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On debt neutrality in life-cycle models
233
Economics Letters 33 (1990) 233-238 North-Holland
ON DEBT NEUTRALITY
IN LIFE-CYCLE
Miguel-Angel LOPEZ-GARCIA Universidad Aut6nomn de Barcelona,
MODELS
*
08193 Bellaterra,
Barcelona,
Spain
Received 22 September 1989 Accepted 15 November 1989
A parallelism is established between pure life-cycle models and those in which altruism is one-sided in the discussion of the public debt neutrality. It is argued that when indiduals behave as pure life-cyclers, the coercive transfers necessary to offset the effects of the debt have the same nature, and take place under the same conditions, as the voluntary intrafamily bequests and gifts which constitute the usual basis for the ‘Ricardian equivalence theorem’.
1. Introduction Since seminal contribution by Samuelson (1958) and the extension to economies with endogenous production by Diamond (1965), there has been great interest in the effects of the public debt in the overlapping generations models. In its pure life-cycle version, the debt has been analyzed in its relationship with taxation [see Diamond (1965), Bierwag et al. (1969) and Feldstein (1985)], as a device to achieve the ‘golden rule’ [see Stein (1969) and Ihori (1978)], and to discuss related questions [see Burbidge (1983a) and L6pez-Garcia (1987)]. On the other hand, Barro (1974) has included the possibility that the intrafamily altruism can give rise to a chain of voluntary intergenerational transfers. The renewed interest in the ‘Ricardian equivalence theorem’ of debt and taxes has generated a body of literature [see Drazen (1978), Buiter (1979, 1980), Carmichael (1982), Burbidge (1983b), Weil (1987), Kimball (1987) and Abel (1987)], and there are even different ways to characterize the model [see Buiter and Carmichael (1984), Burbidge (1984), Hillier and Lunati (1987), Bernheim and Bagwell (1988) and L6pez-Garcia (1989)]. Although the literature on pure life-cycle models has pointed out that the public debt is equivalent to an intergenerational redistribution of resources, and that certain changes in the distribution of taxes among generations can offset the effects of the debt policy, the ultimate reasons for this result and its relationship with the models that allow for voluntary transfers, have not been fully clarified. This paper argues that when individuals are life-cyclers, the coercive transfers necessary to offset the effects of the public debt have the same nature, and take place under the same conditions, as the voluntary intrafamily bequests and gifts which constitute the standard basis for the ‘Ricardian equivalence theorem’. Section 1 describes both models and section 2 establishes a parallelism between them. Section 3 summarizes some final remarks. * Financial support from the FundacGn 0165-1765/90/$3.50
Fondo para la Investigacibn Jkon6mica
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
y Social (Spain) is gratefully acknowledged.
234
M.-A. Lopez-Garcia
/ On debt neutrality in life-cycle models
2. Public debt, life cycle and voluntary intergenerational
transfers
The basic framework is the overlapping generations model developed by Diamond (1965). The population is composed of identical individuals who live for two periods. In the first one they work and supply one unit of labour and in the second one they are retirees. Therefore, two generations coexist in a given time period, the younger one and the older one, and the population growth rate is denoted by n. The technology is represented by means of a linear homogenous production function in intensive terms, f(k,), with f’ > 0 and f” < 0, where k, is the capital-labour ratio in period t. Under competitive conditions each factor will receive its marginal product. The public debt issued in period t and to be repaid at the interest rate r,,, in period t + 1 is denoted by O,+ 1. The government maintains a constant per worker stock of debt, d [i.e., D,+I = (1 + n)D,], so that the amount it spends in excess of the amount it borrows is (r, - n)D,, or (rt - n)d in per worker terms. When r, is greater (less) than n, the government levies (grants) taxes (transfers), which can be divided between the two generations. 2. I. A pure life-cycle model
cf
When individuals are life-cyclers, they maximize a lifetime utility function q = u( c:, ), where c: and c: are first- and second-period consumption by an individual born in period t, subject to their budget constraint. If the government levies in period t a tax (1 - q)(r, - n)D, from the younger generation and q(r, - n)D, from the older one, where 0 4 q I 1, the individuals will not only save for life-cycle reasons, but also for a tax provision. Under these assumptions, the following four equations characterize a steady state:
‘*+P=f(k)-rk-(l-q)(r-n)d-q(r-n)(l+n)d/(l+r), (1Cr)
(1)
r=f’(k),
(2)
[WC’,4/ac~]/[au(c~,
c2)/ac2]
= (1 + r),
(3)
f(k)-rk-(l-q)(r-n)d-c’=(l+n)(k+d).
