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

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

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

$X_{t}=Xz_{t},$

for some constant $X>0$.

Then, the instantaneous utility simplifies to

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

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