Question 3.9.3: Computing Elasticity of Demand and Changes in Revenue Suppos...

The elasticity of demand is given by

E=\frac{p}{f(p)} f^{\prime}(p)=\frac{p}{400(20-p)}(-400)=\frac{p}{p-20}.

We show a graph of E=\frac{p}{p-20} in Figure 3.98. Observe that E < −1 if

or p>-(p-20) \text {. } \quad \text { Since } p-20<0 \text {. }

Solving this gives us  2 p>20

or  p>10 \text {. }

To analyze revenue, we compute R=p f(p)=p(8000-400 p)=8000 p-400 p^2. Revenue decreases if R^{\prime}(p)<0 . \text { From } R^{\prime}(p)=8000-800 p, we see that R^{\prime}(p)=0 if p=10 \text { and } R^{\prime}(p)<0 \text { if } p>10. Of course, this says that the revenue decreases if the price exceeds 10.