Long-Run Implications of Social Security Taxation for Growth and Fertility.

Jie Zhang [*]

This paper compares long-run implications for growth and fertility of four types of taxation for social security with positive bequests. A tax rise under lump-sum taxation enhances growth but lowers fertility, while other types of taxation do so under additional restrictions. A tax rise under consumption taxation is less likely to stimulate growth and to reduce fertility than under payroll taxation. A rise in an interest income tax raises fertility, reduces both savings and human capital investment, and hence is harmful for growth. The case with zero bequests is also discussed.

1. Introduction

All developed countries and most developing countries have social security programs for their retired population. These programs are widely divergent in formulation in terms of how to collect social security contributions and how to allocate social security benefits (U.S. Department of Health and Human Services 1992). On the spending side, social security programs are distinguished by whether they are funded or unfunded and by whether benefits are linked to individuals' own contributions. On the taxation side, social security programs differ in the sources of their revenues. In some countries (e.g., France and the United States), social security benefits come (almost) exclusively from taxes on labor earnings. In some countries (e.g., Australia), the benefits depend only on governments' general revenue from levying direct or indirect taxes. In many countries (e.g., Canada, Germany, Italy, and the United Kingdom), the benefits come from both payroll and general tax revenues.

In the literature on social security, however, lump-sum contributions are widely assumed (Barro 1974; Becker and Barro 1988; Nishimura and Zhang 1992) even if they are rare in practice. The emphases of the existing work have been laid on how to spend on social security programs. In particular, the impact of a pay-as-you-go program (i.e., an unfunded plan) on savings has been the focus of the debate. (In practice, unfunded social security is much more popular than funded social security.) Feldstein (1974), for example, argued that unfunded social security depresses savings and hence has a negative impact on growth. Barro (1974) showed that in a dynastic family model incorporating operative intergenerational transfers, social security is neutral. When fertility is endogenous, Becker and Barro (1988) found that increasing social security benefits reduces fertility and raises capital intensity because more transfers from the working generation to the coexisting retired generation cause a rise in bequests per chi ld and hence a rise in the cost of raising a child. Using an endogenous growth model, Zhang (1995) found that unfunded social security benefits promote growth by reducing fertility and increasing human capital investment if parents value their children's welfare sufficiently.

This paper considers long-run implications for growth and fertility of different types of taxation for social security: a lump-sum tax, a consumption tax, a payroll tax, and an interest income tax. In doing so, we assume operative bequests as in Barro (1974) and Becker and Barro (1988). The main results are the following. A tax rise under lump-sum taxation enhances growth but lowers fertility, while other types of taxation do so under additional restrictions. A tax rise under consumption taxation is less likely to stimulate growth and to reduce fertility than under payroll taxation. A rise in an interest income tax raises fertility, reduces both savings and human capital investment, and hence is harmful for growth. I also discuss results with exogenously fixed fertility or with zero bequests, which are substantially different from those with endogenous fertility and positive bequests except for the case with the interest income tax.

The remainder of the paper is organized as follows. The next section introduces the model. Section 3 examines and compares the effects of using a lump-sum tax, a consumption tax, or a payroll tax to finance social security by assuming positive bequests. Section 4 discusses the results first with interest income taxation for social security and then with zero bequests. The last section provides some concluding remarks.

2. The Model

This model has an infinite number of overlapping generations of three-period-lived agents. Let subscript t denote a period in time and superscript t the generation born in period t - 1. Let [L.sub.t] be the number of middle-aged agents living in period t. Each parent has 1 + [n.sub.t]] (identical) children at the beginning of middle age. Agents learn when young, live in retirement in old age, and are each endowed in middle age with one unit of time that can be supplied to the labor market or spent on rearing children. Let v denote the units of time needed to rear a child (0 [less than] v [less than] 1).

The utility of a middle-aged agent, [V.sub.t], depends separately on own middle-age consumption, [[c.sup.t].sub.t]; own old-age consumption, [[c.sup.t].sub.t+1]; the number of children, 1 + [n.sub.t]; and the utility of each child, [V.sub.t+1]:

[V.sub.t] = ln [[c.sup.t].sub.t] + [beta] ln [[c.sup.t].sub.t+1] + p ln (l+[n.sub.t]) + [alpha][V.sub.t+1], 0 [less than] [alpha] [less than] 1, 0 [less than] [beta] [less than] 1, 0[less than] [rho] [less than] 1.

Here, [beta] is the discount factor on utility from old-age consumption, [rho] the taste for the number of children, and [alpha] the taste for the welfare of each child.

The production function for goods has the form

[Y.sub.t] = D[[K.sup.0].sub.t],[([L.sub.t][l.sub.t][h.sub.t]).sup.1-[theta]], D [greater than] 0, 0 [less than] [theta] [less than] 1,

where [K.sub.t] is (aggregate) physical capital, [l.sub.t], the input of labor per middle-aged agent, and [h.sub.t] a middle-aged agent's human capital.

The human capital of a child, [h.sub.t+1], depends positively on the investment of goods per child, [q.sub.t], and the human capital of his parent, [h.sub.t]:

[h.sub.t+1] = A[[q.sup.[delta]].sub.t][[h.sup.l-[delta]].sub.t], A [greater than] 0, 0 [less than] [delta] [less than] 1.

