Question 11.8.1: Find the (partial) elasticities of z w.r.t. x when: (a) z = ......

(a) When finding the elasticity of Ax^{a}y^{b} w.r.t. x, the variable y, and thus Ay^{b}, is held constant. From Example 7.7.1 we obtain El_{x} z = a. In the same way, El_{y} z = b.

(b) It is convenient here to use Eq. (11.8.2).

\mathrm{El}_{x}z={\frac{\partial\ln z}{\partial\ln x}},{\mathrm{~and~}}\mathrm{~El}_{y}z={\frac{\partial\ln z}{\partial\ln y}}      (11.8.2)

Assuming all variables are positive, taking appropriate natural logarithms gives \ln z=\ln x+\ln y+x+y=\ln x+\ln y+e^{\ln x}+y. Hence \operatorname{El}_{x}z=\partial\ln z/\partial\ln x=1+e^{\ln x}=1+x.