Consumer flexibility, data quality and location choice

Abstract

We analyze firms’ location choices in a Hotelling model with two-dimensional consumer heterogeneity, along addresses and transport cost parameters (flexibility). Firms can price discriminate based on perfect data on consumer addresses and (possibly) imperfect data on consumer flexibility. We show that firms’ location choices depend on how strongly consumers differ in flexibility. Precisely, when consumers are relatively homogeneous, equilibrium locations are socially optimal regardless of the quality of customer flexibility data. However, when consumers are relatively differentiated, firms make socially optimal location choices only when customer flexibility data becomes perfect. These results are driven by the optimal strategy of a firm on its turf, monopolization or market-sharing, which in turn depends on consumer heterogeneity in flexibility. Our analysis is motivated by the availability of customer data, which allows firms to practice third-degree price discrimination based on both consumer characteristics relevant in spatial competition, addresses and transport cost parameters.

Appendix A

In Appendix A we first provide the derivations of the formulas stated in the Section “Equilibrium Analysis” and then we present the proofs of the Lemmas and Propositions omitted in the text (excluding those from the Section “Extensions,” which are presented in the online Appendix B).

Derivations of the formulas. When consumers are relatively homogeneous, in equilibrium every firm charges on any address on its turf the highest price, which allows to monopolize a given flexibility segment. This price is proportional to the difference in the distances between the consumer and each of the firms. Following Lederer and Hurter (1986, in the following: LH) and using the results of Lemma 2 we can state the equilibrium prices of firm $i=\left\{ A,B\right\}$ as

$$\begin{aligned} p_{im}^{h}(d_{i},d_{j};x,k)= & {} \underline{t}^{m}\left( k\right) \left( D_{j}\left( d_{j};x\right) -D_{i}\left( d_{i};x\right) \right) \text { if } D_{j}\left( d_{j};x\right) >D_{i}\left( d_{i};x\right) \text { and} \\ p_{im}^{h}(d_{i},d_{j};x,k)= & {} 0\text { if }D_{j}\left( d_{j};x\right) \le D_{i}\left( d_{i};x\right) , \end{aligned}$$

where $D_{i}\left( d_{i};x\right) :=\left| x-d_{i}\right|$. Then the equilibrium profit of firm i for given locations $d_{i}$ and $d_{j}$ can be expressed as

$$\begin{aligned} \Pi _{i}^{h}(d_{i},d_{j};k)= & {} \frac{1}{2^{k}}\sum \limits _{1}^{2^{k}} \underline{t}^{m}\left( k\right) \int \limits _{D_{j}\left( d_{j};x\right)>D_{i}\left( d_{i};x\right) }\left[ D_{j}\left( d_{j};x\right) -D_{i}\left( d_{i};x\right) \right] dx \\= & {} \frac{\underline{t}\left[ \left( 2^{k}+1\right) +l\left( 2^{k}-1\right) \right] }{2^{k+1}}\int \limits _{D_{j}\left( d_{j};x\right) >D_{i}\left( d_{i};x\right) }\left[ D_{j}\left( d_{j};x\right) -D_{i}\left( d_{i};x\right) \right] dx, \end{aligned}$$

where $i\ne j$ and $j=\left\{ A,B\right\}$. Similar to Lemma 4 in LH we can rewrite $\Pi _{i}^{h}(d_{A},d_{B};k)$ as

$$\begin{aligned}&\frac{\Pi _{i}^{h}(d_{i},d_{j};k)\times 2^{k+1}}{\underline{t}\left[ \left( 2^{k}+1\right) +l\left( 2^{k}-1\right) \right] }\\&\quad =\int \limits _{D_{j}\left( d_{j};x\right)>D_{i}\left( d_{i};x\right) }D_{j}\left( d_{j};x\right) dx-\int \limits _{D_{j}\left( d_{j};x\right)>D_{i}\left( d_{i};x\right) }D_{i}\left( d_{i};x\right) dx \\&\quad =\int \limits _{D_{j}\left( d_{j};x\right)>D_{i}\left( d_{i};x\right) }D_{j}\left( d_{j};x\right) dx+\int \limits _{D_{j}\left( d_{j};x\right) \le D_{i}\left( d_{i};x\right) }D_{j}\left( d_{j};x\right) dx \\&\qquad -\,\int \limits _{D_{j}\left( d_{j};x\right) \le D_{i}\left( d_{i};x\right) }D_{j}\left( d_{j};x\right) dx-\int \limits _{D_{j}\left( d_{j};x\right) >D_{i}\left( d_{i};x\right) }D_{i}\left( d_{i};x\right) dx \\&\quad =\int \limits _{0}^{1}D_{j}\left( d_{j};x\right) dx-\int \limits _{0}^{1}\min \left\{ D_{j}\left( d_{j};x\right) ,D_{i}\left( d_{i};x\right) \right\} dx. \end{aligned}$$

Second-best locations solve the optimization problem

$$\begin{aligned} \min _{d_{A},d_{B}}\left\{ \frac{\left( \overline{t}+\underline{t}\right) }{2 }\int \limits _{D_{A}\left( d_{A};x\right) <D_{B}\left( d_{B};x\right) }D_{A}\left( d_{A};x\right) dx+\frac{\left( \overline{t}+\underline{t} \right) }{2}\int \limits _{D_{B}\left( d_{B};x\right) \le D_{A}\left( d_{A};x\right) }D_{B}\left( d_{B};x\right) dx\right\} , \end{aligned}$$

which is equivalent to the problem

$$\begin{aligned} \min _{d_{A},d_{B}}\left\{ \frac{\left( \overline{t}+\underline{t}\right) }{2 }\int \limits _{0}^{1}\min \left\{ D_{A}\left( d_{A};x\right) ,D_{B}\left( d_{B};x\right) \right\} dx\right\} . \end{aligned}$$

