Spatial competition and flexible manufacturing with spatially discriminatory pricing

Abstract

This paper develops a two-dimensional spatial framework, in which firms have the technique of flexible manufacturing and engage in spatially discriminatory pricing, in order to explore the firms’ optimal locations and optimal attributes of basic products under linear transportation costs. The paper shows that the two firms will agglomerate at the center of the location line and the optimal attributes of the two basic products will be located at the first and third quartiles of the attribute line, respectively, when the ratio of the marginal modification rate to the transport rate is high. It also shows that the two firms will locate separately on the location line and that the optimal attributes of the two basic products will remain at the first and third quartiles, when this ratio is moderate. Moreover, this paper proves that the two firms will locate at the first and third quartiles of the location line, respectively, and that the attributes of the basic products will agglomerate at the center of the attribute line, when this ratio is low.

Appendix

By assuming that the inequality of $t(x_2 -x_1 )>r(y_{b2} -y_{b1} )$ holds, we can solve for the threshold $x_{a}$ from (3) that $z(x_{a})=[t(x_2 +x_1 -2x_a )+r(y_{b1} +y_{b2} )]/2r=y_{{b}2}$ and the threshold $x_{b}$ that $z(x_{b})=[t(x_2 +x_1 -2x_b )+r(y_{b1} +y_{b2} )]/2r=y_{{b}1}$ for $x_1 \le x\le x_2$ as follows:

$$\begin{aligned} x_a&= [t(x_1 +x_2 )-r(y_{b2} -y_{b1} )]/(2t),\end{aligned}$$

(14)

$$\begin{aligned} x_b&= [t(x_1 +x_2 )+r(y_{b2} -y_{b1} )]/(2t). \end{aligned}$$

(15)

Since both the modification cost and the transport cost functions are linear in distance, it is obvious that the consumers residing above $y_{{b}2}$ at the site $x_{a}$ along the attribute line are also marginal consumers. Likewise, the consumers residing beneath $y_{{b}1}$ at the site $x_{b}$ along the attribute line are also marginal consumers. We can derive from (2) and (3) that firm 1 becomes a local monopolist in the region $x\in [0,x_a ]$, while firm 2 is a local monopolist in the region $x\in [x_b,1]$ due to having a cost advantage.

For the region $x\in [x_a,x_b ]$, both firms can serve part of the consumers at each site $x$ and the marginal consumer can be expressed as:

$$\begin{aligned} z(x)=t(x_2 +x_1 -2x)+r(y_{b1} +y_{b2})]/2r,\quad \hbox {if } x_a \le x\le x_b. \end{aligned}$$

(16)

The equilibrium prices solved in stage 3 are identical to those in (2). In stage 2, since firm 1 (2) becomes a local monopolist in the rectangle $[0, x_{a}]\times [0, 1] ([x_{b}, 1]\times [0,1])$, firm $i$’s aggregate profit function $\Pi _{i}$ can be expressed as follows:

$$\begin{aligned} \Pi _1&= \int \limits _0^{x_b } {\pi _1 (x)} \hbox {d}x-k=\int \limits _0^{x_1 } {\pi _1 (x)} \hbox {d}x+\int \limits _{x_1 }^{x_a } {\pi _1 (x)} \hbox {d}x+\int \limits _{x_a }^{x_b } {\pi _1 (x)} \hbox {d}x-k,\end{aligned}$$

(17)

$$\begin{aligned} \Pi _2&= \int \limits _{x_a }^1 {\pi _2 (x)} \hbox {d}x-k=\int \limits _{x_a }^{x_b } {\pi _2 (x)} \hbox {d}x+\int \limits _{x_b }^{x_2 } {\pi _2 (x)} \hbox {d}x+\int \limits _{x_2 }^1 {\pi _2 (x)} \hbox {d}x-k. \end{aligned}$$

(18)

Substituting (14) – (16) into (17) and (18), and then differentiating them with respect to $y_{bi}$, respectively, we derive the following profit-maximizing conditions:

$$\begin{aligned} \partial \Pi _1 /\partial y_{b1}&= r[-(1/2)(r/t)(y_{b2} -y_{b1} )(1+y_{b1} -y_{b2} )\nonumber \\&+(x_1 +x_2 )((1/2)-y_{b1} )]=0,\end{aligned}$$

(19)

$$\begin{aligned} \partial \Pi _2 /\partial y_{b2}&= r[(1/2)(r/t)(y_{b2} -y_{b1} )(1+y_{b1} -y_{b2} )\nonumber \\&-(2-x_1 -x_2 )(y_{b2} -(1/2))]=0. \end{aligned}$$

(20)

