Strategic input outsourcing and equilibrium location choice

Abstract

This study incorporates the strategic behavior of outsourcing with a variant Hotelling model to explore the role of input outsourcing in determining equilibrium locations for firms under quadratic transportation. Given that strategic input outsourcing occurs, we show that the cost-efficient integrated firm will locate as far away from its rival as possible, so as to increase its rival’s input price when its own cost advantage is small; at the same time, the cost-inefficient downstream firm likes to locate closer to its rival to lower the input price. Hence, there is an interior locational equilibrium, and the principle of maximum differentiation does not hold. When an integrated firm’s cost advantage is large, the principle of maximum differentiation is valid. However, when the integrated firm’s own cost advantage is even larger, the integrated firm can become a monopolist through strategic input outsourcing. Under this case, the equilibrium location depends on the magnitude of the input homemade ratio when the input homemade ratio is small. Otherwise, the integrated firm would like to locate at the middle of the market.

Appendices

Appendix 1: Equilibrium prices of final goods under a monopoly market

Under the monopoly downstream market case, we have the price relationship between two final goods by using $\hat{{x}}=1$:

$$\begin{aligned} p_d -p_v =t(x_d -x_v )(2-x_d -x_v ). \end{aligned}$$

(17)

Substituting (4.2) and (4.4) into (2.3), we derive the demand of final goods for firm d:

$$\begin{aligned} q_d^D =[-\alpha (w-c_v )+t(x_d -x_v )(4-x_d -x_v )]/6t(x_d -x_v ). \end{aligned}$$

(18)

Letting $q_{d}^{D }= 0$, we have $\alpha (w-c_{v})=t(x_{d}-x_{v}) (4 - x_{d}-x_{v})$. Substituting $\alpha ( w-c_{v})$ into (4.2), we arrive at the equilibrium price of firm v under the monopoly case:

$$\begin{aligned} p_{v}^{M }=w -t(x_{d}-x_{v}) (2 - x_{d}-x_{v}). \end{aligned}$$

(19)

Substituting (19) into (17), we have the equilibrium price of firm d: $p_{d}^{M }=w$. Substituting these prices into (3.2), we then have the profits of firm v when it monopolizes the whole downstream market: $\pi _{v}^{M} = \alpha$ $(w-c_{v})-t(x_{d}-x_{v}) (2 - x_{v}-x_{d})$.

Appendix 2: Equilibrium input price under a duopoly market

Under the duopoly market case, the total profits of upstream firm u are: $\pi _{u}^{D}= \pi _{u}^{D}(q_{v}^{D} (w), q_{d}^{D} (w)$, w). Differentiating it with respect to w, we obtain the equilibrium input price, which is (8.1). From (7.1), we find that ( $\partial \pi _{u}^{D}/ \partial q_{v}^{D}) = (w-c_{u})(1 - \alpha ) > 0, ( \partial \pi _{u}^{D}/ \partial q_{d}^{D}) = (w-c_{u}) > 0$, and $( \partial \pi _{u}^{D}/ \partial w)=q_{d}+ (1 - \alpha )q_{v}$. Using (5.1) and (5.3), we have $(\partial q_{v}^{D}/ \partial w)= - ( \partial q_{d}^{D}/ \partial w) = \alpha /6t(x_{d}-x_{v})$. We then can derive the equilibrium input price under the duopoly market case when $0 < \alpha < 1$, which is (9) in the context.

Equilibrium input price under a monopoly market

Under the monopoly market case, upstream firm u’s profit is (7.2). Differentiating (7.2) with respect to w, we find that $(\partial \pi _{u}^{M }/ \partial w) = (1 - \alpha ) > 0$. It implies that the upstream firm would like to set the wholesale price as high as possible to increase its profit. However, we should consider that all of the consumers are served in the market; it requires that the utility of the farthest consumer in the market be positive. When $0 \le x_{v} \le 1/2$, the farthest consumer locates at 1, his utility should be positive after purchasing the final good: $U(x =1) = k - p_{v}^{M}-t(1 - x_{v})^{2} \ge 0$. When $1/2 < x_{v} \le 1$, the farthest consumer locates at 0, his utility should be positive after purchasing the final good: $U(x=0) = k - p_{v}^{M} - tx_{v}^{2 } \ge 0$.

Substituting (19) into $U(x =1)$ and $U(x=0)$, respectively, we find that the input price has an upper bound, which is $w^{M} \le k -t(1 - x_{d})^{2}$ if 0 $\le x_{v} \le 1/2$. Similarly, we also can have an upper bound of the input price, which is $w^{M} \le k + t [2(x_{d}-x_{v})-x_{d}^{2}]$ if $1/2 < x_{v} \le 1$.

Considering the above conditions, we obtain the equilibrium input price under the monopoly market case: $w^{M}=k -t(1 - x_{d})^{2}$ when $0 \le x_{v}\le 1/2$ and $w^{M}=k + t[2(x_{d}-x_{v})-x_{d}^{2}]$ when $1/2 < x_{v} \le 1$ respectively.

