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

Derivation of AR and MR from TR under Monopoly Market

In a monopoly market, a firm is the sole seller of a product with no close substitutes. The firm faces the entire industry demand curve as its Average Revenue (AR) curve. The derivation of Marginal Revenue (MR) from Total Revenue (TR) and the relationship between AR and MR can be explained as follows:

1. Total Revenue (TR)

TR is the total income received by the firm from selling a certain quantity of a product. It is calculated as: $$TR=P\times Q$$ where $P$ is the price per unit 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 is a price maker, the AR curve is the same as the demand curve faced by the firm. Mathematically: $$AR=\frac{TR}{Q}=P$$

3. Marginal Revenue (MR)

MR is the additional revenue earned from selling one more unit of the product. It is derived from the change in TR when one additional unit is sold: $$MR=\frac{\Delta TR}{\Delta Q}$$ In a monopoly, the firm must lower the price to sell more units (due to the downward-sloping demand curve). This means that the MR curve lies below the AR curve.

Derivation with Numerical Example

Assume the following demand schedule for a monopolist:

• TR Calculation: For example, at $Q=3$, $TR=7\times 3=21$.
• MR Calculation: When moving from $Q=2$ to $Q=3$, $TR$ increases from 16 to 21, so $MR=21-16=5$.

Key Observations

• The AR curve is downward-sloping because the monopolist must reduce the price to sell more units.
• The MR curve lies below the AR curve because the additional revenue from selling one more unit is less than the price (due to the need to lower the price for all units).
• The MR curve intersects the AR curve at the midpoint of the AR curve (in this case, at $Q=2.5$).

Graphical Representation

The figure above illustrates the relationship:

• The TR curve starts at the origin, rises, reaches a maximum, and then falls.
• The AR curve is a straight line (demand curve) from $P=10$ to $P=0$.
• The MR curve is below AR and intersects it at its midpoint.

This derivation shows that under monopoly, MR < AR, and the firm maximizes profit where MR = MC.