Mixed duopoly in a Hotelling framework with cubic transportation costs

Abstract

We examine a two-stage location-price model of a mixed duopoly where a private profit-maximizing firm competes with a public welfare-maximizing firm in a Hotelling-type framework. A noteworthy result in this model is that, with quadratic transportation costs, which has become the usual assumption in the literature, the socially optimal locations are obtained in equilibrium. We show here that under the alternative assumption of cubic transportation costs this result no longer holds: equilibrium locations are socially suboptimal. We find that just as in the case of linear transportation costs, previously studied in the literature, for some locations there is price equilibrium in the second stage of the game and for other locations there is not. But, in contrast with such a case, there is a location pair for which there is price equilibrium in the second stage of the game and neither firm has an incentive to marginally change its location. We also find that, in contrast with the case of quadratic transportation costs, this location pair is not socially optimal.

1 Introduction

There are markets in which public firms coexist in competition with private firms. For example, in the banking sector Caixa Geral de Depositos in Portugal, Banco Nacional in Costa Rica, BancoEstado in Chile and Banco de la República Oriental in Uruguay, all government-owned banks, are major players in their respective countries.¹ Similar examples can be found in sectors like telecommunications, energy, health, postal services or education. In consonance with the importance of this kind of markets, which have come to be known as mixed markets, a growing literature² has developed studying their different facets, among them their spatial dimension. A noteworthy result³ is that competition in a mixed duopoly between a profit-maximizing private firm and a welfare-maximizing public firm⁴ yields the socially optimal outcome in the standard location-price Hotelling model. This result is obtained under the assumption that transportations costs are quadratic, and contrasts with the socially suboptimal equilibrium locations in a purely private duopoly.⁵

Here we study if the result that a mixed duopoly yields the socially optimal locations continues to hold when we abandon the assumption of quadratic transportation costs-which has become the standard assumption in Hotelling-type models of mixed markets-by examining the alternative assumption of cubic transportation costs. We find that under this extension the result no longer holds: firms’ locations are socially suboptimal.

The most noticeable examples of steeply increasing transportations costs, such as cubic transportation costs, may be those where the Hotelling model is interpreted as referring to consumers that have heterogeneous tastes over some characteristic of a product, such as the sweetness of cereal.⁶ In this interpretation, the location of a consumer refers to his or her most preferred specification of this characteristic (the most preferred degree of sweetness) and the transportation costs refer to the disutility from consuming a product with a different specification. This disutility may steeply increase as a product characteristic gets increasingly different from the consumer’s preferred choice, which is the equivalent of a store being located at a longer distance from a consumer. Similarly, in the interpretation of actual physical locations, the transportation costs may be also steeply increasing if they include not only the monetary cost of travelling but also the disutility from the time spent in the travel and from the uncertainty in its length.⁷ One can easily envisage situations where as the distance, and thus the time spent in the journey and the uncertainty in its duration, increases, the disutility steeply increases.⁸ More importantly, since there is no certainty in the exact functional form of transportation costs, it is of relevance to examine if and how different assumptions on them change the outcomes that we obtain. Cubic transportation costs provide us with a natural alternative assumption to perform such an investigation.

To explain our results let us mention that Lu (2006) finds that there is no equilibrium in the two-stage location-price model of a mixed duopoly under linear transportation costs, while Cremer et al. (1991) show that an equilibrium does exist and yields the socially optimal locations under the alternative assumption of quadratic transportation costs. Here we show that under cubic transportation costs, just as in Lu (2006), for some locations there is price equilibrium in the second stage of the game and for other locations there is not. But, in contrast to Lu, there is a location pair for which there is price equilibrium in the second stage of the game and neither firm has an incentive to marginally change its location and, moreover, in contrast to Cremer et al. (1991), this location pair is not socially optimal.

By comparing the incentives that firms face when choosing their locations under linear, quadratic and cubic transportation costs, our analysis also sheds light on why the result that a mixed duopoly yields the socially optimal outcome is not robust. To minimize transportation costs, the public firm always chooses its location such that it is half as far away from the closest edge of the Hotelling line as it is from the location of the private firm, irrespective of the convexity of these costs. In contrast, the location choice of the private firm involves a trade-off between getting closer to the public firm to increase its market share given fixed prices, and getting away from the public firm to increase prices,⁹ and the convexity of the transportation costs affects the relative strength of these two forces. As the degree of convexity changes, the resolution of the trade-off also changes. This results in the private firm wanting to get as close as possible to its competitor under linear transportation costs,¹⁰ locating at just the right distance from it—from a social point of view-under quadratic transportation costs, and moving too far away from it, to the extent that it locates at one edge of the line, under cubic transportation costs.

The robustness of the result that a mixed duopoly yields the socially optimal outcome has been studied with respect to other assumptions of the model. Matsumura and Matsushima (2004) show that it continues to hold when the private and public firms have different marginal costs, and Matsumura and Matsushima (2003) show that it also holds if there is a sequential choice of locations with the public firm acting as the leader. On the other hand, Kitahara and Matsumura (2013) show that the result no longer holds when the assumption of inelastic demand is dropped, and Benassi et al. (2017) show that it also fails to hold when the assumption of a uniform consumers’ distribution is replaced by other distributions.

Yet, none of these papers examines the role of transportation costs, which is the focus of our paper.

2 The model

We use a standard location-price model of a mixed duopoly as studied by Cremer et al. (1991). Consumers are uniformly distributed with unit density on the interval [0,1] and inelastically demand one unit of a product. Two firms produce this product with zero marginal cost: firm 0, a welfare-maximizing public firm located at $x_{0}$, and firm 1, a profit-maximizing private firm located at $x_{1}$. We assume that $0 \le x_{0} < x_{1} \le 1$. The firms compete in the following two-stage game: in the first stage, they simultaneously choose their locations.¹¹ In the second stage, they simultaneously choose prices. We also assume, contrary to the standard assumption of quadratic transportation costs, that a consumer located at y incurs a transportation cost of $\left| {y - x_{0} } \right|^{3}$ from buying firm 0’s product and $\left| {y - x_{1} } \right|^{3}$ from buying firm 1’s product.

