<p>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.</p>

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

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

<p><strong>The intuition here is that the agent needs to care about the future enough.</strong></p>

CC BY-SA 3.0
2018-02-26T15:23:23.153Z
2018-02-26T15:23:23.153Z
dsmithecon
<p>Consider the utility function $ν(c_1, c_2) = u(c_1) + \beta u(c_2)$, $0 &lt; \beta &lt; 1$, defined for $c_1 ≥0$ and $c2 ≥0$. Assume $ν′(c)&gt;0$ and $ν′′(c)&lt;0$ for all $c&gt;0$ ; 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.</p>

<p>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 $c^′_2$. </p>

<p>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 $c^′_2=c−\delta^′$</p>

<p>Then the first profile is smoother than the second if $|\delta| &lt; |\delta^′|$. 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? </p>

<p>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 $c_1+c_2=constant$.</p>

<macroeconomics><asset-pricing>
asset-pricing problem
user487392
# asset-pricing problem

(Asked by: username_0 on 2018-02-25)

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 $c\_1+c\_2=constant$.

**Tags:** macroeconomics, asset-pricing

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## 1 answers

### Answer by username_1 on 2018-02-26. Score: 1

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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