<p>In growth models, in order to obtain a steady-state in growth rates (i.e. a constant growth rate in long-run equilibrium), the CRRA utility function when labor-leisure choice is present must have a specific functional form.</p>
<p>The mathematical proof can be found in <a href="http://down.cenet.org.cn/upfile/8/200751171644184.pdf" rel="nofollow noreferrer">Barro &amp; Sala-i-Martin book (2nd ed)</a> , Appendix 9.4, pp 427-428.</p>
<p>It is shown that the CRRA utility function must take the form (<span class="math-container">$\ell$</span> represents <strong>labor</strong> here)</p>
<p><span class="math-container">$$u(c,\ell) = \frac {c^{1-\theta}\cdot \exp\{(1-\theta)\cdot \omega(\ell)\}-1}{1-\theta}$$</span></p>
<p>where <span class="math-container">$\omega(\ell)$</span> is some function of labor, and <span class="math-container">$\omega'(\ell)&lt;0$</span>.</p>
<p>For <span class="math-container">$\theta=1$</span> this simplifies to <span class="math-container">$u(c,\ell)=\ln(c) + \omega(\ell)$</span>.</p>
<hr />
<p>After we impose additive separability, the exact functional form of the component related to leisure/labor is sometimes formed so as to have a constant <strong><a href="https://en.wikipedia.org/wiki/Frisch_elasticity_of_labor_supply" rel="nofollow noreferrer">Frisch elasticity of labor supply</a></strong>, For example</p>
<p><span class="math-container">$$u(c,\ell) = \ln(c) +\alpha \frac {(1-\ell)^{1+1/v}}{1+1/v}$$</span></p>
<p>has Frisch elasticity equal to <span class="math-container">$v$</span>, free of <span class="math-container">$\ell$</span>. See <strong><a href="https://economics.stackexchange.com/q/5722/61">this thread</a></strong> for more details on this.</p>
<p>The Frisch elasticity of labor supply is defined as <strong>the elasticity of labor while keeping the marginal utility of wealth constant</strong>. Its expression is derived <em>given</em> the solution to the utility maximization problem (i.e. by using the relations emerging from the optimaztion conditions).  After some manipulations this expression is</p>
<p><span class="math-container">$$\eta_F = \frac {U_{\ell}\cdot U_{cc}}{\ell\cdot [U_{cc}\cdot U_{\ell \ell}-U^2_{c \ell}]}$$</span></p>
<p>Under separability between consumption and labor/leisure,</p>
<p><span class="math-container">$$U^2_{c \ell} = 0 \implies \eta_F = \frac {U_{\ell}}{\ell\cdot U_{\ell \ell}}$$</span></p>
<p>To use one of the functional forms of the OP</p>
<p><span class="math-container">$$U(c,N)=\dfrac{c^{1-θ}}{1-θ}-\alpha \dfrac{N^{1-γ}}{1-γ}$$</span></p>
<p>we have</p>
<p><span class="math-container">$$U_{N} = -\alpha N^{-\gamma},\;\;\; U_{NN} = \alpha \gamma N^{-\gamma-1}$$</span></p>
<p>so</p>
<p><span class="math-container">$$\eta_F = \frac {-\alpha N^{-\gamma}}{N\cdot \alpha \gamma N^{-\gamma-1}} = -\frac 1 {\gamma}$$</span></p>
<p>So functional forms that lead to a constant Frisch elasticity, treat it as a &quot;deep&quot; preference parameter. Functional forms that lead to a Frisch elasticity that includes (the <em>optimal</em>) level of labor (as is the first functional form of the OP), treat it as a derived measure, dependent on the optimization framework also.</p>

