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?^[1]

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

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

where

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

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

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

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

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

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

corresponding to the KPR preferences.

GHH preferences and balanced growth path

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

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

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

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

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

which implies that

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

for some constant ${\displaystyle X>0}$.

Then, the instantaneous utility simplifies to

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

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

References