# Determine price elasticity of demand and marginal revenue

Determine price elasticity of demand and marginal revenue if $q = 30-4p-p^2$, where q is quantity demanded and p is price and p=3.

I solved it for first part-

Price elasticity of demand = $-\frac{p}{q} \frac{dq}{dp}$

on solving above i got answer as $\frac{10}{3}$

But on solving for Marginal revenue i am getting -10. But the correct answer given is $\frac{21}{10}$

• $\frac{dq}{dp} = -4 -2p$, which gets the value $-16$ when $p=3$. Also $q(p=3) = 30-12-9 = 9$. Now $-\frac{p}{q}\frac{dq}{dp} = -\frac{3}{9}(-16) = \frac{16}{3}$. That's what I get at least, if the equations are correct. – Matti P. Apr 17 '18 at 6:42
• Revenue is $pq$, or $30p-4p^2-p^3$, so marginal revenue is $30-8p-3p^2$ or $30-24-27=-21$. How the denominator of 10 gets in there I'm not sure. – Trurl Apr 18 '18 at 14:09
Try this. Total Revenue TR is $pq$. Marginal revenue is the change in TR with change in $quantity$ (not price, as I incorrectly stated in my comment) so marginal revenue is $\frac{\partial TR}{\partial q}$ or $$\frac{\partial (pq)}{\partial p}\frac{\partial p}{\partial q}$$ Revenue is $pq$, or $30p−4p^2−p^3$ so marginal revenue is $30−8p−3p^2$ = 30−24−27=−21. But then, as OP correctly calcualted, $\frac{\partial q}{\partial p}=-10$ so $$\frac{\partial (pq)}{\partial p}\frac{\partial p}{\partial q}= -21/-10.$$
marginal revenue $$= p(1+1/elasticity) = 3(1-3/10)= 21/10$$.