(4)
In general, for given values of d, q and n, this system can be solved in the four unknowns, cl, c2, r and k, giving rise to a steady-state utility level U = u( cl, c*). Equation (1) denotes the lifetime budget constraint; (2) expresses the relationship between the interest rate and the capital-labour ratio; (3) is the usual equality between the marginal rate of substitution and the relative prices; and (4) equates the life-cycle saving and the tax provision of the younger generation with the capital plus debt. It is known that in this model the steady-state interest rate can be greater or less than the population growth rate [Diamond (1965)]. 2.2. A model with voluntary intergenerational
transfers
In the Barro (1974) model, the individuals derive utility not only from their own consumption, but also from the welfare enjoyed by their descendants or ancestors. This intergenerational altruism can generate bequests from parents to children or gifts from children to parents. The utility function can now be written as U, = u(ci, c:, U,:,) or U, = u(c:, cf, U,?l), where U,:, (U,?,) is the maximum utility level attainable by a representative descendant (ancestor) given the bequest (gift) received. ’ We shall only analyze one-sided altruism, either from parents to children or from children dealt with in Buiter (1980), Burbidge (1983b), Kimball (1987) and Abel (1987).
to parents.
Two-sided
altruism
is
M. -A. Lopez-Garcia
/ #I debt neutrality in lije-cycle models
235
For the sake of simplicity, we shall assume that only the younger generation pays (receives) taxes (transfers), i.e., that q = 0. 2 In order to obtain concrete results we shall adopt the additively separable utility function which has been extensively used. In the case of altruism from parents to children, this reduces to u, = u(c;, c;) + h&t,, where h > 0 is a discount factor. A steady state with non-negative bequests can be characterized by means of the system of five equations given by (2) (3) and
c’+(15) +---f(k)-rk-(r-n)d+&, (1L) f(k)-rk-(r-n)d-c’+b/(l+n)=(l (1 + r) I (1 + n)/h,
+n)(k+d),
(6) (7)
where the unknowns are those of the previous model plus the bequest, b 2 0. Equation (5) is the budget constraint under the assumption that the bequest is equally divided among the (1 + n) children; (6) is the new asset identity extended to allow for the bequests; and (7) is a result of the intertemporal optimization problem. In particular, (7) holds as an equality if b > 0 and as a strict inequality when b = 0. The steady-state utility level is given by U = [l/(1 - h)]u(c’, c2), from which it follows that h < 1. Using (7) this implies that a steady state with b > 0 is characterized by an interest rate greater than the population growth rate [see Buiter (1980) and Carmichael (1982)]. This, in turn, implies that (r - n)d > 0, i.e., positive taxes. 3 The situation of altruism from the children to the parents can be modelled as U, = u(c:, c,?) +jU,?,, where j > 0 is a discount factor. A steady state with non-negative gifts can be characterized by the system of five equations given by (2), (3) and c1+c2/(1+r)+g=f(k)-rk-(r-n)d+(l+n)g/(l+r),
(8)
f(k)-rk-(r-n)d-g-cl=(l+n)(k+d),
(9)
(1+ r) 2j(1+
n),
(IO)