CC BY-SA 3.0
2017-07-24T09:30:33.023Z
Thanks for your answer. Just want to precise that the requirement of constant Frisch elasticity of substitution for functional form of CRRA with consumption and labor is to match the empirically observed constant labor supply despite the increasing trend in wage and capital-output ratio. And this requirement is not satisfied by the 3 examples of specifications I provided above, unless each of the CRRA coefficents ($$\theta, \gamma, \rho$$)  is 1.
ludo 
CC BY-SA 3.0
2017-07-24T17:19:27.433Z
@ludo This is not correct. See my expanded answer.
Alecos Papadopoulos
CC BY-SA 3.0
2017-07-26T10:17:26.690Z
@ludo I am treating solely the mathematical properties of specific functional forms as regards the Frisch elasticity, so I do not see in what sense is there a mistake in what I write since these are standard mathematical calculations. All other information you provide appears to have to do with how well these functional forms match real-world data, which is a totally different matter.
Alecos Papadopoulos
CC BY-SA 3.0
2017-07-26T10:37:44.167Z
The above is not correct in the sense a preference specification may lead to a constant Frisch elasticity of substitution without delivering a constant labor supply in the BGP. An illustration is as follows: $$u(c,N)=\dfrac{c^{1-θ}}{1-θ}-\alpha \dfrac{N^{1-γ}}{1-γ}$$, $$\eta_F = -\frac 1 {\gamma}$$ and $$N^{*}=1-\left(\dfrac{\alpha. C^{\theta}}{w}\right)^{\dfrac{1}{\gamma}}$$ \\ $$u(c, n) = \ln(c) -\alpha \frac {n^{1+1/v}}{1+1/v}$$, $$\eta_F = {\gamma}$$ and $$N^{*}=\left(\dfrac{w}{\alpha. C}\right)^{\gamma}$$.
ludo 
CC BY-SA 3.0
2017-07-26T10:38:22.947Z
Along the BGP (w and C growing at the same rate)Only the second can deliver constant labor supply that long run facts support (Cooley and Prescott (1995)).Then, the last specification is preferred over the first one. However it seems a decreasing trend in hours worked is being documented.
ludo 
CC BY-SA 3.0
2017-07-26T10:43:02.107Z
@ alecos . Thanks again. What you provided is indeed correct. However, given my initial question (what factors determine the choice of the CRRA specification form for HH preferences in a macro model), I think extra clarifications I provided above turn out to be helpful.
ludo 
CC BY-SA 3.0
2017-07-26T10:43:48.010Z
@ludo Certainly, they usefully enhance the discussion here.
Alecos Papadopoulos
CC BY-SA 3.0
2017-07-21T18:46:17.360Z
2017-07-24T17:28:45.567Z
2020-06-17T09:40:54.643Z
Alecos Papadopoulos
<p>What determines the choice of the functional form to be adopted for the household CRRA utility function, when utility depends on consumption and leisure?</p>

<p>For examples, we have these three functional specifications:</p>

<ul>
<li><p>$U(c,l)=\left[\dfrac{c^φ l^{1-φ}}{1-θ}\right]^{1-θ}$</p></li>
<li><p>$U(c,l)=\dfrac{c^{1-φ}}{1-φ}+\alpha \dfrac{(1-N)^{1-γ}}{1-γ}$</p></li>
<li><p>$U(c,l)=\dfrac{c^{1-θ}}{1-θ}-\alpha \dfrac{N^{1-γ}}{1-γ}$</p></li>
</ul>

<p>c: consumption, l: leisure, N: labor supplied. Total time endowment: unity</p>

CC BY-SA 3.0
2017-07-21T02:53:40.970Z
The primary choice one makes is separability. The only difference between the last two function is that one models leisure and the other models the disutility of supplying labor.
123
<microeconomics><utility>
Specification form for household utility function
ludo 
# Specification form for household utility function

(Asked by: username_0 on 2017-07-19)

What determines the choice of the functional form to be adopted for the household CRRA utility function, when utility depends on consumption and leisure?

For examples, we have these three functional specifications:

* $U(c,l)=\left[\dfrac{c^φ l^{1-φ}}{1-θ}\right]^{1-θ}$
* $U(c,l)=\dfrac{c^{1-φ}}{1-φ}+\alpha \dfrac{(1-N)^{1-γ}}{1-γ}$
* $U(c,l)=\dfrac{c^{1-θ}}{1-θ}-\alpha \dfrac{N^{1-γ}}{1-γ}$

c: consumption, l: leisure, N: labor supplied. Total time endowment: unity

**Tags:** microeconomics, utility
> **Comments:**
> **username_1**: The primary choice one makes is separability. The only difference between the last two function is that one models leisure and the other models the disutility of supplying labor.  

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## 1 answers

### Answer by username_2 on 2017-07-21. Score: 2

In growth models, in order to obtain a steady-state in growth rates (i.e. a constant growth rate in long-run equilibrium), the CRRA utility function when labor-leisure choice is present must have a specific functional form.

The mathematical proof can be found in [Barro & Sala-i-Martin book (2nd ed)](http://down.cenet.org.cn/upfile/8/200751171644184.pdf) , Appendix 9.4, pp 427-428.

