<p>As Grada Gukovic has commented under the question, the wiki is wrong and misleading. The KPR preference has the form <span class="math-container">$$\begin{array}{c}
u\left(c_{t}, n_{t}\right)=\frac{\left(c_{t} v\left(1-n_{t}\right)\right)^{1-\sigma}-1}{1-\sigma} \quad \text { if } \quad \sigma \neq 1 \\
u\left(c_{t}, n_{t}\right)=\ln c_{t}+\ln v\left(1-n_{t}\right) \quad \text { if } \quad \sigma=1
\end{array}$$</span>.</p>
<p>Actually a commonly used preference like <span class="math-container">$u\left(c_{t}, 1-n_{t}\right)=\frac{c_{t}^{1-\sigma}-1}{1-\sigma}+\theta \frac{\left(1-n_{t}\right)^{1-\xi}-1}{1-\xi}$</span> is not consistent with Balanced Growth except for the case <span class="math-container">$\sigma=1$</span> or for the case that <span class="math-container">$\theta_{t}$</span> is shifting up at exactly the rate of consumption growth. An easy way to see this is through the intratemporal FOC for labor supply: <span class="math-container">$$\theta\left(1-n_{t}\right)^{-\xi}=c_{t}^{-\sigma} w_{t}$$</span>. It is clear that we can only have <span class="math-container">$\sigma=1$</span> to make wage and consumption grow at same rate as well as to have a constant labor supply.</p>

CC BY-SA 4.0
2021-06-22T20:56:46.387Z
2021-06-22T21:16:16.200Z
2021-06-22T21:16:16.200Z
Alalalalaki
<p>The <a href="https://en.wikipedia.org/wiki/King%E2%80%93Plosser%E2%80%93Rebelo_preferences" rel="nofollow noreferrer">wiki of King–Plosser–Rebelo preferences</a> says that the utility function has the multiplicatively separable form <span class="math-container">$$u(C, L)=\frac{1}{1-\sigma_{c}} C^{1-\sigma_{c}} v(L)$$</span> and &quot;in the limit case of <span class="math-container">$\sigma_{c}=1$</span> the resulting preferences specification is additively separable&quot; is <span class="math-container">$$u(C, L)=\ln C_{t}+v(L)$$</span>.</p>
<p>This confused me for a while as I think it should be <span class="math-container">$$u(C, L)=\ln C_{t}+ \ln v(L)$$</span>. But this may not be a big deal as we can change the definition of the function <span class="math-container">$v$</span>.</p>
<p>Then the wiki continues to say that &quot;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 <span class="math-container">$u(C, L)=\frac{1}{1-\sigma_{c}} C^{1-\sigma_{c}}+z^{1-\sigma_{c}} \frac{(1-L)^{1+\kappa}}{1+\kappa}$</span>.&quot;</p>
<p>Now I am really confused because that it seems that the additively separable specification is irrelevant to <span class="math-container">$\sigma_{c}=1$</span>, since we can have both additively separable utility and intertemporal elasticity of substitution in consumption <span class="math-container">$\sigma_{c} \neq 1$</span>. Then why don't we directly set the general utility function as <span class="math-container">$u(C, L)=\frac{1}{1-\sigma_{c}} C^{1-\sigma_{c}}+v(L)$</span>?</p>

CC BY-SA 4.0
2021-06-19T09:36:26.493Z
wikipedia is wrong as allways. The correct functional form is in the top answer to this question:  https://economics.stackexchange.com/questions/5716/king-plosser-rebello-preferences-scale-leisure
with the correct utility function it becomes trivial to see that it is additively separable
Grada Gukovic
<macroeconomics><utility><preferences>
King–Plosser–Rebelo Preferences and Additively Separable
Alalalalaki
# King–Plosser–Rebelo Preferences and Additively Separable

(Asked by: username_0 on 2021-06-12)

The [wiki of King–Plosser–Rebelo preferences](https://en.wikipedia.org/wiki/King%E2%80%93Plosser%E2%80%93Rebelo_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 $u(C, L)=\frac{1}{1-\sigma\_{c}} C^{1-\sigma\_{c}}+v(L)$?

**Tags:** macroeconomics, utility, preferences
> **Comments:**
> **username_1**: wikipedia is wrong as allways. The correct functional form is in the top answer to this question:  https://economics.stackexchange.com/questions/5716/king-plosser-rebello-preferences-scale-leisure with the correct utility function it becomes trivial to see that it is additively separable  

---

## 1 answers

### Answer by username_0 on 2021-06-22. Score: 1 | Accepted answer

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.


---

