Money and Pay-As-you-Go Pension

In an aging society with fewer children, a pay-as-you-go pension system presents severe diﬃcul-ties. A decrease in the share of working people among the population raises the burden for pensions per capita to maintain a constant replacement ratio of pensions. This burden reduces capital accumulation. Therefore, income growth is prevented. The analyses in this paper demonstrate that if the replacement rate of pension is high, a decrease in population growth reduces the income growth rate even if a decrease in population growth can raise the income growth rate per capita because the capital stock that the workers can use increases. However, by setting an appropriate monetary policy for decreasing population growth, the income growth is not prevented by an increase in the burdens for pensions. The negative eﬀect of the burden for pensions on income growth can be eliminated by the change of the money supply rate in the long run.


Introduction
In an aging society with fewer children, pension sustainability is an important topic. With pay-as-you-go pension systems, incoming funds paid by younger people are distributed to older people as pensions in a single period. Therefore, fewer children implies a decrease in the number of working people in the future.
The per-capita pension burden on younger people becomes larger or the per-capita pension benefit for older people becomes smaller.
[Insert Fig. 1 around here.] Fig. 1 shows the elderly population rate in some OECD countries. As shown in Fig. 1, the elderly population rate continues to increase. Because of an increase in the elderly population rate, younger people must pay for their government to provide a social security system that includes programs such as public pensions and medical services. As shown in Fig. 2, social security contributions continue increasing in some OECD countries. Especially, in Japan, which is highest the elderly population late in the world, social security contributions are steeply increasing.

around here.]
To maintain the pension benefit, the government must raise the contribution rate for pensions: the burden for younger people should be raised. However, without increasing the contribution rate, the pension benefit can be raised. Fanti and Gori (2010) report that a decrease in the contribution rate for pensions raises the pension benefit because capital accumulation is facilitated by increased saving. As shown by the overlapping-generations model, a pay-as-you-go pension reduces savings, thereby inhibiting capital accumulation. A decrease in the contribution rate increases capital accumulation. Consequently, the wage rate increases. The government can collect sufficient revenue to provide pension benefits.
Other ways to increase the pension benefit without increasing the burden for younger people might be used. A child-care policy such as a child allowance might be used. Van Groezen, Leers, and Meijdam (2003) derive that child allowance can raise fertility. An increase in fertility increases the future number of working people. Therefore, the pension sustainability is strong. Abío, Mahieu and Patxot (2004) and Fenge and Meier (2005) consider an incentive policy by which the pension benefit depends on the number of children in the household. This policy has a positive effect on population growth.
As these related studies have demonstrated, some means to maintain or raise pension benefits in an aging society with fewer children society exist. However, other policies can be considered for pensions: monetary policies. This paper sets an endogenous growth model with money in the utility and pay-asyou-go pension and examines how a decrease in population growth affects the income growth rate and the inflation rate.
The main results derived by the analyses described in this paper are presented as follows. The effect of a population growth rate decrease depends on the pension contribution rate. With a low contribution rate, a decrease in the population growth rate raises the income growth rate. Even if a decrease in the number of working people impedes capital accumulation because of an increase in the per-capita burden for pension benefits, a decrease in the number of working people raises the per-capita capital stock and raises the rate of productivity. The income growth rate increases if the latter effect is large.
This paper presents examination of the effect of monetary policy on the income growth rate and the inflation rate. An increase in the money stock raises both the income growth rate and the inflation rate on a balanced growth path. An increase in the inflation rate reduces the demand for money stock and increases the demand for investment. An increase in investment raises the income growth rate because of capital accumulation. Moreover, this paper presents derivation of an endogenous monetary policy.
Given a high replacement rate of the pension, a decrease in the population growth rate reduces the income growth rate. Nevertheless, by virtue of an increase in money stock, the income growth rate can be constant. Even if a decrease in population growth and the burden for pensions increases, appropriate monetary policy measures can cancel the negative effects on the income growth rate.
Some studies related to monetary policy exist. Mino and Shibata (1995)  derive an optimal monetary policy. Gregorio (1993) also refers to an optimal monetary policy.
As derived herein, Corneo and Marquardt and Ono (2010) show that a pay-as-you-go pension reduces the income growth rate, too. These papers do not consider monetary policy measures. Yakita (2006) sets an endogenous growth model with monetary policy and examines how increased life expectancy affects the income growth rate and the inflation rate. Depending on the parameter conditions, the effects of increased life expectancy on the income growth rate and the inflation rate are determined.
No report in the relevant literature describes consideration of monetary policy in a pay-as-you-go pension model. This paper presents a derivation showing that the effect of fewer children on the income growth rate depends on the contribution rate for pensions.
The remainder of this paper presents the following. Section 2 explains the model setting. Section 3 derives the equilibrium. Section 4 presents examination of the respective effects of fewer children on the income growth rate and the inflation rate. Section 5 describes derivation of monetary policy measures that might be used to maintain the income growth rate or the inflation rate. The final section concludes this paper.

