Relationship Between (Mutual Determination) AR, MR, and Elasticity of Demand

• Firms use the relationship between AR, MR and elasticity of demand to decide the best price and output under different market conditions.​
• Since the firm’s AR curve is the same as its demand curve, elasticity measured on the AR curve helps to know how MR will behave at any output level.

On a downward‑sloping AR (demand) curve with an MR curve lying below it:

• At any point on the AR curve, the price elasticity of demand (E) is related to AR and MR by the formula:

  \[E=\frac{AR}{AR-MR}\]
  

  where

• E = elasticity of demand at that point,
• AR = average revenue (price) at that output,
• MR = marginal revenue at that output.​

Using

\[E=\frac{AR}{AR-MR}\]

we can understand three important cases:​

• Case 1: Demand is elastic (E > 1).
  (i) E − 1 > 0, so MR > 0.
  (ii) Total Revenue (TR) rises when the firm increases output and slightly lowers price.

• Case 2: Demand is unit elastic (E = 1).
  (i) E − 1 = 0 so MR = 0.  
  (ii) TR is at its maximum at this point on the demand curve.

• Case 3: Demand is inelastic (E<1).
  (i) E − 1 < 0 so MR < 0.
  (ii) If the firm increases output by reducing price further, TR will fall.

 Thus, the sign of MR shows whether the firm is on the elastic, unit‑elastic, or inelastic portion of the demand curve.​

Assume at some output level:

• AR (price) = ₹10
• Elasticity of demand, E = 2 (elastic)

Using the formula:

\[MR=AR\cdot\frac{E-1}{E}=10\cdot\frac{2-1}{2}=10\cdot\frac{1}{2}=\mathrm{₹}5\]

• MR is positive (₹5), so TR increases if output is increased slightly.​

If E = 1:

\[MR=10\cdot\frac{1-1}{1}=0\]

TR is maximum.

If E = 0.5:

\[MR=10\cdot\frac{0.5-1}{0.5}=10\cdot(-1)=-\mathrm{₹}10\]

MR is negative; extra output reduces TR.​

• The firm’s AR curve is its demand curve; elasticity of demand is measured on this curve.
• If E > 1 (elastic demand): MR is positive, and TR rises with more output.​
• If E = 1 (unit elastic): MR is zero and TR is maximum.​
• If E < 1 (inelastic): MR is negative and TR falls when output increases.