Consider now the case of relatively differentiated consumers. As shown in Lemma 1, given any locations every firm serves on its own turf the more loyal consumers and the less loyal consumers on the rival’s turf. In a similar way as above, following LH and using the results of Lemma 1 we can state the equilibrium prices of firm $i=\left\{ A,B\right\}$ as

$$\begin{aligned} p_{im}^{d}(d_{i},d_{j};x,k)= & {} \underline{t}^{m}\left( k\right) \left[ D_{j}\left( d_{j};x\right) -D_{i}\left( d_{i};x\right) \right] \text { if } D_{j}\left( d_{j};x\right)>D_{i}\left( d_{i};x\right) \text { and }\\&m\ge 2 , \\ p_{im}^{d}(d_{i},d_{j};x,k)= & {} 2\overline{t}^{m}\left( k\right) \left[ D_{j}\left( d_{j};x\right) -D_{i}\left( d_{i};x\right) \right] /3\text { if } D_{j}\left( d_{j};x\right) >D_{i}\left( d_{i};x\right) \\&\text {and }m=1, \\ p_{im}^{d}(d_{i},d_{j};x,k)= & {} 0\text { if }D_{j}\left( d_{j};x\right) \le D_{i}\left( d_{i};x\right) \text { and }m\ge 2, \\ p_{im}^{d}(d_{i},d_{j};x,k)= & {} \overline{t}^{m}\left( k\right) \left[ D_{i}\left( d_{j};x\right) -D_{j}\left( d_{i};x\right) \right] /3\text { if } D_{j}\left( d_{j};x\right) \le D_{i}\left( d_{i};x\right) \\&\text {and }m=1 \text {. } \end{aligned}$$

Then the profit of firm i for given locations $d_{i}$ and $d_{j}$ can be expressed as

$$\begin{aligned} \Pi _{i}^{d}(d_{i},d_{j};k)= & {} \left[ \frac{1}{2^{k}}\sum \limits _{2}^{2^{k}}\underline{t}^{m}\left( k\right) +\frac{4\overline{t}}{ 9\times 2^{2k}}\right] \\&\times \int \limits _{D_{j}\left( d_{j};x\right)>D_{i}\left( d_{i};x\right) }\left[ D_{j}\left( d_{j};x\right) -D_{i}\left( d_{i};x\right) \right] dx \\&+\frac{\overline{t}}{9\times 2^{2k}}\int \limits _{D_{j}\left( d_{j};x\right) \le D_{i}\left( d_{i};x\right) }\left[ D_{i}\left( d_{i};x\right) -D_{j}\left( d_{j};x\right) \right] dx \\= & {} \left[ \frac{\overline{t}\left( 2^{k}-1\right) }{2^{k+1}}+\frac{4 \overline{t}}{9\times 2^{2k}}\right] \int \limits _{D_{j}\left( d_{j};x\right) >D_{i}\left( d_{i};x\right) }\left[ D_{j}\left( d_{j};x\right) -D_{i}\left( d_{i};x\right) \right] dx \\&+\frac{\overline{t}}{9\times 2^{2k}}\int \limits _{D_{j}\left( d_{j};x\right) \le D_{i}\left( d_{i};x\right) }\left[ D_{i}\left( d_{i};x\right) -D_{j}\left( d_{j};x\right) \right] dx. \end{aligned}$$

Similar to Lemma 4 in LH we can rewrite $\Pi _{i}^{d}(d_{A},d_{B};k)$ as

$$\begin{aligned} \frac{\Pi _{i}^{d}(d_{i},d_{j};k)}{\left[ \frac{\overline{t}\left( 2^{k}-1\right) }{2^{k+1}}+\frac{4\overline{t}}{9\times 2^{2k}}\right] }= & {} \int \limits _{0}^{1}D_{j}\left( d_{j};x\right) dx+\frac{2}{9\times 2^{k}\left( 2^{k}-1\right) +8}\int \limits _{0}^{1}D_{i}\left( d_{i};x\right) dx \\&-\left[ 1+\frac{2}{9\times 2^{k}\left( 2^{k}-1\right) +8}\right] \\&\times \int \limits _{0}^{1}\min \left\{ D_{j}\left( d_{j};x\right) ,D_{i}\left( d_{i};x\right) \right\} dx. \end{aligned}$$

Hence, when firm i chooses location $d_{i}$, its optimization problem is equivalent to

$$\begin{aligned} \min _{d_{i}}\left[ \left( 1+\alpha \left( k\right) \right) \frac{\overline{t} }{2}\int \limits _{0}^{1}\min \left\{ D_{j}\left( d_{j};x\right) ,D_{i}\left( d_{i};x\right) \right\} dx-\alpha \left( k\right) \frac{ \overline{t}}{2}\int \limits _{0}^{1}D_{i}\left( d_{i};x\right) dx\right] , \end{aligned}$$

where $\alpha \left( k\right) =2/\left[ 9\times 2^{k}\left( 2^{k}-1\right) +8 \right]$.

Second-best locations minimize the transport costs

$$\begin{aligned}&\left[ \int \limits _{D_{A}\left( d_{A};x\right)<D_{B}\left( d_{B};x\right) }D_{A}\left( d_{A};x\right) dx+\int \limits _{D_{B}\left( d_{B};x\right) <D_{A}\left( d_{A};x\right) }D_{B}\left( d_{B};x\right) dx \right] \int \limits _{\frac{\overline{t}}{3\times 2^{k}}}^{\overline{t} }f_{t}tdt \\&+\left[ \int \limits _{D_{A}\left( d_{A};x\right)>D_{B}\left( d_{B};x\right) }D_{A}\left( d_{A};x\right) dx+\int \limits _{D_{B}\left( d_{B};x\right) >D_{A}\left( d_{A};x\right) }D_{B}\left( d_{B};x\right) dx \right] \int \limits _{0}^{\frac{\overline{t}}{3\times 2^{k}}}f_{t}tdt \\&\quad =\frac{\overline{t}\left[ 1-\beta (k)\right] }{2}\int \limits _{0}^{1}\min \left\{ D_{A}\left( \cdot \right) ,D_{B}\left( \cdot \right) \right\} dx+ \frac{\overline{t}\beta (k)}{2}\int \limits _{0}^{1}\max \left\{ D_{A}\left( \cdot \right) ,D_{B}\left( \cdot \right) \right\} dx, \end{aligned}$$

where $\beta (k)=1/\left[ 9\times 2^{2k}\right]$.

