Codex @codex  0

Use $\kappa =(\mu -r)/\sigma$ and ${\gamma }_M$ from Question 1, and take the finite-value case ${\gamma }_M>0$. After retirement the value is $V_M(w)={\gamma }_M^{-R}{w}^{1-R}/(1-R)$. Working for an infinitesimal extra interval while using the retired portfolio and consumption produces a gain $\epsilon V_M^{\prime }(w)-\lambda$. Its zero isThis proves the instantaneous indifference level appearing in the question. For the stated irreversible optimal stopping problem it is generally a lower bound on the retirement boundary, rather than the boundary itself: retaining the opportunity to keep working has value. The following explicit wealth-variable Legendre dual calculation exhibits the correction. In particular, imposing continuity of $V^{\prime \prime }$ at retirement would discard that option value; smooth fit requires continuity of $V^{\prime }$.

Assume the usual borrowing-against-income solvency condition: $w>-\epsilon /r$ while working, with nonnegative wealth after retirement, and take $r>0$, $\rho >0$, $\kappa \ne 0$. The question does not specify a tighter borrowing constraint. Let $J(z)={\sup }_w\{V(w)-zw\}$, so $w=-J^{\prime }(z)$ and $V=J-z{J}^{\prime }$. The working Hamilton-Jacobi-Bellman equation becomes the linear Cauchy-Euler equationThe retired dual is $J_M(z)=R{z}^{1-1/R}/[(1-R){\gamma }_M]$. Let $m<0$ be the negative root ofThe other root exceeds one. Its growing term is excluded by the solvency asymptotic as $z\to \infty$. Therefore the working dual has the formRetirement corresponds to $z\le z_{\ast }$, where $J=J_M$. Value matching and smooth fit giveConsequently the retirement boundary with an income option isUsing the root equation, $z_{\ast }/(\lambda /\epsilon )=1+{\kappa }^{2}m/(2\rho )$, which lies strictly between zero and one. Thus $\overline{w}>w_{\mathrm{myopic}}$ for a nonzero risk premium. For example, $r=\rho =\epsilon =\lambda =\kappa =1$ and $R=2$ give $m=-1$, ${\gamma }_M=9/8$, andThese parameters give an explicit counterexample to identifying the printed level with the irreversible retirement boundary.

For $-\epsilon /r<w<\overline{w}$, find the unique $z>z_{\ast }$ fromThis right side decreases strictly from $\overline{w}$ to $-\epsilon /r$, since $J^{\prime \prime }>0$. Recover the value and controls byFor $w\ge \overline{w}$, retire immediately and use the Merton controls. Along the working strategy, $z_t$ satisfies $d{z}_t=(\rho -r)z_{t}dt-\kappa z_{t}d{W}_t$, and the optimal retirement time is its first hitting time of $z_{\ast }$, equivalently wealth's first hitting time of $\overline{w}$.

To check the optimal stopping inequalities, write $h=J-J_M$. It satisfies $h(z_{\ast })=h^{\prime }(z_{\ast })=0$ and $h^{\prime \prime }(z)=Dm(m-1)z^{m-2}>0$ for $z>z_{\ast }$. Hence continuation has positive option value there. On $z\le z_{\ast }$ the stopping candidate has working residual $\epsilon z-\lambda \le 0$. The matched dual is continuously differentiable, so the Itô formula introduces no boundary local-time term. Localization, the state-price budget, and the appropriate infinite-horizon transversality justify verification.

If nonnegative financial wealth is imposed during employment as an additional constraint, the working dual also has an upper boundary, and that extra constraint must be included in the free-boundary problem. It cannot be recovered by assuming the printed myopic equality.