STOCHASTIC

CONSUMPTION UNDER UNCERTAINTY 331
APPENDIX B 24
Let f{x, y) have the property that fjf, = a(x) + bix)y, fv ^ o. We
wish to show that/(x, y) = Figix) + hix) • y).
Forf(x, y) = constant, we have/, +fvy' = 0, with y' = dy/dx 1/constant.
or y' = —ifJfv), so that:
y' = -aix) - b(x)y. (B.l)
The solution of this ordinary differential equation is readily verified to
be
y = e- Bi '\~ j'a(jc) e B ^dx + C), (B.2)
where B(x) = J b(x) dx\ we may write (B.2) as
gix) + hix) -y^c, (B.3)
with hix) = e Blx> , gix) = $ aix) e BW dx; since (B.3) is equivalent to
"fix, y) = constant," our hypothesis is verified.
APPENDIX C
LEMMA C.l. Let hix) = f{gix)}, where f is different table in g and g is
continuous in x; let furthermore (x) be any density such that J" hix) d0ix)
and jgix)d&ix) exist and are finite. Define x° inot necessarily unique)
implicitly by \gix) d&ix) = gix"). Then
concave
f{g(x)} linear in g over the range of implies j hix) d