Consider the optimal consumption Problem (PO3) for an agent characterized by a time additive state independent utility function of the generalized power form \(u(x)=\frac{1}{b-1}(\gamma+b x)^{\frac{b-1}{b}}\), with \(b otin\{0,1\}\), and discount factor \(\delta\). Letting \(\bar{e}:=e_{0}+\sum_{s=1}^{S} p_{s} e_{s}\) (i.e., the present value of the endowment, where \(p_{1}, \ldots, p_{S}\) denote the prices of the \(S\) Arrow securities) and \(x_{0}^{*}\) the optimal consumption at \(t=0\), the quantity \(\bar{e}-x_{0}^{*}\) represents intertemporal saving. Show that intertemporal saving is an affine function of \(\bar{e}\) (compare also with Lengwiler [1182, Section 5.4.1]).