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Modigliani's life-cycle hypothesis: rational agents smooth consumption over their entire lifespan by saving during working years and dissaving in retirement.

Derivation

Step-by-step mathematical derivation with typeset equations and expandable detail.

Sections

Modigliani framework and lifetime budget

The life-cycle hypothesis, developed by Franco Modigliani and Richard Brumberg, models a consumer who lives for $T$ periods, earns labor income $Y$ for $L$ working years, and then retires for the remaining $T - L$ years with no labor income. The consumer faces a lifetime budget constraint: the present value of consumption over the entire lifespan must equal the present value of lifetime resources, which consist of initial wealth $W_0$ plus the stream of labor income. In the simplest version with zero real interest rates, this reduces to total lifetime consumption equaling total lifetime income plus initial wealth.

The key departure from Keynesian consumption theory is that the consumer plans over the entire lifetime, not period by period. Wealth $W_0$ and expected future income jointly determine today's spending. A young worker with low current income but high expected future earnings will borrow against that future income rather than consume only what is currently available. This forward-looking behavior produces consumption patterns that depend on lifetime resources rather than current income alone.

\sum_{t=0}^{T} C_t = W_0 + \sum_{t=0}^{L} Y_t

Lifetime budget constraint (zero interest rate case): total consumption over $T$ periods equals initial wealth plus total labor income earned over $L$ working years.

R = W_0 + L \cdot Y

Lifetime resources when income is constant at $Y$ per period: initial wealth plus total earnings across the working life.

Optimal consumption and the smooth path

With a zero real interest rate and no bequest motive, the consumer maximizes lifetime utility $\sum_{t=0}^{T} u(C_t)$ subject to the budget constraint. Under standard concavity of $u(\cdot)$ , the first-order conditions equalize marginal utility across all periods, which with identical period utility functions yields perfect consumption smoothing: $C_t = C$ for all $t$ . The constant consumption level is simply lifetime resources divided by the number of periods lived.

This result produces the life-cycle consumption function $C = \alpha W + \beta Y$ , where $\alpha = 1/T$ is the marginal propensity to consume out of wealth and $\beta = L/T$ is the marginal propensity to consume out of income. A 60-year-old with 20 years remaining consumes a larger fraction of wealth per year than a 30-year-old with 50 years remaining. The aggregate implication is that the economy-wide saving rate depends on the demographic structure: economies with a larger share of working-age adults save more than economies dominated by retirees.

C = \frac{W_0 + L \cdot Y}{T} = \frac{1}{T}W_0 + \frac{L}{T}Y

Optimal consumption per period: lifetime resources divided equally across $T$ periods, producing constant consumption.

C = \alpha W + \beta Y, \quad \alpha = \frac{1}{T},\; \beta = \frac{L}{T}

Life-cycle consumption function: separate marginal propensities out of wealth and out of income, both depending on remaining lifetime $T$ and working years $L$ .

Hump-shaped saving and wealth accumulation

The saving pattern implied by consumption smoothing is distinctly hump-shaped. During working years, income exceeds the smooth consumption level ( $Y > C$ when $L < T$ ), so the household saves and accumulates wealth. At retirement, labor income drops to zero but consumption continues at the same rate $C$ , so the household dissaves by drawing down accumulated wealth. Peak wealth occurs at the moment of retirement, after which it declines linearly to zero (or to a bequest level) at the end of life.

The peak wealth at retirement equals the total saving during the working life: $W_{peak} = L \cdot (Y - C) = L \cdot Y \cdot (1 - L/T) = L(T-L)Y / T$ . This expression is maximized when $L = T/2$ , meaning peak wealth is greatest when the working life is exactly half the total lifespan. For the aggregate economy, if cohorts overlap and population grows at rate $n$ , the young savers outnumber the old dissavers, generating positive net national saving even though each individual's lifetime saving is zero.

S_t = Y - C = Y\left(1 - \frac{L}{T}\right) - \frac{W_0}{T}

Per-period saving during working years: income minus smooth consumption. Positive as long as $L < T$ and initial wealth is not too large.

W_{peak} = \frac{L(T - L)}{T} Y

Wealth at retirement: accumulated saving over the working life, maximized when $L = T/2$ .

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