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A constant relative risk aversion utility has $-c{U}^{\prime \prime }(c)/U^{\prime }(c)=R>0$. Up to a positive scale and an additive constant, it is $U(c)=c^{1-R}/(1-R)$ for $R\ne 1$ and $U(c)=\log c$ for $R=1$. Power utility gives multiplicative homogeneity of an investment value; logarithmic utility gives additive scaling.

Two related properties of an extensive quantity must be distinguished. Additivity says that a quantity $X$ for independent subsystems A and B obeysThe subsystems may be in different states. Extensivity expressed through a homogeneous function says that the equilibrium fundamental relation obeyswith composition held fixed. This is degree-one homogeneity along a replication ray; it does not say that internal energy is a linear function of arbitrary independent changes of entropy, volume and amount of substance.

Additivity gives integer replication by repeated addition. Subdividing a uniform bulk sample into $b$ equivalent parts gives rational scaling $a/b$; continuity then extends it to positive real factors. This argument requires negligible boundary contributions and a consistently prepared state. Conversely, a degree-one homogeneous function need not be additive between different rays: $f(x,y)=\sqrt{x^2+y^2}$ has $f(\lambda x,\lambda y)=\lambda f(x,y)$, but $f(1,0)+f(0,1)=2\ne \sqrt{2}=f(1,1)$. Thus a scaling identity alone cannot justify summing arbitrary states.

For finite samples, the surface correction to thermodynamic free energy can matter. Under geometric enlargement with volume multiplied by $\lambda$, surface area grows as ${\lambda }^{2/3}$, so a surface energy term is not a bulk extensive quantity. Replicating separated droplets retains every original surface, whereas merging them changes the total surface area. Likewise, appreciable mutual gravitational potential energy makes interacting subsystems fail the independent-copy premise. The thermodynamic limit explains when bulk scaling becomes accurate; it does not remove these qualifications for every small or strongly interacting system.

With the binary Brownian drift filter, a stock of observed excess drift $b(x)=\mu -r+\sigma a\tanh (ax)$ is driven by the same Brownian motion as its information state $X$. The Hamilton-Jacobi-Bellman equation for portfolio wealth $w$, dollar stock holding $\theta$ and value $J(w,x)$ includes $\sigma \theta J_{wx}$. Consequently ${\theta }^{\ast }=-[b(x)J_w+\sigma J_{wx}]/({\sigma }^2{J}_{ww})$. For CRRA utility, homogeneity gives $J=K(x)w^{1-R}/(1-R)$ and hence the displayed stock fraction. The second term is intertemporal hedging demand; replacing the unknown drift by its posterior mean while omitting this term does not generally give the optimal policy.

Multiplicative habit utility modifies current consumption utility through its ratio to habit $h>0$. With $U(c)=c^{1-R}/(1-R)$, $R>1$, and $\alpha >0$, it is $h^{\alpha }{c}^{1-R-\alpha }/(1-R)$. Its homogeneity in $(c,h)$ has degree $1-R$, while consumption holding habit fixed has effective relative risk aversion $R+\alpha$.

Choose homogeneous polynomials $F,G$ defining the two projective plane curves. Their absence of a common irreducible component makes them coprime. For a point $O=[1:0:0]$ outside both curves, they have degrees $n,m$ in $X$ and nonzero constant leading coefficients. The resultantis therefore nonzero. Indeed, a vanishing resultant over $k(Y,Z)$ would give a common factor there, and Gauss's lemma would give a common component of the original curves.

The homogeneity of $F,G$ makes $R$ a homogeneous polynomial of degree $nm$: scaling $(Y,Z)$ by $a$ scales their $X$-roots by $a$, and the product of the $nm$ differences of roots scales by $a^{nm}$. Every point in the intersection projects from $O$ to a zero of $R$ in the projective line. Each such projection line contains only finitely many points of either curve, since it passes through $O$ outside them. Thus the intersection is finite already.

We may now choose $O$ also outside the finitely many joining lines of distinct intersection points. Projection is then injective on the intersection. A nonzero degree-$nm$ homogeneous polynomial on the projective line has at most $nm$ distinct zeros, so the resultant bound for intersections of plane curves givesThis derives the required cardinality bound without assuming the full intersection-multiplicity statement of Bézout's theorem.

Put $p=1-R$, $b=\mu -r1$, $A=\sigma {\sigma }^T$ and $q=R+\epsilon$. First assume the covariance matrix $A$ is invertible and initial portfolio wealth is positive. Introduce the accumulated fee processThe Itô formula and the self-financing portfolio equation giveThis converts the path-dependent payment into a terminal-utility problem with state $y$. The volatility-penalized terminal fee changes its drift as well as its spot volatility.

