Recursive utility models

What Are Recursive Utility Models?

Recursive utility models are a class of economic models that describe how individuals make intertemporal choice decisions under uncertainty, particularly when their preferences over future uncertain outcomes are not necessarily time-additive. These models are a significant development within economic theory, offering a more flexible framework for analyzing sequential decision making compared to traditional approaches. Unlike standard models where total utility is simply a sum of utilities from current and future consumption discounted over time, recursive utility models express current utility as a function of current consumption and the certainty equivalent of future utility. This structure allows for a distinct separation between an individual's attitudes toward risk and their time preference for consumption across different periods, a critical feature for understanding complex behaviors in financial markets.

History and Origin

The foundational concept of recursive utility can be traced back to Koopmans (1960) in a setting without uncertainty, with later extensions incorporating uncertainty by David Kreps and Evan Porteus in 1978.¹¹ Their work, "Temporal Resolution of Uncertainty and Dynamic Choice Theory," introduced the idea of "temporal lotteries," allowing individuals to express preferences about the timing of the resolution of uncertainty.⁹, ¹⁰ This was a departure from standard expected utility theory, which often implicitly links an individual's risk aversion to their willingness to smooth consumption over time.⁸

A pivotal advancement in the field came with the development of Epstein-Zin preferences by Larry Epstein and Stanley Zin, notably detailed in their 1989 paper, "Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework." This model provides a widely adopted specification of recursive utility that explicitly disentangles the elasticity of intertemporal substitution from the coefficient of relative risk aversion.⁷ This separation was crucial for addressing several empirical puzzles in macroeconomics and finance that traditional models struggled to explain, such as the equity premium puzzle and the risk-free rate puzzle.⁶

Key Takeaways

Recursive utility models offer a flexible framework for modeling dynamic preferences, allowing for the separation of risk aversion from the elasticity of intertemporal substitution.
These models are crucial in situations involving uncertainty and multi-period decision making, providing a more realistic representation of how economic agents evaluate consumption and investment opportunities over time.
The recursive structure lends itself well to techniques like dynamic programming, enabling the analysis of complex sequences of choices.
Key examples include Kreps-Porteus and Epstein-Zin utility models, which have significantly impacted asset pricing and macroeconomics.

Formula and Calculation

The general form of a recursive utility function for an agent making choices over consumption stream ({c_t}_{t=0}^\infty) can be expressed recursively. For Epstein-Zin preferences, the utility at time (t), denoted (U_t), is defined by:

U_t = \left[ (1-\beta) c_t^\rho + \beta \mu_t (U_{t+1})^\rho \right]^{1/\rho}

Where:

(U_t) represents the utility at time (t).
(c_t) is current consumption at time (t).
(\beta) is the time preference parameter (discount factor), with (0 < \beta < 1).
(\rho) is a parameter related to the elasticity of intertemporal substitution (EIS), where (EIS = 1/(1-\rho)).
(\mu_t(\cdot)) is a certainty equivalent operator, which aggregates the risk associated with future utility. For Epstein-Zin, a common choice is the CES certainty equivalent: $\mu_t(U_{t+1}) = [E_t U_{t+1}^\alpha]^{1/\alpha}$ Where (E_t) is the expectation conditional on information available at time (t), and (\alpha) is a parameter related to risk aversion, with the coefficient of relative risk aversion (CRRA) given by (1-\alpha).

This recursive formulation, which relies on a concept of expected value of future utility, allows for a precise decomposition of the influences of consumption smoothing and risk taking.

Interpreting Recursive Utility Models

Interpreting recursive utility models involves understanding how they reflect an economic agents preferences across time and under uncertainty. The key insight is the ability to separate the elasticity of intertemporal substitution (EIS) from the coefficient of relative risk aversion.

In traditional models, a high desire for consumption smoothing (high EIS) implies a low aversion to risk, and vice-versa. Recursive utility models break this linkage. For example, an agent with Epstein-Zin preferences can exhibit a strong desire for consumption smoothing (high EIS) while simultaneously being highly risk-averse. This allows economists to better calibrate models to empirical observations in financial markets, such as the historically high equity premium, which has been difficult to explain with standard utility functions. The parameters (\rho) and (\alpha) in the Epstein-Zin framework allow for independent adjustment, providing a richer set of behaviors. This helps in understanding investment and savings decisions over long horizons where both temporal preferences and attitudes toward risk play distinct roles.

Hypothetical Example

Consider an individual, Sarah, who is planning her consumption and savings over two periods. In a traditional model, her preference for smoothing consumption between today and tomorrow would be directly tied to how much she dislikes risk. If she likes to smooth consumption, she's automatically assumed to be less risk-averse.

With a recursive utility model, Sarah's preferences can be more nuanced. Suppose she strongly desires to maintain a stable consumption path (high elasticity of intertemporal substitution). She wants to avoid large fluctuations in her spending, regardless of how risky her future income might be. Simultaneously, she might be extremely cautious about gambles and losses (high risk aversion).

