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What should the encroaching supplier do?: A Stackelberg Game Approach
Supplier encroachment refers to a strategic move where suppliers bypass traditional distribution channels, set-up an in-house production unit to sell directly to end consumers while continuing supplying to downstream or lower echelon manufacturers. Such strategies have become increasingly prevalent as firms seek to capture larger margins, strengthen brand identity, reduce dependence on downstream intermediaries and thereby gain more control over the supply chain (see e.g., arya2007bright,ha2022supplier,yoon2016supplier). This trend is evident across various industries, transforming conventional supply chain (SC) dynamics. For instance, in the automotive industry, major suppliers such as \href{https://www.boschautoservice.com/} Bosch and \href{https://www.continental-aftermarket.com/us-en} Continental now market parts and services directly to consumers, thereby enhancing their brand visibility and fostering closer customer relationships. Acer Inc., initially a supplier for IBM and Apple, leveraged this strategy to become one of the big computer manufacturers worldwide by 2007 (nystedt2007acer).
The arrangement is closely linked to another aspect namely `vertical integration' studied in SC literature (e.g., ursino2015supply,simchi1999designing). Vertical integration typically implies the integration of various units (across various echelons) into a single unit that controls multiple stages of production and distribution (e.g., simchi1999designing,wadhwapartition,zheng2021willingness). The idea in most of this literature is to illustrate the advantages of a centralized SC formed by complete integration of all the manufacturers and the supplier. Recently in wadhwapartition, we showed that, for essential products, a partially integrated supplier--manufacturer coalition facing competition from another manufacturer is more stable than the fully centralized SC (a structure that is not opposed by other collaborative arrangements). Extending this analysis to more general settings, including non-essential products, remains an open problem. A key challenge is to characterize the worths of coalitions under different partition structures, with the main hurdle being the characterization of worths of the individual coalitions in a partition involving partial vertical integration (see wadhwa2025should,wadhwapartition).
The common feature in both the aspects mentioned above, is a single unit that has capacity spanning across multiple echelons. In this study, we investigate one such SC with one supplier and two manufacturers, where the supplier collaborates with one of the manufacturers resulting in a partial vertical integration, while competing with the other. We explore the optimal operating strategies for the vertical collaborating unit and thereby derive it's worth---when it acts as a leader by setting the wholesale price for the raw materials (to the out-house manufacturer) and by quoting another price to the end customers---while anticipating the optimal response of the out-house, that acts as a follower. We also derive the worth of the out-house manufacturer.
r operating costs or the option to cease operations). Accounting for these factors reveals several surprising market-dependent optimal configurations for the supplier--manufacturer duo---including loss-making in-house operation, forcing out-house to operate at break-even, shutdown of one of the two units, and the conventional regime of profitable co-existence.
As noted earlier, our SC can also be interpreted as a supplier encroachment model, where the supplier operates an in-house production unit while simultaneously supplying materials to an independent outsourced production unit. Analyzing this arrangement in detail, as mentioned above, helps address a central managerial question:
On the other hand, this paper also derives the coalition worths for partitions involving partial vertical integration under substantially more general conditions than wadhwapartition, which primarily focuses on essential products. The results of this paper hence can facilitate a more comprehensive future analysis of coalition formation game in a one-supplier, two-manufacturer SC.
A preliminary conference version of this work wadhwa2025should, that restricted attention only to co-existence possibilities, illustrated that strong customer loyalty (towards individual manufacturers) sustains profitable co-existence of both units; while for more essential products (where customers are desperate to buy from any) the coalition can push the out-house manufacturer to break-even, or even the in-house unit into a loss. The analysis in wadhwa2025should stopped short of asking whether co-existence is optimal at all, relative to shutting down the in-house unit or eliminating the out-house manufacturer entirely.
The present paper closes the gaps in analysis of wadhwa2025should by investigating the coalition's choice across all major operating regimes. Our contributions are as follows.
Together, these results show that profitable co-existence, far from being the generic outcome of partial vertical integration, is confined to conditions of high customer loyalty (and for weakly-substitutable products); when products become more essential, the supplier optimally moves through a sequence of increasingly aggressive strategies---disciplining the out-house manufacturer to break-even, sustaining its own in-house unit at a loss to forestall downstream monopoly, or, at the extreme, shutting down one of the two units altogether---with the specific choice pinned down by simple, computable market-strength scores at the two ends of the essentialness spectrum and by direct numerical comparison in between. Overall, our results demonstrate that partial vertical integration is not merely an intermediate organizational form between outsourcing and full integration, but an active strategic lever for shaping downstream market structure.
Supplier encroachment, whereby an upstream supplier sells directly to end customers while continuing to supply downstream firms, has been extensively studied in the supply chain literature. Early studies primarily emphasized its adverse effects, arguing that encroachment intensifies channel competition and may reduce downstream profitability frazier1996determinants,fein1997patterns. Subsequent research, however, demonstrated that supplier encroachment can also improve channel performance by mitigating double marginalization and better aligning upstream and downstream incentives. For example, chiang2003direct showed that introducing a direct sales channel may reduce wholesale prices and improve retailer profitability when consumers exhibit sufficient acceptance of the direct channel, while arya2007bright identified conditions under which supplier encroachment benefits both suppliers and retailers. More recent studies have extended this literature by incorporating information asymmetry, digital channels, and platform-mediated competition, highlighting the influence of market structure and information availability on the profitability of encroachment ha2022supplier. Collectively, these studies show that supplier encroachment simultaneously involves upstream cooperation and downstream competition, making it a fundamental strategic issue in modern supply chains.
Despite these advances, much of the analytical literature focuses primarily on pricing and channel coordination within a predetermined operating structure. In many existing models, firms optimize prices while the market configuration itself is assumed to remain fixed throughout the game. Comparatively less attention has been devoted to settings in which an integrated supplier can strategically determine the operating structure by choosing whether to sustain co-existence, maintain a downstream competitor at break-even, temporarily accept losses, shut down its own production unit, or eliminate downstream competition. Furthermore, the joint analytical treatment of dedicated customer bases, price-based demand substitution, and non-negligible production and operating costs remains limited. In our formulation, incorporating these features results in piecewise-defined payoff functions with multiple feasible operating regimes, motivating a regime-by-regime analytical framework rather than a conventional single-regime pricing analysis.
A related stream of research investigates the trade-off between outsourcing and in-house production. For example, kaya2011outsourcing studies sourcing decisions under effort-dependent demand, while wang2013advantage analyzes competition between an original equipment manufacturer and a contract manufacturer within a Stackelberg framework. These studies demonstrate that coexistence between outsourcing and internal production can emerge under suitable contractual arrangements. However, the production structure is generally specified exogenously, with relatively little attention devoted to settings in which an upstream supplier simultaneously supplies and competes with a downstream manufacturer while optimally determining its mode of operation.
Another closely related stream considers vertical integration as a mechanism for improving coordination and mitigating double marginalization simchi1999designing,ursino2015supply,arora2025vertical. While this literature demonstrates the efficiency benefits of integration, many models either assume complete integration or treat the integration structure as fixed. Consequently, they provide limited insight into how a partially integrated coalition strategically interacts with an independent downstream manufacturer when multiple operating regimes are feasible.
Game-theoretic models, particularly Stackelberg games, have been widely employed to analyze leader--follower interactions in supply chains li2006channel,yan2011managing,lin2011dual,das2022integration,taleizadeh2016pricing. These studies have generated important insights into pricing, channel power, and competitive behaviour. Nevertheless, supplier encroachment, outsourcing, and partial vertical integration have largely developed as related but distinct research streams, and comparatively few analytical frameworks integrate these decisions while allowing the operating regime itself to be determined endogenously.
Recent behavioural studies further recognize that firms may deliberately sacrifice short-term profitability to preserve long-term strategic objectives zheng2021willingness. Although these studies provide valuable insights into strategic behaviour, they do not explicitly examine such decisions within a supplier encroachment framework that simultaneously captures partial vertical integration, asymmetric dependence, and endogenous operating regimes.