The Model
Three types of agents exist in this model economy: households, firms, and a government. This model economy is assumed as a two-period overlapping-generations model.

Household
Individuals in the household live in two periods: young and old. In any t period, younger people and the older people exist simultaneously. Younger people supply labor inelastically according to units of time. The individuals in the household care about consumption during the younger period c 1t , the consumption in older period c 2t+1 , and the real money stock per capita m t . These variables are assumed as real variables. The utility function in this paper is assumed as the following log utility function with money in the utility as assumed by Sidrauski (1967), Walsh (2010) and others In that equation, α denotes the preference for money. Small α shows that the preference for money is large. Also, ρ denotes the discount factor. During the young period, households supply labor to obtain the real wage income w t and pay the burden for public pension τ t w t , where τ t denotes the contribution rate for pension. 1 In the older period, the older people can receive the real pension benefit Z t+1 . Then, 1 The analyses in this paper assume an inelastic labor supply and that the wage tax does not distort household allocations.
the budget constraint in the young period is where s t denotes the real savings. 2 The budget constraint in old period is where 1+π t and 1+r t+1 respectively denote inflation rate between t and t+1 period and the real interest rate. 3 Then, using (2) and (3), lifetime budget constraint can be derived as The optimal allocations to maximize the utility function (1) subject to budget constraint (4) are derived as We assume (1 + r t+1 )(1 + π t ) > 1 to hold a positive demand for money stock.

Firms
This model includes the assumption of the final goods production function as the following equation: where Y t denotes final goods. Final goods Y t are produced by inputting capital stock K t and labor L t .
A t denotes the productivity. Assuming a perfectly competitive market, the real wage rate w t and the real interest rate 1 + r t+1 are given as 2 Assuming Pt as the price level, Ptst = (1 − τt)Ptwt − Ptc 1t − Ptmt is obtained. This real budget constraint in the young period is derived by omitting the equation by P t . 3 The nominal budget constraint is P t+1 (Fisher equation) and omitting P t+1 , the budget constraint in the older period is derived. This paper presents consideration of the Romer type externality given by Romer (1986) is assumed. This setting is given by Grossman and Yanagawa (1993). a denotes a positive parameter.
Then, the wage rate and real interest rate are given as where k t ≡ K t L t . The capital stock is assumed to be depreciated in a single period.

Government
The government supplies money at the rate of µ(> 0). Assuming an aggregate money stock as M t , then the money stock in the next period is where P t denotes the price level in t period and N t denotes the total population size of younger people in t period. Then, the older population size is N t−1 in t period. Assuming the population growth rate as Nt+1 N t = 1 + n, then the dynamics of money stock m t is The government can use seigniorage when issuing currency. The analyses in this paper subsume that the seigniorage is used for non-productive government expenditure.
In addition to the money supply, the government runs a pay-as-you-go pension system. The government collects revenue from younger people at the contribution rate τ t and provides pension benefit Z t for older people. Now, considering balanced budget constraint, Z t is assumed as presented below: Therein, represents the replacement rate of pension (0 < < 1). The government collects the tax revenue to provide fixed pension benefit w t , such as a defined benefit pension. Then, the pension benefit i.e., the following equation is reduced as The analyses in this paper assume that the contribution rate is constant over time. The contribution must be pulled up to provide a fixed pension benefit if low fertility brings about a decrease in population growth rate. This paper presents consideration of the defined benefit pension. Such an increase in the burden might occur in OECD countries. In Japan, the contribution rate for pension continues increasing because of an aging society. 4

