Derivation Of Equation Which Depicts The Relationship Between A Monopolist Marginal Revenue And Price Elasticity Of Demand.

In most of the text books, the proof for a monopolist marginal revenue seems too hard to understand by students of economics. This may be due to their way of explanation.

In this write-up, I give a well explained means of proving such an equation, with the objective of Making economics students of various institutions happy.

See the equation in the mathematical capsule 2.0.

Mr = P( 1 - 1 / e) ................2.0

Here, Mr represents marginal revenue ; P represents price of a good ; e denotes elasticity of demand.

Proof:

Step 1 : we establish the revenue (total revenue) equation which is given as,

Tr = Pq.............1

Here, Tr is total revenue ; P is price of a good ; q is quantity bought.

Also, for a continuous function, elasticity of demand is given as,

E = - (dp/dq) x (p/q)...........2

It is possible to create a proper linear equations by manipulating the equations given above.

Step 2:

Totally differentiate equation one. That is, differentiating the revenue equation with respect to quantity q. Thus we have,

dTr / dq =( pdq/dq ) + ( qdp/dq )

dTr/dq = P + q( dp/dq)

Recall! Marginal revenue is given as change in total revenue divided by change in quantity. So, for a continuous function, marginal revenue is the first derivative of a total revenue with respect to quantity.

Thus,

Mr = dTr / dq =

Now, we replace dTr/dq with Mr. That is,

Mr = P + q(dp/dq)

Mr - q ( dp/dq) = P...........3

Here, P is treated as a constant ; (dp/dq) is treated as a variable with 'q' as its coefficient ; Mr is treated as a variable with a unity coefficient.

Step 3:

Take the reciprocal of equation 2. That's the elasticity equation. Thus we have,

1/e = - (dp/dq) x (q/p)

Multiply by - p/q to have

- P / eq = dp/dq

dp / dq = - p / eq...........4

Here, dp / dq is a variable while -p / eq is a constant.

At this point, it is very obvious that we have managed to form a proper first degree equation which is kept in a standard form.

Since the equations are two, we can then apply any method of our choice such as substitution method,elimination method,etc.

For me, I prefer matrix transformation.

To do this, I create an augmented matrix,i.e,

( 1 -q) ( -p ).....R1
( 0 1) ( - p/eq ).....R2

So, our aim is to transform the square regular matrix into a reduced row echelon form called a total triangular matrix. After I have derived such a matrix, I take the entries of the column matrix as the values of Mr and dp/dq.

Now I start row reduction.

Since the first entry of R2 is 0, and the second entry is one,then dp/dq is got already,i.e,

dp/dq = - p / eq.

But this value is ignored since it has no definition in economics.

Now, make -q ( the second element of R1) zero.
We can achieve this by multiplying R2 by q and adding the result to R1. That is,

R1 + qR2

( 1 0 ) ( p + (- pq /eq) )
( 0 1 ) ( -p/eq )
At this point, the square matrix is in a reduced row echelon form by Gauss-Jordan method.
The first entry of the column vector is now the value of Mr. That is,

Mr = P - ( pq / eq)
We cancel out q to have

Mr = P - P/e

We factor out P to have

Mr = P ( 1 - 1 / e )
Q.E.D

I know this helps!

By Makasi Chinedu (Bsc economics)
University Of portharcourt

SlimShaddy10:
Derivation Of Equation Which Depicts The Relationship Between A Monopolist Marginal Revenue And Price Elasticity Of Demand.

In most of the text books, the proof for a monopolist marginal revenue seems too hard to understand by students of economics. This may be due to their way of explanation.

In this write-up, I give a well explained means of proving such an equation, with the objective of Making economics students of various institutions happy.

See the equation in the mathematical capsule 2.0.

Mr = P( 1 - 1 / e) ................2.0

Here, Mr represents marginal revenue ; P represents price of a good ; e denotes elasticity of demand.

Proof:

Step 1 : we establish the revenue (total revenue) equation which is given as,

Tr = Pq.............1

Here, Tr is total revenue ; P is price of a good ; q is quantity bought.

Also, for a continuous function, elasticity of demand is given as,

E = - (dp/dq) x (p/q)...........2

It is possible to create a proper linear equations by manipulating the equations given above.

Step 2:

Totally differentiate equation one. That is, differentiating the revenue equation with respect to quantity q. Thus we have,

dTr / dq =( pdq/dq ) + ( qdp/dq )

dTr/dq = P + q( dp/dq)

Recall! Marginal revenue is given as change in total revenue divided by change in quantity. So, for a continuous function, marginal revenue is the first derivative of a total revenue with respect to quantity.

Thus,

Mr = dTr / dq =

Now, we replace dTr/dq with Mr. That is,

Mr = P + q(dp/dq)

Mr - q ( dp/dq) = P...........3

Here, P is treated as a constant ; (dp/dq) is treated as a variable with 'q' as its coefficient ; Mr is treated as a variable with a unity coefficient.

Step 3:

Take the reciprocal of equation 2. That's the elasticity equation. Thus we have,

1/e = - (dp/dq) x (q/p)

Multiply by - p/q to have

- P / eq = dp/dq

dp / dq = - p / eq...........4

Here, dp / dq is a variable while -p / eq is a constant.

At this point, it is very obvious that we have managed to form a proper first degree equation which is kept in a standard form.

Since the equations are two, we can then apply any method of our choice such as substitution method,elimination method,etc.

For me, I prefer matrix transformation.

To do this, I create an augmented matrix,i.e,

( 1 -q) ( -p ).....R1
( 0 1) ( - p/eq ).....R2

So, our aim is to transform the square regular matrix into a reduced row echelon form called a total triangular matrix. After I have derived such a matrix, I take the entries of the column matrix as the values of Mr and dp/dq.

Now I start row reduction.

Since the first entry of R2 is 0, and the second entry is one,then dp/dq is got already,i.e,

dp/dq = - p / eq.

But this value is ignored since it has no definition in economics.

Now, make -q ( the second element of R1) zero.
We can achieve this by multiplying R2 by q and adding the result to R1. That is,

R1 + qR2

( 1 0 ) ( p + (- pq /eq) )
( 0 1 ) ( -p/eq )
At this point, the square matrix is in a reduced row echelon form by Gauss-Jordan method.
The first entry of the column vector is now the value of Mr. That is,

Mr = P - ( pq / eq)
We cancel out q to have

Mr = P - P/e

We factor out P to have

Mr = P ( 1 - 1 / e )
Q.E.D

I know this helps!

By Makasi Chinedu (Bsc economics)
University Of portharcourt

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