This is “The Life-Cycle Model of Consumption”, section 16.23 from the book Theory and Applications of Macroeconomics (v. 1.0).
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The life-cycle model of consumption looks at the lifetime consumption and saving decisions of an individual. The choices made about consumption and saving depend on income earned over an individual’s entire lifetime. The model has two key components: the lifetime budget constraint and individual choice given that constraint.
Consider the consumption/saving decision of an individual who expects to work for a known number of years and be retired for a known number of years thereafter. Suppose his disposable income is the same in every working year, and he will also receive an annual retirement income—again the same in every year. According to the life-cycle model of consumption, the individual first calculates the discounted present value (DPV) of lifetime income:DPV of lifetime income = DPV of income from working + DPV of retirement income.
(If the real interest rate is zero, then the DPV calculation simply involves adding income flows across years.)
We assume the individual wants to consume at the same level in each period of life. This is called consumption smoothing. In the special case of a zero real interest rate, we have the following:
Suppose an individual expects to work for a total of N years and to be retired for R years. Suppose his disposable income is equal to Yd in every year, and he receives annual retirement income of Z. Then lifetime income, assuming a zero real interest rate, is given as follows:lifetime income = NYd + RZ.
If we suppose that he wants to have perfectly smooth consumption, equal to C in each year, then his total lifetime consumption will beC × (N + R).
The lifetime budget constraint says that lifetime consumption equals lifetime income:C × (N + R) = NYd + RZ.
To obtain his consumption, we simply divide this equation by the number of years he is going to live (N + R):
Provided that income during working years is greater than income in retirement years, the individual will save during his working years and dissave during retirement.
If the real interest rate is not equal to zero, then the basic idea is the same—an individual smooths consumption based on a lifetime budget constraint—but the calculations are more complicated. Specifically, the lifetime budget constraint must be written in terms of the discounted present values of income and consumption.