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The Choice of the Functional Form in the Consumption Euler Equation: a Critical View

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The Choice of the Functional Form in the Consumption Euler Equation: a Critical View DOTTORATO DI RICERCA IN ECONOMIA E FINANZA INTERNAZIONALE XXIV CICLO The Choice of the Functional Form in the Consumption Euler Equation: a Critical View Daria Pignalosa A.A. 2015/2016 Supervisor: Prof. Luigi Ventura Contents Abstract ............................................................................................................................................................................. 1 The Euler equation approach: theory and evidence 1. Introduction ............................................................................................................................. 2 2. The consumption Euler equation ......................................................................................... 3 3. Early tests of the Euler equation: the emergence of empirical puzzles ....................... 6 4. Reconciling theory and empirical evidence: extensions of the standard model ....... 10 5. Conclusions ............................................................................................................................. 14 6. References ............................................................................................................................... 18 The choice of the functional form in the consumption Euler equation approach: a critical view 1. Introduction ........................................................................................................................... 22 2. Preference Parameters ......................................................................................................... 25 2.1. How to summarize risk attitudes: the coefficients of risk aversion ................................. 25 2.2. Measuring the precautionary motive for saving: relative and absolute prudence ....... 27 2.3. The aversion to temporal fluctuations of consumption: the elasticity of intertemporal substitution ........................................................................................................................................... 28 2.4. The relationship between preference parameters ................................................................ 29 3. Utility Functions ................................................................................................................... 33 3.1. Quadratic preferences: certainty equivalence ........................................................................ 33 3.2. Exponential preferences: constant absolute risk aversion ................................................. 35 3.3. Isoelastic preferences: constant relative risk aversion ........................................................ 36 3.4. Nonexpected utility: recursive Epstein-Zin-Weil preferences ......................................... 38 4. The empirical evidence on the structural parameters of the utility function .......... 39 5. Conclusions ............................................................................................................................. 41 6. References ............................................................................................................................... 42 7. Appendix ................................................................................................................................. 47 The choice of the functional form in the consumption Euler equation approach: a simulation exercise 1. Introduction ........................................................................................................................... 51 2. The misleading role of quadratic preferences: a simple numerical example ............ 53 3. Preference parameters and utility functions ................................................................... 55 4. A decomposition of the saving rate ................................................................................... 59 5. The choice of the functional form: a simulation exercise ............................................. 66 5.1 A numerical example with different preference specifications ........................................... 68 5.2 The interplay of precautionary and intertemporal motives in a two period model ..... 69 6. Final remarks .......................................................................................................................... 72 7. References ............................................................................................................................... 73 8. Appendix A: Preference specifications.............................................................................. 75 9. Appendix B: Simulation results .......................................................................................... 76 Abstract This thesis deals with some analytical questions that arise in the modern theory of con- sumption based on the intertemporal utility maximization model, also known as the Euler equation approach. It thus builds on the theoretical and empirical literature which has stemmed from the seminal contribution by Hall (1978), who extended the basic life cycle – permanent income model of consumption to the case of uncertainty. In reviewing this literature and the theoretical developments and extensions that the original model has undergone due to the need of accounting for the puzzling empirical evidence, the thesis aims at highlighting the crucial role that the preference parameters play in the specifica- tion of the theoretical content of the model and its predictions, and in the definition of the policy implications that may be drawn from it. Crucial as the role of preference parameters may be both in theory and practice, our analysis reveals, also by making use of some simulation exercises, the substantial inability of the Euler equation approach to give definite content to such parameters, due to the heavy dependence of the results of parameter estimation on the specification of the utility function. The first paper offers a survey of the literature which, following Hall (1978), has en- gaged in the task of testing the model against empirical evidence and proposing successive extensions and refinements of the original model ending up in the current enriched ver- sions of it. The paper draws on other surveys but focuses in particular on an overall as- sessment of the theoretical implications of the extensions which the literature has proposed. The second paper focuses on the literature directly aiming to estimate the pa- rameters characterizing preferences – offering in this respect a specific survey which seems to be lacking in the literature – and highlights the crucial role of the specification of the utility function in such estimates. Drawing on this result, the third paper proposes a number of original simulation exercises aimed at showing that the same time profile of consumption and saving may give rise to the estimation of rather different values of the crucial preference parameters depending on the particular utility function adopted. 1 The Euler equation approach: theory and evidence 1. Introduction The modern theory of consumption rests on the idea that individuals maximize lifetime utility subject to an intertemporal budget constraint. The idea is based on the Life Cycle – Permanent Income Hypothesis developed by Modigliani and Brumberg (1954) and Friedman (1957). In his seminal contribution, Hall (1978) extended the model to the case of uncertainty with the introduction of the rational expectations assumption and proposed to use the first order conditions of the intertemporal optimization problem faced by the consumer to derive a set of orthogonality conditions. In the framework considered by Hall, the basic implication of the equilibrium condition of the model – the Euler equation – is that, conditional on current consumption, other current variables, including income, do not help in predicting future consumption. Starting with Hall, the literature focused on testing the model of intertemporal utility maximization relying on the Euler equation and a new approach to consumption, often referred to as the Euler equation approach, has been established. The early empirical tests of the formulation proposed by Hall found several results that apparently contradicted theoretical predictions. A number of empirical puzzles thus arose. These are known in the literature as: a) excess sensitivity; b) excess smoothness; c) hump in the age profile of consumption; d) retirement puzzle; and d) equity premium puzzle. The subsequent literature tried to provide an interpretation of these empirical puz- zles and to progressively modify and enrich the original version of the model so as to ren- der it able to explain the data. This paper is devoted to the reconstruction of the evolution of the theoretical view of consumer behaviour stemming from the Euler equation approach to consumption. A sur- vey will be proposed of the contributions highlighting the various empirical puzzles and proposing extensions and refinements of the basic model aimed at reconciling theoretical predictions and empirical evidence. In the literature on consumption, the link between em- pirically-oriented contributions and theoretical developments is in fact quite strong, given the fact that the theoretical evolution of the approach has been strongly influenced by the need to face the empirical puzzles and to find more sophisticated versions of the model that could account for observed facts. This has implied a number of relevant theoretical changes with respect to the original formulation. 2 Following some well-documented
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