In period t, each middle-aged agent spends v(1 + [n.sub.t]) units of time rearing children, works for the remaining 1 - v(1 + [n.sub.t]) units of time, and earns [1 - v(1 + [n.sub.t])][W.sub.t]. This agent receives a bequest, [b.sub.t], from his parent at the beginning of period t and leaves a bequest, [b.sub.t+1], to each child at the beginning of period t + 1, where bequests have no direct contribution to physical capital accumulation. [1] The middle-aged agent spends the earning and inheritance on own middle-age consumption, [[c.sup.t].sub.t], life-cycle savings, [s.sub.t][1 - v(1 + [n.sub.t])][w.sub.t]; bequests to children, [b.sub.t+1](1 + [n.sub.t]); and investments in children, [q.sub.t](1 + [n.sub.t]). To finance social security benefits, [B.sub.t+1] per retiree, there is a lump-sum tax at the amount [T.sub.t] per worker, a payroll tax at rate [[tau].sub.1], and a consumption tax at rate [[tau].sub.c]. (The case with an interest income tax has no closed-form solution and will be discussed in section 4.) Then, an individual's budget constraints are

(1 + [[tau].sub.c])[[c.sup.t].sub.t] = [b.sub.t] + [1 - v(1 + [n.sub.t])][w.sub.t](1 - [s.sub.t] - [[tau].sub.t]) - [T.sub.t] - [q.sub.t](1 + [n.sub.t]), (1)

(1 + [[tau].sub.c])[[c.sup.t].sub.t+1] = (1 + [r.sub.t+1])[s.sub.t][1 - v(1 + [n.sub.t])][w.sub.t] + [B.sub.t+1] - [b.sub.t+1](1 + [n.sub.t]), (2)

where w and r denote the wage rate and interest rate, respectively. The government budget constraint is given by

[B.sub.t] = (1 + [n.sub.t-1]){[T.sub.t] + [[tau].sub.1][w.sub.t][1 - v(1 + [n.sub.t])] + [[tau].sub.c][[c.sup.t].sub.t]} + [[tau].sub.c][[c.sup.t-1].sub.t], (3)

where a bar over a variable refers to its average.

Firms maximize profits on perfectly competitive markets. Let [e.sub.t] [equivalent] [K.sub.t]/([L.sub.t][l.sub.t][h.sub.t]) be the physical capital-effective labor ratio where h is average human capital and l the average labor demand per worker. For simplicity, I assume that physical capital lasts for one period in the production of goods. The first-order conditions of firms maximizing profits are

[w.sub.t] = (1 - [theta])D[[e.sup.[theta]].sub.t][h.sub.t], (4)

1 + [r.sub.t] = [theta]D[(l/[e.sub.t]).sup.1-[theta]]. (5)

Equation 4 implies that a middle-aged agent's wage rate depends positively on his own human capital. Labor and capital markets clear when

[l.sub.t] = 1 - v(1 + [n.sub.t]), (6)

[K.sub.t] = [L.sub.t-1][s.sub.t-1][1 - v(1 + [n.sub.t-1])][w.sub.t-1]. (7)

Constant returns to scale and perfect competition imply a zero profit. By Walras's law, the goods market clears as well. Since agents within the same generation are identical, we have the following symmetric conditions: c = c, h = h, l = l, n = n, and w = w.

3. Results

Given the initial state ([b.sub.t], [h.sub.t]), the tax/benefit variables ([[tau].sub.c] [[tau].sub.l], [T.sub.t], [B.sub.t+1]), and the sequence of the physical capital-effective labor ratio [e.sub.t], the problem of a middle-aged agent maximizing utility corresponds to the following concave programming:

V([h.sub.t], [b.sub.t]; [[tau].sub.c], [[tau].sub.l], [T.sub.t], [B.sub.t+1], [e.sub.t]) = [max.sub.[b.sub.t+1],[h.sub.t+1],[n.sub.t],[s.sub.t]] X {ln [b.sub.t] + [ 1 - v(1 + [n.sub.t])](1 - [theta])D[[e.sup.[theta]].sub.t][h.sub.t](1 - [s.sub.t] - [[tau].sub.l]) - [T.sub.t] - [([h.sub.t+1]/A).sup.1/[delta][[h.sub.t].-(1-[delta])/[delta]](1 + [n.sub.t])/1 + [[tau].sub.c] + [beta] ln[theta]D[(1/[e.sub.t+1]).sup.1-[theta][s.sub.t][1 - v(1 + [n.sub.t])](1 - [theta])D[[e.sup.[theta]].sub.t][h.sub.t] + [B.sub.t+1] - [b.sub.l+1](1 + [n.sub.t])/1 + [[tau].sub.c] + [rho] ln(1 + [n.sub.t]) + [alpha]V([h.sub.t+1], [b.sub.t+1]; [[tau].sub.c], [[tau].sub.l, [T.sub.t+1], [B.sub.t+2], [e.sub.t+1])}.

