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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.
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