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The problem asks to prove:

[tex]e_{Q,P} = \frac{AR}{AR-MR}[/tex]

In which AR is average revenue and MR is marginal revenue.

Then verify this for demand equation [tex] p = a-bx[/tex]

I developed several steps:

[tex]

\begin{flalign*}AR = \frac{TR(Q)}{Q} ; MR = \frac{dTR(Q)}{dQ} \\*

\frac{AR}{AR-MR} = 1 - \frac{AR}{MR} = 1 - \frac{TR(Q)}{Q} . \frac{dQ}{dTR(Q)}\end{flalign*}[/tex] (??)

Then I also found that this proof brings me somewhat closer to e(Q,P):

[tex]

\begin{flalign*} Q=a-bP\\*

Thus: ~ TR = (a-bP).P = aP-bP^2 (1)\\*

Taking~the~derivatives:

\frac{\partial TR}{\partial P} = a-2bP (2)\\*

From~(1): P = \frac{a-Q}{b} \\*

Thus~ (a),(b):\\*

\frac{TR}{\partial P} = a+ \frac{a-Q}{b} = 3a-2Q\\*

Elastic~function: E(P) = \frac{\partial Q}{\partial P} . \frac{P}{Q} \\*

= b . \frac{a-Q}{Qb} (from~(b))\\*

We~have: Q(1+E) = Q(1=\frac{Q-a}{Q} = ... = a + b(2-a)\\*

\end{flalign*}[/tex]

I am a bit stuck here - I am attempting to prove that ∂TR/∂P=Q(1+E) is true (which it is I believe and may go from there.

Am I overcomplicating this? Can you give some hints?

Thanks.

Omaron

Note: Apology for the weird indentation - still trying to figure out LaTeX

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# Using calculus to prove a relation between demand elasticity and AR, MR

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