The first-order conditions for an interior solution for this problem are as follows:

[beta](1 + [n.sub.t])/[[c.sup.t].sub.t+1] = [alpha]/[[c.sup.t+1].sub.t+1], (8)

1/[[c.sup.t].sub.t] = [beta](1 + [r.sub.t+1])/[[c.sup.t].sub.t+1], (9)

[q.sub.t](1 + [n.sub.t])/[[c.sup.t].sub.t] = [alpha][delta][1 - v(1 + [n.sub.t+1])][w.sub.t+1] (1 - [[tau].sub.t]) + [alpha](1 - [delta])[q.sub.t+1](1 + [n.sub.t+1])/[[c.sup.t+1].sub.t+1], (10)

[w.sub.t]v(1 - [[tau].sub.l]) + [q.sub.t]/(1 + [[tau].sub.c])[[c.sup.t].sub.t] + [beta][b.sub.t+1]/(1 - [[tau].sub.c])[[c.sup.t].sub.t+1] = [rho]/1 + [n.sub.t]. (11)

Equation 8 means that the utility forgone by leaving one more unit of bequests to children is equal to the utility obtained from improving the welfare of each child by the bequest. Equation 9 says that the loss in utility from saving one unit of goods now equals the gain in utility from receiving [r.sub.t+1] units more in the next period. By Equation 10, the utility forgone from investing one more unit in each child's human capital equals the utility gained from increasing the welfare of each child by the investment. Equation 11 means that the utility forgone from consuming less to have one more child (less earnings, more investment in children's human capital, and more bequests given to children) is equal to the utility obtained from enjoying the child.

Steady-state balanced growth means that [Y.sub.t]/[L.sub.t], [K.sub.t]/[L.sub.t], [h.sub.t], [[c.sup.t].sub.t], [[c.sup.t].sub.t+1], [b.sub.t], [q.sub.t], and [w.sub.t], grow at the same rate denoted as 1 + g. Then, bequests, investment in human capital, savings, and consumption are all proportional to income of middle-aged agents (or labor earnings) in such an equilibrium. Let

[[gamma].sub.b] = [b.sub.t+1](1 + [n.sub.t])/(1 + [r.sub.t+1])[1 - v(1 + [n.sub.t])][w.sub.t],

[[gamma].sub.[c.sub.1]] = [[c.sup.t].sub.t]/[1 - v(1 + [n.sub.t])][w.sub.t],

[[gamma].sub.[c.sub.2]] = [[c.sup.t].sub.t+1]/(1 - [r.sub.t+1])[1 - v(1 + [n.sub.t])[w.sub.t],

[[gamma].sub.q] = [q.sub.t]/[1 - v(1 + [n.sub.t])][w.sub.t] be the ratios of bequests, middle-age consumption, old-age consumption, and investment in human capital to income of middle-aged agents, respectively, where [b.sub.t+1](1 + [n.sub.t])/(1 + [r.sub.t+1]) and [[c.sup.t].sub.t+1]/(1 + [r.sub.t+1]) are the present values of next period bequests and next period old-age consumption.

Note that since labor income is a constant fraction, 1 - [theta], of total income, changes in these ratios and the saving rate also represent changes in the ratios of bequests, consumption, human capital investment, and savings to total income. Also, in a growing economy taxes and transfers rise with income, and hence the government sets the lump-sum tax as a fraction, [tau], of the average labor earning per worker:

[T.sub.t] = [tau][1 - v(1 + [n.sub.t])][w.sub.t].

When bequests are positive, Equations 1 to 11 and the symmetric conditions characterize the equilibrium. Solving these equations under the steady-state balanced growth conditions gives the analytical solution for s, [[gamma].sub.b] n, [[gamma].sub.q] and g:

S = [alpha][theta]/(1 - [theta]), (12)

[[gamma].sub.b] = [[alpha].sup.2][theta](1 + [[tau].sub.c]) + [[alpha].sup.2](1 - [theta])(1 + [[tau].sub.c])([tau] + [[tau].sub.l]) + [alpha]([beta] - [alpha][[tau].sub.c])[[alpha][theta] - (1 - [theta])(1 - [tau] - [[tau].sub.l])]/(1 - [theta])([alpha] + [beta]) + [[alpha].sup.2][delta]([beta] - [alpha][[tau].sub.c])(1 - [[tau].sub.l])/([alpha] + [beta])[1 - [alpha](1 - [delta])]' (13)

1 + n = [1 - [alpha](1 - [delta])][[rho][[gamma].sub.c1](1 + [[tau].sub.c]) - [[gamma].sub.b]] - [alpha][delta](1 - [[tau].sub.t])/v{[1 - [alpha](1 - [delta])](1 - [[tau].sub.t]) + [1 - [alpha](1 - [delta])][p[[gamma].sub.c1](1 + [[tau].sub.c]) - [[gamma].sub.b]] - [alpha][delta](1 - [[tau].sub.l])}' (14)

where [[gamma].sub.c1](1 + [[tau].sub.c] = [[gamma].sub.b]/[alpha] + 1 - [tau] - [[tau].sub.l] - s - [alpha][delta](1 - [[tau].sub.t])/[1 - [alpha](1 - [delta])],

[[gamma].sub.q] = [alpha][delta](1 - [[tau].sub.t])/(1 + n)[1 - [alpha](1 - [delta])]' (15)

1 + g = [[[(A[{[[gamma].sub.q][1 - v(1 + n)]}.sup.[delta]]).sup.1-[theta]][[D(1 - [theta])].sup.[delta]][(s/1 + n).sup.[delta][theta]]].sup.1/[1 - [theta](1 - [delta])]. (16)

Some steps of the derivation of the steady-state balanced growth solution are provide in the Appendix. We now examine and compare the effects of the three types of taxation for social security.

