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Part 5 The purpose of this exercise is to study Betrand-Nash equilibria when some consumers do not pay attention to small price differences and to relate the "Bertrand paradox" to consumers' sensitivity to price differences. There are two firms that produce a homogeneous product. Let p, be the price of firm i (i = 1, 2). Assume for simplicity that the firms have zero costs. There are N consumers with unit demands, each with the same reservation price for the product, denoted by r. That is, each consumer buys one unit at most from one of the two firms if and only if at least one of the prices is less than or equal to r. When pi = p2 5 r, 50% of the consumers go to firm 1 and 50% to firm 2. What happens when the two prices are different? Some consumers are very sensitive to price differences, while others are not. For example, if pi = p2 + 0.01 then some consumers might prefer firm 2 because they save 1 penny, but probably most consumers would be indifferent between the two firms. Let f : [0, r] - R+ be a density function (thus f(z) 2 0, for all r E [0, r], and / f(x)da = 1) that measures the price-difference sensitivity of consumers. For example, suppose that p1 0 where r; is the consumption of private good, y the quantity of public good, and y