Respuesta :
Answer:
Demand is inelastic at p = 9 and therefore revenue will increase with
an increase in price.
Step-by-step explanation:
Given a demand function that gives q in terms of p, the elasticity of demand is
[tex]E=|\frac{p}{q}\cdot \frac{dq}{dp} |[/tex]
- If E < 1, we say demand is inelastic. In this case, raising prices increases revenue.
- If E > 1, we say demand is elastic. In this case, raising prices decreases revenue.
- If E = 1, we say demand is unitary.
We have the following demand equation [tex]D(p)=-\frac{3}{4}p+29[/tex]; p = 9
Applying the above definition of elasticity of demand we get:
[tex]E(p)=\frac{p}{q}\cdot \frac{dq}{dp}[/tex]
where
- p = 9
- q = [tex]-\frac{3}{4}p+29[/tex]
- [tex]\frac{dq}{dp}=\frac{d}{dp}(-\frac{3}{4}p+29)[/tex]
[tex]\frac{d}{dp}\left(-\frac{3}{4}p+29\right)=-\frac{d}{dp}\left(\frac{3}{4}p\right)+\frac{d}{dp}\left(29\right)\\\\\frac{d}{dp}\left(-\frac{3}{4}p+29\right)=-\frac{3}{4}[/tex]
Substituting the values
[tex]E(9)=\frac{9}{-\frac{3}{4}(9)+29}\cdot -\frac{3}{4}\\\\E(9)=\frac{36}{89}\cdot -\frac{3}{4}\\\\E(9)=-\frac{27}{89}\approx -0.30337[/tex]
[tex]|E(9)|=|\frac{27}{89}| < 1[/tex]
Demand is inelastic at p = 9 and therefore revenue will increase with an increase in price.