PROPOSITION 1.

(a) For [[gamma].sub.b] [greater than] 0 and [[tau].sub.c] = [[tau].sub.t] = 0, [partial]s/[partial][tau] = 0; [partial]n/[partial][tau] [less than] 0; and [partial]g/[partial][tau] [greater than] 0.

(b) For [[gamma].sub.b] [greater than] 0 and [tau] = [[tau].sub.l] = 0, [partial]s/[partial][[tau].sub.c] = 0; [partial]n/[partial][[tau].sub.c] [less than] 0 and [partial]g/[partial][[tau].sub.c] [greater than] 0 if and only if [rho] [less than] [alpha].

(c) For [[gamma].sub.b] [greater than] 0 and [tau] = [[tau].sub.c] = 0, [partial]s/[partial][[tau].sub.l] = 0; [partial]n/[partial][[tau].sub.l] [less than] 0 if [rho] [less than] [alpha](1 + [theta][beta])/(1 - [alpha][theta]); [partial]g/[partial][[tau].sub.l] [greater than] 0 if [rho] [less than or equal to] [alpha]{[1 - [alpha](1 - [delta])][1 + [theta] + [theta][beta](1 + [alpha]) + [alpha](1 - [theta])[[tau].sub.l] - [beta](1 - [theta])(1 - [[tau].sub.l)] + (1 - [theta])(1 - [[tau].sub.l])[[alpha](1 + [beta]) + [beta]}/[(1 - [alpha])(1 - [alpha][theta])(1 + [alpha]) + [alpha][delta](1 - [alpha][theta])(1 + [alpha]) - [[alpha].sup.2][delta](1 - [theta])].

PROOF. The results follow Equations 12 to 16. QED.

According to Proposition 1 or Equation 12, the saving rate is independent of all the three types of taxation as long as bequests are positive. The bequest ratio increases with [gamma], [[gamma].sub.c], and [[gamma].sub.l] in Equation 13. As in Barro (1974), when bequests are positive, agents respond to a rise in taxes for social security by keeping saving rates constant and by leaving more bequests to children to offset the public intergenerational transfer.

The responses of fertility to tax changes differ under different types of taxation. On the one hand, increasing any one of the three taxes has a negative effect on fertility because such a tax rise raises the cost of rearing a child through increasing bequests as in Becker and Barro (1988). On the other hand, a rise in either the consumption tax or the payroll tax also reduces the cost of a child and hence may raise fertility. A rise in the payroll tax rate reduces the after-tax wage rate (the opportunity cost of spending time on a child), and a rise in the consumption tax rate lowers the cost of a child relative to the cost of consumption. [2] But such a positive effect of a tax rise on fertility through reducing the cost of a child is absent under the lump-sum tax. Therefore, a rise in the lump-sum tax reduces fertility, whereas rises in other taxes introduce opposing forces on fertility, and their net effects depend on tastes for the number versus the welfare of children. Under the consumption tax, the ne t effect on fertility of a tax rise is negative if and only if the taste for the welfare of children is stronger than that for the number of children. Under the payroll tax, the net effect on fertility of a tax rise is negative if the taste for the welfare of children is not much weaker than that for the number of children. Substantial declines in fertility in recent decades in many countries may suggest strong tastes for children's welfare relative to their numbers. [3]

Under the lump-sum tax with positive bequests, human capital investment per child relative to per family income, [[gamma].sub.q], increases with the ratio of the lump-sum tax to income, [tau], because fertility decreases with [tau] because of the well-known trade-off between the quality and the quantity of children. With positive bequests, a rise in the consumption tax rate raises human capital investment per child as a fraction of per family income if and only if it reduces fertility since the consumption tax affects human capital investment only indirectly through fertility as under the lump-sum tax. Under the payroll tax with positive bequests, a tax rise has an indirect effect via fertility as well as a directly negative effect through reducing the after-tax wage rate; the net effect is more likely to be positive if the tax rise reduces fertility more substantially.

Responses of the growth rate of per capita income to tax rises for social security under each of the three types of taxation depend on how fertility and human capital investment respond to the tax rises, given that the saving rate is independent of the taxes when bequests are positive. By Equation 16, the (steady-state) growth rate is higher if the saving rate or human capital investment per child relative to per family income is higher or if fertility is lower. Under the lump-sum tax with positive bequests, a tax rise raises the growth rate because it stimulates human capital investment but reduces fertility without changing the saving rate. Similarly, a rise in the consumption tax raises the growth rate when it raises the fraction of income spent on children's education and lowers fertility under the condition that the taste for the welfare of children is stronger than that for the number of children. With positive bequest, a rise in the payroll tax can lead to a higher growth rate through reducing fertili ty even though it may reduce human capital investment.

It is interesting to note the different restrictions for the three types of taxation to have negative fertility effects and positive growth effects in the long run. First of all, the lump-sum tax has the least restriction. This is not surprising since the lump-sum tax has the least distortions on the relative cost of children, human capital investment, and consumption. It is less obvious but important in practice to know which of the consumption tax and the payroll tax has less restriction. The conventional view is in favor of a consumption tax over a payroll tax in the literature on the implication of taxation for growth.

