Explaining the Difference between Male and Female in Pension System

Transcription

1 Explaining the Difference between Male and Female in Pension System Yusuke KINAI February 16, 2014 <Please Do NOT Cite; Very preliminary> Abstract There exist some gender differences in regard to pension, consumption profile, wage, and so forth. In this paper, we construct the dynamic general equilibrium model in which households consist of husband and wife, using a model of family life-cycle model developed by Apps and Rees (2010). Then, we attempt to clarify such gender differences using a numerical aid. Keywords: PAYG-type Pension; Gender Difference; Wealth Inequality; Family life-cycle. JEL Classification: E61; H55; D72. The author would like to thank... for helpful comments. The usual disclaimer applies. Correspondence to: Osaka University ykinai@js8.so-net.ne.jp

2 Figure 1 Source: Ministry of Internal Affairs and Communications, Labour Force Survey Figure 2 Source: Ministry of Internal Affairs and Communications, Labour Force Survey 1 Introduction The purpose of the present paper is to clearly the gender difference in pension system. Specifically we establish the model that explains the gender difference in pension system, which is rooted in wealth inequality and the difference in labor supply. First, let us show the summarise the facts regarding consumption profile of male and female. As is well-known, in Japan, the income of household has been strongly dependent on the income of husband. Behind this fact, this is due to the difference in labor supply of male and female. As a result, wage inequality between male and female has also engendered. This observation is related with the Douglas-Arisawa s (second) law. Regarding this fact, recent study ( Bredemeier and Juessen (2013)) clarified that the income of husband and wife are mutually substitute, that is, if the wage of husband lowers, the income of wife becomes higher, and vice versa. Then, we find that it is necessary to clearly how this relationship has emerged in Japan. Motivated by this findings, this paper tries to establish the model that can explain such a fact. Here, let us explain the relationship the past studies and this paper. Since the seminal paper, Bewley (1986), Aiyagari (1994), Krusell and Smith Jr. (1998), there are many studies those study policy effect under the existence of incomplete market. Among others, Krusell and Smith Jr. (1998, 2006) propose an idea that aggregate risk can be approximated by the method of first order moment. Huggett (1996) expanded Aiyagari (1994) into multi-period overlap- 1

3 ping generations model. Since his pioneering studies, there are many studies explained as follows: Because the model of this paper is a variant of overlapping generations (hereafter, OLG) model, we limit the survey to those use that model. As the studies those expand the model of Huggett (1996), we can cite as follows: Weinzierl (2011) shows that by introduction of agedependent taxation is welfare-improving. Kitao (2010) analyzes the effect of consumption taxation and shows that this taxation can be Pareto-improving. Imrohoroğlu, Imrohoroğlu and Joines (2001), Storesletten, Telmer and Yaron (2004), and Heathcote, Storesletten and Violante (2010) investigate the effect of social security on wealth inequality in an OLG model similar to that of Huggett (1996) 1). Yamada (2006) extended the model of Huggett (1996) into a case of Kreps-Porteus type utility function. Nishiyama and Smetters (2011) analyze the effect the difference the cases in which financial resource is covered with consumption tax and labor-income. Benhabib, Bisin and Zhu (2011) show that taxation on capital-income and wealth decreases inequality of income using a stochastic OLG model. In those studies, policy variables are exogenous. On the other hand, as an attempt to endogenize it by introducing politico-economic approach 2), we can cite Mateos-Planas (2008, 2010), They investigate the effect of demographic structure on the tax rate of labor-income and capital -income. With detail, they show that demographic change affects the policy determination. From those past studies, we find that the above studies do not investigate the following topics: Introduction of gender difference Transitional path to steady state with the process of aging and falling birth rate Introduction of Catastrophic (uninsurable) Shock The effect on both intra- and inter-generational link Nardi (2004) The effect of change in fund-raising (from pay-as-you-go to perfect funded) Policy-determination with political factor Extension to two or dimensional variables. (Ex; Retirement Age, Pension Benefit, 1) Note that the difference between Storesletten et al. (2004) and Heathcote et al. (2010) is that the former assumes inelastic labor supply and the latter elastic labor supply. 2) Corbae, D Erasmo and Kuruscu (2009) introduce the setting of Meltzer and Richard (1981) and show the relationship between policy determination and the voter s distribution. 2

