$$ u(c)=\frac{c^{1-\sigma}-1}{1-\sigma}, \quad \sigma \in [0,1) $$
$$ u'(c)=c^{-\sigma} $$
$$ u”(c)=-\sigma c^{-\sigma -1} $$
Risk Aversion is \( – \frac{u”}{u’}\). So,
$$ – \frac{u”}{u’}=\frac{ -\sigma c^{-\sigma -1} }{ u'(c)=c^{-\sigma} }=\frac{\sigma}{c_t} \leftarrow CRRA$$
How does the isoelastic utility function work?
Recall a Euler equation \(u'(c_t)=\beta (1+r)u'(c_{t+1})\).
$$ c_t^{-\sigma}=\beta (1+r) c_{t+1}^{-\sigma} $$
$$ \frac{c_t}{c_{t+1}}=( \beta (1+r) )^{-\frac{1}{\sigma}} =e^{ -\frac{1}{\sigma} ln(\beta(1+r)) }$$
That implies the consumption as a ratio over time is a constant, depending on \(\beta, r, \sigma\). Also, as \(\beta (1+r)\) is a very small number, \(ln(\beta (1+r))\approx \beta(1+r)\). Thus, \(\frac{c_t}{c_{t+1}}<0\).
In microeconomics, we always think of factors that grow constantly over time, e.g. constant saving rate.
Further study.