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# payoffs for an advertising game between Coke and Pepsi

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A monopolist has demand and cost curves given by:
QD = 10,000 - 20P
TC = 1,000 + 10Q + .05Q2

a. Find the monopolist's profit-maximizing quantity and price.
b. Find the monopolist's profit.

The following matrix shows the payoffs for an advertising game between Coke and Pepsi. The firms can choose to advertise or to not advertise. Numbers in the matrix represent profits; the first number in each cell is the payoff to Coke. (Numbers in millions.)

Don't Advertise (-50, 500) (100, 100)

a. Explain why this would be described as a Prisoner's Dilemma game.
b. Explain the probable outcome of this game.

##### Solution Summary

Prisoner's Dilemma is depicted in this case. A payoff matrix is given for an advertising game between Coke and Pepsi.

##### Solution Preview

Question 1
a. First, let's write the demand functions as a function of Q:

Q = 10000 - 20P
P = - Q/20 + 10000/20 = -Q/20+ 500

Now, we have that total revenue for the monopolist is PQ (price times quantity sold), and the total cost is given by the TC function. Therefore, total profits are:

Profit = PQ - 1,000 - 10Q - .05Q^2

Replacing P with the one we found in the demand function:

Profit = (-Q/20+ 500)Q - 1,000 - 10Q - .05Q^2
Profit = -(Q^2)/20+ 500Q - 1,000 - 10Q - .05Q^2

Now we want to maximize profits. The usual procedure for doing this is to find the derivative of the profits function with ...

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