CHAPTER 3.
3.1. Mathematical interpretation:
We can use Jensen’s inequality, which states that if f(.) is concave, then
XEfXfE
Economic interpretation:
Under uncertainty, the important quantities are risk aversion coefficients, which depend
a) L = ( B, M, 0.50) = 0.50U(B) + 0.50U(M) = 55 > U(P) = 50. Lottery L is preferred
to the ”sure lottery” P.
b) f(U(X)) = a+bU(X)
= a + b50. Again, L is preferred to P under transformation f.
is preferred to L under transformation g.
3.2. Lotteries:
We show that (x,z,) = (x,y, + (1-)) if z = (x,y, ).
The total probabilities of the possible states are
11)y(
1)x(
Of course,
.1111)y()x(
Hence we obtain lottery (x,y, +
(1-)).
Could the two lotteries (x,z,) and (x,y, + (1-)) with z = (x,y, ) be viewed as non-
equivalent ? Yes, in a non-expected utility world where there is a preferences for
3.3 U is concave. By definition, for a concave function f(.)
1,0,bf1afb1af
Use the definition with f = U, a =
1
c
, b =
2
c
, = 1/2
21
21
21
2121
ccVc,cV
cUcUcU2
cU
2
1
cU
2
1
cU
cU
2
1
cU
2
1
c
2
1
c
2
1
U