<p>If you specify</p>

<p>$$ U(c, l) =\left(c\cdot [(v(l))^A]\right)^{1-\sigma}$$</p>

<p>then </p>

<p>$$\ln U(c,l) = (1-\sigma)\ln c + A(1-\sigma)\ln[v(l)]$$</p>

<p>...and $A$ can be seen as regulating the relative weight of leisure, for any $v(l)$.</p>

CC BY-SA 3.0
2015-05-19T18:21:06.980Z
2015-05-19T18:21:06.980Z
Alecos Papadopoulos
<p>One approach: Look at the unscaled preferences</p>

<p>$$U(c,l) = \left(c v(l)\right)^{1-\sigma}$$</p>

<p>A log transformation reveals</p>

<p>$$ (1-\sigma)\log\left(c \right) + (1-\sigma)\log v(l)$$</p>

<p>Let $v(l) = l^\gamma$. Then, $\log v(l) = \gamma \log l$. As $(1-\sigma)$ premultiplies both terms, the degree of curvature in $v(l)$ is the only way to shift the <em>level</em> of leisure.</p>

CC BY-SA 3.0
2015-05-19T15:19:18.030Z
2015-05-19T15:19:18.030Z
FooBar
<p>KPR preferences are given by</p>

<p>$$ U(c, l) = \frac{\left(cv(l)\right)^{1-\sigma}-1}{1-\sigma}$$</p>

<p>with concave increasing $v$ and $c$, $l$ denoting consumption and leisure. In the limiting case of $\sigma\to 1$, we receive the standard additively-separable preferences </p>

<p>$$ U(c,l) = \log c + v(l)$$</p>

<p>In the latter case, if I want to scale the relevance of leisure in total preferences, I can rewrite the preferences as $\log c + Av(l)$, and use $A$ for this matter.</p>

<p><strong>How do I do that - scale leisure - in the general case?</strong></p>

<p>If I rewrite the nominator as $(c A v(l))^{1-\sigma}$, it is not clear to me whether $A$ is scaling $c$ or $l$ (probably neither). </p>

<utility><labor-economics><preferences>
King-Plosser-Rebello Preferences: Scale leisure
FooBar
# King-Plosser-Rebello Preferences: Scale leisure

(Asked by: username_0 on 2015-05-19)

KPR preferences are given by

$$ U(c, l) = \frac{\left(cv(l)\right)^{1-\sigma}-1}{1-\sigma}$$

with concave increasing $v$ and $c$, $l$ denoting consumption and leisure. In the limiting case of $\sigma\to 1$, we receive the standard additively-separable preferences

$$ U(c,l) = \log c + v(l)$$

In the latter case, if I want to scale the relevance of leisure in total preferences, I can rewrite the preferences as $\log c + Av(l)$, and use $A$ for this matter.

**How do I do that - scale leisure - in the general case?**

If I rewrite the nominator as $(c A v(l))^{1-\sigma}$, it is not clear to me whether $A$ is scaling $c$ or $l$ (probably neither).

**Tags:** utility, labor-economics, preferences

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## 2 answers

### Answer by username_1 on 2015-05-19. Score: 4

If you specify

$$ U(c, l) =\left(c\cdot [(v(l))^A]\right)^{1-\sigma}$$

then

$$\ln U(c,l) = (1-\sigma)\ln c + A(1-\sigma)\ln[v(l)]$$

...and $A$ can be seen as regulating the relative weight of leisure, for any $v(l)$.


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### Answer by username_0 on 2015-05-19. Score: 2

One approach: Look at the unscaled preferences

$$U(c,l) = \left(c v(l)\right)^{1-\sigma}$$

A log transformation reveals

$$ (1-\sigma)\log\left(c \right) + (1-\sigma)\log v(l)$$

Let $v(l) = l^\gamma$. Then, $\log v(l) = \gamma \log l$. As $(1-\sigma)$ premultiplies both terms, the degree of curvature in $v(l)$ is the only way to shift the *level* of leisure.


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