n lectures, we considered two models of duopoly competition: Cournot(quantity) competition and Bertrand (price) competition. It seems morerealistic to think of firms’ competing in prices than in quantities, but theCournot outcome seems more ‘realistic’ than the Bertrand outcome. Thisproblem considers a third model of duopoly competition. Like Bertrand, thetwo firms will compete in prices rather than quantities. Unlike the Bertrandmodel, however, the products of the two firms are not identical. In economicsjargon, the products are di§erentiated. Instead of my solving the model onthe board in class, you will solve it in this problem set. But don’t panic: Iwill walk you through the model step by step.The Game.• We can think a ‘city’ as a line of length one.• There are two firms, 1 and 2, at either end of this line.— The firms simultaneously set prices p 1 and p 2 respectively.— Both firms have constant marginal costs, c.— Each firm’s aim is to maximize its profit.• Potential customers are evenly distributed along the line, one at eachpoint.— Let the total population be one (or, if you prefer, think of demandin terms of market shares).• Each potential customer buys exactly one unit, buying it either fromfirm 1 or from firm 2. So total demand is always exactly one.• Consider a customer at a position y on the line. She is distance y fromfirm 1 and distance (1  y) from firm 2.3— The customer at position y on the line is assumed to buy fromfirm 1 ifp 1 + ty 2 < p2 + t(1  y)2 ; (a)to buy from firm 2 ifp 1 + ty 2 > p2 + t(1  y)2 ; (b)and to toss a fair coin if this is an exact equality.Interpretation. Customers care about both price and about the ‘distance’they are from the firm. If we think of the line as representing geographicaldistance, then we can think of the t(distance)2 term as the ‘transport cost’of getting to the firm. Alternatively, if we think of the line as representingsome aspect of product quality – say, fat content in ice-cream – thenthis term is a measure of the inconvenience of having to move away fromthe customer’s most desired point. As the transport-cost parameter t getslarger, we can think of products becoming more di§erentiated from the pointof view of the customers. If t = 0 then the products are perfect substitutes.What happens?(a) (2 points) Will either firm i ever set its price p i < c? Why?(b) (3 points) Suppose that firm 2 sets price p 2 . At what price can firm1 capture the entire market (that is, given p 2 , at what p 1 will all thecustomers buy from firm 1)?