Motivated by these observations, this paper develops a unified Stackelberg framework for supplier encroachment under partial vertical integration. The model jointly incorporates customer loyalty, demand substitution through fallback behaviour, production and operating costs, and endogenous regime selection. Rather than optimizing prices within a predetermined operating structure, the integrated supplier optimally chooses among multiple operating regimes, including profitable coexistence, break-even operation of the downstream manufacturer, loss-making operation of the in-house unit, shutdown of the in-house unit, and elimination of the downstream manufacturer.
The analysis contributes in four directions. First, it characterizes the conditions under which different operating regimes become optimal. Second, it derives closed-form analytical characterizations for high- and low-essentialness scenarios together with a numerical procedure for determining the optimal regime in intermediate parameter regions. Third, it demonstrates how customer loyalty, product essentialness, demand fallback, and market asymmetry jointly influence the supplier's optimal operating strategy. Finally, numerical experiments over economically meaningful parameter ranges illustrate how optimal operating regimes evolve as product essentialness and demand transfer intensify. Overall, the paper contributes to the supplier encroachment and partial vertical integration literature by explicitly treating the operating regime as an endogenous strategic decision rather than assuming a predetermined operating structure.
Consider a partially integrated SC with one supplier $S$ that collaborates with a manufacturer $M$, by forming the coalition ${{\mathbb V}} = \{M, S\}$. The coalition competes with an out-house manufacturer, referred by $M_e$.
We consider a Stackelberg framework, where the members of the coalition ${{\mathbb V}}$ quote their prices first: (a) the supplier $S$ quotes wholesale price $q$ to $M_e$ for the raw materials; and (b) the manufacturer $M$ quotes price $p$ for its final product to the end-customers. Thus the coalition ${\mathbb V}$ forms the leader of the Stackelberg game, while the manufacturer $M_e$ is the follower and responds to the quoted prices $(q,p)$. The latter can choose to operate by quoting a price $p_e$ to the end customers for its own finished product. It can also choose not to operate represented by action $n_o$, if the resultant market response is not conducive. Any unit can choose $n_o$ and with such a choice, the corresponding unit is completely shut-down and incurs zero utility (zero profit and cost). The out-house manufacturer $M_e$ uses the raw material supplied by ${\mathbb V}$ for producing the finished products, see Figure (ref) for the flow of materials and the prices.
The demand attracted by any manufacturer depends upon the price quoted for the finished product, for example, that attracted by manufacturer $M$ is given by (see wadhwapartition,zheng2021willingness for similar models):
where the different influencing factors are as below:
The parameter $\varepsilon$ represents the essentialness of the product. When $\varepsilon \approx 1$, the product is essential implying that the manufacturers (or their products) are substitutable and the customers can buy the product from any of the manufacturers. On the other hand, when $\varepsilon \approx 0$, the product is not essential, i.e., the customers are loyal and choose to buy the product only from `their' manufacturers.
We begin with the utility of out-house manufacturer $M_e$. When it does not operate, represented by indicator ${\cal F}^c_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} = \mathds{1}_{\left\{ p_e= n_{o} \right \}} $, it derives zero utility. When it operates (represented by ${\cal F}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$), it attracts demand as in (ref) and then the revenue derived equals the demand times the price minus the expenses (the raw material price $q$ plus the production cost). Thus the utility of the manufacturer $M_e$ equals:
where $C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ represents the production cost per unit and $O_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ represents the operating cost. The profit of manufacturer $M_e$ is zero when the supplier does not operate (represented by indicator ${\cal F}_{{\mbox{\fontsize{5.2}{5.2}\selectfont{${\mathbb V}$}}}}^c$). Some dependencies are suppressed, when there is clarity, to keep the notations simple.
The utility of ${\mathbb V}$ due to demand $D_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} $ attracted by its manufacturer will have similar structure. Additionally, the demand $D_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ attracted by $M_e$ also contributes towards the revenue of ${\mathbb V}$ (as it supplies raw material). In all, the utility of the coalition ${\mathbb V}$ is given by,
{
} where $C_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}}$ represents the raw material procurement cost (per unit) and $C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$, $O_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}}$ and $O_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$ have similar interpretations. The coalition ${\mathbb V}$ can choose to shut in-house production (or it's manufacturer $M$) if it deems advantageous, represented by action $(p, q)$ with $p=n_o$, and hence the inclusion of the flag ${\cal F}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} := \mathds{1}_{\left\{ p \ne n_o \right \}} $ in (ref); alternatively it might find it beneficial to not operate at all, indicated by $ {\cal F}^c_{{{\mbox{\fontsize{5.2}{5.2}\selectfont{${\mathbb V}$}}}}} = 1-\mathds{1}_{\left\{ q \ne n_o \right \}} $.
We need to choose an upper bound for the prices without loosing generality, as compact domains significantly simplify the analysis. Towards this, first observe that the demand attracted by any manufacturer (say $m$) gets zero, even after considering that maximum possible fraction {$\varepsilon \bar{d}_{-m}$} is received from the other manufacturer (say $-m$), if $ p_m > \nicefrac{ (\bar{d}_m + \varepsilon \bar{d}_{-m})} {\alpha_m}. $ Thus we set $p_{mx} =\nicefrac{(\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} + \varepsilon\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}})}{\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}}$ and ${p_e}_{_{mx}} =\nicefrac{(\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} + \varepsilon\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}})}{\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$ as the maximum prices respectively for $M$ and $M_e.$ We assume that if any agent is indifferent between the action $a = n_o$ and an $a \ne n_o$, the agent prefers to operate. This consideration is inspired from the practical scenarios (see wadhwapartition). We further assume the following as in wadhwapartition, which ensures all the agents find it `beneficial to operate':
Assumption {{\bf A.1}} ensures that the market potentials of both the manufacturers are sufficiently high compared to production, procurement and the operating costs (see wadhwapartition for similar details). We will observe that ${\mathbb V}$ finds it optimal to operate (i.e ${\cal F}^*_{{\mbox{\fontsize{5.2}{5.2}\selectfont{${\mathbb V}$}}}} =1$) under {{\bf A.1}}, which is important for meaningful analysis. Assumption {{\bf A.2}} is required for some technical reasons in the proof of Theorem (ref) mentioned in (ref); besides, in general the operating and the production costs are significantly large and hence the assumption would automatically be satisfied (note here $\varepsilon\le 1$).
We begin by obtaining the best response of the follower, the out-house manufacturer $M_e$, when the Stackelberg leader (coalition ${{\mathbb V}}$) declares $ (p, q)$. In particular we consider the case with ${\cal F}_{{\mbox{\fontsize{5.2}{5.2}\selectfont{${\mathbb V}$}}}} =1$, or when ${{\mathbb V}}$ decides to operate. This response of $M_e$ is governed by the following optimization problem (observe from (ref) that $D_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ depends upon $(p,q)$):
{
} Such a problem is considered in wadhwapartition. By similar concavity arguments, the best response exists and equals:
{
} In the above $p_{sw}$ represents a switching point---if the price of in-house $M$ is above $p_{sw}$, the optimal price of the out-house $M_e$ is clamped at the maximum possible value ${p_e}_{_{mx}}$. Further, $M_e$ may not find it beneficial even to operate if the price $q$ quoted for raw materials is high (this happens when $q > \theta (p) $ in (ref)). Interestingly, this also depends upon the price $p$ quoted by the in-house manufacturer $M$ towards the end-product. More interestingly $M_e$ can tolerate a larger $q$ if the price $p$ is higher (observe $\theta(p)$ increases with $p$)---a large part of loyal customer-base of in-house $M$ can improve market opportunities for $M_e$ (observe from (ref) that $\varepsilon \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} p$ fraction of customers seek products from $M_e$, and it is increasing in $p$).
The coalition ${\mathbb V}$ comprising of in-house manufacturer $M$ and supplier $S$ has several advantages, as the vertical cooperation (VC) provides it multiple choices as discussed below.
