Hotelling was right with decreasing returns to scale and a coalition-proof refinement

Abstract

This paper provides a simple, realistic, and very slightly modified version of the production technology in Hotelling’s (Econ J 39:41–57, 1929) spatial model with linear transportation costs to overcome the nonexistence problem of equilibrium—decreasing returns to scale. It is shown that a pure strategy Nash equilibrium in price competition always exists for all location pairs and guarantees uniqueness if we utilize a coalition-proof refinement introduced by Bernheim et al. (J Econ Theory 42:1–12, 1987). Decreasing returns to scale reduce the profit a firm can capture through price undercutting and stabilize the price equilibrium due to the increasing average production cost of firms. As a consequence, duopoly firms agglomerating at the center of a line are shown to be at the unique location equilibrium. This paper confers a new validity to the so-called principle of minimum differentiation, in some sense, with the least deviation from the original Hotelling (Econ J 39:41–57, 1929) model.

Appendix

1.1 Definition of a coalition-proof Nash equilibrium

Following Chowdhury and Sengupta (2004), let $N$ be the set of firms in the market, where $\left|N\right|=n\ge 2$. A coalition $T\subseteq N$ has a profitable deviation from the price vector $p$ if there exists ${p}^{\prime }(T)=({p}_{i}^{\prime })_{i\in T}$ such that:

$$\begin{aligned} \pi _{i}({p}^{\prime }(T),p(N/T))>\pi _{i}(p), \forall i\in T, \end{aligned}$$

where $p(N/T)$ is the restriction of $p$ on $N/T$.

A coalition $T$ with $\left| T \right| =1$ has a self-enforcing profitable deviation from the price vector $p$ if $T$ has a profitable deviation from $p$.

Suppose now we have defined a self-enforcing profitable deviation $\forall S\subset N$, with $\left| S \right| \le m\le n-1$ and for all price vectors $p$. Now consider a coalition $T\subseteq N$ such that $\left| T \right| =m+1$.

Coalition $T$ is said to have a self-enforcing profitable deviation from $p$ if there exists a vector ${p}^{\prime }(T)$ such that:

• 1. Coalition $T$ has a profitable deviation from $p$ using the price vector ${p}^{\prime }(T)$.

• 2. For any $S\subset T$, $S\ne T$, the coalition $S$ has no self-enforcing profitable deviation from $({p}^{\prime }(T),p(N/T))$.

Definition 1

A vector of prices $\hat{p}$ is said to be a CPNE if no coalition $T$ has a self-enforcing profitable deviation from $\hat{p}$.

Proof of Lemma 1

We define a new function $m_{1}(x_{1},x_{2})$ as:

$$\begin{aligned} m_{1}(x_{1},x_{2})&= p_{1}^{*}(x_{1},x_{2})-{C}^{\prime }\left(Q_{1}^{*} (x_{1},x_{2})\right)\nonumber \\&= \frac{t(x_{1}+x_{2}+2)}{3}+\frac{{C}^{\prime }(Q_{2}^{* }\left(x_{1},x_{2})\right)}{3}-\frac{{C}^{\prime }\left(Q_{1}^{*}(x_{1},x_{2})\right)}{3}, \end{aligned}$$

where the second equality is derived from the first-order condition for the maximizing profit of a firm in Eq. (2). Differentiating the function $m_{1}(x_{1},x_{2})$ with respect to $x_{i}$, for $i=1,2$, and taking into account Eqs. (5) and (6) yield:

$$\begin{aligned} \frac{\partial m_{1}(x_{1},x_{2})}{\partial x_{i}}&= \frac{\partial p_{1}^{*}(x_{1},x_{2})}{\partial x_{i}}-{C}^{\prime \prime }\left(Q_{1}^{* }(x_{1},x_{2})\right)\cdot \frac{\partial Q_{1}^{*}(x_{1},x_{2})}{\partial x_{i}}\\&= \frac{t\left[2t\!+\!{C}^{\prime \prime }(Q_{1}^{\!*\!})\right]}{6t\!+\!{C}^{\prime \prime } (Q_{1}^{\!*\!})\!+\!{C}^{\prime \prime } (Q_{2}^{\!*\!})}\!-\!{C}^{\prime \prime } \left(Q_{1}^{\!*\!}(x_{1},x_{2})\right)\cdot \frac{t}{6t\!+\!{C}^{\prime \prime }(Q_{1}^{\!*\! })\!+\!{C}^{\prime \prime }(Q_{2}^{\!*\!})}\\&= \frac{2t^{2}}{6t+{C}^{\prime \prime } (Q_{1}^{*})+{C}^{\prime \prime } (Q_{2}^{*})}>0, \end{aligned}$$

