Elasticity of intertemporal substitution

What Is Elasticity of Intertemporal Substitution?

The elasticity of intertemporal substitution (EIS), often referred to as intertemporal elasticity of substitution (IES), is a fundamental concept in macroeconomics and behavioral economics. It quantifies how readily individuals or households are willing to adjust their consumption patterns across different time periods in response to changes in the real interest rate. In essence, it measures the responsiveness of the growth rate of consumption to changes in the return on savings. A higher elasticity of intertemporal substitution indicates that consumers are more willing to shift consumption between the present and future when faced with varying returns, while a lower elasticity suggests that their consumption choices are less sensitive to such changes. This parameter is crucial for understanding how individuals allocate their resources over their lifetime and how various economic policies might influence their consumer behavior.

History and Origin

The concept of intertemporal substitution has been central to economic theory for decades, particularly in the study of consumption and saving decisions. Early work by economists such as Robert E. Hall significantly contributed to the understanding and empirical estimation of the elasticity of intertemporal substitution. His 1978 paper, "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence," and subsequent work, explored how consumption growth should respond to changes in expected real interest rates, with the EIS as a key determinant.¹³ These foundational contributions laid the groundwork for modern economic models that aim to explain dynamic consumption choices. Despite its importance, empirical efforts to precisely estimate the value of the elasticity of intertemporal substitution have faced significant challenges, leading to a wide range of estimates in academic literature over the years.¹¹, ¹²

Key Takeaways

• The elasticity of intertemporal substitution (EIS) measures the willingness of individuals to reallocate consumption over time in response to changes in the real interest rate.
• A high EIS implies that consumers are highly responsive to changes in returns, while a low EIS suggests less responsiveness.
• It is a critical parameter in macroeconomics, asset pricing, and policy analysis.
• Empirical estimates of the EIS vary widely, posing challenges for model calibration and policy implications.
• The EIS is inversely related to the coefficient of relative risk aversion under standard utility specifications, though this connection is a subject of ongoing debate.

Formula and Calculation

The elasticity of intertemporal substitution ($\sigma$) is often derived from an individual's utility function that exhibits constant relative risk aversion (CRRA). In its continuous-time form, for a utility function $u(c)$, where $c$ is consumption, the EIS is defined as:

$\sigma(c) = - \frac{u'(c)}{c u''(c)}$

Where:

• $u'(c)$ is the first derivative of the utility function with respect to consumption (marginal utility).
• $u''(c)$ is the second derivative of the utility function with respect to consumption.

For a CRRA utility function, given by $u(c) = \frac{c^{1-\theta}}{1-\theta}$ (where $\theta > 0$ and $\theta \neq 1$), the elasticity of intertemporal substitution is $\sigma = \frac{1}{\theta}$. Here, $\theta$ represents the coefficient of relative risk aversion. A special case is when $\theta=1$, which corresponds to a logarithmic utility function ($u(c) = \ln(c)$), where the EIS is 1.

Interpreting the Elasticity of Intertemporal Substitution

Interpreting the elasticity of intertemporal substitution involves understanding how sensitive an individual's consumption path is to changes in the cost of consuming today versus tomorrow, which is largely driven by the real interest rate.

• EIS > 1 (High Elasticity): If the EIS is greater than 1, it means that the substitution effect dominates the income effect. In this scenario, a rise in the real interest rate (making future consumption relatively cheaper) leads to a more than proportional increase in the growth rate of consumption. Consumers are highly willing to defer current consumption to gain higher returns in the future, thus significantly increasing their investment decisions.
• EIS < 1 (Low Elasticity): If the EIS is less than 1, the income effect tends to be stronger than the substitution effect. A rise in the real interest rate would still encourage some saving, but consumers are less willing to drastically alter their current consumption for future gains. Their desire to smooth consumption over time is very strong, even if it means foregoing higher returns.
• EIS = 1 (Unit Elasticity): A unit elasticity implies that the percentage change in the growth rate of consumption is equal to the percentage change in the real interest rate. This is often associated with logarithmic utility functions and is a common assumption in some financial markets models for simplicity.

The magnitude of the elasticity of intertemporal substitution directly impacts how individuals respond to incentives to save or borrow, influencing their overall wealth accumulation over their lifetime.

Hypothetical Example

Consider two individuals, Alice and Bob, both with an initial income of $50,000 per year for two periods and no initial savings. There is a capital market where they can borrow or lend at a real interest rate.

• Alice (High EIS = 1.5): Alice has a high elasticity of intertemporal substitution. When the real interest rate increases from 2% to 5%, Alice sees a strong incentive to save more today to enjoy significantly higher consumption tomorrow. She might cut her current year's consumption from $45,000 to $40,000, saving an extra $5,000. This additional saving, benefiting from the higher interest rate, allows her to increase her consumption in the second period much more substantially than if the interest rate hadn't risen. Her willingness to shift consumption across time is pronounced because the cost of deferring consumption is low for her.
• Bob (Low EIS = 0.3): Bob has a low elasticity of intertemporal substitution. When the real interest rate increases from 2% to 5%, Bob is less inclined to significantly alter his current consumption habits. While he might save a little more, perhaps reducing his current consumption from $45,000 to $44,500, he prioritizes a relatively stable consumption path. The higher interest rate makes him feel slightly wealthier, but his strong preference for smoothing consumption means he won't drastically cut current spending, even for significantly higher future returns. His budget constraint allows for more saving, but his preferences limit his response.

