Retailer voluntary investment against a threat of manufacturer encroachment

Abstract

To elucidate supply chain cooperation between a manufacturer and a retailer, this study examines a model in which the retailer makes voluntary investments to reduce the marginal production cost of the manufacturer. The manufacturer is allowed to introduce a direct selling channel in addition to the indirect channel through the retailer (i.e., manufacturer encroachment), which however dampens the retailers’ investment incentives. The retailer can leverage its voluntary investments as a means of deterring manufacturer encroachment. We demonstrate that manufacturer encroachment is strategically deterred when the retailer’s cost-reduction technology is sufficiently effective. This strategic encroachment deterrence encourages the retailer to invest more, but it narrows the variety of channels from which consumers can select. When the latter effect dominates the former effect, consumer surplus declines with strategic encroachment deterrence.

Appendices

Appendix A: Proofs of results derived in the main model

Proof of Lemma 1

First, it is apparent that

$$ \left. {x_{R}^{E}} \right|_{\theta=0} = \frac{a-c}{8bk-1} = {x_{R}^{N}} . $$

(A.1)

Next, the partial derivative of ${x_{R}^{E}}$ with respect to 𝜃 is given as

$$ \begin{array}{@{}rcl@{}} \frac{\partial {x_{R}^{E}}}{\partial \theta}=-\frac{16 b (1-\theta ) \left( 64 -48 \theta+18 \theta^{3} -9 \theta^{4}\right) k (a-c)}{\left[b \left( 8-3 \theta^{2}\right)^{2} k-8 (1-\theta )^{2}\right]^{2}}<0. \end{array} $$

(A.2)

□

Proof of Proposition 1

Computing ${\pi _{R}^{N}} - {\pi _{R}^{E}}$ yields

$$ \begin{array}{@{}rcl@{}} {\pi_{R}^{N}} - {\pi_{R}^{E}}=\frac{b \theta \left( 128-112 \theta +9 \theta^{3}\right) k^{2} (a-c)^{2}}{2 (8 b k-1) \left[b \left( 8-3 \theta^{2}\right)^{2} k-8 (\theta -1)^{2}\right]}, \end{array} $$

(A.3)

which is greater than zero under Assumption 1. □

Proof of Proposition 2

Solving ${\pi _{M}^{N}} > {\pi _{M}^{E}}$ yields

$$ \frac{\Phi-\theta(128-112\theta+9\theta^{3})\sqrt{\Psi}} {8b(8-3\theta^{2})^{3}(16-16\theta+5\theta^{2})} <k< \frac{\Phi+\theta(128-112\theta+9\theta^{3})\sqrt{\Psi}} {8b(8-3\theta^{2})^{3}(16-16\theta+5\theta^{2})} \equiv \tilde{k}, $$

(A.4)

where Φ = 2(8 − 3𝜃²)²(64 − 60𝜃² + 24𝜃³ − 3𝜃⁴) > 0 and Ψ = 2(2 − 𝜃)(6 − 𝜃)(8 − 3𝜃²)³ > 0.

Moreover, $\underline {k}$ lies within the interval presented in Inequality Eq. A.4. Therefore, it holds that ${\pi _{M}^{N}}>{\pi _{M}^{E}}$ if and only if $\underline {k}<k<\tilde {k}$. □

Proof of Corollary 1

The partial derivative of $\tilde {k}$ with respect to 𝜃 is given as

$$ \frac{\partial \tilde{k}}{\partial \theta} = \frac{\sqrt{2}(1-\theta) \cdot {\Xi}} {2b(8-3\theta^{2})^{2}(16-16\theta+5\theta^{2})^{2}\sqrt{\Psi}} , $$

(A.5)

where ${\Xi } = 2(8-3\theta ^{2})(49152-85016\theta +74752\theta ^{2}-54016\theta ^{3}+29824\theta ^{4}-8688\theta ^{5}+840\theta ^{6}+27\theta ^{7})+4(1024-768\theta +192\theta ^{2}-288\theta ^{3}+192\theta ^{4}-27\theta ^{5})\sqrt {\Psi }$. It is noteworthy that Ξ consists of a parameter, which is 𝜃 only. We can demonstrate that Ξ takes a positive value for all 𝜃 ∈ [0, 1]. □

Proof of Proposition 3

Solving CS^N > CS^E gives

$$ \frac{{\Phi}_{CS}-\theta(8-3\theta)\sqrt{{\Psi}_{CS}}} {4b(8-3\theta^{2})(16+8\theta-9\theta^{2})} <k< \frac{{\Phi}_{CS}+\theta(8-3\theta)\sqrt{{\Psi}_{CS}}} {4b(8-3\theta^{2})(16+8\theta-9\theta^{2})} \equiv \tilde{k}_{CS}, $$

(A.6)

where Φ_CS = 2(2 − 𝜃)(2 + 3𝜃)(8 − 3𝜃²) > 0 and Ψ_CS = 80 − 76𝜃² + 12𝜃³ + 9𝜃⁴ > 0.

