Asymmetric Payoffs and Spatial Competition

Abstract

We investigate the location choice of two firms whose objectives are the weighted average of their own profit and social welfare, in which they simultaneously decide their locations before setting their prices. The purpose of this paper is to examine whether the asymmetric locations are influenced by the asymmetry of the firms’ objectives or by the asymmetry of firms’ marginal costs. We show that, when both firms have the same marginal cost, the equilibrium locations are always symmetric even in the case of the asymmetric objectives. On the other hand, the cost differences lead the asymmetric locations in equilibrium. That is, the asymmetric locations are a result of the cost asymmetry, but not the asymmetry of the firms’ objectives. We also demonstrate that the pursuit of profit by the cost-inefficient firm may increase consumer surplus.

Appendix

Proof of Proposition 1

If both firms maximize social welfare ( α ₁ = α ₂=0), it is well known that the social optimal location ($\phantom {\dot {i}\!}l_{i}^{\ast }=1/4$) is achieved, in equilibrium. We analyze the equilibrium locations for the rest of the parameter space below. The first-order condition with respect to the location l _i is given by:

$$\begin{array}{@{}rcl@{}} \frac{\partial \hat O_{i}}{\partial l_{i}}&=&\frac{t (1+\alpha_{i}) A_{i} B_{i}}{4 (\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2}}=0, \\ \text{where}&& A_{i} = (1+l_{i}-l_{j})(\alpha_{i}+\alpha_{j})+(1-l_{i}+l_{j})\alpha_{i}\alpha_{j} -\frac{(\alpha_{i} + \alpha_{j} - \alpha_{i} \alpha_{j})(c_{i}-c_{j})}{t(1-l_{i}-l_{j})}, \\ &&B_{i} = (1-3l_{i} - l_{j})(\alpha_{i}+\alpha_{j})-(3-3l_{i}-l_{j})\alpha_{i}\alpha_{j} -\frac{(\alpha_{i} + \alpha_{j} - \alpha_{i} \alpha_{j})(c_{i}-c_{j})}{t(1-l_{i}-l_{j})}. \end{array} $$

In the case of $\phantom {\dot {i}\!}\hat x_{i} >0$, it is easy to show that A _i >0. Therefore, the equilibrium locations $\phantom {\dot {i}\!}(l_{1}^{\ast },l_{2}^{\ast })$ have to satisfy the following system of equations:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} B_{1} = (1-3l_{1} - l_{2})(\alpha_{1}+\alpha_{2})-(3-3l_{1}-l_{2})\alpha_{1} \alpha_{2} -\frac{(\alpha_{1} + \alpha_{2} - \alpha_{1} \alpha_{2})(c_{1}-c_{2})}{t(1-l_{1}-l_{2})} =0, \\ B_{2} = (1-l_{1} - 3l_{2})(\alpha_{1}+\alpha_{2})-(3-l_{1}-3l_{2})\alpha_{1} \alpha_{2} -\frac{(\alpha_{1} + \alpha_{2} - \alpha_{1} \alpha_{2})(c_{2}-c_{1})}{t(1-l_{1}-l_{2})} =0. \end{array}\right. \end{array} $$

Solving these for the firms’ locations yields the following symmetric equilibrium:

$$\begin{array}{@{}rcl@{}} l_{i}^{\ast}=\frac{\alpha_{i}+\alpha_{j}-3\alpha_{i} \alpha_{j}}{4(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})} + \frac{(c_{j}-c_{i}) (\alpha_{i} +\alpha_{j}-\alpha_{i} \alpha_{j})}{t(\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})},~~(i=1,2,~j \neq i). \end{array} $$

