A function f:X->R concave if forall a in [0,1] forall x,y in X f(ax+(1-a)y)=>af(x)+(1-a)f(y)
A function f:X->R strictly concave if forall a in (0,1) forall uniqes x,y in X f(ax+(1-a)y)>af(x)+(1-a)f(y)

u concave on an open interval => u continuous and u differentiable almost everywhere.

Theorem:
A DM's expected utility is maximized with u:X->R => DM (strictly) risk averse <=> u (strictly) concave

Theorem:(concavity and risk premiums)
u = g(v(x)) with u,g,v concave and u,v C^2 with strictly positve u', v' on X=>

• forall x in X s.t Ex=0 Piu(x)=>Piv(x0
• forall x in X -u''(x)/u'(x) => -v''(x)/v'(x)
• there exists a concave g on range of v to satisfy u(x)=g(v(x))