On the effects of supplier encroachment under endogenous quantity leadership

Abstract

Prior literature has shown that, for exogenously prescribed sequence of quantity decisions, supplier encroachment into a retailer’s market can mitigate double marginalization and thereby benefit both the supplier and the retailer. This study extends existing understanding of supplier encroachment to the contexts in which the Stackelberg leadership/followership preferences of the two firms are endogenous under exogenous and endogenous wholesale prices. In our model, the two firms can engage in one of the three quantity competition games: simultaneous game; sequential game with the retailer being the Stackelberg leader; and sequential game with the retailer being the Stackelberg follower. Accordingly, this study reexamines the effects of encroachment in tandem with the Stackelberg leadership/followership decisions of the two firms. Under exogenous wholesale price, either firm may choose the Stackelberg leader or follower role contingent on their relative operational costs. Under endogenous wholesale price, the supplier strategically relies less on wholesale price concessions to induce the retailer-as-leader sequential game as an equilibrium. In either scenario, encroachment always benefits consumers, but it cannot secure Pareto gains. Encroachment leads to either a “win-lose” or a “lose-lose” outcome for the supplier and retailer because endogenizing the Stackelberg leadership/followership preferences radically influences the interplay between the wholesale price effect and competition effect of encroachment. We find that the ineluctable need to cultivate the reselling channel is likely to cause the encroaching supplier to write a contingent contract characterized by per-unit wholesale markups and lump-sum slotting allowances (i.e., fixed transfer from the supplier to the retailer).

Appendices

Appendix A: No encroachment benchmark

1.1 A.1. Exogenous wholesale price

In the benchmark, the manufacturer does not have the option to encroach and reaches end consumers only through its downstream retailer. In this setting, the retailer chooses its output $q_R$ to maximize its monopoly profit from retail sales, taking the per-unit wholesale price w as given. The retailer’s problem is $\max \limits _{q_R\ge 0} \pi _R= [a-bq_R]q_R-wq_R$. Given $\frac{\partial ^2\pi _R}{\partial q_R^2}=-2b<0$ for any $b>0$, the retailer’s profit function is strictly concave in $q_R$, and solving the first-order condition $\frac{\partial \pi _R}{\partial q_R}=a-2bq_R-w=0$ yields ${\overline{q}}_R(w)$, the retailer’s output in the no-encroachment setting given per-unit wholesale price w:

$$\begin{aligned} {\overline{q}}_R(w)=\frac{a-w}{2b}. \end{aligned}$$

(A.1)

Substituting ${\overline{q}}_R(w)$ into the profit functions of the retailer, $\pi _R(w)=[a-bq_R(w)]q_R(w)-wq_R(w)$, and manufacturer, $\pi _M(w)=wq_R(w)$, yields:

$$\begin{aligned} {\overline{\pi }}_R(w)=\frac{[a-w]^2}{4b} \text{ and } {\overline{\pi }}_M(w)=\frac{w[a-w]}{2b}. \end{aligned}$$

(A.2)

The resulting consumer surplus in the no-encroachment setting is

$$\begin{aligned} \overline{\textsc {cs}}(w)=\int _{0}^{{\overline{q}}_R(w)}b[{\overline{q}}_R(w)-q]dq=\frac{b}{2}[{\overline{q}}_R(w)]^2=\frac{[a-w]^2}{8b}. \end{aligned}$$

(A.3)

Given $b>0$, $a>0$ and $w>0$ (by definition), ${\overline{q}}_R(w)>0$ and therefore ${\overline{\pi }}_M(w)>0$ if and only if $a>w$. To omit cases in which no production occurs, the upper bound on the market size a is assumed to be sufficiently large relative to the wholesale price: $a>w$.

1.2 A.2. Endogenous wholesale price

Under endogenized wholesale price, anticipating the retailer’s response to the wholesale price it sets, the manufacturer chooses w to maximize its profit given in (A.2): $\max \limits _{w\ge 0} {\overline{\pi }}_M(w)=\frac{w[a-w]}{2b}$. Given $\frac{\partial ^2 {\overline{\pi }}_M}{\partial w^2}=-\frac{1}{b}<0$ for any $b>0$, the manufacturer’s profit function is strictly concave in w, and solving the first-order condition $\frac{\partial {\overline{\pi }}_M(w)}{\partial w}=\frac{a-2w}{2b}=0$ yields ${\overline{w}}$, the wholesale price of the product in the no-encroachment setting: ${\overline{w}}=\frac{a}{2}$. Substituting ${\overline{w}}$ into (A.1), (A.2) and (A.3) yields the retailer’s order quantity, the profits of the retailer and the manufacturer, and the consumer surplus as follows: ${\overline{q}}_R = \frac{a}{4b}$; ${\overline{\pi }}_R = \frac{a^2}{16b}$; ${\overline{\pi }}_M = \frac{a^2}{8b}$; and $\overline{\textsc {cs}}=\frac{a^2}{32b}$. $\square$

Appendix B: Proof of Lemma 1

In the encroachment setting, the profit functions of the retailer and manufacturer are respectively as follows:

$$\begin{aligned} \pi _R= & {} [a-b(q_R+kq_M)]q_R-wq_R \end{aligned}$$

(B.1)

$$\begin{aligned} \pi _M= & {} [a-b(q_M+kq_R)-c]q_M+wq_R \end{aligned}$$

(B.2)

For the simultaneous game, the manufacturer and retailer choose their respective retail output levels $q_M$ and $q_R$ simultaneously. The second-order derivative of each firm’s profit function with respect to its quantity is $\frac{\partial ^2 \pi _M}{\partial q_M^2}=\frac{\partial ^2 \pi _R}{\partial q_R^2}=-2b<0$ for any $b>0$. The best response functions are $q_M(q_R)=\frac{a-c-bkq_R}{2b}$ and $q_R(q_M)=\frac{a-w-bkq_M}{2b}$. Solving these two equations yields the equilibrium quantities $q_M^S$ and $q_R^S$ as follows:

$$\begin{aligned} q_M^S = \frac{2[a-c]-k[a-w]}{b[4-k^2]}, \text{ and } q_R^S = \frac{2[a-w]-k[a-c]}{b[4-k^2]}. \end{aligned}$$

