(See also short version below). Consider two lotteries $A$ and $B$ \begin{align} L_A&= ( (1/2),w_0-h;\ (1/2),w_0+h )\\ L_B&= ( (1/2),w_0-2h;\ (1/2),w_0+2h ) \end{align} where $0<h<w_0/2$. The idea here is that both A and B are fair bets, i.e. they offer a 50-50 odds of winning or losing the same amount of money. Of course lottery B is more risky (it is a MPS of A). I have to show that any risk-averse agent will prefer A to B. An agent is risk verse if he has strictly concave utility function $u(\cdot)$. It is easy to prove this graphically, but I want a formal proof. I have to show that
$EU(L_A)>EU(L_B)$ i.e. that
$(1/2) u(w_0+h)+(1/2)u(w_0-h) \geq (1/2)u(w_0+2h)+(1/2)u(w_0-2h)$
This is easy if I replace u() with e.g. $u=\log$, but I would like to do it for a generic utility function $u$, strictly increasing and striclty concave.
Short question Suppose $u$ is a strictly increasing and strictly concave function defined over positive real numbers. Let $w_0>0$ and $0<h<w_0/2$. Show that the following holds true:
$(1/2) u(w_0+h)+(1/2)u(w_0-h) \geq (1/2)u(w_0+2h)+(1/2)u(w_0-2h)$
Any help is appreciated!