King–plosser–rebelo preferences

What Is King–Plosser–Rebelo Preferences?

King–Plosser–Rebelo (KPR) preferences refer to a specific type of utility function widely employed in macroeconomic models to analyze how individuals make intertemporal choice regarding consumption and leisure. This class of preferences is a cornerstone in dynamic stochastic general equilibrium (DSGE) models, falling under the broader category of economic modeling. KPR preferences are particularly noted for their ability to allow for a consistent economic growth path, even when individuals face decisions under uncertainty, making them highly useful for understanding long-term economic dynamics and cyclical fluctuations.

His³⁷tory and Origin

King–Plosser–Rebelo preferences were introduced by economists Robert G. King, Charles I. Plosser, and Sergio T. Rebelo in their seminal 1988 paper, "Production, Growth, and Business Cycles I: The Basic Neoclassical Model." This work l³⁵, ³⁶aid crucial groundwork for modern real business cycle (RBC) theory. The authors developed this particular utility function to ensure that their neoclassical growth model could generate a "balanced growth path," a state where key macroeconomic variables like consumption, output, and capital grow at constant, proportional rates. The explici³⁴t derivation of these preferences was later detailed in a technical appendix published in 2002. The goal was to create a framework that was consistent with observed patterns of sustained economic growth over long periods while also providing a laboratory for analyzing economic fluctuations driven by factors such as technology shocks.

Key Tak³², ³³eaways

• King–Plosser–Rebelo preferences are a type of utility function used in macroeconomics to model individual choices between consumption and leisure over time.
• They are a key component of DSGE models, allowing for the analysis of economic fluctuations and long-term growth.
• A distinguishing feature is their compatibility with a "balanced growth path," meaning economic variables can grow proportionally in the long run.
• KPR preferences enable researchers to study the effects of various policies, such as monetary policy and fiscal policy, on aggregate economic activity.
• They provid³¹e flexibility to incorporate varying degrees of risk aversion and intertemporal elasticity of substitution.

Formula and³⁰ Calculation

King–Plosser–Rebelo preferences are typically represented by a specific functional form of the utility function. One common multiplicatively separable form for utility derived from consumption (C_t) and leisure (L_t) at time (t) is:

Where:

• (U(C_t, L_t)) represents the individual's total utility at time (t).
• (C_t) is current consumption.
• (L_t) is current leisure.
• (\sigma_c) is the inverse of the intertemporal elasticity of substitution in consumption. Strict concavity of the utility function implies (\sigma_c > 0).
• (v(L_t)) is an increasing and concave function of leisure, often specified as (\frac{L_t^{1-\sigma_L}}{1-\sigma_L}) or (\ln(L_t)).

In the limit case where (\sigma_c = 1), the utility function becomes additively separable and takes a logarithmic form:

These functional forms are chosen because they ensure that a model built upon them can exhibit a balanced growth path, where the growth rates of consumption and capital remain constant over time, consistent with empirical observations of long-run economic trends.

Interpreting Ki²⁸ng–Plosser–Rebelo Preferences

The interpretation of King–Plosser–Rebelo preferences centers on how individuals trade off consumption today versus consumption in the future, as well as the allocation of time between work (for income) and leisure. In models featuring KPR preferences, agents are assumed to maximize their overall utility across different periods.

A key element in interpret²⁷ing KPR preferences is the concept of intertemporal substitution. These preferences allow for the analysis of how individuals adjust their labor supply and consumption patterns in response to changes in interest rates and wages. For example, if interest rates increase, a household might be incentivized to save more and consume less today, deferring consumption to the future when it can yield a higher return. Similarly, changes in real wages can influence an individual's decision to work more or enjoy more leisure. The specific structure of K²⁶PR preferences ensures that, along a steady-state equilibrium path, the income and substitution effects related to rising labor productivity cancel each other out, leading to a stable proportion of time allocated to work.

Hypothetical Example

C²⁴, ²⁵onsider a hypothetical macroeconomist developing a DSGE model to understand how a sustained increase in technological progress affects a country's economy. The economist chooses to incorporate King–Plosser–Rebelo preferences into their model's household sector.

Scenario: The economy experiences a consistent 2% annual increase in labor-augmenting technology.

Step-by-step analysis with KPR preferences:

1. Household Optimization: Households in the model, endowed with KPR preferences, continuously make decisions to maximize their lifetime utility, balancing current consumption and future consumption, as well as choosing between working hours and leisure.
2. Productivity Growth Impact: As technology improves, the real wage rate (the reward for labor) tends to increase. With simpler utility functions, this might lead to a significant increase in leisure (an income effect) or a significant increase in labor supply (a substitution effect).
3. Balanced Growth Consistency: However, because of the specific structure of KPR preferences, the income effect (households can afford more leisure due to higher wages) and the substitution effect (higher wages make leisure more expensive in terms of foregone income) precisely offset each other along a balanced growth path. This means that, despite growing productivity and wages, the optimal proportion of time spent working remains relatively constant in the long run.
4. Model Output: The model, incorporating KPR preferences, would project that the economy achieves a new balanced growth path where output, consumption, and capital stock all grow at the same 2% rate as the technological progress, while the aggregate labor supply remains stable. This result aligns with observed macroeconomic facts about industrialized economies.