(B.3)

The corresponding profits can be obtained by substituting $q_M^S$ and $q_R^S$ into the function in (B.1) and (B.2):

$$\begin{aligned} \pi _M^S= & {} \frac{\left[ 2(a-c)-ak\right] ^2+aw\left[ 8-k^2(4-k)\right] -w^2[8-3k^2]-wck^3}{b[4-k^2]^2}, \text{ and } \pi _R^S \nonumber \\= & {} \frac{\left[ 2(a-w)-k(a-c)\right] ^2}{b[4-k^2]^2}. \end{aligned}$$

(B.4)

The resulting consumer surplus is calculated as follows:

$$\begin{aligned} \textsc {cs}^S= & {} \frac{b}{2}\left[ q_M^2+q_R^2+2kq_Mq_R\right] \nonumber \\= & {} \frac{[4-3k^2][w^2+c^2]+2wck^3+2a[a-w-c][1+k][2-k]^2}{2b[4-k^2]^2}. \end{aligned}$$

(B.5)

Given $a>w>0$, $a>c>0$, $b>0$ and $k\in (0,1)$, $q_R^S>0$ if and only if $a>\max \left\{ c,\frac{2w-ck}{2-k}\right\}$ and $q_M^S>0$ if and only if $a>\max \left\{ w,\frac{2c-wk}{2-k}\right\}$. Taken together, both firms are non-trivial participants in the retail market if and only if $a>\max \left\{ \frac{2c-wk}{2-k},\frac{2w-ck}{2-k}\right\}$, or equivalently if and only if $\frac{a[2-k]+ck}{2}>w>\max \left\{ 0,\frac{ak-2[a-c]}{k}\right\}$.

For the retailer-as-leader game, knowing the retailer’s quantity $q_R$, the manufacturer chooses its quantity $q_M$ to maximize its total profit in (B.1). Solving the first-order condition yields

$$\begin{aligned} q_M(q_R) = \frac{a-c-bkq_R}{2b}. \end{aligned}$$

(B.6)

Substituting $q_M(q_R)$ into the retailer’s profit function in (B.1) yields $\pi _R=\frac{q_R}{2}\big [2(a-w)-k(a-c)-b(2-k^2)q_R\big ]$. The second-order derivative of $\pi _R$ with respect to $q_R$ is $\frac{\partial ^2 \pi _R}{\partial q_R^2}=-b[2-k^2]<0$ for any $b>0$ and $k\in (0,1)$. Solving the first-order condition yields the equilibrium quantity of the retailer along with the manufacturer as follows:

$$\begin{aligned} q_R^L = \frac{2[a-w]-k[a-c]}{2b[2-k^2]}, \text{ and } q_M^F = \frac{[a-c][4-k^2]-2k[a-w]}{4b[2-k^2]}. \end{aligned}$$

(B.7)

The corresponding profits can be obtained by substituting $q_R^L$ and $q_M^F$ into the function in (B.1) and (B.2):

$$\begin{aligned} \pi _R^L= & {} \frac{[2(a-w)-k(a-c)]^2}{8b[2-k^2]}, \text{ and } \pi _M^F\nonumber \\= & {} \frac{[(4-k^2)(a-c)-2ka]^2+4w[(8-5k^2)(a-w)+k^3(a-c)-ak^2]}{16b[2-k^2]^2}. \end{aligned}$$

(B.8)

The resulting consumer surplus is obtained as follows:

$$\begin{aligned} \textsc {cs}^L= & {} \frac{[2(2a-w-c)(2-k)-(a-c)k^2]^2}{32b[2-k^2]^2}\nonumber \\{} & {} +\frac{[1-k][4(2a-w-c)k-8(a-w)(a-c)(1+k)-(a-c)^2k^3]}{8b[2-k]^2}. \end{aligned}$$

(B.9)

Given $a>w>0$, $a>c>0$, $b>0$ and $k\in (0,1)$, $q_R^L>0$ if and only if $a>\max \left\{ c,\frac{2w-ck}{2-k}\right\}$ and $q_M^F>0$ if and only if $a>\max \left\{ w,\frac{c[4-k^2]-2wk}{4-2k-k^2}\right\}$. Taken together, both firms are non-trivial participants in the retail market if and only if $a>\max \left\{ \frac{2w-ck}{2-k},\frac{c[4-k^2]-2wk}{4-2k-k^2}\right\}$, or equivalently if and only if $\frac{a[2-k]+ck}{2}>w>\max \left\{ 0,\frac{c[4-k^2]-a[4-2k-k^2]}{2k}\right\}$.

For the manufacturer-as-leader game, knowing the manufacturer’s quantity $q_M$, the retailer chooses its quantity $q_R$ to maximize its profit in (B.1). Solving the first-order condition yields

$$\begin{aligned} q_R(q_M) = \frac{a-w-bkq_M}{2b} \end{aligned}$$

(B.10)

Substituting $q_R(q_M)$ into the manufacturer’s profit function in (B.2) yields $\pi _M=\frac{w[a-w]+bq_M[a(2-k)-2c-b(2-k^2)q_M]}{2b}$. The second-order derivative of $\pi _M$ with respect to $q_M$ is $\frac{\partial ^2 \pi _M}{\partial q_M^2}=-b[2-k^2]<0$ for any $b>0$ and $k\in (0,1)$. Solving the first-order condition yields the equilibrium quantity of the manufacturer along with the retailer as follows:

$$\begin{aligned} q_M^L = \frac{2[a-c]-k\alpha }{2b[2-k^2]}, \text{ and } q_R^F = \frac{[a-w][4-2k^2]-2[a-c]k+ak^2}{4b[2-k^2]}. \end{aligned}$$

(B.11)

The corresponding profits can be obtained by substituting $q_R^F$ and $q_M^L$ into the function in (B.1) and (B.2):

$$\begin{aligned} \pi _R^F= & {} \frac{[2(a-w)(2-k^2)-2k(a-c)+ak^2]^2}{16b[2-k^2]^2}, \text{ and } \pi _M^L \nonumber \\ {}= & {} \frac{4w[a-w][2-k^2]-[2(a-c)-ak]^2}{8b[2-k^2]}. \end{aligned}$$