4 Contribution to Pension) Among these topics, we specifically focus on the case of extension to gender difference. The above studies have treated households are the representative agent, but those has ignored the gender difference. What is the problem of ignoring the gender gap is summarize into two points; First, it is important to distinguish the kinds of risks although male and female are faced to different risks. Second, the dimension of policy variables tends to be limited although it is necessary to make the result of numerical simulation similar to that of real world. In other words, in my opinion, the analysis based on the single-gender is a bit far from real world in the respect that such an analysis cannot explain the movement in income of households. Regarding this issue, the most close to the present paper is Guner, Kaygusuz and Ventura (2012), which differs from the present paper in the respect that they consider both one-earner case and two earner case. The former corresponds to the case in which households are single and the latter does to the case in which both husband and wife work. In his study, there are two kinds of earner while this paper consider the situation in which husband and wife are faced to different risks. Here, let us present a narrative explanation of the model. In this economy, there are many heterogeneous households and they are faced to income risks. In other words, market is incomplete in the sense that risk-sharing is proceeded only through safety asset. Through this process, wealth inequality emerges. As a result, we show that 1. The rise in income tax has an effect in the sense that it expands the degree in equality. 2. If the weight of pension is higher, the degree of inequality becomes smaller. In what follows, the rest of this paper is organised as follows: In the next section, we set up the model and in the section 3, we describe the parameter value and specification of some functions. In the section 4, we explain the results. Section 5 is the conclusion. 2 Model setup 2.1 Behaviors We consider J-period overlapping generations model. Time is indexed by t and... The population growth rate is n, which is exogenous. We introduce incomplete market into the 3

5 variant of family-life cycle model, which is developed by Apps and Rees (2010). Households consist of husband (male) and wife (female). Unlike an ordinary overlapping generations model in which there are two stages; working and retirement stages, there are four stages as follows: Phase 1. Pre-children Phase Phase 2. Child-Making and Rearing Phase Phase 3. Post-children Phase Phase 4. Retirement Phase In the phase 1, both husband and wife work and do not have any children. In the second stage, they have children and bring them up. Under this phase, only male work and female stop working. In the third phase, as in phase 3, both husband and wife work, although wife has a risks of being unemployed. In the final phase, both husband and wife are retired. There exist heterogeneity among households (ex ante identical agent). In this paper, we consider the case in which policy determination is exogenous. We assume the existence of incomplete market, that is, risk-sharing is achieved only through safety assets. 3) We consider constant population growth and households are faced to mortality risks, and we assume exogenous retirement age. Specifically, we assume that households retire at the end of third stage. The government has two kinds of policies, that is, PAYG-type Pension and child allowance Households Households consist of male (husband) and female (wife). Households live J periods. For both male and female, agent s utility is denoted as U(c t,1 l t ), which means utility depends on consumption and leisure. There are two kinds of constraints, that is, budget constraint and time constraint, explained in the later. In this paper, we assume households hold asset a A R + for future decision making. Moreover, we assume that only male works in the second phase, while in the first and third stage, both husband and wife work. However, as explained with detail in the later, both husband and wife face to two kinds of risks, respectively. By letting β denote the discount factor, the whole objective function of household is written 3) If Arrow securities are available, households can share risks, therefore, the inequality does not emerge. 4

6 as { } J E t β j 1 Π t j=1 s tu[u m (c j,1 l j ),u f (c j,1 l j )] t=1 (1) where c t, l t and s t denote consumption, labor supply, and survival rate respectively. The subscript m and f denote male and female, respectively. Here, note that this problem is recursive, Then, using the method of dynamic programming, the decision making is written as v(x t ) = max c,a {U(c,1 l) + βev (x )} (2) where x t (a t,z t ) X = A Z and the subscript means the variable at the next period. Here, Z is also subset of R +, that is, z Z R +. As already stated, households face four phases. Then, budget constraints and time constraints of each phases are explained as follows: First Phase :0 t 1 In this phase, both husband and wife choose to working and make saving 4), and they do not have children, and both labor-income and saving are subject to taxation. Therefore, the budget constraints is written as follows: c m t + c f t + a t+1 = (1 η)w t (l m t + l f t ) + (1 τ(1 + r))a t (3) Therein, η and τ denote labor-income and capital-income taxes, respectively. Second phase : j 1 t j 2 Child is born and risen up by parents in the second phase. Under this phase, they receive child allowance. In this period, wife does not work and only husband works. Then, the budget constraint is expressed as c m t + c f t + a t+1 = (1 η)w t l m t + (1 τ(1 + r))a t + P t, (4) where p t denotes child allowance. Moreover, because in this phase, wife has to bring up children and husband has to work, the time constraint is written as l m t + e t n t = 1 (5) where l m t, n t, and e t denote labor time, the number of children and educational time, respectively. 4) This type of saving is categorized as precautionary savings. 5