$ \bullet${\bf [Eliminate downstream competition (E$\ell$)]} The existence of in-house manufacturer in ${\mathbb V}$ provides it an option to operate in monopolistic manner when it is possible to attract a large fraction of `unhappy' loyal customers of the out-house $M_e$; this is possible probably when the manufacturers are substitutable to a good extent, i.e., if $\varepsilon$ is large. In this case, it can completely eliminate out-house manufacturer $M_e$ (by quoting exorbitantly large $q$) and operate in the monopolistic manner in the downstream market with the combined market potential, $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} + \epsilon \bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$.
$ \bullet${\bf [Shut down the in-house (Sh)]} If either the market potential of the in-house manufacturer is low or when its reputation is not very good (when $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$ is more, its customers are highly sensitive to price $p$), or when these factors of the out-house are significantly better, then ${\mathbb V}$ has an option to completely shut its in-house production unit $M$. Such a choice can reduce the competition for out-house $M_e$ which in turn can become beneficial for ${\mathbb V}$---it may have an option to sell large amount of raw material (as market $D_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ attracted by $M_e$ can be large) at good/optimal prices and without expending on operating costs of in-house.
However it may not be beneficial to allow the out-house to operate in a monopolistic manner; like-wise it may not be beneficial to completely eliminate out-house $M_e$ unless the two production units are completely substitutable (in an ideal world with $\varepsilon=1$). In such cases, there are other choices for ${{\mathbb V}}$ which we describe next and are the focus of this paper.
$ \bullet${\bf [Co-existence (Co)]} In this scenario, both ${{\mathbb V}}$ and out-house $M_e$ operate; rather ${{\mathbb V}}$ allows both to operate. By virtue of this, it can charge sufficiently large (optimal) price $q$ for raw materials, which (probably) leaves few choices for $M_e$---the latter then has to quote larger prices $p_e$ to survive in the downstream market. This facilitates ${\mathbb V}$ to benefit from both the worlds, because of the `unhappy' loyal customers ($\varepsilon \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} p$) of $M_e$ that seek product from $M$ as well as from the high profits derived by selling the raw material to $M_e$ at large $q$. Basically it chooses optimal $( p, q)$ that provides the best combined utility as a Stakelberg leader, while competing with the out-house manufacturer $M_e$ in the downstream market. There are several sub-possibilities for ${{\mathbb V}}$ here:
The objective is to identify $(p^*,q^*)$ that maximizes the coalition’s payoff across all the above regimes—--co-existence (in one of the three modes), shutdown of the in-house unit, and elimination of the downstream competition. As shown in Theorem (ref) (provided in later sections), the optimal choice is driven by essentialness, relative market strengths and other parameters. The results of the current paper significantly extend those provided in conference paper wadhwapartition, where the dominance of co-existence is established only under restricted conditions like under essentialness assumption (near $\varepsilon \approx 1$). We now proceed towards detailed analysis and begin with the co-existence regime.
e - \alpha_{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}} p_e^{*} + \varepsilon \alpha_{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}} p \right)^{+} \left( p_e^{*} - C_{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}} - q \right) - O_{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}} \big ] \mathds{1}_{\left\{ p_e^{*} \ne n_o \right \}} , \end{eqnarray}}
where $p_e^{*}=p_e^*(p,q)$, the optimizer of $M_e$, is given by (ref). In this section, we are interested in obtaining the optimal utility of ${{\mathbb V}}$ under co-existence, hence consider those $(p,q)$ for which $p_e^* \ne n_o$. Thus the feasible region for co-existence using (ref)-(ref) is given by:
For $(p,q) \in {\cal F}_{co}$ or in the co-existence regime, the utility of ${\mathbb V}$ is given by:
{
} and the aim in this section is to optimize the above over $(p,q) \in {\cal F}_{co}$ of (ref). Towards this, first consider the following `unconstrained' optimization problem, which resembles (ref) but for $(\cdot)^+$ operators, and when $p_e^*(p,q) < {p_e}_{_{mx}}$:
{
}
It is easier to analyze the above unconstrained problem. Now define the following pair of prices,
{
} which become the optimizers of (ref), when there exists one (proved in (ref))---the above pair $(p_{co}^*, q_{co}^*)$ will also become the optimizer for the original co-existence objective function $U_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\mathbb V}$}}}$ given in (ref), under the same existence condition---this pair and the equivalence is obtained in steps (g.1)-(g.3) of the proof of Theorem (ref) provided in (ref). For now, we continue with discussing the other details related to the `optimal co-existence policy' or the optimizer of (ref) in ${\cal F}_{co}$.
It is clear that, (ref) is different from the `unconstrained' function (ref) in some sub-regimes of the co-existence regime ${\cal F}_{co}$. So in the quest towards the optimal co-existence policy, one also needs to find the optimizer(s) in the sub-regimes where the two differ. In all, we will partition ${\cal F}_{co}$ into many sub-regimes, such that the objective functions (ref) and (ref) match in the first sub-regime, while they differ in the remaining sub-regimes (see Figure (ref)). We now consider them one after the other. Interestingly three of these sub-regimes align with our initial discussion on the choices of ${{\mathbb V}}$, however, an extra boundary line $\{p=p_{mx}\} \cap {\cal F}_{co}$ of ${\cal F}_{co}$ also becomes important and requires separate attention.
We begin with operate both profitably (where functions (ref) and (ref) match) in the immediate next, while the sub-regime with in-house at loss is analyzed in subsection (ref); operate at maximum price (i.e.$\{p = p_{mx}\}$) is provided in subsection (ref), and the sub-regimes $\{p_e^* = {p_e}_{_{mx}}\} $ and operate out-house at par (i.e., $\{ q= \theta(p)\}$) are considered together in subsection (ref).
The BP regime is the set/region of prices $(p,q)$ where both the manufacturers derive strictly positive utilities. It is interesting to identify the conditions under which the coalition ${\mathbb V}$ finds it beneficial to operate in Bp regime and we answer this partially in this subsection. As we will see, such a regime consists of two sub-regimes: a) the interior and some boundaries of a certain region, identified in this subsection; and b) a certain part of the boundary with price $p = p_{mx}$, discussed in subsection (ref).
