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:


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



 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


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


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

La demanda residual de la empresa 2 es


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


 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



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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