u(c_1, c_2)= \dfrac{\sigma}{\sigma-1}(c_1^{ \frac{\sigma-1}{\sigma}}-1)+\dfrac{\rho\sigma}{\sigma-1}(c_2^{ \frac{\sigma-1}{\sigma}}-1)
To determine the elasticity of substitution, we’ll first find the marginal rate of substitution:
\text{MRS}=\dfrac{\frac{\partial u}{\partial c_1}}{\frac{\partial u}{\partial c_2}} = \dfrac{c_1^{\frac{-1}{\sigma}}}{\rho c_2^{\frac{-1}{\sigma}}}=\dfrac{1}{\rho}\left(\dfrac{c_1}{c_2}\right)^{\frac{-1}{\sigma}}
Now we’ll take log both sides and get:
\ln(\text{MRS})=-\ln(\rho)-{\dfrac{1}{\sigma}}\ln\left(\dfrac{c_1}{c_2}\right)
which can be rewritten as
\sigma\ln(\text{MRS})=-\sigma \ln(\rho)+\ln\left(\dfrac{c_2}{c_1}\right)
Elasticity of substitution is simply:
\dfrac{d\ln\left(\dfrac{c_2}{c_1}\right)}{d\ln(\text{MRS})}=\sigma