3 Results

Let ${\text{p}}_{\text{i}}$ be firm i’s price and $q_{i}$ firm i’s demand, i = 0, 1. A consumer located at y incurs a total cost (price plus transportation cost) of $p_{0} + \left| {y - x_{0} } \right|^{3}$ to buy from firm 0 and of $p_{1} + \left| {y - x_{1} } \right|^{3}$ to buy from firm 1. Let $y^{*}$ be the solution of the equation¹²:

$$p_{0} + \left| {y^{*} - x_{0} } \right|^{3} = p_{1} + \left| {y^{*} - x_{1} } \right|^{3}$$

(1)

If $0 \le y^{*} \le 1$, $y^{*}$ is the location of the consumer indifferent between buying from either firm. Moreover, since the consumers located to the left of $y^{*}$ prefer to buy from firm 0, while those located to the right of $y^{*}$ prefer to buy from firm 1, $y^{*}$ is also firm 0’s demand. On the other hand, if $y^{*} < 0\,\left( {y^{*} > 1} \right)$ then all consumers prefer to buy from firm 1 (firm 0). We therefore have:

$$q_{0} = \left\{ {\begin{array}{*{20}l} 0 \hfill & {\quad if \;y^{*} \le 0} \hfill \\ {y^{*} } \hfill & {\quad if \;0 \le y^{*} \le 1} \hfill \\ 1 \hfill & { \quad if\; 1 \le y^{*} } \hfill \\ \end{array} } \right.$$

It is easy to check that $q_{0}$ satisfies¹³:

$$q_{0} = \left\{ {\begin{array}{*{20}l} { = 0} \hfill & {\quad if\;p_{1} - p_{0} \le - \left( {x_{1} ^{3} - x_{0} ^{3} } \right)} \hfill \\ { \in \left[ {0,x_{0} } \right]} \hfill & {\quad if\; - \left( {x_{1} ^{3} - x_{0} ^{3} } \right) \le p_{1} - p_{0} \le - \left( {x_{1} - x_{0} } \right)^{3} } \hfill \\ { \in \left[ {x_{0} ,x_{1} } \right]} \hfill & { \quad if\; - \left( {x_{1} - x_{0} } \right)^{3} \le p_{1} - p_{0} \le \left( {x_{1} - x_{0} } \right)^{3} } \hfill \\ { \in \left[ {x_{1} ,1} \right]} \hfill & {\quad if\;\left( {x_{1} - x_{0} } \right)^{3} \le p_{1} - p_{0} \le \left( {1 - x_{0} } \right)^{3} - \left( {1 - x_{1} } \right)^{3} } \hfill \\ { = 1} \hfill & { \quad if\;\left( {1 - x_{0} } \right)^{3} - \left( {1 - x_{1} } \right)^{3} \le p_{1} - p_{0} } \hfill \\ \end{array} } \right.$$

(2)

while $q_{1}$ is given by

$$q_{1} = 1 - q_{0}$$

(3)

Since demand is completely inelastic, maximization of social surplus is equivalent to minimization of transportation costs, which are given by:

$$TC = \mathop \int \limits_{0}^{{q_{0} }} \left| {y - x_{0} } \right|^{3} dy + \mathop \int \limits_{{q_{0} }}^{1} \left| {y - x_{1} } \right|^{3} dy$$

(4)

and can be written as:

$$TC = \left\{ {\begin{array}{*{20}l} {\frac{{\left\{ {x_{0}^{4} - \left( {x_{0} - q_{0} } \right)^{4} + \left( {x_{1} - q_{0} } \right)^{4} + \left( {1 - x_{1} } \right)^{4} } \right\}}}{4}} \hfill & {\quad if \;0 \le q_{0} \le x_{0} } \hfill \\ {\frac{{\left\{ {x_{0}^{4} + \left( {q_{0} - x_{0} } \right)^{4} + \left( {q_{0} - x_{1} } \right)^{4} + \left( {1 - x_{1} } \right)^{4} } \right\}}}{4}} \hfill & {\quad if\; x_{0} \le q_{0} \le x_{1} } \hfill \\ {\frac{{\left\{ {x_{0}^{4} + \left( {q_{0} - x_{0} } \right)^{4} - \left( {q_{0} - x_{1} } \right)^{4} + \left( {1 - x_{1} } \right)^{4} } \right\}}}{4}} \hfill & { \quad if\; x_{1} \le q_{0} \le 1} \hfill \\ \end{array} } \right.$$

(5)

Firm 0 chooses $p_{0}$ to minimize TC. It follows from (1) and (5) that to solve this problem it sets

$$p_{0} = p_{1}$$

(6)

To see this notice first that, from (5), transportation costs are minimized when $q_{0} = \frac{{x_{0} + x_{1} }}{2}$ because TC is strictly decreasing (increasing) in $q_{0}$ when $0 \le q_{0} \le x_{0}$ ($x_{1} \le q_{0} \le 1)$, $\frac{dTC}{{dq_{0} }} = 0$ when $q_{0} = \frac{{x_{0} + x_{1} }}{2}$ and $\frac{{d^{2} TC}}{{dq_{0}^{2} }} > 0$ in the interval $x_{0} \le q_{0} \le x_{1}$. Notice then that, from (1), firm 0 must set $p_{0} = p_{1}$ to achieve $q_{0} = \frac{{x_{0} + x_{1} }}{2}$.