7 Third phase : j 2 t j 3 Children leave home and have no demand to parents. Only parents work. However, wife has a risk of unemployment. However, at the beginning of this phase, wife has a risk of unemployment. (The behavior is same as in the first phase.) Budget constraint and time constraint are written as follows: c m t + c f t + a t+1 = (1 η)w t (l m t + πl f t ) + (1 τ(1 + r))a t (6) As already said, wife is faced to unemployment risk at beginning of this phase. Letting pi denote probability of unemployment, the transitional equation is denoted as follows 5) : π = ( π uu π ul ), (7) in which two subscripts u and l denote the states of being unemployed and employed, respectively. Forth phase : j 3 t J In the final phase, neither husband nor wife work and retire. They receive pension. Then, the budget constraint is written as, where d t denote transfer from the government, that is, pension. c m t + c f t + a t+1 = (1 + r)a t + d t, (8) Here, note that in our setting, state variable is asset (a t ), time(t), and productivity (z t ) therefore, the policy function is denoted as a function of asset, therefore, the policy function is a function of a t and z t. Finally, it is necessary to impose the borrowing constraint: a t 0. (9) This equation means that household cannot borrow as much as they want. In other words, there exists upper bound with regard to the amount of borrowing. To summarize, the problem for household is written as V (x) = max c t,l m.l f,a {u m(c,1 l) + u f (c,1 l t ) + βev (x )}, (10) 5) Note that wife do not work, that is, unemployed in the second phase. 6

8 where V ( ) is the value function. Constraints are summarised as follows: First (0 t j 1 ) : c m t + c f t + a t+1 = (1 η)w t (l m t + l f t ) + (1 τ(1 + r))a t (11a) Second ( j 1 t j 2 ) : c m t + c f t + a t+1 = (1 η)w t l m t + (1 τ(1 + r))a t (11b) Third ( j 2 t j 3 ) : c m t + c f t + a t+1 = (1 η)w t (l m t Forth ( j 3 t J) : ct m + ct f + a t+1 = (1 + r)a t + d t Borrowing Constraint : a t 0. + πl f t ) + (1 τ(1 + r))a t (11c) (11d) (11e) Time Constraint : l m t + n t e t = 1 (11f) Demographic Structure Its transition equation is denoted as In this economy, the demographic structure alters as time passes. N t+1 = s t N t. (12) where s t denotes survival rate, which means the probability that alive agents at t 1 period, which transitional equation is denoted as µ j+1 = s j n µ j, (13) µ = [µ 1,..., µ J ] R J, in which µ i denotes the number of population. Total number of population is assumed to be 1, that is, J µ j = 1. (14) j= Firms Production Function is denoted as Y t = z t K α t L 1 α t (α (0,1)), where z t z is productivity shock, which follows Markov process. Then, factor prices are denoted as w t = z t αkt α 1 (15) R t = z t (1 α)kt α 1. (16) As already denoted, we assume that firms are exposed to idiosyncratic risk, that is, productivity shock. 7

9 2.1.3 Government The government collects the tax revenue and gives it in forms of pension and child allowance. Exogenous public expenditures G, which is covered with labor-income and capitalincome taxes. The income tax code is endogenous T (y L,y K ), that balances the overall government budget. In the policy experiment, the function distinguishes between sources of income. We assume Exogenous social security system self-financed by pay capital- and labor-income tax. Therefore, two budget constraints are imposed on saving and labor-income of alive workers, which is distributed to younger generation in a form of pay-as-you-go type pension and child allowance. Letting θ (0, 1) denote the exogenous parameters, which also is a policy variable, the budgets are written as follows: Pension child allowance G t = θ (ηw t L t + τra t )dγ (17) A Z ( P t = (1 θ) τr t a t + A Z ηw t l m t ) dγ (18) Remark If we assume that determination is endogenous, that is determination through political process, the main key point is distribution of voters. In other words, the content of inequality matters. In this paper, we assume that policy determination is based on majority voting. Regarding that point, distribution of voters matters. The tax rate of next period is contingent on the today s distribution, namely, the distribution and policy determination is mutually related. 2.2 Market Equilibrium We have to define each market equilibrium, that is, capital, labor, and goods market. Capital Market: Labor Market: J µ t a t dγ = K (19) j=1 A Z L = T t=0 l m t + T t=0 l f t (20) 8

