I'm supposed to find the subgame Nash equilibrium but I'm stuck. Would appreciate the help.
We were presented with a Cournot duopoly problem (select quantities) and are basically asked to solve it as a Bertrand problem (select prices). Firm 1 and Firm 2 compete to sell a good, and they have advertising budgets.
$Q = a - p$ where a is advertising and p is the price. The firms' costs are $2a^3 / 81$
Therefore, total profits are: $(a-p)p - 2a^3 / 81$
Given the information, I'm asked to find the subgame perfect equilibrium. With this I'm supposed to derive and set equal to 0.
Because the lowest price receives all the customers, I'm going to assume that each firm will sell at the same price- is this wrong to do? Therefore, I'll get:
$(a-p)0.5p - 2a^3 / 81$ and the derivative with respect to p should be: $0.5a - p=0$ This makes the optimal $ p = 2a$ right?
To find the optimal a, we must therefore do: $4a^2 -2a^3/81=0$ but this makes $a = 162$
That doesn't really make sense at all. What did I do wrong? Appreciate the help