Problem 3 Consider a two-date economy and a consumer with utility function over consumption at each period $u(c) = \frac{c^{1-\gamma}}{1-\gamma}$, where $\gamma > 0$. Define the intertemporal utility function as $v(c_1, c_2) = u(c_1) + u(c_2)$. Show that the consumer will always prefer a smooth consumption stream to a more variable one with the same mean, i.e., $v(\frac{c_1+c_2}{2}, \frac{c_1+c_2}{2}) > v(c_1, c_2)$.

          Problem 3
Consider a two-date economy and a consumer with utility function over consumption at each
period
$u(c) = \frac{c^{1-\gamma}}{1-\gamma}$,
where $\gamma > 0$. Define the intertemporal utility function as
$v(c_1, c_2) = u(c_1) + u(c_2)$.
Show that the consumer will always prefer a smooth consumption stream to a more variable
one with the same mean, i.e.,
$v(\frac{c_1+c_2}{2}, \frac{c_1+c_2}{2}) > v(c_1, c_2)$.

Problem 3
Consider a two-date economy and a consumer with utility function over consumption at each
period
u(c) = (c^1-γ)/(1-γ),
where γ > 0. Define the intertemporal utility function as
v(c1, c2) = u(c1) + u(c2).
Show that the consumer will always prefer a smooth consumption stream to a more variable
one with the same mean, i.e.,
v((c1+c2)/(2), (c1+c2)/(2)) > v(c1, c2).

Added by Eddie F.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Chasen Shaw

Step 1

For the smooth consumption stream, we have c1 = c2 = (c1 + c2)/2. Substituting this into the intertemporal utility function, we get: v((c1 + c2)/2, (c1 + c2)/2) = u((c1 + c2)/2) + u((c1 + c2)/2) Show more…

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Consider a two-date economy and a consumer with a utility function over consumption at each period u(c) = c^(1-Î³)/(1-Î³), where Î³ > 0. Define the intertemporal utility function as v(c1, c2) = u(c1) + u(c2). Show that the consumer will always prefer a smooth consumption stream to a more variable one with the same mean, i.e., v((c1 + c2)/2, (c1 + c2)/2) > v(c1, c2).