Practical Applications

Kin²³g–Plosser–Rebelo preferences are a foundational tool in modern macroeconomics, primarily within the framework of dynamic stochastic general equilibrium (DSGE) models. These models are employed by central banks, such as the Federal Reserve, and international organizations like the International Monetary Fund (IMF) for a variety of purposes:

• Policy Analysis: KPR preferences are crucial for analyzing the effects of monetary policy and fiscal policy on aggregate economic activity, labor supply, and consumption patterns. Researchers can assess how changes ²²in interest rates, taxes, or government spending might impact the economy. The Federal Reserve Bank of New Yor¹⁹, ²⁰, ²¹k, for example, utilizes a DSGE model that incorporates such preferences for forecasting and policy analysis.
• Business Cycle Analysis: Th¹⁸ese preferences help construct models that can replicate observed macroeconomic fluctuations. By allowing for a balanced growth path, KPR preferences enable economists to disentangle long-run trends from short-run business cycles.
• Forecasting: DSGE models wi¹⁷th KPR preferences are used to generate forecasts for key macroeconomic variables like GDP growth and inflation, contributing to the economic outlooks provided by institutions. The Chicago Fed, Cleveland Fed, and¹⁵, ¹⁶ St. Louis Fed also employ DSGE models for policy analysis and forecasting.
• Academic Research: KPR pref¹², ¹³, ¹⁴erences are a standard assumption in academic papers exploring topics such as economic growth, capital accumulation, and the behavior of households and firms in dynamic settings.

Limitations and Criticisms

Whi¹¹le King–Plosser–Rebelo preferences offer significant advantages for macroeconomic modeling, particularly their consistency with balanced growth, they also face certain limitations and criticisms:

One primary criticism relates to the strong assumptions required for KPR preferences to generate a constant labor supply along a balanced growth path. This implies that the income effect (where higher wages might lead to more leisure) and the substitution effect (where higher wages make leisure more costly, encouraging work) must exactly offset each other. While convenient for modeling, this exa⁹, ¹⁰ct offset may not fully capture the complexities of real-world labor supply decisions. Some alternative preference specifications aim to relax this strong restriction, allowing for varying trends in hours worked.

Another limitation stems from the gene⁸ral nature of representative agent models, which KPR preferences are often embedded within. These models assume a single, "representative" household that acts rationally to maximize utility, which may not adequately reflect the heterogeneity of actual households, their diverse financial situations, or their varying degrees of risk aversion. This simplification can sometimes limit⁷ the models' ability to fully capture certain real-world phenomena or predict outcomes during periods of significant economic disruption.

Despite these criticisms, KPR preferences remain a widely used and valuable tool for understanding the fundamental dynamics of economic systems and providing a benchmark against which more complex models can be compared.

King–Plosser–Rebelo Preferences vs. Epstein–Zin Preferences

King–Plosser–Rebelo (KPR) preferences and Epstein–Zin preferences are both functional forms of utility function used in dynamic macroeconomic models, but they differ significantly in their structure and what they allow economists to model.

KPR preferences are often chosen for their compatibility with a "balanced growth path," meaning that macroeconomic variables can grow at a constant rate in the long run while maintaining a stable proportion of time devoted to labor supply. They typically feature a multiplicative separability between consumption and leisure within the utility function. A key implication of standard KPR preferences is that the intertemporal elasticity of substitution (how willing agents are to substitute consumption over time) is directly linked to the coefficient of relative risk aversion.

In contrast, Epstein–Zin preferences, developed by Larry Epstein and Stanley Zin, are a type of recursive utility. This means that current utility depends on current consumption and the certainty equivalent of future utility. A distinguishing feature of Epstein–Zin preferences is their ability to disentangle the coefficient of relative risk aversion from the intertemporal elasticity of substitution. This separation is particularly valuable in financial e⁶conomics and asset pricing models, as it allows researchers to explore asset price puzzles (like the equity premium puzzle) by independently adjusting risk aversion without altering the willingness to substitute consumption over time. While KPR preferences are standard in many [real busine⁵ss cycle (RBC) theory](https://diversification.com/term/real-business-cycle-theory) models, Epstein–Zin preferences offer greater flexibility in modeling agents' attitudes toward risk and intertemporal tradeoffs, especially in environments with uncertainty.

FAQs

What is a utility function in economics?

A utility function in economics is a mathematical expression that represents the satisfaction or "utility" an individual derives from consuming goods and services or engaging in activities like leisure. It helps economists understand and model how individuals make choices to maximize their overall well-being.

Why are King–Plosser–Rebelo preferences used in macroeconomic models?

King–Plosser–Rebelo preferences are used because they enable macroeconomic models, particularly DSGE models, to exhibit a "balanced growth path." This means that key economic variables like consumption, output, and capital accumulation can grow at consistent rates over the long term, which aligns with observed historical trends in developed economies.

How do King–Plosser–Rebelo preferences handle risk?

King–Pl⁴osser–Rebelo preferences incorporate aspects of risk aversion through their functional form. The parameter (\sigma_c) (inverse of the intertemporal elasticity of substitution in consumption) influences the degree to which individuals dislike variability in their consumption over time. However, this structure typically links risk aversion and intertemporal substitution.

What is a balanced growth path?

A balanced growth path in macroecon³omics refers to a theoretical state where an economy's key aggregate variables—such as output, consumption, and investment—grow at the same, constant rate. This path represents a long-run equilibrium where the economy's structure remains stable despite ongoing growth.

Are King–Plosser–Rebelo preferences still relevant in modern economic re²search?

Yes, King–Plosser–Rebelo preferences remain highly relevant. They continue to be a standard benchmark in many macroeconomic models, especially those used by central banks and academics for policy analysis and understanding long-run economic dynamics. While more complex preferences exist, KPR preferences provide a robust and analytical¹ly tractable framework for many research questions.