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Proof of Proposition 2

Differentiating $\phantom {\dot {i}\!}l_{i}^{\ast }$ and $\phantom {\dot {i}\!}l_{j}^{\ast }$ with respect to α _i yields:

$$\begin{array}{@{}rcl@{}} \frac{\partial l_{i}^{\ast}}{\partial \alpha_{i}} &=& -\frac{{\alpha_{j}^{2}}}{2(\alpha_{i}+\alpha_{j}-\alpha_{i}\alpha_{j})^{2}} - \frac{2(c_{j} -c_{i}) {\alpha_{j}^{2}}}{t(\alpha_{i}+\alpha_{j}+\alpha_{i}\alpha_{j})^{2}}, \\ \frac{\partial l_{j}^{\ast}}{\partial \alpha_{i}} &=& -\frac{{\alpha_{j}^{2}}}{2(\alpha_{i}+\alpha_{j}-\alpha_{i}\alpha_{j})^{2}} + \frac{2(c_{j} -c_{i}) {\alpha_{j}^{2}}}{t(\alpha_{i}+\alpha_{j}+\alpha_{i}\alpha_{j})^{2}}. \end{array} $$

In the case of the same marginal cost, c _i = c _j , it is easy to see that $\partial l_{i}^{\ast } / \partial \alpha _{i} < 0$ and $\partial l_{j}^{\ast } / \partial \alpha _{i} < 0$. When firm i has a smaller marginal cost, c _i <c _j , it can be shown that $\phantom {\dot {i}\!}\partial l_{i}^{\ast } / \partial \alpha _{i} < 0$. The sign of $\partial l_{j}^{\ast } / \partial \alpha _{i}$ is as follows:

$$\begin{array}{@{}rcl@{}} \frac{\partial l_{j}^{\ast}}{\partial \alpha_{i}} &=& -\frac{{\alpha_{j}^{2}}}{2(\alpha_{i}+\alpha_{j}-\alpha_{i}\alpha_{j})^{2}} + \frac{2(c_{j} -c_{i}) {\alpha_{j}^{2}}}{t(\alpha_{i}+\alpha_{j}+\alpha_{i}\alpha_{j})^{2}} \\ &<& -\frac{{\alpha_{j}^{2}}}{2(\alpha_{i}+\alpha_{j}-\alpha_{i}\alpha_{j})^{2}} + \frac{2 {\alpha_{j}^{2}}}{(\alpha_{i}+\alpha_{j}+\alpha_{i}\alpha_{j})^{2}} \cdot \frac{1}{4} \left( \frac{\alpha_{i}+\alpha_{j}+\alpha_{i}\alpha_{j}}{\alpha_{i}+\alpha_{j}-\alpha_{i}\alpha_{j}} \right)^{2} ~~\left( t > \underline t \right) \\ &=& 0. \end{array} $$

Furthermore, in the case of c _i > c _j , it can be similarly proved that $\phantom {\dot {i}\!}\partial l_{i}^{\ast } / \partial \alpha _{i} < 0$ and $\phantom {\dot {i}\!}\partial l_{j}^{\ast } / \partial \alpha _{i} < 0$. □

Proof of Proposition 3

$$\begin{array}{@{}rcl@{}} 0< x_{i}^{\ast} < 1 &\iff& -\frac{1}{2} < \frac{2(c_{j} -c_{i})(\alpha_{i} +\alpha_{j} -\alpha_{i} \alpha_{j})^{2}}{t(\alpha_{i} +\alpha_{j} +\alpha_{i} \alpha_{j})^{2}} < \frac{1}{2} \\ &\iff& 0 \leq \frac{2|c_{i} -c_{j} |(\alpha_{i} +\alpha_{j} -\alpha_{i} \alpha_{j})^{2}}{t(\alpha_{i} +\alpha_{j} +\alpha_{i} \alpha_{j})^{2}}< \frac{1}{2} \\ &\iff& t > 4|c_{i} -c_{j} | \left( \frac{\alpha_{i} +\alpha_{j} -\alpha_{i} \alpha_{j}}{\alpha_{i} +\alpha_{j} +\alpha_{i} \alpha_{j}} \right)^{2} \equiv \underline t \end{array} $$

