Dominant retailers’ incentives for product quality in asymmetric distribution channels

Abstract

This paper investigates the diverging incentives for product quality in a channel with two asymmetric retailers and a common supplier. When retailers differ in terms of service provision and channel power, changes in manufactured quality cause channel conflicts. In particular, our results show that if the low service retailer becomes dominant in the channel, it may induce a low level of quality that is detrimental for the other members of the channel. The low service retailer benefits from quality reduction first by improving its competitive standing against its rival retailer by lessening the importance of quality for consumer choice and second by strengthening its relative bargaining position vis-à-vis its supplier. Our results also show that consumer surplus may increase as a result of quality reduction.

Appendix

Proof of proposition 1

Pricing reactions in the second stage, derived from the first order conditions of the maximization in (3), are given by:

$$ {p}_1\left({w}_1,{w}_2\right)=\frac{1}{3}\left(3t+2{w}_1+{w}_2-s- qh\right) $$

(A1)

$$ {p}_2\left({w}_1,{w}_2\right)=\frac{1}{3}\left(3t+{w}_1+2{w}_2+s+ qh\right) $$

The second order conditions for the maximizations of (3) are satisfied because

$$ \frac{\partial^2{\varPi}_i}{\partial {\left({p}_i\right)}^2}=-\frac{1}{t}<0,i=1,2 $$

The first-order condition for the maximization of (5) is:

$$ qh+s+3t+2\left({w}_1-{w}_2\right)=0 $$

(A2)

In the case of a disagreement between the manufacturer and retailer 1, retailer 2 maximizes its profits Π ^− 1₂  = D ^− 1₂ p ^− 1₂ . This maximization implies

$$ {p}_2^{-1}=\frac{v+s+ hq+{w}_2}{2} $$

(A3)

Using (A3) in (12) implies the following first-order condition

$$ \left(3t- qh-s+{w}_2-{w}_1\right)\left\{\begin{array}{l}{\left(s+q-3t\right)}^2+4{w}_1^2\\ {}+{w}_1\left[5\left(s+ qh-3t\right)-8{w}_2\right]+{w}_2\left(s+ qh+3t+6v\right)-2{w}_2^2\end{array}\right\}=0 $$

(A4)

Solving (A2) and (A4) simultaneously we get the wholesale prices given in the proposition.¹⁸ As mentioned above, second order condition for the maximization of (5) is satisfied since: $\frac{\partial^2{\varPi}_M}{\partial {\left({w}_2\right)}^2}=-\frac{1}{3t}<0.$ Denoting the Nash product in (12) by F, we also verify the second order condition for the maximization at (w ^*₁ ,w ^*₂ ): $\frac{\partial^2F}{\partial {\left({w}_1\right)}^2}=-\frac{{\left(s+ qh\right)}^2+\left(243t-36s-36 qh\right)t}{216{t}^2}<0$. Equilibrium retail prices are found by using (w ^*₁ ,w ^*₂ ) in (A1) and the optimal profits by using (p ^*₁ ,p ^*₂ ) and (w ^*₁ ,w ^*₂ ) in (3) and (5).

Note that the market is covered when the negotiations are successful since under Assumption 1: ${U}_1\left({D}_1\right)=\frac{1}{2}\left(2v+s+ qh-{p}_2-{p}_1-t\right)=\frac{1}{12}\left(6v+3s+3 qh+3t-2\sqrt{3}\alpha \right)>0$. And retailer 2 does not sell to the entire market in the case of a breakdown in negotiations since for the consumer at x = 0 under Assumption1: ${U}_2^{-1}(0)=v+s+ qh-{p}_2^{-1}-t\cdot =\frac{1}{12}\left(3v+3s+3 qh-6t-\sqrt{3}\alpha \right)<0$. This justifies our demand specifications in (2) and (11). Q.E.D.

Proof of proposition 2

Using the equilibrium values in Proposition 1: $\frac{\partial {\varPi}_1^{*}}{\partial q}=-\frac{h\left(9t-s- qh\right)}{36t}<0,$ $\frac{\partial {\varPi}_2^{*}}{\partial q}=\frac{h\left(9t+s+ qh\right)}{36t}>0\;\mathrm{and}\;\frac{\partial \left[{\varPi}_M^{*}+K(q)\right]}{\partial q}=\frac{h}{12 t\alpha}\left[4\sqrt{3}t\left(s+ qh-3t\right)+6\sqrt{3} tv+\alpha \left(s+ qh+3t\right)\right]>0$ for v > 2t which is implied by the threshold given for v in Assumption 1. Q.E.D.

