Specification form for household utility function 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
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 &amp; Sala-i-Martin book (2nd ed) , 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 $\omega'(\ell)&lt;0$.
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, 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 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 &quot;deep&quot; 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.