We now focus on the first sub-regime which includes the complete interior of the Bp regime; this sub-regime is identified using the following steps (as we will see, such a choice also ensures (ref) equals (ref)):
$\bullet$ includes the pair of prices $(p,q)$, for which the optimal price $p_e^*(p,q)$ of the out-house manufacturer is strictly less than ${p_e}_{_{mx}} = \nicefrac{(\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} + \varepsilon \bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}})}{\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$; such a regime from (ref)-(ref) is given by $\{ (p,q) : q < \phi (p) \}$ where $\phi (\cdot) $ is defined below,
the boundary of such a regime is the straight line, $$ \mathbb{L}_1 := \{ q = \phi (p)\}; $$ this condition ensures $p_e^*$ in (ref) matches with its counterpart in (ref);
$\bullet$ includes the pair of prices $(p,q)$, for which the ${{\mathbb V}}$ coalition derives strict positive utility from in-house production unit also; this is the sub-regime where $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} + \varepsilon\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} p_e^{*}(p,q) - \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} p > 0$ (see (ref) and (ref)); such a regime, further within $\{ q < \phi (p) \}$, is given by $\{ q < \phi (p) \mbox{ and } p < \psi (q) \}$, with $\psi (\cdot)$ defined below:
observe here that the straight line, $$ \mathbb{L}_2 := \{p = \psi (q) \}, $$bounds the regime of interest only when it is also bounded by $\mathbb{L}_1$ and these constraints ensure both $(\cdot)^+$ terms in (ref) are positive and hence match with the correspnding terms in (ref);
$\bullet$ the pair of prices $(p,q)$ which ensure co-existence, along with out-house operating, belong to $\{q \le \theta(p)\}$ with $\theta(\cdot)$ as in (ref); when $p \ge p_{sw}$ in (ref) by simple computations one can show that\footnote{By directly substituting the terms, for any $p > p_{sw}$, we have, $\theta(p)-\phi(p) > \theta(p_{sw}) -\phi(p_{sw}) = 0 $.} $\phi(p ) \le \theta(p)$ and so this constraint is already satisfied by bounding with $\mathbb{L}_1$; so it is sufficient to ensure bounding by $\theta(p)$ for $p < p_{sw}$, which is provided in the first row of (ref); hence, in all, it is sufficient to bound by the following additional straight line within the regime bounded by the lines $\mathbb{L}_1$ and $\mathbb{L}_2$,
{{ $$ \mathbb{L}_3 = \left \{q = \frac{\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} + \varepsilon\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} p -\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} - 2\sqrt{\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} O_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}}{\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}} \right \}; $$}}
$\bullet$ and finally bounded by the horizontal line of maximum price, $$ \mathbb{L}_4 =\{ p = p_{mx} \}.$$
Such a sub-regime (actually its closure), represented by ${\cal F}^+_{co}$, is the regime in the positive quadrant, bounded by all the lines $\mathbb{L}_1, \mathbb{L}_2, \mathbb{L}_3$ and $\mathbb{L}_4$ (see polygon ABCDEF in Figure (ref) for one representative scenario). To summarize:
Now the BP regime, where both the manufacturers obtain strictly positive profits, denoted by ${\cal F}_{_{Bp}}$, is a subset of the above region. More precisely, ${\cal F}_{_{Bp}} = \{ (p, q) \in {\cal F}^+_{co}: q < \theta(p), p < \psi(q) \} $---because with $p = \psi(q)$ or $q = \theta(p)$ we respectively have $U_{{\mbox{\fontsize{4.7}{5}\selectfont{${\mathbb M}$}}}}(p,q) = 0$ in (ref) or $U^*_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}(p,q) = 0$ in (ref). Thus the ${\cal F}_{Bp}$ region spans the interior of ${\cal F}^+_{co}$ and the boundary lines:
Hence ${\cal F}_{Bp}$ is not a closed set and may not have an optimizer and therefore it is more convenient to analyze the closed region ${\cal F}^+_{co}$ (ref); we proceed with the same for now, and later discuss the possible optimality of Bp regime.
Further ${\cal F}^+_{co}$ is also a sub-region of the co-existence regime---clearly ${\cal F}^+_{co} \subseteq {\cal F}_{co} $---for example, when non-empty, the I$\ell$ regime $\{p > \psi(q)\} \subset {\cal F}_{co}\setminus {\cal F}^+_{co}$. Hence the optimal utility in co-existence regime is given by:
Thus with the dual purpose of analyzing the co-existence and the BP regimes, we begin with analyzing the first term of (ref):
{\bf Proof} is provided in (ref). { $\Box$}
Thus by part (i), if $(p^*_{co}, q^*_{co})$ of (ref) is in the interior of ${\cal F}_{co}^+$, this pair in the Bp regime has the potential to become a global optimizer of the co-existence region ${\cal F}_{co}$; and then that of the overall problem. From (ref), one may also have a Bp optimizer on lines ${\mathbb L}_1$ (i.e., with $q = \phi(p)$) or on ${\mathbb L}_4$ (i.e., with $p = p_{mx}$); we will show in later subsections that the former is not possible but one can have an optimizer in BP regime with $p = p_{mx}$ in subsection (ref). Eventually in section (ref), we derive some conditions for global optimality of a pair where both manufacturers operate profitably (i.e., optimality of Bp regime), by building upon the results of Theorem (ref) and that of subsection (ref). We also provide more insights on this aspect along with others in section (ref) using several numerical examples.
Next, we analyze the remaining co-existence regions at which at least one of the manufacturers derive zero utility (but still operates at marginal costs)---these comprise of ${\cal F}_{co} \setminus {\cal F}_{co}^+ $ and the boundary lines of $ {\cal F}_{co}^+ $---as already mentioned, there is however a small exception, one may find an optimizer in Bp regime while analyzing $\mathbb{L}_4 =\{ p = p_{mx} \}$.
This sub-region corresponds to the case in which the coalition ${\mathbb V}$ allows its in-house manufacturer $M$ to operate while incurring losses. In this scenario, the coalition ${\mathbb V}$ strategically quotes a very high price $p$, leading to zero demand for its in-house production. Such a strategy can still be advantageous to ${\mathbb V}$, as it may enhance the overall market potential captured via the out-house manufacturer $M_e$ (I$\ell$ is the best among all possible regimes in Figure (ref) of section (ref), depicted by green regions).
We denote this sub-region by ${\cal F}_{_{I\ell}} $, and define it using (ref) (see also (ref)):
From (ref), $\psi(\cdot)$ is an increasing function in $q$, immediately implying, I$\ell$ regime is non-empty only if $\psi(0) < p_{mx}$. Within this sub-region, the co-existence utility $U_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\mathbb V}$}}}$ in (ref), simplifies to the following function (see (ref)):
This is basically the regime, where ${\mathbb V}$ keeps the presence of its in-house alive with negligible production and profits, whose presence molds the demand of the out-house in a much more profitable manner for the supplier component of ${\mathbb V}$ (and this happens, for example, when I$\ell$ is the optimal regime).
Let the optimal utility under this regime be denoted by $U_{_{I\ell}}^{*}$. Using (ref) and (ref), it immediately follows that the optimizer of $U_{_{I\ell}} $ (which is strictly increasing in $p$ for any fixed $q$) is given by $(p_{mx},q^*_{_{I\ell}})$, where $q^*_{_{I\ell}}$ is the solution to the following optimization problem (recall $\psi(\cdot)$ is increasing),
Thus the optimal in I$\ell$-regime (if non-empty) is along ${\mathbb L}_4 = \{p = p_{mx}\}$ line. As we will see in the next sub-section the Mp-Bp regime along the same line starts after I$\ell$-regime, when both the regimes are non-empty (clearly, we have $ r_{_{I\ell}} \le l_{_{Mp}} $, the left boundary of Mp regime given in (ref)).
Further, using (ref), for any $q \le r_{_{_{I\ell}}}$ (with the convention that $[a,b]=\emptyset$ when $a>b$), we have:
{
The first term can be optimized by ignoring the indicator ${\mathbb I}_1(q)$ to obtain a candidate optimizer ${\tilde q}^*$ (given below). The overall optimizer for this sub-case then becomes (first term in $U_{_{_{I\ell}}}$ is concave, while the second is linear in $q$):
By direct substitution (when $\varepsilon > 0$, the inequality '$a$' is strict, in the below),
{
} and since $r_{_{_{I\ell}}} \le \max\{0, \psi^{-1} (p_{mx})\}$ we have $ \min\{ r_{_{_{I\ell}}}, \phi(p_{mx})\} = r_{_{_{I\ell}}}, $ and so,
Hence the optimal pair in this sub-regime (when non-empty, i.e., when $\psi(0) < p_{mx}$) is $(p_{mx},q^*_{_{I\ell}})$ and the corresponding (sub) optimal utility equals:
{ \scalebox{0.82}{
} }
The above indicates the possibility of an interesting optimizer for ${\mathbb V}$---when $r_{_{_{I\ell}}} = \theta(p_{mx}) $ and ${\tilde q}^* \ge r_{_{_{I\ell}}}$ in (ref), then ${\mathbb V}$ might find the optimizer in the combined I$\ell$-Op regime, where the in-house incurs losses and the out-house operates at par simultaneously. However, we could not observe the optimality of such a pair in the exhaustive numerical examples of section (ref); we did not observe such an optimality even in the other exhaustive set of numerical examples not included in the paper.