Equilibrium
The equilibrium of this model economy can be specified by three market equilibria of the monetary market, labor market, and capital market. Because of the inelastic labor supply, N t = L t and (11) provide the labor market equilibrium.
The capital market equilibrium is derived as K t+1 = N t s t , i.e., (1 + n)k t+1 = s t . s t is given as (5), (7), (11), (12), (14), and (15), the dynamics of capital stock per capita k t is derived as shown below: This paper presents consideration of the equilibrium of balanced growth path. Now, when the growth rate is defined as 1 + g (g > 0), both the growth rate of capital stock per capita kt+1 k t given by (16) and that of mt+1 mt given by (13) are equal to 1 + g. 5 Considering (7), (9), (14), and (15), the ratio of mt kt is given as In the balanced growth path with constant inflation rate π, m t k t is also constant. Then, the growth rate of m t+1 m t is 1 + g.

Fewer Children Society
This section presents an examination of how fewer children, i.e., a decrease in population growth rate affects the income growth rate and inflation rate. The income growth rate 1 + g and the inflation rate π 4 In Japan, the contribution rate continues increasing to 18.3% by 2017.(Data: Ministry of Health, Labour and Welfare, Japan) 5 Considering (13) and (16) is reduced. The growth rate 1 + g can be obtained uniquely to hold the equation.
in balanced growth path are given as Total differentiation of (18) and (19) by g, π and n at the balanced growth path reduces the following equations as 6 The effects of decreased population growth (fewer children) on the income growth rate and inflation rate in the balanced growth path are where det = 1+g 1+π φ 1 + φ 2 > 0. The signs of these equations are ambiguous because the sign of φ 3 is ambiguous. The first term of φ 3 presents the positive effect of an increase in population growth on the income growth. An increase in population growth reduces the burden of pensions for younger people. Then, the younger people can increase the savings. Therefore an increase in capital accumulation facilitates income growth in an increase in population growth n. However, the second term of φ 3 shows the negative effect of an increase in population growth on the income growth. An increase in population growth makes the population size large and reduces the capital stock per capita. Then, the income growth per capita 1 + g is reduced. The positive effect on income growth shows the effect of a decrease in burdens for pension. Without , an increase in population growth brings about negative effects on income growth. 6 See Appendix for a detail proof.
If φ 3 > 0, i.e., the positive effect on income growth is larger than the negative effect on income growth, then an increase in population growth brings about dπ dn < 0. However, even if the sign of dπ dn is determined, the sign of dg dn remains ambiguous because of the change in demand for money m t . This point demonstrates that the inflation rate decreases in an increase in population growth n if dπ dn < 0 is obtained. A decrease in the inflation rate increases the demand for money, as shown by (7), because of an increase in benefits from holding money. The household decreases savings s t . Capital accumulation decreases because they hold more cash. Actually, φ 2 shows the effect of the demand for money on income growth.
Then, how does a decrease in population growth affect the income growth and the inflation rate with a pay-as-you-go pension? I define * to hold φ 3 = 0. 7 φ 3 > 0 and dπ dn < 0 is obtained if * < < 1. That is, a decrease in population growth increases the inflation rate. However, the effect on the income growth is ambiguous because φ 3 > 0 and φ 2 > 0.
The sign of dg dn is always negative because φ 3 < 0 if < * . Then, a decrease in population growth increases income growth because the workers can use more capital stock per capita. However, the sign of dπ dn is ambiguous because φ 1 > 0 and φ 3 < 0. Then, the proof above reduces to the following proposition.

Monetary Policy
This section presents an examination of how monetary policy affects the income growth and inflation rate.
Moreover, the following subsection presents examination of endogenous monetary policy. Concretely, we derive the appropriate monetary policy to maintain the income growth rate or the inflation rate in an environment with a decreasing population growth rate. As derived in the preceding section, a decrease in the population growth rate might give negative effects on the income growth rate. This research shows how the government should provide the money stock in a society with fewer children.

Effect of Monetary Policy
This subsection presents effects of monetary policy on the income growth rate and the inflation rate.
Total differentiation of (13) and (16) by g, π, and µ presents the following signs: These results are intuitive. An increase in the money supply increases the inflation rate based on the 'quantity theory of money'. An increase in the inflation rate decreases the demand for money stock.
Then, households increase savings or investment for goods. Therefore, capital accumulation is facilitated.

