= Dual equation for multiplicative habit investment
{title2=$\mathcal D=z-\lambda RzJ^\prime+\lambda(R-1)J$}
After scaling wealth by habit and applying the <wealth-variable Legendre dual>, the effective consumption shadow price is proportional to $\mathcal D=z-\lambda RzJ^\prime+\lambda(R-1)J$. 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 $R+\alpha$.
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