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## Intertemporal Complementarity and the New Keynesian model

**Introduction**

This paper explores an alternative specification of the utility function and its theoret-ical implications for the new keynesian model.

A recent strand of literature has introduced wealth in the utility function. Krish-namurthy and Vissing- Jorgensen (2012) use bonds in the utility function and infer demand functions to explain the behavior of the bond market. Michaillat and Saez (2014, 2018) explore consequences for the New Keynesian model, especially at the zero lower bound. Saez and Stantcheva (2017) look at implications for optimal capital taxation. Kumhof, Ranciere and Winant (2015) use it to match profiles of income distribution in a model of financial crisis. Michau (2017) introduces a similar spec-ification in a model with downward nominal wage rigidity and zero lower bound on nominal interest rate. A related class of models comes from the overlapping gener-ations literature. Yaari (1964) introduced the joy of giving specification to explain bequests. Under this specification, households care about the amount of wealth their children will inherit but do not care about the consumption of their children, a model refined by Abel and Warshawsky (1988).

My contribution is to study the case of nonseparability between consumption and future wealth. The utility function of a representative agent depends on three vari-ables, consumption, leisure and next period wealth. Allowing for wealth in the utility function and nonseparability introduces two new parameters in the model. The first parameter, denoted κ, governs the discount rate in the linear Euler equation. The second parameter ν reflects the degree of complementarity between consumption and future wealth. A positive value of ν allows me to obtain a low elasticity of intertem-poral substitution along a moderate income eﬀect on labor supply whereas, under the standard specification, the former is the inverse of the latter. I call such complemen-tarity between consumption and future wealth the intertemporal complementarity. Disentangling the income eﬀect and the intertemporal substitution eﬀect has impor-tant implications for the model. The elasticity of hours worked with respect to real wages becomes diﬀerent from the elasticity with respect to the real interest rate, mod-ifying the response of real wages and unemployment to a monetary policy shock and the response of output gap to a demand shock.

I extend the analysis to a medium scale DSGE model. I focus on implications for labor market variables, especially real wages and unemployment. Following Gali(2011), I in-troduce sticky wages and identify unemployment as the diﬀerence between the desired labor supply, given by the first order condition of a family behaving competitively, and the eﬀective labor demand. In the standard model, an expansionary monetary policy shock generates a relatively large response of real wages and a very large re-sponse of unemployment. This large response of unemployment is caused by the shift in labor demand but also by a large shift in labor supply. Introducing intertemporal complementarity allows me to reduce substantially this response of labor supply.

Then, I provide an estimation of parameters κ and ν. I estimate the medium scale model using Bayesian techniques. I find a large value for the parameter κ, suggesting a substantial discount rate in the Euler equation. I also find a large and positive value for the parameter ν, supporting intertemporal complementarity. This high value of ν is the consequence of the diﬀerence it creates between the income eﬀect on labor supply and the intertemporal substitution eﬀect. When the elasticity of intertemporal substitution is the inverse of the income eﬀect, the estimation gives a low EIS and a large income eﬀect, leading to implausible fluctuations in labor supply. A positive parameter ν allows for a low EIS and a moderate income eﬀect providing a more plausible serie for labor supply. High values for ν and κ seem a robust result. I reestimate the model with several alternative specifications, like alternative prior for κ, habits consumption, a longer sample, and labor force participation instead of hours worked in observables. Outcomes are consistent with my baseline estimation.

Substituability between consumption and leisure time is an alternative way to separate the income eﬀect and the intertemporal substitution eﬀect. This has been extensively analyzed by Bilbiie (2009) to explain the response of consumption to fiscal policy shocks. Some degree of substituability seems plausible but the evolution of consump-tion at retirement provides an upper bound to it (See Kimball and Shapiro 2008 for some quantitative exercise). Compatibility with balanced growth is also a concern. It seems interesting to complement this approach by exploring an alternative specifica-tion focusing on the intertemporal choice.

