Let firm A face demand curve $Q _ { A } = 100 - P _ { A } + .5 P _ { B }$ and firm B face demand curve $Q _ { B } = 100 -$ $P _ { B } + .5 P _ { A }$. Products $\mathrm { A }$ and $\mathrm { B }$ both have constant marginal cost of production of 10 per unit (and no fixed cost). Each firm acts as a Bertrand competitor. What are the Bertrand Equilibrium prices in this market?
A) $P _ { B } = 72.5 ; P _ { A } = 70$
B) $P _ { B } = P _ { A } = 73.33$
C) $P _ { B } = 74 ; P _ { A } = 87$
D) $P _ { B } = P _ { A } = 74$

Question

Quizplus · Accepted Answer

In a Bertrand equilibrium, both firms will set their price equal to their marginal cost of production (assuming no capacity constraints). Therefore, we need to solve for the values of P that satisfy the condition that each firm's marginal cost is equal to the other firm's price.

Starting with firm A:
MC = P
P = 100 - QA + 0.5PB
10 = 100 - QA + 0.5PB (since P = MC = 10)
90 = QA - 0.5PB
QA = 90 + 0.5PB

Now, plugging this into firm B's demand curve:
QB = 100 - PB + 0.5PA
QB = 100 - PB + 0.5(90 + 0.5PB)
QB = 145 - 0.25PB

Setting QB equal to MC and solving for PB:
10 = 145 - 0.25PB
PB = 580/9 = 64.44

Plugging this back into the equation for QA above:
QA = 90 + 0.5(64.44) = 122.22

Therefore, the Bertrand equilibrium prices are P_A = P_B = MC = $10, or answer choice B.

Let Firm a Face Demand Curve QA=100−PA+.5PBQ _ { A } = 100 - P _ { A } + .5 P _ { B }QA​=100−PA​+.5PB​

Let Firm a Face Demand Curve $Q _ { A } = 100 - P _ { A } + .5 P _ { B }$