EF5472 Chap.3 Intertemporal Choice and Euler Logic
Intertemporal Choice and Euler Logic
Balance utility across dates
Dynamic choice begins with feasible consumption and saving, then compares the marginal utility cost of giving up one unit today with the discounted marginal utility benefit delivered tomorrow. Patience enters through the discount factor, curvature governs willingness to substitute consumption over time and technology or asset markets provide the gross return.
The Euler equation gives a local consumption-growth relation, not complete levels. Resource constraints close levels, while borrowing or non-negativity limits must be checked because a binding constraint converts the interior equality into an inequality.
Solve timing before algebra
Draw two dates and place the saving decision between them.
One less unit of current consumption costs current marginal utility; the asset created by that sacrifice earns the next-period gross return and is valued using next-period marginal utility and the discount factor. Writing those units under the Euler equation makes the timing visible and prevents using the return on inherited capital.
Under CRRA preferences, rearrange the condition into a consumption-growth factor and interpret patience, curvature and return separately. A larger discount factor or return tilts consumption toward the future, while greater curvature dampens the growth response. The ratio is not a complete allocation.
Combine it with every resource constraint or lifetime budget equation to recover levels, then check positivity and any borrowing limit. If a candidate asks for negative saving beyond the permitted bound, replace the interior equality with the appropriate inequality and explain the corner. For tax or policy questions, locate the exact wedge between the private return and the technological return before predicting behaviour.
Close by substituting the proposed consumption path into both optimality and feasibility; satisfying only one of them does not establish an equilibrium.
What this chapter covers
- 01
Patience and discounting
- 02
CRRA curvature
- 03
Two-period feasibility
- 04
Gross return timing
- 05
Euler conditions
- 06
Binding constraints
Worked example · free
Choose saving across two dates
- 3Write both dated resource constraints.
- 3Use the Euler equation to obtain the consumption ratio.
- 3Solve levels and test the constraint.
Key terms
- Discount Factor
- The weight placed on next-period utility relative to current utility.
- CRRA Utility
- A constant-relative-risk-aversion utility form whose curvature controls intertemporal substitution in this setting.
- Gross Return
- One plus the net rate earned by carrying a unit of resources across dates.
- Euler Equation
- The interior optimality condition equating marginal utility costs and discounted return-weighted benefits.
- Interior Solution
- An optimum away from a constraint boundary where marginal equalities hold.
Intertemporal Choice and Euler Logic FAQ
Why does the Euler equation not determine consumption levels by itself?
It equates a local trade-off and usually yields a ratio between consumption dates. Total resources, production feasibility or a lifetime budget is needed to determine the scale. In longer horizons, initial wealth and a terminal or transversality condition also help select the complete path.
What does higher curvature do to consumption growth?
With CRRA preferences, the consumption-growth response to a return gap is raised to the power one over curvature. Greater curvature reduces that elasticity and pulls growth closer to one. It changes willingness to tilt consumption, while the discount factor separately represents patience.
When does the Euler equality become an inequality?
If a borrowing, saving or non-negativity constraint binds, the household cannot execute the marginal transfer assumed by an interior optimum. Current marginal utility can then remain above the discounted future benefit. Always solve the unconstrained candidate first and test whether it violates the feasible set.
Exam move
For each intertemporal problem, draw a timeline and place resources, choices and returns above the correct dates. Write feasibility before taking derivatives. Practise log utility first because it turns the Euler condition into a transparent ratio, then repeat with general CRRA and explain the role of curvature. Build a two-column error log separating net from gross returns and inherited assets from newly chosen saving.
After solving an interior candidate, deliberately tighten a borrowing constraint and state how the equality changes. The goal is not only to calculate consumption but to recognise which marginal trade is available, which is blocked and what additional condition is needed to select levels. Recheck the terminal condition in finite-horizon exercises because it frequently supplies the missing equation for consumption levels.
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