Two supermarkets, A and B, are located 6 miles apart along a road on which an unit mass of consumers are evenly distribu

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Two supermarkets, A and B, are located 6 miles apart along a
road on which an unit mass of consumers are evenly distributed. It
costs consumers 50p per mile to travel to the supermarket. Each
consumer wants to buy a basket of goods that costs the supermarkets
£21 to purchase. The supermarkets have no other variable costs.
a) Suppose that the supermarkets compete in a Bertrand-Nash
manner; that is they each set price without knowing the other
supermarket’s price. What are the optimal prices they will set?
What are the supermarkets’ profit in this scenario?
b)Before competing in price, supermarkets can simultaneously
invest to re- duce the marginal costs of each basket. For each
supermarket, to reduce the marginal cost to c, firms need to invest
(21−c)^2. What are the optimal investment decisions for each
firm?