which is unambiguously positive. In this sense, the function $m_{1} (x_{1},x_{2})$ reaches its minimum at $(x_{1}=0,x_{2}=1/2)$ and reaches its maximum at $(x_{1}=1/2,x_{2}=1)$:

$$\begin{aligned}&m_{1}(x_{1}=0,x_{2}=1/2)\nonumber \\&\quad \quad \quad \quad =\frac{5t}{6}+\frac{{C}^{\prime }\left(Q_{2}^{*}(x_{1}=0,x_{2}=1/2)\right)}{3} -\frac{{C}^{\prime }\left(Q_{1}^{*}(x_{1}=0,x_{2}=1/2)\right)}{3}\nonumber \\&\quad \quad \quad \quad \ge \frac{5t}{6}+\frac{{C}^{\prime }(Q_{2}^{*}=1/2)}{3}-\frac{{C} ^{\prime }(Q_{1}^{*}=1/2)}{3}=\frac{5t}{6}>0, \end{aligned}$$

(16)

$$\begin{aligned}&m_{1}(x_{1}=1/2,x_{2}=1)\nonumber \\&\quad \quad \quad \quad =\frac{7t}{6}+\frac{{C}^{\prime }\left(Q_{2}^{*}(x_{1}=1/2,x_{2}=1)\right)}{3} -\frac{{C}^{\prime }\left(Q_{1}^{*}(x_{1}=1/2,x_{2}=1)\right)}{3}\nonumber \\&\quad \quad \quad \quad \le \frac{7t}{6}+\frac{{C}^{\prime }(Q_{2}^{*}=1/2)}{3}-\frac{{C} ^{\prime }(Q_{1}^{*}=1/2)}{3}=\frac{7t}{6}<2t, \end{aligned}$$

(17)

where the first inequalities in Eqs. (16) and (17) are derived from the assumption of a convex production cost function (${C}^{\prime \prime } (Q_{i})\ge 0)$. From Eqs. (16) and (17), it follows that $0<m_{1} (x_{1},x_{2})<2t$ for all $x_{1}\in [0,1/2]$ and $x_{2}\in [1/2,1]$.

We next define a new function $m_{2}(x_{1},x_{2})$ as:

$$\begin{aligned} m_{2}(x_{1},x_{2})&= p_{2}^{*}(x_{1},x_{2})-{C}^{\prime }\left(Q_{2}^{*} (x_{1},x_{2})\right)\\&= \frac{t(4-x_{1}-x_{2})}{3}+\frac{{C}^{\prime }\left(Q_{1}^{* }(x_{1},x_{2})\right)}{3}-\frac{{C}^{\prime }\left(Q_{2}^{*}(x_{1},x_{2})\right)}{3}, \end{aligned}$$

where the second equality is derived from the first-order condition for the maximizing profit of a firm in Eq. (2). The first-order derivative of $m_{2}(x_{1},x_{2})$ with respect to $x_{i}$, for $i=1,2$, can be calculated as:

$$\begin{aligned} \frac{\partial m_{2}(x_{1},x_{2})}{\partial x_{i}}&= \frac{\partial p_{2}^{*}(x_{1},x_{2})}{\partial x_{i}}-{C}^{\prime \prime }\left(Q_{2}^{*}(x_{1},x_{2})\right)\cdot \frac{\partial Q_{2}^{*}(x_{1},x_{2})}{\partial x_{i}}\\&= \frac{\!-\!t\left[2t\!+\!{C}^{\prime \prime }(Q_{2}^{\!*\!})\right]}{6t\!+\!{C}^{\prime \prime }(Q_{1}^{\!*\!}) \!+\!{C}^{\prime \prime }(Q_{2}^{\!*\!})}\!-\!{C}^{\prime \prime } \left(Q_{2}^{\!*\!}(x_{1},x_{2})\right)\cdot \frac{\!-\!t}{6t\!+\!{C}^{\prime \prime } (Q_{1}^{*})\!+\!{C}^{\prime \prime }(Q_{2}^{\!*\!})}\\&= \frac{-2t^{2}}{6t+{C}^{\prime \prime }(Q_{1}^{*}) +{C}^{\prime \prime }(Q_{2}^{*})}<0. \end{aligned}$$

