Misplaced Pages

Jaimovich–Rebelo preferences

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

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

u ( C t , N t ) = ( C t − ψ N t θ X t ) 1 − σ − 1 1 − σ , {\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

X t = C t γ X t − 1 1 − γ . {\displaystyle X_{t}=C_{t}^{\gamma }X_{t-1}^{1-\gamma }.}

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

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

U = E 0 ∑ t = 0 ∞ β t u ( C t , N t ) , {\displaystyle U=E_{0}\sum _{t=0}^{\infty }\beta ^{t}u\left({C_{t},N_{t}}\right),}

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

u ( C t , N t ) = ( C t ( 1 − ψ N t θ ) ) 1 − σ − 1 1 − σ , {\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 γ → 0 {\displaystyle \gamma \rightarrow 0} , and if the economy does not present exogenous growth, then the scaling variable X t {\displaystyle X_{t}} reduces to a constant X t = X > 0 , {\displaystyle X_{t}=X>0,} and the instantaneous utility simplifies to

u ( C t , N t ) = ( C t − ψ X N t θ ) 1 − σ − 1 1 − σ , {\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 0 < γ ≤ 1 {\displaystyle 0<\gamma \leq 1} . To reconcile these facts, first note that the Jaimovich-Rebelo preferences are compatible with a balanced growth path for 0 < γ ≤ 1 {\displaystyle 0<\gamma \leq 1} because the scaling variable, X t {\displaystyle X_{t}} , grows at the same rate as the labor augmenting technology.

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

X t z t = X t − 1 z t − 1 z t − 1 z t , {\displaystyle {\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 , {\displaystyle X_{t}=Xz_{t},}

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

Then, the instantaneous utility simplifies to

u ( C t , N t ) = ( C t − z t ψ X N t θ ) 1 − σ − 1 1 − σ , {\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. CiteSeerX 10.1.1.172.1551. doi:10.1257/aer.99.4.1097. S2CID 8238010.
Category: