What's the joint profit-maximizing prices for the stores.?

Question

cost of bread=1
cost of cheese=2

equations:
Q1=10 - P1 - 0.5P2
Q2=12 - 0.5P1 - P2
1. wt is the joint profit-maximizing prices for the stores
2. wt's the diff. between the nash equilibrium prices and the maximize joint profit. Why is joint profit maximization not a nash equilibrium.

Answer 1

The context of your question is lost, but I am guessing by joint profit-maximizing you mean a cartel with the alternative being a Cournot duopoly.

A monopoly cartel is optimal pricing, BUT because the change of profit versus the change in quantities is not optimized for two firms at the monopoly point, because the other player's quantity is in the optimizer's derivative. An optimizer, knowing the other player will play cartel, will optimize profit by playing Cournot. Since both players know this, Cournot becomes the optimum for both players. This is a variant on the Prisoner's delima.

You will have to solve the calculus. But from the game theory side, you have two choices, supply the joint monopoly or cartel quantity OR the Cournot duopoly quantity.

Solve the profit function and then use calculus to optimize. You will find the partial derivative of profit will include the other player's quantity.