King–Plosser–Rebelo Preferences and Additively Separable

Question

The wiki of King–Plosser–Rebelo preferences says that the utility function has the multiplicatively separable form $$u(C, L)=\frac{1}{1-\sigma_{c}} C^{1-\sigma_{c}} v(L)$$ and "in the limit case of $\sigma_{c}=1$ the resulting preferences specification is additively separable" is $$u(C, L)=\ln C_{t}+v(L)$$.

This confused me for a while as I think it should be $$u(C, L)=\ln C_{t}+ \ln v(L)$$. But this may not be a big deal as we can change the definition of the function $v$.

Then the wiki continues to say that "To have additively separable preferences along with balanced growth, some studies use the shortcut of introducing a scaling factor containing the level of labor augmenting technology before the leisure term. An example of such a utility function would be $u(C, L)=\frac{1}{1-\sigma_{c}} C^{1-\sigma_{c}}+z^{1-\sigma_{c}} \frac{(1-L)^{1+\kappa}}{1+\kappa}$."

Now I am really confused because that it seems that the additively separable specification is irrelevant to $\sigma_{c}=1$, since we can have both additively separable utility and intertemporal elasticity of substitution in consumption $\sigma_{c} \neq 1$. Then why don't we directly set the general utility function as $u(C, L)=\frac{1}{1-\sigma_{c}} C^{1-\sigma_{c}}+v(L)$?

Answer 1

As Grada Gukovic has commented under the question, the wiki is wrong and misleading. The KPR preference has the form $$\begin{array}{c} u\left(c_{t}, n_{t}\right)=\frac{\left(c_{t} v\left(1-n_{t}\right)\right)^{1-\sigma}-1}{1-\sigma} \quad \text { if } \quad \sigma \neq 1 \\ u\left(c_{t}, n_{t}\right)=\ln c_{t}+\ln v\left(1-n_{t}\right) \quad \text { if } \quad \sigma=1 \end{array}$$.

Actually a commonly used preference like $u\left(c_{t}, 1-n_{t}\right)=\frac{c_{t}^{1-\sigma}-1}{1-\sigma}+\theta \frac{\left(1-n_{t}\right)^{1-\xi}-1}{1-\xi}$ is not consistent with Balanced Growth except for the case $\sigma=1$ or for the case that $\theta_{t}$ is shifting up at exactly the rate of consumption growth. An easy way to see this is through the intratemporal FOC for labor supply: $$\theta\left(1-n_{t}\right)^{-\xi}=c_{t}^{-\sigma} w_{t}$$. It is clear that we can only have $\sigma=1$ to make wage and consumption grow at same rate as well as to have a constant labor supply.