Firm 1 chooses $p_{1}$ to maximize $\Pi_{1}$, where

$$\Pi_{1} = p_{1} \left( {1 - q_{0} } \right)$$

(7)

Suppose first that firm 1 chooses a price $p_{1}$ in the interval $- \left( {x_{1} - x_{0} } \right)^{3} \le p_{1} - p_{0} \le \left( {x_{1} - x_{0} } \right)^{3}$, which corresponds to $x_{0} \le q_{0} \le x_{1}$. The first-order condition to maximize profits in the interior of this interval is given by:

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = - p_{1} \frac{{\partial q_{0} }}{{\partial p_{1} }} + 1 - q_{0} = 0$$

It is convenient to obtain $\frac{{\partial q_{0} }}{{\partial p_{1} }}$ directly from the condition (1), which takes the form

$$p_{0} + \left( {q_{0} - x_{0} } \right)^{3} = p_{1} + \left( {x_{1} - q_{0} } \right)^{3}$$

when $x_{0} \le q_{0} \le x_{1}$. This condition implies that

$$\frac{{\partial q_{0} }}{{\partial p_{1} }} = \frac{1}{{3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {x_{1} - q_{0} } \right)^{2} }}$$

and therefore firm 1’s first-order condition is:

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{ - p_{1} }}{{3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {x_{1} - q_{0} } \right)^{2} }} + 1 - q_{0} = 0$$

(8)

Let us solve for $p_{0}$ and $p_{1}$ in (6) and (8). Condition $p_{0} = p_{1}$ from (6) implies $q_{0} = \frac{{x_{0} + x_{1} }}{2}$ which in turn implies, using (8), that:

$$p_{0} = p_{1} = \frac{{3\left( {2 - x_{0} - x_{1} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4}$$

(9)

Suppose instead that firm 1 chooses a price $p_{1}$ such that $p_{1} - p_{0} \le - \left( {x_{1} - x_{0} } \right)^{3}$. We will then have $p_{1} < p_{0}$ and firm 0 will not minimize TC (which is done by choosing $p_{0} = p_{1}$). Similarly, if firm 1 chooses a price $p_{1}$ such that $\left( {x_{1} - x_{0} } \right)^{3} \le p_{1} - p_{0}$ we will have $p_{1} > p_{0}$ and, again, firm 0 will not minimize TC.

Therefore, if there is an equilibrium in the second stage of the game, it is given by the prices in Eq. (9). To know if these prices are indeed an equilibrium, we need to make sure that neither firm can improve its objective function by deviating to another price.

As argued above, given any $p_{1}$ [and therefore also firm 1’s price in Eq. (9)], firm 0 minimizes transportation costs by setting $p_{0} = p_{1}$. Thus, firm 0 cannot lower transportation costs by deviating to another price. However, given $p_{0}$ as in Eq. (9), firm 1’s response, according to condition (8), only tells us that $p_{1}$ satisfies a necessary condition to maximize $\Pi_{1}$ in the interior of the range $p_{0} - \left( {x_{1} - x_{0} } \right)^{3} \le p_{1} \le p_{0} + (x_{1} - x_{0} )^{3}$. Therefore, we have to examine if firm 1 can increase its profits by deviating to another price either in this range or outside of it. It turns out that for some locations $(x_{0} ,x_{1} )$, firm 1 is able to increase its profits by deviating from the price in Eq. (9). For these locations, there is no equilibrium in the second stage of the game. Technically, the reason for the non-existence of equilibrium is the lack of quasi-concavity of firm 1’s profit function. For these locations, given $p_{0}$ as in Eq. (9) although firm 1’s profit function has one local maximum at $p_{1} = p_{0}$, it has another one that yields higher profits at another value of $p_{1}$.

Proposition 1

For every $x_{0}$, there exists $x_{1}^{*} \in \left( {\frac{{3x_{0} + 2}}{5},\frac{{\left( {\sqrt {13} - 2} \right)x_{0} + 5 - \sqrt {13} }}{3}} \right)$such that there exists a Nash equilibrium in prices in the second stage of the game if $x_{1} > x_{1}^{*}$while there does not exist equilibrium if $x_{1} < x_{1}^{*}$. Whenever it exists, the equilibrium is given by Eq. (9).

Proposition 1 tells us that there exists price equilibrium in the second stage of the game if firms are sufficiently far apart, but not otherwise. It tells us that, for each $x_{0}$, there is a cutoff point $x_{1}^{*}$ such that $x_{1}$ needs to be further away from $x_{0}$ than this point to have equilibrium.

It follows from Proposition 1 that the restriction $x_{0} \le 0.34$, $x_{1} \ge 0.66$, which amounts to banning firms from locating at distance smaller than 0.16 from the city center, is a sufficient condition for a price equilibrium to exist in the second stage of the game. We impose such assumption on the locations of firms before proceeding with the analysis of the first stage. This restriction may be justified by zoning regulations [see for example Lai and Tsai (2004), Chen and Lai (2008) or Matsumura and Matsushima (2012)]. A particular type of zoning regulation bans some businesses from locating around the city center for several reasons. One of them is pollution, which tends to be particularly intense in this area¹⁴ and thus calls for the prohibition of polluting businesses. Another reason is traffic congestion, for which businesses that by their nature generate a lot of traffic may also be banned around the city center.¹⁵

Let us consider the first stage of the game. Replacing the equilibrium prices $p_{0}$ and $p_{1}$ in (9), and the implied firm 0’s demand $q_{0}$, into the transportation costs in (5) and firm 1’s profits in (7), we obtain both transportation costs and firm 1’s profits as a function of the first-stage locations $x_{0}$ and $x_{1}$ as follows:

$$TC = \frac{{x_{0}^{4} + 2\left( {\frac{{x_{1} - x_{0} }}{2}} \right)^{4} + \left( {1 - x_{1} } \right)^{4} }}{4}$$

(10)

$$\Pi_{1} = \frac{{3\left( {x_{1} - x_{0} } \right)^{2} \left( {2 - x_{1} - x_{0} } \right)^{2} }}{8}$$

(11)

In the first stage of the game, firm 0 chooses $x_{0}$ to minimize TC as given in (10) and firm 1 chooses $x_{1}$ to maximize $\Pi_{1}$ as given in (11). Minimization of TC with respect to $x_{0}$ leads to the following first-order condition:

$$\frac{\partial TC}{{\partial x_{0} }} = x_{0}^{3} - \left( {\frac{{x_{1} - x_{0} }}{2}} \right)^{3} = 0$$

(12)

The second-order condition is:

$$\frac{{\partial^{2} TC}}{{\partial x_{0}^{2} }} = 3x_{0}^{2} + \frac{3}{2}\left( {\frac{{x_{1} - x_{0} }}{2}} \right)^{2} > 0$$

(13)

Maximization of $\Pi_{1}$ with respect to $x_{1}$ leads to the following first-order condition:

$$\frac{{\partial \Pi_{1} }}{{\partial x_{1} }} = \left( {\frac{3}{2}} \right)\left( {2 - x_{0} - x_{1} } \right)\left( {x_{1} - x_{0} } \right)\left( {1 - x_{1} } \right) = 0$$

