamma _{m}\right) \left[ \tilde{\pi}_{s}+MF_{m}-\bar{\pi} _{s}\right] ^{-M\gamma _{m}}\left[ \tilde{\pi}_{m}-F_{m}\right] ^{M\gamma _{m}} \\ &\displaystyle &\displaystyle -M\gamma _{m}\left[ \tilde{\pi}_{s}+MF_{m}-\bar{\pi}_{s}\right] ^{\left( 1-M\gamma _{m}\right) }\left[ \tilde{\pi}_{m}-F_{m}\right] ^{(M\gamma _{m}-1)} \\ \Rightarrow &\displaystyle &\displaystyle \left( 1-M\gamma _{m}\right) /\gamma _{m}=(\tilde{\pi} _{s}+MF_{m}-\bar{\pi}_{s})/(\tilde{\pi}_{m}-F_{m}) \end{array} \end{aligned} $$


$$\displaystyle \begin{aligned} F_{m}=\tilde{\pi}_{m}-\gamma _{m}\left( M\tilde{\pi}_{m}+\tilde{\pi}_{s}- \bar{\pi}_{s}\right) . {} \end{aligned} $$

If we substitute \(\tilde {\pi }_{m}=[p-c_{m}-w_{m}]q_{m}\), \(\tilde {\pi } _{s}=Mw_{m}q_{m}+N(F_{n}+w_{n}q_{n})\) and \(\bar {\pi }_{s}=N(F_{n}+w_{n}q_{c})\) in (1.15) where q c = (a − w n)∕(k + bN) is the quantity sold at the market clearing retail price p c = (ak + bNw n)∕(k + bN) we end up with (1.11).

In order to find w m that solves (1.14), we introduce (1.15) in the objective function in (1.14) and we rearrange terms so that

$$\displaystyle \begin{aligned} &\left[ \tilde{\pi}_{s}+MF_{m}-\bar{\pi}_{s}\right] ^{\left( 1-M\gamma _{m}\right) }\left[ \tilde{\pi}_{m}-F_{m}\right] ^{M\gamma _{m}}\\ &\quad =[(1-M\gamma _{m})^{\left( 1-M\gamma _{m}\right) }\gamma _{m}^{M\gamma _{m}}]\left( M \tilde{\pi}_{m}+\tilde{\pi}_{s}-\bar{\pi}_{s}\right). \end{aligned} $$

Consequently, the maximization problem can be written as

$$\displaystyle \begin{aligned} \max_{w_{m}}M\tilde{\pi}_{m}+\tilde{\pi}_{s}-\bar{\pi}_{s} {} \end{aligned} $$

because \((1-M\gamma _{m})^{\left ( 1-M\gamma _{m}\right ) }\gamma _{m}^{\ M\gamma _{m}}\) is a constant. Notice also that \(\bar {\pi }_{s}\) does not depend on w m, so, in fact, w m maximizes the multilateral profits of the supplier with the M Cournot retailers (efficiency of Nash bargaining solution). So

$$\displaystyle \begin{aligned} \begin{array}{rcl} M\tilde{\pi}_{m}+\tilde{\pi}_{s}-\bar{\pi}_{s} &\displaystyle =&\displaystyle M(p-c_{m}-w_{m})q_{m}+Mw_{m}q_{m}+N(F_{n}+w_{n}q_{n})\\ &\displaystyle &\displaystyle -N(F_{n}+w_{n}q_{c}) \\ &\displaystyle =&\displaystyle M[p-c_{m}]q_{m}+Nw_{n}(q_{n}-q_{c}) \\ &\displaystyle =&\displaystyle M\left[ a-b(Mq_{m}(w_{m})+Nq_{n}(w_{m})-c_{m}\right] q_{m}(w_{m})\\ &\displaystyle &\displaystyle +Nw_{n}(q_{n}-q_{c}) \\ &\displaystyle =&\displaystyle NF_{n}+\frac{1}{bk(1+M)^{2}(k+bN)}Z \end{array} \end{aligned} $$

where \( Z=a^{2}k^{2}M+M(k+bN)^{2}(c_{m}+w_{m})(c_{m}-Mw_{m})+bMN(k+bN)[c_{m}(M-1)+2Mw_{m}]w_{n}-bN[k(1+M)^{2}+bM^{2}N]w_{n}^{2}+ak[M(k+bN)((M-1)w_{m}-2c_{m})+b(1+3M)Nw_{n}] \). The maximization of (1.16) with respect to w m will give (1.12).

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Geronikolaou, G., Papadopoulos, K.G. (2018). Countervailing Power with Large and Small Retailers. In: Petrosyan, L., Mazalov, V., Zenkevich, N. (eds) Frontiers of Dynamic Games. Static & Dynamic Game Theory: Foundations & Applications. Birkhäuser, Cham.

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