where the unknowns are cl, c2, r, k plus the gift, g 2 0. The interpretation of (8)-(10) is in essence the same as that of (5)-(7) allowing for the fact that an old individual receives a gift from each of his (1 + n) children. Again, (10) is a consequence from the intertemporal optimizing behaviour; it will be satisfied as an equality if g > 0 and as a strict inequality when g = 0. The steady-state utility level is U = [l/(1 -j)]u(c’, c’), from which j < 1. Using (10) it can be seen that this implies that a steady state with g > 0 will be characterized by an interest rate less than the rate of population growth [see Buiter (1980) and Carmichael (1982)]. This means that (r - n)d -C 0, i.e., negative taxes. 4 The fact that taxes (transfers) are paid (received) by the younger generation does not affect the analysis. In contrast to the pure life-cycle model, in which neutrality requires variations in the intergenerational distribution of taxes, when individuals are ‘dynastic savers’ they can undo the effects of the debt for any parameter of tax distribution. If h =l/(l + 6), i.e., a parents takes care of (and discounts) the welfare of a representatiue child [Buiter (1979, 1980) and Carmichael (1982)], eq. (7) becomes (l+ r) d (l+ S)(l + n). If h = (lt n)/(l + 6) i.e., a parent cares about the total welfare of all his children [Burbidge (1983b)] we have that (1 + r) d (1 + 8). In any case, the quolitatiue conclusions do not change [Lopez-Garcia (1989)]. When j = l/(1 + p), i.e., a child discounts the welfare of a representatiue parent [Buiter (1979, 1980) and Carmichael (1982)], eq. (10) becomes (1 + r) > (1+ a)/(1 + p). On the other hand, if j = (1 + p)/(l + n), i.e., there is ‘reverse’ discount of the total utility of the l/(1 + n) ‘parents’ [the one-sided counterpart of Burbidge’s (1983b) two-sided analysis], we obtain that (1 + r) > (1 + p). Again, the qualitative results are the same [Lopez-Garcia (1989)].
236
M.-A. Loper-Garcia / On debi neutrality in life-cycle models
3. The neutrality of the public debt: Voluntary and coercive intergenerational
transfers
The non-effects of the variations in the amount of public debt in the presence of operative voluntary intergenerational transfers can easily be shown. An increase from d to d’ could be offset by an increase in the bequests from b to b’, where b’ = b + (1 + r)(l + n)(d’ - d), and nothing else would change in (5)-(7) including the steady-state utility level. The same conclusion can be obtained in @-(lo) if the gifts are reduced from g to g’, where g’=g-(1 +r)(d’-d). Hence, the voluntary intergenerational transfers would reestablish the initial equilibrium insofar as the nonnegativity constraints are not violated. The result does not depend on the bonds being net wealth, i.e., it holds regardless of whether r is greater or less than n [see Carmichael (1982)]. The indications that arise from the pure life-cycle model are quite different. In the case analyzed by Diamond (1965) in which (as before) q = 0, the public debt is no longer neutral but crowds our capital accumulation. However, the conclusions change when the tax distribution parameter is modified to 0 < q I 1 [see Bierwag, Grove, and Khang (1969) Diamond (1973) and Ihori (1978)], thus suggesting the possibility of using variations in the intergenerational distribution of tax or transfers in order to neutralize the effects of the debt policy. In effect, if the increase from d to d’, along with the modification from q to q’, has to maintain unchanged all the variables in (l)-(4) (and hence the utility level), the following expression must be verified:
(11)
d’/d=[(l+r)-(r-n)q],‘[(l+r)-(r-n)q’].