Various explanations have been put forward to justify the inclusion of wealth in the utility function. Wealth can provide an important social status leading consumers to have a preference for it. Alternatively, it may capture several saving motivations. Under the standard specification, a representative agent only saves for consumption smoothing. Households also save to insure themselves against negative income shock, to increase their income at retirement or to hand their estate to their children. A model integrating explicitly all these motivations would be better. However, the cost in complexity would be very high. Wealth in the utility function may generate in some extent a similar behavior for aggregate consumption and leisure whereas keeping the convenience of the representative agent framework.

Several pieces of literature have recently cast doubts on the standard model of in-tertemporal choice. The model implies that theoretical responses to expected mone-tary policy shocks are much larger than their empirical counterparts (Del Negro et al. 2013). A simple way to solve the forward guidance puzzle is to introduce a discount in the linear Euler equation (see McKay, Nakamura and Steinsson 2016 and Gabaix 2017). Such discount rate emerges immediately when wealth is in the utility function. It is interesting to note that intertemporal complementarity increases the discount in the linear Euler equation. Heterogeneous agents models suggest that a substantial part of the response of consumption to monetary policy could come from an indirect eﬀect, through the increase of the income of « hand to mouth » households, and not from the direct eﬀect through intertemporal substitution (Auclert 2017, Kaplan, Moll and Violante 2017).

The paper is organized as follows. The first section explores the household choice in a simple optimization problem in finite horizon with wealth in the utility. The second section extends the results to infinite horizon. The third gives several implications for the macroeconomic model using both a simple model to highlight intuition and a medium scale model to confirm these insights in a more « realistic » environment. I estimate the model in the fourth. I discuss some assumptions and implications in the fifth. I examine additional consequences for forward guidance in the sixth. The seventh section introduces time varying wealth in the model.

### Intratemporal Household choice

When wealth enters into the utility function, households have two motives to accumu-late it. First, because it increases the « income » of the next period. Second, because it increases their current utility. To get a better intuition of the household’s behavior, it is useful to start by only considering this second motive. To do so, I consider an house-hold which only cares about its current utility whose wealth is one of the arguments. I label this model as the « Wealth Targeting Model ».

**The household program**

I consider the optimization problem of a consumer who does not care about future util-ity streams but whose current utility function accepts its future wealth as an argument. At period t, the objective function of the consumer is

U(Ct, Lt, At+1) (2.1)

under the budget constraint

QtAt+1 + WtLt + Ct = At + Wt + Πt (2.2)

where C is the consumption, L is leisure time, W is real wage, and Π are profits distributed by firms. A is an asset which gives the right to receive one unit of con-sumption good at the next period. A does not provide utility through a continuation value but directly provides some utility, hence there is a positive demand for assets even if the optimization program of the household is purely static. Households still have a choice to make between current consumption and future assets.

To buy one unit of this asset, the consumer should pay a price Q. This price is the inverse of the interest factor. Qt = 1 (2.3) where rr is the real interest rate. Qt is the price of future consumption goods.

For the moment, I identify wealth with safe bonds. The point is that consumers may buy in period t a « promise » on final good of period t + 1. The amount of this promise enters into the utility function. I consider alternative interpretations and some of their consequences in the section dedicated to the model with varying wealth. Until then, I will use « wealth » and « assets » as synonyms.

**First order conditions**

I now solve for first order conditions.

**Proposition 2.1.** The utility function reaches its local maximum under the budget constraint if the following first order conditions are fulfilled

UC (Ct, Lt, At+1) = Λt (2.4a)

UL(Ct, Lt, At+1) = WtΛt (2.4b)

UA(Ct, Lt, At+1) = QtΛt (2.4c)

Where Uc (resp. UL and UA) is the first derivative of the utility function with respect to consumption(resp. leisure and wealth). Λt is the Lagrange multiplier.

