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 is firm B's profit-maximizing price when firm A sets a price of $\$ 70$ for its good?
A) $\$ 70$
B) $\$ 72.5$
C) $\$ 74$
D) $\$ 76.5$

Question

Quizplus · Accepted Answer

Given that firm A sets its price at $70, we substitute $P_A = 70$ into firm B's demand curve to find firm B's profit-maximizing price. The demand curve for firm B becomes $Q_B = 100 - P_B + 0.5(70)$, which simplifies to $Q_B = 135 - P_B$. To maximize profit, firm B needs to equate marginal revenue (MR) to marginal cost (MC). Since MC is given as $10 and assuming linear demand, MR can be found by doubling the slope of the demand curve. However, in this case, we look for the price $P_B$ that maximizes profit given the MC and the demand. Setting MC equal to the derivative of the revenue function (which is not directly provided but implied by the cost structure and demand), we find that firm B will set a price slightly above firm A's price to capture some of the market without starting a price war. Given the choices, the closest strategic price above $70, considering the demand curve's sensitivity, is $72.5, which allows firm B to maximize its profit by attracting consumers sensitive to price differences between the two products without lowering its price too much below its perceived value or the cost of production.

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 $Q _ { A } = 100 - P _ { A } + .5 P _ { B }$