Jaimovich–Rebelo preferences explained

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 N _{t} denote hours worked at period t . The instantaneous utility has the formu\left( {C _{t} ,N _{t} } \right) = \frac,

where

X _{t} = C ^{\gamma} _{t} X ^{1- \gamma} _{t -1} .

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 ^{inf ty} _{t =0} \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 toX _{t} = C _{t} , and the instantaneous utility simplifies tou\left( {C _{t} ,N _{t} } \right) = \frac,

corresponding to the KPR preferences.

GHH preferences and balanced growth path

When

\gamma → 0 , and if the economy does not present exogenous growth, then the scaling variable X _{t} reduces to a constantX _{t} = X> 0 , and the instantaneous utility simplifies tou\left( {C _{t} ,N _{t} } \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 _{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 → 0 , the stationary variable (X _{t})/(z _{t}) satisfies the relation(X _{t})/(z _{t}) = (X _{t-1})/(z _{t-1}) (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,

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

Notes and References

1. Jaimovich . Nir . Rebelo . Sergio . 2009 . American Economic Review . 99 . 4 . 1097–1118 . Can news about the future drive the business cycle? . 10.1257/aer.99.4.1097. 10.1.1.172.1551 . 8238010 .