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. -

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Consider the following scenario in a duopoly with homogeneous products: Marginal cost: $21 Market demand: 972 units Competitor's price: $41 Your price: $44 Assuming your competitor maintained their price, what would be your pricing response, and how many units would you expect to sell at that new price? (Enter dollar amounts to the nearest penny and units to the nearest whole number.) Pricing response: Sales: $ units
3 In a Cournot market with two firms, the inverse market demand curve is P=50-2Q, where Q=q1+q2(Firm 1’s output is ; Firm 2’s output is ). Both firms have a constant marginal cost of 14. If Firm 2 produces 12 units of output, how much should Firm 1 produce? Group of answer choices 3 6 0 12