10 Goods Market: J J µ t c t dγ + µ t a t+1 dγ = Y t (21) j=1 A Z j=1 A E Then, let us define the equilibrium concept. Defnition 1 Recursive Competitive Equilibrium: RCE (Note that taxation is given in this case.) Households: Under eqs.(11), It satisfies the Bellman equation eq.(10) and g(a, z) is a policy function. Firms: Factor prices are determined as eqs.(15) and (16). Government: The budget constraint of the government; eq.(17) and eq.(18) hold. Market-clearing condition: Each market clears. Distribution stationarity: Distributive function; Γ is stationary. 3 Parameters and Functions Before entering into calibration, we have to set up some parameter values and specify the some functions. In this section, let me describe how we set up some values and functions. Utility Function 6) : Production Function U(c,l) = c(1 lm t ) 1 σ 1 1 σ Y t = K α t L 1 α t (α (0,1)) + c(1 l f t ) 1 σ 1 1 σ Setup-of-Parameters The parameter value is set as table.1. Following Hayashi and Prescott (2002), we specify capital share; α = 0.382, and the discount factor, β = The rate of employment is set as The value of σ is adopted from Yamada (2006). Regarding τ, η, and θ are policy variable, therefore, these values are treated as experimental (i,e, tentative) values. Regarding the ages ( j 1, j 2, j 3,J), we set as in Table 1. In the next section, we investigate the effect of policy (especially, τ and θ)on inequality. 6) Yamada (2006) employed the Kreps-Porteus type utility function. 9

11 Table1 The Value of Parameters β σ 0.38 α τ 0.25 n 0.05 η 0.20 θ 0.3 π 0.95 j 1 20 j 2 45 j 3 65 J 85 Remark: Property of Variables In this model, there are many variables. Then, it is useful to distinguish the property of variables. First, control variables are c, a t+1 and lt(i{m, i f }). On the other hand, State variables are age; t, asset:a t, and labor productivity z t. Moreover, policy variables are τ, η and θ. 4 Computation: Policy Experiment 4.1 Algorithm In this paper, we employee the method of VFI (Value function Iteration). This method is based on the property of convergence. The algorithm is summarised as follows 7) : Step1: Choose Policy variables. Step2: Make initial guess with regard to a 0, R and w. Step3: Solve the household s optimisation problem as a function of the individual and state variables using VFI method. Step4: Compute the steady-state distribution. Step5: Compute the variables those satisfy the solution of households decision problem and the budget constraint of the government. Step6: Iterate Step.4 and 5 until the parameters converge. 7) This is based on the (Heer and Maussner, 2009, Chapter 10). 10

12 Figure 3 τ = 0.05 Figure 4 τ = 0.2 Figure 5 µ = 0.15 Figure 6 µ = Results The figure of results are listed fig.3 6. From these figures, in summary, we find that The rise in capital-income tax has an effect in the sense that it expands the degree in equality. If the weight of pension is higher, the degree of inequality becomes smaller. The first result means that capital-income has a role of expanding the inequality. On the other hand, the second result means that the rise in mu do not worsen the degree of inequality. 11

13 5 Conclusion In this paper, we study the effects of social security taxation using a variant of family life-cycle model. What we showed in this paper is summarised as follows: The rise in income tax causes expands the degree of inequality. If the weight of pension is higher, the degree of inequality is smaller. Because the content of pension system in Japan is very complicated, it is difficult to calibrate the Japanese economy, but this is challenging. Moreover, in this paper, policy determination is exogenous. As a next step, it is meaningful to make it endogenous. Appendix A Appendix In this appendix, let us explain the case in which policy determination is endogenous. In our setting, alive agents at t period determines the tax rate on t +1 period. From another angle, the determination of tax rate and distribution are mutually determined as in Corbae et al. (2009) or Azzimonti, de Francisco and Krusell (2008). Namely, the following relation holds. The tax rate of next period is dependent on the today s distribution. τ = Ψ(Γ,τ) (22) On the other hand, the distribution of next period depends on today s distribution and tax rate: Γ = Φ(Γ,τ) (23) By repeating this, the problem is written as follows: maxv ( ;τ) = max{u(c t,l t ) + βev (;τ )} s.t. (11) anda t+1 0. Defnition 2 Politico-Economics Recursive Competitive Equilibrium: PERCE RCE Condition: The conditions of RCE hold. One-shot Deviation Principle: If the government deviates from..., the solution to the following problem is optimal. 12

