Answer:
the Cournot-Nash equilibrium, Simon's production is 82 units
Explanation:
The Cournot-Nash Equilibrium for Simon's production is calculated as follows:
[tex]P = 400 - Q \\ \\ Q = Q_s + Q_c[/tex]
Reaction function of Carl is as follows:
Carl maximize profit at [tex]HR_c = HC_c[/tex]
[tex]TR_c = P*Q_c[/tex]
[tex]TR_c = (400 -Q_s -Q_c)Q_c[/tex]
[tex]TR_c = 400Q_c -Q_sQ_c -Q_c^2[/tex]
⇒ [tex]HR_c = \delta TR_c/ \delta Q_c[/tex]
[tex]HR_c =400 -Q_s -2 Q_c[/tex]
[tex]C_c = 30 Q_c + Q_c^2[/tex]
⇒ [tex]HC_c = \delta C_c/ \delta Q_c[/tex]
[tex]HC_c =30+2Q_c[/tex]
Set [tex]HR_c = HC_c[/tex]
[tex]400 - Q_s - 2 Q_c = 30 - 2Q_c \\ \\ 400 - Q_s -30 = 2Q_c + 2Q_c \\\\(370 Q_s) = 4 Q_c \\ \\ Q_c = (370-Q_s)/4 \\ \\ Q_c = 92.5 - 0.25 Q_s \to Reaction \ function \ of \ Carl --- equation (1)[/tex]
Reaction function of Simon
Since Simon maximize profit at [tex]HR_s = HC_s[/tex]
[tex]TR_s = PQ_s \\ \\ TR_s = (400-Q_c -Q_s)Q_s \\ \\ TR_s = 400 Q_s - Q_cQ_s - Q_s^2[/tex]
[tex]HR_s = \delta TR_s/ \delta Q_s[/tex]
[tex]HR_s =400 - Q_c -2Q_s[/tex]
[tex]C_s = Q_s^2[/tex]
[tex]HC_s= \delta C_s/ \delta Q_s[/tex]
[tex]HC_s=2Q_s[/tex]
Set [tex]HR_s = HC_s[/tex]
[tex]400- Q_c - 2Q_s = 2Q_s \\ \\ 400 - Q_c = 2Q_s+2Q_s \\ \\ 4Q_s = 400 - Q_c \\ \\ Q_s = (4000- Q_c)/4 \\ \\ Q_s = 100 -0.25 Q_c --- Reaction \ function \ of \ Simon \ -- equation (2)[/tex]
Substituting equation (1) into equation (2)
[tex]Q_s =100 -0.25Q_c \\ \\ Q_s = 100 - 0.25(92.5-0.25 Q_s) \\ \\ Q_s = 100 -23.125 +0.0625Q_s \\ \\ (Q_s-0.0625Q_s) = 76.375 \\ \\ 0.9375 Q_s = 76.875 \\ \\ Q_s = 76.375/0.9375 \\ \\ Q_s = 82[/tex]
Thus; the Cournot-Nash equilibrium, Simon's production is 82 units
