GOVERNMENT BONDS denoted by r and is assumed to be paid out once per period. Expectations on r for future periods are assumed to be static at the current value. A member of the ith generation holds the amount of assets ay while young nd the amount a while old The asset holding while old constitutes the provision of a bequest, which is assumed to go to the immediate descen ant, a member of generation i 1. Since the focus of the analysis concerns shifts in tax liabilities and government debt for a given level of government expenditure, it is assumed for convenience that the govern ment neither demands commodities nor provides public services. In this section,it is also assumed that the amounts of government debt and taxes are zero. Using the letter c to denote consumption, and assuming that consumption and receipt of interest income both occur at the start of the period, the budget equation for a member of generation l, who is currently old, is A+A48=c+(1-r)A The total resources available are the assets held while young, Ai, plus the bequest from the previous generation, Ao. The total expenditure is con- sumption while old, ci, plus the bequest provision, Ai, which goes to a member of generation 2, less interest earnings at rate r on this asset holding The budget equation for members of generation 2(and, more generally, for members of any generation i 22) is, assuming that wage payments occur at the start of the young period =C+(1-r)A2 and, for the old period A2+A2=c2+(1-r)A2 a portion of the lifetime resources of a member of generation i goes to a bequest provision, Ai, which I assume is motivated by a concern for a member of generation i 1. This concern could be modeled by intro- ducing either the(anticipated)consumption levels or attainable utility of a member of generation i I into the utility function for a member of the ith generation. For the purpose of the present analysis, the crucial condition is that this utility depend on the endowment of a member of +I rather than, per se, on the gros distinction between the gross bequest and the net bequest, which deter- mines the endowment of i l, will be discussed below. So long as a member of generation i can transfer resources to a member of generation l only through the transfer of unrestricted purchasing power(which rules out the"merit good"case discussed in n 8 below), the two types of models of interdependent preferences--concern with consumption levels and concern with attainable utility-will be equivalent in the sense of
IIOP JOURNAL OF POLITICAL ECONOMY indirectly implying a concern for the endowment of a member of generation i+ I For present purposes, it is convenient to assume that the utility of a member of generation i depends solely on own two-period consumption cI and ci, and on the attainable utility of his immediate descendant, Ui+ I The asterisk denotes the maximum value of utility, conditional on giver values of endowment and prices. Hence, the utility function for a member f the ith generation has the form U;=U4(c,cU*1) Subsequently, I consider the implications of entering the attainable utility of a member of the previous generation, Ui-I, as an additional ent Each member of generation I determines his allocation of resources to maximize U,, subject to equations (1)-(4)and to the inequality conditions,(ci, ca, 4920 for all i. The key restriction here is that the bequest to the member of the next generation cannot be negative. The choice of bequest, subject to this restriction, takes o v1, and the chain of Ai on generation 2's resources, the impact of U> or dependence of U2 on U3, of U3 on U4, etc. The solution to this problem will take the general for c"=c(A+A8,,r) (A1+A8-c)=A(4+A8,c,r) Similarly, for members of generation 2(and, more generally, for members of any generation i 22), the solution would take the form c2= c2(A1, w,r), Ai c=c(42+AB,c,r), (A2+A-c2)=A2(4 A member of get with the at generation i can attach a ce of his descendant. further it indiffer pposed that hich makes it comparable to cl and f in terms of generating U, in the form of ca rface dependent preferences in Becker(1974, sec. 3.A I have not imposed the condition, A120, so that young individuals are allowed issue interest-bearing debt on themselves. Ifissued, these debts are assumed to be perfec substitutes for equity capital. These debts correspond to the consumption loans which have been discussed by Samuelson (1958)
GOVERNMENT BONDS The model can be closed, as in Diamond(1965, pp. 1130-35), b specifying a constant-returns-to-scale production function that depend of capital and labor the products of capital and labor to r and w, respectively. The value of r for the current period would then be determined in order to equate the supply of assets to the demand-that is K where K(r, w) is such as to equate the marginal product of capital to r The current demand for assets, A1+ A,, depends, from equations (5 and(6), on r, w, and the previous periods value of K, which is equal to A1+ Ao. Since the number of people in each generation is assumed to equal a fixed number N, it is not necessary to enter this number explicitly into the aggregate asset demand in equation (7). Similarly, N is omitted from the aggregate formulations below. Since N is constant and technical change is not considered, the current and previous periods'values of K would be equal in a steady state With the marginal product of labor equated to we and with constant returns to scale, output is given by y=rk+w Equations(2),(3),(7), and(8) imply a commodity market clearing where Ak denotes the change in capital stock from the previous to the current period. The value of AK would be zero in a steady state, but the present analysis is not restricted to steady-state situations Suppose now that the government issues an amount of debt, B, which can be thought of as taking the form of one-period, real-valued bonds. These bonds pay the specified amount of real interest, rB, in the current period and the specified real principal, B, in the next period. It supposed that asset holders regard equity and government bonds perfect substitutes. It can be assumed, for simplicity, that the government bond issue takes the form of a helicopter drop to currently old (generation 1)households. Equivalently, it could be assumed that the bonds were sold on a competitive capital market, with the proceeds from this sale used to effect a lump-sum transfer payment to generation I households amount of bond issue would be limited by the government's collateral, in the ense of its taxing capacity to finance the interest and principal payments(see n. 12