 Solution

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.