Household Bequest Behaviour and the National Distribution of Wealth

Much of the theoretical literature on the distribution of wealth among households concentrates on identifying continuous-time Markov process models which have specific stationary distributions (see, for example, Sargan (1957), Wold and Whittle (1957), and Steindl (1972)). That research, which is highly technical, devotes little attention to the role of economic behavioural factors. Studies which are more concerned with the exact character of the wealth transfer process, on the other hand, often eliminate all interpersonal income differences so that the stationary distribution of wealth for each generation collapses to a single value (see Meade (1966), Stiglitz (1969), and Atkinson (1971)). The purpose of this paper is to prove the existence and uniqueness of a non-trivial stationary distribution of wealth for a simple growth model the key component of which is a utility-maximization theory of the bequest behaviour of individual families. The growth model, an adaptation of Samuelson's (1958) consumption-loan model, assumes that exogenous factors determine the intragenerational distribution of labour incomes and that the distribution remains unchanged over time. The theory of household bequest behaviour modifies the familiar tenet that each family's utility comes solely from its own consumption to the following: a family's utility is a function not only of its own consumption, but also of the consumption of its descendants in all future generations. For simplicity we assume that individual families live only one period; that there is only one output good; that either the good is storable or that there is a fixed supply of money (and money can be transferred between generations); that all existing family lines will continue forever; and, that all families have identical preference orderings. These assumptions cause each household's direct utility function to have an infinite number of arguments. If the interest rate on holdings of money is constant because of permanent, steady-state growth, however, each family's indirect utility function depends only on that interest rate and present family wealth. Assuming that preferences are additively separable and stationary in the Koopmans' (1960) sense, Bellman's (1957) " principle of optimality " enables us to define the indirect function with the aid of a single recursive functional equation. We can, in turn, use the indirect utility function to determine each family's optimal bequest. Although every household's utility-maximizing bequest is a single-valued function of its wealth, the function will not be linear. In fact, although we can establish many of its mathematical characteristics, we cannot represent the function explicitly. Nevertheless, using Doob's (1953) theorems on discrete-time, continuous-state-space Markov processes, we can prove that provided each household's indirect utility function is finite, associated with each fixed distribution of labour incomes and steady-state interest rate there is a unique stationary distribution of wealth. After presenting our model of family bequest behaviour in Section 1, in Section 2 we study in greater detail the properties of the function relating each family's wealth and optimal bequest. Section 3 proves the existence of a unique stationary distribution of wealth, and Section 4 constructs a complete growth model. In Section 4 we prove the surprising proposition that in the case of an economy with perishable goods and money