(14)

It follows from (14) that, since $\frac{{\partial \Pi_{1} }}{{\partial x_{1} }} > 0$ for all $x_{0} < x_{1} < 1$ and $\frac{{\partial \Pi_{1} }}{{\partial x_{1} }} = 0$ for $x_{1} = 1$, firm 1’s optimal choice is $x_{1} = 1$. It follows from (12) and (13) that firm 0’s optimal choice is $x_{0} = \frac{{x_{1} }}{3}$. Simultaneous optimization of (10) and (11) therefore yields:

$$x_{0} = \frac{1}{3} ,\;\;x_{1} = 1$$

(15)

which is a location pair outside the restricted zone and, thus, for which there is equilibrium in the second stage. This is the unique subgame-perfect equilibrium, or location equilibrium, of the whole game.

The problem of non-existence of price equilibrium for some locations in the second stage of two-stage location-price models also arises in private oligopolies. We follow the approach of this literature (Economides 1984, 1986, 1989)¹⁶ and define the direction in which $\frac{{\partial \Pi_{1} }}{{\partial x_{1} }}$ is positive as the ‘relocation tendency’ of firm 1. Similarly, we define the direction in which $\frac{\partial TC}{{\partial x_{0} }}$ is negative as the ‘relocation tendency’ of firm 0.¹⁷

We have shown that with cubic transportation costs, $\frac{{\partial \Pi_{1} }}{{\partial x_{1} }} > 0$ for all $x_{0} < x_{1} < 1$ and thus firm 1’s relocation tendency is toward the right edge, away from firm 0, for all interior points, and it is zero at the edge of the line: $\frac{{\partial \Pi_{1} }}{{\partial x_{1} }} = 0$ for $x_{1} = 1$. When $x_{1} = 1$, the relocation tendency of the public firm is towards the private firm when $x_{0} < \frac{1}{3}$$\left( {\frac{\partial TC}{{\partial x_{0} }} < 0} \right)$, away from the private firm when $x_{0} > \frac{1}{3}$$\left( {\frac{\partial TC}{{\partial x_{0} }} > 0} \right)$ and it is zero when $x_{0} = \frac{1}{3}$. Therefore, when $(x_{0} ,x_{1} ) = \left( {1/3, 1} \right)$ the relocation tendency is zero for both firms. At this location pair, neither firm improves its objective function by marginally changing its location.

We can compare the results of linear, quadratic and cubic transportation costs as follows. The reaction function of the public firm is the same with all three transportation costs. This firm always minimizes transportation costs by setting $x_{0} = \frac{{x_{1} }}{3}$. For location pairs satisfying this condition, the public firm will have no incentives to relocate.

The private firm, in contrast, will behave differently as we change the convexity of the transportation costs. There are two forces that drive firm 1’s behavior. First, there is an incentive for firm 1 to move closer to firm 0 because with fixed prices, this movement increases its demand. Second, there is an incentive for firm 1 to move away from firm 0 to increase second-stage prices. An increase in the convexity of the cost function increases the strength of the second effect relative to the first one. With linear transportation costs, the trade-off of the two effects results in a relocation tendency for firm 1 towards firm 0’s location, which results in turn in the absence of equilibrium. If we increase the convexity of transportation costs and consider quadratic transportation costs, firm 1 has a relocation tendency towards firm 0 if it is far from this firm, but away from firm 0’s location if it is close to it. The relocation tendency vanishes exactly at the social optimum. If we further increase the convexity of transportation costs and consider cubic transportation costs, firm 1 has a relocation tendency away from firm 0. Since the existence zone of second-stage price equilibrium is precisely the zone of locations where firms are not too close, the tendency of firm 1 to move away from firm 0 keeps firms in the existence zone and leads to an equilibrium with firm 1 at the right edge and firm 0 one-third of the line away from the other edge.

Importantly, for the socially optimal locations $\left( { 1/4, 3/4} \right)$ there is also equilibrium in the second stage of the game, but the private firm exhibits a tendency to move away from the public firm: $\frac{{\partial \Pi_{1} }}{{\partial x_{1} }} > 0$. Given the public firm’s location $x_{0} = 1/4$, the private firm increases its profits by marginally changing its location from $x_{1} = 3/4$ to the right.

Proposition 2

$(x_{0} ,x_{1} ) = \left( {1/3, 1} \right)$is the unique location equilibrium of the location-price game with cubic transportation costs. In this equilibrium locations are different from the socially optimal locations $(x_{0} ,x_{1} ) = \left( {1/4, 3/4} \right)$.

4 Conclusion

We have studied a standard Hotelling location-price model of a mixed duopoly with cubic transportation costs. We have shown that, in contrast with the case of linear transportation costs, there exists a location pair for which there exists a unique price equilibrium in the second stage and neither firm has incentives to marginally relocate. We have also shown that, in contrast with the case of quadratic transportation costs, this location pair is not socially optimal.

Appendix

1.1 Proof of Proposition 1

Let $p^{*} = \frac{{3\left( {2 - x_{0} - x_{1} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4}$ be the candidate equilibrium common price in Eq. (9) and $\Pi_{1}^{*} = \frac{{3\left( {2 - x_{0} - x_{1} } \right)^{2} \left( {x_{1} - x_{0} } \right)^{2} }}{8}$ be the candidate equilibrium firm 1’s profits as given in Eq. (11). We now examine if given $p_{0} = p^{*}$, firm 1 can increase its profits above $\Pi_{1}^{*}$ by choosing a price $p_{1} \ne p^{*}$.