Thus, in a situation in which r is greater (less) than n, an increase in d requires a rise (fall) in q. Therefore, debt issue could be considered as equivalent to a redistribution between generations, with the consequence that it could only be claimed that the debt entails a ‘burden’ when the government is constrained in the use of taxes [see Atkinson and Stiglitz (1980) and Kotlikoff (1984)]. The question now is why the conclusions about debt neutrality are the same as those arising from the models with intergenerational altruism. The answer is that in the life-cycle model, when r > n and therefore taxes are positive, the tax provision q(r - n)(l + n)d can be considered as an effective bequest. A rise in q would increase the ‘bequest’ and thus the saving, thus offsetting the effects of the new debt. The expressions (1) and (4) can be rewritten as
=f(k)-rk-(r-n)d+q(r-n)d,
(12)
f(k)-rk-(r-n)d-c’+q(r-n)d=(l+n)(k+d),
(13)
whose similarity with (5) and (6) is clear if q(r - n)(l + n)d is interpreted as the counterpart of b and (r - n)d is considered ‘as if’ it fell on the younger generation. In a similar way, when r < n and hence taxes are negative, the transfer q(n - r)(l + n)d received by the older generation has the same consequences as receiving an effective gift. A fall in q would decrease the ‘gifts’ and thus would increase the saving, thus neutralizing the effects of the new debt issue. Rewriting (1) and (4) gives:
“+
----+q(n-r)d=f(k)-rk-(r-n)d+ (l:r)
f(k)-rk-(r-n)d-cl-q(n-r)d=(l+n)(k+d),
q(n - r)(l (l+r)
+ n)d ’
(14 (15)
M.-A. Lopez-Garcia / On debt neutrality in life-cycle models
237
where, again, the comparison with (8) and (9) is direct when q(n - r)d is interpreted as the equivalent of g. The neutrality arises in Barro (1974) because the intrafamily altruism generates uoluntary intergenerational transfers. In the pure life-cycle model these kind of transfers are excluded by assumption. However, in contrast to the case analyzed by Diamond (1965), in which q = 0 and the taxes only fall on the younger generation, the existence of taxes on the two generations may give rise to coerciue intergenerational transfers, which have the same result of neutralizing the effects of the public debt. The symmetry between the conditions for debt neutrality with and without voluntary transfers can also be recognized in another way. In effect, the relationships between r and n associated with coercive ‘bequests’ and ‘gifts’ in the pure life-cycle model are the same as those that reveal the existence of voluntary bequests or gifts [see Buiter (1980) Carmichael (1982) and Lopez-Garcia (1989)]. In terms of the previous section, a situation with operative bequests (gifts) is associated with r greater (less) than n. An increase (decrease) in these voluntary bequests (gifts) offsets the effects of the debt, just in the same way as a rise (fall) in q increases (decreases) the forced ‘bequests’ (‘gifts’) when r > ( <)n in the pure life-cycle model.
4. Concluding comments The parallelism between pure life-cycle models and those in which altruism is one-sided reveals that the coercive transfers necessary to neutralize the effects of the public debt in the former have in essence the same nature, and take place under the same conditions, as the offsetting voluntary gifts and bequests in the latter. Although the ability of pure life-cycle models to explain, by themselves, capital accumulation in actual economies remains a controversial topic, they are firmly rooted in the analyses of questions in which an intertemporal perspective must be adopted. Which model offers a more adequate representation of the real world is, of course, an empirical question, although each one may well describe the behaviour of different individuals.
References Abel, A.B., 1987, Operative gift and bequest motives, American Economic Review 77, 1037-1047. Atkinson, A.B. and J.E. Stiglitz, 1980, Lectures on public economics (McGraw-Hill, New York). Barro, R.J., 1973, Are government bonds net wealth?, Journal of Political Economy 82, 1095-1117. Bemheim, B.D. and K. Bagwell, 1988, Is everything neutral?, Journal of Political Economy 96, 308-338. Bierwag, G.O., M.A. Grove and C. Khang, 1969, National debt in a neoclassical growth model: Comment, American Economic Review 59, 205-210. Buiter, W.H., 1979, Government finance in an overlapping generations model with gifts and bequests, in: G.M. Von Furstenberg, ed., Social security versus private saving (Ballinger, Cambridge, MA) 395-429. Buiter, W.H., 1980, ‘Crowding’ out of private capital formation by government borrowing in the presence of intergenerational gifts and bequests, Greek Economic Review 2, 111-142. Buiter, W.H. and J. Carmichael, 1984, Government debt: Comment, American Economic Review 74, 762-765. Burbidge, J.B., 1983a, Social security and savings plans in overlapping-generations models, Journal of Public Economics 21, 79-92. Burbidge, J.B., 1983b, Government debt in an overlapping-generations model with bequests and gifts, American Economic Review 73, 222-227. Burbidge, J.B., 1984, Government debt: Reply, American Economic Review 74, 766-767. Carmichael, J., 1982, On Barro’s theorem of debt neutrality: The irrelevance of net wealth, American Economic Review 72, 202-213. Diamond, P.A., 1965, National debt in a neoclassical growth model, American Economic Review 55, 1126-1150.
