Throughout we work with the assumption that there is no force of interest (i.e. the interest rate is $i=1$). Consider a lottery $X$ that pays $1000$ with probability $e^{-1}$ and $0$ otherwise. Then its expected net present value is $1000e^{-1} + 0\cdot (1-e^{-1}) = 1000e^{-1}$.
Now fix a positive real number $\gamma$ and consider a utility function $u(w)$ with $u'(w) = w^{-\gamma}$. By definition, the certainly equivalent $CE$ of an agent with initial wealth $w>0$ is determined by $\mathbb{E}[u(w+X)]= u(w+CE)$. In our case we get $e^{-1}\cdot u(w+1000) + (1-e^{-1}) \cdot u(w) = u(w+CE)$. The possibilities for $u$ are $u(w) = \frac{w^{1-\gamma}}{1-\gamma}$ and $u(w) = \log w$ and we get the certainty equivalents $ CE = \big(e^{-1}(w+1000)^{1-\gamma} + (1-e^{-1})w^{1-\gamma}\big)^{\frac{1}{1-\gamma}} - w $ and $CE = e^{e^{-1}\log(w+1000)+(1-e^{-1})\log w} - w$.
One can check that $\lim_{w\to \infty} CE(w) = 1000e^{-1}$ and that $CE(w) < 1000e^{-1}$. Any intuitive reason why does this happen? How much does it have to do with lack of interest or risk aversion or perhaps constant relative risk aversion (note that $u(w)$ has this property)?
Any help appreciated!
