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After scaling wealth by habit and applying the wealth-variable Legendre dual, the effective consumption shadow price is proportional to $\mathcal{D}=z-\lambda Rz{J}^{\prime }+\lambda (R-1)J$. Its power in the utility conjugate keeps the dual equation nonlinear. When $\lambda =0$, habit is fixed and the dual equation is a linear Euler differential equation with effective relative risk aversion $R+\alpha$.

For the wealth-variable Legendre dual $J(z)={\sup }_{w\ge 0}[V(w)-zw]$, a finite slope $z_{\ast }=V^{\prime }(0+)$ at an absorbing zero-wealth boundary makes $J$ identically zero for $z\ge z_{\ast }$. Matching gives $J(z_{\ast })=J^{\prime }(z_{\ast })=0$. It does not generally give $J^{\prime \prime }(z_{\ast }^{-})=0$: killing at ruin permits the dual curvature to jump.

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.

Available portfolio wealth and the ruin boundary. Put $h=\bar{r}D$, the constant interest payment on the fixed loan. The loan principal is already included in available portfolio wealth; it is not a growing portfolio holding. ThereforeIn particular the interest outflow is $\bar{r}D$, not $(\bar{r}-r)D$. Writing $w=x+D$ would instead give net portfolio wealth drift $rx+(\mu -r)\theta -(\bar{r}-r)D-c$, which explains the distinction.

Let ${\tau }_0$ denote the ruin time, to avoid confusing it with a fixed terminal horizon. The objective stops at ${\tau }_0$; consequently the absorbing boundary is $V(0)=0$, without an obligation to keep financing the loan after ruin. Dynamic programming gives, for $w>0$,For increasing strictly concave value, put $z=V^{\prime }(w)>0$ and use inverse marginal utility $I$. The optimal controls and the optimized HJB equation arewhere $\tilde{U}(z)={\sup }_{c\ge 0}[U(c)-zc]$. For CRRA utility with $0<R<1$, write $p=1-R$ and $q=1-1/R<0$. Then

Dualization and the printed constant. Use the convex wealth-variable Legendre dualAt an interior maximizing portfolio wealth, $J^{\prime }=-w$, $J^{\prime \prime }=-1/V^{\prime \prime }>0$ and $V=J-z{J}^{\prime }$. The dual HJB equation is the linear Euler differential equationA trial power $z^t$ givesDirect substitution givesThe PDF prints an additional factor $1/2$ before ${\kappa }^2/(2R)$ in its definition of ${\gamma }_M$. That printed definition is inconsistent with its own identity for $Q(q)$. The expression above is the one used here; assume this corrected ${\gamma }_M>0$.

Solution when $\kappa \ne 0$. Assume positive discount $\rho$, nonzero $\sigma$, and $h>0$. Let $m>0$ and $n<0$ be the two roots of $Q$:Since $Q(q)<0$, $n<q<0<m$. Put $C=R/(p{\gamma }_M)$. For $r\ne 0$, the general interior solution isThe appropriate large-wealth condition is the Merton consumption-investment problem boundIndeed any original control consumes $c+h$ in the debt-free comparison model until ruin, and $U(c+h)\ge U(c)$. Dualizing this bound gives $0\le J(z)\le C{z}^q$. Because $n<q$, convexity and this upper bound force $B=0$ as $z\downarrow 0$.

At the other endpoint portfolio wealth reaches zero. If $z_{\ast }=V^{\prime }(0+)$, the dual ruin boundary with debt service requiresSolving these two equations givesFor $r>0$, $m>1$, and for $r<0$, $0<m<1$, so the quantity defining $z_{\ast }$ is positive in either case. These formulas determine the entire value. For each $w>0$, choose the unique $z\in (0,z_{\ast })$ satisfyingThenBoth terms in the bracket are positive, even when $r<0$ and $A<0$. Thus $J^{\prime \prime }>0$, $-J^{\prime }$ decreases from infinity to zero as $z$ increases, and the portfolio wealth inversion really is unique. The extended dual is continuously differentiable and convex.

There is no additional condition $J^{\prime \prime }(z_{\ast }^{-})=0$. In factAvailable portfolio wealth is killed at zero; the portfolio can have a nonzero limiting volatility immediately before ruin. Imposing a reflecting-boundary or zero-curvature condition would solve a different problem.

