# Habit formation ala Constantinides (1990)

Consider the following problem, from Constantinides (1990).

\begin{align} V(W_0, x_0) \equiv \max_{c, \alpha} \mathrm{E}_0 \int_0^\infty e^{-\rho s}\gamma^{-1}[c(s) - x(s)]^\gamma \mathrm{d}s, \end{align} subject to $$\mathrm{d} W(t) =[(\mu-r) \alpha(t)+r] W(t)\mathrm{d} t -c(t) \mathrm{d} t+\sigma \alpha(t) W(t) \mathrm{d} w(t),$$ and $$\mathrm{d}x(t) = [bc(t) - ax(t)]\mathrm{d}t,$$ where $$c$$ is consumption, $$x$$ is the "habit stock" with $$x(0) = x_0$$ as initial condition, $$\alpha$$ is the share of wealth $$W$$ invested in the risky asset with drift $$\mu$$ and variance $$\sigma^2$$, and $$w(t)$$ is a standard Brownian motion.

Theorem 1 in the paper establishes the closed form expressions for consumption, the risky share and the value function, using a standard guess and verify approach. They guess that for some constant $$B$$ $$V(W_0, x_0) = B\left(W_0 - \frac{x_0}{r + a - b}\right).$$

My question is: how to guess?

The standard way is to use some kind of homogeneity argument. For example, scaling $$W, c, x$$ by a constant $$\gamma > 0$$, we get

$$V(\gamma W_0, \gamma x_0) = \gamma^{\alpha}V(W_0, x_0),$$ which shows that the value function is homogenous of degree $$\alpha$$. Then, as one would do for the special case of a standard Merton model ($$a = b = 0$$), by an appropriate choice of $$\gamma$$, one gets a guess for the value function, which can then be verified by an application of a verification theorem (see the symmetry section in Chang (2004)). However, since there are two states here, one would need another homogeneity to get something, and I am not seeing anything that can lead me to the guess above. Any suggestions would be greatly appreciated.