We now derive the formulas necessary for the comparison of Anderson and de Palma (1988, in the following: AP) and the version of our model with relatively differentiated consumers. In that case the equilibrium prices on segment $m=1$ (with relatively differentiated consumers) can be derived from Proposition 1 in AP. Consider, for example, the interval $x\le d_{A}$.²⁸ We need to set $c_{A}=c_{B}=0$ and replace $F_{1}$ with the demand of firm A on segment $m=1$:

$$\begin{aligned} d_{A1}\left( \left. p_{A1}\left( x\right) -p_{B1}\left( x\right) ,d_{B}-d_{A},\overline{t}^{1}\left( k\right) \right| x\le d_{A}\right) :=1-\tfrac{p_{A1}\left( x\right) -p_{B1}\left( x\right) }{\overline{t} ^{1}\left( k\right) \left( d_{B}-d_{A}\right) }. \end{aligned}$$

(8)

In AP (in the logit model) equilibrium locations depend on parameter $\mu \ge 0$, which is interpreted as a measure of consumer/product heterogeneity. In our case it makes sense to define $\mu$ as $\mu _{1}\left( k\right) :=\overline{t}\left( d_{B}-d_{A}\right) /2^{k}$. Then from (8) we get that $\left| \partial d_{i1}\left( \cdot \right) /\partial p_{j1}\left( x\right) \right| =1/\mu _{1}\left( k\right)$ ($i,j=\left\{ A,B\right\}$), such that with an increase in $\mu _{1}\left( k\right)$ firms’ prices become less important in determining their demands. Parameter $\mu _{1}\left( k\right)$ is inversely related to the quality of customer flexibility data. If $k=0$, then for any x, $\mu _{1}\left( k\right)$ gets its highest value of $\mu _{1}\left( 0\right) = \overline{t}\left( d_{B}-d_{A}\right)$.

Proof of Lemma 1

As firms are symmetric, we will derive equilibrium on the two intervals on the turf of firm A.

• (i) Interval 1: $x\le d_{A}$. Consider some $x\le d_{A}$ and some m. The transport cost parameter of the indifferent consumer is

  $$\begin{aligned}&\widetilde{t}\left( \left. d_{A},d_{B},p_{Am}(x),p_{Bm}(x);x\right| x\le d_{A}\right) :=\tfrac{p_{A}(x)-p_{B}(x)}{d_{B}-d_{A}}, \quad \text {provided}\nonumber \\&\quad \widetilde{t}(\cdot )\in \left[ \underline{t}^{m}(k),\overline{t}^{m}(k) \right] , \end{aligned}$$

  (9)

  such that consumers with $t\ge \widetilde{t}(\cdot )$ buy at firm A. Consider first $m=1$. Maximization of firms’ expected profits yields the best-response functions

  $$\begin{aligned}&p_{A1}\left( \left. d_{A},d_{B},p_{B1}(x);x,k\right| x\le d_{A}\right) \nonumber \\&\quad =\left\{ \begin{array}{lll} \tfrac{\overline{t}^{1}(k)\left( d_{B}-d_{A}\right) +p_{B1}(x)}{2} &{} \text {if } &{} p_{B1}(x)<\overline{t}^{1}(k)\left( d_{B}-d_{A}\right) \\ p_{B1}(x) &{} \text {if} &{} p_{B1}(x)\ge \overline{t}^{1}(k)\left( d_{B}-d_{A}\right) \end{array} \right. \end{aligned}$$

  (10)

  and

  $$\begin{aligned} p_{B1}\left( \left. d_{A},d_{B},p_{A1}(x);x,k\right| x\le d_{A}\right) = \end{aligned}$$
  $$\begin{aligned} \left\{ \begin{array}{lll} \tfrac{p_{A1}(x)}{2} &{} \text {if} &{} p_{A1}(x)\le 2\overline{t}^{1}(k)\left( d_{B}-d_{A}\right) \\ p_{A1}(x)-\overline{t}^{1}(k)\left( d_{B}-d_{A}\right) &{} \text {if} &{} p_{A1}(x)>2\overline{t}^{1}(k)\left( d_{B}-d_{A}\right) . \end{array} \right. \end{aligned}$$

  (11)

  Given the best-response functions (10) and (11) we conclude that two types of equilibria are possible, where either firm A monopolizes segment m or where both firms serve consumers. Only the latter equilibrium exists where firms charge prices $p_{A1}^{d}(d_{A},d_{B};x,k)=2\overline{t}\left( d_{B}-d_{A}\right) /\left( 3\times 2^{k}\right)$ and $p_{B1}^{d}(d_{A},d_{B};x,k)=\overline{t}\left( d_{B}-d_{A}\right) /\left( 3\times 2^{k}\right)$. Firm A serves consumers with $t\ge \overline{t}^{1}(k)/3$. Consider now segments $m\ge 2$, where the best-response function of firm A is

  $$\begin{aligned} p_{Am}(\left. d_{A},d_{B},p_{Bm}(x);x,k\right| x\le d_{A})=p_{Bm}(x)+ \underline{t}^{m}(k)\left( d_{B}-d_{A}\right) . \end{aligned}$$

  (12)