Let $J(t,y)$ be the manager's value function. Its Hamilton-Jacobi-Bellman equation, with terminal condition $J(T,y)=y^p/p$, isThe homogeneity of CRRA utility suggests $J(t,y)=f(t)y^p/p$. Since $y{J}_y=f(t)y^p$ and $y^2{J}_{yy}=-Rf(t)y^p$, the portfolio-dependent expression isIts coefficient is positive even when $p<0$. Completing the square givesThus the stock fractions are constant. Substitution into the Hamilton-Jacobi-Bellman equation yieldsIn particular, multiplying the fee by $a$ does not affect the optimal investment portfolio.

To prove global optimality rather than merely solve a formal Hamilton-Jacobi-Bellman equation, write $v_t={\sigma }^T{\pi }_t$, $v^{\ast }={\sigma }^T{\pi }^{\ast }$ and let $y^{\ast }$ be the fee under the proposed constant policy. Then $b^T\pi =q{v}^{\ast }\cdot v$. Define a probability $Q$ byThis density is a true martingale because $v^{\ast }$ is constant. By the Girsanov theorem, $W_t^Q=W_t-p{v}^{\ast }t$ is a Brownian motion under $Q$. The geometric Brownian motion formula for the fee ratio givesFor admissible, locally square-integrable exposures with finite accumulated quadratic variation, setThis stochastic exponential is a positive local martingale, so $E_Q{Z}_T\le 1$. Moreover,For $p>0$, the exponent $p/(1+\epsilon )$ lies between zero and one. The Jensen inequality for this concave function gives $E_Q[Z_T^{p/(1+\epsilon )}]\le 1$. For $p<0$, the power is convex and decreasing, so the Jensen inequality gives $E_Q[Z_T^{p/(1+\epsilon )}]\ge (E_Q{Z}_T{)}^{p/(1+\epsilon )}\ge 1$; an infinite moment only makes this inequality stronger. Dividing by $p$, with its corresponding sign, proves $E[U(y_T)]\le E[U(y_T^{\ast })]$ in both cases. Equality is achieved by the constant policy.

For an ordinary terminal-wealth agent with relative risk aversion $q$, the same Hamilton-Jacobi-Bellman equation calculation, with no fee penalty, maximizes $b^T\pi -q{\pi }^{T}A\pi /2$. It therefore gives exactly the same optimal investment portfolio. For $q\ne 1$, the preceding stochastic exponential and Jensen inequality verification applies with penalty zero and exponent $1-q$, proving global optimality for this agent too. The spot volatility penalty adds $\epsilon$ to effective relative risk aversion. If $q=1$, the printed expression for $U_0$ has a zero denominator; its appropriate limit, after subtracting an irrelevant constant, is $U_0(x)=\log x$, which still gives ${\pi }^{\ast }=A^{-1}b$. In this logarithmic case, the terminal log-wealth difference from the candidate equals $\log Z_T$ for a positive stochastic exponential $Z$, so $E[\log Z_T]\le \log E[Z_T]\le 0$ proves its optimality.

If $A$ is singular, the same conclusion holds in the absence of arbitrage, provided $b$ lies in the image of a linear map $x\mapsto Ax$: replace $A^{-1}$ by the Moore-Penrose inverse, and add any vector in the kernel of $A$ to ${\pi }^{\ast }$. These additions change neither returns nor Brownian portfolio exposures. If $b$ has a nonzero component in that kernel, a zero-volatility position has a nonzero excess return. Scaling its profitable sign gives arbitrage, so a finite attained optimum as claimed requires this degeneracy to be excluded.

Interpret the exponential distribution change time as an exogenous time independent of the stock's Brownian motion, or, equivalently for this calculation, as having constant conditional intensity $\lambda$ while the rate remains high. Its marginal distribution alone would not imply that conditional intensity if the change time carried information about stock returns. Write $r_1=r+\epsilon$, $p=1-R$, $B_1=B(r_1)$, and let $V_1$ be the prechange value function.

Over a short interval $dt$, a rate change occurs with probability $\lambda dt+o(dt)$ and changes continuation value from $V_1(w)$ to $V_0(w)$. The dynamic programming equation is consequentlyThis jump term distinguishes the exponential interest-rate switch from a permanent higher bank rate.

