Multiplicative habit utility ($g(c,h)=U(c)(c/h{)}^{-\alpha }$)

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 $R+\alpha$.

After scaling wealth by habit and applying the wealth-variable Legendre dual, the effective consumption shadow price is proportional to $\mathcal{D}=z-\lambda Rz{J}^{\prime }+\lambda (R-1)J$. Its power in the utility conjugate keeps the dual equation nonlinear. When $\lambda =0$, habit is fixed and the dual equation is a linear Euler differential equation with effective relative risk aversion $R+\alpha$.

When habit evolves by $dh=\lambda (c-h)dt$, increasing consumption both costs wealth and increases future habit. Consequently the consumption coefficient in the Hamilton-Jacobi-Bellman equation is $V_w-\lambda V_h$. A finite interior optimum requires this effective shadow price to be positive.

An exponentially weighted consumption habit is $h_t=e^{-\lambda t}{h}_0+{\int }_0^t\lambda e^{-\lambda (t-s)}{c}_{s}ds$. Differentiation gives $d{h}_t=\lambda (c_t-h_t)dt$. The state remains positive from positive initial habit and nonnegative consumption.