  As $\overline{t}^{m}(k)-2\underline{t}^{m}(k)\le 0$ for any $m\ge 2$, there is no $p_{Bm}(x)\ge 0$ under which it is optimal for firm A to share the market with firm B. (12) yields $p_{Bm}^{d}(d_{A},d_{B};x,k)=0$. Indeed, assume that in equilibrium $p_{Bm}^{d}(d_{A},d_{B};x,k)>0$ holds. Firm B gets in equilibrium the profit of zero, because (12) implies that firm B serves no consumers. But firm B can increase its profit through slightly decreasing its price. Hence, $p_{Bm}^{d}(d_{A},d_{B};x,k)=0$ must hold. Firm A charges the price $p_{Am}^{d}(d_{A},d_{B};x,k)=\underline{t} ^{m}(k)\left( d_{B}-d_{A}\right)$ and serves all consumers on segment m on address x. On the interval $x\le d_{A}$ firms realize profits

  $$\begin{aligned} \Pi _{A}\left( \left. d_{A},d_{B};k\right| x\le d_{A}\right)= & {} \int \limits _{0}^{d_{A}}f_{t}\left( \left( \tfrac{2\overline{t}}{3\times 2^{k}} \right) ^{2}\left( d_{B}-d_{A}\right) +\tfrac{\overline{t}\left( d_{B}-d_{A}\right) }{2^{k}}\sum \limits _{2}^{2^{k}}\underline{t} ^{m}(k)\right) dx \\= & {} \tfrac{\overline{t}\left( d_{B}-d_{A}\right) d_{A}\left( 9\times 2^{2k}-9\times 2^{k}+8\right) }{9\times 2^{2k+1}} \end{aligned}$$

  and

  $$\begin{aligned} \Pi _{B}(\left. d_{A},d_{B};k\right| x\le d_{A})=\tfrac{\overline{t} \left( d_{B}-d_{A}\right) d_{A}}{9\times 2^{2k}}. \end{aligned}$$

  Using symmetry we conclude on firms’ profits on the interval $x\ge d_{B}$:

  $$\begin{aligned} \Pi _{A}(\left. d_{A},d_{B};k\right| x\ge & {} d_{B})=\tfrac{\overline{t} \left( d_{B}-d_{A}\right) \left( 1-d_{B}\right) }{9\times 2^{2k}}\text { and} \\ \Pi _{B}(\left. d_{A},d_{B};k\right| x\ge & {} d_{B})=\tfrac{\overline{t} \left( d_{B}-d_{A}\right) \left( 1-d_{B}\right) \left( 9\times 2^{2k}-9\times 2^{k}+8\right) }{9\times 2^{2k+1}} \end{aligned}$$

• (ii) Interval 2: $d_{A}<x\le \left( d_{A}+d_{B}\right) /2$. Consider some $d_{A}<x\le \left( d_{A}+d_{B}\right) /2$ and segment m. The transport cost parameter of the indifferent consumer is

  $$\begin{aligned} \widetilde{t}\left( \left. d_{A},d_{B},p_{Am}(x),p_{Bm}(x);x\right| d_{A}<x\le \tfrac{d_{A}+d_{B}}{2}\right) :=\tfrac{p_{A}(x)-p_{B}(x)}{ d_{A}+d_{B}-2x}, \end{aligned}$$

  provided $\widetilde{t}(\cdot )\in \left[ \underline{t}^{m}(k),\overline{t} ^{m}(k)\right]$. Firm A serves consumers with $t\ge \widetilde{t}(\cdot )$. Consider first $m=1$. Maximization of firms’ profits yields the best-response functions

  $$\begin{aligned} p_{A1}\left( \left. d_{A},d_{B},p_{B1}(x);x,k\right| d_{A}<x\le \tfrac{ d_{A}+d_{B}}{2}\right) = \end{aligned}$$
  $$\begin{aligned} \left\{ \begin{array}{lll} \frac{\overline{t}^{1}(k)\left( d_{A}+d_{B}-2x\right) +p_{B1}(x)}{2} &{} \text { if} &{} p_{B1}(x)<\overline{t}^{1}(k)\left( d_{A}+d_{B}-2x\right) \\ p_{B1}(x) &{} \text {if} &{} p_{B1}(x)\ge \overline{t}^{1}(k)\left( d_{A}+d_{B}-2x\right) \end{array} \right. \end{aligned}$$

  (13)

  and

  $$\begin{aligned} p_{B1}\left( \left. d_{A},d_{B},p_{A1}(x);x,k\right| d_{A}<x\le \tfrac{ d_{A}+d_{B}}{2}\right) = \end{aligned}$$
  $$\begin{aligned} \left\{ \begin{array}{lll} \frac{p_{A1}(x)}{2} &{} \text {if} &{} p_{A1}(x)\le 2\overline{t}^{1}(k)\left( d_{A}+d_{B}-2x\right) \\ p_{A1}(x)-\overline{t}^{1}(k)\left( d_{A}+d_{B}-2x\right) &{} \text {if} &{} p_{A1}(x)>2\overline{t}^{1}(k)\left( d_{A}+d_{B}-2x\right) . \end{array} \right. \end{aligned}$$

  (14)

  Given the best-response functions (13) and (14) we conclude that two types of equilibria are possible where either firm A serves all consumers on segment m or shares it with the rival. Only the latter equilibrium exists, where firms charge prices

  $$\begin{aligned} p_{A1}^{d}(d_{A},d_{B};x,k)= & {} 2\overline{t}\left( d_{A}+d_{B}-2x\right) /\left( 3\times 2^{k}\right) \text { and} \\ p_{B1}^{d}(d_{A},d_{B};x,k)= & {} \overline{t}\left( d_{A}+d_{B}-2x\right) /\left( 3\times 2^{k}\right) . \end{aligned}$$