It is shown that the CRRA utility function must take the form ($\ell$ represents **labor** here)

$$u(c,\ell) = \frac {c^{1-\theta}\cdot \exp\{(1-\theta)\cdot \omega(\ell)\}-1}{1-\theta}$$

where $\omega(\ell)$ is some function of labor, and $\omega'(\ell)<0$.

For $\theta=1$ this simplifies to $u(c,\ell)=\ln(c) + \omega(\ell)$.

---

After we impose additive separability, the exact functional form of the component related to leisure/labor is sometimes formed so as to have a constant **[Frisch elasticity of labor supply](https://en.wikipedia.org/wiki/Frisch_elasticity_of_labor_supply)**, For example

$$u(c,\ell) = \ln(c) +\alpha \frac {(1-\ell)^{1+1/v}}{1+1/v}$$

has Frisch elasticity equal to $v$, free of $\ell$. See **[this thread](https://economics.stackexchange.com/q/5722/61)** for more details on this.

The Frisch elasticity of labor supply is defined as **the elasticity of labor while keeping the marginal utility of wealth constant**. Its expression is derived *given* the solution to the utility maximization problem (i.e. by using the relations emerging from the optimaztion conditions). After some manipulations this expression is

$$\eta\_F = \frac {U\_{\ell}\cdot U\_{cc}}{\ell\cdot [U\_{cc}\cdot U\_{\ell \ell}-U^2\_{c \ell}]}$$

Under separability between consumption and labor/leisure,

$$U^2\_{c \ell} = 0 \implies \eta\_F = \frac {U\_{\ell}}{\ell\cdot U\_{\ell \ell}}$$

To use one of the functional forms of the OP

$$U(c,N)=\dfrac{c^{1-θ}}{1-θ}-\alpha \dfrac{N^{1-γ}}{1-γ}$$

we have

$$U\_{N} = -\alpha N^{-\gamma},\;\;\; U\_{NN} = \alpha \gamma N^{-\gamma-1}$$

so

$$\eta\_F = \frac {-\alpha N^{-\gamma}}{N\cdot \alpha \gamma N^{-\gamma-1}} = -\frac 1 {\gamma}$$

So functional forms that lead to a constant Frisch elasticity, treat it as a "deep" preference parameter. Functional forms that lead to a Frisch elasticity that includes (the *optimal*) level of labor (as is the first functional form of the OP), treat it as a derived measure, dependent on the optimization framework also.

> **Comments:**
> **username_0**: Thanks for your answer. Just want to precise that the requirement of constant Frisch elasticity of substitution for functional form of CRRA with consumption and labor is to match the empirically observed constant labor supply despite the increasing trend in wage and capital-output ratio. And this requirement is not satisfied by the 3 examples of specifications I provided above, unless each of the CRRA coefficents ($$\theta, \gamma, \rho$$)  is 1.  
> **username_2**: @ludo This is not correct. See my expanded answer.  
> **username_2**: @ludo I am treating solely the mathematical properties of specific functional forms as regards the Frisch elasticity, so I do not see in what sense is there a mistake in what I write since these are standard mathematical calculations. All other information you provide appears to have to do with how well these functional forms match real-world data, which is a totally different matter.  
> **username_0**: The above is not correct in the sense a preference specification may lead to a constant Frisch elasticity of substitution without delivering a constant labor supply in the BGP. An illustration is as follows: $$u(c,N)=\dfrac{c^{1-θ}}{1-θ}-\alpha \dfrac{N^{1-γ}}{1-γ}$$, $$\eta_F = -\frac 1 {\gamma}$$ and $$N^{*}=1-\left(\dfrac{\alpha. C^{\theta}}{w}\right)^{\dfrac{1}{\gamma}}$$ \\ $$u(c, n) = \ln(c) -\alpha \frac {n^{1+1/v}}{1+1/v}$$, $$\eta_F = {\gamma}$$ and $$N^{*}=\left(\dfrac{w}{\alpha. C}\right)^{\gamma}$$.  
> **username_0**: Along the BGP (w and C growing at the same rate)Only the second can deliver constant labor supply that long run facts support (Cooley and Prescott (1995)).Then, the last specification is preferred over the first one. However it seems a decreasing trend in hours worked is being documented.  
> **username_0**: @ alecos . Thanks again. What you provided is indeed correct. However, given my initial question (what factors determine the choice of the CRRA specification form for HH preferences in a macro model), I think extra clarifications I provided above turn out to be helpful.  
> **username_2**: @ludo Certainly, they usefully enhance the discussion here.  

---