In Proposition 1, the condition for a negative net effect on fertility of a tax rise under the consumption tax is more restrictive than under the payroll tax because [rho] [less than] [alpha] [less than][alpha](l + [theta][beta])/(1 -[alpha][theta]). Namely, all possible parameter configurations that allow a negative effect on fertility of a rise in the consumption tax rate will also allow a negative effect on fertility of a rise in the payroll tax rate but not vice versa.

The sufficient (nonnecessary) condition in Proposition 1 for a tax rise under payroll taxation to enhance growth is also less restrictive than the condition under consumption taxation, unless the payroll tax rate exceeds some unlikely high level. This is obtained by noting that [rho] [less than] [alpha][less than] [alpha] {[1 -[alpha](1 - [delta])][1 + [theta] + [theta][beta](1 + [alpha]) + [alpha](1 - [theta])[[tau].sub.l] -[beta](1 - [theta])(1 - [[tau].sub.l])] + (1 - [theta])(1 - [[tau].sub.l])[[alpha](1 + [beta]) + [beta]]}/[(1 - [alpha])(1 - [alpha][theta])(1 + [alpha]) + [alpha][theta](1 - [alpha][theta])(l + [alpha]) - [[alpha].sup.2][delta](1 - [theta])] under a sufficient but nonnecessary condition [theta][1 - [alpha](1 - [delta])][1 + ([alpha] + [beta])(1 + [alpha])] + [alpha](1 - [delta]) - [[tau].sub.t] + [beta](1 - [[tau].sub.t]) [greater than] 0. This condition should be satisfied in practice since the payroll tax rate for social security, [[tau].sub.t], is below or around 20% and the discount factors ([beta] and [alpha]) sh ould be near or above 0.5.

The differences in restrictions for different taxes to raise the growth rate are intuitive. The lump-sum tax has no direct effect on human capital investment and lowers fertility as long as bequests are positive, while the other two types of taxation need more restrictions to lower fertility and may have directly negative effects on human capital investment. Thus, the lump-sum tax does better for growth than other taxes for social security. Compared to the consumption tax, a rise in the payroll tax rate is more likely to lower fertility but has a directly negative effect on human capital investment. The higher the payroll tax rate, the stronger its directly negative effect on human capital investment. As a result, when the payroll tax is extremely high, its further rise may be less likely to raise the growth rate than a rise in the consumption tax rate.

If bequests are positive but fertility were fixed at some exogenous level, then both the lump-sum tax and the consumption tax would have no effect on human capital investment and growth since they impinge on the economy only through fertility. In contrast, if bequests are positive but fertility were exogenous, a rise in the payroll tax rate would be harmful for growth through its directly negative effect on human capital investment. These cases yield very different results by abstracting from responses of fertility to changes in taxes for social security and from the interactions between fertility and human capital investment.

4. Discussions: Interest Income Taxation and Zero Bequests

In this section, we first look at the impacts of an interest income tax for social security. Then we will see what happens to the results if bequests are zero.

Interest Income Taxation and Social Security

As mentioned earlier, some countries use general tax revenues, which include the revenue from an interest income tax, to finance social security. It is thus interesting to see how a tax on interest income to finance social security affects the economy. In this case, unlike previous ones with other taxes, there is no closed-form solution. However, we could still draw conclusions by comparing the case with a positive interest income tax for social security to that with a zero tax.

Let [T.sub.r] be the rate of the interest income tax. The government budget constraint is [B.sub.t] = [T.sub.r][r.sub.t][s.sub.t-1][1 - v(1 + [n.sub.t-1])][w.sub.t-1], and the agent's budget constraints are

[[c.sup.t].sub.t] = [b.sub.t] + [1 - v(1 + [n.sub.t])][w.sub.t](1 - [s.sub.t]) - [q.sub.t](1 + [n.sub.t]),

[c.sup.[t.sub.t+1]] = [1 + [r.sub.t+1](1 - [T.sub.r])][s.sub.t][1 - v(1 + [n.sub.t])][w.sub.t] + [B.sub.t+1] - [b.sub.t+1](1 + [n.sub.t]).

The first-order condition with respect to saving is given by

1/[[c.sup.t].sub.t] = [beta][1 + [r.sub.t+1](1 - [[tau].sub.r])]/[[c.sup.t].sub.t+1],

which differs from that in Equation 9. Note that a rise in the rate of the interest income tax lowers the after-tax return to saving. The other first-order conditions are the same as before.

Define [[gamma].sub.c2] = [[c.sup.t].sub.t+1]/{[1 + r(1 - [[tau].sub.r])][1 - v(1 + [n.sub.t])][w.sub.t]} and [[gamma].sub.b] = [b.sub.t+1](1 + [n.sub.t])/{[1 + r(1 - [[tau].sub.r])][1 - v(1 + [n.sub.t])][w.sub.t]}. We can then solve the system of equations as functions of the interest rate r in the steady-state balanced growth equilibrium in a way similar to that in previous cases, but now r is only implicitly determined. The saving function is

s = [alpha][theta][1 + r(1 - T)]/(1 - [theta])(1 + r), (17)

implying that the saving rate is lower with a positive interest income tax than that with a zero tax on interest income. The result is intuitive: Taxing interest income for social security lowers the rate of return to savings.

