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

[idempotency]

Hi all,

The problem asks to prove:
$$e_{Q,P} = \frac{AR}{AR-MR}$$

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

Then verify this for demand equation $$p = a-bx$$

I developed several steps:
$$\begin{flalign*}AR = \frac{TR(Q)}{Q} ; MR = \frac{dTR(Q)}{dQ} \\* \frac{AR}{AR-MR} = 1 - \frac{AR}{MR} = 1 - \frac{TR(Q)}{Q} . \frac{dQ}{dTR(Q)}\end{flalign*}$$ (??)

Then I also found that this proof brings me somewhat closer to e(Q,P):
$$\begin{flalign*} Q=a-bP\\* Thus: ~ TR = (a-bP).P = aP-bP^2 (1)\\* Taking~the~derivatives: \frac{\partial TR}{\partial P} = a-2bP (2)\\* From~(1): P = \frac{a-Q}{b} \\* Thus~ (a),(b):\\* \frac{TR}{\partial P} = a+ \frac{a-Q}{b} = 3a-2Q\\* Elastic~function: E(P) = \frac{\partial Q}{\partial P} . \frac{P}{Q} \\* = b . \frac{a-Q}{Qb} (from~(b))\\* We~have: Q(1+E) = Q(1=\frac{Q-a}{Q} = ... = a + b(2-a)\\* \end{flalign*}$$

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

 

[snipez90]

Well your first mistake is in thinking that

$$\frac{AR}{AR-MR} = 1 - \frac{AR}{MR}$$

which is an algebra error that renders each of those equalities false. I might take a look at this later. Since it's econ and not really math, I expect that an abuse of notation will probably be employed.

 

[idempotency]

o you - that's dumb... :S

 

[snipez90]

All right I worked out the details. It's always a good idea to work out the specific case first, so I'll start with that. Well the other part of what you wrote that you don't want to do is to treat the derivatives as fractions (this may be confusing because econ professors like to translate discrete cases into continuous cases by abusing differentials).

Now notice that you immediately worked with the inverse of the demand function by treating price as a function of quantity (defined by P(Q)). This is good, since AR is price, so in place of AR, you should have (a-Q)/b. To calculate MR, note we have chosen to work with price as a function of quantity, so simply multiply P = (a-Q)/b by Q and then differentiate with respect to Q. This is the wise choice since we already wrote down dTR/dQ for MR so we need to write things in terms of quantity. Well we don't need to, since we could work with quantity as a function of price, but I think this way is more intuitive. You should check that this works.

I'll let you figure out the general case some more. It's really simply a matter of not getting confused about the functional relationship between P and Q, namely that they are obviously inverse functions. If you understood the specific case, you should be able to do this. There is however, no way around not using the formula for the derivative of an inverse function because elasticity has a dQ/dP factor but on the right hand side we have MR = dTR/dQ.