(1) if firm 1 chooses a price in the interval $p_{0} - \left( {x_{1}^{3} - x_{0}^{3} } \right) \le p_{1} \le p_{0} - \left( {x_{1} - x_{0} } \right)^{3} ,$ then $q_{0} \in \left[ {0 ,x_{0} } \right]$ and we have

$$p_{0} + \left( {x_{0} - q_{0} } \right)^{3} = p_{1} + \left( {x_{1} - q_{0} } \right)^{3}$$

(16)

from where

$$\frac{{\partial q_{0} }}{{\partial p_{1} }} = \frac{1}{{3\left( {x_{1} - q_{0} } \right)^{2} - 3\left( {x_{0} - q_{0} } \right)^{2} }}$$

and therefore:

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{ - p_{1} }}{{3\left( {x_{1} - q_{0} } \right)^{2} - 3\left( {x_{0} - q_{0} } \right)^{2} }} + 1 - q_{0}$$

(17)

Replacing $p_{1}$ from (16) into (17) and using $p_{0} = p^{*}$ we obtain

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{36q_{0}^{2} - 24\left( {x_{0} + x_{1} + 1} \right)q_{0} + 7x_{1}^{2} + 4x_{0} x_{1} + 6x_{1} + x_{0}^{2} + 18x_{0} }}{{12\left( { - 2q_{0} + x_{0} + x_{1} } \right)}}$$

(18)

The denominator in the RHS of (18) is positive because $q_{0} \le x_{0} < x_{1}$. The numerator is also positive because (a) it is decreasing in $q_{0 }$ (its derivative with respect to $q_{0}$ is equal to $72q_{0} - 24\left( {x_{0} + x_{1} + 1} \right) \le 72x_{0} - 24\left( {x_{0} + x_{1} + 1} \right) = - 24\left( {x_{1} - x_{0} } \right) - 24\left( {1 - x_{0} } \right) < 0$) and (b) it is positive when $q_{0} = x_{0}$ (it is then equal to $\left( {x_{1} - x_{0} } \right)\left( { - 13x_{0} + 7x_{1} + 6} \right) > 0).$

It follows that $\Pi_{1}$ is increasing over the whole range $p_{0} - \left( {x_{1}^{3} - x_{0}^{3} } \right) \le p_{1} \le p_{0} - \left( {x_{1} - x_{0} } \right)^{3}$. Let $\Pi_{1}^{a}$ be firm 1’s profits when $p_{1} = p_{0} - \left( {x_{1} - x_{0} } \right)^{3}$-which implies $q_{0} = x_{0}$- and $p_{0} = p^{*}$. Then

$$\Pi_{1}^{a} = \left( {\frac{{3\left( {2 - x_{1} - x_{0} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4} - \left( {x_{1} - x_{0} } \right)^{3} } \right)\left( {1 - x_{0} } \right)$$

If we substract from these profits the candidate equilibrium profits $\Pi_{1}^{*}$ we obtain, after some simplifications,

$$\Pi_{1}^{a} - \Pi_{1}^{*} = \frac{{\left( {x_{1} - x_{0} } \right)^{3} \left( { - 3x_{1} + 5x_{0} - 2} \right)}}{8} < 0$$

which implies that firm 1 will not deviate to such a price, and will thus neither deviate to any price $p_{1}$ with $p_{0} - \left( {x_{1}^{3} - x_{0}^{3} } \right) \le p_{1} \le p_{0} - \left( {x_{1} - x_{0} } \right)^{3}$.

(2) If firm 1 chooses a price in the interval $p_{0} - \left( {x_{1} - x_{0} } \right)^{3} \le p_{1} \le p_{0} + \left( {x_{1} - x_{0} } \right)^{3}$, then $q_{0} \in \left[ {x_{0} ,x_{1} } \right]$ and we will have:

$$p_{0} + \left( {q_{0} - x_{0} } \right)^{3} = p_{1} + \left( {x_{1} - q_{0} } \right)^{3}$$

(19)

and thus

$$\frac{{\partial q_{0} }}{{\partial p_{1} }} = \frac{1}{{3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {x_{1} - q_{0} } \right)^{2} }}$$

Therefore:

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{ - p_{1} }}{{3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {x_{1} - q_{0} } \right)^{2} }} + 1 - q_{0}$$

(20)

Replacing $p_{1}$ from (19) into (20) and using $p_{0} = p^{*}$ we obtain, after some simplifications:

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{\left[ {2q_{0} - x_{0} - x_{1} } \right]\left[ {16q_{0}^{2} - \left( {10\left( {x_{0} + x_{1} } \right) + 12} \right)q_{0} + \left( {7x_{1} - 10x_{0} + 6} \right)x_{1} + 7x_{0}^{2} + 6x_{0} } \right]}}{{ - 12\left( {q_{0} - x_{0} } \right)^{2} - 12\left( {x_{1} - q_{0} } \right)^{2} }}$$

Therefore, $\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = 0$ if $q_{0} = \frac{{x_{0} + x_{1} }}{2}$, which implies $p_{1} = p_{0} = p^{*} .$ Notice also that, from (20):

$$\frac{{\partial^{2} \Pi_{1} }}{{\partial p_{1}^{2} }} = \frac{{ - \left( {3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {x_{1} - q_{0} } \right)^{2} } \right) + 6p_{1} \left( {\left( {q_{0} - x_{0} } \right) - \left( {x_{1} - q_{0} } \right)} \right)\frac{{\partial q_{0} }}{{\partial p_{1} }}}}{{\left( {3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {x_{1} - q_{0} } \right)^{2} } \right)^{2} }} - \frac{{\partial q_{0} }}{{\partial p_{1} }}$$

and that this second derivative evaluated at $p_{1} = p_{0} = p^{*}$ (and thus at $q_{0} = \frac{{x_{0} + x_{1} }}{2}$) is negative:

$$\frac{{\partial^{2} \Pi_{1} }}{{\partial p_{1}^{2} }} = \frac{ - 4}{{3\left( {x_{1} - x_{0} } \right)^{2} }} < 0$$

Therefore, $p_{1} = p^{*}$ is a local maximum, and it is also the only value where $\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}$ vanishes (in the segment $q_{0} \in \left[ {x_{0} ,x_{1} } \right]$) unless the following equation has roots in $q_{0} \in \left[ {x_{0} ,x_{1} } \right]:$

$$16q_{0}^{2} - \left( {10\left( {x_{0} + x_{1} } \right) + 12} \right)q_{0} + \left( {7x_{1} - 10x_{0} + 6} \right)x_{1} + 7x_{0}^{2} + 6x_{0} = 0$$

(21)

