Question

A monopolist faces $Q=\dfrac{100}{(P-1)}$ with $P>1$. Average variable cost is $AVC=\dfrac{4}{\sqrt{Q}}$ and fixed cost is $10$. Find the profit–maximizing price (integer).

Hint:

For non-linear demand, first write \(P(Q)\) to get \(R(Q)=P(Q)Q\) and \(MR=\dfrac{dR}{dQ}\). If \(AVC\) is given, get \(VC=AVC . Q\) and then \(MC=\dfrac{dVC}{dQ}\).

Answer 1

Correct Answer: 26

Step 1: Express price and revenue as functions of $Q$.
From demand, \(P=1+\dfrac{100}{Q}\). Hence revenue \(R(Q)=P . Q=\left(1+\dfrac{100}{Q}\right)Q=Q+100\).
Marginal revenue \(MR=\dfrac{dR}{dQ}=1\).
Step 2: Obtain MC from AVC.
\(AVC=\dfrac{4}{\sqrt{Q}}\Rightarrow VC(Q)=AVC . Q=\dfrac{4}{\sqrt{Q}}\,Q=4\sqrt{Q}\).
Thus marginal cost \(MC=\dfrac{dVC}{dQ}=\dfrac{d}{dQ}(4\sqrt{Q})=\dfrac{2}{\sqrt{Q}}\).
Step 3: Profit maximization $MR=MC$.
\(1=\dfrac{2}{\sqrt{Q}}\Rightarrow \sqrt{Q}=2\Rightarrow Q^\ast=4\).
Step 4: Compute the price.
\(P^\ast=1+\dfrac{100}{Q^\ast}=1+\dfrac{100}{4}=26\).
Check: \(P^\ast>AVC(4)=\dfrac{4}{2}=2\) so operating is optimal.
Final Answer: 26