The bequest function is

[[gamma].sub.b] = [alpha]/[alpha] + [beta]{[alpha][beta][theta][1 + r(1 - [[tau].sub.r])]/(1 - [theta])(1 + r) + [alpha][theta]/1 - [theta] - [beta](1 - [alpha])/1 - [alpha](1 - [delta])}. (18)

Here, bequests as a fraction of income are lower with a positive interest income tax than that with a zero interest income tax. This is because of the following reasons: (i) Such a tax is not a public intergenerational transfer from the young to the old generation, so there is no need to raise bequests, as opposed to the previous three types of taxation, and (ii) as the interest income tax reduces savings, old-age disposable income relative to that in middle age falls, and a reduction in bequests helps to smooth life-cycle consumption.

Investment in children's education as a fraction of a family's income, [[gamma].sub.q](1 + n), is equal to [alpha][delta]/[1 - [alpha](1 - [delta])], which is a constant. Then, the fraction of a family's income used for middle-age consumption, a function of the interest rate, is

[[gamma].sub.ct] = - [[[alpha].sup.2][theta]/(1 - [theta])([alpha] + [beta])] [1 + r(1 - [[tau].sub.r])/1 + r] + [alpha][theta]/(1 - [theta])([alpha] + [beta]) + [alpha](1 - [alpha])/([alpha] + [beta])[1 - [alpha](1 - [delta])]' (19)

which is higher with a positive tax on interest income than with a zero tax.

The fertility function is

1 + n = [1 - [alpha](1 - [delta])([rho][[gamma].sub.c1] - [[gamma].sub.b]) - [alpha][delta]/v{1 - [alpha](1 - (delta)](1 + ([rho][[gamma].sub.c1] - [[gamma].sub.b]) - [alpha][delta]}. (20)

From the previous line of argument, [rho][[gamma].sub.c1] - [[gamma].sub.b], and thereby fertility, is higher with a positive tax on interest income than with a zero tax on interest income. Obviously, human capital investment per child as a fraction of a family's income, [[gamma].sub.q] is lower with a positive tax on interest income than with a zero tax. Consequently, the growth rate, given in Equation 16, is lower with a positive tax on interest income than with a zero tax.

When fertility is exogenous, the interest income tax for social security has no effect on investment in education per child but lowers the saving rate. Thus, it is still harmful for growth.

Zero Bequests

When bequests are zero, what are the impacts of the four types of taxation for social security? We first give the results regarding the lump-sum tax and the payroll tax where we can find closed-form solutions. Later, we investigate the results with consumption and interest income taxes, respectively, when there is no closed-form solution.

With lump-sum and payroll taxes and with zero bequests, the saving rate is given by

S = [beta][theta]{(1 - [tau] - [[tau].sub.l])[1 - [alpha](1 - [delta])] - [alpha][delta](1 - [[tau].sub.l])}/[1 - [alpha](1 - [delta])][[theta](1 + [beta]) + (1 - [theta])([tau] + [[tau].sub.l])] (21)

(See the derivation in the Appendix.) Evidently, [partial]s/[partial][tau] [less than] 0 and [partial]s/[partial][[tau].sub.l] [less than] 0; that is, the saving rate falls as either the lump-sum tax or the payroll tax rises relative to income. The intuition is that when bequests are zero (because of a sufficiently weak taste for the welfare of children), agents reduce savings in anticipation of a reduction in middle-age after-tax income and a rise in social security benefits through a rise in lump-sum or payroll taxes.

The solution for fertility, children's education, and the growth rate are the same as in Equations 14 to 16, respectively, with [[gamma].sub.b] = 0. Thus, a rise in the rate of the payroll tax raises fertility but reduces education investment per child, in addition to its negative effect on saving, and hence is harmful for growth. On the other hand, a rise in the rate of the lump-sum tax has ambiguous effects on fertility and education investment per child as opposed to its negative effect on saving. As a result, a rise in the lump-sum tax rate has an ambiguous (possibly negative) net effect on the growth rate.

When consumption taxes are positive and bequests are zero, the saving rate is implicitly given by

s = [beta](1 - [alpha])/(1 + [beta])[1 - [alpha](1 - [delta])] - [[[tau].sub.s]/(1 + [beta])(1 - [[tau].sub.c])(1 + [[tau].sub.c])][(1 - [alpha])(1 - [theta])/[theta][1 - [alpha](1 - [delta])] + 1 + [[tau].sub.c - s(1 - [theta])/[theta]]. (22)

Here, the last factor in the second term on the right-hand side of Equation 22, (1 - [alpha])(l - [theta]){[theta][1 - [alpha](1 - [delta])]} + [[tau].sub.c - s(1 - [theta])/[theta], is positive at least for s [less than or equal to] [theta]/(1 - [theta]). If [theta] = 1/3 as widely used, then s [less than or equal to] 1/2, which is surely satisfied in reality, is sufficient for s [less than or equal to] [theta](1 - [theta]). Thus, the saving rate is lower with a positive consumption tax than with a zero consumption tax. [4] This is due to the fact that such a tax collects part of middle-age consumption expenditures of the working generation and transfers it as social security benefits to the generation in retirement. When bequests are zero, agents naturally save less to counteract the reduction in middle-age consumption relative to old-age consumption under the consumption tax-financed social security.