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M.-A. Lopez-Garcia
/ On debt neutrality in life-cycle models
Diamond, P.A., 1973, Taxation and public production in a growth setting, in: J.A. Mirrlees and N.H. Stem, eds., Models of Economic Growth (Macmillan, New York). Drazen, A., 1978, Government debt, human capital and bequests in a life-cycle model, Journal of Political Economy 86, 505-516. Feldstein, MS., 1985, Debt and taxes in the theory of public finance, Journal of Public Economics 28, 233-245. Hillier, B. and M.T. Lunati, 1987, On Nash versus Stackelberg strategies and the conditions for operative intergenerational transfers, Scottish Journal of Political Economy 34, 91-96. Ihori, T., 1978, The golden rule and the role of government in a life cycle growth model, American Economic Review 68, 389-396. Kimball, M.S., 1987, Making sense of two-sided altruism, Journal of Monetary Economics 20, 301-326. Kotlikoff, L.J., 1984, Taxation and savings: A neoclassical perspective, Journal of Economic Literature 22, 1576-1629. Lopez-Garcia, M.A., 1987, Public debt and demographic growth in an overlapping generations model, Economics Letters 24, 197-201. Lopez-Garcia, M.A., 1989, Public debt, voluntary intergenerational transfers and overlapping generations, Economics Letters 29, 115-120. Samuelson, P.A., 1958, An exact consumption-loan model of interest with or without the social contrivance of money, Journal of Political Economy 66, 467-482. Stein, J.L., 1969, A minimal role of government in achieving optimal growth, Economica 36, 139-150. Weil, P., 1987, Love thy children. Reflections on the Barro debt neutrality theorem, Journal of Monetary Economics 19, 377-391.
Economics Letters 33 (1990) 233-238 North-Holland
ON DEBT NEUTRALITY
IN LIFE-CYCLE
Miguel-Angel LOPEZ-GARCIA Universidad Aut6nomn de Barcelona,
MODELS
*
08193 Bellaterra,
Barcelona,
Spain
Received 22 September 1989 Accepted 15 November 1989
A parallelism is established between pure life-cycle models and those in which altruism is one-sided in the discussion of the public debt neutrality. It is argued that when indiduals behave as pure life-cyclers, the coercive transfers necessary to offset the effects of the debt have the same nature, and take place under the same conditions, as the voluntary intrafamily bequests and gifts which constitute the usual basis for the ‘Ricardian equivalence theorem’.
1. Introduction Since seminal contribution by Samuelson (1958) and the extension to economies with endogenous production by Diamond (1965), there has been great interest in the effects of the public debt in the overlapping generations models. In its pure life-cycle version, the debt has been analyzed in its relationship with taxation [see Diamond (1965), Bierwag et al. (1969) and Feldstein (1985)], as a device to achieve the ‘golden rule’ [see Stein (1969) and Ihori (1978)], and to discuss related questions [see Burbidge (1983a) and L6pez-Garcia (1987)]. On the other hand, Barro (1974) has included the possibility that the intrafamily altruism can give rise to a chain of voluntary intergenerational transfers. The renewed interest in the ‘Ricardian equivalence theorem’ of debt and taxes has generated a body of literature [see Drazen (1978), Buiter (1979, 1980), Carmichael (1982), Burbidge (1983b), Weil (1987), Kimball (1987) and Abel (1987)], and there are even different ways to characterize the model [see Buiter and Carmichael (1984), Burbidge (1984), Hillier and Lunati (1987), Bernheim and Bagwell (1988) and L6pez-Garcia (1989)]. Although the literature on pure life-cycle models has pointed out that the public debt is equivalent to an intergenerational redistribution of resources, and that certain changes in the distribution of taxes among generations can offset the effects of the debt policy, the ultimate reasons for this result and its relationship with the models that allow for voluntary transfers, have not been fully clarified. This paper argues that when individuals are life-cyclers, the coercive transfers necessary to offset the effects of the public debt have the same nature, and take place under the same conditions, as the voluntary intrafamily bequests and gifts which constitute the standard basis for the ‘Ricardian equivalence theorem’. Section 1 describes both models and section 2 establishes a parallelism between them. Section 3 summarizes some final remarks. * Financial support from the FundacGn 0165-1765/90/$3.50
Fondo para la Investigacibn Jkon6mica
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
y Social (Spain) is gratefully acknowledged.
