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

by idempotency
Tags: calculus, demand, elasticity, prove, relation
 P: 2 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
 P: 1,104 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.