Consider a two-period consumption savings model under

Question

Consider a two-period consumption savings model under uncertainty, i.e. the income in period 1 (Y₁) is known, but the income in period 2 is a random variable that has the following distribution:

• Y₂ = μ - ε with probability 1/2,

• Y₂ = μ + ε with probability 1/2.

Suppose that β = 1/(1+r), and U (C) = In (C), prove that:

(a) Precautionary savings exist. Hint: prove that Uc (C) is a convex function, then that C₁ < E₁ [C₂].

(b) The higher the uncertainty (ε), the larger the saving (or the smaller C₁.

Hint: using the expected life time budget constraint to compute C₁ as a function of ε. Then show that -∂C₁/∂ε < 0.