Moreover, $\underline {k}$ lies within the interval presented in Inequality Eq. A.6. Therefore, it holds that CS^N > CS^E if and only if $\underline {k}<k<\tilde {k}_{CS}$.

Additionally, comparison between $\tilde {k}$ and $\tilde {k}_{CS}$ relies on parameter 𝜃 only. One can show that $\tilde {k}>\tilde {k}_{CS}$ holds for any 𝜃 ∈ [0, 1]. □

Proof of Proposition 4

Solving SW^N > SW^E gives

$$ \frac{{\Phi}_{SW}-\theta(4+\theta)(8-3\theta)(4-3\theta)\sqrt{{\Psi}_{SW}}} {8b(8-3\theta^{2})^{2} \cdot {\Xi}_{SW}} <k< \frac{{\Phi}_{SW}+\theta(4+\theta)(8-3\theta)(4-3\theta)\sqrt{{\Psi}_{SW}}} {8b(8-3\theta^{2})^{2} \cdot {\Xi}_{SW}} \equiv \tilde{k}_{SW}, $$

(A.7)

where Φ_SW = 49152 − 8192𝜃− 60416𝜃²− 7168𝜃³+ 49024𝜃⁴− 10368𝜃⁵− 7344𝜃⁶+ 2160𝜃⁷+ 27𝜃⁸ > 0, Ψ_SW = 348160 − 163840𝜃− 413696𝜃²+ 171008𝜃³+ 192064𝜃⁴− 78336𝜃⁵− 28656𝜃⁶+ 10800𝜃⁷+ 621𝜃⁸ > 0, and Ξ_SW = 768 − 768𝜃 − 64𝜃² + 240𝜃³ − 51𝜃⁴ > 0.

Moreover, $\underline {k}$ lies within the interval presented in Inequality Eq. A.7. Therefore, it holds that SW^N > SW^E if and only if $\underline {k}<k<\tilde {k}_{SW}$.

Additionally, comparison between $\tilde {k}$ and $\tilde {k}_{SW}$ relies solely on parameter 𝜃. One can demonstrate that $\tilde {k}<\tilde {k}_{SW}$ holds if and only if $\tilde {\theta }_{SW} \leq \theta \leq 1$, where $\tilde {\theta }_{SW} \simeq 0.689911$. □

Appendix B: Analysis for Section 5.1

We reverse the order of decisions of stages 1 and 2 in the original game. Specifically, the timing of the game is the following. In stage 1, the retailer chooses the level of voluntary investment. In stage 2, the manufacturer decides whether to open its direct channel. In stage 3, the manufacturer charges a wholesale price. In stage 4, Cournot competition takes place in the retail market. Then, we solve the game using backward induction.

Analyses for stages 3 and 4 are the same as those of the main model. That is, without encroachment, the manufacturer sets w = (a + c − x_R)/2 in stage 3; then, the retailer chooses q_R = (a − c + x_R)/(4b). The resulting profits of the manufacturer and the retailer are computed as

$$ \begin{array}{@{}rcl@{}} \pi_{M} &=& \frac{(a-c+x_{R})^{2}}{8b}, \end{array} $$

(B.1)

$$ \begin{array}{@{}rcl@{}} \pi_{R} &=& \frac{(a-c+x_{R})^{2}}{16b} - \frac{k}{2} {x_{R}^{2}}. \end{array} $$

(B.2)

By contrast, with encroachment, the manufacturer sets its wholesale price at the one presented in Eq. 8 in stage 3. In stage 4, Cournot competition occurs. The quantities sold by the manufacturer and the retailer are therefore computed as q_M = (a − c + x_R)(2 − 𝜃)(4 + 𝜃)/2b(8 − 3𝜃²) and q_R = 2(a − c + x_R)(1 − 𝜃)/b(8 − 3𝜃²). Then, the resulting profits are given as

$$ \begin{array}{@{}rcl@{}} \pi_{M} &=& \frac{(a-c+x_{R})^{2}(12-8\theta+\theta^{2})}{4b(8-3\theta^{2})}, \end{array} $$

(B.3)

$$ \begin{array}{@{}rcl@{}} \pi_{R} &=& \frac{4(a-c+x_{R})^{2}(1-\theta)^{2}}{b(8-3\theta^{2})^{2}} - \frac{k}{2} {x_{R}^{2}}. \end{array} $$

(B.4)

In stage 2, given x_R, the manufacturer decides whether to encroach or not. Comparison between Eqs. B.1 and B.3 reveals that the manufacturer certainly chooses to encroach for any x_R.

In stage 1, the retailer anticipates manufacturer encroachment; then the retailer chooses its investment level to maximize the profit presented in Eq. B.4, implying that $x_{R} = {x_{R}^{E}}$. The resulting investment level and other equilibrium outcomes are equal to the values derived in the right column of Table 2.