Wealth-to-satisfaction reduction ($V(\xi ,w)={\xi }^{1-R}v(w/\xi )$)

For CRRA utility of a consumption satisfaction stock, jointly scaling initial portfolio wealth, consumption satisfaction and controls gives homogeneity of degree $1-R$. The value function therefore depends on one dimensionless ratio after factoring out ${\xi }^{1-R}$. The effective consumption coefficient becomes ${\xi }^{-R}[(1-R)v-(x+1)v^{\prime }]$.