Q25 — Relation between AR, MR and Price Elasticity of Demand

Introduction

In the theory of the firm, the two pillars of analysis are revenue and cost. On the revenue side, three concepts are fundamental: Total Revenue (TR), Average Revenue (AR) and Marginal Revenue (MR). On the demand side, the key concept is price elasticity of demand, which measures the responsiveness of quantity demanded to changes in price.

In the prescribed text “Microeconomics” by T.R. Jain & V.K. Ohri for B.Com Semester I (Panjab University), a very important theoretical result is derived which directly links AR, MR and price elasticity of demand. This question asks you to state and explain that relationship. A complete answer must: (i) recall the definitions of AR, MR and price elasticity, (ii) derive the mathematical relation between them, (iii) interpret the relation for different degrees of elasticity, and (iv) support the explanation with a neat diagram and economic implications.

I. Basic Concepts: AR, MR and Price Elasticity of Demand

1. Average Revenue (AR)

Average Revenue is the revenue per unit of output. It is obtained by dividing total revenue by the quantity sold:

AR = TR / Q

Since total revenue is price multiplied by quantity (TR = P × Q), it follows that:

AR = (P × Q) / Q = P

Thus, AR is numerically equal to price. Therefore, the firm’s demand curve is also its AR curve.

2. Marginal Revenue (MR)

Marginal Revenue is the addition to total revenue when one extra unit of output is sold. It is the change in total revenue per unit change in quantity:

MR = ΔTR / ΔQ

In calculus form, MR is the first derivative of TR with respect to Q:

MR = d(TR) / dQ

3. Price Elasticity of Demand (e_p)

Price elasticity of demand measures the responsiveness of quantity demanded to changes in price. For small changes, point elasticity at a point on the demand curve is defined as:

e_p = − (dQ/dP) × (P/Q)

The minus sign is taken because the demand curve generally slopes downward (price and quantity move in opposite directions). In practical use, we usually refer to the absolute value of elasticity, i.e. |e_p|.

II. Derivation of the Relation between AR, MR and Price Elasticity

We now derive the standard textbook formula that links marginal revenue (MR) with average revenue (AR) and price elasticity of demand (e_p), as given in T.R. Jain & V.K. Ohri.

Step 1: Start from Total Revenue

Total Revenue is given by:

TR = P × Q

Step 2: Express MR as the Derivative of TR

By definition,

MR = d(TR)/dQ = d(PQ)/dQ

Using the product rule of differentiation:

MR = P + Q·(dP/dQ)

Step 3: Use the Definition of Price Elasticity

Point elasticity of demand is:

e_p = − (dQ/dP) × (P/Q)

Taking reciprocal on both sides:

1 / e_p = − (dP/dQ) × (Q/P)

Rearranging, we get:

dP/dQ = − (P / Q) × (1 / e_p)

Step 4: Substitute dP/dQ into the Expression for MR

From Step 2 we have:

MR = P + Q·(dP/dQ)

Substituting the value of dP/dQ from above:

MR = P + Q × [ − (P / Q) × (1 / e_p) ]

Simplifying:

MR = P − P × (1 / e_p)
MR = P × (1 − 1 / e_p)

Since P = AR, we finally obtain the standard relation:

Basic Relation (with sign of e):
MR = AR × (1 − 1 / e_p)

In practice, because the price elasticity of demand for a downward sloping demand curve is negative, many authors (including your textbook) work with the absolute value of elasticity, |e_p|. Then the relation is written as:

Standard Exam Formula (using absolute elasticity):
MR = AR × ( 1 − 1 / |e_p| )

This is the required theoretical relation between Average Revenue (AR), Marginal Revenue (MR) and price elasticity of demand (e_p).

III. Interpretation of the Relation for Different Elasticities

The formula MR = AR × ( 1 − 1 / |e_p| ) allows us to relate the sign of MR to the degree of price elasticity.

1. When Demand is Elastic (|e_p| > 1)

Suppose |e_p| > 1. Then 1 / |e_p| < 1, so:

( 1 − 1 / |e_p| ) > 0

Since AR is positive (price cannot be negative in normal cases), it follows that:

When |e_p| > 1, then MR > 0.

So, in the elastic range of the demand curve, marginal revenue is positive. In this range, a fall in price (and rise in output) causes total revenue to increase.

2. When Demand is Unitary Elastic (|e_p| = 1)

If |e_p| = 1, then:

1 − 1 / |e_p| = 1 − 1 = 0

So:

When |e_p| = 1, then MR = 0 and TR is maximum.

At unitary elasticity, total revenue neither increases nor decreases when price changes; it is at its highest point. This is a very important result.

3. When Demand is Inelastic (|e_p| < 1)

If |e_p| < 1, then 1 / |e_p| > 1 and therefore:

( 1 − 1 / |e_p| ) < 0

So:

When |e_p| < 1, then MR < 0.

In the inelastic range of the demand curve, marginal revenue is negative. In this range, a fall in price (and rise in output) actually reduces total revenue.