(B.12)

The resulting consumer surplus is obtained as follows:

$$\begin{aligned} \textsc {cs}^F= & {} \frac{[a(8-4k-k^2)-2w(2-k^2)-2c(2-k)]^2}{16b[2-k^2]}\nonumber \\{} & {} +\frac{[1-k][2(a-c)-ak][a(4-2k-k^2)-2w(2-k^2)+2ck]}{8b[2-k^2]^2}. \end{aligned}$$

(B.13)

Given $a>w>0$, $a>c>0$, $b>0$ and $k\in (0,1)$, $q_R^F>0$ if and only if $a>\max \left\{ c,w,\frac{2w[2-k^2]-2ck}{4-2k-k^2}\right\}$ and $q_M^L>0$ if and only if $a>\max \left\{ w,\frac{2c}{2-k}\right\}$. Taken together, both firms are non-trivial participants in the retail market if and only if $a>\max \left\{ \frac{2c}{2-k},\frac{2w[2-k^2]-2kc}{4-k[2+k]}\right\}$, or equivalently if and only if $w<\frac{a[4-2k-k^2]+2kc}{2[2-k^2]}$. $\square$

Table 2 Effect of the wholesale price and the manufacturer’s selling cost on equilibrium retail output

Appendix C: Proof of Proposition 1

When $a>\max \left\{ \frac{2w-kc}{2-k},\frac{2c}{2-k},\frac{c[4-k^2]-2kw}{4-k[2+k]}\right\}$, or equivalently $\frac{2a-k[a-c]}{2}>w>\max \left\{ 0,\frac{2ak-[a-c][4-k^2]}{2k}\right\}$, all three basic games exist. In the following, we compare the performance of the retailer and the manufacturer under the three basic games derived in Appendix B. First, we show that

$$\begin{aligned} \pi _R^L-\pi _R^S= & {} \frac{[2(a-w)-k(a-c)]^2}{8b[2-k^2]}-\frac{\left[ 2(a-w)-k(a-c)\right] ^2}{b[4-k^2]^2}\nonumber \\= & {} \frac{k^4[2(a-w)-k(a-c)]^2}{8b[4-k^2]^2[2-k^2]}>0, \text{ and } \end{aligned}$$

(C.1)

$$\begin{aligned} \pi _M^L-\pi _M^S= & {} \frac{4w[a-w][2-k^2]-[2(a-c)-ak]^2}{8b[2-k^2]}-\frac{[2(a-c)-k(a-w)]^2}{b[4-k^2]^2}\nonumber \\{} & {} -\frac{w[2(a-w)-k(a-c)]}{b[4-k^2]} \nonumber \\= & {} \frac{k^2[2k(a-c)-k^2(a-w)-w(4-k^2)]^2}{8b[4-k^2]^2[2-k^2]}>0. \end{aligned}$$

(C.2)

Therefore, $\pi _R^L-\pi _R^S>0$ and $\pi _M^L-\pi _M^S>0$ given $b>0$ and $k\in (0,1)$. Next, we show that

$$\begin{aligned} \pi _M^F - \pi _M^S= & {} \frac{[(4-k^2)(a-c)-2k(a-w)]^2}{16b[2-k^2]^2}+\frac{w\left[ 2(a-w)-k(a-c)\right] }{2b\left[ 2-k^2\right] }\nonumber \\{} & {} -\frac{[2(a-c)-k(a-w)]^2}{b[4-k^2]^2}-\frac{w[2(a-w)-k(a-c)]}{b[4-k^2]} \nonumber \\= & {} \frac{k^2[32(1-k^2)+7k^4][2(a-w)-k(a-c)]}{8b[4-k^2]^2[2-k^2]^2} \nonumber \\{} & {} \left[ w-\frac{k(a-c)(32-16k^2+k^4)-2ak^2(8-3k^2)}{64(1-k^2)+14k^4}\right] . \end{aligned}$$

(C.3)

Then, the sign of $\pi _M^F - \pi _M^S$ depends on the value of w relative to $\frac{k(a-c)(32-16k^2+k^4)-2ak^2(8-3k^2)}{64(1-k^2)+14k^4}$, denoted by ${\widehat{w}}_L(c,k)$. Specifically, for values of $w>{\widehat{w}}_L(c,k)$, $\pi _M^F-\pi _M^S>0$, and $\pi _M^F-\pi _M^S<0$ for values of $w<{\widehat{w}}_L(c,k)$. This means that for values of $w<{\widehat{w}}_L(c,k)$, we have both $\pi _M^L>\pi _M^S$ and $\pi _M^S>\pi _M^F$ and therefore L is a strict dominant strategy for the manufacturer. The equation ${\widehat{w}}_L(c,k)-\frac{2ka-[a-c][4-k^2]}{2k}=0$ has a unique solution of ${\overline{c}}_M(k)=a-\frac{4ak[4-k^2]}{32-16k^2+3k^4}>0$, with ${\widehat{w}}_L(c,k)<\frac{2ka-[a-c][4-k^2]}{2k}$ if $c>{\overline{c}}_M(k)$. Thus, for values of $c>{\overline{c}}_M(k)$, we have $\pi _M^F-\pi _M^S>0$ regardless of the value of $w>0$. Since $\frac{\partial {\overline{c}}(k)}{\partial k}=-\frac{4a[128-32k^2-20k^4+3k^6]}{[32-16k^2+3k^4]^2}<0$, the cutoff ${\overline{c}}_M(k)$ is a decreasing function of k.

