Explain the derivation of AR and MR from TR under monopoly market. [5]

Under monopoly, the firm is the sole seller of a product with no close substitutes. The demand curve faced by the monopolist is downward sloping, meaning the firm must reduce the price to sell more units. The relationship between Total Revenue (TR), Average Revenue (AR), and Marginal Revenue (MR) is derived as follows:

1. Total Revenue (TR)

TR is the total income received from selling a given quantity of output. It is calculated as: $$TR=P\times Q$$ where $P$ is the price and $Q$ is the quantity sold.

2. Average Revenue (AR)

AR is the revenue per unit sold, which is equal to the price of the product. Since the monopolist faces a downward-sloping demand curve, AR decreases as output increases. Mathematically: $$AR=\frac{TR}{Q}=P$$

3. Marginal Revenue (MR)

MR is the additional revenue earned from selling one more unit of output. It is derived by taking the change in TR divided by the change in quantity: $$MR=\frac{\Delta TR}{\Delta Q}$$ In a monopoly, MR is always less than AR (price) because the firm must lower the price on all previously sold units to sell an additional unit. This relationship is expressed as: $$MR=AR+\frac{d(AR)}{dQ}\times Q$$ or, more simply: $$MR=P+\left(\frac{dP}{dQ}\right)Q$$ where $\frac{dP}{dQ}$ is the slope of the demand curve.

Derivation with an Example

Assume the demand function is: $$P=20-2Q$$ Then: $$TR=P\times Q=(20-2Q)\times Q=20Q-2Q^2$$

• AR is simply the price: $$AR=20-2Q$$
• MR is the derivative of TR with respect to Q: $$MR=\frac{d(TR)}{dQ}=20-4Q$$ This shows that MR is twice as steep as AR (since the slope of demand is $-2$, MR has a slope of $-4$).

Key Observations

• AR = MR at the midpoint of the demand curve (where $Q=5$ in this example).
• MR is always below AR because the monopolist must cut the price on all units to sell an extra unit.
• The profit-maximizing output occurs where MR = MC (Marginal Cost), but this is not derived here.

This derivation explains why monopolists produce less and charge higher prices compared to competitive markets.