Endogenous Monetary Policy
This subsection derived the endogenous monetary policy to maintain the income growth rate or the inflation rate. The income growth rate and inflation rate might change if the population growth rate decreases. This subsection presents consideration of a policy to maintain the income growth rate or the inflation rate by changing the money supply rate µ with a decreasing population growth rate n.
First, we derive the appropriate monetary policy to maintain the income growth rate dg dµ = 0. A decrease in the income growth rate reduces the income level in the future period. The welfare of future generations might decrease. Consequently, there exists an appropriate reason to maintain the income growth rate. From total differentiation of (18) and (19) by π, n and µ, the following signs are obtained: These signs are ambiguous. However, without the burden for pension , φ 3 is invariably negative. Therefore, dµ dn > 0 and dπ dµ > 0. However, considering a pay-as-you-go pension, the sign of dµ dn is ambiguous. Now, this paper presents consideration of a case for which φ 3 > 0, by which a decrease in population growth negatively affects the income growth rate by an increase in the per-capita burden for pensions. A decrease in population growth is expected to raise the supply rate of money stock to maintain the income growth rate if φ3 φ2 > 1+µ 1+n . A decrease in the population growth negatively affects the income growth rate because of an increase in the pension burden. The inflation rate must be raised to recover the income growth.
However, if φ 3 φ2 < 1+µ 1+n , then a decrease in population growth can be expected to reduce the supply rate of money stock to maintain income growth. A decrease in population has a positive effect on the inflation rate, as shown by (19) for given 1 + g. A high inflation rate accelerates investment. Then the income growth is higher than the initial growth rate. The supply rate of the money stock should be reduced to prevent income growth. The inflation rate is given as (25) to maintain the income growth rate.
Second, we derive an endogenous monetary policy to maintain the inflation rate dπ dµ = 0. An increase in the inflation rate entails higher costs such as shoe costs and menu costs. Therefore, the inflation rate should not be raised. Total differentiation of (18) and (19) by g, n, and µ gives the following signs: Here, φ 3 > 0 is assumed. Then, both dg dµ and dµ dn are positive. A decrease in population growth is expected to reduce the supply rate of money stock, as shown by (26). Considering (19), a decrease in population growth increases the inflation rate. The money supply rate is expected to decrease, not to increase the inflation rate. Because of a decrease in the money supply, the income growth rate is reduced because the burden for pension increases. Then, the following proposition is established.
Proposition 2 With a decrease in population growth rate n, the government sets the monetary policy to hold (24) to maintain the income growth rate. Then, the inflation rate is changed, as shown by (25).
The monetary policy is set as (26) if the government wants to maintain the inflation rate. Then, the income growth rate is given as (27).
Considering (24)-(27) in φ 3 > 0, the government should adopt some policy to maintain the income growth rate because the inflation rate might decrease. The effects of these policies are determined by the pension replacement ratio .

Conclusion
This paper presents an examination of how a decrease in the population growth as shown in an aging society with fewer children affects the income growth rate and the inflation rate in the balanced growth path in the model with money in the utility and pay-as-you-go pension. A decrease in population growth has a positive effect on income growth because the workers can use more capital stock per capita. However, considering the pension burden, decreased population growth negatively affects income growth. Public pensions have a negative effect on capital accumulation, as shown by many related studies. The per-capita pension burden decreases. Capital accumulation might increase if younger people increase in number as a result of child care policies. The government might decrease the pension burden for younger people and cut benefits for older people to maintain the income growth rate. However, considering monetary policy, even if a decrease in population growth decreases the income growth rate, income growth can be expected to recover by virtue of an appropriate monetary policy.
Defining L = 1 + g and R = 1+ρ−α− , L and R has a unique intersect and the growth rate in balanced growth path 1 + g * is obtained, as shown in Fig. 3.

around here.]
The sign of φ 3 This appendix shows the sign of φ 3 . I define the following equations: L increases with an increase in .R decreases with an increase in because an increase in reduces g as shown in Fig. 1. The sign of φ 3 is negative becauseL <R if = 0. I think that the range of is between 0 to 1 + n not to be negative saving.L >R becauseR becomes zero if = 1 + n. I can obtain φ 3 > 0 and * to hold φ 3 = 0 in 0 < < 1 + n.