A firm with market power faces the following estimated demand and average variable cost functions: $Q _ { d } = 39,000 - 500 P + 0.4 M - 8,000 P _ { R }$ $A V C = 30 - 0.005 Q + 0.0000005 Q ^ { 2 }$ where $Q _ { d }$ is quantity demanded,P is price,M is income,and $P _ { R }$ is the price of a related good.The firm expects income to be $40,000 and $P _ { R }$ to be $2.Total fixed cost is $100,000.What is the estimated marginal revenue function for the firm?

Question

Quizplus · Accepted Answer

The formula for calculating marginal revenue (MR) is MR = Change in Total Revenue/Change in Quantity. We can find total revenue (TR) by multiplying price (P) and quantity (Q). So, TR = PQ. 
Differentiating TR with respect to Q, we get d(TR)/d(Q) = P + Q(dP/dQ)
We know that d(TR)/d(Q) = MR, and from the demand function, we can find dP/dQ as - 0.0005Q + 500. 
Substituting the known values, we have MR = (39,000 - 500P + 0.4M - 8,000PR)(-0.0005Q + 500) + P
At M = 40,000 and PR = 2, the above equation becomes MR = (39,000 - 500P - 7200)(-0.0005Q + 500) + P
On simplification, we get MR = 78 - 0.004Q. Thus, the answer is C.

A Firm with Market Power Faces the Following Estimated Demand Qd=39,000−500P+0.4M−8,000PRQ _ { d } = 39,000 - 500 P + 0.4 M - 8,000 P _ { R }Qd​=39,000−500P+0.4M−8,000PR​

A Firm with Market Power Faces the Following Estimated Demand $Q _ { d } = 39,000 - 500 P + 0.4 M - 8,000 P _ { R }$