}
We now consider the sub-regime located along the boundary $\mathbb{L}_4$ of ${\cal F}_{co}^+$ and find the corresponding sub-optimizer. As mentioned in subsection (ref), this optimizer can also be in the BP regime, where both the manufacturer s derive strictly positive utility. Define the following using (ref),(ref)-(ref), to reflect the boundary points corresponding to $p=p_{mx}$:
{
} It is clear that Mp regime corresponds to $\left \{ (p,q) : p = p_{mx}, q \in [\, l_{_{Mp}}, r_{_{Mp}} \,] \right \}$ and is non-empty only when $l_{_{Mp}} < r_{_{Mp}} $. Using (ref)-(ref), $r_{_{Mp}}$ equals:
When $p_{mx} \ge p_{sw}$ (i.e., when $\varepsilon^2 \bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} \ge \sqrt{\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} O_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$ in the above), it is immediate that $l_{_{Mp}} < r_{_{Mp}} $, however the same is not always guaranteed for $p_{mx} < p_{sw}$. For non-empty Mp regime, the corresponding sub-optimizer (for $p = p_{mx}$) is obtained as in the proof of Theorem (ref) and is given by (function $h$ is defined in (ref) of (ref)):
Clearly, when $q^{*}(p_{mx}) = h(p_{mx})$ or when $q^{*}(p_{mx}) = r_{_{Mp}} = \phi(p_{mx})$, this sub-optimizer is in Bp regime. If the overall optimal pair for ${\mathbb V}$ is at such points, we say the optimal regime is Mp-Bp regime. Likewise, the sub-optimizer is in the OP regime if $q^{*}(p_{mx}) = r_{_{Mp}} = \theta(p_{mx})$ (recall in Op regime, the out-house derives zero utility and hence is not in Bp regime).
We are now left with two additional sub-regimes: one in which the out-house’s optimal price equals ${p_e}_{_{mx}}$ in (ref), and another in which the out-house $M_e$ is forced to operate at break-even. In the latter case, the optimal utility of $M_e$ is exactly zero. We first discuss the former case.
From (ref) and (ref), when the optimal price of the out-house saturates at ${p_e}_{_{mx}}$, then the utility function $U_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\mathbb V}$}}}$ of coalition ${\mathbb V}$ modifies to the following:
The set of $(p,q) \in {\cal F}_{co}$ where such a saturation occurs is given by (see (ref) (ref)):
{
} Comparing section wise, once again across $q$, one can easily verify that
Further, it is not difficult to see that if there exists a $p$ such that $(p,q) \in {\cal F}_{_{St}}$, then $(p,\theta(p)) \in {\cal F}_{_{St}}$. Thus the optimal co-existence utility in ${\cal F}_{_{St}}$ is given by the optimal across all points in which the out-house operates at par and at saturation, i.e., in $ {\cal F}_{_{St}} \cap \{ (p, \theta(p) ) \}.$ As a result, towards finding the global optimization point, it is sufficient to consider the optimal across Op regime (and compare with others). This is discussed in the immediate next. Before proceeding, we would also like to note here that an optimal pair in BP regime does not exist with out-house operating at maximum price. This also implies Mp-Bp regime is optimal (if at all) only when $q^{*}_{p_{mx}} = h(p_{mx})$ in (ref).
This is the regime, where the out-house operates, however is forced to do so at par. As just discussed in (ref), towards finding the optimal among the Op and the out-house-price-saturation regions, it is sufficient to consider optimal across the Op regime---the relevant optimization problem is:
Towards solving the above, we need to proceed separately depending upon the sign of $ (p_{sw}-p) $ (see (ref)). The following optimization problem is relevant for $p \le p_{sw}$
The optimizer of the above by strict concavity is at $p^{1, *} $ given below:
The second optimization for $p > p_{sw}$ is given by the following and is applicable only when $p_{sw} \le p_{mx}$:
{
} The optimizer of the above by strict concavity is at $p^{2, *}$, given below:
{
}
In all, the optimal value in the Op sub-regime is given by:
{
} The sub-optimal pair is either $(p^{1,*}\theta(p^{1,*})) $ or $(p^{2,*}, \theta(p^{2,*}) ) $, depending upon the bigger of the two in the above.
We finally have the following result using Theorem (ref) and the sub-optimal utilities in (ref), (ref), (ref):
{\bf Remarks:} Thus, the optimal operating pair for ${\mathbb V}$ within the co-existence regime can occur in one of the following four (actually three) sub-regimes:
Taken together, these three co-existence sub-regimes capture the coalition’s strategic flexibility in balancing cooperation and competition. From a managerial perspective, they demonstrate that profitability for a vertically integrated supplier need not always depend on the in-house unit’s direct success. In certain market conditions, deliberately constraining or influencing the out-house manufacturer through strategic pricing, for its in-house, can yield superior overall outcome.
ha_{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}} (2-\varepsilon^2))}$ and thus when,
{
} operating the in-house production unit at loss is never optimal. In other words, when the product is sufficiently essential, I$\ell$ is never an optimal regime.
For subsequent theoretical analysis, in the regime $\varepsilon \to 1$, we require $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}\ne\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ for a technical reason (for example results in Lemma (ref) and Theorem (ref))). One can separately analyze the case with equal $\alpha$'s in a similar manner, however we skip them for keeping the paper short and because the case with $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ close to $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$ would provide a good picture about the equal case, by regular continuity arguments (one does require some technical proof for saying this formally). We begin with the following lemma which first shows that the optimal among the co-existence regimes near $\varepsilon \to 1$ is either Mp-Bp or Op, and also finds the best among the two.
The proof is in (ref). { $\Box$}
Lemma (ref) characterizes the asymptotic structure of the optimal co-existence regime as the product essentialness parameter approaches unity, i.e., as $\varepsilon \to 1$. In this regime, product becomes highly essential and the customer loyalty weakens. As a result, the coalition ${\mathbb V}$ no longer finds it beneficial to operate it's in-house unit at loss (I$\ell$ regime)---basically there is no requirement to explicitly force the customers towards the out-house manufacturer as they automatically switch gears because of essentialness (and non-loyalty). Instead, the optimal strategy necessarily reduces to one of two boundary regimes: the operate-at-par or Op regime or the maximum-price profitable co-existence or Mp-Bp regime (here opponent also derives positive profit and ${\mathbb V}$ operates at maximum price, latter is facilitated again by essentialness).
More specifically, condition (ref) serves as the threshold criterion governing this choice. The score $\eta_{co}$ of (ref) can be seen as a consolidated relative strength indicator, near essentialness regime, which is constructed by a special combination of the three levers or the characteristics of the two production houses: market potentials ($\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}, \bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$), price sensitivities ( $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}, \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$) and the production costs ($C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}, C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$). When $\eta_{co} < 0$, the out-house is a significant player and can't be bullied by ${\mathbb V}$---at the optimal choice of ${\mathbb V}$, the out-house also derives positive profit; here the optimal pair for ${\mathbb V}$ is $(p_{mx},h(p_{mx}))$.
If $C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} > C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$ and $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} > \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$, then the out-house is clearly weaker---here $\eta_{co} > 0$ and ${\mathbb V}$ manages to force the out-house to operate at par---its optimal pair is $(p_{mx},\theta(p_{mx}))$. If one or both of the above inequalities are not true, i.e., when the out-house is supremum either in customer reputation (with $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} < \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$) or in production cost (with $C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} < C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$), there is a possibility of it operating with profit in the essentialness regime (as $ \eta_{co} $ can become negative under such conditions).
Also observe the market potentials $\bar{d}_{\mbox{\fontsize{4.7}{5}\selectfont{${\mathbb M}$}}}$ and $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ play a role in determining the zero/non-zero profits of out-house, only when the out-house is strictly superior in one of the two quantities, reputation factor or the production costs. More strikingly, with good reputation, for the sake of better explanation consider $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} \approx 0$, we observe that a higher market potential of out-house can force it to operate at par: note with $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} \approx 0$, $\eta_{co} \approx - \bar{d}_{\mbox{\fontsize{4.7}{5}\selectfont{${\mathbb M}$}}} + 2 \bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$. In other words, having a higher market potential in the essentialness regime along with good reputation, can only become hazardous for the out-house manufacturer. We provide an elaborate numerical case study to clearly illustrate this surprising dependency in subsection (ref), see Figures (ref)-- (ref).
