Questions: Euler Equation and Intertemporal Substitution

Question 1

According to the Euler equation with CRRA utility, the real interest rate rises unexpectedly. What does the Euler equation directly predict?

• A. Current consumption falls immediately, because the higher interest rate reduces the present value of future income
• B. Consumption growth increases — future consumption rises relative to current consumption — but the Euler equation alone does not determine whether current consumption rises or falls
• C. Both current and future consumption rise proportionally, because households are wealthier when returns to saving are higher
• D. Consumption is unchanged, because rational households smooth consumption across all interest rate fluctuations

Answer: (B)

The Euler equation c_{t+1}/c_t = [β(1+r)]^(1/σ) governs consumption *growth*, not the consumption *level*. When r rises, the right-hand side increases, meaning households optimally tilt their consumption path toward the future: consumption growth accelerates. Whether current consumption c_t rises or falls depends on the full budget constraint (income, wealth, borrowing limits) — the Euler equation alone is silent on the level. This is the most common source of confusion: students conflate the Euler equation's prediction about the growth rate with a claim about current consumption.

Question 2

What does the elasticity of intertemporal substitution (EIS = 1/σ in CRRA utility) measure in the context of the Euler equation?

• A. How much a household's lifetime income changes when the interest rate changes by 1 percentage point
• B. How responsive the growth rate of consumption is to changes in the real interest rate — households with high EIS strongly tilt consumption toward periods with higher returns
• C. The fraction of income saved at any given interest rate, holding wealth constant
• D. The degree of risk aversion, which determines how much consumption volatility the household will tolerate

Answer: (B)

The EIS (= 1/σ) is the elasticity of consumption growth with respect to the gross real interest rate. When EIS is high (σ small), a modest rise in r produces a large tilt toward future consumption — these households are flexible substituters willing to defer gratification for higher returns. When EIS is low (σ large), households have a strong preference for smooth consumption and barely change their consumption growth in response to interest rate variation. Note: σ also governs risk aversion in CRRA utility, but these two interpretations are conceptually distinct — EIS governs deterministic intertemporal substitution, while risk aversion governs responses to uncertainty about consumption levels.

Question 3

The Euler equation determines the optimal *level* of consumption in each period, given the household's budget constraint and preferences.

• T. True
• F. False

Answer: False

The Euler equation is a first-order condition that pins down the *ratio* of consumption across periods — the rate of growth — not the absolute level. The equation u'(c_t) = β(1+r)u'(c_{t+1}) says the marginal utility of consuming a dollar today must equal the discounted marginal utility of saving and consuming that dollar tomorrow. It is the intertemporal optimality condition. The absolute level of consumption in any period is determined by combining the Euler equation with the household's lifetime budget constraint (which pins down total spending given total resources). The Euler equation provides one equation in two unknowns (c_t and c_{t+1}); the budget constraint provides the second.

Question 4

The Euler equation holds under uncertainty, where the relevant condition becomes: the marginal utility of consuming today equals the discounted expected marginal utility of consuming tomorrow.

• T. True
• F. False

Answer: True

The stochastic Euler equation is: u'(c_t) = β(1+r) E_t[u'(c_{t+1})], where E_t is the expectation conditional on information available at time t. This extends naturally from the deterministic case: the household equates the certain marginal utility of consuming today with the *expected* discounted marginal utility of saving and consuming tomorrow (uncertain because future income, prices, or interest rates may vary). This stochastic version is the workhorse of modern macroeconomics and asset pricing, where it can be combined with asset return data to test whether household consumption behavior is consistent with rational optimization.

Question 5

Explain the role of the elasticity of intertemporal substitution (1/σ) in the Euler equation, and contrast the consumption behavior of a household with σ = 0.1 versus one with σ = 10 when the real interest rate rises.

Model answer: The EIS (1/σ) measures how much the growth rate of consumption responds to a change in the real interest rate. In the CRRA Euler equation, c_{t+1}/c_t = [β(1+r)]^(1/σ). A household with σ = 0.1 (EIS = 10) responds very strongly: a modest increase in r dramatically increases the consumption growth rate, meaning the household sharply tilts consumption toward the future to take advantage of higher returns. A household with σ = 10 (EIS = 0.1) barely changes its consumption growth rate in response to the same interest rate increase — it has a strong preference for smooth consumption across time. The first household acts like a flexible financial optimizer; the second behaves closer to a rule-of-thumb consumer with near-rigid consumption smoothing.

This parameter has enormous macroeconomic implications. If σ is small (high EIS), monetary policy — which works partly through the real interest rate — powerfully affects consumption timing. If σ is large (low EIS), the intertemporal substitution channel of monetary policy is weak. Empirical estimates of σ are debated, but most macroeconomic models assume σ around 1–2 (EIS of 0.5–1), implying moderate responsiveness of consumption growth to interest rates.