By the homogeneity of CRRA utility, put $V_1(w)=K_1{w}^p/p$, with $K_1>0$. Differentiating the strictly concave expressions in $c$ and $\pi$ givesThe optimized consumption terms contribute $R{K}_1^{1-1/R}{w}^p/p$, and the optimized investment terms contribute $K_1{B}_1{w}^p$. With $K_0={\gamma }_0^{-R}$, the Hamilton-Jacobi-Bellman equation reduces toEquivalently,The left side is continuous and strictly increasing for ${\gamma }_1>0$, with limits zero and infinity. Thus $D_1>0$ gives one and only one positive root. This scalar equation is an explicit characterization for arbitrary real $R$; elementary closed forms are not generally available. For example, when $R=2$,Use $({\pi }_1^{\ast },{\gamma }_1)$ until the rate changes, and $({\pi }_0^{\ast },{\gamma }_0)$ afterwards. If the two rates coincide, the root is ${\gamma }_1={\gamma }_0$; as $\lambda$ tends to infinity, it also tends to ${\gamma }_0$.

For a direct attainability check, use the proposed prechange constant fractions. The geometric Brownian motion moment is $E[w_t^p]=w^p\exp [p(B_1-{\gamma }_1)t]$ on the prechange evolution. Conditioning on the independent exponential distribution time therefore givesThe denominator is positive because the scalar equation gives $D_1+p{\gamma }_1={\gamma }_1+\lambda K_0/K_1>0$. Hence the candidate policy actually achieves the stated value.

One can also verify the upper bound for both signs of $p$ without relying on a sign-sensitive terminal-value argument. Let $j_t$ be the current regime, $w^{\ast }$ the candidate portfolio wealth, andIf $N_t=1_{\{\tau \le t\}}$ and ${\tilde{N}}_t=N_t-\lambda (t\wedge \tau )$, the Itô formula and the scalar equation givewhere the jump term stops after the switch and continuous coefficients are understood predictably. Thus $M$ is a positive martingale deflator: for any admissible portfolio wealth $w$, the process $M_t{w}_t+{\int }_0^t{M}_s{c}_{s}ds$ is a nonnegative local martingale. It follows thatFor the candidate policy, $E[M_t{w}_t^{\ast }]\to 0$. Indeed its surviving prechange term decays at rate $D_1+p{\gamma }_1>0$, and its postchange terms decay at rate ${\gamma }_0>0$; integrating over the switch time preserves convergence to zero. The finite-time moment computations also make its budget equality exact. Consequently $E{\int }_0^{\infty }{M}_t{c}_t^{\ast }dt=M_0{w}_0$. Apply the concave supporting-tangent inequalityand integrate in time and expectation. The budget inequality proves that no admissible policy exceeds the candidate. This verifies the value function, including when $R>1$.

Both ${\gamma }_0>0$ and $D_1>0$ are needed for a finite prechange answer. If the postchange value is already infinite, that problem persists after the almost surely finite switch. If ${\gamma }_0>0$ but $D_1\le 0$, then for $p>0$ a small constant prechange consumption fraction makes the expected utility function before the switch diverge, or makes its supremum diverge when $D_1=0$. For $p<0$, the optimal no-consumption terminal moment before the switch is $w^p{e}^{p{B}_{1}t}$; withdrawals only increase this negative-power moment. The expected magnitude of the negative postchange continuation utility function therefore contains the divergent integral $\lambda K_0{w}^p{\int }_0^{\infty }{e}^{-D_{1}t}dt/|p|$. Thus these cases yield respectively $+\infty$ or $-\infty$, rather than a positive-root solution.

Choose the utility function primitive $U(c)=c^{1-R}/(1-R)$. Any other primitive with the same derivative differs by a constant $C$; this adds $C/\rho$ to the value function and does not alter the policy. Put $p=1-R$ and let ${\pi }_t$ be the stock fraction. The self-financing portfolio equation, including consumption withdrawals, isFor a time-independent infinite-horizon value function $V$, the Hamilton-Jacobi-Bellman equation isScaling initial portfolio wealth and all consumption rates by the same positive number preserves admissibility and scales utility function by its $p$th power. This homogeneity gives $V(w)=K{w}^p/p$. In particular,The consumption first-order condition $c^{-R}=V^{\prime }(w)$ and stock first-order condition $w(\mu -r)V^{\prime }(w)+{\sigma }^2\pi w^2{V}^{\prime \prime }(w)=0$ giveThe second derivatives of the optimized expressions are negative, so these are their global maximizers. Writing ${\gamma }_M=K^{-1/R}$ and substituting into the Hamilton-Jacobi-Bellman equation givesThereforeThe given inequality is precisely ${\gamma }_M>0$, so the Merton consumption-investment problem value is $V(w)={\gamma }_M^{-R}{w}^{1-R}/(1-R)$. This derives the requested policy; no additional verification argument is needed here.

