I am preparing for an exam and I have come across this question in my textbook but I am not quite sure how to solve it so would really appreciate some help!
Suppose that company L produces left shoes and company R produces right shoes. If L charges $p_L$ for a left shoe, and R charges $p_R$ for the right shoe, then the price for a pair of shoes is $p=p_L + p_R$. The quantity of pairs of shoes purchased is determined by the demand function $q = 100 - p$. The cost of production is $c>0$ per shoe. Both firms choose prices simultaneously and independently of each other. Formulate this situation as a game (specify the players, strategies and payoff functions).
So obviously there are two players in the game, company L and company R. In terms of strategies I am slightly confused but I assume they are the following three:
- $p_R < p_L$
- $p_R = p_L$
- $p_R > p_L$
Please can you tell me is this is correct? Secondly, I am extremely confused by the payoff. I understand it would be optimal for both firms to price their shoes at the same level. And I know that to calculate the payoff we need to calculate profit.
Profit = $(100-p)\cdot q - c\cdot q$ Where I am stuck is whether it is necessary to sub in the expression that we have for q?
Finally, I believe that the payoff when prices are equal is just half the total profit for each firm. However, I have absolutely no idea what the payoff would be when the prices don't align. Any hint would be really useful! Thank you!