It follows that the function $m_{2}(x_{1},x_{2})$ reaches its minimum at $(x_{1}=1/2,x_{2}=1)$ and reaches its maximum at $(x_{1}=0,x_{2}=1/2)$:

$$\begin{aligned}&m_{2}(x_{1}=1/2,x_{2}=1)\nonumber \\&\quad \quad \quad \quad =\frac{5t}{6}+\frac{{C}^{\prime }\left(Q_{1}^{*}(x_{1}=1/2,x_{2}=1)\right)}{3} -\frac{{C}^{\prime }\left(Q_{2}^{*}(x_{1}=1/2,x_{2}=1)\right)}{3}\nonumber \\&\quad \quad \quad \quad \ge \frac{5t}{6}+\frac{{C}^{\prime }(Q_{1}^{*}=1/2)}{3}-\frac{{C} ^{\prime }(Q_{2}^{*}=1/2)}{3}=\frac{5t}{6}>0, \end{aligned}$$

(18)

$$\begin{aligned}&m_{2}(x_{1} =0,x_{2}=1/2)\nonumber \\&\quad \quad \quad \quad =\frac{7t}{6}+\frac{{C}^{\prime }\left(Q_{1} ^{*}(x_{1}=0,x_{2}=1/2)\right)}{3}-\frac{{C}^{\prime } \left(Q_{2}^{*}(x_{1}=0,x_{2}=1/2)\right)}{3}\nonumber \\&\quad \quad \quad \quad \le \frac{7t}{6}+\frac{{C}^{\prime }(Q_{1}^{*}=1/2)}{3}-\frac{{C} ^{\prime }(Q_{2}^{*}=1/2)}{3}=\frac{7t}{6}<2t, \end{aligned}$$

(19)

where the first inequalities in Eqs. (18) and (19) are derived from the assumption of a convex production cost function (${C}^{\prime \prime } (Q_{i})\ge 0)$. It follows that $0<m_{2}(x_{1},x_{2})<2t$ for all $x_{1} \in [0,1/2]$ and $x_{2}\in [1/2,1]$.

We next show that a corner solution, $Q_{i}=0$ for $i\in \{1,2\}$, will never be an equilibrium outcome in the price subgame.

We shall assume that this is not so, and then $\bar{Q}_{j}=1$ is a candidate equilibrium demand only if firm $j$ quotes a price $\bar{p}_{j}\ge C(1)$. Given such a price strategy of firm $j$, firm $i$ can always get positive demand by quoting the same price (i.e., $p_{i}=\bar{p}_{j})$ and get a profit:

$$\begin{aligned} \pi _{i}&= \bar{p}_{j}{\cdot }Q_{i}-C(Q_{i})\ge \left[ {C(1)\cdot Q_{i}}\right] -C(Q_{i})\nonumber \\&= \left[ {Q_{i}\cdot C(1)+(1-Q_{i})\cdot C(0)}\right] -C(Q_{i} \cdot 1+(1-Q_{i})\cdot 0)\ge 0, \end{aligned}$$

(20)

for $i,j\in \{1,2\}$ and $i\ne j, \forall 0<Q_{i}<1$, where the last inequality in Eq. (20) is derived from the definition of a convex function, and equality is satisfied only if production cost is linear. It follows that for a strictly convex production cost (${C}^{\prime \prime } (Q_{i})>0)$, firm $i$ can always guarantee a positive profit by mimicking its opponent’s strategy. Now, combine the fact that for a linear production cost (${C}^{\prime \prime }(Q_{i})=0)$, firm $i$ can guarantee positive demand and profit by quoting a slightly higher price than marginal cost. We thus conclude that a corner solution $Q_{i}=0$ is never an equilibrium outcome for a convex production cost function (${C}^{\prime \prime }(Q_{i})\ge 0)$. $\square$

Proof of Corollary 2

The derivations of a linear production cost function $C(Q_{i})=c\cdot Q_{i}$ are omitted. With a quadratic (square) production cost function $C(Q_{i})=c\cdot Q_{i}^{2}$, routine manipulation yields the results in the price subgame:

$$\begin{aligned} \begin{aligned} p_{1}^{*}&=\frac{t(cx_{1}+tx_{1}+cx_{2}+tx_{2}+4c+2t)+2c^{2}}{3t+2c},\\ p_{2}^{*}&=\frac{t(-cx_{1}-tx_{1} -cx_{2}-tx_{2}+6c+4t)+2c^{2}}{3t+2c}. \end{aligned} \end{aligned}$$

