King-Plosser-Rebello Preferences: Scale leisure

Question

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).

Answer 1

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 $v(l)$.

Answer 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.