# 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

• 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
Jul 21 '17 at 2:53

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) , 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, 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 "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.

• 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
Jul 24 '17 at 9:30
• @ludo This is not correct. See my expanded answer. Jul 24 '17 at 17:19
• @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. Jul 26 '17 at 10:17
• 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
Jul 26 '17 at 10:37
• 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
Jul 26 '17 at 10:38