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Proof of Proposition 4

First, it can be shown as follows that both firms increase their prices as the cost-efficient firm puts a greater weight on its own profit.

$$\begin{array}{@{}rcl@{}} \frac{\partial p_{i}^{\ast}}{\partial \alpha_{i}} &=& \frac{\alpha_{j} \left[ t(\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2} +4(c_{j} -c_{i})(1+\alpha_{j})(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2} \right]}{(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2} (\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2}} >0~~~~~~~ \\ \frac{\partial p_{j}^{\ast}}{\partial \alpha_{i}} &=& \frac{\alpha_{j} \left[ t(\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2} +4(c_{j} -c_{i})(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2} \right]}{(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2} (\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2}} >0 \end{array} $$

Next, we can derive the following results regarding the signs of the derivatives of prices with respect to the weight the cost-inefficient firm puts on its profit:

$$\begin{array}{@{}rcl@{}} \frac{\partial p_{i}^{\ast}}{\partial \alpha_{j}} &=& \frac{\alpha_{i} \left[ t \alpha_{i} (\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2} - 4(c_{j} -c_{i})(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2} \right]}{(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2} (\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2}} >0, \\ &\iff& t > \underline t \cdot \frac{1}{\alpha_{i}}, \\ \frac{\partial p_{j}^{\ast}}{\partial \alpha_{j}} &=& \frac{\alpha_{i} \left[ t \alpha_{i} (\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2} - 4(c_{j} -c_{i})(1+\alpha_{i})(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2} \right]}{(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2} (\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2}} >0, ~~~~~~~\\ &\iff& t > \underline t \cdot \frac{1+\alpha_{i}}{\alpha_{i}}. \end{array} $$

Finally, we summarize these results as follows:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lll} \frac{\partial p_{i}^{\ast}}{\partial \alpha_{j}}>0 ~~ \& ~~ \frac{\partial p_{j}^{\ast}}{\partial \alpha_{j}}>0 &\text{if}~~\underline t \cdot \frac{1+\alpha_{i}}{\alpha_{i}} < t, \\ \frac{\partial p_{i}^{\ast}}{\partial \alpha_{j}}>0 ~~ \& ~~ \frac{\partial p_{j}^{\ast}}{\partial \alpha_{j}}<0 &\text{if}~~\underline t \cdot \frac{1}{\alpha_{i}} < t < \underline t \cdot \frac{1+\alpha_{i}}{\alpha_{i}}, \\ \frac{\partial p_{i}^{\ast}}{\partial \alpha_{j}}<0 ~~ \& ~~ \frac{\partial p_{j}^{\ast}}{\partial \alpha_{j}}<0 &\text{if}~~\underline t < t < \underline t \cdot \frac{1}{\alpha_{i}} . \end{array}\right. \end{array} $$

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Proof of Proposition 5

Suppose that firm i is more efficient than firm j, c _i <c _j . The derivatives of the equilibrium demands with respect to the parameter α _i and α _j are given by:

$$\begin{array}{@{}rcl@{}} \frac{\partial x_{i}^{\ast}}{\partial \alpha_{i}} &=& - \frac{8(c_{j}-c_{i})(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j}){\alpha_{j}^{2}}}{t(\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{3}} <0, \\ \frac{\partial x_{j}^{\ast}}{\partial \alpha_{i}} &=& - \frac{\partial x_{i}^{\ast}}{\partial \alpha_{i}} >0, \\ \frac{\partial x_{i}^{\ast}}{\partial \alpha_{j}} &=& - \frac{8(c_{j}-c_{i})(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j}){\alpha_{i}^{2}}}{t(\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{3}} <0, \\ \frac{\partial x_{j}^{\ast}}{\partial \alpha_{j}} &=& - \frac{\partial x_{i}^{\ast}}{\partial \alpha_{j}} >0. \end{array} $$