Thus, Lemma (ref) is instrumental in characterizing the high-essentialness asymptotics of the co-existence regime. It also provides the foundation for the subsequent analysis of global optimality that includes regimes other than co-existence, or more precisely the single-existence regimes, where one of the production units is shut or is forced to shut completely; these regimes are discussed in the next section. The complementary low-essentialness scenario (results when $\varepsilon \to 0$), where the customers remain highly loyal to their respective manufacturers, is directly analyzed along with single-existence regimes after analyzing the latter.
In the single-existence regimes, the coalition ${\mathbb V}$ allows only one production unit to operate. Accordingly, we have two regimes: i) Shut down its in-house unit (Sh regime); and ii) Eliminate the downstream competition (E$\ell$ regime). We begin with the study of the Sh regime.
In contrast to the co-existence regimes analyzed in Section (ref), the coalition ${\mathbb V}$ may choose to completely shut down its in-house production unit (Sh) and operate solely as an upstream supplier, if the choice is optimal. By withdrawing from the downstream competition, the coalition eliminates internal cannibalization and effectively consolidates the downstream market in favor of the out-house $M_e$. Although this results in the loss of direct retail revenue, it may enhance the upstream profitability---a stronger (and monopoly) downstream presence of $M_e$ may expand the total demand captured, enabling the supplier to extract probably a much higher wholesale revenue.
We model the demand attracted by the out-house manufacturer, in this $M$-absent-regime, by first considering that the market potential of out-house increases to $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} + \varepsilon \bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$ (keeping in view of substituitability and essentialness factors). We then model the demand after customer response as below:
using an additional parameter $r \in [0,1]$, which we refer to as the secondary fallback rate, to capture the double fold-back behavior of customers in the absence of $M$. This modeling is considered because of the following reasons: a) among the $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} p_e$ fraction of customers dissatisfied with $M_e$, the sub-fraction $\varepsilon \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} p_e$ would have resorted to $M$, had $M$ been operational; b) however, since $M$ is not operational, they attempt to fold back to $M_e$ due to lack of options; and c) we model this fraction as $r\varepsilon \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} p_e$, using the additional parameter $r$. In effect, due to the absence of competition, the effective price sensitivity reduces to $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}(1-r \varepsilon)$ and the market potential increases to $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} + \varepsilon \bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$. And the parameter $r$ quantifies the intensity of the fallback effect:
We study a general problem by analyzing the system for different values of $r$ and other system parameters. We also have a special numerical case-study where we set $ r =\varepsilon$ in subsection (ref)---such a study is important as both the factors represent a kind of fallback or substituting nature of the customers.
The utilities of the coalition ${\mathbb V}$ and the out-house manufacturer $M_e$ under this regime are given by (see (ref)-(ref)):
Formally, the shutdown regime is characterized by the set of wholesale prices $q$ that ensure the out-house operates in the absence of in-house and hence is given by:
where $ \theta(n_o)$ is defined in a similar manner as in (ref)---this represents the maximum wholesale price $q$ beyond which $M_e$ prefers not to operate even in the monopoly regime with $r$-fold-back---we obtain it by solving $U_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}^{*}(n_o,q) = 0$, with $D_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ as in (ref) (see also (ref)):
For finding the optimal utilities of the coalition ${\mathbb V}$ and the out-house $M_e$, we need to solve the Stackelberg game as we did in section (ref), but here the utilities are given by (ref)-(ref). For Sh regime, the Stackelberg game is simplified to optimization problem at both the levels---with $p=n_o$, coalition ${\mathbb V}$ just has to optimize it's upstream wholesale price $q$ while $M_e$ has to optimize it's downstream retail price $p_e$ (as a monopoly). Thus the optimal or equilibrium strategies are obtained by solving the Stackelberg game as in the following:
Further with $p_e^{*} := p_e^{*}(q^{*})$,
represent the utilities at the equilibrium. We obtain the same in the following:
{\bf Proof } follows from wadhwapartition .
The I$\ell$ and Sh regimes differ fundamentally in both market structure and strategic objective. In the \(I\ell\) regime, the coalition \({\mathbb V}\) keeps the in-house unit operational but prices it sufficiently high that the unit incurs some losses. Its role is therefore not profit generation, but strategic discipline: by remaining active, the in-house unit constrains the downstream market power of the out-house manufacturer \(M_e\) and preserves the coalition’s bargaining position in wholesale pricing.
In contrast, under the Sh regime the coalition completely shuts down the in-house production unit \((p=n_o)\), allowing \(M_e\) to become the sole downstream producer. The \(Sh\) regime thus represents a qualitative departure from I$\ell$---rather than maintaining downstream presence to influence market outcomes, the coalition deliberately relinquishes downstream control and relies exclusively on upstream value extraction.
{\color{blue} Additionally, the two regimes differ fundamentally in the mechanism through which the out-house manufacturer captures demand. In the Sh regime, the complete shutdown of the in-house unit leaves the out-house manufacturer as the sole downstream seller. Consequently, its effective market potential increases from $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ to $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}+\varepsilon\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$, while the effective price sensitivity changes from $1$ to $(1-r\varepsilon)$. In contrast, under the I$\ell$ regime, the in-house manufacturer remains operational, albeit at a loss. Consequently, dissatisfied customers continue to have an alternative seller and can switch directly to the out-house manufacturer instead of facing a monopolistic market. Thus, the additional demand captured by the out-house arises through customer substitution rather than through monopoly expansion. Since the coalition ${\mathbb V}$ may prefer to retail exclusively through the out-house manufacturer, it compares these two fundamentally different operating modes and selects the one yielding the higher profit. For ease of comparison, the equilibrium demands of the out-house manufacturer in the two regimes are summarized below (see (ref), (ref)):
In all, the Sh and the I$\ell$ regimes typically differ in customer response towards the out-house manufacturer. }
This distinction raises a broader strategic question: under what market conditions should the coalition derive value through upstream wholesale extraction, downstream retail participation, or a combination of both? Section (ref) showed that the I$\ell$ regime is not optimal when the product essentialness is high (see (ref)), irrespective of the relative powers ($\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}},\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$, $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}, \bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ and $C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}, C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$) of the two production units. An immediate question in this regard is, will ${\mathbb V}$ find Sh regime to be optimal in such scenarios? The problem becomes more nuanced when product essentialness is moderate or low. Our aim in this paper is to derive answers to such questions in complete generality (however still under the minimal assumptions of this paper). For now, we would like to mention that one can have scenarios under which I$\ell$ is optimal or Sh regime is optimal among all possible choices for ${\mathbb V}$ (see Figure (ref) of section (ref) where we depict such possibilities, after considering the overall comparison).
Similar to the exclusionary strategy followed by the coalition ${\mathbb V}$ to shut down it's in-house unit which led to the Sh regime analyzed in the previous subsection, the coalition ${\mathbb V}$ may choose to completely eliminate the out-house manufacturer (E$\ell$) and operate solely through its in-house production unit, if such a choice is optimal. By strategically quoting a sufficiently high wholesale price, the coalition renders the operation of $M_e$ economically infeasible, thereby forcing its exit from the downstream market (recall we are considering a stylized model with one exclusive supplier, which can be close to real-world applications with monopoly in supply segment---probably where the competitor suppliers are significantly inferior).
This strategic choice is not motivated solely by short-term profit considerations. By driving $M_e$ out of the market, the supplier reshapes the competitive environment and attains full vertical control: the in-house unit $M$ becomes the exclusive downstream producer, effectively creating a monopolistic configuration at both levels. Such a structure enables the supplier to govern the entire value chain and this can become profitable under certain conditions on the system parameters or the relative powers of the two production units.
The payoff function of the coalition ${\mathbb V}$ in this regime is
where the demand $D_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}(p,n_o)$ faced by the in-house manufacturer $M$ is given by the $M_e$-absent demand model which follows a similar functional form as the previously introduced $M$-absent demand model in (ref):
We again increase the potential to $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} + \varepsilon \bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ and decrease the price sensitivity to $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} + r\varepsilon \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$, due to the absence of $M_e$, as in the previous subsection--- basically the term $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} p$ captures the demand loss caused by the retail price $p$, while the final term $r\varepsilon \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} p$ models the fallback of some customers who still consume the product of $M$ due to lack of options, despite being dissatisfied with the quoted price $p$.
