Consider an investor with power utility and an infinite horizon. Assume the capital market line is constant, so we can write J(w) instead of J(x,w)for the value function.

(a) Define

ξ = δ −(1− ρ)r

ρ − (1− ρ)κ2 2ρ2 .

Assume (14.26) holds, so ξ > 0. Show that J(w) = ξ −ρ

 1 1−ρ

w1−ρ



solves the HJB equation (14.25). Show that c = ξw and π = (1/ρ)−1 (μ−rι) achieve the maximum in the HJB equation.

(b) Show that, under the assumption ξ > 0, the transversality condition limT→∞ E e−δt J(W∗
T)

= 0 holds, where W∗ denotes the wealth process generated by the consumption and portfolio processes in Part (a).