8. Two firms competing a la Hotelling with fixed locations and symmetric constant marginal costs. Suppose there is a line between 0 and 1 in which two rms locate themselves, firm 0 being at point xo = 0 and firm 1 being at point x1 = 1. Each firm has a linear cost function C (q) = cg,where q.represents output level of firm/product i = 0,1, parameter is such that c > 0. Consumers are located between 0 and 1 and are distributed uniformly according the the graph below. 1 Buy from 0 Buy from 1 x* 1 x Consumers value the product firm i = 0,1 in r;> 0 and incur a disutility from travelling to the location of the product which is assume to be a linear function of distance. The mass of consumers is normalized to 1 since that is the area in the graph. Indirect utility function for consumers is then V (rup;x) = ri-t|Xi- x|- p¡ where pi denotes the price of product/ firm i = 0,1 and x is the consumer (point) that is located in the line between 0 and 1. Each consumer wants at most one unit of one of the products, the one that is closest in distance. Note that the term |x¡-x| represents the distance between consumer x and the location x; of firm i = 0,1. For example the consumer located at x = 0.5 has a distance to either firm of 0.5 since |xo- x| = | 0.5| = 0.5 and |x1. x| = |1- 0.5| = 0.5. AssumeT > § r1 – ro| > 0 where t is the location differentiation of the products among the firms. The demand for each firm depends on the existence of an "indifferent" consumer denoted x* who is the one point between 0 and 1 that achieves the same utility regardless of the product purchased. This means that this indifferent x* consumer satisfies ro- T Xo -x |-po = r1. T |X1-x* |-p1 or replacing xo = 0 and x1 = 1 the indifferent consumer is determined to be after rearranging ro – r1 + P1 – po 1 x" 27 2 Hence the demand for firm 0 is the mass of consumers to the left of x* (base of x* times the height of one): Qo (Po, P1) = ro-ri+Pi-po 27 2 while the demand for firm 1 is the mass of consumers to the right of x* (base Q1 (Po,P1) = } T0-T1+p1-Po 2T of 1 -x* times the height of one): each firm is 0.5 which is equal to x* as represented in the graph above. For different values of ro + r1 and/or po + P1 the indifferent consumer is not located at 0.5 but will be a point between 0 and 1. . Note that if ro = r1 and po = Pi then the demand for а. Find the best response price pi (p;) firm i = 0,1 for any p; that firm j = 0,1 chooses (Hint: maximize profits choosing pifor firm i = 0,1 taking as given p;). b. Graph the two best response functions in a space where the vertical axis is pi and the horizontal axis is po. Find the Nash equilibrium (pı*p2*). Are the equilibrium prices above marginal cost when r1 = ro? Are they increasing in t? If so, what does that mean? Explain. с. What happens to the equilibrium prices if ri increases while ro remains constant? Interpret what you find. d. Find the profits at equilibrium. Are they positive? What happens to profits whent increases? Interpret what you find.