Under the elimination regime, the supplier’s pricing decision therefore centralizes all downstream activity within its own production unit. Formally, the elimination regime is represented by the feasible action set
where $\bar{\theta}$ denotes the critical wholesale-price threshold beyond which the out-house manufacturer $M_e$ finds downstream operation economically infeasible and therefore optimally exits the market. In this regime, the Stackelberg game like in section (ref) (see also the game in subsection (ref)) collapses to the optimization problem (because of absence of $M_e$) where ${\mathbb V}$ maximizes the downstream retail price $p$. Thus we have the following:
We prove that the above optimizer $p^*$ exists and obtain $U_{El}^*$ in the following:
{\bf Proof } follows from wadhwapartition.
The E$\ell$ and Op regimes again differ fundamentally in both market structure and strategic objective. In the E$\ell$ regime, the coalition ${\mathbb V}$ completely eliminates the out-house manufacturer $M_e$ and operates in a monopolistic manner in both the segments---it banks on the effective market demand captured due to the absence of $M_e$, which includes the transferable part of $M_e$, captured with the help of parameters $(\varepsilon, r)$. In contrast, under the Op regime, it allows $M_e$ to remain operational while strategically maximizing its own revenue---this is achieved by setting a sufficiently high wholesale price that renders $M_e$ operate at break-even point (or derive zero profit). The Op regime therefore represents a controlled co-existence structure in which downstream competition is preserved operationally, but all economic surplus generated by the out-house manufacturer is extracted upstream by the coalition.
As observed in Lemma (ref) and also illustrated by numerical experiments in subsection (ref) (see cyan regions in Figure (ref)), even a relatively strong out-house manufacturer can be forced into break-even operation when the coalition strategically leverages its dominant upstream position, more so when the substituitability or the essentialness factors are high. Interestingly, we find many scenarios in which E$\ell$ regime can also become the optimal choice for coalition ${\mathbb V}$ (see yellow regions in Figure (ref)).
The final question is regarding the optimal or the best choice among all the available configurations, which were discussed in the previous sections. Clearly the optimal value of ${\mathbb V}$ is given by the following, using Theorem (ref):
{
} and the corresponding optimizer (including the configuration) provides the overall optimal choice for ${\mathbb V}$. The final solution of the above problem characterizes the optimal strategy of the encroaching supplier among the three strategic regimes (and their sub-regimes): co-existence (with Bp, I$\ell$ and Op as sub-choices), shutdown of the in-house production unit, and elimination of downstream competition. We next compare the equilibrium utilities achieved under each regime to identify the coalition's optimal strategy for the given market conditions and the relative strengths of the agents involved.
In the previous sections we derived the theoretical performance---the closed form expressions for optimizers and the utilities---in various sub-regimes. We now characterize the overall optimal choice of coalition ${\mathbb V}$, using these performance expressions. One can compute the optimal utilities of various regimes numerically using the derived expressions and easily obtain a numerical comparative study for any given set of system parameters, the strength indicators of the two production units $(\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}, \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}), $ $(\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}, \bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}),$ $(C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}, C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}, C_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}})$, $(O_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}, O_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}, O_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}})$ and the system parameters $(\varepsilon, r)$. One can also derive theoretical characterization of the optimal choice of ${\mathbb V}$ under some asymptotic regimes. We begin with the theoretical study in the immediate next, while a more complete numerical study is considered later.
We study two asymptotic scenarios primarily based on the essentialness factor $\varepsilon$---the scenario characterized by $\varepsilon \to 0$ is referred to as the Low-Essentialness (LE) scenario, while that near $\varepsilon \to 1$ is the High-Essentialness (HE) scenario. We begin with characterization of the optimal choice in the LE scenario (proof is in (ref)).
Thus when the product is not sufficiently essential and when the customer loyalty is high (when the consumers are loyal towards the preferred brands and would only purchase when the prices are not too high), coalition ${\mathbb V}$ finds Bp as the optimal strategy---where both the units operate profitably. High customer loyalty towards individual manufacturers ensures that the coalition ${\mathbb V}$ finds it neither beneficial to shut down a unit nor optimal to choke one of them (i.e., operate in I\(\ell \) or Op regimes). Interestingly, this is true irrespective of the fallback rate $r$.
We now derive the optimal configuration for the HE scenario (the proof is in (ref)).
The above result provides theoretical insights into optimal choice of ${\mathbb V}$ when $\varepsilon \approx 1$, see also Figure (ref). One can further evaluate the two scores $\eta_{co}$ and $\eta_{se}$ analytically, to exactly determine the the optimal configuration, for two interesting case studies, which we consider in the immediate next.
Consider a scenario where the in-house is superior both in terms of price-sensitivity and production technology: $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} < \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ and $C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} < C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$. Then $\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} (C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}+ C_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}}) < \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} (C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}+ C_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}})$ and so $ \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} \left(\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}+\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}-\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}(1-r)(C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}+C_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}})\right)^2 $ is strictly less than $ \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}\left(\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}+\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}-\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}(1-r)(C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}+C_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}})\right)^2 $, implying the score $\eta_{se}$ is negative in (ref). Further $\eta_{co}$ of (ref) is positive, using {\bf A}.1:
{
}
Hence E$\ell$ regime is optimal for $r\ge\bar r$ and Op is optimal when $r<\bar r$. In other words, when in-house unit is superior to out-house solely in terms of price sensitivity and production cost, it is optimal for the coalition to keep the out-house manufacturer at par when the fallback rate is low (not many customers of a manufacturer \ exiting the market are interested in buying from the other) and eliminate the downstream competition once the fallback rate exceeds $\bar r$---this is true irrespective of the market potentials of the two production units.
We next consider the scenario where the out-house is superior with, $2\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}<\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$ and $C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}<C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$. Then $ \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}(C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}+C_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}})> \alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}(C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}+C_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}}), $ and so {
} Clearly, from (ref), $\eta_{se} > 0$, as $8\alpha_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}\alpha_{\mbox{\fontsize{4.7}{5}\selectfont{${\mathbb M}$}}}(1-r)O_{\mbox{\fontsize{4.7}{5}\selectfont{${\mathbb M}$}}}>0$. Thus in scenarios with higher fallback rates, the Sh regime is optimal. However, the sign of $\eta_{co}$ of (ref) depends upon the relative values of the third characteristic of the manufacturers' market strengths, the market potentials $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$ and $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$---the score $\eta_{co} >0$ only if $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} $ is sufficiently bigger than $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}$. In other words, in scenarios with smaller fallback rates, Op regime is optimal if $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} $ is sufficiently large, else the coalition finds it optimal to let the out-house operate profitably.
Thus the surprising negative dependency of the optimal configuration on the market potential of the out-house, among the co-existence scenarios, discussed immediately after Lemma (ref), continues to hold even for the overall optimal choice, albeit only when the fallback rates are small. More precisely, the coalition facing a strong out-house (in all three market characteristics) will find it beneficial to shut-down its in-house unit, only if the fallback rate is higher; however will force the mighty out-house to operate at par when the fallback rate is small---and such a choice becomes optimal as the out-house market potential $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ increases beyond a threshold.
Thus the following is the summary of our theoretical results.
The overall theoretically derived optimal strategy of the coalition is shown in Figure (ref). Now we move to numerical examples starting with the optimal regime characterization using Algorithm (ref) to obtain the complete picture .
with an aim to derive more insights into the problem.