Use the primitive $U(c)=c^{1-R}/(1-R)$, as in Question 3, and the usual positive spot volatility convention $\sigma >0$. An additive utility function constant shifts both agents' value functions equally and does not affect their policies or the comparison. SetThe Itô formula applied to the stock price and the binary Brownian drift filter givesLet ${\theta }_t$ denote the dollar stock holding. The separated state equations areBoth states use the same observed Brownian motion. In particular, their quadratic covariation is $d[w,X{]}_t=\sigma {\theta }_{t}dt$.

Let $J(w,x)$ be the Bayesian value function. Its Hamilton-Jacobi-Bellman equation isThe cross derivative is necessary: learning and portfolio wealth are correlated through the same observation shock. At a smooth, finite, strictly concave value, the optimal consumption and stock holding areThe number of shares is ${\theta }^{\ast }/S$, and the wealth fraction is ${\theta }^{\ast }/w$.

More explicitly, homogeneity gives $J(w,x)=K(x)w^p/p$, where $p=1-R$. ThenThe additional term is the learning hedge in a binary-drift investment model, a form of intertemporal hedging demand. Merely putting the Bayesian posterior mean drift into a constant-coefficient investment formula would omit it. Eliminating the controls gives the coefficient equationThe economic solution is selected by admissibility and the infinite-horizon value condition, rather than by an arbitrary solution of this ordinary differential equation.

For the comparison, now suppose $0<R<1$ and $\mu >r$. Write $m=\mu -r>0$, $b_{+}=m+\sigma a$, andWhen ${\gamma }_{+}>0$, the informed agent's Merton consumption-investment problem has valueThese are the policies available to an agent who knows the true drift is $+a$.

For any admissible Bayesian policy, not only its optimum, defineThe integration variable is $s$ throughout the accumulated utility function. Applying the Itô formula in the Bayesian observation filtration giveswhereThe informed-agent Hamilton-Jacobi-Bellman equation says ${\sup }_{c,\theta }{H}_{b_{+}}=0$. Maximizing the stock term at any excess drift $b$ givesSince $V_{+}^{\prime \prime }<0$, this maximum increases with $b^2$. The binary Brownian drift filter satisfies $|h(x)|\le a$, and thereforeIt follows that $H_{b(x)}(w,c,\theta )\le {\sup }_{c,\theta }{H}_{b(x)}\le {\sup }_{c,\theta }{H}_{b_{+}}=0$. Thus $Y$ is a local supermartingale. Because $0<R<1$, both accumulated utility function and $V_{+}$ are nonnegative. A nonnegative local supermartingale is a genuine supermartingale, by localization and the conditional Fatou lemma. This proves the requested assertion under the Bayesian probability law, without conditioning that law on the favorable drift.

ConsequentlyThe monotone convergence theorem lets $t$ tend to infinity. Taking the supremum over Bayesian policies proves the maximal squared Sharpe ratio value boundIn particular this holds at the equal-prior state $x=0$. No vanishing-terminal-value assumption is needed for this upper bound, because the terminal term is nonnegative and can simply be dropped.

The finite-value assumption ${\gamma }_{+}>0$ is missing from the printed comparison. Without it, $V_{+}=+\infty$ and the value inequality is trivial, but $Y$ is not a finite supermartingale. For a concrete allowed example, take $r=0$, $\mu =1$, $\sigma =a=1$, $R=1/2$ and $\rho =1$. Then $b_{+}=2$ and ${\gamma }_{+}=-2$. For the informed agent, the constant stock fraction $\pi =4$ and consumption $c=w/2$ giveThe time integral diverges, so his value is infinite. This is why the nontrivial finite-supermartingale proof above needs the stated qualification.

Wealth and admissibility. The dollar holding $\theta$ earns the risky return, while $w-\theta$ earns the continuous-time bank account return. Removing consumption therefore givesHere $\theta$ is a dollar amount, not a number of shares; the number of shares is ${\theta }_t/S_t$. Both controls must use available information: take $\theta$ predictable and $c\ge 0$ progressively measurable, withalmost surely on every finite interval. Require a well-defined objective and an admissible trading strategy satisfying $w_t\ge 0$. At zero portfolio wealth this excludes continued risky gambling or positive consumption. The state-price budget constraint rules out doubling strategies. Throughout the diffusion calculations take $\sigma \ne 0$, positive discount $\rho$, positive decay $\lambda$, and finite value; degeneracies are discussed where they affect the conclusions.

