- First find the firms' best response functions. Firm 1's profit is
*y*_{1}(120*y*_{1}*y*_{2}) 30*y*_{1}.*y*_{1}(holding*y*_{2}constant) and setting the derivative equal to zero we obtain120 2

or*y*_{1}*y*_{2}30 = 0,*y*_{1}= (90*y*_{2})/2.*b*_{1}(*y*_{2}) = (90*y*_{2})/2. This function is shown in the following figure.Similarly, we find that the best response function of firm 2 is given by

*b*_{2}(*y*_{1}) = (90*y*_{1})/2. This function is superimposed on the best response function of firm 1 in the following figure. - We now need to find a pair (
*y*_{1},*y*_{2}) of outputs with the property that*y*_{1}=*b*_{1}(*y*_{2}) and*y*_{2}=*b*_{2}(*y*_{1}).*y*_{1}= (90*y*_{2})/2 and*y*_{2}= (90*y*_{1})/2.*y*_{1}= (90 (90*y*_{1})/2)/2, so that*y*_{1}= 30; substituting in the equation for*y*_{2}we get*y*_{2}= 30.

- First find the firms' best response functions. If firm 1 chooses the output
*y*_{1}its profit is*y*_{1}(120*y*_{1}*y*_{2})*y*_{1}^{2}.*y*_{1}(holding*y*_{2}constant) and setting the derivative equal to zero we obtain120 2

or*y*_{1}*y*_{2}2*y*_{1}= 0,*y*_{1}= (120*y*_{2})/4.*b*_{1}(*y*_{2}) = (120*y*_{2})/4.Similarly, we find that the best response function of firm 2 is given by

*b*_{2}(*y*_{1}) = (120*y*_{1})/4. - We now need to find a pair (
*y*_{1},*y*_{2}) of outputs with the property that*y*_{1}=*b*_{1}(*y*_{2}) and*y*_{2}=*b*_{2}(*y*_{1}).*y*_{1}= (120*y*_{2})/4 and*y*_{2}= (120*y*_{1})/4.*y*_{1}= (120 (120*y*_{1})/4)/4, so that*y*_{1}= 24; substituting in the equation for*y*_{2}we get*y*_{2}= 24.

[Solution]

Copyright © 1997 by Martin J. Osborne