(21)

$$\begin{aligned} \begin{aligned} Q_{1}^{*}&=\frac{(tx_{1}+tx_{2}+2c+2t)}{2(3t+2c)},\\ Q_{2}^{*}&=1-Q_{1}^{*}=\frac{(-tx_{1}-tx_{2}+2c+4t)}{2(3t+2c)}. \end{aligned} \end{aligned}$$

(22)

The equilibrium profits of the duopoly firms are calculated as:¹¹

$$\begin{aligned} \begin{aligned} \pi _{1}^{*}&=\frac{(tx_{1}+tx_{2}+2c+2t)^{2}(2t+c)}{4(3t+2c)^{2} }>0,\\ \pi _{2}^{*}&=\frac{(tx_{1}+tx_{2}-2c-4t)^{2}(2t+c)}{4(3t+2c)^{2}}>0. \end{aligned} \end{aligned}$$

(23)

The price-cost margin of firm 1 is calculated as:

$$\begin{aligned} p_{1}^{*}-{C}^{\prime }(Q_{1}^{*})=\frac{t(2c+2t+tx_{1} +tx_{2} )}{3t+2c}. \end{aligned}$$

Differentiating $p_{1}^{*}-{C}^{\prime }(Q_{1}^{*})$ with respect to $x_{i}$, for $i=1,2$, yields:

$$\begin{aligned} \frac{\partial \left[p_{1}^{*}-{C}^{\prime }(Q_{1}^{*})\right]}{\partial x_{i} }=\frac{t^{2}}{3t+2c}>0, \end{aligned}$$

which is unambiguously positive. Substituting $(x_{1} =0,x_{2} =1/2)$ into $p_{1}^{*}-{C}^{\prime }(Q_{1}^{*})$ yields:

$$\begin{aligned} p_{1}^{*}(x_{1} =0,x_{2} =1/2)-{C}^{\prime }\left(Q_{1}^{*}(x_{1} =0,x_{2} =1/2)\right)=\frac{t(5t+4c)}{2(3t+2c)}>0. \end{aligned}$$

Substituting $(x_{1} =1/2,x_{2} =1)$ into $p_{1}^{*}-{C}^{\prime } (Q_{1}^{*})-2t$ yields:

$$\begin{aligned} p_{1}^{*}(x_{1} =1/2,x_{2} =1)-{C}^{\prime }\left(Q_{1}^{*}(x_{1} =1/2,x_{2} =1)\right)-2t=-\frac{t(5t+4c)}{2(3t+2c)}<0. \end{aligned}$$

It follows that $0<p_{1}^{*}-{C}^{\prime }(Q_{1}^{*})<2t$ for all separated location pairs.

The price-cost margin of firm 2 is calculated as:

$$\begin{aligned} p_{2}^{*}-{C}^{\prime }(Q_{2}^{*})=\frac{t(2c+4t-tx_{1} -tx_{2} )}{3t+2c}. \end{aligned}$$

Differentiating $p_{2}^{*}-{C}^{\prime }(Q_{2}^{*})$ with respect to $x_{i}$, for $i=1,2$, yields:

$$\begin{aligned} \frac{\partial \left[p_{2}^{*}-{C}^{\prime }(Q_{2}^{*})\right]}{\partial x_{i} }=-\frac{t^{2}}{3t+2c}<0, \end{aligned}$$

which is unambiguously negative. Substituting $(x_{1} =1/2,x_{2} =1)$ into $p_{2}^{*}-{C}^{\prime }(Q_{2}^{*})$ yields:

$$\begin{aligned} p_{2}^{*}(x_{1} =1/2,x_{2} =1)-{C}^{\prime }\left(Q_{2}^{*}(x_{1} =1/2,x_{2} =1)\right)=\frac{t(5t+4c)}{2(3t+2c)}>0. \end{aligned}$$

Substituting $(x_{1} =0,x_{2} =1/2)$ into $p_{2}^{*}-{C}^{\prime } (Q_{2}^{*})-2t$ yields:

$$\begin{aligned} p_{2}^{*}(x_{1} =0,x_{2} =1/2)-{C}^{\prime }\left(Q_{2}^{*}(x_{1} =0,x_{2} =1/2)\right)-2t=-\frac{t(5t+4c)}{2(3t+2c)}<0. \end{aligned}$$

It follows that $0<p_{2}^{*}-{C}^{\prime }(Q_{2}^{*})<2t$ for all separated location pairs.

