Question 7.7.1: Find the elasticity of f (x) = Ax^b, where A and b are const......

Essential Mathematics for Economic Analysis [1185377]

Find the elasticity of $f (x) = Ax^{b},$ where A and b are constants, with $A ≠ 0.$

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In this case, $f^{\prime}(x)=A b x^{b-1}.\mathrm{Hence},\mathrm{E}\mathrm{l}_{x}(A x^{b})=(x/A x^{b})A b x^{b-1}=b,\mathrm{so}$

$f(x)=A x^{b}~\Rightarrow~\mathrm{El}_{x}\,f(x)=b$ (7.7.3)

The elasticity of the power function $Ax^{b}$ w.r.t. x is simply the exponent b. So this function has constant elasticity. In fact, it is the only type of function which has constant elasticity. This is shown in Exercise 9.9.6.

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