The reaction functions for the two duopolists, firms X and Y are given by the equations P t^X = 45 + 0.8 \, P t 1^Y and P t^Y = 45 + 0.8P t 1^X , respectively. If the assumptions of the Bertrand…

The reaction functions for the two duopolists, firms X and Y are given by the equations $P_t^X=45+0.8P_{t-1}^Y$ and $P_t^Y=45+0.8P_{t-1}^X$, respectively. If the assumptions of the Bertrand model hold, derive a difference equation for $P_t^X$ and calculate what $P_t^X$ will be in time period 10 if firm X starts off in time period 0 by setting a price of 300.