Similarly, we show that

$$\begin{aligned} \pi _R^F - \pi _R^S= & {} \frac{[2(a-w)(2-k^2)-2k(a-c)+ak^2]^2}{16b[2-k^2]^2}-\frac{\left[ 2(a-w)-k(a-c)\right] ^2}{b[4-k^2]^2} \nonumber \\= & {} \frac{4b[4-k^2]^2}{k^2[8-k^2]}\left[ \frac{2ck(8-3k^2)+a(2-k)(16-8k^2-k^3)}{2(8-k^2)(2-k^2)}-w\right] \nonumber \\{} & {} \times \left[ w-\frac{2k(a-c)-ak^2}{2(2-k^2)}\right] . \end{aligned}$$

(C.4)

Since $\frac{2a-k[a-c]}{2}-\frac{2ck(8-3k^2)+a(2-k)(16-8k^2-k^3)}{2(8-k^2)(2-k^2)}=-\frac{k^2[a-k(a-c)][4-k^2]}{2[8-k^2][2-k^2]}<0$, given that $w<\frac{2a-k[a-c]}{2}$ for all three basic games to exist, $w<\frac{2ck(8-3k^2)+a(2-k)(16-8k^2-k^3)}{2(8-k^2)(2-k^2)}$. Then, the sign of $\pi _R^F-\pi _R^S$ depends on the value of w relative to $\frac{2k(a-c)-ak^2}{2(2-k^2)}$, denoted by ${\widehat{w}}_U(c,k)$. Specifically, for values of $w>{\widehat{w}}_U(c,k)$, $\pi _R^F-\pi _R^S>0$, and $\pi _R^F-\pi _R^S<0$ for values of $w<{\widehat{w}}_U(c,k)$. This means that for values of $w<{\widehat{w}}_U(c,k)$, we have both $\pi _R^L>\pi _R^S$ and $\pi _R^S>\pi _R^F$ and thereby L is a strict dominant strategy for the manufacturer. The equation ${\widehat{w}}_L(c,k)-\frac{2ka-[a-c][4-k^2]}{2k}=0$ has a unique solution of ${\overline{c}}_R(k)=a-\frac{ak[4-k^2]}{8-4k^2+k^4}>{\overline{c}}_M(k)>0$, with ${\widehat{w}}_U(c,k)<\frac{2ka-[a-c][4-k^2]}{2k}$ if $c>{\overline{c}}_R(k)$. Thus, for values of $c>{\overline{c}}_R(k)$, we have $\pi _R^F-\pi _R^S>0$ regardless of the value of $w>0$.

Comparison of two w-cutoffs reveals that ${\widehat{w}}_U(c,k)-{\widehat{w}}_L(c,k)=\frac{k^4[a-k(a-c)][4-k^2]}{2[2-k^2][32(1-k^2)+7k^4]}>0$ for any $a>c>0$ and $k\in (0,1)$, with $\frac{\partial [{\widehat{w}}_U(c,k)-{\widehat{w}}_L(c,k)]}{\partial c}=\frac{k^5[4-k^2]}{2[64-96k^2+46k^4-7k^6]}>0$ and $\frac{\partial [{\widehat{w}}_U(c,k)-{\widehat{w}}_L(c,k)]}{\partial k}=\frac{k^3[a(1024-1152k^2+384k^4-36k^6)-(a-c)k(1280-1600k^2+664k^4-110k^6+7k^8)]}{2[64-96k^2+46k^4-7k^6]^2}>0$. Since $\frac{\partial {\overline{c}}_R(k)}{\partial k}=-\frac{a[32-8k^2-8k^4+k^6]}{[8-4k^2+k^4]^2}<0$, the cutoff ${\overline{c}}_R(k)$ is a decreasing function of k. In addition, $\frac{\partial {\widehat{w}}_U(c,k)}{\partial c}=-\frac{k[32-16k^2+k^4]}{64[1-k^2]+14k^4}<0$ for any $k\in (0,1)$, with ${\widehat{w}}_U(c,k)=\frac{ak[2-k]}{2[2-k^2]}$ at $c=0$. Then, for values of $w>\frac{ak[2-k]}{2[2-k^2]}$, we have $w>{\widehat{w}}_U(c,k)$ and $w>{\widehat{w}}_L(c,k)$ for any $c \in (0,a)$ and $k\in (0,1)$.

In sum, given $a>c>0$, $k\in (0,1)$ and $\frac{2a-k[a-c]}{2}>w>\max \left\{ 0,\frac{2ak-[a-c][4-k^2]}{2k}\right\}$, for $w<{\widehat{w}}_L(c,k)$, we have that $\pi _i^L-\pi _i^S>0$ and $\pi _i^S-\pi _i^F>0$ for $i=\{R,M\}$. Thus, L is a strict dominant strategy for both the manufacturer and the retailer, and (L, L) is the unique equilibrium under encroachment.

Given $a>c>0$, $k\in (0,1)$ and $\frac{2a-k[a-c]}{2}>w>\max \left\{ 0,\frac{2ak-[a-c][4-k^2]}{2k}\right\}$, for $w>{\widehat{w}}_L(c,k)$ but $w<{\widehat{w}}_U(c,k)$, we have that $\pi _i^L-\pi _i^S>0$ for $i=\{R,M\}$, $\pi _R^S-\pi _R^F>0$, and $\pi _M^S-\pi _M^F<0$. Therefore, L is a strict dominant strategy for the retailer and, given such play, the manufacturer chooses strategy F. In other words, the game is dominance solvable and yields (L, F) as the unique equilibrium.

Given $a>c>0$, $k\in (0,1)$ and $\frac{2a-k[a-c]}{2}>w>\max \left\{ 0,\frac{2ak-[a-c][4-k^2]}{2k}\right\}$, for $w>{\widehat{w}}_U(c,k)$, we have that $\pi _i^L-\pi _i^S>0$ for $i=\{R,M\}$, $\pi _R^S-\pi _R^F<0$, and $\pi _M^S-\pi _M^F<0$. Therefore, (L, F) and (F, L) are the two pure strategy equilibria of the game. $\square$

Appendix D: Proof of Corollary 1

In simultaneous play equilibrium (L, L), characterized by $w<{\widehat{w}}_L(c,k)$, we have

$$\begin{aligned} q_R^S - {\overline{q}}_R= & {} -\frac{k[2(a-c)+k(a+w)]}{2b[4-k^2]}<0 \end{aligned}$$

