Solution to a functional equation arising from different ways of measuring utility (Q1290970)

The authors study the functional equation \(H(x,y)z= H[xz,y P (x, z)]\), where \(P\) maps \([0,1)\times[0,1]\) into \([0,1]\), \(H(x,y)=G^{-1}[G(x) G(y)]\) \((x \in[0,1)\), \(y\in [0,1)\), \(z\in[0,1))\) and \(G\) maps \([0,1)\) onto \((0,1]\), with \(G\) strictly decreasing describing the conditions under which four measures of utility of gains are the same. It is shown that \(P(x,z)= z(1-x^b)^{1/b}(1-x^b z^b)^{-1/b}\) and \(H(x,y)= (x^b+y^b- x^by^b)^{1/b}\) satisfy the functional equations with arbitrary positive constant \(b\), and that there are no other solutions. In the reviewer's opinion, results (3) and (4) are incorrect becasue then one has \(P(0,z)=z=0\) for all \(z\in]0,1[\).