For $r=0$, the linear forcing resonates with the root $m=1$. Put $b=\rho +{\kappa }^2/2=Q^{\prime }(1)$. The dual and boundary constants instead areExtend by zero for $z\ge z_{\ast }$. Hereand the controls remain $c=z^{-1/R}$ and $\theta =\kappa z{J}^{\prime \prime }/\sigma$, with $J^{\prime \prime }=h/(bz)+Cq(q-1)z^{q-2}>0$.

Verification and transversality. The candidate is nonnegative, increasing, strictly concave and zero at ruin. Its HJB equation makes the discounted value plus accrued utility a local supermartingale for every admissible control, and a local martingale for the stated feedback. Localization at positive lower and finite upper portfolio wealth levels gives the finite-horizon comparison. The investment value transversality condition follows from the same debt-free bound: applying the Itô formula to $w^p$ and maximizing its risky term givesThe nonnegative consumption and debt-service drifts only decrease this bound. Thus the expected terminal candidate tends to zero. The feedback has at most linear growth, including a finite limit as portfolio wealth decreases to zero; stopping it at ruin gives an admissible policy. Letting localization levels and then the horizon tend to their limits proves that the candidate is the value, rather than just a formal dual solution. If $D=0$, the absorbing-debt boundary disappears and the ordinary Merton consumption-investment problem formula is recovered.

Zero market price of risk. With $\sigma \ne 0$ and $\kappa =0$, the dual equation is first order. If $\rho >r$, the preceding formulas remain valid with $m=\rho /(\rho -r)$ for $r\ne 0$, and the logarithmic formula with $b=\rho$ for $r=0$; there is no $B$ term and ${\theta }^{\ast }=0$. The same boundary and transversality argument verifies this deterministic consumption policy.

The remaining finite-value case has $r>0$ and $pr<\rho \le r$. Put ${\gamma }_0=(\rho -pr)/R$, $w_c=h/(rR)$ andThe correct convex dual and its corresponding value areThe two value branches have the same value and derivative at $w_c$. Above $w_c$, hold no stock and consume ${\gamma }_0(w-h/r)$; the surplus over $h/r$ grows at rate $r-{\gamma }_0\ge 0$. If $\rho =r$, the lower branch is attained by zero stock holding and constant consumption $ph/R$: portfolio wealth solves $\dot{w}=r(w-w_c)$ until ruin, and direct integration gives $z_{c}w$.

If $\rho <r$ and $0<w<w_c$, the lower branch is a supremum attained in a limit of increasingly rapid fair stock lotteries between zero and $w_c$, followed by the upper-branch policy on success. The success probability tends to $w/w_c$ and the fixed service cost during the lottery tends to zero. This is possible because unrestricted dollar holdings in the nonzero-volatility stock produce a fair Brownian motion exposure even when its excess drift is zero. The supporting linear branch has optimized waiting residual $(r-\rho )z_c(w-w_c)<0$; the fast lotteries, rather than a finite feedback optimizer, supply the missing control limit. The piecewise candidate is concave, has nonpositive waiting residual everywhere, and the preceding moment bound still supplies an upper-bound verification. This degenerate case can have a supremum without an ordinary maximizing strategy.

Normalize the constant relative risk aversion utility as $U(c)=c^{1-R}/(1-R)$. The printed specification of $U^{\prime }$ alone also permits an additive constant $K$; that would add $K/\rho$ to the value, so the stated homogeneity holds for the normalized value. SetMultiplying initial wealth, the historical maximum, dollar investments, and consumption by $a>0$ multiplies every term of the wealth equation by $a$, including the tax term. It maps admissible policies bijectively and multiplies the normalized reward by $a^{1-R}$. Therefore the value scales asThe historical maximum means ${\bar{w}}_t=\max ({\bar{w}}_0,{\sup }_{0\le s\le t}{w}_s)$ when initially $w<\bar{w}$.

Inside the state region $w<\bar{w}$, the maximum is locally constant. The Hamilton-Jacobi-Bellman equation isAt $w=\bar{w}$, the finite-variation contribution in the Itô formula is $(V_{\bar{w}}-\tau V_w)d\bar{w}$. The high-water mark tax boundary condition is consequentlyFor an increasing, strictly concave function of wealth, optimizing the two controls givesThe consumption convex conjugate isSubstitution of the homogeneity derivatives gives the reduced equation and boundary conditionThe controls in scaled variables are $c^{\ast }/\bar{w}=(v^{\prime }{)}^{-1/R}$ and ${\theta }^{\ast }/\bar{w}=-(\mu -r)v^{\prime }/({\sigma }^2{v}^{\prime \prime })$.