To validate the analytical results and extract more practical insights, we conduct numerical experiments across a wide range of market environments by varying the product essentialness parameter $(\varepsilon)$ and the secondary fallback rate $(r)$. We also consider experiments with varying potentials. These numerical examples illustrate how customer loyalty, market potential asymmetries, and fallback behavior jointly determine the supplier’s optimal strategy among Bp, I$\ell$ ,Op, E$\ell$ and Sh regimes. The following parameters are kept fixed, while others are fixed/varied based on the experiment: \[ C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} = C_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} = 4, \quad C_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}} = 3, \quad O_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}} = O_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} = O_{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}} = 10. \]
Our first set of results are provided in Figure (ref) and its five sub-figures. These results summarize the coalition’s optimal regime as a function of $\varepsilon$ and $r$ under five representative market structures---recall the in-house and out-house can be compared in terms of all three levers, price-sensitivity $\alpha$ factors, production $C_{{\mbox{\fontsize{4.7}{5}\selectfont{${\mathbb M}$}}}}$ costs and $\bar{d}$ the market potentials and our studies are focused on $(\alpha, \bar{d})$ for equal production costs---the combinations of these parameters define the five market structures as described below:
In all the figures and sub-figures we provide color patches in a two dimensional square, where each color at any point represents the optimal configuration for $(r, \varepsilon)$ pair representing the point. The following are the observations:
01$, $\alpha_{{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}=0.1$, $C_{{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}=4$, $C_{{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}=1$, $C_{{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}}=3$, and $O_{{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}=O_{{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}=O_{{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}}}=10$, and plot the coalition's optimal regime as a function of the out-house market potential $\bar d_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$ and the fallback rate $r$ for $\varepsilon\in\{0.35,0.65,0.95\}$ in Figures (ref)--(ref).
At low essentialness ($\varepsilon=0.35$), the co-existence option dominates when the fallback for smaller values of rate $r$: the coalition finds it optimal to operate in Bp for small values of $\bar d_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$ and shifts to $I\ell$ as $\bar d_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$ increases; the coalition finds it beneficial to operate in Sh mode once $r$ is sufficiently large.
For intermediate values of essentialness $\varepsilon=0.65$ in Figure (ref), the Bp region shrinks sharply, with Op being optimal for most values of $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ when $r$ is small; however Sh or shutdown in-house is the best option once fallback rate $r$ is sufficiently large---interestingly this option is optimal for larger market potentials of out-house only for very high values of $r$. Thus once the essentialness factor is sufficiently high, the out-house is forced to operate mostly at break-even potential as its potential increases---this is because of the combined effect of monopoly of supplier in upper echelon and it's ability to enter downstream market via in-house.
At high essentialness ($\varepsilon=0.95$) in Figure (ref), Bp almost disappears and the coalition's optimal choice alternates only between Op (below a fallback threshold) and Sh (above it). Thus even at higher values of $\varepsilon$, as the out-house's market potential increases, the coalition forces it to operate at par rather than ceding profits, unless the fall-back rate is too high. Such high fall-back rates may not be realistic, and hence one can again conclude that higher market potential is not a good news for out-house in the presence of a monopolistic supplier with in-house production house, unless the product is of luxury catagory.
There is a natural correlation between the essentialness factor $\varepsilon$ and $r$, the fallback rate---both these parameters represent a kind of fallback or substituting nature of the customers to an alternate production unit. We now consider a third case study, where we set $r=\varepsilon$, to obtain more focused insights for the scenarios that reflect the said correlation.
We again investigate how the coalition's optimal regime changes with the out-house manufacturer's characteristics $(\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}, C_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}, \alpha_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}})$ and with $r$ (which now equals $\varepsilon$) in Figures (ref)--(ref). The remaining parameters are set at: $\bar d_{{\mbox{\fontsize{4.7}{5}\selectfont{${\mathbb M}$}}}}=800$, $\alpha_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}}=0.01$, $C_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}}=4$, $C_{{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}}}=3$, and $O_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf M}$}}}}=O_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}=O_{{\mbox{\fontsize{5.2}{5.2}\selectfont{${\bf S}$}}}}=10$.
We first fix $\alpha_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}=0.001$ and $C_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}=1$ and vary the out-house market potential $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ and $r$ in Figure (ref). For small values of $r$, once again the co-existence-oriented regimes dominate: the coalition operates in the $Bp$ regime when $\bar d_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$ is small and transitions to the loss-making regime I$\ell$ as $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}$ increases. As $r$ further increases, the coalition relies on a more strategic market discipline Op, where it curbs the out-house to operate at par. For even large values of $r$, co-existence ceases to be optimal and the coalition finds it optimal to shutdown the in-house (Sh regime), irrespective of the out-house market potential.
Next, fixing $\alpha_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}=0.001$ and $\bar{d}_{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}} =1000$, we vary the out-house production cost $C_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$ and $r$ in Figure (ref). The regime boundaries are nearly insensitive to changes in $C_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$. As $r$ increases, the coalition sequentially transitions through the $Bp$, I$\ell$, $Op$, and $Sh$ regimes, indicating that the fallback intensity plays a much more significant role than the out-house production cost in determining the coalition's optimal strategy.
Finally, we fix $C_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}=1$ and $\bar d_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}=1000$ and vary the out-house manufacturer's price sensitivity $\alpha_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$ and $r$ in Figure (ref). In contrast to the production cost, the coalition's optimal regime is highly sensitive to $\alpha_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$. For very small values of $\alpha_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$, the coalition transitions from I$\ell$ to $Bp$ and then to $Op$ as $r$ increases. However, even a modest increase in $\alpha_{{\mbox{\fontsize{4.7}{4.5}\selectfont{${\bf Me}$}}}}$ causes the elimination regime E$\ell$ to rapidly dominate the parameter space. The shutdown regime $Sh$ appears only for high values of $r$ when the out-house manufacturer has a very strong reputation.
This paper studies a partially vertically integrated supply chain in which a supplier simultaneously operates an in-house production unit and supplies raw material to an independent out-house manufacturer that competes with it in the downstream. Using a Stackelberg game framework with dedicated yet cross-influenced customer bases, we characterize how the coalition of supplier and in-house manufacturer can jointly set wholesale and retail prices to shape the entire downstream market to their advantage based on system parameters, rather than merely coordinate it.
Our analysis shows that the presence of an in-house unit fundamentally alters the competitive dynamics: it gives the supplier an instrument to discipline the out-house manufacturer by forestalling its downstream monopoly. It can optimally select among co-existence with profits for both, force the opponent to operate at break-even point, in-house at losses, shutdown of in-house, or outright exclusion of the rival---depending on customer loyalty, product essentialness, demand fallback (in the absence of a particular production-unit), and the relative strengths of the two units. We derive closed-form optimal prices and utilities for each of these regimes and provide a numerical procedure that identifies the coalition's optimal choice for any given set of market and system parameters.
For two limiting, yet important, cases, we also provide provable optimal characterizations. When the customer loyalty is high and the products are not so essential (i.e., for the case with luxury products), we prove that profitable co-existence is optimal for the coalition. At the opposite extreme, when essentialness is high, we prove that the optimal regime collapses to a choice between at most two alternatives, governed by two closed-form relative-strength scores: for low demand fallback, the coalition either forces the out-house manufacturer to operate at break-even or sustains profitable co-existence at the maximum retail price; for high fallback, it either shuts down its own in-house unit or eliminates the out-house manufacturer entirely. Outside these two limiting regimes, our numerical study---validated against the theoretical asymptotics---shows that the transition between regimes is generally smooth and monotone in essentialness factor and fallback rate, with one notable exception: when the in-house unit is distinctly inferior to its rival, the coalition can find it optimal to operate the in-house unit at a loss over an intermediate range of essentialness, using it purely as a competitive instrument rather than a profit center.
The most surprising outcome of this study is that a higher market potential can become detrimental to the out-house---the coalition forces the otherwise superior out-house to operate at par when the market potential of the latter is higher than a threshold and optimally prefers profitable operation of both units otherwise---this is true especially when the out-house is superior in terms of price-sensitivity (or market reputation) and production costs.
This work opens several avenues for further research. A natural extension is to study the same encroachment problem when the supplier is no longer a monopoly in the upstream market. Another promising direction is to embed the model in a dynamic setting incorporating inventory decisions, demand fluctuations, learning, and inter-period strategic adaptation. Yet another interesting study is to quantify the incremental value of the vertical encroachment by comparing supplier profit with and without an in-house production capability.