  On $m=1$ firm A serves consumers with $t\ge \overline{t}/\left( 3\times 2^{k}\right)$. We now consider $m\ge 2$, where the best-response function of firm A takes the form

  $$\begin{aligned}&p_{Am}\left( \left. d_{A},d_{B},p_{Bm}(x);x,k\right| d_{A}<x\le \frac{ d_{A}+d_{B}}{2}\right) \\&\quad =p_{Bm}(x,k)+\underline{t}^{m}(k)\left( d_{A}+d_{B}-2x\right) , \end{aligned}$$

  such that firm A never finds it optimal to share segment m with firm B. Applying the logic described in part (i) of the proof, we conclude that $p_{Bm}^{d}(d_{A},d_{B};x,k)=0$ and $p_{Am}^{d}(d_{A},d_{B};x,k)= \underline{t}^{m}(k)\left( d_{A}+d_{B}-2x\right)$. Firm A serves all consumers on any segment $m\ge 2$ on address x. On the interval $d_{A}<x\le \left( d_{A}+d_{B}\right) /2$ firms realize profits

  $$\begin{aligned}&\Pi _{A}\left( \left. d_{A},d_{B};k\right| d_{A}<x\le \tfrac{ d_{A}+d_{B}}{2}\right) \\&\quad =\int \limits _{d_{A}}^{\frac{d_{A}+d_{B}}{2}}f_{t}\left[ \left( \tfrac{2 \overline{t}}{3\times 2^{k}}\right) ^{2}\left( d_{A}+d_{B}-2x\right) +\tfrac{ \overline{t}\left( d_{A}+d_{B}-2x\right) }{2^{k}}\sum \limits _{2}^{2^{k}} \underline{t}^{m}(k)\right] dx \\&\quad =\tfrac{\overline{t}\left( d_{B}-d_{A}\right) ^{2}\left( 9\times 2^{2k}-9\times 2^{k}+8\right) }{9\times 2^{2k+3}}\text { and } \end{aligned}$$
  $$\begin{aligned} \Pi _{B}\left( \left. d_{A},d_{B};k\right| d_{A}<x\le \tfrac{ d_{A}+d_{B}}{2}\right) =\tfrac{\overline{t}\left( d_{B}-d_{A}\right) ^{2}}{ 9\times 2^{2k+2}}. \end{aligned}$$

  Using symmetry, we can conclude on firms’ profits on the interval $\left( d_{A}+d_{B}\right) /2<x\le d_{B}$:

  $$\begin{aligned} \Pi _{A}\left( \left. d_{A},d_{B};k\right| \tfrac{d_{A}+d_{B}}{2}<x\le d_{B}\right)= & {} \tfrac{\overline{t}\left( d_{B}-d_{A}\right) ^{2}}{9\times 2^{2k+2}}\text { and} \\ \Pi _{B}\left( \left. d_{A},d_{B};k\right| \tfrac{d_{A}+d_{B}}{2}<x\le d_{B}\right)= & {} \tfrac{\overline{t}\left( d_{B}-d_{A}\right) ^{2}\left( 9\times 2^{2k}-9\times 2^{k}+8\right) }{9\times 2^{2k+3}}. \end{aligned}$$

  Summing up firms’ profits on all the four intervals we get

  $$\begin{aligned} \Pi _{A}^{d}(d_{A},d_{B};k)= & {} \tfrac{\overline{t}\left( d_{B}-d_{A}\right) \left( 9\times 2^{k}\left( 2^{k}-1\right) (3d_{A}+d_{B})+2\left( 11d_{A}+d_{B}\right) +8\right) }{9\times 2^{2k+3}}\text { and } \\ \Pi _{B}^{d}(d_{A},d_{B};k)= & {} \tfrac{\overline{t}\left( d_{B}-d_{A}\right) \left( 32-9\times 2^{k}(2^{k}-1)\left( 3d_{B}+d_{A}-4\right) -2\left( 11d_{B}+d_{A}\right) \right) }{9\times 2^{2k+3}}. \end{aligned}$$

$\square$

Proof of Lemma 2

As firms are symmetric, we will only derive equilibrium on the two intervals on the turf of firm A and then conclude on the equilibrium on the turf of firm B.

• (i) Interval 1: $x\le d_{A}$. Consider some $x\le d_{A}$ and some $m\ge 1$. The transport cost parameter of the indifferent consumer is

  $$\begin{aligned}&\widetilde{t}\left( \left. d_{A},d_{B},p_{Am}(x),p_{Bm}(x);x\right| x\le d_{A}\right) :=\tfrac{p_{A}(x)-p_{B}(x)}{d_{B}-d_{A}}, \quad \text {provided}\\&\quad \widetilde{t}(\cdot )\in \left[ \underline{t}^{m}(k),\overline{t}^{m}(k) \right] . \end{aligned}$$

  The best-response function of firm A is

  $$\begin{aligned} p_{Am}(\left. d_{A},d_{B},p_{Bm}(x);x,k\right| x\le d_{A})=p_{Bm}(x)+ \underline{t}^{m}(k)\left( d_{B}-d_{A}\right) , \end{aligned}$$

  (15)

  such that for any price of the rival firm A monopolizes segment m on address x. As $\overline{t}^{m}(k)-2\underline{t}^{m}(k)\le 0$ for any $m\ge 2$, there is no $p_{Bm}(x)\ge 0$ under which it is optimal for firm A to share the market with firm B. (15) yields $p_{Bm}^{h}(d_{A},d_{B};x,k)=0$. Indeed, assume that in equilibrium $p_{Bm}^{h}(d_{A},d_{B};x,k)>0$ holds. Firm B gets in equilibrium the profit of zero, because (15) implies that firm B serves no consumers. But firm B can increase its profit through slightly decreasing its price. Hence, $p_{Bm}^{h}(d_{A},d_{B};x,k)=0$ must hold. Firm A charges the price $p_{Am}^{h}(d_{A},d_{B};x,k)=\underline{t} ^{m}(k)\left( d_{B}-d_{A}\right)$ and serves all consumers on segment m on address x. Hence, $\Pi _{B}(\left. d_{A},d_{B};k\right| x\le d_{A})=0$ and the profit of firm A is computed as

  $$\begin{aligned} \Pi _{A}\left( \left. d_{A},d_{B};k\right| x\le d_{A}\right) =\sum \limits _{1}^{2^{k}}\underline{t}^{m}(k)\int \limits _{0}^{d_{A}}\left( \tfrac{\left( d_{B}-d_{A}\right) }{2^{k}}\right) dx=\tfrac{d_{A}\left( d_{B}-d_{A}\right) \left( \overline{t}\left( 2^{k}-1\right) +\underline{t} \left( 2^{k}+1\right) \right) }{2^{k+1}}. \end{aligned}$$

  (16)