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Proof of Proposition 6

Suppose that firm i is more efficient than firm j, c _i <c _j . The derivative of $\pi _{i}^{\ast } = (p_{i}^{\ast } -c_{i}) x_{i}^{\ast }$ with respect to the parameter α _i is given by:

$$\begin{array}{@{}rcl@{}} \frac{\partial \pi_{i}^{\ast}}{\partial \alpha_{i}} &=& \frac{\alpha_{j} N_{ii}}{2t(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2}(\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{4}}, \\ \text{where}~~N_{ii} &=& t^{2} \alpha_{j} (\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{4} \\ &&+ 4t (c_{j}-c_{i})(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2}(\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2} \\ && + 16(c_{j}-c_{i})^{2}(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{3} \left\{ \alpha_{i}(1-\alpha_{j}) + \alpha_{j} (1-\alpha_{i}) + 3(1-\alpha_{i}) {\alpha_{j}^{2}} \right\}.~~~~~~~ \end{array} $$

It is easy to see that N _{i i} >0, which in turn provides $\phantom {\dot {i}\!}\partial \pi _{i}^{\ast } / \partial \alpha _{i} >0$. Similarly, the derivative of $\phantom {\dot {i}\!}\pi _{j}^{\ast }$ with respect to the parameter α _i is given by:

$$\begin{array}{@{}rcl@{}} \frac{\partial \pi_{j}^{\ast}}{\partial \alpha_{i}} &=& \frac{\alpha_{j} N_{ji}}{2t(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2}(\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{4}}, \\ \text{where}~~N_{ji} &=& t^{2} \alpha_{j} (\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{4} \\ &&+ 4t (c_{j}-c_{i})(1+\alpha_{j})(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{2}(\alpha_{i}+\alpha_{j}+\alpha_{i} \alpha_{j})^{2} \\ && - 16(c_{j}-c_{i})^{2}(\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})^{3} \left\{ \alpha_{i} + \alpha_{j} -3 \alpha_{i} \alpha_{j} +2(1+\alpha_{i}){\alpha_{j}^{2}} \right\}.~~~~~~~ \end{array} $$

As N _{j i} is increasing in t and $\phantom {\dot {i}\!}t> \underline t$, it can be shown that $N_{ji} > N_{ji}|_{t= \underline t}=64(c_{j}-c_{i})^{2}\alpha _{i} \alpha _{j} (1-\alpha _{j})(\alpha _{i}+\alpha _{j}-\alpha _{i} \alpha _{j})^{3} \geq 0$. Thus, we can show that $\phantom {\dot {i}\!}\partial \pi _{j}^{\ast } / \partial \alpha _{i} > 0$. □

Proof of Proposition 7

The partial derivative of the equilibrium consumer surplus with respect to α _i is given by:

$$\begin{array}{@{}rcl@{}} \frac{\partial CS^{\ast}}{\partial \alpha_{i}} &=& -\frac{t {\alpha_{j}^{2}} (2\alpha_{i}+2\alpha_{j}-\alpha_{i} \alpha_{j})}{2(\alpha_{i}+\alpha_{j}-\alpha_{i}\alpha_{j})^{3}} - \frac{2(c_{j}-c_{i})\alpha_{j} (2+\alpha_{j})}{(\alpha_{i}+\alpha_{j}+\alpha_{i}\alpha_{j})^{2}} \end{array} $$

(1)

$$\begin{array}{@{}rcl@{}} && -\frac{8(c_{i}-c_{j})^{2} {\alpha_{j}^{2}} (\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})\{ \alpha_{i} (1-\alpha_{j})+\alpha_{j} (1-\alpha_{i}) \}}{t(\alpha_{i} +\alpha_{j} +\alpha_{i} \alpha_{j})^{4}}. \end{array} $$

