In 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)?Let’s consider if Firm 1 can do better by setting a price higher than thesolution to question (b). The downside of firm 1’s setting a higher price isthat it will lose some of the market. The upside is that it will charge moreto any customer it keeps. The next question gets you to work out just howmany customers buy from firm 1 when the prices are ‘close’.(c) (10 points) Suppose that prices p 1 and p 2 are close enough that themarket is split between the two firms. Use expressions (a) and (b)above to find the location of the customer who is exactly indi§erentbetween buying from firm 1 and buying from firm 2. Use your answerto argue that, when the market is split, firm 1’s demand is given by:D1 (p 1 , p2 ) = p 2 + t  p 12t (1)4We now have all the information we need to calculate firm 1’s best responseto each p 2 . When the market is split, firm 1’s profits are given byu 1 (p 1 , p2 ) = (p 1  c) D1 (p 1 , p2 )= p 1 (p 2 + t + c)  p 21  c (p 2 + t)2t (2)Notice it follows from expressions (1) and (2), that for intermediate levelsof p 2 , the best response of firm 1 to firm 2 setting some intermediate pricep 2 is to set a price p 1 that solvesmaxp1p 1 (p 2 + t + c)  p 21  c (p 2 + t)2t(d) (5 points) Given that for intermediate levels of p 2 , that the (par-tial) derivative of u 1 (p 1 , p2 ) with respect to p 1 isp 2 + t + c  2p 12tshow that the best response for firm 1 for intermediate levels of p 2 ,can be expressed asBR 1 (p 2 ) = p 2 + t + c2(e) (15 points) Draw a picture of the best responses of firms 1 and 2.Be careful to indicate in your picture what happens to BR 1 (p 2 ) whenp 2 < ct, and when p 2 > 3t+c. [Hint: recall your answers to parts (a)and (b) above]. Draw the best response BR 2 (p 1 ) on the same picture.(f) (10 points) Use algebra to find the Nash equilibrium.(g) (5 points) What is the equilibrium price when t = 0? Interpret youranswer. People sometimes say ‘competition gets less fierce as productsbecome less similar and more di§erentiated’. How does this show upin our model?