We next define two new functions $f(x_{1},x_{2})$ and $g(x_{1},x_{2})$ as:

$$\begin{aligned} f(x_{1},x_{2})&= Q_{1}^{*}-x_{1}.\\ g(x_{1},x_{2})&= x_{2}-Q_{1}^{*}. \end{aligned}$$

The first-order derivatives of $f(x_{1},x_{2})$ and $g(x_{1},x_{2})$ with respect to $x_{2}$ are, respectively, calculated as:

$$\begin{aligned} \frac{\partial f(x_{1},x_{2})}{\partial x_{2}}&= \frac{t}{2(3t+2c)}>0.\\ \frac{\partial g(x_{1},x_{2})}{\partial x_{2}}&= \frac{5t+4c}{2(3t+2c)}>0. \end{aligned}$$

Substituting $x_{2}=1/2$ into $f(x_{1},x_{2})$ and $g(x_{1},x_{2})$, respectively, yields:

$$\begin{aligned} f(x_{1},x_{2} =1/2)&= \frac{(1-2x_{1})(5t+4c)}{4(3t+2c)}>0,\quad \forall x_{1}\in [0,1/2).\\ g(x_{1},x_{2}=1/2)&= \frac{t(1-2x_{1})}{4(3t+2c)}>0,\quad \forall x_{1}\in [0,1/2). \end{aligned}$$

It follows that $f(x_{1},x_{2})>0$ and $g(x_{1},x_{2})>0$, that is, $x_{1}<Q_{1}^{*}<x_{2}$ (and thus $0<Q_{1}^{*},Q_{2}^{*}<1)$ for all separated location pairs.

The first-order derivative of each firm’s equilibrium profit with respect to its own location is, respectively, calculated as:

$$\begin{aligned} \frac{\partial \pi _{1}^{*}}{\partial x_{1}}&= \frac{t(2t+2c+tx_{1} +tx_{2})(2t+c)}{2(3t+2c)^{2}}>0,\\ \frac{\partial \pi _{2}^{*}}{\partial x_{2}}&= \frac{-t(4t+2c-tx_{1} -tx_{2})(2t+c)}{2(3t+2c)^{2}}<0, \end{aligned}$$

for all separated location pairs. The optimal undercutting profit of firm 1 is:

$$\begin{aligned} \pi _{1}^{u}(x_{1},x_{2})=p_{2}^{*}- t(x_{2}-x_{1})-c=\frac{t(4t+3c+cx_{1} +2tx_{1}-3cx_{2}-4tx_{2})}{3t+2c}. \end{aligned}$$

Differentiating the profit difference $\pi _{1}^{*}(x_{1},x_{2})-\pi _{1} ^{u}(x_{1},x_{2})$ with respect to $x_{1}$ and $x_{2}$, respectively, we yield:

$$\begin{aligned} \frac{\partial (\pi _{1}^{*}-\pi _{1}^{u})}{\partial x_{1}}&= \frac{-t(4t+2c-tx_{1}-tx_{2})(2t+c)}{2(3t+2c)^{2}}<0,\\ \frac{\partial (\pi _{1}^{*}-\pi _{1}^{u})}{\partial x_{2}}&= \frac{t\left[t(40c+28t+cx_{1}+2tx_{1}+cx_{2}+2tx_{2})+14c^{2}\right]}{2(3t+2c)^{2}}>0, \end{aligned}$$

for all separated location pairs.

The profit difference $\pi _{1}^{*}(x_{1},x_{2})-\pi _{1}^{u}(x_{1},x_{2})$ of a limit as both firms approach the market center is calculated as:

$$\begin{aligned} \mathop {\lim }\limits _{x_{1}\rightarrow 1/2^{-} }\left[ {\pi _{1}^{*}(x_{2}\!=\!1/2)\!-\!\pi _{1}^{u}(x_{2}\!=\!1/2)}\right]&= \mathop {\lim }\limits _{x_{2}\rightarrow 1/2^{+} }\left[ {\pi _{1}^{*}(x_{1}=1/2)\!-\!\pi _{1}^{u}(x_{1}\!=\!1/2)}\right] \\&= \!-\!t/2\!+\!c/4>0,\;\text{ if}\;c>2t. \end{aligned}$$

It follows that if $c>2t$, then duopoly firms agglomerating at the center, $(x_{1}=1/2,x_{2}=1/2)$, and choosing the same price, $(p_{1}=3c/2,p_{2} =3c/2)$, are the unique PCPNE outcome. $\square$