234
M.-A. Lopez-Garcia
/ On debt neutrality in life-cycle models
2. Public debt, life cycle and voluntary intergenerational
transfers
The basic framework is the overlapping generations model developed by Diamond (1965). The population is composed of identical individuals who live for two periods. In the first one they work and supply one unit of labour and in the second one they are retirees. Therefore, two generations coexist in a given time period, the younger one and the older one, and the population growth rate is denoted by n. The technology is represented by means of a linear homogenous production function in intensive terms, f(k,), with f’ > 0 and f” < 0, where k, is the capital-labour ratio in period t. Under competitive conditions each factor will receive its marginal product. The public debt issued in period t and to be repaid at the interest rate r,,, in period t + 1 is denoted by O,+ 1. The government maintains a constant per worker stock of debt, d [i.e., D,+I = (1 + n)D,], so that the amount it spends in excess of the amount it borrows is (r, - n)D,, or (rt - n)d in per worker terms. When r, is greater (less) than n, the government levies (grants) taxes (transfers), which can be divided between the two generations. 2. I. A pure life-cycle model
cf
When individuals are life-cyclers, they maximize a lifetime utility function q = u( c:, ), where c: and c: are first- and second-period consumption by an individual born in period t, subject to their budget constraint. If the government levies in period t a tax (1 - q)(r, - n)D, from the younger generation and q(r, - n)D, from the older one, where 0 4 q I 1, the individuals will not only save for life-cycle reasons, but also for a tax provision. Under these assumptions, the following four equations characterize a steady state:
‘*+P=f(k)-rk-(l-q)(r-n)d-q(r-n)(l+n)d/(l+r), (1Cr)
(1)
r=f’(k),
(2)
[WC’,4/ac~]/[au(c~,
c2)/ac2]
= (1 + r),
(3)
f(k)-rk-(l-q)(r-n)d-c’=(l+n)(k+d).
(4)
In general, for given values of d, q and n, this system can be solved in the four unknowns, cl, c2, r and k, giving rise to a steady-state utility level U = u( cl, c*). Equation (1) denotes the lifetime budget constraint; (2) expresses the relationship between the interest rate and the capital-labour ratio; (3) is the usual equality between the marginal rate of substitution and the relative prices; and (4) equates the life-cycle saving and the tax provision of the younger generation with the capital plus debt. It is known that in this model the steady-state interest rate can be greater or less than the population growth rate [Diamond (1965)]. 2.2. A model with voluntary intergenerational
transfers
In the Barro (1974) model, the individuals derive utility not only from their own consumption, but also from the welfare enjoyed by their descendants or ancestors. This intergenerational altruism can generate bequests from parents to children or gifts from children to parents. The utility function can now be written as U, = u(ci, c:, U,:,) or U, = u(c:, cf, U,?l), where U,:, (U,?,) is the maximum utility level attainable by a representative descendant (ancestor) given the bequest (gift) received. ’ We shall only analyze one-sided altruism, either from parents to children or from children dealt with in Buiter (1980), Burbidge (1983b), Kimball (1987) and Abel (1987).
to parents.
