Intertemporal elasticity of substitution

What Is Intertemporal Elasticity of Substitution?

Intertemporal elasticity of substitution (IES) is a fundamental concept in financial economics that quantifies how willing individuals are to shift their consumption patterns between different periods in time, particularly in response to changes in the real interest rate. It measures the responsiveness of the growth rate of consumption to changes in the return on saving or borrowing⁴⁵, ⁴⁶. A higher intertemporal elasticity of substitution indicates that consumers are more flexible and willing to adjust their current versus future spending based on financial incentives. Conversely, a lower IES suggests that individuals prefer a smoother consumption path, exhibiting less responsiveness to interest rate fluctuations. This concept is crucial for understanding household behavior regarding savings and investment decisions over their lifetime.

History and Origin

The concept of intertemporal substitution has roots in early economic thought concerning how individuals allocate resources across time. However, its formalization and widespread application in modern macroeconomics are often associated with the work of economist Robert E. Hall. In his influential 1988 paper, "Intertemporal Substitution in Consumption," published in the Journal of Political Economy, Hall explored the empirical relationship between consumption growth and expected real interest rates, examining how changes in these rates influence consumers' decisions to shift consumption between periods⁴², ⁴³, ⁴⁴. Hall's work, building on earlier theories like the life cycle hypothesis, became a cornerstone for subsequent research on intertemporal choices, even as his findings suggested a lower elasticity than some prior expectations⁴⁰, ⁴¹.

Key Takeaways

• Intertemporal elasticity of substitution (IES) measures how sensitive an individual's consumption choices are to changes in the real interest rate across different time periods.
• A high IES means consumers are more willing to shift consumption from the present to the future (or vice versa) if the real interest rate changes, making saving more or less attractive.
• A low IES indicates that consumers prioritize a smooth consumption path, showing less responsiveness to interest rate fluctuations.
• IES is a crucial parameter in economic models for understanding savings behavior, capital accumulation, and the impact of fiscal and monetary policy.
• Empirical estimates of IES vary widely across studies, presenting challenges for economic modeling and policy implications.

Formula and Calculation

The intertemporal elasticity of substitution is derived from an individual's utility function, which represents their preferences for consumption over time. For a utility function (u(c)), where (c) denotes the consumption level, the IES, often denoted as (\sigma) (sigma) or EIS, is typically defined as the inverse of the elasticity of the marginal utility of consumption³⁹.

A common representation for a constant intertemporal elasticity of substitution (constant EIS) utility function, often referred to as the Constant Relative Risk Aversion (CRRA) utility function, is:

and for (\gamma = 1):

In this formulation, the intertemporal elasticity of substitution is given by:

Where:

• (U(c)) is the utility derived from consumption (c).
• (c) is the level of consumption.
• (\gamma) (gamma) is the coefficient of relative risk aversion.

This relationship implies an inverse link between the degree of risk aversion and the intertemporal elasticity of substitution, particularly in simpler models³⁷, ³⁸. A higher (\gamma) (more risk-averse) corresponds to a lower (\sigma) (less willing to substitute consumption over time).

Interpreting the Intertemporal Elasticity of Substitution

Interpreting the value of the intertemporal elasticity of substitution provides insight into how consumers manage their finances over time.

• IES > 1 (Elastic): If IES is greater than 1, individuals are highly responsive to changes in the real interest rate. A 1% increase in the real interest rate would lead to a more than 1% increase in the growth rate of future consumption relative to current consumption. This suggests a strong willingness to postpone gratification and increase saving when returns are attractive³⁶. In such a scenario, even small changes in interest rates could lead to significant shifts in consumption and saving patterns.
• IES = 1 (Unit Elastic): When IES equals 1, the percentage change in consumption growth is equal to the percentage change in the real interest rate. This is often associated with logarithmic utility functions.
• IES < 1 (Inelastic): If IES is less than 1, individuals are relatively insensitive to changes in the real interest rate. A 1% increase in the real interest rate would result in less than a 1% increase in the growth rate of future consumption. This indicates a stronger preference for consumption smoothing, meaning consumers prioritize a stable consumption path over time, even if it means foregoing higher returns on savings³⁵. Empirical studies often find IES estimates in this range, sometimes close to zero³³, ³⁴.

The magnitude of IES has significant implications for understanding macroeconomic phenomena, including aggregate consumption dynamics, the effectiveness of interest rate policies, and long-term economic growth.

Hypothetical Example

Consider an individual, Sarah, who is deciding how much to consume today versus how much to save for future consumption.

Scenario 1: High IES (e.g., IES = 1.5)
The real interest rate offered on savings accounts increases from 2% to 3%. Sarah, having a high intertemporal elasticity of substitution, is very responsive to this change. The higher return on her saving makes future consumption significantly more attractive. She decides to cut back substantially on her current discretionary spending, perhaps delaying the purchase of a new car and eating out less frequently. Instead, she increases her monthly contributions to her investment portfolio by a large margin, aiming to capitalize on the higher returns and enjoy a much higher level of consumption in the future. Her consumption growth rate significantly accelerates due to this interest rate change. This demonstrates her high willingness to substitute current consumption for future consumption.

Scenario 2: Low IES (e.g., IES = 0.2)
Now, suppose the same real interest rate increase (from 2% to 3%) occurs, but Sarah has a low intertemporal elasticity of substitution. She values a stable stream of current consumption very highly. While she acknowledges the slightly better returns on saving, her preference for consumption smoothing means she doesn't drastically alter her current spending habits. She might slightly increase her savings, perhaps by a small percentage, but her overall consumption patterns remain largely unchanged. Her consumption growth rate is only marginally affected, reflecting her inelastic response to the change in the real interest rate.

