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Households optimize consumption across periods: maximizing lifetime utility subject to a lifetime budget constraint. The marginal rate of intertemporal substitution (consumption tradeoff between periods) equals the real interest rate in optimum. Higher interest rates induce substitution (save more now) but also income effects (savers are better off). The Euler equation links current and future marginal utility: MU_t = β(1+r) MU_{t+1}. Life-cycle consumption depends on lifetime income, not current income.
You've already studied consumption smoothing and the Euler equation as individual building blocks. This topic connects them into a unified framework for understanding *why* households save and how saving decisions respond to interest rates and income shocks. The key shift in thinking: your current paycheck is almost irrelevant to rational consumption — what matters is your entire lifetime income stream.
The lifetime budget constraint is the starting point. If you live two periods, earn income Y₁ now and Y₂ later, and face a real interest rate r, your constraint is: C₁ + C₂/(1+r) = Y₁ + Y₂/(1+r). The right-hand side is your lifetime wealth — present value of all earnings. Saving in period 1 allows you to consume more than Y₂ in period 2; borrowing allows you to consume more than Y₁ now. The budget line has slope -(1+r), which is the relative price of consuming today versus tomorrow: giving up one unit of consumption now buys you (1+r) units in the future. The optimal point on this budget line is where the marginal rate of intertemporal substitution — how many future consumption units you'd trade for one more now — equals (1+r). This is exactly the Euler equation condition you've already derived: MU_t = β(1+r) MU_{t+1}.
The interest rate effect on saving has two components that work in opposite directions. The substitution effect of a higher r makes future consumption cheaper relative to current consumption, so you shift toward saving more today and consuming more tomorrow — savings rise. But the income effect works differently for net savers versus net borrowers: if you're a saver, a higher interest rate makes you richer (your savings yield more), which tends to *increase* current consumption — savings fall. The net effect of r on aggregate savings is theoretically ambiguous, and empirical estimates are small. This is a recurring example of why comparative statics requires careful separation of substitution and income effects.
The deepest insight here is the life-cycle hypothesis: rational households plan consumption over their entire lifetime, not paycheck to paycheck. A young worker expecting a higher future income (say, after finishing a degree) optimally borrows against that future income to maintain smooth consumption now. A retiree draws down savings accumulated during working years. Temporary income fluctuations — a one-time bonus or a brief unemployment spell — have small effects on optimal consumption, because they're small relative to lifetime wealth. Permanent income shocks, however, shift consumption strongly because they change the entire lifetime wealth calculation. This distinction — temporary versus permanent shocks — is one of the most empirically powerful predictions of the intertemporal framework, and it underpins the Permanent Income Hypothesis developed by Milton Friedman.