• (ii) Interval 2: $d_{A}<x\le \left( d_{A}+d_{B}\right) /2$. Consider some $d_{A}<x\le \left( d_{A}+d_{B}\right) /2$ and some $m\ge 1$. The transport cost parameter of the indifferent consumer is

  $$\begin{aligned} \widetilde{t}\left( \left. d_{A},d_{B},p_{Am}(x),p_{Bm}(x);x\right| d_{A}<x\le \tfrac{d_{A}+d_{B}}{2}\right) :=\tfrac{p_{A}(x)-p_{B}(x)}{ d_{A}+d_{B}-2x}, \end{aligned}$$

  provided $\widetilde{t}(\cdot )\in \left[ \underline{t}^{m}(k),\overline{t} ^{m}(k)\right]$. Firm A serves consumers with $t\ge \widetilde{t}(\cdot )$. The best-response function of firm A is

  $$\begin{aligned}&p_{Am}\left( \left. d_{A},d_{B},p_{Bm}(x);x,k\right| d_{A}<x\le \tfrac{ d_{A}+d_{B}}{2}\right) \nonumber \\&\quad =p_{Bm}(x)+\underline{t}^{m}(k)\left( d_{A}+d_{B}-2x\right) . \end{aligned}$$

  Following the logic applied in part (i) of the proof we conclude that

  $$\begin{aligned} p_{Bm}^{h}(x,k)= & {} 0\text { and} \\ p_{Am}^{h}(d_{A},d_{B};x,k)= & {} \underline{t}^{m}(k)\left( d_{A}+d_{B}-2x\right) . \end{aligned}$$

  Hence, $\Pi _{B}(\left. d_{A},d_{B};k\right| d_{A}<x\le \left( d_{A}+d_{B}\right) /2)=0$ and the profit of firm A is computed as

  $$\begin{aligned} \Pi _{A}\left( \left. d_{A},d_{B};k\right| d_{A}<x\le \tfrac{ d_{A}+d_{B}}{2}\right)= & {} \sum \limits _{1}^{2^{k}}\underline{t}^{m}(k)\int \limits _{d_{A}}^{\left( d_{A}+d_{B}\right) /2}\left( \tfrac{d_{A}+d_{B}-2x}{ 2^{k}}\right) dx \nonumber \\= & {} \tfrac{\left( d_{B}-d_{A}\right) ^{2}\left( \overline{t}\left( 2^{k}-1\right) +\underline{t}\left( 2^{k}+1\right) \right) }{2^{k+3}}. \end{aligned}$$

  (17)

  Summing up the profits (16) and (17) we get the profits of firm A as stated in the lemma. The profits of firm B are derived using symmetry.

$\square$

Proof of Lemma 3

We first derive first-best locations and prices. We will proceed in two steps. We will first derive first-best prices for any given locations and then will find first-best locations. Assume that firms are located at $d_{A}\le d_{B}$. Social welfare is maximized when every consumer buys from the closer firm. Prices, which yield such a distribution of consumers between the firms are $p_{im}^{FB}\left( x\right) ,p_{jm}^{FB}\left( x\right) \ge 0$ such that

$$\begin{aligned} p_{im}^{FB}\left( x\right) \le t\left| x-x_{j}\right| -t\left| x-x_{i}\right| +p_{jm}^{FB}\left( x\right) \text { if }\left| x-x_{i}\right| \le \left| x-x_{j}\right| . \end{aligned}$$

(18)

Given (18), the first-best locations, $d_{A}^{FB}$ and $d_{B}^{FB}$, have to minimize the transport costs

$$\begin{aligned}&TC^{FB}\left( d_{A},d_{B};k\right) \nonumber \\&\quad =\tfrac{\left( \overline{t}+\underline{t}\right) }{2}\left( \int \limits _{0}^{d_{A}}\left( d_{A}-x\right) dx+\int \limits _{d_{A}}^{\left( d_{A}+d_{B}\right) /2}(x-d_{A})dx+\int \limits _{\left( d_{A}+d_{B}\right) /2}^{d_{B}}\left( d_{B}-x\right) dx\right. \nonumber \\&\qquad \left. +\int \limits _{d_{B}}^{1}\left( x-d_{B}\right) dx\right) , \end{aligned}$$

(19)

which yields the locations $d_{A}^{FB}=1/4$ and $d_{B}^{FB}=3/4$. Note that SOCs are fulfilled.

We now turn to the second-best locations. Here we have to distinguish between the cases of relatively homogeneous and differentiated consumers, because for any locations firms charge different prices in equilibrium depending on the case. We start with the case of relatively homogeneous consumers. Note that the equilibrium prices stated in Lemma 2 satisfy (18). Indeed, in equilibrium every consumer buys from the closer firm. Then second-best locations should also minimize (19 ), which yields $d_{A}^{SB,h}=1/4$ and $d_{B}^{SB,h}=3/4$.

We now consider the case of relatively differentiated consumers. According to Lemma 1, on its own turf each firm serves consumers with $t\ge \overline{ t}/\left( 3\times 2^{k}\right)$, while consumers with $t<\overline{t} /\left( 3\times 2^{k}\right)$ switch to the rival. Then second-best locations have to minimize the transport costs

