In many oligopolistic markets, firms offer similar but not exactly the same products (called differentiated products). Examples include products offered by Coles and Woolworths, Coke and Pepsi, Pizza Hut and Domino's, Apple and Samsung, Qantas and Virgin, just to name a few.

In such markets, price interdependence is prevalent: When one firm adjusts its prices, other firms tend to adjust their prices in response due to the substitution effect. This exercise helps you understand the phenomenon of price interdependence and how the objective of profit maximization guides changes in firms pricing strategies when market conditions change. Consider the following highly stylized model on price competition.

Suppose there are only two supermarkets in Australia: Coles and Woolworths, and they sell only one product: milk. Let us further assume that after graduating from Deakin, you work at Coles as a business analyst, and you are in charge of its pricing strategy. Based on historical sales data, you estimate the demand function of Coles milk is

Q^c (P^C, P^W) = 7.6 10P^C + 8P^W

Where P^C is the price of Coles milk and P^W is the price of Woolworths milk. You view Woolworths as an equivalent rival of Coles. Accordingly, you estimate the demand function of Woolworths milk is

Q^W (P^C, P^W) = 7.6 10P^W + 8P^C

Coles faces the following total cost function:

TC (Q^C) = (FMP + V) Q^C + 1

where FMP represents the average farmgate milk price (the wholesale cost of milk before processing) and V represents all other variable costs per unit. Similarly, you estimate that Woolworths faces the following total cost function

TC (Q^W) = (FMP + V) Q^W + 1

Suppose currently we have FMP = $0.54 and V = $0.26. Each Firm chooses one of the following two price points for its milk: $1.3 or $1.5. Your objective as a Coles business analyst is to set the price of Coles milk correctly to maximize Coles profits.

Construct the payoff matrix for this pricing game.

Suppose Coles and Woolworths set their prices simultaneously and independently. Find all Nash equilibria of this game. Based on the notion of Nash equilibrium, determine the price point you should set for Coles milk. Explain.

Based on your answers in (a)-(b), discuss whether the price competition between Coles and Woolworths is strategically equivalent to the prisoners dilemma. If Coles and Woolworths are able to agree on playing the best equilibrium (from the firms perspective), what would be the price set by each firm?