(D.1)

$$\begin{aligned} \pi _R^S - {\overline{\pi }}_R= & {} \frac{\left[ 2(a-w)-k(a-c)\right] ^2}{b[4-k^2]^2}-\frac{[a-w]^2}{4b} \nonumber \\= & {} -\frac{k[2(a-c)-k(a-w)][(8-k^2)(a-w)-2k(a-c)]}{4b[4-k^2]^2}<0, \end{aligned}$$

(D.2)

$$\begin{aligned} \pi _M^S - {\overline{\pi }}_M= & {} \frac{[2(a-c)-k(a-w)]^2}{b[4-k^2]^2}+\frac{w[2(a-w)-k(a-c)]}{b[4-k^2]}-\frac{w[a-w]}{2b} \nonumber \\= & {} \frac{[2(a-c)-k(a-w)][4(a-c)-2k(a+w)+k^3w]}{2b[4-k^2]^2}>0, \end{aligned}$$

(D.3)

$$\begin{aligned} \textsc {cs}^S-\overline{\textsc {cs}}= & {} \frac{[4-3k^2][w^2+c^2]+2wck^3+2a[a-w-c][1+k][2-k]^2}{2b[4-k^2]^2}\nonumber \\{} & {} -\frac{[a-w]^2}{8b} \nonumber \\= & {} \frac{[2(a-c)-k(a-w)][(a-c)(8-6k^2)+k(a-w)(4+k^2)]}{8b[4-k^2]^2}>0, \text{ and } \end{aligned}$$

(D.4)

$$\begin{aligned} \pi _R^S+\pi _M^S-[{\overline{\pi }}_R+{\overline{\pi }}_M]= & {} \frac{k[4+k^2][2(a-c)-k(a-w)]}{4b[4-k^2]^2}\nonumber \\{} & {} \left[ w-\frac{ak[12-k^2]-[8-2k^2][a-c]}{k[4+k^2]}\right] . \end{aligned}$$

(D.5)

Channel profit increases in this equilibrium if and only if $w> \frac{ak[12-k^2]-[8-2k^2][a-c]}{k[4+k^2]}$, denoted by $w_I^S(c,k)$, or equivalently $c<\frac{2a[4-8k+k^2]+k[4+k^2][a+w]}{2[4+k^2]}$, denoted by $c_I^S(w,k)$. In addition, the equation $w_I^S(c,k)-{\widehat{w}}_L(c,k)=0$ has a unique solution, given by $c=a\left[ 1-\frac{68k}{5(4+k^2)}+\frac{8k(38-21k^2)}{5(32-16k^2-k^4)}\right]$, so that for values $c> a\left[ 1-\frac{68k}{5(4+k^2)}+\frac{8k(38-21k^2)}{5(32-16k^2-k^4)}\right]$, channel profit always decreases with the simultaneous encroachment.

In sequential play equilibrium (L, F), characterized by ${\widehat{w}}_L(c,k)<w$, we have

$$\begin{aligned}{} & {} \pi _R^L-{\overline{\pi }}_R = \frac{[2(a-w)-k(a-c)]^2}{8b[2-k^2]}-\frac{[a-w]^2}{4b} \nonumber \\{} & {} \quad = -\frac{4k[a-w][a-c]-k^2[2(a-w)^2-(a-c)^2]}{8b[2-k^2]}<0, \end{aligned}$$

(D.6)

$$\begin{aligned}{} & {} \pi _M^F-{\overline{\pi }}_M = \frac{\left[ (4-k^2)(a-c)-2k(a-w)\right] ^2}{16b\left[ 2-k^2\right] ^2}\nonumber \\{} & {} \qquad +\frac{w\left[ 2(a-w)-k(a-c)\right] }{2b\left[ 2-k^2\right] }-\frac{w[a-w]}{2b}, \nonumber \\{} & {} \quad = \frac{[(a-c)(4-k^2)-ak^2]^2+4wk^2[k(a-c)+2a(1-k^2)]-4w^2k^2[3-2k^2]}{16b[2-k^2]^2}>0, \end{aligned}$$

(D.7)

$$\begin{aligned}{} & {} \textsc {cs}^L - \overline{\textsc {cs}} = \frac{[2(2a-w-c)(2-k)-(a-c)k^2]^2}{32b[2-k^2]^2}\nonumber \\{} & {} \qquad +\frac{[1-k][4(2a-w-c)k-8(a-w)(a-c)(1+k)-(a-c)^2k^3]}{8b[2-k]^2}\nonumber \\{} & {} \qquad -\frac{[a-w]^2}{8b}>0, \text{ and } \end{aligned}$$

(D.8)

$$\begin{aligned}{} & {} \pi _R^L+\pi _M^F-[{\overline{\pi }}_R+{\overline{\pi }}_M] \nonumber \\{} & {} \quad =\frac{k[(a-c)(4-k^2)-2k(a+w)-2wk^3]+k\sqrt{a^2[8-12k+5k^2]-2ac[4-k][2-k]+c^2[8+k^2]}}{8b[2-k^2]^2} \nonumber \\{} & {} \qquad \times \left[ w-\frac{k(4-k^2)(a-c)+2ak^2-k(2-k^2)\sqrt{a^2(8-12k+5k^2)-2ac(4-k)(2-k)+c^2(8+k^2)}}{2k^2[1-k^2]}\right] ,\nonumber \\ \end{aligned}$$

(D.9)

for values of $w<{\widehat{w}}_U(c,k)$.

Channel profit increases in this equilibrium if and only if $w>$ $\frac{k(4-k^2)(a-c)+2ak^2-k(2-k^2)\sqrt{a^2(8-12k+5k^2)-2ac(4-k)(2-k)+c^2(8+k^2)}}{2k^2[1-k^2]}$, denoted by $w_I^L(c,k)$, or $c<\frac{[2-k][a(8-4k-4k^2+k^3)+2kw(2+k)]+2k[2-k^2]\sqrt{a^2[4-k^2]-8aw+w^2[8+k^2]}}{16-4k^2+k^4}$, denoted by $c_I^{L1}(w,k)$. The equation $w_I^L(c,k)-{\widehat{w}}_U(c,k)=0$ has a unique solution, given by $c=$ $\frac{-ak[2-k^2]^2\sqrt{64-80k^2+32k^4-3k^6}+a[64-64k-48k^2+72k^3-28k^5+8k^6+3k^7-k^8]}{64-48k^2+8k^6-k^8}$, so that for values of $c>\frac{-ak[2-k^2]^2\sqrt{64-80k^2+32k^4-3k^6}+a[64-64k-48k^2+72k^3-28k^5+8k^6+3k^7-k^8]}{64-48k^2+8k^6-k^8}$, channel profit always decreases with the retailer-as-leader sequential encroachment.