14 Majority voting: A E,τ dγ > 1 2 By numerical method, all we have to do is to derive the variables those satisfy the above condition. References Aiyagari, S Rao (1994) Uninsured Idiosyncratic Risk and Aggregate Saving, The Quarterly Journal of Economics, Vol. 109, No. 3, pp Apps, Patricia and Ray Rees (2010) Family labor supply, taxation and saving in an imperfect capital market, Review of Economics of the Household, Vol. 8, No. 3, pp , September. Azzimonti, Marina, Eva de Francisco, and Per Krusell (2008) Aggregation and Aggregation, Journal of the European Economic Association, Vol. 6, No. 2-3, pp , Benhabib, Jess, Alberto Bisin, and Shenghao Zhu (2011) The Distribution of Wealth and Fiscal Policy in Economies With Finitely Lived Agents, Econometrica, Vol. 79, No. 1, pp , 01. Bewley, Truman F. (1986) Stationary Monetary Equilibrium with a Continuum of Independently Fluctuating Consumers, in W. Hildenbrand and A. Mas-Colell eds. Contributions to Mathematical Economics in Honor of Gerard Debreu: Amsterdam: North-Holland. Bredemeier, Christian and Falko Juessen (2013) Assortative Mating and Female Labor Supply, Journal of Labor Economics, Vol. 31, No. 3, pp Corbae, Dean, Pablo D Erasmo, and Burhanettin Kuruscu (2009) Politico-economic consequences of rising wage inequality, Journal of Monetary Economics, Vol. 56, No. 1, pp Guner, Nezih, Remzi Kaygusuz, and Gustavo Ventura (2012) Taxation and Household Labour Supply, Review of Economic Studies, Vol. 79, No. 3, pp Hayashi, Fumio and Edward C. Prescott (2002) The 1990s in Japan: A Lost Decade, Review of Economic Dynamics, Vol. 5, No. 1, pp , January. Heathcote, Jonathan, Kjetil Storesletten, and Giovanni L. Violante (2010) The Macroeconomic Implications of Rising Wage Inequality in the United States, Journal of Political Economy, Vol. 118, No. 4, pp , 08. Heer, Burkhard and Alfred Maussner (2009) Dynamic General Equilibrium Modelling, NewYork and London: Springer. Huggett, Mark (1996) Wealth distribution in life-cycle economies, Journal of Monetary Economics, Vol. 38, No. 3, pp Imrohoroğlu, Ayşe, Selahattin Imrohoroğlu, and Douglas H. Joines (2001) Computing Models of Social Security, in R.Marimon and B.Scott eds. Computational Methods for the Study of Dynamic Economies, Cambridge: Oxford University Press, Chap. 10, pp Kitao, Sagiri (2010) Short-run fiscal policy: Welfare, redistribution and aggregate effects in the short and long-run, Journal of Economic Dynamics and Control, Vol. 34, No. 10, pp Krusell, Per and Anthony A. Smith Jr. (1998) Income and Wealth Heterogeneity in the Macroecon- 13

15 omy, Journal of Political Economy, Vol. 106, No. 5, pp (2006) Quantitative Macroeconomic Models with Heterogeneous Agents, in W. Newey R. Blundell and T. Persson eds. Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress: Cambridge University Press, London, pp Mateos-Planas, Xavier (2008) A quantitative theory of social security without commitment, Journal of Public Economics, Vol. 92, No. 3-4, pp (2010) Demographics and the Politics of Capital Taxation in a Life-Cycle Economy, American Economic Review, Vol. 100, No. 1, pp Meltzer, Allan H and Scott F Richard (1981) A Rational Theory of the Size of Government, Journal of Political Economy, Vol. 89, No. 5, pp Nardi, Mariacristina De (2004) Wealth Inequality and Intergenerational Links, Review of Economic Studies, Vol. 71, pp , 07. Nishiyama, Shinichi and Kent Smetters (2011) Analyzing Fiscal Policies in a Heterogeneous-Agent Overlapping-Generations Economy. forthcoming in Handbook of Computational Economics. Storesletten, Kjetil, Christopher I. Telmer, and Amir Yaron (2004) Consumption and risk sharing over the life cycle, Journal of Monetary Economics, Vol. 51, No. 3, pp , April. Weinzierl, Matthew (2011) The Surprising Power of Age-Dependent Taxes, Review of Economic Studies, Vol. 78, No. 4, pp Yamada, Tomoaki (2006) On the Effect of Income Distributive Policy by PAYG-type Pnesion on Wealth Inequlity (in Japanese), JCER, Vol. 55, No. 1, pp