Use the wealth-variable Legendre dualSince $v^{\prime \prime }<0$, the inverse map satisfies $J^{\prime }=-x$ and $J^{\prime \prime }=-1/v^{\prime \prime }>0$. Inserting $v=J-z{J}^{\prime }$ into the reduced Hamilton-Jacobi-Bellman equation gives a linear Euler equationFor the Euler differential equation operator on the left, its action on $z^t$ is $Q(t)z^t$. Write $b=1-1/R$. Under ${\gamma }_M=-Q(b)>0$, the particular solution is $-\tilde{U}(z)/Q(b)$. For the nondegenerate case $\kappa \ne 0$, the two characteristic roots are $-\alpha <0<t_{+}$, with $t_{+}>1$ because $Q(0)=-\rho$ and $Q(1)=-r$. Thus the general solution also contains $A(z/z_{\ast }{)}^{-\alpha }+B{z}^{t_{+}}$.

Here $z_{\ast }=v^{\prime }(1)$ and the relevant dual domain is $z\ge z_{\ast }$. The inverse wealth ratio must have $x=-J^{\prime }(z)\to 0$ as $z\to \infty$. The particular solution and the negative-root term have derivatives tending to zero, whereas $B{z}^{t_{+}}$ has an unbounded derivative unless $B=0$. Hence the admissible dual solution has the claimed formThe quadratic-root formulation presupposes a nonzero market price of risk. If $\kappa =0$, the dual equation becomes first order; it is treated directly or by an appropriate nondegenerate limit rather than by assuming two quadratic roots.

The inverse map and the tax boundary provide two equations at $z_{\ast }$:Because ${\tilde{U}}^{\prime }(z)=-z^{-1/R}$, these becomeLet $D=\alpha R+R-1$. Eliminating $z_{\ast }^{1-1/R}/{\gamma }_M$ gives $-DA=\tau z_{\ast }$. Also $D=R(\alpha +b)>0$, since $Q(b)<0$ puts $b$ strictly between the two roots. The derivative boundary then gives $z_{\ast }^{-1/R}/{\gamma }_M=(D+\alpha \tau )/D$. Therefore the constants areThere is an important admissibility qualification in the printed conclusion. With $x(z)=-J^{\prime }(z)$,The second term is negative, but decays faster than the first because $\alpha +1>1/R$. Direct differentiation at the boundary givesA wealth-variable Legendre dual of a concave value must be a convex function. Thus the printed smooth tax-paying solution requires $\alpha \tau <1$, with the limiting case $\alpha \tau =1$ allowed as a degenerate boundary. The assumptions ${\gamma }_M>0$ and $0<\tau <1$ do not imply this: for example $R=2$, $r=0.03$, $\rho =0.1$, $\kappa =0.2$, and $\tau =0.5$ give $\alpha =(5+\sqrt{105})/4>2$ and ${\gamma }_M=0.07$, hence negative boundary dual curvature.

For higher tax the investor can avoid raising the historical maximum. The wealth-cap investment boundary replaces the tax-paying equality by $J^{\prime \prime }(z_{\ast })=0$, alongside $J^{\prime }(z_{\ast })=-1$. Solving these equations givesThese are the same constants with $\tau$ replaced by ${\tau }_{\mathrm{eff}}=1/\alpha$. More generally the admissible two-regime expression usesin the constants, while the claimed formulas themselves describe the tax-paying regime. In the cap regime the portfolio volatility vanishes at $x=1$ and the consumption rate there is $r+{\kappa }^2(\alpha +1)/(2R)>r$, so the wealth drift points inward and no new maximum is required. The remaining boundary inequality is $(V_{\bar{w}}-\tau V_w)/V_w=1/\alpha -\tau \le 0$, consistent with avoiding costly maximum increases. For $z>z_{\ast }$, dual curvature is positive because the negative term in $J^{\prime \prime }$ decays faster than the positive term. Thus this qualification repairs an actual missing parameter restriction rather than a TeX transcription error.

The PDF prints $w_0=0$ in the conditioning of the value function. Taken literally, nonnegative wealth and the state-price budget constraint force both wealth and consumption to remain zero, and the utility for $R>1$ has value $-\infty$. The meaningful value function underlying the subsequent requests uses $w_0=w>0$; the following calculation makes that source correction explicit.

