Marginal Revenue 

1.0What Is Marginal Revenue?

Marginal Revenue (MR) is defined as the additional revenue a firm earns by selling one more unit of a product. In mathematical terms, it is the derivative of the total revenue function with respect to output (x).

If the total revenue function is given as:

$R(x)=p(x).x$

Where:

• R(x) = total revenue,
• p(x) = price per unit (which may depend on x),
• x = number of units sold,

Then:

$MR=\frac{dR}{dx}$

Thus, MR tells us how much additional revenue a firm earns if it sells one more unit.

2.0Marginal Revenue Example 

Problem: A firm faces the demand function p(x)=50−2x. Find the marginal revenue when 10 units are sold.

Solution:

Total revenue:

$R(x)=p(x).x=(50-2x)x=50x-2x^2$

Differentiate to find MR:

$MR=\frac{dR}{dx}=50-4x$

At x=10:

$MR=50-40=10$

Thus, the marginal revenue at 10 units = 10.

This means if the firm sells one more unit after 10 units, it will earn an additional revenue of 10.

3.0How Marginal Revenue Works

To understand how marginal revenue functions in real scenarios, consider two cases:

Perfect Competition:

• Price remains constant regardless of quantity sold.
• Total revenue increases proportionally with output.
• MR=AR=Price.

Imperfect Competition (Monopoly/Oligopoly):

• Price decreases when more units are sold.
• Marginal revenue decreases faster than average revenue.
• MR<AR.

Key Rule:

• When MR is positive, total revenue increases.
• When MR is zero, total revenue reaches its maximum point.
• When MR is negative, total revenue starts to decline.

Thus, marginal revenue helps in identifying whether selling more units increases or decreases overall revenue.

4.0Marginal Revenue Curve

The Marginal Revenue Curve represents the relationship between MR and output (x).

Characteristics of MR Curve

Under Perfect Competition:

• MR curve is a horizontal straight line, coinciding with the price line.
• MR = AR = Price.

Under Monopoly or Imperfect Competition:

• MR curve is downward sloping, lying below the average revenue (AR) curve.
• The slope of the MR curve is usually twice as steep as the AR curve if the demand function is linear.

Example:

Let the demand function be: $p(x)=a-bx$

• Total Revenue: $R(x)=(a-bx)x=ax-b{x}^2$
• Marginal Revenue: $MR=\frac{dR}{dx}=a-2bx$

Thus, MR is a straight line with slope −2b, showing its faster decline compared to AR.

5.0How to Calculate Marginal Revenue

The calculation depends on the given information:

Case 1: Using Total Revenue Function

If R(x) is given directly: $MR=\frac{dR}{dx}$

Example:

If $R(x)=100x-5x^2$, then:

$MR=\frac{dR}{dx}=100-10x$

At x=5:

MR=100−50=50

Case 2: Using Demand Function

If the demand function is given as p(x):

$R(x)=p(x)\cdot x$

$MR=\frac{d}{dx}[p(x).x]=p(x)+x.\frac{dp}{dx}$

Example:

If p(x)=50−2x:

$R(x)=50x-2x^2,MR=50-4x$

At x=10:

$MR=50-40=10$

Thus, marginal revenue can be calculated either by differentiating total revenue or by applying the MR formula directly to the demand function.

6.0Marginal Revenue vs. Marginal Cost

In optimisation problems, people often compare Marginal Revenue (MR) to Marginal Cost (MC) to find the output that makes the most money.

Definitions:

• Marginal Revenue (MR): Extra revenue from selling one more unit. $MR=\frac{dR}{dx}$
• Marginal Cost (MC): Extra cost of producing one more unit. $MC=\frac{dC}{dx}$

Decision Rule:

• If MR > MC, producing more increases profit.
• If MR < MC, producing more decreases profit.
• Profit is maximized when:
  MR=MC

7.0Solved Examples on Marginal Revenue

Example 1: The demand function for a product is given by: P = 100 - 2Q where P is the price and Q is the quantity demanded. Find the marginal revenue when Q = 10.

Solution:

• Total Revenue (TR): $TR=P\times Q=(100-2Q)Q=100Q-2Q^2$
• Marginal Revenue (MR): $MR=\frac{d(TR)}{dQ}=100-4Q$
• At Q = 10: $MR=100-4(10)=60$

Answer: The marginal revenue is 60.

Example 2: The total revenue of a firm is given by: $TR=50Q-Q^2$. Find the marginal revenue when Q = 20.

Solution:

• Marginal Revenue: $MR=\frac{d(TR)}{dQ}=\frac{d}{dQ}(50Q-Q^2)=50-2Q$
• At Q = 20: $MR=50-2(20)=10$

Answer: The marginal revenue is 10.

Example 3: If the total revenue function is: $TR=200Q-5Q^2$. Find the output level at which marginal revenue becomes zero.

Solution:

• Marginal Revenue: $MR=\frac{d(TR)}{dQ}=200-10Q$
• Set MR = 0: $200-10Q=0\Rightarrow Q=20$

Answer: Marginal revenue is zero at Q = 20.

Example 4: The demand function is P = 80 - Q.  Find the marginal revenue function.

Solution:

• Total Revenue: $TR=P\times Q=(80-Q)Q=80Q-Q^2$
• Marginal Revenue: $MR=\frac{d(TR)}{dQ}=80-2Q$

Answer: The marginal revenue function is MR = 80 - 2Q.

8.0Practice Questions on Marginal Revenue

1. If the total revenue function is ($TR=100Q-5Q^2$), find the marginal revenue when (Q = 6).
2. For a demand function (P = 80 - 2Q), derive the marginal revenue function and find MR at (Q = 10).
3. If the price elasticity of demand is 2 and the price is ₹50, what is the marginal revenue.
4. A firm’s marginal cost function is (MC = 20 + Q), and its marginal revenue function is (MR = 60 - 2Q). Find the profit-maximizing output.
5. For a perfectly competitive market, if the price is ₹40, what is the marginal revenue for any output level?