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Un problema de Oligopolio: modelo Cournot
February 20th, 2009 by Guillermo Pereyra

 Rubén nos envía este problema:

 

Un mercado presenta una función de demanda

P = 1000 - Q

En él trabajan dos oligopolistas cuyas funciones de costes respectivas son las siguientes:
 
CT_{1}=10+130Q_{1}

CT_{2}=10+170Q_{2}

Se pide calcular las respectivas funciones de reacción, y la cantidad que cada uno suministra si siguen un modelo de Cournot.
 

Los costos marginales de cada duopolista son constantes

CMg_{1}=130

 CMg_{2}=170

 Para encontrar la solución a la Cournot, vamos a analizar primero, la conducta estratégica de la empresa 1. Como la demanda del mercado es

P = 1000 - Q

La demanda residual de la empresa 1 es

P=1000-Q_{2}-Q_{1}

Y el ingreso total es

 IT_{1} = 1000Q_{1} - Q_{1} Q_{2}-Q_{1}^{2}

Y el ingreso marginal es

 IMg_{1} = 1000- Q_{2}-2Q_{1}

Ahora igualamos el ingreso marginal con el costo marginal y obtenemos

 1000- Q_{2}-2Q_{1}=130

Y de aquí obtenemos la función de reacción de la empresa 1

 Q_{1}=435-\frac{Q_{2}}{2}

 Seguimos el mismo procedimiento para obtener la función de reacción de la empresa 2:

La demanda residual de la empresa 2 es

P=1000-Q_{1}-Q_{2}

Y el ingreso total es

 IT_{2} = 1000Q_{2} - Q_{1} Q_{2}-Q_{2}^{2}

Y el ingreso marginal es

 IMg_{2} = 1000- Q_{1}-2Q_{2}

Ahora igualamos el ingreso marginal con el costo marginal y obtenemos

 1000- Q_{1}-2Q_{2}=170

Y de aquí obtenemos la función de reacción de la empresa 1

 Q_{2}=415-\frac{Q_{1}}{2}

 En el siguiente grafico se muestran las funciones de reacción de los duopolistas. El punto de intersección entre ellas es el equilibrio Cournot. Resolviendo las ecuaciones de las funciones de reacción

 Q_{1}=435-\frac{Q_{2}}{2}

 Q_{2}=415-\frac{Q_{1}}{2}

Se obtiene Q_{1}=303.33Q_{2}=263.33Q=566.66 y el precio de venta sería P=1000-566.66=433.33


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