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

$u\left(C_t,N_t\right)=\frac{{(C_t-\psi N_t^{\theta }{X}_t)}^{1-\sigma }-1}{1-\sigma },$

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,

$U=E_0\sum _{t=0}^{\mathrm{\infty }}{\beta }^{t}u\left(C_t,N_t\right),$

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

$u\left(C_t,N_t\right)=\frac{{\left(C_t(1-\psi N_t^{\theta })\right)}^{1-\sigma }-1}{1-\sigma },$

corresponding to the KPR preferences.

GHH preferences and balanced growth path

When $\gamma \to 0$, 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

$u\left(C_t,N_t\right)=\frac{{(C_t-\psi X{N}_t^{\theta })}^{1-\sigma }-1}{1-\sigma },$

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 $0<\gamma \le 1$. To reconcile these facts, first note that the Jaimovich-Rebelo preferences are compatible with a balanced growth path for $0<\gamma \le 1$ 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 $\gamma \to 0$, 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

$u\left(C_t,N_t\right)=\frac{{(C_t-z_t\psi X{N}_t^{\theta })}^{1-\sigma }-1}{1-\sigma },$

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