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 $N_$ denote hours worked at period $t$. The instantaneous utility has the form

$u\left( \right) = \frac,$

where

$X_ = C_^X_^.$

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,

$U = E_ \sum_^ \beta^u\left( \right),$

where $E_$ 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 $X_$ reduces to
$X_ = C_,$
and the instantaneous utility simplifies to

$u\left( \right) = \frac,$

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 $X_$ reduces to a constant
$X_ = X>0,$
and the instantaneous utility simplifies to

$u\left( \right) = \frac,$

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, $X_$, grows at the same rate as the labor augmenting technology.

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

$\frac = \frac\frac,$

which implies that

$X_ = X z_,$

for some constant $X>0$.

Then, the instantaneous utility simplifies to

$u\left( \right) = \frac,$

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

References

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Business cycle theories