In sequential play equilibrium (F, L), characterized by $w>{\widehat{w}}_U(c,k)$, we have

$$\begin{aligned}{} & {} q_R^F -{\overline{q}}_R = -\frac{k[2(a-c)-ak]}{4b[2-k^2]}<0 \end{aligned}$$

(D.10)

$$\begin{aligned}{} & {} \pi _R^F - {\overline{\pi }}_R = \frac{[2(a-w)(2-k^2)-2k(a-c)+k^2a]^2}{16b[2-k^2]^2}-\frac{[a-w]^2}{4b} \nonumber \\{} & {} \quad = -\frac{k[2(a-c)-ak][a(2+k)(4-3k)-4w(2-k^2)+2ck]}{16b[2-k^2]^2}<0, \end{aligned}$$

(D.11)

$$\begin{aligned}{} & {} \pi _M^L-{\overline{\pi }}_M = \frac{4w[a-w][2-k^2]-[2(a-c)-ka]^2}{8b[2-k^2]}\nonumber \\{} & {} \qquad -\frac{w[a-w]^2}{8b}=\frac{[2(a-c)-ak]^2}{8b[2-k^2]}>0, \end{aligned}$$

(D.12)

$$\begin{aligned}{} & {} \textsc {cs}^F-\overline{\textsc {cs}}\nonumber \\{} & {} \quad = \frac{[2(a-c)-ak][(a-c)(8-6k^2)+ak(4-k^2)-4kw(2-k^2)]}{32b[2-k^2]^2}>0, \text{ and } \end{aligned}$$

(D.13)

$$\begin{aligned}{} & {} \pi _R^F+\pi _M^L-[{\overline{\pi }}_R+{\overline{\pi }}_M]\nonumber \\{} & {} \quad =\frac{k[2(a-c)-ak]}{4b[2-k^2]}\left[ w-\frac{5ak(2-k^2)-4(2-k^2)(a-c)+2k(a(1-k)+ck)}{4k(2-k^2)}\right] .\nonumber \\ \end{aligned}$$

(D.14)

Channel profit increases in this equilibrium if and only if $w>$ $\frac{5ak(2-k^2)-4(2-k^2)(a-c)+2k(a(1-k)+ck)}{4k(2-k^2)}$, denoted by $w_I^F(c,k)$, or equivalently $c<$ $\frac{a[8-12k-2k^2+5k^3]+4kw[2-k^2]}{2[4-k^2]}$, denoted by $c_I^F(w,k)$. In addition, the equation $w_I^F(c,k)-\frac{2a-k[a-c]}{2}=0$ has a unique solution, given by $c=a-\frac{ak[4-k^2]}{2[4-3k^2+k^4]}$, so that for values of $c>a-\frac{ak[4-k^2]}{2[4-3k^2+k^4]}$, channel profit always decreases with retailer-as-follower sequential encroachment.

When the retailer-as-leader subgame arises in equilibrium for values of $w>{\widehat{w}}_U(c,k)$, we still have $\pi _R^L-{\overline{\pi }}_R<0$ and $\textsc {cs}^L-\overline{\textsc {cs}}>0$. However, we have $\pi _M^F-{\overline{\pi }}_M>0$ if and only if $w<\frac{[a-c]k^3+2ak^2[1-k^2]+[2-k^2]k\sqrt{a^2(6-6k+k^2)+c^2(6-k^2)-2ac(6-3k-k^2)}}{2k^2[3-2k^2]}$. Also, channel profit increases with encroachment if $w>\frac{k(4-k^2)(a-c)+2ak^2-k(2-k^2)\sqrt{a^2(8-12k+5k^2)-2ac(4-k)(2-k)+c^2(8+k^2)}}{2k^2[1-k^2]}$ or $w<\frac{k(4-k^2)(a-c)+2ak^2+k(2-k^2)\sqrt{a^2(8-12k+5k^2)-2ac(4-k)(2-k)+c^2(8+k^2)}}{2k^2[1-k^2]}$, which translate into $c<c_I^{L1}(w,k)$ or $c>c_I^{L2}(w,k)$, where $c_I^{L2}(w,k)$ denotes $\frac{[2-k][a(8-4k-4k^2+k^3)+2kw(2+k)]-2k[2-k^2]\sqrt{a^2[4-k^2]-8aw+w^2[8+k^2]}}{16-4k^2+k^4}$. $\square$

Appendix E: Proof of Proposition 2

In the simultaneous play equilibrium (L, L), the manufacturer chooses the wholesale price w to maximize its total profit $\pi _M^S(w) =\frac{[2(a-c)-k(a-w)]^2}{b[4-k^2]^2}+\frac{w[2(a-w)-k(a-c)]}{b[4-k^2]}$ subject to $w\le {\widehat{w}}_L(c,k)$. The second-order derivative of the manufacturer’s profit function with respect to the wholesale price is $\frac{\partial ^2 \pi _M^S(w)}{\partial w^2}=-\frac{2[8-3k^2]}{b[4-k^2]^2}<0$ for any $b>0$ and $k\in (0,1)$. Solving the first-order condition yields

$$\begin{aligned} w = \frac{a}{2} -\frac{k^2[a(1-k)+ck]}{2[8-3k^2]}, \end{aligned}$$