Proof of proposition 3

Let q* be a maximizer of $G(q)\equiv {\varPi}_1^{*}\left({\varPi}_M^{*}-{\tilde{\varPi}}_M\right)$, which uniquely exists for K(q) sufficiently convex. Then q* must satisfy the first order condition of this maximization:

$$ {\left.\frac{\partial G}{\partial q}\right|}_{q={q}^{*}}={\left.\frac{\partial {\varPi}_1^{*}}{\partial q}\right|}_{q={q}^{*}}\left[{\varPi}_M^{*}\left({q}^{*}\right)-{\tilde{\varPi}}_M\right]+{\varPi}_1^{*}\left({q}^{*}\right){\left.\frac{\partial \left({\varPi}_M^{*}-{\tilde{\varPi}}_M\right)}{\partial q}\right|}_{q={q}^{*}}=0. $$

By Proposition 2 and the fact that ${\varPi}_M^{*}\left({q}^{*}\right)-{\tilde{\varPi}}_M>0,$ the first additive term above is negative. Therefore, the second term is positive. Also observe that

$$ \operatorname{sgn}\left\{{\left.\frac{\partial {\varPi}_M^{*}}{\partial q}\right|}_{q={q}^{*}}\right\}=\operatorname{sgn}\left\{{\varPi}_1^{*}\left({q}^{*}\right){\left.\frac{\partial \left({\varPi}_M^{*}-{\tilde{\varPi}}_M\right)}{\partial q}\right|}_{q={q}^{*}}\right\} $$

since Π ^*₁ (q*) > 0 and ${\tilde{\varPi}}_M$ does not depend on q. Hence, Π ^* _M (q) is increasing in q at q ^*. Finally, since ∂ ² Π ^* _M /∂ q ² < 0 for K(q) sufficiently convex, we can conclude that M’s optimal quality $\widehat{q}$ must exceed the negotiated quality q ^*. Q.E.D.

Proof of proposition 4

Computing the consumer surplus in equilibrium as defined in (13) gives the expression: $C{S}^{*}=\frac{72 tv+{s}^2+{q}^2{h}^2+30t+81{t}^2+2 qh\left(s+15t\right)-24\sqrt{3} t\alpha}{144t}\operatorname{}$ Thus,

$$ \frac{\partial C{S}^{*}}{\partial q}= hz,\frac{\partial C{S}^{*}}{\partial s}=z,\mathrm{and}\kern0.24em \frac{\partial C{S}^{*}}{\partial h}= qz, $$

(A8)

where $z=\left[-24\sqrt{3}t\left(s+ qh-3t\right)-36\sqrt{3} tv+\alpha \left( qh+s+15t\right)\right]/\left(72 t\alpha \right)\operatorname{}$

Note that $\frac{\partial^2C{S}^{*}}{\partial v\partial q}= hy,\frac{\partial^2C{S}^{*}}{\partial v\partial s}=y,$ and $\frac{\partial^2C{S}^{*}}{\partial v\partial h}= qy$ with $y=\left[\sqrt{3}\left(2 st-81{t}^2+ qh\left(2t-v\right)\right)- sv\right]/\left(2{\alpha}^3\right)$ and that y < 0 if v > 2 t which holds under Assumption 1. Therefore, the partial derivatives in (A8) are monotonic and decreasing in v. Solving z = 0 for v, we get $\begin{array}{c}\overline{v}=\frac{1}{3\left(1071{t}^2-{q}^2{h}^2-{s}^2-30 st-2 qh\left(s+15t\right)\right)}\left\{\sqrt{3}\right[{\left( qh+s+15t\right)}^2\Big({q}^4{h}^4+{s}^4+30{s}^3t-720{s}^2{t}^2\\ {}-2430s{t}^3+86751{t}^4+2{q}^3{h}^3\left(2s+15t\right)+6{q}^2{h}^2\left({s}^2+15 st-120{t}^2\right)+2 qh\left(2{s}^3+45{s}^2t-720s{t}^2-1215{t}^3\right)\Big){\Big]}^{0.5}\\ {}+\left[3{q}^3{h}^3+3{s}^3+84{s}^2t-2097s{t}^2+6426{t}^3+3{q}^2{h}^2\left(3s+28t\right)+3 qh\right(3{s}^2+56 st-699{t}^2\left)\right]\Big\}\end{array}$

Hence, the expressed signs of the derivatives in the proposition depend on the relative order of v and $\overline{v}$. To see that $\overline{v}$ is real observe that the term within the square root in $\overline{v}$ $\begin{array}{c}m\equiv {\left( qh+s+15t\right)}^2\Big[{q}^4{h}^4+{s}^4+30{s}^3t-720{s}^2{t}^2\\ {}-2430s{t}^3+86751{t}^4+2{q}^3{h}^3\left(2s+15t\right)+6{q}^2{h}^2\left({s}^2+15 st-120{t}^2\right)+2 qh\left(2{s}^3+45{s}^2t-720s{t}^2-1215{t}^3\right)\Big]\end{array}$ is positive when t = 0 (i.e., m _t = 0 = (qh + s)⁵ > 0). Notice also that $\frac{\partial m}{\partial t}\left|{}_{t=0}=60{\left( qh+s\right)}^2>0\right.$ and ∂ (∂ m/∂ t)/∂ t = 36(28917t ² − 40(qh + s)² − 405t(s + qh)) > 0 under Assumption 1. Therefore, ∂ m/∂ t > 0 for all t ≥ 0. This implies that m > 0 and ensures that $\overline{v}\in \mathfrak{R}$. Furthermore, if s + hq < 3t then $\overline{v}>0$. Q.E.D.