Two-sided
altruism
is
M. -A. Lopez-Garcia
/ #I debt neutrality in lije-cycle models
235
For the sake of simplicity, we shall assume that only the younger generation pays (receives) taxes (transfers), i.e., that q = 0. 2 In order to obtain concrete results we shall adopt the additively separable utility function which has been extensively used. In the case of altruism from parents to children, this reduces to u, = u(c;, c;) + h&t,, where h > 0 is a discount factor. A steady state with non-negative bequests can be characterized by means of the system of five equations given by (2) (3) and
c’+(15) +---f(k)-rk-(r-n)d+&, (1L) f(k)-rk-(r-n)d-c’+b/(l+n)=(l (1 + r) I (1 + n)/h,
+n)(k+d),
(6) (7)
where the unknowns are those of the previous model plus the bequest, b 2 0. Equation (5) is the budget constraint under the assumption that the bequest is equally divided among the (1 + n) children; (6) is the new asset identity extended to allow for the bequests; and (7) is a result of the intertemporal optimization problem. In particular, (7) holds as an equality if b > 0 and as a strict inequality when b = 0. The steady-state utility level is given by U = [l/(1 - h)]u(c’, c2), from which it follows that h < 1. Using (7) this implies that a steady state with b > 0 is characterized by an interest rate greater than the population growth rate [see Buiter (1980) and Carmichael (1982)]. This, in turn, implies that (r - n)d > 0, i.e., positive taxes. 3 The situation of altruism from the children to the parents can be modelled as U, = u(c:, c,?) +jU,?,, where j > 0 is a discount factor. A steady state with non-negative gifts can be characterized by the system of five equations given by (2), (3) and c1+c2/(1+r)+g=f(k)-rk-(r-n)d+(l+n)g/(l+r),
(8)
f(k)-rk-(r-n)d-g-cl=(l+n)(k+d),
(9)
(1+ r) 2j(1+
n),
(IO)
where the unknowns are cl, c2, r, k plus the gift, g 2 0. The interpretation of (8)-(10) is in essence the same as that of (5)-(7) allowing for the fact that an old individual receives a gift from each of his (1 + n) children. Again, (10) is a consequence from the intertemporal optimizing behaviour; it will be satisfied as an equality if g > 0 and as a strict inequality when g = 0. The steady-state utility level is U = [l/(1 -j)]u(c’, c’), from which j < 1. Using (10) it can be seen that this implies that a steady state with g > 0 will be characterized by an interest rate less than the rate of population growth [see Buiter (1980) and Carmichael (1982)]. This means that (r - n)d -C 0, i.e., negative taxes. 4 The fact that taxes (transfers) are paid (received) by the younger generation does not affect the analysis. In contrast to the pure life-cycle model, in which neutrality requires variations in the intergenerational distribution of taxes, when individuals are ‘dynastic savers’ they can undo the effects of the debt for any parameter of tax distribution. If h =l/(l + 6), i.e., a parents takes care of (and discounts) the welfare of a representatiue child [Buiter (1979, 1980) and Carmichael (1982)], eq. (7) becomes (l+ r) d (l+ S)(l + n). If h = (lt n)/(l + 6) i.e., a parent cares about the total welfare of all his children [Burbidge (1983b)] we have that (1 + r) d (1 + 8). In any case, the quolitatiue conclusions do not change [Lopez-Garcia (1989)]. When j = l/(1 + p), i.e., a child discounts the welfare of a representatiue parent [Buiter (1979, 1980) and Carmichael (1982)], eq. (10) becomes (1 + r) > (1+ a)/(1 + p). On the other hand, if j = (1 + p)/(l + n), i.e., there is ‘reverse’ discount of the total utility of the l/(1 + n) ‘parents’ [the one-sided counterpart of Burbidge’s (1983b) two-sided analysis], we obtain that (1 + r) > (1 + p). Again, the qualitative results are the same [Lopez-Garcia (1989)].
236
M.-A. Loper-Garcia / On debi neutrality in life-cycle models
3. The neutrality of the public debt: Voluntary and coercive intergenerational
transfers
The non-effects of the variations in the amount of public debt in the presence of operative voluntary intergenerational transfers can easily be shown. An increase from d to d’ could be offset by an increase in the bequests from b to b’, where b’ = b + (1 + r)(l + n)(d’ - d), and nothing else would change in (5)-(7) including the steady-state utility level. The same conclusion can be obtained in @-(lo) if the gifts are reduced from g to g’, where g’=g-(1 +r)(d’-d). Hence, the voluntary intergenerational transfers would reestablish the initial equilibrium insofar as the nonnegativity constraints are not violated. The result does not depend on the bonds being net wealth, i.e., it holds regardless of whether r is greater or less than n [see Carmichael (1982)]. The indications that arise from the pure life-cycle model are quite different. In the case analyzed by Diamond (1965) in which (as before) q = 0, the public debt is no longer neutral but crowds our capital accumulation. However, the conclusions change when the tax distribution parameter is modified to 0 < q I 1 [see Bierwag, Grove, and Khang (1969) Diamond (1973) and Ihori (1978)], thus suggesting the possibility of using variations in the intergenerational distribution of tax or transfers in order to neutralize the effects of the debt policy. In effect, if the increase from d to d’, along with the modification from q to q’, has to maintain unchanged all the variables in (l)-(4) (and hence the utility level), the following expression must be verified:
(11)
d’/d=[(l+r)-(r-n)q],‘[(l+r)-(r-n)q’].