The problem is a basic consumer choice problem. whose solution is straightforward. A proof is given in appendix B.1.1. Now, suppose that consumption C, wealth A an leisure L reach steady state values. I can linearize conditions from proposition 2.1 around the steady state. I defined ct, at and lt and λt as percentage deviation from their steady state value. More generally, small letters will denote percentage deviation from steady state or deviation from steady state.

Proposition 2.2. A linear approximation of the system of equations defined in proposition 2.1 is UCC C ct + UCAA at+1 + UCLL lt = λt (2.5a)

UCLC ct + ULAA at+1 + ULLL lt = wt + λt (2.5b)

UCAC ct + UAAA at+1 + ULAL lt = qt + λt (2.5c)

Terms UC (resp UL, UA), and UCC (and other similar terms), denote the steady state value of first and second order derivatives of the utility function.

Proof The linear approximation is a first order Taylor expansion of first order condi-tions defined in proposition 2.1. Detailed computations are given in appendix B.1.2

**Separable preferences**

Before considering the case of intertemporal complementarity, I show that, under separable preferences, first order conditions defined in proposition 2.2 are related to the first order conditions of the standard model.

**Hypothesis 2.1.** Preferences are separable. Cross derivative of the utility function are equal to zero : UCL=0

UAL=0

UCA=0

To allow a proper comparison with the standard model, I combine first order conditions with several general equilibrium conditions. I assume that the supply of assets is fixed. Thus, the deviation from steady state is equal to zero. I also introduce a relation between leisure time and hours worked.

**Hypothesis 2.2.** the asset supply equation is given by At+1 = A where A is a constant, hence the percentage deviation from steady state is at+1 = 0 (2.7)

This assumption states that asset supply is not sensitive to the asset price and thus to the demand of assets by consumers. In the last section of the paper, I relax this assumption and study the model with a varying asset supply.

With assumptions 2.1 and 2.2, The system of linear equations considered in proposition 2.2 becomes UCCC ct = λt (2.8a) ULLL lt = wt + λt (2.8b) qt + λt = 0 (2.8c)

To obtain more friendly equations, I substitute leisure with hours worked and the price of assets with real interest rate

Hours worked are given by lt = ηnt (2.9) where η is the ratio between the steady state working time and the steady state leisure time1 . 1 Often denoted N

The deviation from the steady state real interest rate rrt is directly related with the percentage deviation from the steady state bond price qt rrt = −qt (2.10)

It is convenient to make notations easier by introducing parameters σ = UCC C and θ = ULLL . I obtain two equations for labor supply and consumption.

**Proposition 2.3.** Labor supply and consumption equation are

θηnt = wt − σct (2.11)

σct = −rrt (2.12)

Both equations are derived from the system 2.8. Labor supply equation (2.11) is common with the standard model fof intertemporal choice. The diﬀerence lies in equation (2.12). Instead of having an equation for consumption growth, I have an equation for consumption levels with respect to interest rate. The parameter σ governs both the intertemporal substitution eﬀect and the income eﬀect on labor supply. This feature is shared with the standard model of intertemporal choice.

#### Intertemporal nonseparability

I now allow the cross derivative between wealth and consumption to be diﬀerent from zero. Hypothesis 2.1 becomes

Hypothesis 2.3.

UCL=0

UAL=0

UCA =6 0

A positive cross derivative between consumption and wealth implies that assets and consumption are complements in the sense of Edgeworth, whereas a negative cross derivative means the two are substitutes.

I keep the separability assumption for leisure. UAL = 0, UCL = 0. This is a strong assumption but the goal is to keep a tractable model and to focus on intertemporal choice. My analysis follows the analysis made by Bilbiie (2009) for the nonseparability between consumption and leisure.

Nonseparability between consumption and assets allows me to disentangle the con-sumption elasticity to interest rate from the income eﬀect on labor supply.

