y1(120 y1 y2) 30y1.Taking the derivative of this profit with respect to y1 (holding y2 constant) and setting the derivative equal to zero we obtain
120 2y1 y2 30 = 0,or
y1 = (90 y2)/2.Thus the best response function of firm 1 is given by b1(y2) = (90 y2)/2. This function is shown in the following figure.
Similarly, we find that the best response function of firm 2 is given by b2(y1) = (90 y1)/2. This function is superimposed on the best response function of firm 1 in the following figure.
y1 = b1(y2) and y2 = b2(y1).That is,
y1 = (90 y2)/2 and y2 = (90 y1)/2.Substituting one equation in the other we obtain y1 = (90 (90 y1)/2)/2, so that y1 = 30; substituting in the equation for y2 we get y2 = 30.
y1(120 y1 y2) y12.Taking the derivative of this profit with respect to y1 (holding y2 constant) and setting the derivative equal to zero we obtain
120 2y1 y2 2y1 = 0,or
y1 = (120 y2)/4.Thus the best response function of firm 1 is given by b1(y2) = (120 y2)/4.
Similarly, we find that the best response function of firm 2 is given by b2(y1) = (120 y1)/4.
y1 = b1(y2) and y2 = b2(y1).That is,
y1 = (120 y2)/4 and y2 = (120 y1)/4.Substituting one equation in the other we obtain y1 = (120 (120 y1)/4)/4, so that y1 = 24; substituting in the equation for y2 we get y2 = 24.
[Solution]