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 $C_{t}$ denote consumption and let ${\displaystyle N_{t}}$ denote hours worked at period $t$. 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 ${\displaystyle \theta >1}$, $\psi >0$, and $\sigma >0$.

The agents in the model economy maximize their lifetime utility, $U$, 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 $E_{0}$ 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 ${\displaystyle X_{t}=C_{t},}$ 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 $X_{{t}}$ 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, $X_{{t}}$, 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 $C_{{t}}$ and the scaling variable $X_{{t}}$ 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 $X>0$.

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

1. Jaimovich, Nir; Rebelo, Sergio (2009). "Can news about the future drive the business cycle?". American Economic Review. 99 (4): 1097–1118. doi:10.1257/aer.99.4.1097.