Firms 1 and 2 produce and sell goods that are perfect substitutes (for instance, both firms produce and sell microprocessors) and face the (market) demand curve D (p) =1000 - 2p; where p is the price. We assume that the firms choose output quantities simultaneously and non-cooperatively (i.e. quantity is the choice variable).
a) Solve for the inverse demand curve.
b) Denote the number of units produced (and sold) by firm 1 by q1 and the number of units produced (and sold) by firm 2 by q2. Express the inverse demand curve as a function of q1 and q2:
c) Write the total revenue function of each firm as a function of q1 and q2:
d) Assume that the total cost function of firm 1 is T C (q1) = 0 and the total cost function of firm 2 is T C (q2) = 0. Write the maximization problem of each firm.
e) Solve for the Reaction Functions of firms 1 and 2 when the firms' costs functions are as given in (d).
f) Solve for the Nash Equilibrium when the firms' costs functions are as given in (d).
g) What would the price be if only one firm produces and sells this good? Are consumers better off when one firm sells the good or when two firms sell the good?
h) Solve for the Best Response functions of firms 1 and 2 when the marginal cost function of firm 1 is MC (q1) = 2q1; where q1 is output of firm 1, and the marginal cost function of firm 2 is MC(q2) = 2q2; where q2 is output of firm 2.
i) Solve for the Nash Equilibrium when the firms' marginal cost functions are as given in (g).