**Proposition 2.4**. The system from proposition 2.2 becomes

UCC C ct = λt (2.13a)

ULLL lt = wt + λt (2.13b)

UCAC ct = qt + λt (2.13c)

Let me define the parameter ν ≡ UCAC .

**Corollary 2.** The intertemporal substitution eﬀect is governed by σ + ν whereas the income eﬀect is is governed by the parameter σ

θηnt = wt − σct (2.14)

(σ + ν)ct = −rrt (2.15)

Intuition Some intuition may be given for this result. The sensitivity of leisure with respect to interest rate is equal to σ+σν . A large and positive ν is obtained if UCA > 0 and thus complementarity between present consumption and future assets. A fall in real rates implies that the marginal utility of assets should rise relative to the marginal utility of consumption and relative to the marginal utility of leisure. With separable preferences, consumption increases and thus reduces the marginal utility of consump-tion. With nonseparable preferences, if consumption and assets are complements, the rise in consumption decreases the marginal utility of consumption and increases the marginal utility of assets. Thus, a much lower rise in consumption may achieve the equality between the relative price of future consumption goods and the marginal rate of substitution between consumption and assets. A similar line of reasoning explains the smaller sensitivity of leisure to real interest rate. The increase in the marginal utility of assets implies a lower fall in the marginal utility of leisure and thus a lower rise in leisure.

**Table of contents :**

**Introduction **

**1 Are consistent expectations better than rational expectations ? **

1.1 Introduction

1.2 Literature

1.3 Framework

1.4 Long Run Equilibrium

1.4.1 Equilibrium definition

1.4.2 The Rational Learner Equilibrium

1.4.3 The Consistent Learner Equilibrium

1.5 Simulations

1.5.1 Overview

1.5.2 Summary of the model and the algorithm

1.5.3 Initialization

1.5.4 Baseline calibration

1.5.5 Experiments

1.6 Results

1.6.1 Summary

1.6.2 Main result

1.6.3 Exploration

1.6.4 The role of initialization

1.6.5 Path dependence

1.7 Extensions: misspecification and structural breaks

1.7.1 Intuition

1.7.2 Adding persistent unobservables

1.7.3 Adding structural breaks

1.7.4 Results

1.7.5 Why do rational learners misspecify their model

1.8 Discussion

1.9 Conclusion

**2 Intertemporal Complementarity and the New Keynesian model **

2.1 Introduction

2.2 Intratemporal Household choice

2.2.1 The household program

2.2.2 First order conditions

2.2.3 Separable preferences

2.2.4 Intertemporal nonseparability

2.3 Intertemporal Household choice

2.4 Implications for the New Keynesian model

2.4.1 Framework

2.4.2 The supply effect of the real interest rate

2.4.3 Monetary shocks

2.4.4 Demand Shock

2.5 A Bayesian estimation of the model

2.5.1 Baseline estimation

2.5.2 Inspecting the mechanism

2.5.3 Robustness

2.5.4 Quantitative implications of the estimated model

2.6 Discussion

2.7 Implications for the forward guidance puzzle

2.8 The model with varying wealth

2.9 Conclusion

**3 Corporate Investment and Adverse Selection, A Reappraisal **

3.1 Introduction

3.2 The adverse selection problem

3.2.1 Firms and lenders

3.2.2 Equilibrium Definition under private information

3.2.3 Characterization of the separating equilibrium

3.2.4 The scope of the equilibrium

3.3 The macroeconomic model

3.3.1 Extension of the equilibrium for time varying interest and discount rate

3.3.2 Dynamic equations for good and bad firms

3.3.3 Investment Dynamics

3.3.4 Steady state

3.4 Implications: Response to changes in borrowing cost

3.4.1 The determinants of corporate investment

3.4.2 Response of leverage to interest rate

3.4.3 Long run effects of changes in borrowing costs

3.4.4 Short run effects

3.4.5 Response to changes in opportunity cost

3.5 Conclusion