1.7. In Section 1.2.B, we analyzed the Bertrand duopoly modelwith differentiated products. The case of homogeneous productsyields a stark conclusion. Suppose that the quantity that con-sumers demand from firm i is a – p¡ when pi < Pj, 0 when Pi > Pj,Pi piand (a − p¡)/2 when p¡ = pj. Suppose also that there are no fixedcosts and that marginal costs are constant at c, where c < a. Showthat if the firms choose prices simultaneously, then the uniqueNash equilibrium is that both firms charge the price c. We consider the case of differentiated products. (See Prob-lem 1.7 for the case of homogeneous products.) If firms 1 and 2choose prices p₁ and p2, respectively, the quantity that consumersdemand from firm i isqi(Pi, Pj) = a − pi + bpj,Piwhere b > 0 reflects the extent to which firm i's product is a sub-stitute for firm j's product. (This is an unrealistic demand functionbecause demand for firm i's product is positive even when firm icharges an arbitrarily high price, provided firm j also charges ahigh enough price. As will become clear, the problem makes senseonly if b < 2.) As in our discussion of the Cournot model, we as-sume that there are no fixed costs of production and that marginalcosts are constant at c, where c < a, and that the firms act (i.e.,choose their prices) simultaneously.

Demand function : qi = a - pi (When pi < pj ) qi = 0 (when pi > pj ) qi = (a-pi )/2…

- 1.7. In Section 1.2.B, we analyzed the Bertrand duopoly model with differentiated products. The case of homogeneous products yields a stark conclusion. Suppose that the quantity that con- sumers demand from firm i is a – p; when p¡ < Pj, 0 when p¡ > Pj, and (a − p;)/2 when p¡ = pj. Suppose also that there are no fixed costs and that marginal costs are constant at c, where c < a. Show that if the firms choose prices simultaneously, then the unique Nash equilibrium is that both firms charge the price c. -

- 1.7. In Section 1.2.B, we analyzed the Bertrand duopoly model with differentiated products. The case of homogeneous products yields a stark conclusion. Suppose that the quantity that con- sumers demand from firm i is a – p; when p¡ < Pj, 0 when p¡ > Pj, and (a − p;)/2 when p¡ = pj. Suppose also that there are no fixed costs and that marginal costs are constant at c, where c < a. Show that if the firms choose prices simultaneously, then the unique Nash equilibrium is that both firms charge the price c. -

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter15: Imperfect Competition

Section: Chapter Questions

Problem 15.3P

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QUESTION 5 Consider the Cournot model of quantity competition. In the Nash equilibrium, firms always end up producing more than a monopolist would produce. O True O False
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