## Abstract

In a spatial competition model, we analyze the stability of the Nash-price equilibrium under horizontal and vertical product differentiation, considering both homogenous and heterogeneous expectations. Regardless of the nature of product differentiation, assuming that firms behave according to an adaptive expectations rule, it is found that the Nash-price equilibrium is asymptotically stable. If at least one firm follows the gradient rule based on marginal profit, an increase in the adjustment speed turns out to be a source of complexity. Moreover, the influence of the locations on price stability depends on the nature of product differentiation and on the expectations scheme.

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## Notes

- 1.
- 2.
The second order conditions that ensure the existence of the maximum are the following: \( \frac{\partial^2{\prod}_1}{\partial {p_1}^2}=\frac{-1}{1-b-a}<0\kern0.5em \mathrm{and}\kern0.5em \frac{\partial^2{\prod}_2}{\partial {p_2}^2}=\frac{-1}{1-b-a}<0 \), for

*a*≥ 0,*b*≥ 0 and*a*+*b*< 1. - 3.
The second order conditions for maximum are the following: \( \frac{\partial^2{\prod}_1}{\partial {p_1}^2}=\frac{-1}{b-a}<0\ \mathrm{and}\ \frac{\partial^2{\prod}_2}{\partial {p_2}^2}=\frac{-1}{b-a}<0 \) for

*b*>*a*. - 4.
If

*a*+*b*> 4, the Nash price equilibrium is given by: \( {p}_{1V}^{\ast }=\left(b-a\right)\left(b+a-2\right),{p}_{2V}^{\ast }=0 \). See Gabszewicz and Thisse (1986). - 5.
- 6.
\( \left(0,0\right),\left(\frac{\left(1-b-a\right)\left(1-b+a\right)}{2},0\right)\ \mathrm{and}\ \left(0,\frac{\left(1-b-a\right)\left(1-b+a\right)}{2}\right) \) are unstable boundary equilibria of (14).

- 7.
The Nash equilibrium never loses its stability through either a transcritical or a Neimark-Sacker bifurcation.

- 8.
- 9.
\( \left(0,0\right)\ \mathrm{and}\ \left(\frac{\left(b-a\right)\left(a+b\right)}{2},0\right) \) are two unstable boundary fixed points of (20).

- 10.
If firm 1 is a gradient player and firm 2 is an adaptive player, the dynamic system will be \( {T}_H^{GA}:\left\{\begin{array}{l}{p}_{1,t+1}={p}_{1,t}+\alpha {p}_{1,t}\frac{p_{2,t}-2{p}_{1,t}+\left(1-b-a\right)\left(1-b+a\right)}{2\left(1-b-a\right)}\\ {}{p}_{2,t+1}={\theta}_2{p}_{2,t}+\left(1-{\theta}_2\right)\frac{p_{2,t}-2{p}_{1,t}+\left(1-b-a\right)\left(1-b+a\right)}{2\left(1-b-a\right)}\end{array}\right. \) being the equilibrium points \( {E}_H^{\ast } \) and the unstable boundary equilibrium\( \left(0,\frac{\left(1-b-a\right)\left(1+b-a\right)}{2}\right) \).

- 11.
\( \left(\frac{\left(1-b-a\right)\left(1-b+a\right)}{2},0\right) \) is an unstable boundary fixed point of (24).

- 12.
If firm 1 is a gradient player and firm 2 is an adaptive player, the dynamic system will be \( {T}_V^{GA}:\left\{\begin{array}{l}{p}_{1,t+1}={p}_{1,t}+\alpha {p}_{1,t}\frac{p_{2,t}-2{p}_{1,t}+\left(b+a\right)\left(b-a\right)}{2\left(b-a\right)}\\ {}{p}_{2,t+1}={\theta}_2{p}_{2,t}+\left(1-{\theta}_2\right)\frac{p_{2,t}-2{p}_{1,t}+\left(b+a\right)\left(b-a\right)}{2\left(b-a\right)}\end{array}\right. \)being the Nash equilibrium \( {E}_V^{\ast } \) the only steady state.

- 13.
\( \left(\frac{\left(b-a\right)\left(b+a\right)}{2},0\right) \) is an unstable boundary fixed point of (28).

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## Funding

This study was funded by the Spanish Ministry of Economics and Competitiveness (ECO2016–74940-P) and the Government of Aragon and FEDER (S10/2016 and S13/2016 Consolidated Groups) and S40-17R Reference Group.

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This paper benefited from comments made by two anonymous referees of this journal.

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Andaluz, J., Jarne, G. On price stability and the nature of product differentiation.
*J Evol Econ* **29, **741–762 (2019). https://doi.org/10.1007/s00191-018-0584-2

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### Keywords

- Horizontal product differentiation
- Vertical product differentiation
- Bounded rationality
- Dynamic stability

### JEL classification

- C62
- D43
- L13