$$\begin{aligned}&TC^{SB,d}\left( d_{A},d_{B};k\right) \\&\quad =\int \limits _{\overline{t}/\left( 3\times 2^{k}\right) }^{\overline{t} }\left( tf_{t}\int \limits _{0}^{d_{A}}\left( d_{A}-x\right) dx\right) dt+\int \limits _{\overline{t}/\left( 3\times 2^{k}\right) }^{\overline{t} }\left( tf_{t}\int \limits _{d_{A}}^{\left( d_{A}+d_{B}\right) /2}\left( x-d_{A}\right) dx\right) dt \\&\qquad +\int \limits _{0}^{\overline{t}/\left( 3\times 2^{k}\right) }\left( tf_{t}\int \limits _{\left( d_{A}+d_{B}\right) /2}^{d_{B}}\left( x-d_{A}\right) dx\right) dt+\int \limits _{0}^{\overline{t}/\left( 3\times 2^{k}\right) }\left( tf_{t}\int \limits _{d_{B}}^{1}\left( x-d_{A}\right) dx\right) dt \\&\qquad +\int \limits _{\overline{t}/\left( 3\times 2^{k}\right) }^{\overline{t} }\left( tf_{t}\int \limits _{d_{B}}^{1}\left( x-d_{B}\right) dx\right) dt+\int \limits _{\overline{t}/\left( 3\times 2^{k}\right) }^{\overline{t} }\left( tf_{t}\int \limits _{\left( d_{A}+d_{B}\right) /2}^{d_{B}}\left( d_{B}-x\right) dx\right) dt \\&\qquad +\int \limits _{0}^{\overline{t}/\left( 3\times 2^{k}\right) }\left( tf_{t}\int \limits _{d_{A}}^{\left( d_{A}+d_{B}\right) /2}\left( d_{B}-x\right) dx\right) dt+\int \limits _{0}^{\overline{t}/\left( 3\times 2^{k}\right) }\left( tf_{t}\int \limits _{0}^{d_{A}}\left( d_{B}-x\right) dx\right) dt \\&\quad =-\tfrac{\overline{t}}{2^{2k}}\left( \tfrac{(d_{A}-d_{B})^{2}}{36}+\tfrac{ (d_{A}-d_{B})}{18}-3\times 2^{2k-3}\left( \left( d_{A}\right) ^{2}+\left( d_{B}\right) ^{2}\right) \right) \\&\qquad -\tfrac{\overline{t}}{2^{2k}}\left( -2^{2k-2}+2^{2k-1}d_{B}+2^{2k-2}d_{A}d_{B}\right) , \end{aligned}$$

which yields

$$\begin{aligned} d_{A}^{SB}\left( k\right) =\tfrac{9\times 2^{2k}}{36\times 2^{2k}-4} \quad \text {and} \quad d_{B}^{SB}\left( k\right) =\tfrac{27\times 2^{2k}-4}{36\times 2^{2k}-4}. \end{aligned}$$

SOCs are fulfilled. Note finally that $\lim _{k\rightarrow \infty }d_{A}^{SB}\left( k\right) =1/4$ and $\lim _{k\rightarrow \infty }d_{B}^{SB}\left( k\right) =3/4$. $\square$

Proof of Proposition 1

• (i) Maximization of the profits

  $$\begin{aligned} \Pi _{A}^{h}(d_{A},d_{B};k)= & {} \tfrac{\underline{t}\left( \left( 2^{k}+1\right) +l\left( 2^{k}-1\right) \right) \left( d_{B}+3d_{A}\right) \left( d_{B}-d_{A}\right) }{2^{k+3}}\quad \text { and } \\ \Pi _{B}^{h}(d_{A},d_{B};k)= & {} \tfrac{\underline{t}\left( \left( 2^{k}+1\right) +l\left( 2^{k}-1\right) \right) \left( 4-d_{A}-3d_{B}\right) \left( d_{B}-d_{A}\right) }{2^{k+3}} \end{aligned}$$

  with respect to $d_{A}$ and $d_{B}$ yields the FOCs: $d_{B}-3d_{A}=0$ and $d_{A}-3d_{B}+2=0$, respectively. Solving them simultaneously we get $d_{A}^{h}=1/4$ and $d_{B}^{h}=3/4$, such that the profit of firm $i=\left\{ A,B\right\}$ is

  $$\begin{aligned} \Pi _{i}^{h}\left( k\right) =\tfrac{3\underline{t}\left( \left( 2^{k}+1\right) +l\left( 2^{k}-1\right) \right) }{2^{k+5}}. \end{aligned}$$

  (20)

  Note that SOCs are fulfilled. To prove that these locations constitute indeed the equilibrium, we have to prove that firm A does not have an incentive to choose a location $d_{A}\ge d_{B}^{h}=3/4$. Due to symmetry, this would also imply that firm B does not have an incentive to choose $d_{B}\le d_{A}^{h}=1/4$. If firm A locates at $d_{A}\ge d_{B}^{h}$, then according to Lemma 2 it realizes the profit

  $$\begin{aligned} \Pi _{B}\left( d_{B},d_{A};k\right) =\tfrac{\underline{t}\left( \left( 2^{k}+1\right) +l\left( 2^{k}-1\right) \right) \left( 4-d_{B}-3d_{A}\right) \left( d_{A}-d_{B}\right) }{2^{k+3}}. \end{aligned}$$

  (21)

  Maximizing (21) with respect to $d_{A}$ yields the FOC: $d_{A}\left( d_{B}^{h}\right) =\left( d_{B}^{h}+2\right) /3=11/12$. Firm A realizes the profit

  $$\begin{aligned} \Pi _{B}\left( \tfrac{3}{4},\tfrac{11}{12};k\right) =\tfrac{\underline{t} \left( \left( 2^{k}+1\right) +l\left( 2^{k}-1\right) \right) \left( 4-d_{B}-3d_{A}\right) \left( d_{A}-d_{B}\right) }{2^{k+3}}=\tfrac{\underline{ t}\left( \left( 2^{k}+1\right) +l\left( 2^{k}-1\right) \right) }{3\times 2^{k+5}}.\quad \end{aligned}$$

  (22)

  Comparing the profits (20) and (22) we conclude that

  $$\begin{aligned} \Pi _{i}^{h}\left( k\right) -\Pi _{B}\left( \tfrac{3}{4},\tfrac{11}{12} ;k\right) =\tfrac{\underline{t}\left( \left( 2^{k}+1\right) +l\left( 2^{k}-1\right) \right) }{3\times 2^{k+2}}>0\quad \text {for any} \quad k\ge 0, \end{aligned}$$

  hence, firm A does not have an incentive to deviate to $d_{A}\ge d_{B}^{h}$. We conclude that the locations $d_{A}^{h}=1/4$ and $d_{B}^{h}=3/4$ constitute indeed the equilibrium.