(E.1)

which is strictly greater than ${\widehat{w}}_L(c,k)$; i.e., $\frac{a}{2} -\frac{k^2[a-k(a-c)]}{2[8-3k^2]}-{\widehat{w}}_L(c,k)=$ $\frac{[4-k^2][a-k(a-c)][32-24k^2+5k^4]}{[8-3k^2][32-32k^2+7k^4]}>0$ given $a>c>0$, $k\in (0,1)$ and $\frac{c}{a}<1-\frac{k[4-k^2]}{4+[2-k^2]^2}$. Recall that the simultaneous play equilibrium does not arise for $\frac{c}{a}>1-\frac{k[4-k^2]}{4+[2-k^2]^2}$ per Proposition 1. Therefore, the optimal solution of the problem $w^S \doteq \mathop {\text {arg\,max}}_w \left\{ \pi _M^S(w) \text{ s.t. } w\le {\widehat{w}}_L(c,k)\right\}$ is given by

$$\begin{aligned} w^S = {\widehat{w}}_L(c,k)= \frac{a}{2}-\frac{[a(1-k)+ck][32-16k^2+k^4]}{64-64k^2+14k^4}. \end{aligned}$$

(E.2)

The resulting quantities, profits and consumer surplus are obtained by substituting $w^S$ into the quantity, profit and consumer surplus functions outlined in Lemma 1 as follows:

$$\begin{aligned} q_M^S= & {} \frac{[a-c][32-16k^2-k^4]-8ak[2-k^2]}{2b[32-32k^2+7k^4]}, \text{ and } q_R^S = \frac{8[a-k(a-c)][2-k^2]}{b[32-32k^2+7k^4]},\end{aligned}$$

(E.3)

$$\begin{aligned} \pi _M^S= & {} \frac{[(a-c)(32-16k^2-k^4)-8ak(2-k^2)]^2}{4b[32-32k^2+7k^4]^2}+ \left[ \frac{a}{2}-\frac{(a-k(a-c))(32-16k^2+k^4)}{64-64k^2+14k^4}\right] \nonumber \\{} & {} \left[ \frac{8(a-k(a-c))(2-k^2)}{b(32-32k^2+7k^4)}\right] , \text{ and } \nonumber \\ \pi _R^S= & {} \frac{64[a-k(a-c)]^2[2-k^2]^2}{b[32-32k^2+7k^4]^2}, \text{ and }\end{aligned}$$

(E.4)

$$\begin{aligned} \textsc {cs}^S= & {} \frac{[a-c]^2[1024-2048k^2+1216k^4-160k^6-31k^8]+16a[2-k^2][k(a-c)(32-48k^2+17k^4)+4a(8-10k^2+3k^4)]}{8b[32-32k^2+7k^4]^2}.\nonumber \\ \end{aligned}$$

(E.5)

Given $a>c>0$, $b>0$ and $k\in (0,1)$, the necessary non-negativity condition is $a>\frac{c[32-16k^2+k^4]}{[2-k][16-8k^2-k^3]}$.

In the sequential play equilibrium (L, F), the manufacturer chooses the wholesale price w to maximize its total profit $\pi _M^F(w)=\frac{\left[ (4-k^2)(a-c)-2k(a-w)\right] ^2}{16b\left[ 2-k^2\right] ^2}+\frac{w\left[ 2(a-w)-k(a-c)\right] }{2b\left[ 2-k^2\right] }$ subject to $w\ge {\widehat{w}}_L(c,k)$. The second-order derivative of the manufacturer’s profit function with respect to the wholesale price is $\frac{\partial ^2 \pi _M^F(w)}{\partial w^2}=-\frac{8-5k^2}{2b[2-k^2]^2}<0$ for any $b>0$ and $k\in (0,1)$. Solving the first-order condition yields

$$\begin{aligned} w = \frac{a}{2}-\frac{k^2[a(1-k)+ck]}{2[8-5k^2]}, \end{aligned}$$

(E.6)

which is strictly greater than ${\widehat{w}}_L(c,k)$; i.e., $\frac{a}{2}-\frac{k^2[a(1-k)+ck]}{2[8-5k^2]}-{\widehat{w}}_L(c,k)=$ $\frac{[2-k^2][a(1-k)+ck][32-24k^2+3k^4]}{[8-5k^2][32-32k^2+7k^4]}>0$ given $a>c>0$ and $k\in (0,1)$. Therefore, the optimal solution of the problem $w^L \doteq \mathop {\text {arg\,max}}_w \left\{ \pi _M^F(w) \text{ s.t. } w\ge {\widehat{w}}_L(c,k)\right\}$ is given by

$$\begin{aligned} w^L = \frac{a}{2}-\frac{k^2[a(1-k)+ck]}{2[8-5k^2]}. \end{aligned}$$

(E.7)

Substituting $w^L$ into the quantity, profit and consumer surplus functions in Lemma 1 yields:

$$\begin{aligned} q_M^F= & {} \frac{[a-c][8-3k^2]-2ak}{2b[8-5k^2]}, \text{ and } q_R^L = \frac{2[a(1-k)+ck]}{b[8-5k^2]},\end{aligned}$$

(E.8)

$$\begin{aligned} \pi _M^F= & {} \frac{[a-c]^2[8-k^2]+4a[a(1-2k)+2ck]}{4b[8-5k^2]}, \text{ and } \pi _R^L = \frac{2[a(1-k)+ck]^2[2-k^2]}{b[8-5k^2]^2}, \text{ and } \end{aligned}$$

(E.9)

$$\begin{aligned} \textsc {cs}^L= & {} \frac{[a-c]^2[64-96k^2+33k^4]+4a[a(2-k)^2(1+k)-ck^3]}{8b[8-5k^2]^2}. \end{aligned}$$

(E.10)

Given $a>c>0$, $b>0$ and $k\in (0,1)$, the necessary non-negativity condition is $a>\frac{c[8-3k^2]}{8-2k-3k^2}$.