Satisfaction and dynamic programming. Write the consumption satisfaction stock asThe product rule gives, almost everywhere in time,This state is a finite-variation process, so it has no quadratic covariation with portfolio wealth. Applying the Itô formula to the discounted value function over a short interval, then using dynamic programming, gives the interior HJB equationAssuming $V_{ww}<0$, completing the square givesThe last supremum is zero if $V_{\xi }-V_w\le 0$ and infinite otherwise. Thus the gradient constraint for unbounded consumption is $V_{\xi }\le V_w$, and $c=0$ wherever the inequality is strict. Since there is no direct penalty for a very large consumption rate, an active boundary can involve singular consumption control. In that relaxed interpretation the HJB equation isWith ordinary rate controls, this describes the supremum and its limiting transfer policy; it does not promise that an instantaneous transfer is attained by a finite rate.

Power reduction. An additive constant in the utility function only adds a control-independent constant divided by $\rho$ to the value, so normalize $U(\xi )={\xi }^{1-R}/(1-R)$. Put $p=1-R$ and $x=w/\xi$ for $\xi >0$. Scaling portfolio wealth, consumption satisfaction and the controls by the same positive number gives the wealth-to-satisfaction reductionConsequently the reduced HJB equation isIn the strict waiting region the first expression vanishes and consumption is zero. On a transfer region the second vanishes; integrating it gives $v(x)=K(1+x{)}^p$. This reflects preservation of $w+\xi$ during an instantaneous wealth-to-satisfaction transfer.

Why a waiting threshold is expected, and its qualification. The gradient constraint for unbounded consumption compares the benefit of increasing consumption satisfaction with the opportunity cost of spending financial portfolio wealth. When consumption satisfaction is already large relative to cash, waiting lets consumption satisfaction decay while financial portfolio wealth earns returns; consuming immediately can be wasteful. Homogeneity makes the comparison depend only on $x$. For fixed $s=w+\xi$, joint concavity of the value function makes $q_s(a)=V(s-a,a)$ concave in financial portfolio wealth $a$. It is nondecreasing because an immediate transfer can reproduce any smaller financial allocation. Consequently its derivative $V_w-V_{\xi }$ is nonnegative and nonincreasing in $a$: a strict waiting region, if present, starts at zero financial portfolio wealth and ends at a single transfer boundary. Homogeneity makes the corresponding boundary a ratio $x_{\ast }$. In the usual finite-boundary regime this gives $c=0$ for $x<x_{\ast }$, and transfers push a larger ratio down towards $x_{\ast }$. The condition is a comparison of marginal values, not the ordinary formula $c=(V_w{)}^{-1/R}$: current utility function here depends on consumption satisfaction, not on current consumption.

A positive threshold is not guaranteed by the printed hypotheses alone. A useful sufficient local test illustrates the intended argument. Let $a=\rho +\lambda p>0$. At zero portfolio wealth,Starting with a small extra portfolio wealth $\epsilon$, holding it in the continuous-time bank account until a fixed time $t$, then transferring it into consumption satisfaction, has right derivative in $\epsilon$ at zero equal toTherefore, if $\rho <r+\lambda R$, the portfolio wealth marginal value is strictly larger than the consumption satisfaction marginal value at zero. With the usual continuity of marginal values, there is a positive interval on which the gradient constraint for unbounded consumption is strict. The fixed-total-resource concavity argument then gives the threshold structure.

For a concrete counterexample to an unconditional positive threshold, take $R=1/2$, $\lambda =1$, $r=\mu =1/10$, $\sigma =1$ and $\rho =2$. There is zero market price of risk. The relaxed value isIndeed $F_{\xi }=F_w$, $F_{ww}<0$, and the optimized waiting residual, after division by ${\xi }^p$, isbecause $(r+\lambda )/a<1$ and the integrand is decreasing. The Itô formula gives an upper bound by $F$, while transferring all portfolio wealth into consumption satisfaction over intervals tending to zero attains that bound in the limit. Hence the transfer boundary is $x_{\ast }=0$ in this example. The positive-threshold explanation needs a parameter regime supporting a genuine waiting region. For $R>1$, even the zero-wealth value is finite only if $\rho >\lambda (R-1)$; otherwise decaying consumption satisfaction gives value $-\infty$.

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.