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 $$C_t$$ denote consumption and let $$N_{t}$$ denote hours worked at period $$t$$. The instantaneous utility has the form

$$ u\left( {C_{t},N_{t}} \right) = \frac{ \left( C_{t} - \psi N_{t}^{\theta}X_{t} \right)^{1-\sigma}-1}{1-\sigma}, $$

where

$$ X_{t} = C_{t}^{\gamma}X_{t-1}^{1-\gamma}. $$

It is assumed that $$\theta>1$$, $$\psi>0$$, and $$\sigma>0$$.

The agents in the model economy maximize their lifetime utility, $$U$$, defined over sequences of consumption and hours worked,

$$ U = E_{0} \sum_{t=0}^{\infty} \beta^{t}u\left( {C_{t},N_{t}} \right), $$

where $$E_{0}$$ denotes the expectation conditional on the information available at time zero, and the agents internalize the dynamics of $$X_t$$ 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 $$\gamma = 1$$, the scaling variable $$X_{t}$$ reduces to $$ X_{t} = C_{t}, $$ and the instantaneous utility simplifies to

$$ u\left( {C_{t},N_{t}} \right) = \frac{ \left( C_{t}\left( 1 - \psi N_{t}^{\theta} \right) \right)^{1-\sigma}-1}{1-\sigma}, $$

corresponding to the KPR preferences.

GHH preferences and balanced growth path
When $$\gamma \rightarrow 0$$, and if the economy does not present exogenous growth, then the scaling variable $$X_{t}$$ reduces to a constant $$ X_{t} = X>0, $$ and the instantaneous utility simplifies to

$$ u\left( {C_{t},N_{t}} \right) = \frac{ \left( C_{t} - \psi X N_{t}^{\theta} \right)^{1-\sigma}-1}{1-\sigma}, $$

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 $$0<\gamma \leq 1$$. To reconcile these facts, first note that the Jaimovich-Rebelo preferences are compatible with a balanced growth path for $$0<\gamma \leq 1$$ because the scaling variable, $$X_{t}$$, grows at the same rate as the labor augmenting technology.

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

$$ \frac{X_{t}}{z_{t}} = \frac{X_{t-1}}{z_{t-1}}\frac{z_{t-1}}{z_{t}}, $$

which implies that

$$

X_{t} = X z_{t},

$$

for some constant $$X>0$$.

Then, the instantaneous utility simplifies to

$$ u\left( {C_{t},N_{t}} \right) = \frac{ \left( C_{t} - z_{t}\psi X N_{t}^{\theta} \right)^{1-\sigma}-1}{1-\sigma}, $$

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