The first order condition for an interior solution to this problem is:
Call the ct which satisfies this equation cf. It is clear that for sufficiently small wt the equation will be satisfied only by choosing a с£ larger than wT, that is, by ending with negative wealth. If we impose the condition that consumers may not die in debt, the solution to the problem is:
If p > a, end-of-period wealth will be a luxury good. Furthermore, if 7 is positive, there will be a range of initial wealth such that the marginal value of an extra dollar of consumption always exceeds the marginal value of an additional dollar of wealth. In this range, the consumer will choose to spend all available resources and end the period (and life) with zero wealth.
The problem can be solved analytically if we choose p = 2 and a = 1. If we set 7=1 the solution is
Define the saving rate as the fraction of beginning-of-period total assets the consumer ends up holding at the end of the period, wt+i/wt’ Figure 6 shows the saving rate of this consumer as initial wealth goes from 0 to 10. For initial wealth between 0 and 1 the consumer saves nothing, but above initial wealth of 1 the saving rate rises monotonically. Furthermore, as wT —► oo the saving rate approaches 100 percent.
The essential insights from this model carry over when the model is extended to many periods and when labor and capital income are incorporated: consumers with permanent income below a certain threshhold will behave like standard Life Cyle consumers and will try to spend all their assets before death, while consumers with permanent incomes above the threshhold will save at ever increasing rates as lifetime income rises.
The idea that bequests (charitable or otherwise) are insignificant for most of the population, but become increasingly important in the upper reaches of the lifetime income distribution, has been informally expressed by several previous authors. Indeed, Modigliani (1986) himself has argued that, to the extent that bequests must be included in the Life Cycle framework, they should be incorporated in precisely this “luxury good” manner. There is also a growing body of empirical evidence in support of the proposition. Dynan, Skinner, and Zeldes (1996) examine data from several micro datasets and find consistent and strong evidence that households with higher lifetime income leave larger bequests; Lillard and Karoly (1997) find similar results.
In theoretical terms, the value added in this paper relative to the previous literature is simply the proposal of a specific and simple functional form for the consumer’s utility function which captures the informal idea that rich people save more in a way that is at least roughly consistent with the empirical evidence marshalled above. But such consistency may not be a high enough standard.