Consider a consumer who has incom$\frac{e}{b}$udget $w$ per period and chooses con$-$

sumption $xin{R}_{+}^L.$ The consumer can make $an$ investment $C>0$ that perma$-$

nently reduces the price $of$ good $1$ from $p_1^{0}to{p}_1^1$ for every unit she wants $to$ buy

$(suchas$ a soda maker that turns tap water into sparkling water$).$ The prices for

all other goods remain unchanged during the period under consideration. $To$

simplify the formulas, let $us$ therefore call $q=q(p_1,w)q(p_1,w)=x_1(p,w)$ where $xis$ the Walrasian demand function