Price Homogeneity

All inequalities that define the Radner budget set are homogeneous. So prices at each node in the date-event tree can be independently rescaled. Let $\alpha_0,\alpha_1,\ldots,\alpha_S$ be positive scalars. Then \begin{equation*} B(\alpha_0 p_0,\alpha_1 p_1,\ldots,\alpha_S p_S,\alpha_0 q)=B(p,q). \end{equation*} Since all assets pay off in good 1, take good 1 to be the numeraire good in each state. For $s=0,1,\ldots,S$, \begin{equation*} p_{s1}=1. \end{equation*}