Thus, in a situation in which r is greater (less) than n, an increase in d requires a rise (fall) in q. Therefore, debt issue could be considered as equivalent to a redistribution between generations, with the consequence that it could only be claimed that the debt entails a ‘burden’ when the government is constrained in the use of taxes [see Atkinson and Stiglitz (1980) and Kotlikoff (1984)]. The question now is why the conclusions about debt neutrality are the same as those arising from the models with intergenerational altruism. The answer is that in the life-cycle model, when r > n and therefore taxes are positive, the tax provision q(r - n)(l + n)d can be considered as an effective bequest. A rise in q would increase the ‘bequest’ and thus the saving, thus offsetting the effects of the new debt. The expressions (1) and (4) can be rewritten as
=f(k)-rk-(r-n)d+q(r-n)d,
(12)
f(k)-rk-(r-n)d-c’+q(r-n)d=(l+n)(k+d),
(13)
whose similarity with (5) and (6) is clear if q(r - n)(l + n)d is interpreted as the counterpart of b and (r - n)d is considered ‘as if’ it fell on the younger generation. In a similar way, when r < n and hence taxes are negative, the transfer q(n - r)(l + n)d received by the older generation has the same consequences as receiving an effective gift. A fall in q would decrease the ‘gifts’ and thus would increase the saving, thus neutralizing the effects of the new debt issue. Rewriting (1) and (4) gives:
“+
----+q(n-r)d=f(k)-rk-(r-n)d+ (l:r)
f(k)-rk-(r-n)d-cl-q(n-r)d=(l+n)(k+d),
q(n - r)(l (l+r)
+ n)d ’
(14 (15)
M.-A. Lopez-Garcia / On debt neutrality in life-cycle models
237
where, again, the comparison with (8) and (9) is direct when q(n - r)d is interpreted as the equivalent of g. The neutrality arises in Barro (1974) because the intrafamily altruism generates uoluntary intergenerational transfers. In the pure life-cycle model these kind of transfers are excluded by assumption. However, in contrast to the case analyzed by Diamond (1965), in which q = 0 and the taxes only fall on the younger generation, the existence of taxes on the two generations may give rise to coerciue intergenerational transfers, which have the same result of neutralizing the effects of the public debt. The symmetry between the conditions for debt neutrality with and without voluntary transfers can also be recognized in another way. In effect, the relationships between r and n associated with coercive ‘bequests’ and ‘gifts’ in the pure life-cycle model are the same as those that reveal the existence of voluntary bequests or gifts [see Buiter (1980) Carmichael (1982) and Lopez-Garcia (1989)]. In terms of the previous section, a situation with operative bequests (gifts) is associated with r greater (less) than n. An increase (decrease) in these voluntary bequests (gifts) offsets the effects of the debt, just in the same way as a rise (fall) in q increases (decreases) the forced ‘bequests’ (‘gifts’) when r > ( <)n in the pure life-cycle model.
4. Concluding comments The parallelism between pure life-cycle models and those in which altruism is one-sided reveals that the coercive transfers necessary to neutralize the effects of the public debt in the former have in essence the same nature, and take place under the same conditions, as the offsetting voluntary gifts and bequests in the latter. Although the ability of pure life-cycle models to explain, by themselves, capital accumulation in actual economies remains a controversial topic, they are firmly rooted in the analyses of questions in which an intertemporal perspective must be adopted. Which model offers a more adequate representation of the real world is, of course, an empirical question, although each one may well describe the behaviour of different individuals.
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