Relationship between Average Revenue (AR), Marginal Revenue (MR), and Elasticity of Demand (e)

The PDF contains the Relationship between Average Revenue (AR), Marginal Revenue (MR), and Elasticity of Demand (e) with geometric proof in a simple understandable form.

1. Microeconomics | Rohan Byanjankar
1

Relationship between AR, MR, and Elasticity of Demand

??????� ∆ ??????�� ??????�� ∆ ���,
∠APE=∠EQC (Right angles)
∠PAE=∠QEC (Corresponding angle)
∠AEP=∠ECQ (Corresponding angle)
So,
∆APE≅∆EQC
Hence,
��
??????�
=
��
??????�
=
��
��

Again,

2. Microeconomics | Rohan Byanjankar
2

??????� ∆ ??????�� ??????�� ∆ ���,
∠APF=∠FED (Alternate angles are equal)
∠AFP=∠EFD (Vertically opposite angles are equal)
∠PAF=∠FDE (Remaining angles of triangles are also equal)
So,
∆ APF ≅∆ FED
Hence,
��
??????�
=
��
��
=
��
??????�

Marginal Revenue being a straight line acts as the median cuts PE in two equal halves, so PF=
FE.
ED
AP
=
PF
FE

ED
AP
=
PF
FE

ED
AP
=1
∴��=??????�
Now,
��??????��??????�??????�?????? �� ���??????�� =
��??????�� �������
����� �������
=
��
??????�

Elasticity of demand (e)=
EC
AE
=
EQ
AP
=
EQ
ED
=
EQ
EQ−DQ
=
AR
AR−MR

e=
AR
AR−MR

e(AR−MR)=AR

3. Microeconomics | Rohan Byanjankar
3

eAR−eMR=AR
(e−1)×AR=eMR
????????????=????????????×(
�−??????
�
)
Elasticity of Demand (e) is always negative. So,
We can write,
MR=AR×(
−e−1
−e
)
MR=AR×(
−(1+e)
−e
)
????????????=????????????×(
??????+�
�
)
AR is also known as demand curve, so
MR=Price (P)×(
1+e
e
)
At equilibrium,
??????�=????????????=��??????�� (�)×(
??????+�
�
)