Transportation costs of vehicle recycling under Hotelling’s Duopoly competition

Abstract

We explore the spatial features of end-of-life product recyclers. Our motivation for this analysis comes from real-world data, which look very different because the change in the marginal transportation cost to distance seems negative. To tackle this question, a theoretical consideration is provided. The result derived from our analysis remains largely the same as that reported in conventional location theory. We utilize this puzzling data set to prove our analytical result, which has positive marginal transportation costs after proper empirical treatment. Combining the theoretical and empirical analyses that support agglomeration due to market competition, we conclude that it is important to prepare a place for recycling firms to foster the industry.

Appendices

Proof of Proposition 1

Eliminating $p_{a}$ and $p_{b}$ from (3) by using $p_{a}^{m}$ and $p_{b}^{m}$, we obtain

$$\begin{aligned} h({\hat{x}},a,b) \equiv \gamma \left( 2{\hat{x}}+z_{b}-z_{a}-1\right) \left( |{\hat{x}}-a|^{\gamma -1}+|b-{\hat{x}}|^{\gamma -1}\right) +|{\hat{x}}-a|^{\gamma }-|b-{\hat{x}}|^{\gamma }=0. \end{aligned}$$

(12)

In our definition, $h({\hat{x}}, a,b)=0$ holds for all ${\hat{x}}$ that correspond to ($p_{a}^{m}, p_{b}^{m}$), the Nash equilibrium prices in the second stage. Now, the profit functions in the first stage for firm i, ${\hat{\pi }}_{i}^{m}$, become

$$\begin{aligned} {\hat{\pi }}_{a}^{m}&=p_{a}^{m} \left( {\hat{x}}(p_{a}^{m},p_{b}^{m})-z_{a}\right) \end{aligned}$$

(13)

$$\begin{aligned} {\hat{\pi }}_{b}^{m}&=p_{b}^{m}\left( 1-{\hat{x}}(p_{a}^{m},p_{b}^{m})-z_{b}\right) \end{aligned}$$

(14)

and each firm attempts to maximize its profit by changing its location on the line. Differentiating the above equations, we obtain

$$\begin{aligned} \frac{\partial {\hat{\pi }}_{a}^{m}}{\partial a}&=\frac{\partial p_{a}^{m}}{\partial a}\left( {\hat{x}}(p_{a}^{m},p_{b}^{m})-z_{a}\right) +\frac{\partial {\hat{x}}(p_{a}^{m},p_{b}^{m})}{\partial a}p_{a}^{m} \end{aligned}$$

(15)

$$\begin{aligned} \frac{\partial {\hat{\pi }}_{b}^{m}}{\partial b}&=\frac{\partial p_{b}^{m}}{\partial b}\left( 1-{\hat{x}}(p_{a}^{m},p_{b}^{m})-z_{b}\right) -\frac{\partial {\hat{x}}(p_{b}^{m},p_{b}^{m})}{\partial b}p_{b}^{m}. \end{aligned}$$

(16)

If (15) is negative and (16) is positive, we have the usual maximal differentiation theorem with our remote market model for an end-of-life product. However, if we observe that (15) is positive and (16) is negative, a result â la Hotelling emerges.

To examine the above relationship, we first differentiate $p_{a}^{m}$ and $p_{b}^{m}$.

$$\begin{aligned} \frac{\partial p_{a}^{m}}{\partial a}&=-\gamma (\gamma -1)({\hat{x}}-z_{a})|{\hat{x}}-a|^{\gamma -2} \end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial p_{b}^{m}}{\partial b}&=\gamma (\gamma -1)(1-{\hat{x}}-z_{b})|b-{\hat{x}}|^{\gamma -2} \end{aligned}$$

(18)

Because we exclude the trivial case in which the nonresidential area exceeds each firm’s demand, the signs of (17) and (18) depend on the curvature of the transportation cost function.

Proof of Proposition 2

The first step to determine the signs of (15) and (16) is to differentiate ${\hat{x}}(p_{a}^{m}, p_{b}^{m})$ with respect to a and b, which can be derived by applying the implicit function theorem to (12),

$$\begin{aligned} \frac{\partial {\hat{x}}(p_{a}^{m},p_{b}^{m})}{\partial a}&= -\frac{-\gamma (\gamma -1)\left( 2 {\hat{x}}+z_{b}-z_{a}-1\right) |{\hat{x}}-a|^{\gamma -2}-\gamma |{\hat{x}}-a|^{\gamma -1}}{3\gamma \left( |{\hat{x}}-a|^{\gamma -1}+|b-{\hat{x}}|^{\gamma -1}\right) +\gamma (\gamma -1)\left( |{\hat{x}}-a|^{\gamma -2}-|b-{\hat{x}}|^{\gamma -2}\right) \left( 2{\hat{x}}+z_{b}-z_{a}-1\right) } \nonumber \\&= \frac{(\gamma -1)(D_{a}-D_{b})X_{a}^{\gamma -2}+X_{a}^{\gamma -1}}{(\gamma -1)(D_{a}-D_{b})(X_{a}^{\gamma -2}-X_{b}^{\gamma -2})+3(X_{a}^{\gamma -1}+X_{b}^{\gamma -1})}, \end{aligned}$$

(19)

and

$$\begin{aligned} \frac{\partial {\hat{x}}(p_{a}^{m},p_{b}^{m})}{\partial b}&= -\frac{\gamma (\gamma -1)(2 {\hat{x}}+z_{b}-z_{a}-1)|b-{\hat{x}}|^{\gamma -2}-\gamma |b-{\hat{x}}|^{\gamma -1}}{3\gamma (|{\hat{x}}-a|^{\gamma -1}+|b-{\hat{x}}|^{\gamma -1})+\gamma (\gamma -1)(|{\hat{x}}-a|^{\gamma -2}-|b-{\hat{x}}|^{\gamma -2})(2{\hat{x}}+z_{b}-z_{a}-1)} \nonumber \\&= \frac{-(\gamma -1)(D_{a}-D_{b})X_{b}^{\gamma -2}+X_{b}^{\gamma -1}}{(\gamma -1)(D_{a}-D_{b})(X_{a}^{\gamma -2}-X_{b}^{\gamma -2})+3(X_{a}^{\gamma -1}+X_{b}^{\gamma -1})}, \end{aligned}$$

(20)

where $D_{a} \equiv {\hat{x}}-z_{a}$ and $D_{b} \equiv 1-{\hat{x}}-z_{b}$, which denote the demand of each firm, and $X_{a} \equiv |{\hat{x}}-a|$ and $X_{b} \equiv |b-{\hat{x}}|$, respectively. These four are positive by definition.

Suppose that $\gamma >1$. Then, (19) is negative if $D_{a}<D_{b}$. Because $X_{b}$ is a part of $D_{b}$, we can state that $X_{a}<X_{b}$.⁹ Therefore, the denominator of (19) is positive. $\frac{\partial {\hat{x}}(p_{a}^{m},p_{b}^{m})}{\partial a}$ becomes negative only when either the real demand of firm B, $D_{b}$, or the curvature of the transportation cost is large enough to make the numerator of (19) negative. If $D_{a}<D_{b}$, we automatically obtain that $\frac{\partial {\hat{x}}(p_{a}^{m},p_{b}^{m})}{\partial b}>0$. In summary, we can state Proposition 2.