Derivation of Jensen's inequality as it applies to expected utility theory in the case of two-point support.

1. If u is strictly concave and 0 < p <
1, then u(E(x)) > E(u(x)).
Jensen’s Inequality
Derivation for the two-point support case

2. x is a random variable with realizations
x1 and x2, x1 < x2.
u is the utility function. u is increasing
and strictly concave.
p = prob{x = x1}, 1 – p = prob{x = x2}
Model Setup

3. E(x) = px1 + (1-p)x2
E(u(x)) = pu(x1) + (1-p)u(x2)
Expectation of x and
Expectation of u(x)

4. Graph of Model Setup

5. Similar Triangles

6. Larger : Base = x2 - x1 Height = u(x2) – u(x1).
Smaller : Base = E(x) - x1 Height = ?
Base and Height

7. Larger : Base = x2 - x1 Height = u(x2) – u(x1).
Smaller : Base = E(x) - x1 Height = ?
E(x) - x1 = px1 + (1 – p)x2 – x1 = (1 – p)(x2 –x1).
A little bit of algebra

8. Larger : Base = x2 - x1 Height = u(x2) – u(x1).
Smaller : Base = (1 – p)(x2 –x1).
Height = (1 – p)[u(x2) – u(x1)].
Conclusion

9. u(x1) + (1 – p)[u(x2) – u(x1)] =
u(x1) - (1 – p)u(x1) + (1 – p)u(x2) =
pu(x1) + (1 – p)u(x2) = E(u(x))
More Algebra

10. Justification of Labeling