This example illustrates how different EIS values lead to distinct responses to changes in economic conditions, impacting how individuals manage their finances over time.

Practical Applications

The elasticity of intertemporal substitution is a critical parameter in various fields of economics and finance:

• Monetary Policy: Central banks, when conducting monetary policy, need to understand how changes in interest rates affect household consumption and saving. A low EIS implies that interest rate changes may have a more muted effect on aggregate consumption, making monetary policy less effective in stimulating or cooling the economy. Conversely, a high EIS suggests that consumers would respond more strongly, enhancing policy effectiveness. Research indicates that heterogeneity in EIS across countries can lead to dissimilar impacts of common monetary policy in regions like the Eurozone.¹⁰
• Fiscal Policy: Governments consider the EIS when designing tax policies or social security reforms. For instance, the impact of a dividend tax reform on household spending behavior is directly linked to the EIS. Studies using dividend tax news shocks have shown that the EIS plays a crucial role in how households respond to such policy changes.⁹
• Asset Pricing Models: In quantitative finance, the EIS is a key input in asset pricing models, particularly those based on consumption. It helps explain observed patterns in asset returns and volatility.
• Household Finance: Understanding an individual's EIS can inform financial planning. For example, it influences optimal saving rates for retirement planning or responses to employer matching contributions in 401(k) plans.⁸

Limitations and Criticisms

Despite its theoretical importance, the elasticity of intertemporal substitution faces several limitations and criticisms, primarily concerning its empirical estimation and theoretical implications:

• Empirical Estimation Challenges: Accurately estimating the EIS has proven to be notoriously difficult, leading to a wide range of empirical estimates, often varying from near zero to significantly above one.⁶, ⁷ This disparity stems from challenges in isolating exogenous variations in interest rates and disentangling the substitution effect from the income effect, as well as potential approximation biases in commonly used econometric methods.⁵
• Homogeneity Assumption: Many economic models assume a constant or homogenous EIS across all individuals. However, evidence suggests that the EIS can vary significantly across different demographic groups, wealth levels, and countries, impacting the effectiveness of broad economic policies.³, ⁴
• Relationship with Risk Aversion: In widely used expected utility frameworks, especially with the constant relative risk aversion (CRRA) utility function, the EIS is the inverse of the coefficient of relative risk aversion. This imposes a restrictive link: if individuals are highly risk-averse, their EIS must be low. This automatic connection is often debated, as it implies that a finding of low intertemporal substitution directly translates to extreme risk aversion, which may not align with observed willingness to take on risk in other contexts.² Some studies caution that models assuming a unit EIS, when the actual elasticity is lower, may overstate consumers' willingness to reallocate consumption and thus mislead policymakers.¹
• Data Quality and Methodology: The reliability of EIS estimates can be sensitive to the quality of consumption data, the specific econometric techniques employed, and the assumptions made about agents' information sets and expectations.

Elasticity of Intertemporal Substitution vs. Risk Aversion

While closely related in many standard expected utility models, the elasticity of intertemporal substitution (EIS) and risk aversion represent distinct aspects of preferences.

In the common constant relative risk aversion (CRRA) utility framework, a higher coefficient of relative risk aversion implies a lower elasticity of intertemporal substitution (they are reciprocals). However, modern preference specifications, such as Epstein-Zin preferences, allow these two parameters to be disentangled, acknowledging that an individual can have a strong desire to smooth consumption over time (low EIS) yet still be willing to take on significant financial risks (low risk aversion), or vice-versa. This distinction is crucial for accurately modeling financial markets and individual investment decisions.

FAQs

What does a high elasticity of intertemporal substitution mean for individuals?

A high elasticity of intertemporal substitution means an individual is very responsive to changes in the real interest rate. If interest rates rise, they are willing to significantly reduce current consumption to save more and enjoy much higher consumption in the future. They find it relatively easy to postpone gratification.

How does the elasticity of intertemporal substitution impact monetary policy?

The elasticity of intertemporal substitution (EIS) is crucial for monetary policy because it determines how much individuals adjust their spending and savings in response to interest rate changes. If the EIS is low, central banks might find their interest rate adjustments have less impact on stimulating or cooling aggregate demand, as consumer behavior is less sensitive to these changes.

Is the elasticity of intertemporal substitution the same as risk aversion?

No, while they are related in some common utility function specifications, they are conceptually distinct. The elasticity of intertemporal substitution measures how individuals trade off consumption across time in response to returns, whereas risk aversion measures their willingness to accept uncertain outcomes. More advanced economic models can separate these two preferences.