Equation (21) is a second degree equation in $q_{0}$ with discriminant equal to

$$12\left( { - 29x_{1}^{2} + \left( {70x_{0} - 12} \right)x_{1} - 29x_{0}^{2} - 12x_{0} + 12} \right)$$

(22)

If the discriminant in (22) is negative $\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}$ will only vanish at $q_{0} = \frac{{x_{0} + x_{1} }}{2}$ (which corresponds to $p_{1} = p^{*}$) in the segment $q_{0} \in \left[ {x_{0} ,x_{1} } \right].$ If this discriminat is positive, then Eq. (21) will have two roots:

$$q_{0}^{H} = \frac{{5x_{0} + 5x_{1} + 6 + \sqrt 3 \sqrt { - 29x_{1}^{2} + \left( {70x_{0} - 12} \right)x_{1} - 29x_{0}^{2} - 12x_{0} + 12} }}{16}$$

(23)

$$q_{0}^{L} = \frac{{5x_{0} + 5x_{1} + 6 - \sqrt 3 \sqrt { - 29x_{1}^{2} + \left( {70x_{0} - 12} \right)x_{1} - 29x_{0}^{2} - 12x_{0} + 12} }}{16}$$

(24)

But, $q_{0}^{H} > x_{1}$, since the positiveness of the discriminant in (22) implies that (a) $q_{0}^{H} > \frac{{5x_{0} + 5x_{1} + 6}}{11}$, and (b) $x_{1} < \frac{{5x_{0} + 6}}{11}$, because this discriminat is strictly decreasing in $x_{1}$ and it is negative when $x_{1} = \frac{{5x_{0} + 6}}{11}$. These two facts imply $q_{0}^{H} - x_{1} \ge \frac{{5x_{0} + 5x_{1} + 6}}{11} - x_{1} = \frac{{5x_{0} - 11x_{1} + 6}}{11} > 0$. Thus, $q_{0}^{L}$ is the only possible root additional to $q_{0} = \frac{{\left( {x_{0} + x_{1} } \right)}}{2}$ in the range $q_{0} \in \left[ {x_{0} ,x_{1} } \right]$, which cannot therefore be a local maximum (since $p_{1} = p^{*}$ is a local maximum and $\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}$ only vanishes at $p^{*}$ and $p^{L}$ (associated to $q_{0}^{L}$) then $\Pi_{1}$ is strictly decreasing in $p_{1} \in \left( {p^{*} , p^{L} } \right)$ if $p^{*} < p^{L}$ and, similarly, it is strictly increasing in $p_{1} \in \left( {p^{L} , p^{*} } \right)$ if $p^{L} < p^{*}$).

Since we proved above that firm 1 will not deviate to a price $p_{1}$ such that $q_{0} = x_{0}$, the only price that remains to be considered in this interval is $p_{1} = p_{0} + \left( {x_{1} - x_{0} } \right)^{3}$ which corresponds to $q_{0} = x_{1}$. If $p_{0} = p^{*}$ and firm 1 chooses such a price, firm 1’s profits will be equal to

$$\Pi_{1}^{b} = \left( {\frac{{3\left( {2 - x_{0} - x_{1} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4} + \left( {x_{1} - x_{0} } \right)^{3} } \right) \left( {1 - x_{1} } \right)$$

Subtracting the candidate equilibrium profits $\Pi_{1}^{*}$ from $\Pi_{1}^{b}$ we obtain:

$$gb\left( {x_{0} ,x_{1} } \right) = \Pi_{1}^{b} - \Pi_{1}^{*} = \left( {\frac{{3\left( {2 - x_{0} - x_{1} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4} + \left( {x_{1} - x_{0} } \right)^{3} } \right)\left( {1 - x_{1} } \right) - \frac{{3\left( {2 - x_{0} - x_{1} } \right)^{2} \left( {x_{1} - x_{0} } \right)^{2} }}{8}$$

with

$$\frac{{\partial gb\left( {x_{0} ,x_{1} } \right)}}{{\partial x_{1} }} = \frac{{ - \left( {x_{1} - x_{0} } \right)^{2} \left( {10x_{1} - 7x_{0} - 3} \right)}}{4}$$

For any given $x_{0}$,$gb\left( {x_{0} ,x_{0} } \right) = 0$, $gb\left( {x_{0} ,x_{1} } \right)$ reaches a maximum at $x_{1} = \frac{{\left( {7x_{0} + 3} \right)}}{10}$ with $gb\left( {x_{0} ,\frac{{\left( {7x_{0} + 3} \right)}}{10}} \right) = \frac{{27\left( {1 - x_{0} } \right)^{4} }}{16,000} > 0$, and $gb\left( {x_{0} ,1} \right) = \frac{{ - 3\left( {1 - x_{0} } \right)^{4} }}{8} < 0$. Since $gb\left( {x_{0} ,x_{1} } \right)$ is strictly increasing in $x_{1}$ for $x_{0} < x_{1} < \frac{{\left( {7x_{0} + 3} \right)}}{10}$, strictly decreasing in $x_{1}$ for $\frac{{\left( {7x_{0} + 3} \right)}}{10} < x_{1} < 1$, and $gb\left( {x_{0} ,\frac{{\left( {3x_{0} + 2} \right)}}{5}} \right) = 0$, it follows that, when $\frac{{3x_{0} + 2}}{5} \le x_{1} \le 1$, $gb\left( {x_{0} ,x_{1} } \right) \le 0,$ and firm 1 does not find this deviation profitable, while for $x_{1} < \frac{{3x_{0} + 2}}{5}$, $gb\left( {x_{0} ,x_{1} } \right) > 0$, firm 1 does deviate from the candidate equilibrium price and there does not exist an equilibrium in the second stage of the game.

