Finance:Jaimovich–Rebelo preferences

Jaimovich-Rebelo preferences refer to a utility function that allows to parameterize the strength of short-run wealth effects on the labor supply, originally developed by Nir Jaimovich and Sergio Rebelo in their 2009 article Can News about the Future Drive the Business Cycle?

Let [math]\displaystyle{ C_t }[/math] denote consumption and let [math]\displaystyle{ N_{t} }[/math] denote hours worked at period [math]\displaystyle{ t }[/math]. The instantaneous utility has the form

[math]\displaystyle{ u\left( {C_{t},N_{t}} \right) = \frac{ \left( C_{t} - \psi N_{t}^{\theta}X_{t} \right)^{1-\sigma}-1}{1-\sigma}, }[/math]

where

[math]\displaystyle{ X_{t} = C_{t}^{\gamma}X_{t-1}^{1-\gamma}. }[/math]

It is assumed that [math]\displaystyle{ \theta\gt 1 }[/math], [math]\displaystyle{ \psi\gt 0 }[/math], and [math]\displaystyle{ \sigma\gt 0 }[/math].

The agents in the model economy maximize their lifetime utility, [math]\displaystyle{ U }[/math], defined over sequences of consumption and hours worked,

[math]\displaystyle{ U = E_{0} \sum_{t=0}^{\infty} \beta^{t}u\left( {C_{t},N_{t}} \right), }[/math]

where [math]\displaystyle{ E_{0} }[/math] denotes the expectation conditional on the information available at time zero, and the agents internalize the dynamics of [math]\displaystyle{ X_t }[/math] in their maximization problem.

Relationship to other common macroeconomic preference types

Jaimovich-Rebelo preferences nest the KPR preferences and the GHH preferences.

KPR preferences

When [math]\displaystyle{ \gamma = 1 }[/math], the scaling variable [math]\displaystyle{ X_{t} }[/math] reduces to [math]\displaystyle{ X_{t} = C_{t}, }[/math] and the instantaneous utility simplifies to

[math]\displaystyle{ u\left( {C_{t},N_{t}} \right) = \frac{ \left( C_{t}\left( 1 - \psi N_{t}^{\theta} \right) \right)^{1-\sigma}-1}{1-\sigma}, }[/math]

corresponding to the KPR preferences.

GHH preferences and balanced growth path

When [math]\displaystyle{ \gamma \rightarrow 0 }[/math], and if the economy does not present exogenous growth, then the scaling variable [math]\displaystyle{ X_{t} }[/math] reduces to a constant [math]\displaystyle{ X_{t} = X\gt 0, }[/math] and the instantaneous utility simplifies to

[math]\displaystyle{ u\left( {C_{t},N_{t}} \right) = \frac{ \left( C_{t} - \psi X N_{t}^{\theta} \right)^{1-\sigma}-1}{1-\sigma}, }[/math]

corresponding to the original GHH preferences, in which the wealth effect on the labor supply is completely shut off.

Note however that the original GHH preferences are not compatible with a balanced growth path, while the Jaimovich-Rebelo preferences are compatible with a balanced growth path for [math]\displaystyle{ 0\lt \gamma \leq 1 }[/math]. To reconcile these facts, first note that the Jaimovich-Rebelo preferences are compatible with a balanced growth path for [math]\displaystyle{ 0\lt \gamma \leq 1 }[/math] because the scaling variable, [math]\displaystyle{ X_{t} }[/math], grows at the same rate as the labor augmenting technology.

Let [math]\displaystyle{ z_{t} }[/math] denote the level of labor augmenting technology. Then, in a balanced growth path, consumption [math]\displaystyle{ C_{t} }[/math] and the scaling variable [math]\displaystyle{ X_{t} }[/math] grow at the same rate as [math]\displaystyle{ z_{t} }[/math]. When [math]\displaystyle{ \gamma \rightarrow 0 }[/math], the stationary variable [math]\displaystyle{ \frac{X_{t}}{z_{t}} }[/math] satisfies the relation

[math]\displaystyle{ \frac{X_{t}}{z_{t}} = \frac{X_{t-1}}{z_{t-1}}\frac{z_{t-1}}{z_{t}}, }[/math]

which implies that

[math]\displaystyle{ X_{t} = X z_{t}, }[/math]

for some constant [math]\displaystyle{ X\gt 0 }[/math].

Then, the instantaneous utility simplifies to

[math]\displaystyle{ u\left( {C_{t},N_{t}} \right) = \frac{ \left( C_{t} - z_{t}\psi X N_{t}^{\theta} \right)^{1-\sigma}-1}{1-\sigma}, }[/math]

consistent with the shortcut of introducing a scaling factor containing the level of labor augmenting technology before the hours worked term.