2.- Each of two firms, firms 1 and 2, has a cost function C(q) = : the demand function. That is, firms choose simultaneously what price each charges. Consumers will buy from the firm offering the lowest price. In case of tying, firms split equally the demand at the (common) price. The firm that charges the higher price sells nothing. (Bertrand model.) (a) Argue formally that there could be no equilibrium in prices other than that where d=doyde6eup saoq.
Now, suppose that firms have to produce (and so incur their costs), prior to the price competition. That is, first firm i chooses q, =1,2 (firms make these choices simultaneously), and then, after observing these levels of output, firms simultaneously choose their prices, p, Pz. Consumers may then be rationed. Indeed, if more than q units are demanded from firm i, only q will be served, and then unsatisfied consumers - ugsog onssepe u go oAnq e uogodaddn fed csuuo sunuoo eg ua demand function - will be served first. Thus, if consumers are rationed in, say, firm 1, then firm 2's demand will be the total demand at price P minus Qs. Let's work this dynamic model backwards to obtain the equilibrium.
(b) Suppose that firms have already chosen their output, and q for both firms Bugos suuy uoqos puebb= goyde Bupsaj woq eg mous all their inventories) is an equilibrium. That is, conjecturing that the other firm should be eg sgod sy e sope ea ood aod soep.
(c) Argue that, indeed, q must be below. To do so, show that, even if the rival did not operate at all (i.e., chose q = O) a firm will never recoup the cost incurred in producing q.
(d) Compute the profits for each firm, which will depend on Qr. s.
(e) Given (d), consider the firms' (simultaneous) decisions on q so as to maximize pndn pd. Find an equilibrium in output q.
(f) Compare this outcome with the outcome in the Cournot model of this industry - oligopolistic competition?