(3) if firm 1 chooses a price in the interval $p_{0} + \left( {x_{1} - x_{0} } \right)^{3} \le p_{1} \le p_{0} + \left( {1 - x_{0} } \right)^{3} - \left( {1 - x_{1} } \right)^{3}$, then $q_{0} \in \left[ {x_{1} ,1 } \right]$ and we have

$$p_{0} + \left( {q_{0} - x_{0} } \right)^{3} = p_{1} + \left( {q_{0} - x_{1} } \right)^{3}$$

(25)

from where

$$\frac{{\partial q_{0} }}{{\partial p_{1} }} = \frac{1}{{3\left( {q_{0} - x_{0} } \right)^{2} - 3\left( {q_{0} - x_{1} } \right)^{2} }}$$

and therefore:

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{ - p_{1} }}{{3\left( {q_{0} - x_{0} } \right)^{2} - 3\left( {q_{0} - x_{1} } \right)^{2} }} + 1 - q_{0}$$

(26)

Replacing $p_{1}$ from (25) into (26) and using $p_{0} = p^{*}$ we obtain

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{ - 36q_{0}^{2} + 24\left( {x_{0} + x_{1} + 1} \right)q_{0} - x_{1}^{2} - 4x_{0} x_{1} - 18x_{1} - 7x_{0}^{2} - 6x_{0} }}{{12\left( {2q_{0} - x_{0} - x_{1} } \right)}}$$

The denominator of $\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}$ is positive because $x_{0} < x_{1} \le q_{0}$. $\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}$ will vanish when the numerator does so, which yields a quadratic equation in $q_{0}$ with discriminant equal to

$$144 \left( {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} \right)$$

(27)

The discriminant in (27) is strictly decreasing in $x_{1}$ and it is equal to zero when $x_{1} = x_{1}^{R}$, with

$$x_{1}^{R} = \frac{{\left( {\sqrt {13} - 2} \right)x_{0} + 5 - \sqrt {13} }}{3} \in \left( {\frac{{3x_{0} + 2}}{5},1} \right)$$

Therefore,

(a) when $x_{1} > x_{1}^{R}$, the discriminant in (27) is negative and $\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}$ never vanishes (it is negative). It then suffices to consider the price deviation associated to $q_{0} = x_{1}$ (which yields profits $\Pi_{1}^{b}$). Since $x_{1}^{R} > \frac{{3x_{0} + 2}}{5}$, we have that $x_{1} > \frac{{3x_{0} + 2}}{5}$ and thus $gb\left( {x_{0} ,x_{1} } \right) < 0$, $\Pi_{1}^{b} < \Pi_{1}^{*}$ and there is equilibrium.

(b) when $x_{1} \le x_{1}^{R}$ the discriminant in (27) is positive and $\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = 0$ has the following roots:

$$\begin{aligned} q_{0}^{c} & = \frac{{2x_{1} + 2x_{0} + 2 + \sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} }}{6} \\ q_{0}^{d} & = \frac{{2x_{1} + 2x_{0} + 2 - \sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} }}{6} \\ \end{aligned}$$

Now, from (26)

$$\frac{{\partial^{2} \Pi_{1} }}{{\partial p_{1}^{2} }} = \frac{{ - 3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {q_{0} - x_{1} } \right)^{2} + p_{1} \left( {6\left( {q_{0} - x_{0} } \right) - 6\left( {q_{0} - x_{1} } \right)} \right)\frac{{\partial q_{0} }}{{\partial p_{1} }}}}{{\left( {3\left( {q_{0} - x_{0} } \right)^{2} - 3\left( {q_{0} - x_{1} } \right)^{2} } \right)^{2} }} - \frac{{\partial q_{0} }}{{\partial p_{1} }}$$

(28)

Also, from (26), $\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = 0$ implies that

$$p_{1} = \left( {1 - q_{0} } \right)\left( {3\left( {q_{0} - x_{0} } \right)^{2} - 3\left( {q_{0} - x_{1} } \right)^{2} } \right)$$

(29)

Replacing $p_{1}$ from (29) and $q_{0}$ with $q_{0}^{c}$ in (28) we get:

$$\frac{{\partial^{2} \Pi_{1} }}{{\partial p_{1}^{2} }} = \frac{{ - 3\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} }}{{\left( {x_{1} - x_{0} } \right)\left( {\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} - x_{0} - x_{1} + 2} \right)^{2} }}$$

Similarly, when $q_{0} = q_{0}^{d}$ we get:

$$\frac{{\partial^{2} \Pi_{1} }}{{\partial p_{1}^{2} }} = \frac{{3\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} }}{{\left( {x_{1} - x_{0} } \right)\left( {\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} + x_{0} + x_{1} - 2} \right)^{2} }}$$

Therefore $\Pi_{1}$ reaches a local maximum at $q_{0}^{c}$ and a local minimum at $q_{0}^{d}$.

Let $\Pi_{1}^{c}$ be firm 1’s profits when $p_{1}$ is such that $q_{0} = q_{0}^{c}$,

$$\Pi_{1}^{c} = \left( {\frac{{3\left( {2 - x_{0} - x_{1} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4} + \left( {q_{0}^{c} - x_{0} } \right)^{3} - \left( {q_{0}^{c} - x_{1} } \right)^{3} } \right)\left( {1 - q_{0}^{c} } \right)$$

and let $gc\left( {x_{0} ,x_{1} } \right) = \Pi_{1}^{c} - \Pi_{1}^{*}$. $gc\left( {x_{0} ,x_{1} } \right)$ is the increase in profits when firm 1 chooses $p_{1}$ such that $q_{0} = q_{0}^{c}$ instead of the candidate equilibrium price $p^{*}$.

Since we know that for $x_{1} < \frac{{3x_{0} + 2}}{5}$ there is no equilibrium (because $\Pi_{1}^{b} > \Pi_{1}^{*}$), while for $x_{1} > x_{1}^{R}$ there is equilibrium, we will focus our attention on the behaviour of $gc\left( {x_{0} ,x_{1} } \right)$ in the interval $x_{1} \in \left[ {\frac{{3x_{0} + 2}}{5},x_{1}^{R} } \right]$.

