Question

demand for light bulbs can be characterized by Q=100-P, where Qis in millions of boxes of lights sold and P iis the price per box; there are two producers of lights, Everglow and Dimlite ; they have identical cost functions; find reaction functions for Everglow and Dimlit; C=10Q+.5Q^2 (where i=E,D) MC=10+Q Q=Qe+Qd P=100-(Qe+Qd)

Answer 1

Answer : Given, P = 100 - (Qe + Qd)

=> P = 100 - Qe - Qd

TRe (Total Revenue) = P×Qe = (100 - Qe - Qd)×Qe

=> TRe = 100Qe - Qe^2 - Qd.Qe

MRe (Marginal Revenue) = TRe /Qe = 100 - 2Qe - Qd

TRd = P×Qd =(100 - Qe - Qd)×Qd

TRd=100Qd - Qe.Qd - Qd^2

MRd = TRd / Qd = 100 - Qe - 2Qd

MC = 10+Qe+Qd [As Q= Qe + Qd]

The equilibrium condition for Everglow producer,

MRe = MC

=> 100 - 2Qe - Qd = 10+ Qe +Qd

=> 100 - 10 - Qd -Qd = Qe + 2Qe

=> 90 - 2Qd = 3Qe

=> Qe = (90 - 2Qd)/3

=> Qe = 30 - 0.667Qd ........ (i)

Equation (i) is the reaction function of Everglow producer.

The equilibrium condition for Dimlit producer,

MRd = MC

=>100 - Qe - 2Qd = 10 +Qe +Qd

=> 100 - 10 - Qe - Qe = Qd + 2Qd

=> 90 - 2Qe = 3Qd

=> Qd = (90 - 2Qe)/3

=> Qd = 30 - 0.667Qe ..... (ii)

Equation (ii) is the reaction function of Dimlit producer.