Like (7a, 7b), the first term in the brackets on the right-hand side of (19) and (20) denotes the competition effect, which is negative in (19) but is positive in (20). The second term represents the hinterland effect, which is positive in (19) but is negative in (20). Following Sydsaeter and Hammond (1995), the optimal attributes of the two firms’ basic products are subject to the second-order and stability conditions as follows:

$$\begin{aligned}&\partial ^{2}\Pi _1 /\partial y_{b1}^2 =r[r(1+2y_{b1} -2y_{b2} )-2t(x_1 +x_2 )]/2t\le 0,\end{aligned}$$

(21)

$$\begin{aligned}&\partial ^{2}\Pi _2 /\partial y_{b2}^2 =r[r(1+2y_{b1} -2y_{b2} )-2t(2-x_1 -x_2 )]/2t\le 0,\end{aligned}$$

(22)

$$\begin{aligned}&\left( {\partial ^{2}\Pi _1 /\partial y_{b1}^2 } \right) \left( {\partial ^{2}\Pi _2 /\partial y_{b2}^2 } \right) -\left( {\partial ^{2}\Pi _1 /\partial y_{b1} \partial y_{b2} } \right) \left( {\partial ^{2}\Pi _2 /\partial y_{b1} \partial y_{b2} } \right) \nonumber \\&\quad =r^{2}[t(x_1 +x_2 )(2-x_1 -x_2 )-r(1+2y_{b1} -2y_{b2} )]/t\ge 0. \end{aligned}$$

(23)

Solving (19) and (20), we derive the optimal attributes of the two firms’ basic products as follows:

$$\begin{aligned} y_{b1}&= y_{b2} =1/2,\end{aligned}$$

(24)

$$\begin{aligned} y_{b1}&= \left\{ {(x_1 +x_2 )[t(x_1 +x_2 )^{2}-4t(x_1 +x_2 )+4t+r]-r} \right\} /(2r),\hbox { and} \nonumber \\ y_{b2}&= \left\{ {(x_1 +x_2 )[t(x_1 +x_2 )^{2}-2t(x_1 +x_2 )+r]+r} \right\} /(2r). \end{aligned}$$

(25)

There are two possible equilibrium attributes of the basic products, namely central agglomeration and spatial dispersion.

In stage 1, we analyze two possible equilibrium locations based on the two possible equilibrium attributes of the basic products in (24) and (25). Firstly, for the case of central agglomeration on the attribute line, by substituting (24) into (14) and (15) we obtain that $x_{a}=x_{b} = (x_{1}+x_{2})/2$. Since the thresholds $x_{a}$ and $x_{b}$ are located at the same site, it follows that the rival region $[x_{a}, x_{b}]$ vanishes. Firm 1 captures the whole market in $[0, x_{a}]$, while firm 2 grasps the entire market in $[x_{b}, 1]$. It is worth pointing out that the third term on the right-hand side of (17) and the first term on the right-hand side of (18) vanish in this case.

Differentiating the reduced aggregate profit functions for firm 1 and 2 with respect to $x_{i}$, respectively, we derive the following profit-maximizing conditions:

$$\begin{aligned}&\partial \Pi _1 /\partial x_1 =2t[(x_1 +x_2 )/4-x_1 ]=0,\end{aligned}$$

(26)

$$\begin{aligned}&\partial \Pi _2 /\partial x_2 =2t[(x_1 +x_2 +2)/4-x_2 ]=0. \end{aligned}$$

(27)

Solving (26) and (27), we obtain the equilibrium location as follows:¹⁸

$$\begin{aligned} x_1 =1/4, x_2 =3/4. \end{aligned}$$

(28)

Equation (28) shows that firms 1 and 2 locate at the first and the third quartiles, respectively, which are the centers of their own respective market areas. Each firm captures half of the market share, and becomes a local monopolist.

By substituting (24) and (28) into (21) – (23), we figure out that the second-order condition is fulfilled when $(r/t) < 2$, the stability condition is satisfied when $(r/t) < 1$ and Eq. (4) is violated when $(r/t)< (1/2)$. By taking into account these three constraints, we conclude that the two firms are located at the first and third quartiles of the location line, when the ratio of the marginal modification rate to the transport rate is low, say, $(r/t) < (1/2)$.

Next, in the case of the dispersion equilibrium for the attributes, by substituting (25), (2) and (14)–(16) into (17) and (18), we can derive the equilibrium locations as follows:

$$\begin{aligned} x_1 =-1/4+(r^{2}+t^{2})/4rt, x_2 =5/4-(r^{2}+t^{2})/4rt. \end{aligned}$$

(29)

Substituting (29) into (25), we obtain the optimal attribute address of the basic product as follows:

$$\begin{aligned} y_1 =t/2r, y_2 =1-(t/2r). \end{aligned}$$

(30)

However, the stability condition in stage 2 contradicts the condition of the violation of Eq. (4). Thus, the results derived in (29) and (30) can never be equilibria.