= Retirement boundary with an income option
{title2=$\overline w=\gamma_M^{-1}\left[\frac{\lambda r}{\varepsilon\rho}\frac{m}{m-1}\right]^{-1/R}$}
Consider irreversible <optimal stopping> of employment paying income $\varepsilon>0$ and imposing disutility $\lambda>0$. Assume $r,\rho>0$, nonzero <market price of risk> $\kappa$, a positive <Merton consumption constant>, and solvency allowing borrowing against perpetual income up to $\varepsilon/r$ before retirement. Let $m<0$ solve $\kappa^2m(m-1)/2+(\rho-r)m-\rho=0$. The retirement threshold is the displayed wealth level.
To derive it, apply the <wealth-variable Legendre dual>. The working dual equals the retired dual plus $h(z)=\varepsilon z/r-\lambda/\rho+Dz^m$. Excluding the growing homogeneous solution enforces the solvency asymptotic. Value matching and <smooth fit> at the retirement marginal value $z_*$ give $h(z_*)=h'(z_*)=0$, hence $z_*=(\lambda r/(\varepsilon\rho))m/(m-1)$ and $D=-\varepsilon z_*^{1-m}/(rm)>0$. The retired inverse marginal value gives $\overline w=\gamma_M^{-1}z_*^{-1/R}$.
The root equation implies $z_*/(\lambda/\varepsilon)=1+\kappa^2m/(2\rho)<1$. Thus the boundary exceeds the instantaneous gain-zero level $\gamma_M^{-1}(\varepsilon/\lambda)^{1/R}$. A tighter borrowing constraint changes this free-boundary problem. Smooth fit matches the first derivative; matching the second derivative would incorrectly remove the income option.
Back to article page