It is easy to show that, when firm i is more efficient ( c _i <c _j ), the right-hand-side of Eq. (1) becomes negative. Similarly, we can derive the partial derivative of the equilibrium consumer surplus with respect to α _j as follows:

$$\begin{array}{@{}rcl@{}} \frac{\partial CS^{\ast}}{\partial \alpha_{j}} &=& -\frac{t {\alpha_{i}^{2}} (2\alpha_{i}+2\alpha_{j}-\alpha_{i} \alpha_{j})}{2(\alpha_{i}+\alpha_{j}-\alpha_{i}\alpha_{j})^{3}} + \frac{2(c_{j}-c_{i})\alpha_{i} (2+\alpha_{i})}{(\alpha_{i}+\alpha_{j}+\alpha_{i}\alpha_{j})^{2}} \end{array} $$

(2)

$$\begin{array}{@{}rcl@{}} && -\frac{8(c_{i}-c_{j})^{2} {\alpha_{i}^{2}} (\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})\{ \alpha_{i} (1-\alpha_{j})+\alpha_{j} (1-\alpha_{i}) \}}{t(\alpha_{i} +\alpha_{j} +\alpha_{i} \alpha_{j})^{4}}. \end{array} $$

The partial derivative of Eq. (2) yields that ∂ C S ^∗/∂ α _j is a decreasing function in the parameter $\phantom {\dot {i}\!}t \in ( \underline t, \infty )$ as follows:

$$\begin{array}{@{}rcl@{}} \frac{\partial}{\partial t} \left( \frac{\partial CS}{\partial \alpha_{j}} \right) &=& -\frac{{\alpha_{i}^{2}} (2\alpha_{i} +2\alpha_{j} -\alpha_{i} \alpha_{j})}{2(\alpha_{i} +\alpha_{j}-\alpha_{i} \alpha_{j})^{3}} \\ && +\frac{8(c_{i}-c_{j})^{2} {\alpha_{i}^{2}} (\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})\{ \alpha_{i} (1-\alpha_{j})+\alpha_{j} (1-\alpha_{i}) \}}{t^{2}(\alpha_{i} +\alpha_{j} +\alpha_{i} \alpha_{j})^{4}} \\ &<& -\frac{{\alpha_{i}^{2}} (\alpha_{i} +\alpha_{j} +\alpha_{i} \alpha_{j})}{2(\alpha_{i} +\alpha_{j} -\alpha_{i} \alpha_{j})^{3}}~~~(t > \underline t) \\ &<& 0. \end{array} $$

We can additionally show the following two inequalities.

$$\begin{array}{@{}rcl@{}} \frac{\partial CS^{\ast}}{\partial \alpha_{j}} \left( t = \underline t \right) &=& \frac{4(c_{j}-c_{i})\alpha_{i} (1+2\alpha_{i})}{(\alpha_{i} + \alpha_{j} +\alpha_{i} \alpha_{j})^{2}}>0 \\ \lim\limits_{t \to \infty} \partial CS^{\ast} / \partial \alpha_{j} &=& -\infty \end{array} $$

Therefore, we can prove that there exists a threshold $\phantom {\dot {i}\!}\hat t$ such that the consumer surplus is increasing in the parameter α _j at the range of $\phantom {\dot {i}\!}t \in (\underline t, \hat t~]$, and is decreasing in the parameter α _j at the range of $\phantom {\dot {i}\!}t \in [\hat t, \infty )$. □

Proof of Proposition 8

The partial derivative of the equilibrium social welfare with respect to α _i is given by:

$$\begin{array}{@{}rcl@{}} \frac{\partial W^{\ast}}{\partial \alpha_{i}}=-\frac{t \alpha_{i} {\alpha_{j}^{3}}}{2(\alpha_{i}+\alpha_{j}-\alpha_{i}\alpha_{j})^{3}} -\frac{24(c_{i}-c_{j})^{2} \alpha_{i} {\alpha_{j}^{3}} (\alpha_{i}+\alpha_{j}-\alpha_{i} \alpha_{j})}{t(\alpha_{i} +\alpha_{j} +\alpha_{i} \alpha_{j})^{4}}<0. \end{array} $$

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