Appendix 3: Interior location equilibrium with a duopoly market condition for firm d

Substituting $x_v^{{D}^{*}} =0$ and $x_v^{{D}^{*}} =[t(5\alpha -3)+\sqrt{A}]/3\alpha t$ (i.e., (12) and (13.1)) into the duopoly market condition $(\varPhi _{3})$, we obtain that the two downstream firms can survive under the interior location equilibrium if $(c_{u}-c_{v}) \le \varPhi _{3}^{I } \equiv (t/^{3})(5 \alpha - 3)^{2}$, where superscript “I” denotes variables involving the interior location equilibrium case. Comparing $\varPhi _{3}^{I }$ and $\theta _{d}$, we find that $\varPhi _{3}^{I} - \theta _{d} = (t/ \alpha ^{3})[(4 \alpha - 3)(8 \alpha - 3)] \ge 0$ if $\alpha \ge (3/4)$. Moreover, $\theta _{d}$ is positive if $\alpha < (6/7)$. Thus, the interior location equilibrium holds and the two downstream firms survive when $0 \le (c_{u}-c_{v}) < \theta _{d}$ and $(3/4) \le \alpha < (6/7)$.

Corner location equilibrium with a duopoly market condition for firm d

From (13.2), the corner location equilibrium occurs when $(c_{u}-c_{v}) \ge \theta _{d} \equiv (t/\alpha ^{2})(6 - 7\alpha )$ and $\alpha < (6/7)$. Substituting $x_{v}^{{D}^{*}}= 0$ and $x_{d}^{{D}^{*}}= 1$ into the duopoly market condition $(\varPhi _{3})$, we have $(c_{u}-c_{v}) \le \varPhi _{3}^{\mathrm{C}}\equiv (t/\alpha ^{2})(9\alpha - 6)$ and $\alpha > (6/9)$ due to $\varPhi _{3}^{\mathrm{C} }> 0$, where superscript “C” denotes variables involving the corner location equilibrium case. Comparing $\varPhi _{3}^{\mathrm{C}}$ and $\theta _{d}$, we find that $\varPhi _{3}^{\mathrm{C}} - \theta _{d}= (3t/\alpha ^{2})(5\alpha - 4) \ge 0$ if $\alpha \ge (4/5)$. Therefore, the principle of maximum differentiation holds and two firms survive in the market when $\theta _{d }\le (c_{u}-c_{v}) \le \varPhi _{3}^{\mathrm{C} }$ and (4/5) $\le \alpha < (6/7)$.

Appendix 4: Equilibrium location under a monopoly market

From (15), we have that the equilibrium location of the integrated firm v is 1/2 for any $\alpha$ when $0 \le x_{v} \le 1/2$ and $1 - \alpha$ for $0 <\alpha < 1/2$ when $1/2 < x_{v} \le 1$. It shows that the equilibrium location of firm v is 1/2 when $1/2 \le \alpha < 1$. However, when $0 <\alpha < 1/2$, we have to compare $\pi _{v}^{{M}^{*}}(x_{v}^{{M}^{*}}= 1/2)$ with $\pi _{v}^{{M}^{*}} (x_{v}^{{M}^{*}}= 1 - \alpha )$ so as to obtain the equilibrium location of firm v. Substituting (16.2) into (15.2), the profits of the monopolist firm v when $x_{v}^{{M}^{*}}= 1/2$ as:

$$\begin{aligned} \pi _{v}^{M^{*}}\left( x_{v}^{M^{*}}=1/2\right) =\alpha (k-c_v )-t[\alpha (1-x_d )^{2}+(x_d -(1/2))((3/2)-x_d )],\nonumber \\ \end{aligned}$$

(20)

Substituting (16.1) into (15.1), the profits of the monopolist firm v when $x_{v}^{{M}^{*}}=1 - \alpha$ as:

$$\begin{aligned} \pi _{v}^{M ^{*}}=(x_{v}^{M^{*}}=1-\alpha )=\alpha (k-c_v )+t(1-\alpha )^{2}-tx_d (1-\alpha )(2-x_d ), \end{aligned}$$

(21)

Comparing $\pi _{v}^{M }(x_{v}^{{M}^{*}}=1/2)$ with $\pi _{v}^{M }(x_{v}^{{M}^{*}} =1 - \alpha$) when $0 <\alpha < 1/2$, we have:

$$\begin{aligned} \pi _{v}^{M^{*}}(x_{v}^{M^{*}}=1/2)-\pi _{v}^{M^{*}}(x_{v}^{M^{*}}=1-\alpha )=t(\alpha -\alpha ^{2}-1/4)<0, \end{aligned}$$

(22)

According to the above analysis, the equilibrium location of firm v is $x_{v}^{{M}^{*}}= 1 - \alpha$ when $0 <\alpha < 1/2$.