The Mathematical Model Of Optimal Debt In Social Security With Altruistic Bequests

Government bond is an important means for macro-controlling. Bonds must be repaid by the state finances. Therefore,the bonds must be issued at an appropriate scale. There is important practical significance to determine the optimal debt scale at pay-as-you-go system. Zhang Jie(2003)supposes that agents live for two periods: young and adulthood,and studies the optimal debt when fertility and education subsidyratio reach the optimal level simultaneously. But he did not consider the consumption and utility of agents' when they retired.In this paper, we divide the life cycle of agents' into three periods:young,adulthood and old,and take the consumption and utility of agents' when they retired into consideration.We study the optimal debt when fertility and education subsidy ratio reach the optimal level simultaneously.We consider the two-sector growth model which has an infinite number of periods and overlapping-generations of agents who live for three periods. Young agents study and consume. In the adult period, each adult agent has one unit time endowment: work and rear children, and choose income allocation and the number of identical children. We assume the preference of an adult agent aswhere C_td^t stands for the adult agent's consumption when he works, C_t+1^t the adult agent's consumption when he retires. V_t+1 children's utility,ρ the taste for the number of children,andαthe taste for per child welfare. The adult agents are altruistic. They only enjoy the process of rearing children and do not seek return.At the beginning of t period, each adult agent receives a bequest plus interest a_tr_t from his parents, and leave a bequest to each child at the end of t period.The wages of adult agents and bequests from his parents spent on consumption, children's education,and bequests to children. [Fuster 1999]show that social security has no influence on saving rate. So we do not consider deposit (the saving rate is 0). The consumption of adult agent when they work and retire aswhereπ_c stands for consumption taxes,τthe rate of social security tax, s the rate education subsidies, and T social security income of each agent when they retire. The old agents live on pension alone.The government issues one-period bonds and consumption taxes to finance debt repayment and education subsidieswhere b_t stands for the amount of outstanding debt per adult agent. For tractability,we assume that the debt-output ratio, the subsidy rate and consumption-tax rate are all stationary.The social security income depend on a common factor g_t and his contribution fr_t (1 - vn_t-1) w_t-1h_t-1. where f is a policy parameter that link social security income with wage when he worked.In the pay-as-you-go system, the total income of social security L(t - 1)T_t are equal to the social security of the next generation paidwhere g_t is the common factor, that is minimum living security generally.The households' problem in adult' utility function by choice is for-mulated asFor national convenience, we set (?)(?). The competitive equilibrium has the following solutionIn addition,when the debt-output ratio is maintained at its initial level, the government budget constraint is balanced by To determine optimal debt,an essential step is to express the welfare in terms of the debt-output ratio and the initial state (k₀, h₀),as followsThe social planner's problem is to maximizeV_tand subject toAccording to the relationship between households" problem and the social planner's problem, we obtain an proposition by letting households' equal to social planner's problem.Proposition 1Debt and education subsidies financed by consumption taxes achieve the first-best outcome such thatif and only ifThe competitive equilibrium solution is first-best at (?). So we obtain the analytic expression of the first-best debt-output ratio.Analog the parameter, we can show: when education externality is very strong(such asβ= 0.1), the optimal education subsidy ratio is about to 70%, education subsidy-output ratio is approximately 6%, and the optimal debt-output ratio is approximately 27%.