# Reconciling Jaimovich–Rebello preferences, habit in consumption and labor supply

Abstract: This note studies a form of a utility function of consumption with habit and leisure that (a) is compatible with long-run balanced growth, (b) hits a steady state observed target for hours worked and (c) is consistent with micro-econometric ev- idence for the inter-temporal elasticity of substitution and the Frisch elasticity of labor supply. Employing Jaimovich-Rebello preferences our results highlight a con- straint on the preference parameter needed to target the steady-state Frisch elastic- ity. This leads to a lower bound for the latter that cannot be reconciled empirically with external habit, but the introduction of a labor wedge solves the problem. We also propose a dynamic Frisch inverse elasticity measure and examine its business cycle properties.

## Summary (2 min read)

### 1 Introduction

- In particular, RBC-DSGE models in which a consumer’s utility level not only depends on her consumption level but also how that level compares to a standard set either by her own past consumption levels (internal habit-formation) or the levels of those in her peerage (catching-up with the Joneses’ or external habit) are now ubiquitous in the literature.
- At the same time to achieve co-movement of output, hours, consumption and investment modellers turn to preferences proposed by Jaimovich and Rebello (2008) (henceforth JR) that control short-run wealth effects.

### 2 The Frisch Elasticity of Labor Supply

- Bilbiie (2011) shows that the constantmarginal-utility inverse elasticity of labor supply is given by δ = ULCH UL ( ULC UCC − ULL ULC ) and it is straightforward to show that g′′ >.
- Microeconomic and macroeconomic estimates of the Frisch elasticity differ significantly, the former typically ranging from 0 to 0.5 and the latter from 2 to 4 (Peterman, 2016).
- Estimations of the elasticity of labor supply found using microeconomic data depend on factors such as gender, age, marital status and dependants.
- Reichling and Whalen (2017) give a thorough review of the estimates found in the literature based on microeconomic data, finding that estimates typically range from 0 to over 1.

### 3 The Household Problem

- Households choose between work and leisure and therefore how much labor they supply.
- They also own the capital stock which is rented to firms at a rental rate rKt and choose an optimal investment path.
- The single-period utility is given by JR preferences (3) and (4).
- Tt where All variables are expressed in real terms relative to the price of output.

### 3.4 Numeral Illustration

- The authors can now assess the empirical plausibility of JR preferences with habit in consumption.
- From their numerical results for the lower bound δJR(ψ), this rules out external habit if the authors are to choose JR preferences that allow for only weak wealth effects (γ very small).
- But even without habit, or with internal habit, it is difficult to reconcile the extreme choice of γ almost zero with a Frisch elasticity within this empirical range.

### 4 A Resolution: Generalized JR Utility and Labor Wedge

- Again for a given H this pins down % given the remaining parameters.
- The second modification is to introduce a labor wedge into the household problem (as in Shimer (2009).
- Figure 1 shows that a combination of their generalized JR utility and an empirically supported wedge, the authors can calibrate parameters to achieve the desired empirical value for δJR.

### 5 Conclusions

- This note has reviewed a utility function commonly used in RBC-DSGE models that is nonseparable in habit-adjusted consumption and leisure, compatible with balanced growth and elim- 7 inates counterfactual wealth effects highlighted by Jaimovich and Rebello (2008).
- The authors main contributions are first, Theorems 1 and 2 that highlight a constraint on the preference parameter ψ needed to target the Frisch elasticity.
- This leads to a lower bound for the latter that cannot be reconciled empirically with external habit.
- Even with internal or no habit, the range of possible values of the Frisch elasticity lie outside empirical results unless the authors allow for a modest wealth effect.

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### Cites background from "Reconciling Jaimovich–Rebello prefe..."

...JR preferences with habits are studied in Schmitt-Groh and Uribe (2012) in the context of news shocks and business cycles, and more recently by Holden et al. (2017) in the context of reconciling these preferences with estimates of the Frisch elasticity of labour supply....

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##### References

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### Additional excerpts

...Two special cases are worth noting: KPR (γ = 1) : δF (ψ, 1) = ψ + (1 + ψ)%H1+ψ(2σ − 1) σ(1− %H1+ψ) , GHH (γ = 0) : δF (ψ, 0) = ψ, where KPR preferences are those proposed by King et al. (1988), and GHH preferences are those proposed by Greenwood et al. (1988)....

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