= Solution
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 $r_1=r+\varepsilon$, $p=1-R$, $B_1=B(r_1)$, 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 consequently
$$
0=\sup_{c>0,\,\pi}\left\{\frac{c^p}{p}+\bigl[r_1w+\pi w(\mu-r_1)-c\bigr]V_1'(w)
+\frac12\sigma^2\pi^2w^2V_1''(w)-\rho V_1(w)+\lambda[V_0(w)-V_1(w)]\right\}.
$$
This jump term distinguishes the <exponential interest-rate switch> from a permanent higher bank rate.
By the <homogeneity> of <CRRA utility>, put $V_1(w)=K_1w^p/p$, with $K_1>0$. Differentiating the strictly <concave> expressions in $c$ and $\pi$ gives
$$
c=K_1^{-1/R}w=\gamma_1w,
\qquad
\pi_1^*=\frac{\mu-r_1}{R\sigma^2}.
$$
The optimized <consumption> terms contribute $RK_1^{1-1/R}w^p/p$, and the optimized investment terms contribute $K_1B_1w^p$. With $K_0=\gamma_0^{-R}$, the <Hamilton-Jacobi-Bellman equation> reduces to
$$
D_1K_1=RK_1^{1-1/R}+\lambda K_0,
\qquad
D_1=\rho+\lambda-pB_1.
$$
Equivalently,
$$
\boxed{R\gamma_1+\lambda\left(\frac{\gamma_1}{\gamma_0}\right)^R=D_1,
\qquad
V_1(w)=\frac{\gamma_1^{-R}w^{1-R}}{1-R}.}
$$
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 $R=2$,
$$
\gamma_1=\frac{\gamma_0^2}{\lambda}\left(\sqrt{1+\frac{\lambda D_1}{\gamma_0^2}}-1\right).
$$
\b[Use $(\pi_1^*,\gamma_1)$ until the rate changes, and $(\pi_0^*,\gamma_0)$ 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 gives
$$
E\left[\int_0^\tau e^{-\rho t}\frac{(\gamma_1w_t)^p}{p}dt+e^{-\rho\tau}V_0(w_\tau)\right]
=\frac{w^p}{p}\frac{\gamma_1^p+\lambda K_0}{D_1+p\gamma_1}
=\frac{K_1w^p}{p}.
$$
The 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, $w^*$ the candidate <portfolio wealth>, and
$$
M_t=e^{-\rho t}K_{j_t}(w_t^*)^{-R}=e^{-\rho t}(c_t^*)^{-R}.
$$
If $N_t=\mathbf1_{\{\tau\leq t\}}$ and $\widetilde N_t=N_t-\lambda(t\wedge\tau)$, the <Itô formula> and the scalar equation give
$$
\frac{dM_t}{M_{t-}}=-r_{j_t}dt-\frac{\mu-r_{j_t}}{\sigma}dW_t
+\left(\frac{K_0}{K_1}-1\right)d\widetilde N_t,
$$
where 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 $M_tw_t+\int_0^tM_sc_sds$ is a nonnegative <local martingale>. It follows that
$$
E\int_0^\infty M_tc_tdt\leq M_0w_0.
$$
For the candidate policy, $E[M_tw_t^*]\to0$. 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 $E\int_0^\infty M_tc_t^*dt=M_0w_0$. Apply the <concave supporting-tangent inequality>
$$
U(c_t)\leq U(c_t^*)+(c_t^*)^{-R}(c_t-c_t^*)
$$
and 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 $D_1\leq0$, 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 $w^p e^{pB_1t}$; withdrawals only increase this negative-power moment. The expected magnitude of the negative postchange continuation <utility function> therefore contains the divergent integral $\lambda K_0w^p\int_0^\infty e^{-D_1t}dt/|p|$. Thus these cases yield respectively $+\infty$ or $-\infty$, rather than a positive-root solution.
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