Again, the solution for fertility, education investment, and the growth rate has the same form as that in Equation 14 to 16 with [[gamma].sub.b] = 0. Observe that the sign of the impact of a rise in the consumption tax rate on (1 + [[tau].sub.c])[[gamma].sub.c1] is the opposite of that on s. Thus, fertility is higher with a positive consumption tax than with a zero consumption tax. It is then obvious that education investment per child and the growth rate are lower with a positive consumption tax than with a zero tax.

When the interest income is taxed for social security with zero bequests, the saving function is

S = [beta](1 - [alpha])/[1 - [alpha](1 - [delta])]{[beta] + (1 + r)/[1 + r(1 - [[tau].sub.r])]}. (23)

Obviously, the saving rate is lower with a positive interest income tax than with a zero tax. Paralleling the argument used earlier, fertility is higher but education investment and the growth rate are lower with a positive tax on interest income than with a zero tax.

5. Concluding Remarks

This paper has shown that various types of taxation for social security have different implications for steady-state balanced growth and fertility. In terms of enhancing growth and reducing fertility, we rank the taxes for social security by the following order: the lump-sum tax first, the payroll tax second, the consumption tax third, and the interest income tax last. While the main results are based on the assumption of operative bequests, we have considered the case with zero bequests. The case with operative bequests may be quite relevant, as found in Kotlikoff and Summers (1981), that bequests are an important element in accounting for capital accumulation in the United States. These results may have some useful policy implications. If lump-sum taxation for social security is not an option in practice, then governments that are mainly concerned about long-run growth should use payroll taxes rather than consumption/interest income taxes to finance social security as we observe in many countries.

Our results are similar to those in the literature on social security in one important aspect: "at least one of the determinants of the economy's growth path--fertility, savings, or human capital accumulation, hence growth, must be adversely affected," as stated in Ehrlich and Lui 1997; see also Ehrlich and Lui 1998). However, our study differs from the previous work by showing the different implications of various ways of collecting tax revenues for social security. In so doing, it echoes some established results in the literature on taxation and growth, such as the advantage of a lump-sum tax over other taxes, but it differs from the work on taxation by yielding two interesting results when social security is concerned: (i) The lump-sum taxation is distortionary as long as fertility is endogenous, and (ii) labor income taxation is likely to be more conducive to economic growth than consumption taxation. We achieve the results by relating various types of taxation to the financing of social security, wherea s many studies on taxation assume that tax revenues are given back to individuals as lump-sum transfers.

(*.) School of Economics and Finance, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand; E-mail [e-mail].

I would like to thank an editor and two anonymous referees for very helpful comments and suggestions. Responsibility for any remaining omissions or errors is my own.

Received April 1999; accepted March 2000.

(1.) This is only a simplifying assumption; there is no change in results if alternatively we assume that physical capital comes from both bequests and savings.

(2.) Define the cost or price of a child as [P.sub.n], and the price of current consumption as [P.sub.c1], which equals unity. Also define the current period utility as U([[c.sup.t].sub.t], [[c.sup.t].sub.t+1], [n.sub.t]) [equivalent] ln [c.sup.t].sub.t] + [beta] ln [[c.sup.t].sub.t+1] + [rho] ln(1 + [n.sub.t]). Then, we can rewrite Equation II as [P.sub.n]/[P.sub.ct] = [[w.sub.t]v(1 - [[tau].sub.1]) + [q.sub.t] + [b.sub.t+1]/(1 + [r.sub.t+1])]/(1 + [[tau].sub.c]) = [rho][[c.sup.t].sub.t]/(1 + [n.sub.t] = [U.sub.n]/[U.sub.ct] since 1/[[c.sup.t].sub.t] = [beta](1 + [r.sub.t+1])/[[c.sup.t].sub.t+1], where [U.sub.n] and [U.sub.c1] are partial derivatives of U(*, *, *) with respect to [n.sub.t] and [[c.sup.t].sub.t], respectively. The left-hand side of this equation is the cost of a child relative to the cost of consumption, while the right-hand side is the marginal rate of substitution between the number of children and consumption. A rise in either [[tau].sub.1] or [[tau].sub.c] has a direct negative effect in addition to an indirect positive effect through [b.sub.t+1], on [P.sub.n]/[P.sub.c1].

(3.) The substantial declines in fertility might also have resulted from relative price changes rather than changes in tastes. In particular, the rise in females' wage rates relative to males' may lead to declines in fertility.

(4.) Note that the saving rate can now have two solutions when the consumption tax is positive. Thus, the impact of a rise in the rate of the consumption tax on savings (and hence on other variables) may not be monotonic.

References

Barro. Robert J. 1974. Are government bonds net wealth? Journal of Political Economy 82:1095-117.

Becker, Gary S., and Robert J. Barro. 1988. A reformulation of the economic theory of fertility. Quarterly Journal of Economics 103:1-26.

Ehrlich, Isaac, and Francis T. Lui. 1997. The problem of population and growth: A review of the literature from Malthus to cotemporary models of endogenous population and endogenous growth. Journal of Economic Dynamics and Control 21:205-42.

