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Consider the utility function $ν(c_1, c_2) = u(c_1) + \beta u(c_2)$, $0 < \beta < 1$, defined for $c_1 ≥0$ and $c2 ≥0$. Assume $ν′(c)>0$ and $ν′′(c)<0$ for all $c>0$ ; if you like, you can also assume that ν(0) = 0. An indifference curve for such an agent consists of points $(c_1, c_2)$ such that $ν(c_1, c_2) = C$, where $C$ is a constant.

Consider two consumption profiles $(c_1 , c_2 )$ and $(c_1^′ , c^′_2 )$, where $c_1 + c_2 = c^′_1 + c^′_2$. The first profile can be described as smoother than the second if there is smaller discrepancy between $c_1$ and $c_2$ than between $c^′_1$ and $c^′_2$.

To make this more precise, let $c = (c_1+c_2)/2 = (c^′_1+c^′_2)/2$, and then define $\delta$ and $\delta^′$ according to $c_1 =c+\delta$ and $c_2 =c−\delta$ and $c^′_1=c+\delta^′$ and $c^′_2=c−\delta^′$

Then the first profile is smoother than the second if $|\delta| < |\delta^′|$. Does an agent with utility function $ν(c_1, c_2) = u(c_1) + \beta u(c_2)$ always prefer consumption profiles that are more smooth rather than profiles that are less smooth?

This agent's preference is strictly convex. I get the idea of smooth profile in this question, but what does that mean? How can I do this problem? I think I need to maximize the utility over the constraint $c_1+c_2=constant$.

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The agent doesn't always prefer consumption profiles that are more smooth. It depends on the curvature of $u$ and the maginitude of $\beta$. Consider the following example.

Let $u(c) = c^{1/2}$. Then $v(c_1,c_2) = c_1^{1/2} + \beta c_2^{1/2}$ satisfies $v^\prime(c)>0$ and $v^{\prime\prime}(c)<0$ for all $c_1,c_2>0$. Note: I'm assuming by $v^{\prime\prime}(c)<0$ you mean that $v$ is concave. It is a little unclear what v^{\prime\prime}(c) is because $c$ is a vector.

Consider the bundles $a=(1,0)$ and $b=(.5,.5)$. $b$ is smoother than $a$, but the agent only prefers $b$ to $a$ if $$1^{1/2} + \beta 0^{1/2} < .5^{1/2} + \beta .5^{1/2}$$ This occurs only if $\beta > .414$. Since the assumption was only that $0<\beta<1$ then the agent doesn't always prefer more smooth profiles to less.

The intuition here is that the agent needs to care about the future enough.

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