For the linear demand function $D(p)=90-p \quad$ used in Practice Problem 3 and graphed on page 314 :
a. Find a formula for the elasticity of demand at any value of $p .$ [Hint: Use the formula on page 311 with the given demand function.]
b. Show that the elasticity is zero at the top of the line (that is, for $p=0$ ).
c. Show that the elasticity approaches infinity as $p$ approaches the bottom of the line.
d. Show that the elasticity is $\frac{1}{2}$ at the midpoint of the line.
e. Show that statements $(\mathrm{b}),(\mathrm{c})$, and $(\mathrm{d})$ are true for any linear demand function $D(p)=a-b p \quad$ for any positive constants $a$ and $b ?[$ Hint $:$ Find an expression for the elasticity at any value of $p$, as you did in part (a).]
Exponential And Logarithmic Functions
Two Applications to Economics: Relativeā¦