How should I find the point elasticity of $x$ with respect to $y$ double prime?

Question

suppose that $y =a x^b$ then what is the elasticity of $x$ with respect to $y$ double prime? I know that the elasticity of $y$ with respect to $x$ is constant here because its an exponential demand curve.

Answer 1

Elasticity of $Y$ with respect to $X$ is defined as

$$\varepsilon_{Y,X}\equiv \frac{\partial \log Y}{\partial \log X}.$$

Convince yourself that for $Y=AX^B$ for constant $A,B$, we have $\varepsilon_{Y,X}=B.$

In your case, assuming $a>0,b>2,$ then $y''=ab(b-1)x^{b-2}\implies x=\left(\frac{1}{ab(b-1)}y''\right)^{1/(b-2)}$ so

$$\varepsilon_{x,y''}=1/(b-2).$$