• (ii) Maximization of the profits

  $$\begin{aligned} \Pi _{A}^{d}(d_{A},d_{B};k)= & {} \tfrac{\overline{t}\left( d_{B}-d_{A}\right) \left( 9\times 2^{k}\left( 2^{k}-1\right) (3d_{A}+d_{B})+2\left( 11d_{A}+d_{B}\right) +8\right) }{9\times 2^{2k+3}}\quad \text { and } \\ \Pi _{B}^{d}(d_{A},d_{B};k)= & {} \tfrac{\overline{t}\left( d_{B}-d_{A}\right) \left( -9\times 2^{k}(2^{k}-1)\left( 3d_{B}+d_{A}-4\right) -2\left( 11d_{B}+d_{A}\right) +32\right) }{9\times 2^{2k+3}} \end{aligned}$$

  with respect to $d_{A}$ and $d_{B}$ yields the FOCs

  $$\begin{aligned} d_{A}\left( d_{B};k\right)= & {} \tfrac{9\times 2^{k}d_{B}\left( 2^{k}-1\right) +10d_{B}-4}{27\times 2^{2k}-27\times 2^{k}+22} \quad \text { and} \\ d_{B}\left( d_{A};k\right)= & {} \tfrac{9\times 2^{k}\left( 2^{k}-1\right) \left( d_{A}+2\right) +10d_{A}+16}{27\times 2^{2k}-27\times 2^{k}+22}, \end{aligned}$$

  respectively. Solving FOCs simultaneously we get the locations

  $$\begin{aligned} d_{A}^{d}\left( k\right)= & {} \tfrac{9\times 2^{2k}-9\times 2^{k}+6}{36\times 2^{2k}-36\times 2^{k}+32} \quad \text {and} \nonumber \\ d_{B}^{d}\left( k\right)= & {} \tfrac{27\times 2^{2k}-27\times 2^{k}+26}{ 36\times 2^{2k}-36\times 2^{k}+32}, \end{aligned}$$

  (23)

  such that firm $i=\left\{ A,B\right\}$ realizes the profit

  $$\begin{aligned} \Pi _{i}^{d}(k)=\tfrac{\overline{t}\left( 9\times 2^{2k}-9\times 2^{k}+10\right) ^{2}\left( 27\times 2^{2k}-27\times 2^{k}+22\right) }{ 9\times 2^{2k+5}\left( 9\times 2^{2k}-9\times 2^{k}+8\right) ^{2}}. \end{aligned}$$

  Note that the SOCs are also fulfilled. To prove that the locations $d_{A}^{d}\left( k\right)$ and $d_{B}^{d}\left( k\right)$ constitute indeed the equilibrium, we have to show that firm A does not have an incentive to locate at $d_{A}\ge d_{B}^{d}\left( k\right)$. As firms are symmetric, firm B then does not have an incentive to locate at $d_{B}\le d_{A}^{d}\left( k\right)$ either. If firm A chooses $d_{A}\ge d_{B}^{d}\left( k\right)$, then it realizes the profit

  $$\begin{aligned}&\Pi _{B}^{d}\left( d_{B}^{d}\left( k\right) ,d_{A};k\right) \nonumber \\= & {} \tfrac{\overline{t}\left( d_{A}-d_{B}^{d}\left( k\right) \right) \left( -9\times 2^{k}(2^{k}-1)\left( 3d_{A}+d_{B}^{d}\left( k\right) -4\right) -2\left( 11d_{A}+d_{B}^{d}\left( k\right) \right) +32\right) }{9\times 2^{2k+3}}. \end{aligned}$$

  (24)

  Maximization of (24) with respect to $d_{A}$ yields

  $$\begin{aligned} d_{A}\left( d_{B}^{d}\left( k\right) ;k\right)= & {} \tfrac{9\times 2^{k}\left( 2^{k}-1\right) \left( d_{B}^{d}\left( k\right) +2\right) +10d_{B}^{d}\left( k\right) +16}{27\times 2^{2k}-27\times 2^{k}+22} \\= & {} \tfrac{2547\times 2^{2k}-1782\times 2^{3k}+891\times 2^{4k}-1656\times 2^{k}+772}{\left( 36\times 2^{2k}-36\times 2^{k}+32\right) \left( 27\times 2^{2k}-27\times 2^{k}+22\right) }, \end{aligned}$$

  such that firm A realizes the profit

  $$\begin{aligned} \Pi _{B}\left( d_{B}^{d}\left( k\right) ,d_{A}\left( d_{B}^{d}\left( k\right) ;k\right) ;k\right) =\tfrac{\overline{t}\left( 9\times 2^{2k}-9\times 2^{k}+10\right) ^{4}}{9\times 2^{2k+5}(9\times 2^{2k}-9\times 2^{k}+8)^{2}(27\times 2^{2k}-27\times 2^{k}+22)}\text {. } \end{aligned}$$

  Comparison of the profits $\Pi _{i}^{d}(k)$ and $\Pi _{B}\left( d_{B}^{d}\left( k\right) ,d_{A}\left( d_{B}^{d}\left( k\right) ;k\right) ;k\right)$ yields

  $$\begin{aligned}&\Pi _{i}^{d}(k)-\Pi _{B}\left( d_{B}^{d}\left( k\right) ,d_{A}\left( d_{B}^{d}\left( k\right) ;k\right) ;k\right) \\= & {} \tfrac{\left( 3\times 2^{2k}-3\times 2^{k}+2\right) \left( 9\times 2^{2k}-9\times 2^{k}+10\right) ^{2}}{3\times 2^{2k+2}\left( 27\times 2^{2k}-27\times 2^{k}+22\right) (9\times 2^{2k}-9\times 2^{k}+8)}>0\quad \text { for any} \quad k, \end{aligned}$$

  hence, firm A does not have an incentive to locate at $d_{A}\ge d_{B}^{d}\left( k\right)$. We conclude that the locations $d_{A}^{d}\left( k\right)$ and $d_{B}^{d}\left( k\right)$ in (23) constitute indeed the equilibrium.

$\square$