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## A framework for testing the Euler equation under limited participation and near rationality

The Euler equation relates households’ expectation for consumption growth and inflation to the interest rate at which they can borrow or save. I first consider the standard CES utility function. Under a log-normality hypothesis of forecasting errors, or a first order approximation on these error, equation (3.1) can be linearised (see Appendix 3.B). it = + EtDct+1 +Ett+1 (3.2) with the relative risk aversion parameter or 1~ the intertemporal elasticity of substitution parameter, EtDct+1 consumption growth expectation (Dlog) and Ett+1 inflation expectations.2 Such an equation is commonly introduced in monetary policy models but is not assumed to holds exactly. A residual to this equation is often interpreted as a preference shock, a demand shock from households part introduced through a time varying discount factor. There are however more reasons for the Euler equation not to hold on aggregate.

First, even if each individual behaves in accordance to this equation, on aggregate it should not be exactly verified because households wealth vary and so does their marginal utility of consumption (Carroll, 2000). Second, under its linear form, the constant captures the risk associated to the forecast error. It is often assumed to be constant but changes in the volatility of the economic environment can imply through this channel a precautionary savings behaviour (?). This volatility can account for a significant share of fluctuations (Bloom, 2009). Third, equation (3.2) is based on a separable utility function, but more general utility functions imply non separability in with labour (Mankiw et al., 1985) or in time through durable consumption (Mankiw, 1982; Bernanke, 1985; Ferson and Constantinides, 1991; Grossman and Laroque, 1990) or through habit formation (Abel, 1990; Gali, 1994; Constantinides, 1990; Campbell and Cochrane, 1999; Ravn et al., 2006).

The latter has received a lot of attention because it provides a solution to the equity premium/risk free rate puzzles (Abel, 1990; Gali, 1994; Constantinides, 1990; Abel, 1999; Campbell and Cochrane, 1999). In the context of monetary policy models, habit formation also implies a conveniently gradual response of private consumption to monetary policy shocks (Fuhrer, 2000). Habit formation can also explain why savings may be high in a growing economy (Carroll et al., 2000).

### The Euler equation with households-specific interest rates

In this section, I consider households-specific interest rates in testing the Euler equation. The interest rates I consider are depicted on Figure 3.1. The correlation of real monetary policy rate and the spreads on households-specific rates are reported in Table 3.1.

It is noteworthy that households-specific spreads are negatively correlated to real monetary policy rate, which could a priori account for the results exposed by Canzoneri et al.

(2007) and also motivate this analysis where this spread is not assumed orthogonal to the monetary policy stance.

#### Expectations in the Euler equation

In section 3.1, I exposed how estimating the VARX of expectations ex ante biases the test of the Euler equation towards its rejection while sticking to the rational expectations hypothesis. To investigate this bias, I estimate jointly the system (3.9). Doing so I now allow expectations to depart from the rational expectation assumption (i.e. VARX model) so as to improve the fit of the Euler equation.9 Because of this bias, the outcome of the estimation in terms of expectations must be looked into before concluding to a validation of the Euler equation. Building on section 3.2, I only consider two interest rates, the one on Treasury bills and personal loans.

Estimating the model and expectations jointly I find a zero weight on the Treasury bill, a discount factor equal to 99% and a much larger CRRA parameter (3.3). As for habit formation or constrained households, the corresponding parameters are estimated to be null.

How close to optimality are expectations? As for the expectation VARX parameters, I can test whether they are significantly different from the estimates in section 3.2 (see Table 3.5). Consumption expectations are significantly different from the unconstrained estimate (rational expectations). However, inflation expectations are not.

**One way of reconciliation**

Based on the previous estimation I assume that households are subject to a combination of 75% personal loans rate and 25% Treasury bill rate. I also assume that these households are inattentive (Reis, 2006) so that aggregate expectations for inflation are smoothed compared to the VARX forecast. This stylised fact is not only one of the results from joint estimations, but found in consumer surveys and an important aspect of the mismatch between the MRS and actual interest rates: volatile expectations feed into a volatile MRS while nominal interest rates are quite smooth. I thus measure expectations through a moving average of past VAR forecast to approximate this inattentiveness hypothesis.

**Table of contents :**

Contents

List of Tables

List of Figures

**1 Résumé **

1.1 Consommation, taux d’intérêt et anticipations : vers une réconciliation

1.2 Le principe de Taylor est vérifié lorsque les salaires sont rigides

1.3 Un compte satellite des ménages français

1.4 Réformes structurelles dans les modèles DSGE

**2 Introduction **

Introduction

Main results

Relation to the literature

Connections between chapters

**3 Consumption, interest rates and expectations: a reconciliation **

3.1 A framework for testing the Euler equation under limited participation and near rationality

3.2 The Euler equation with households-specific interest rates

3.3 Expectations in the Euler equation

3.4 One way of reconciliation

3.5 Conclusion

Appendices

3.A Data and the VAR of expectations

3.B The conditional log-normality hypothesis

3.C On French dat

**4 The Taylor principle is valid under wage stickiness: an analytical proof **

4.1 Introduction

4.2 A monetary model with sticky wages and prices

4.3 Outline of the proof

4.4 Conclusion

Appendices

4.A Proof

**5 Households Satellite Account for France. Methodological issues on the assessment of domestic production. **

5.1 Introduction

5.2 Domestic production amounts to 30 to 50% of GDP in most studies

5.3 The accounting and valuation of hours of domestic work

5.4 From TUS to HHSA

5.5 A households satellite account for France in 1998 and 2010

5.6 Conclusion

Appendices

5.A Additional tables for international comparisons

5.B Activity, time and wage

5.C Comparisons on the main issue: the frontier of production

**6 Structural reforms in DSGE models: a plead for sensitivity analyses **

6.1 Introduction

6.2 Model

6.3 Understanding the mechanisms

6.4 A sensitivity analysis

6.5 Conclusion

Appendices

6.A Sensitivity analysis

6.B Some calculations

**Bibliography**