I cannot figure out this problem, if anyone can just point me in the right direction that would be great.
The price is $\$20$ and the price elasticity changes from $-2$ to $-3.$ If $\$20$ was the optimal price at $-2$, then what is it at $-3$?
If no one can help that is fine, I just thought this might be a good place to ask.
The optimal pricing rule for a profit maximizing monopolist is $MR = MC$ (marginal revenue = marginal cost). However, we know the following relationship between $MR$ and the elasticity $\epsilon$: $$MR = P \left( 1- \dfrac{1}{|\epsilon|} \right)$$
Equating the two we get $$MC = P \left( 1- \dfrac{1}{|\epsilon|} \right) \Rightarrow MC = 20(1-0.5) = 10. $$ If the elasticity shifts, we now have $$ P = \left(\dfrac{|\epsilon|}{|\epsilon|-1} \right)MC = \left(\dfrac{3}{3-1} \right)(10)=15. $$
