Government Debt in an Overlapping-Generations Model with Bequests and Gifts

Abstract

In 1974, Robert Barro argued that if individuals in successive generations were linked by bequests, changes in the stock of government debt or in Social Security programs would have no effect on the steady-state capital stock. Barro assumed that individuals held static expectations and that the size of the population was constant. Martin Feldstein (1974, 1977) had concluded that Social Security would reduce the capital-labor ratio in models that did not admit a bequest motive, and, in his 1976 comment on Barro, argued that the introduction of government debt or Social Security into a perfect foresight, dynamic growth model with bequests would still reduce the capital-labor ratio. Barro replied that Feldstein's criticisms were invalid if the steady-state capital-labor ratio were smaller than the Golden Rule level, but would be correct if the opposite were true. Barro did suggest, however, that the latter possibility, with overaccumulation of capital, might be ruled out in his model, as it was in Miguel Sidrauski (1967), but he was unable to demonstrate that the required behavior would be consistent with utility maximization by finite-lived individuals (1976, p. 345). Although recent work by Willem Buiter (1979), Jeffrey Carmichael (1979, 1982), Truman Bewley (1981 a, b), and others has helped to elucidate the nature of overlapping-generations models with bequests and gifts, and the issues in the Barro-Feldstein debate, they remain far from clear. The purpose of this paper is to shed a little more light on the subject. In what follows, I contend that some of the authors cited above either have specified the individual's optimization problem in an asymmetric way, or have failed to impose necessary conditions for a sensible optimization problem, or both. There is a natural specification of the Barro model1 that emphasizes the similarity between this model and optimal growth models (see, for example, David Cass, 1965; Peter Diamond, 1973). Here, in steady-state equilibrium, the (aftertax) interest rate, r, must equal the rate at which individuals discount the utility of their heirs, p (r = p has become known as the modified Golden Rule) and a meaningful individual optimization problem requires that p exceed n, the natural growth rate of the economy. My results confirm Barro's hunch about his model; in a Barro model, r must exceed n. Much of the Barro-Feldstein 1976 interchange, which assumed that r could have any relation to n, is wrong-headed. Barro and Feldstein reach different conclusions about the effects of government debt and Social Security because they assume different individual utility functions. Also, my results contrast sharply with those of Buiter and Carmichael. For example, Carmichael concluded that with intergenerational transfers from parents to children (bequests), r must exceed n, but with transfers in the opposite direction, n must exceed r (1982, pp. 205-06). Indeed, I believe that the Buiter-Carmichael representation of the Barro model is logically faulty (see Section II). In Section I, I outline an overlapping-generations model with gifts and bequests, and derive its steady-state properties. In the next section, I discuss the shortand long-run effects of introducing government debt into the model and contrast my results with those

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