Q.A monopolist faces the demand function $P=100-2Q$ and has the total cost function $TC=Q^{2}+10Q+50$. Find the profit-maximising output and price, and calculate the monopolist's profit.

Derive TR and MR from the demand function, derive MC from TC, set MR=MC to find Q, then find P from the demand function and profit from TR-TC.

From the demand function $P=100-2Q$, Total Revenue is:

Marginal Revenue:

From $TC=Q^{2}+10Q+50$, Marginal Cost:

Setting $MR=MC$ for profit maximisation:

Equilibrium price, read off the demand function at $Q=15$:

Verification: at $Q=15$, $MR=100-4(15)=40$ and $MC=2(15)+10=40$ — the two are equal, confirming the equilibrium. …

An alternative check: build a small output table around Q=15 (Q=14, 15, 16), compute TR-TC (profit) at each, and confirm profit peaks exactly at Q=15 — at Q=14, TR=100(14)-2(196)=1400-392=1008, TC=196+140+50=386, profit=622; at Q=16, TR=1600-512=1088, TC=256+160+50=466, …

A frequent error is setting Price equal to MC (the perfect-competition rule) instead of MR equal to MC — under monopoly, MR is NOT equal to price, so using P=MC would giv …