Consider the following version of the Bertrand model with differentiated products. Specifically, if player I sets price $p_1$ and player II sets price $p_2$ for goods, then the demand is given by
$$q_1(p_1,p_2)=(14−p_1+p_2)+$$ for player I, and $$q_2(p_1,p_2)=(14−p_2+p_1)+$$ for player II.
Thus, the profit to each player, $u_i$, is given by $$u_i(p_1,p_2)=(p_i−9)⋅q_i(p_1,p_2)\qquad\;(i=1,2).$$
Further, we assume that player I sets his price first; player II knows this price, and sets his price accordingly. Give the strategic equilibrium strategy for this model.
I keep getting 23 for $p_1$ and $p_2$ but this answer isn't correct.
To find the subgame perfect equilibrium, we first look at, given a price $p_1$, what price player 2 would choose. This is what player 2 would play in the subgame after they know player 1's price.
Player 2 maximizes: $$(p_2-9)(14-p_2+p_1)$$ which has FOC $$14-2p_2+p_1+9=0$$ So, given $p_1$, player 2 plays $\frac{23+p_1}{2}$.
Now, player 1 anticipates this, he maximizes $$(p_1-9)\left(14-p_1+\frac{23+p_1}{2}\right)$$ which gives us the FOC $$14-p_1+\frac{23+p_1}{2}-\frac{1}{2}p_1 +\frac{9}{2}=0$$ So, in equilibrium, $p_1=30$ and $p_2(p_1)=\frac{23+p_1}{2}$. On the path of play, $p_2=26.5$.
Edit: Sorry, mixed up a + and a -