This example illustrates how an individual's IES can dictate the degree to which financial incentives, such as changes in interest rates, influence their intertemporal allocation of resources.

Practical Applications

The intertemporal elasticity of substitution is a crucial parameter in various areas of economics and finance, influencing policy decisions and theoretical models:

• Macroeconomic Modeling: IES is a key input in dynamic stochastic general equilibrium (DSGE) models, which economists use to understand business cycles, calibrate macroeconomic shocks, and forecast the economy's response to various policies. The assumed value of IES significantly impacts the predictions of these models regarding consumption and saving behavior³².
• Monetary and Fiscal Policy: Central banks and governments consider IES when designing monetary policy (e.g., interest rate adjustments) and fiscal policy (e.g., tax changes on savings or consumption). A high IES implies that interest rate changes can be very effective in stimulating or dampening economic activity by encouraging or discouraging current consumption. Conversely, a low IES suggests that such policies might have a weaker impact on consumer spending.
• Asset Pricing: In asset pricing models, IES plays a role in determining how individuals value risky assets based on their consumption preferences. It helps explain phenomena like the equity premium puzzle, where equity returns historically exceed risk-free rates by a larger margin than standard models predict³¹.
• Retirement Planning and Household Finance: For individuals and financial advisors, understanding IES—even if implicitly—can inform decisions about retirement savings, debt management, and wealth accumulation over a lifetime. It highlights the trade-offs involved in current versus future consumption and the sensitivity of these choices to prevailing financial markets conditions. Research, such as a paper using mortgage data, has been used to estimate IES based on how households respond to interest rate thresholds in loan-to-value ratios, providing insights into their intertemporal choices.

#³⁰# Limitations and Criticisms

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

One major challenge is the difficulty in obtaining precise and consistent empirical estimates of IES. Studies using aggregate macroeconomic data often yield estimates that are small, sometimes close to zero, or even negative, which has been a puzzle in economics. Th²⁴, ²⁵, ²⁶, ²⁷, ²⁸, ²⁹ese low estimates suggest that consumers are very insensitive to changes in the real interest rate, which contradicts the strong intertemporal substitution effects implied by some theoretical models. The discrepancy may arise from measurement errors in consumption data, difficulty in identifying exogenous variations in interest rates, or the presence of other factors like liquidity constraints.

F²¹, ²², ²³urthermore, many models assume a constant IES across all levels of wealth or consumption, which may not hold in reality. Research suggests that the elasticity of intertemporal substitution might vary across different demographic groups or income levels, with richer households potentially exhibiting a higher IES for certain luxury goods that are easier to postpone. Th¹⁸, ¹⁹, ²⁰e assumption of time-separable utility functions, where current utility is independent of past or future consumption decisions, is another simplification that may not fully capture complex household behavior. Critics also point out that the inverse relationship between IES and risk aversion, often seen in standard utility functions, may not hold empirically, adding complexity to its interpretation and estimation. Th¹⁵, ¹⁶, ¹⁷e estimation methods themselves can introduce biases, and failure to account for factors like labor supply decisions can lead to significantly different IES estimates.

#¹³, ¹⁴# Intertemporal Elasticity of Substitution vs. Risk Aversion

While both the intertemporal elasticity of substitution (IES) and risk aversion describe aspects of individual preferences, they address distinct behaviors and are often confused due to their mathematical relationship in certain utility function specifications.

In simple models, particularly those using the Constant Relative Risk Aversion (CRRA) utility function, the intertemporal elasticity of substitution is the inverse of the coefficient of relative risk aversion ((\sigma = 1/\gamma)). Th¹⁰, ¹¹, ¹²is mathematical linkage means that if an individual is highly risk-averse, they are also modeled as having a low willingness to substitute consumption across time, preferring a very smooth consumption path.

However, in more complex models and empirical studies, these two concepts can be disentangled. An individual can be highly risk-averse (disliking uncertainty) but still have a high IES (willing to shift consumption across time if the rewards are there), or vice versa. The empirical relationship between IES and risk aversion is a subject of ongoing research, with some studies finding evidence that supports the inverse relationship while acknowledging significant heterogeneity.

#⁷, ⁸, ⁹# FAQs

Q: Why is Intertemporal Elasticity of Substitution important in economics?
A: IES is crucial because it helps economists understand and predict how individuals and aggregate economies respond to changes in interest rates, taxes, and other policies that affect the trade-off between current and future consumption. It informs models of economic growth, asset pricing, and the effectiveness of monetary policy.

Q: What does a high Intertemporal Elasticity of Substitution mean for saving?
A: A high IES means that individuals are very sensitive to the real interest rate. If the interest rate rises, making future consumption relatively cheaper or more rewarding, these individuals are likely to significantly increase their current saving in anticipation of higher future returns.

⁵, ⁶Q: Can Intertemporal Elasticity of Substitution be negative?
A: Theoretically, IES can be negative if individuals increase current consumption when the real interest rate rises, or decrease it when the interest rate falls. This is unusual but can occur if income effects outweigh substitution effects, such as a strong desire to meet a future target (like retirement income) that becomes harder to reach with lower returns, thus necessitating more current saving. Em⁴pirically, some studies have found estimates close to zero or even negative, though these are often associated with measurement challenges.

², ³Q: How does IES relate to consumer behavior?
A: IES directly reflects how flexible and forward-looking consumers are in their spending and saving decisions. Consumers with a high IES are more likely to adjust their financial plans significantly based on economic incentives, while those with a low IES tend to maintain a more consistent spending pattern regardless of changes in investment returns.¹