Q.A monopolist faces the linear demand curve $P = 20 - 2Q$ (P in ₹, Q in units). Derive the TR, AR and MR functions, and find their values at Q = 4. Comment on the AR-MR relationship you obtain.

For a linear demand curve, average revenue is the price line itself and marginal revenue has the same intercept but twice the slope.

Step 1 — Total Revenue. $TR = P \times Q = (20 - 2Q)Q$

Step 2 — Average Revenue. $AR = TR/Q = (20Q - 2Q^2)/Q$

This is just the demand (price) equation — confirming AR = price.

Step 3 — Marginal Revenue. Differentiating TR with respect to Q, $MR = \dfrac{d(TR)}{dQ}$:

Compared with $AR = 20 - 2Q$, marginal revenue has the same intercept (20) but double the slope, so it falls twice as fast.

Step 4 — Values at Q = 4.

Step 5 — Comment. MR (₹4) is well below AR (₹12), the standard result for a price-maker: to sell the 4th unit the monopolist lowers price on all units, so the extra revenue is much less than the price received.

Dual-check MR by the increment method around Q = 4. $TR(3) = 20(3)-2(9) = 60-18 = 42$; $TR(4) = 48$; $TR(5) = 100-50 = 50$. The incremental MR from Q = 3→4 is $48-42 = 6$ and from Q = 4→5 is $50-48 = 2$; their average is $(6+2)/2 = 4$, matching the calculus value $MR = 20-4(4) = ₹4$ at Q = 4. Also $AR\times Q = 12\times4 = ₹48 = TR$, confirming AR.

Writing $MR = 20 - 2Q$ (copying the AR slope) — the slope must double for a linear demand curve. Forgetting to multiply demand by Q before differentiating. Reporting AR and MR as equal here, which is only true under perfect competition, not for a monopolist's downward-sloping demand.