1. a) Suppose the person indifferent between purchasing from either firm is located at point x* which lies x* (<1) KM from the left endpoint of the street. The indifference implies for this person, U1(x*) = U2(x*) => 5 – P1- x* = 5 – P2 – (1- x*) => x* = ? + (P2 – P1)/2 1. b) Since the consumer located x* KM to the right of the origin is the marginal consumer who is indifferent in purchasing from either firm, all consumers lying to the left shall prefer buying from firm one while those to the right will prefer the other firm.
Now, all individuals lying to the left of x1 shall purchase the bottle of water from firm 1. Assuming the marginal consumer also makes the purchase, we have: D1 = x1 However in case of firm 2, all firms lying to the right of x2 shall make the purchase from firm 2. Thus D2 = (1 – x2) = (5 – P2) / 20 Now, ? 1 = (P1 – 3) [(5 – P1) / 20] and, ? 2 = (P2 – 3) [(5 – P2) / 20]. Observe that in this case, the determination of equilibrium price of either seller is not contingent upon the price determination of the other, a fact reflected in the profit functions of each seller ignoring the rival’s price.
Proceeding as before, we obtain P1* = P2* = 4. 1. g) Though we find in both cases the sellers charge a per unit price of 4, it has to be noted that the mechanisms are entirely different for the two cases. While in the first case, given the significantly lower travel costs which in essence represent the extent of differentiation perceived by the consumers for the products of the two sellers, the products from either seller was affordable for each buyer along the street.
So, the competition was strenuous and the prices of either seller entered the profit function of the other. But in the second case, due to the heightened travel cost reflecting a significantly higher differentiation, for a segment of buyers, purchasing from either seller becomes an untenable option. For the rest, purchasing from a particular seller remains as an option with the cost of acquiring the product from the other seller becoming greater than the benefit as measured by derived utility.
Thus, in effect, the market becomes segmented with each seller catering to a particular group of customers as a monopolist. Thus in setting the price the sellers act as monopolists and determine the profit maximizing price without regard to the price set by the other seller. While in the first case the equilibrium prices are set equal to the sum of the marginal cost and the travel cost borne per unit of distance, in the second case, the prices are determined very much in the fashion of individual monopoly pricing.