IV. Summary Table: AR–MR–Elasticity Relation

Elasticity of Demand (\|e_p\|)	Value of (1 − 1/\|e_p\|)	Sign of MR	Behaviour of TR	Region of Demand Curve
\|e_p\| > 1	Positive	MR > 0	TR increases as output increases	Elastic region (upper part of AR)
\|e_p\| = 1	Zero	MR = 0	TR is maximum	Mid-point of straight-line AR
\|e_p\| < 1	Negative	MR < 0	TR decreases as output increases	Inelastic region (lower part of AR)

V. Diagram: AR, MR and Elasticity along a Straight-line Demand Curve

In the traditional geometric presentation (also used in T.R. Jain & V.K. Ohri), the relation between AR, MR and elasticity is shown on a straight-line demand (AR) curve. The MR curve is also a straight line starting from the same price-axis intercept but cutting the quantity axis at half the distance.

Fig. — Straight-line AR and MR Curves: Upper half of AR is elastic (MR > 0), mid-point is unitary elastic (MR = 0), lower half is inelastic (MR < 0).

Explanation of the Diagram

In the figure:

AR is a straight downward sloping line from the price axis to the quantity axis.
MR is also a straight line from the same price intercept but meets the quantity axis at half the distance.
The mid-point E of the AR curve divides it into two parts: the upper half where |e_p| > 1 (elastic region) and the lower half where |e_p| < 1 (inelastic region).
At point E (mid-point), |e_p| = 1 and corresponding MR = 0; TR is maximum here.

VI. Economic Implications of the AR–MR–Elasticity Relation

1. Output and Pricing Decisions of a Monopolist

For a monopolist (or any firm under imperfect competition), the equilibrium output is determined by the condition MR = MC. Using the relation MR = AR × (1 − 1/|e_p|), we can show that:

At equilibrium, MR must be positive (equal to MC, which is positive).
Therefore, the firm will never produce in the inelastic range of the demand curve where MR is negative.
A monopolist will always operate on the elastic portion of the demand curve; otherwise, reducing output and raising price would increase TR and reduce cost, contradicting profit maximisation.

2. Behaviour of Total Revenue (TR)

The relation between MR and elasticity tells us how TR behaves:

When |e_p| > 1, MR > 0 → TR rises with increase in output (or fall in price).
When |e_p| = 1, MR = 0 → TR is maximum.
When |e_p| < 1, MR < 0 → TR falls with increase in output (or fall in price).

Thus, the TR curve is rising, flat, or falling according to whether elasticity is greater than, equal to, or less than unity.

3. Taxation and Public Policy

Governments often impose indirect taxes (excise, GST, etc.) on goods. Whether a tax increase will raise or reduce total revenue depends on the elasticity of demand:

If demand is inelastic, higher tax (higher price) may increase TR (and tax revenue).
If demand is elastic, higher price may reduce TR.

The MR–AR–elasticity relation is thus useful in understanding revenue effects of taxation and pricing policies.

4. Practical Pricing Strategy for Firms

For a firm engaged in price-setting:

Price cuts are advisable only in the elastic region where MR is positive and TR rises.
In the inelastic region, further price cuts reduce TR; firms should generally avoid this region.
Knowledge of elasticity helps firms identify the output range where MR is positive and profits can be increased.

5. Elasticity as a Bridge between Demand and Revenue

Conceptually, the relation MR = AR(1 − 1/|e_p|) acts as a bridge between the consumer side and the producer side:

On the demand side, elasticity summarises how consumers react to price changes.
On the firm’s side, MR summarises how the firm’s total revenue changes with output.
This formula shows how consumer behaviour (elasticity) determines the shape and position of the MR curve.

6. Examination Significance (Panjab University)

In the B.Com (Sem I) Microeconomics paper of Panjab University, the question “Explain the relation between AR, MR and price elasticity of demand.” is a classic 15-marks theory question. A high-quality answer should:

Define AR, MR and price elasticity of demand.
Derive the relation MR = AR(1 − 1/|e_p|) step by step.
Explain the sign of MR for different ranges of elasticity with a summary table.
Support the explanation with a neat diagram showing AR, MR and the elastic/unitary/inelastic regions.
Mention economic implications for monopolist’s pricing, TR behaviour and policy aspects.

Exam Tip (for 15 marks): In your answer booklet, use the following sequence: (i) brief introduction, (ii) definitions of AR, MR and price elasticity, (iii) algebraic derivation of MR = AR(1 − 1/|e|) from TR = PQ and the elasticity formula, (iv) interpretation for |e| > 1, |e| = 1 and |e| < 1 with a small table, (v) a neat diagram of linear AR with MR and marking of elastic, unitary and inelastic segments, and (vi) a short paragraph on implications for monopolist and TR behaviour. This structure is exactly in line with the presentation given in T.R. Jain & V.K. Ohri and is highly appreciated by Panjab University examiners.

Conclusion

To conclude, the relation between Average Revenue (AR), Marginal Revenue (MR) and price elasticity of demand is captured in the compact and powerful formula MR = AR(1 − 1/|e_p|). This formula shows that the sign and size of MR at any point on the firm’s demand (AR) curve are completely determined by the price elasticity of demand at that point. In the elastic range of demand MR is positive and TR rises with output; at unitary elasticity MR is zero and TR is maximised; in the inelastic range MR is negative and TR falls with further expansion. This elegant relationship unifies demand analysis and revenue analysis and forms one of the most important theoretical results in the B.Com (Sem I) Microeconomics syllabus of Panjab University.

These notes form part of a carefully curated set of important questions which have frequently appeared in past university examinations and therefore carry a high probability of being reflected, in whole or in part, in future question papers. However, they are intended as high-quality academic support material only and should not be treated as a guarantee or assurance of any specific questions being asked in forthcoming exams.