In a hypothetical scenario, if Sarah receives an unexpected windfall today, a traditional model might predict she would smooth it significantly across both periods, implying a low aversion to the risk of future income variability. A recursive utility model, however, could show that while she still smooths consumption (due to her high EIS), she also saves a larger portion of the windfall as a buffer against future uncertainty due to her distinct high risk aversion. This nuanced behavior, where the desire for smooth consumption doesn't force a specific level of risk tolerance, makes recursive utility models more powerful in describing real-world financial behaviors.

Practical Applications

Recursive utility models are extensively applied in various areas of economics and finance due to their enhanced ability to model complex preferences. Their primary use cases include:

Asset Pricing: These models have been instrumental in explaining empirical puzzles in asset pricing, such as the equity premium puzzle and the risk-free rate puzzle. By decoupling risk aversion from intertemporal substitution, recursive utility models allow for more realistic calibration of investor preferences, leading to stochastic discount factors that can better match observed asset returns.⁵
Macroeconomics: They are widely used in dynamic general equilibrium models to study phenomena like business cycles, long-run risk, and the impact of government policies on consumption and savings behavior. Their recursive structure facilitates analysis using dynamic programming techniques.
Optimal Portfolio and Consumption Choice: Recursive utility models provide a more accurate framework for economic agents to make optimal investment and consumption decisions over their lifetime, especially when facing stochastic investment opportunities or time-varying risk. Researchers use these models to derive approximate analytical solutions for consumption/investment problems, even with finite horizons.⁴
Climate Change Economics: The flexibility of recursive utility in disentangling attitudes towards risk and time preferences makes them valuable in evaluating long-term policies, such as those related to climate change, where decisions involve significant uncertainty over extended periods.

Limitations and Criticisms

Despite their advantages, recursive utility models are not without limitations or criticisms. One significant challenge lies in the analytical tractability and numerical solution of these models. While they offer greater flexibility, the complexity of the recursive structure, especially when combined with realistic stochastic processes for economic variables, can make deriving closed-form solutions difficult or impossible.³ Researchers often rely on numerical methods or approximations, which can introduce their own complexities and potential errors.

Another area of debate concerns the existence and uniqueness of solutions for recursive utility functions under certain conditions. Establishing that a recursive equation actually yields a unique, well-defined utility function can be non-trivial.² For instance, some specific formulations, particularly in life-cycle models, may lead to scenarios where non-constant utility solutions only exist under very restrictive and potentially unrealistic assumptions about mortality rates.¹ This highlights that while recursive utility models offer increased realism, their application requires careful consideration of the underlying mathematical properties and assumptions.

Furthermore, while recursive utility models excel at separating risk aversion from intertemporal substitution, some criticisms suggest they still may not fully capture all facets of real-world decision making, such as ambiguity aversion or model misspecification concerns, which are areas of ongoing research in behavioral finance.

Recursive Utility Models vs. Discounted Utility Models

The primary distinction between recursive utility models and discounted utility models lies in their treatment of future utility function and uncertainty.

Discounted Utility Models (also known as time-additive expected utility models) are based on the premise that total utility over time is the sum of discounted utilities from consumption in each period. The utility derived from consumption in one period is independent of utility from other periods, and the discount factor is constant. Crucially, in these models, the coefficient of risk aversion is inversely linked to the elasticity of intertemporal substitution. This means a high desire for consumption smoothing inherently implies a low aversion to risk, and vice versa. This strong linkage has been a significant point of contention in matching empirical data, particularly in asset pricing.

Recursive Utility Models, conversely, define current utility as a function that aggregates current consumption and the certainty equivalent of future utility. This "recursive" structure means that the utility from future periods feeds back into the current period's calculation, but it does so through an aggregation function that can be chosen to disentangle risk aversion from the elasticity of intertemporal substitution. This allows for greater flexibility, enabling economic agents to exhibit preferences where they might be highly risk-averse but also have a strong desire for consumption smoothing, or vice versa, aligning more closely with observed behaviors and resolving some of the empirical shortcomings of the simpler discounted utility framework.

FAQs

What is the main advantage of recursive utility models?

The main advantage is their ability to separate an individual's attitude towards risk (risk aversion) from their willingness to substitute consumption across different time periods (elasticity of intertemporal substitution). This allows for a more nuanced and realistic representation of preferences than traditional expected utility theory.

How do recursive utility models handle uncertainty?

Recursive utility models handle uncertainty by incorporating a "certainty equivalent" of future utility. Instead of simply taking the expected value of future utility, they use an operator that accounts for the individual's risk attitudes, thus allowing preferences over the timing of resolution of uncertainty to be modeled.

Are recursive utility models used in practice?

Yes, recursive utility models are widely used in advanced economic and financial research, particularly in asset pricing models, macroeconomics, and optimal consumption and savings problems. They provide a more robust framework for analyzing dynamic decisions under uncertainty.

What is the Epstein-Zin utility function?

The Epstein-Zin utility function is a popular specific form of a recursive utility model. It is designed to explicitly separate the coefficient of relative risk aversion from the elasticity of intertemporal substitution, making it highly influential in modern finance and macroeconomics.

What are the criticisms of recursive utility models?

Criticisms of recursive utility models often center on their mathematical complexity, which can make them challenging to solve analytically or numerically. Additionally, ensuring the existence and uniqueness of solutions for these models under various economic conditions can be a complex theoretical task.