In the sequential play equilibrium (F, L), the manufacturer chooses the wholesale price w to maximize its total profit $\pi _M^L=\frac{4w[a-w][2-k^2]-[2(a-c)-ka]^2}{8b[2-k^2]}$ subject to $w>{\widehat{w}}_U(c,k)$. The second-order derivative of the manufacturer’s profit function with respect to the wholesale price is $\frac{\partial ^2 \pi _M^L(w)}{\partial w^2}=-\frac{1}{b}<0$ for any $b>0$. Solving the first-order condition yields

$$\begin{aligned} w=\frac{a}{2}, \end{aligned}$$

(E.11)

which is strictly greater than ${\widehat{w}}_U(c,k)$; i.e., $\frac{a}{2}-{\widehat{w}}_U(c,k)=\frac{a[1-k]+ck}{2-k^2}$ given $a>c>0$ and $k\in (0,1)$. Therefore, the optimal solution of the problem $w^F \doteq \mathop {\text {arg\,max}}_w\left\{ \pi _M^L(w) \text{ s.t. } w>\ge {\widehat{w}}_U(c,k)\right\}$ is given by

$$\begin{aligned} w^F = \frac{a}{2}. \end{aligned}$$

(E.12)

Substituting $w^F$ into the quantity, profit and consumer surplus functions in Lemma 1 yields:

$$\begin{aligned} q_M^L= & {} \frac{2[a-c]-ak}{2b[2-k^2]}, \text{ and } q_R^F = \frac{a[1-k]+ck}{2b[2-k^2]}, \end{aligned}$$

(E.13)

$$\begin{aligned} \pi _M^L= & {} \frac{a^2[3-2k]-2ac[2-k]+2c^2}{4b[2-k^2]}, \text{ and } \pi _R^F = \frac{[a(1-k)+ck]^2}{4b[2-k^2]^2}, \text{ and } \end{aligned}$$

(E.14)

$$\begin{aligned} \textsc {cs}^F= & {} \frac{[a-c]^2[4-3k^2]+a[1-k^2][a-2k(a-c)]}{8b[2-k^2]^2}. \end{aligned}$$

(E.15)

Given $a>c>0$, $b>0$ and $k\in (0,1)$, the necessary non-negativity condition is $a>\frac{2c}{2-k}$. $\square$

Appendix F: Proof of Corollary 2

In the retailer-as-leader sequential game (L, F), the manufacturer’s desired wholesale price is obtained by solving $\frac{\partial \pi _M^F(w)}{\partial w}=0$, where $\pi _M^F(w)=\frac{\left[ (4-k^2)(a-c)-2k(a-w)\right] ^2}{16b\left[ 2-k^2\right] ^2}+\frac{w\left[ 2(a-w)-k(a-c)\right] }{2b\left[ 2-k^2\right] }$ as presented in Lemma 1(b). The solution of this equation yields $w^*=\frac{a}{2}-\frac{k^2[a(1-k)+ck]}{2[8-5k^2]}$. Thus, the retailer-as-leader sequential play is the manufacturer’s preferred equilibrium to induce if and only if the following inequalities hold: $\pi _M^F(w^*)-\max _w\left\{ \pi _M^S(w) \text{ s.t. } w \le {\widehat{w}}_L(c,k)\right\} \ge 0$; and $\pi _M^F(w^*)-\max _w\left\{ \pi _M^L(w) \text{ s.t. } w \ge {\widehat{w}}_U(c,k)\right\} \ge 0$, where $\pi _M^S(w)=\frac{[2(a-c)-k(a-w)]^2}{b[4-k^2]^2}+\frac{w[2(a-w)-k(a-c)]}{b[4-k^2]}$ and $\pi _M^L(w)=$ $\frac{4w[a-w][2-k^2]-[2(a-c)-ka]^2}{8b[2-k^2]}$. Accounting for the constraints, we have

$$\begin{aligned} \pi _M^F(w^*)-\max _w\pi _M^S(w)= & {} \frac{[a(1-k)+ck]^2[32-3k^2(8-k^2)]^2}{b[8-5k^2][32-32k^2+7k^4]^2}>0, \text{ and } \end{aligned}$$

(F.1)

$$\begin{aligned} \pi _M^F(w^*)-\max _w\pi _M^L(w)= & {} \frac{k^2[a(1-k)+ck]^2}{4b[2-k^2][8-5k^2]}>0. \end{aligned}$$

(F.2)

Therefore, $w^*=\frac{a}{2}-\frac{k^2[a(1-k)+ck]}{2[8-5k^2]}=w^L$ is the desired wholesale price to offer for the manufacturer and (L, F) is the preferred equilibrium to induce.

The effects of encroachment on firm-level profits, consumer surplus and channel profit in (L, F) equilibrium are summarized as follows:

$$\begin{aligned}{} & {} \pi _M^F(w^*)-{\overline{\pi }}_M = \frac{a^2[4-k][4-3k]+2c^2[8-k^2]-4ac[8-4k-k^2]}{8b[8-5k^2]}>0,\end{aligned}$$

(F.3)

$$\begin{aligned}{} & {} \pi _R^L(w^*)-{\overline{\pi }}_R \nonumber \\{} & {} \quad = -\frac{k[a^2(4-3k)(32-4k-19k^2)-32c^2(2-k^2)-64ac(1-k)(2-k^2)]}{16b[8-5k^2]^2}<0, \end{aligned}$$

(F.4)

$$\begin{aligned}{} & {} \textsc {cs}^L(w^*) - \overline{\textsc {cs}}\nonumber \\{} & {} \quad = \frac{4[64-96k^2+33k^4][a-c]^2+ak^2[a(32-25k^2)+16k(a-c)]}{32b[8-5k^2]^2}>0, \text{ and } \end{aligned}$$

(F.5)

$$\begin{aligned}{} & {} \pi _M^F(w^*)+\pi _R^L(w^*)-[{\overline{\pi }}_M+{\overline{\pi }}_R]\nonumber \\{} & {} \quad = \frac{4[64-32k^2-3k^4][a-c]^2-ak[32(a-c)(12-7k^2)+ak(128-75k^2)]}{16b[8-5k^2]^2}. \end{aligned}$$

(F.6)

Channel profit increases in this equilibrium if and only if $c<a-\frac{8ak[12-7k^2]+ak[8-5k^2]\sqrt{16-9k^2}}{2[64-32k^2-3k^4]}$, denoted by $c_I^L(k)$.