I have two years of sales data for a product and would like to justify a price for the upcoming year.
$$\text {Year 2017}: \text {Price}_1 = $135, \text {Quantity}_1 \text{ Sold} = 2239$$
$$\text {Year 2018}: \text {Price}_2 = $150, \text {Quantity}_2 \text { Sold} = 2414$$
$$\text {Marginal Cost} = $101$$
I can calculate the elasticity by:
$$\text {Price elasticity} = \dfrac {\text {% change in demand}} {\text {% change in price}}$$
or $$\dfrac {\dfrac{(Q_2-Q_1)}{(Q_2+Q_1)/2}}{\dfrac {(P_2-P_1)}{(P_2+P_1)/2}}$$ and $$0.71 = \eta$$
However, when I use the formula to calculate optimal price:
$$P= MC*\dfrac {\eta}{\eta+1} = \$42$$
I am given a price of $\$42$ which can't be right? My inelastic demand gives me a much lower price than what I am currently using, and a resulting "max" revenue that is also lower than what I'm currently seeing.
If my demand is inelastic, shouldn't I be able to raise prices? I'm not sure where I'm going wrong. If I use $-0.71 = \eta$ in the second equation (because everything I'm seeing says to make it negative), I get $\$-235$ as the optimal price.
Positive $\$235$ makes sense here, but I'm guessing the wrong sign means this is incorrect as well?