We have that $gc\left( {x_{0} ,\frac{{3x_{0} + 2}}{5}} \right) > 0$, $gc\left( {x_{0} ,x_{1}^{R} } \right) < 0$,and we will now prove that $\frac{{\partial gc\left( {x_{0} ,x_{1} } \right)}}{{\partial x_{1} }} < 0$ for $x_{1} \in \left[ {\frac{{3x_{0} + 2}}{5},x_{1}^{R} } \right]$. This implies that there exists $x_{1}^{*} \in \left( {\frac{{3x_{0} + 2}}{5},x_{1}^{R} } \right)$ such that $\Pi_{1}^{c} = \Pi_{1}^{*}$ if $x_{1} = x_{1}^{*}$ and $\Pi_{1}^{c} > \Pi_{1}^{*}$$\left( {\Pi_{1}^{c} < \Pi_{1}^{*} } \right)$ if $x_{1} < x_{1}^{*}$$\left( {x_{1} > x_{1}^{*} } \right)$. This in turn implies that there does not exist (there exists) equilibrium if $x_{1} < x_{1}^{*}$$\left( {x_{1} > x_{1}^{*} } \right).$

We have:

$$\frac{{\partial gc\left( {x_{0} ,x_{1} } \right)}}{{\partial x_{1} }} = \frac{{f\left( {x_{0} ,x_{1} } \right)\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} + h\left( {x_{0} ,x_{1} } \right)}}{{36\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} }}$$

with

$$f\left( {x_{0} ,x_{1} } \right) = - \left( {34x_{1}^{3} + \left( { - 12x_{0} - 90} \right)x_{1}^{2} + \left( { - 18x_{0}^{2} + 60x_{0} + 60} \right)x_{1} + 4x_{0}^{3} + 6x_{0}^{2} - 36x_{0} - 8} \right)$$

and

$$\begin{aligned} h\left( {x_{0} ,x_{1} } \right) & = 36x_{1}^{4} + \left( {51x_{0} - 195} \right)x_{1}^{3} + \left( { - 59x_{0}^{2} - 35x_{0} + 310} \right)x_{1}^{2} \\ & \quad + \,\left( { - 39x_{0}^{3} + 235x_{0}^{2} - 200x_{0} - 140} \right)x_{1} + 27x_{0}^{4} - 69x_{0}^{3} \\ & \quad - \,14x_{0}^{2} + 76x_{0} + 16 \\ \end{aligned}$$

Now, $f\left( {x_{0} ,x_{1} } \right) < 0$ because, as can be easily checked, $\frac{{\partial^{2} f}}{{\partial x_{1}^{2} }} > 0$, $\frac{\partial f}{{\partial x_{1} }} < 0$ when $x_{1} = \frac{{3x_{0} + 2}}{5}$, $\frac{\partial f}{{\partial x_{1} }} > 0$ when $x_{1} = x_{1}^{R}$ and $f < 0$ when both $x_{1} = \frac{{3x_{0} + 2}}{5}$ and $x_{1} = x_{1}^{R}$.

Similarly, $h\left( {x_{0} ,x_{1} } \right) < 0$ because $\frac{{\partial^{2} h}}{{\partial x_{1}^{2} }} > 0$, $\frac{\partial h}{{\partial x_{1} }} > 0$ when $x_{1} = \frac{{3x_{0} + 2}}{5}$ and $h = 0$ when $x_{1} = x_{1}^{R}$.

1.2 Proof of socially optimal locations in Proposition 2

Since demand is inelastic, the socially optimal locations are those that minimize transportation costs as given in (5). To find them, remember first that, as shown above, $q_{0} = \frac{{x_{0} + x_{1} }}{2}$ minimizes transportation costs for any locations $x_{0} ,x_{1}$. Replacing this value back in (5) yields TC as a function of $x_{0}$ and $x_{1}$ as given in (10). The first-order conditions to minimize TC in (10) with respect to both $x_{0}$ and $x_{1}$ are:

$$\frac{\partial TC}{{\partial x_{0} }} = x_{0}^{3} - \left( {\frac{{x_{1} - x_{0} }}{2}} \right)^{3} = 0$$

$$\frac{\partial TC}{{\partial x_{1} }} = \left( {\frac{{x_{1} - x_{0} }}{2}} \right)^{3} - \left( {1 - x_{1} } \right)^{3} = 0$$

from where we can obtain, respectively, $x_{0} = \frac{{x_{1} }}{3}$ and $x_{0} = 3x_{1} - 2$ and, therefore, $x_{0} = \frac{1}{4}$, $x_{1} = \frac{3}{4}$. The second order conditions are also satisfied.

1.3 Firm 0’s demand $q_{0}$ written as an explicit function of $p_{0}$ and $p_{1}$

If $- \left( {x_{1}^{3} - x_{0}^{3} } \right) \le p_{1} - p_{0} \le - \left( {x_{1} - x_{0} } \right)^{3}$ then:

$$q_{0} = \frac{{x_{0} + x_{1} }}{2} - \sqrt {\frac{{4\left( {p_{0} - p_{1} } \right) - \left( {x_{1} - x_{0} } \right)^{3} }}{{12\left( {x_{1} - x_{0} } \right)}} }$$

If $- \left( {x_{1} - x_{0} } \right)^{3} \le p_{1} - p_{0} \le \left( {x_{1} - x_{0} } \right)^{3}$ then:

$$\begin{aligned} q_{0} & = \frac{{\left( {x_{0} + x_{1} } \right)}}{2} + \left( {\frac{{\sqrt {4\left( {p_{1} - p_{0} } \right)^{2} + \left( {x_{1} - x_{0} } \right)^{6} } }}{8} - \frac{{p_{0} - p_{1} }}{4}} \right)^{1/3} \\ & \quad - \,\frac{{\left( {x_{1} - x_{0} } \right)^{2} }}{{4\left( {\frac{{\sqrt {4\left( {p_{1} - p_{0} } \right)^{2} + \left( {x_{1} - x_{0} } \right)^{6} } }}{8} - \frac{{p_{0} - p_{1} }}{4}} \right)^{1/3} }} \\ \end{aligned}$$

If $\left( {x_{1} - x_{0} } \right)^{3} \le p_{1} - p_{0} \le \left( {1 - x_{0} } \right)^{3} - \left( {1 - x_{1} } \right)^{3}$ then:

$$q_{0} = \frac{{x_{0} + x_{1} }}{2} + \sqrt {\frac{{4\left( {p_{1} - p_{0} } \right) - \left( {x_{1} - x_{0} } \right)^{3} }}{{12\left( {x_{1} - x_{0} } \right)}} }$$