\(L_ p\)-Fréchet differentiable preference and ``local utility'' analysis
We denote by \(D[0,M]\) the space of all probability distribution functions on \([0,M]\), and a preordering \(\lesssim\) on \(D[0,M]\) is assumed to be given. If there exists a real-valued function \(V: D[0,M]\to \mathbb{R}\) such that \[ F\lesssim G(F,G\in D[0,M])\Leftrightarrow V(F)\leq V(G), \] then \(V\) is called a utility function which represents \(\lesssim\). The space \(D[0,M]\) endowed with the weak convergence topology can be embedded naturally as a subset of \(L_ p[0,M]\) endowed with the norm topology. Hence, by regarding \(D[0,M]\) as a subset of \(L_ p[0,M]\), we may discuss the differentiability of the utility function \(V\). Assume that \(V\) is differentiable (in the sense of Fréchet) at \(F\in D[0,M]\) and denote the derivative of \(V\) at \(F\) by \(dV(\cdot,F)\). Since \(dV(\cdot, F)\) is a bounded linear operator of \(L_ p[0,M]\) into \(\mathbb{R}\), there exists \(f(\cdot,F)\in L_ q[0,M]\) \((1/p+ 1/q= 1)\) such that \[ dV(G- F,F)= \int^ M_ 0 \{G(x)- F(x)\} f(x,F) dx. \] We define an absolutely continuous ``local utility'' function \(U(\cdot,F)\) by \(U(x,F)=-\int^ x_ 0 f(t,F)dt\). Making use of integration by parts, we obtain the relation: \[ dV(G- F,F)= \int^ M_ 0 U(x,F) d(G(x)- F(x)). \] Although there seems no particularly novel discovery in these reasonings from the viewpoint of mathematics, it deserves well of its constructive work to prepare and broaden the ``infinitesimal'' way to analyzing the variation of \(V\) caused by a parametric change of \(F\in D[0,M]\).
absolutely continuous local utility function
scientific article; zbMATH DE number 438780