Write $h_t={\bar{c}}_t>0$. Differentiating the exponentially weighted consumption habit gives the habit-state dynamicsMultiplying initial wealth, initial habit, investments, and consumption by $a>0$ multiplies the wealth and habit paths by $a$. The ratio $c/h$ is unchanged, while $U(ac)=a^{1-R}U(c)$. Thus the homogeneous value isThis is a multiplicative habit utility model: higher habit makes utility more negative at fixed consumption.

Set $S=R+\alpha >1$, $K=(S-1)/(R-1)>0$, and $\kappa =(\mu -r)/\sigma$. The instantaneous reward isFor smooth increasing, strictly concave wealth value, the Hamilton-Jacobi-Bellman equation isThe effective consumption shadow price with habit changes from $V_w$ to $D=V_w-\lambda V_h$. For $D>0$, consumption maximization givesFor $D<0$ the supremum is infinite, and for $D=0$ its zero supremum is approached only as consumption tends to infinity; a finite interior optimum therefore requires $D>0$. The portfolio maximum is $-{\kappa }^2{V}_w^2/(2V_{ww})$.

Let $x=w/h$, $z=v^{\prime }(x)$, and defineThe homogeneity derivatives areConsequently $D=h^{-R}d(x)$, and the reward conjugate is $h^{1-R}H(d(x))$. The reduced habit equation isFor completeness its feedback controls are $c^{\ast }/h=(K/d{)}^{1/S}$ and ${\theta }^{\ast }/h=-(\mu -r)v^{\prime }/({\sigma }^2{v}^{\prime \prime })$.

The wealth-variable Legendre dual $J(z)=v(x)-xz$ satisfies $J^{\prime }=-x$, $J^{\prime \prime }=-1/v^{\prime \prime }$, and $v=J-z{J}^{\prime }$. The effective shadow price becomesTherefore the dual equation for multiplicative habit investment iswith $\mathcal{D}>0$. The dependence of the reward conjugate on $J$ and $J^{\prime }$ is the remaining nonlinearity.

When $\lambda =0$, habit is fixed and the equation becomes a linear Euler equationThe forcing is a pure power $z^{1-1/S}$. Thus the Euler differential equation method gives a power particular solution plus the two homogeneous characteristic powers in the nondegenerate case. Economically, this is the Merton consumption-investment problem with effective relative risk aversion $S=R+\alpha$ and a constant reward multiplier. Writingits value and controls areIndeed $H(z)/{\delta }_S$ solves the dual equation, since its characteristic polynomial at $1-1/S$ equals $-{\delta }_S$. Unlike the $\lambda >0$ case, the fixed habit creates no feedback coupling between the dual value and the shadow price of consumption.

Consider irreversible optimal stopping of employment paying income $\epsilon >0$ 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 $\epsilon /r$ before retirement. Let $m<0$ solve ${\kappa }^{2}m(m-1)/2+(\rho -r)m-\rho =0$. 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 $h(z)=\epsilon z/r-\lambda /\rho +D{z}^m$. Excluding the growing homogeneous solution enforces the solvency asymptotic. Value matching and smooth fit at the retirement marginal value $z_{\ast }$ give $h(z_{\ast })=h^{\prime }(z_{\ast })=0$, hence $z_{\ast }=(\lambda r/(\epsilon \rho ))m/(m-1)$ and $D=-\epsilon z_{\ast }^{1-m}/(rm)>0$. The retired inverse marginal value gives $\overline{w}={\gamma }_M^{-1}{z}_{\ast }^{-1/R}$.

The root equation implies $z_{\ast }/(\lambda /\epsilon )=1+{\kappa }^{2}m/(2\rho )<1$. Thus the boundary exceeds the instantaneous gain-zero level ${\gamma }_M^{-1}(\epsilon /\lambda {)}^{1/R}$. 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.

A wealth-cap investment boundary prevents wealth from exceeding a fixed historical maximum. In the wealth-variable Legendre dual for wealth normalized by that maximum, $J^{\prime }(z_{\ast })=-1$ and vanishing optimal portfolio volatility gives $J^{\prime \prime }(z_{\ast })=0$. For power utility and a high-water tax, the smooth tax-paying solution has boundary dual curvature $(1-\alpha \tau )/(R{z}_{\ast })$, so it cannot be admissible when $\alpha \tau >1$.