Ehrlich, Isaac, and Francis T. Lui. 1998. Social security, the family, and economic growth. Economic inquiry 36:390-409.

Feldstein, Martin S. 1974. Social security, induced retirement, and aggregate capital formation. Journal of Political Economy 82:905-26.

Kotlikoff, Laurence J., and Lawrence H. Summers. 1981. The role of intergenerational transfers in aggregate capital accumulation. Journal of Political Economy 89:706-32.

Lucas, Robert E., Jr. 1988. On the mechanics of economic development. Journal of Monetary Economics 22:3-42.

Nishimura, Kazuo, and Junsen Zhang. 1992. Pay-as-you-go public pensions with endogenous fertility. Journal of Public Economics 48:239-58.

U.S. Department of Health and Human Services. 1992. Social security programs throughout the world--1991. Washington, DC: U.S. Government Printing Office.

Zhang, Jie. 1995. Social security and endogenous growth. Journal of Public Economics 58:185-213.

Appendix

Derivation of the Steady-State Balanced Growth Solution with Positive Bequests

Updating Equations 4 to 7 by one period, we have [w.sub.t+1]/(1 + r) = [(1 - [theata])/[theta]][K.sub.t+1]/{[L.sub.t+1][1 - v(1 + n)[h.sub.t+1]]} and [K.sub.t+1] = [L.sub.t]S][1 - v(1 + n)][w.sub.t]. Thus, [w.sub.t+1]/(1 + [r.sub.t+1]) = (1 - [theta])[SW.sub.t]/[[theta](1 + n)] or

(1 + g)(1 + n)/(1 + r) = (1 - [theta])s/[theta]. (A1)

With positive bequests, Equations S and 9 imply

(1 + g)(1 + n)I(1 + r) = [alpha]. (A2)

According to Equations A1 and A2, s = [alpha][theta]/(1 - [theta]) as in Equation 12, which is independent of the rates of the lump-sum tax, the consumption tax, and the payroll tax.

The solution for [[gamma].sub.q] in Equation 15 follows Equation 10 by noting that [q.sub.t]/[[c.sup.t].sub.1] = [q.sub.t+1]/[[c.sup.t+1].sub.r+1] = [[gamma].sub.q]/[[gamma].sub.c1] and [[gamma].sub.c1] cancels out from both sides of Equation 10. By Equation 9, [[gamma].sub.c2] = [beta][[gamma].sub.c1], which, together with Equations 1, 2, 12, and 15, provides the solution for [[gamma].sub.b] in Equation 13 by observing that [b.sub.t] = [b.sub.t+1](1 + n)(1 + r)/[(1 + g)(1 + n)(1 + r)] = [[gamma].sub.b][1 - v(1 + n)][w.sub.t](1 + r)/[(1 + g)(1 + n)] = ([[gamma].sub.b]/[alpha])[1 - v(1 + n)[[w.sub.t]. The solution for 1 + n in Equation 14 then follows Equations 9 and 11.

The rate of human capital accumulation is determined by the education technology, Equation 4, and Equation 15:

1 + g = [h.sub.t+1]/[h.sub.t] = A[{[[gamma].sub.q][1 - v(1 + n)](1 - [theta])D[e.sup.[theta]]}.sup.[delta]]. (A3)

Similarly, Equations 4 and 7 determine the rate of physical capital accumulation per worker:

1 + g = [K.sub.t+1]/[L.sub.t+1]/[K.sub.t]/[L.sub.t] = [s(1 - [theta])D/(1+n)][e.sup.[theta]-1]. (A4)

Equations A3 and A4 produce the steady-state physical capital/effective labor ratio:

e = [{s[[D(1 - [theta])].sup.1-[delta]]/A(1 + n)[[[gamma].sup.[delta]].sub.q][[1 - v(1 + n)].sup.[delta]]}.sup.1/[1-[theta](1-[delta])]]. (A5)

Substituting Equation A5 into either Equation A3 or Equation A4 yields the solution for the steady-state balanced growth rate in Equation 16.

Derivation of the Steady-State Balanced Growth Solution with Zero Bequests

When bequests are zero, the first-order condition with respect to bequests cannot hold in strict equality, and thus we use the other equations for the equilibrium. In particular, the solution for [[gamma].sub.q](1 + n) is the same as that in Equation 15.

As in the first part of the Appendix, we have [w.sub.t+1]/(1 + [r.sub.t+1]) = (1 - [theta])[sw.sub.t]/[[theta](1 + n)] and Equation A1. We use them to rewrite [B.sub.t+1]/{(1 + r)[1 + r)[1 - v(1 - v(1 + n)][w.sub.t]} [equiv] [[gamma].sub.B] = (1 + g)(1 + n)([tau] + [[tau].sub.1])/(1 + r) as (1 - [theta]([tau] + [[tau].sub.1])s/[theta]. Aslo Equations 1 to 3 with b = 0, [[gamma].sub.c2] = [beta][[gamma].sub.c1], and Equation 15 mean s + [[gamma].sub.B] = [beta](1 - [tau] - [[tau].sub.l] - s) - [alpha][beta][delta](1 - [[tau].sub.1])/[1 - [alpha](1 - [delta])]. The solution for s follows. The solution for 1 + n is the same as in Equation 14 with [[gamma].sub.b] = 0 and the growth rate the same as in Equation 16.