| Social Security is one of the most important social economic institutions of modern country. To establish perfect social security system which fits with the level of economic development, is necessary requirement of the harmonious development economy and society, and is important insurance of social stability and country's security. Thus, researching social security is very important.In the introduction, the paper firstly introduces the historical background, significance of research and recent research condition of social security system. In the modern sense, social security system began in Germany in the 1880s. In the early of 20th century, theorists began to research in this field, and after that, the scope and methods of research deepened. Since now, it has formed a specific research field which arouses great interest of mathematicians.If the study aims to establish mathematic model for social security system, it is necessary to know the features of various social security systems. So the next chapter, the paper discuss the social security systems of different countries.Finally,the paper concludes four kinds of social security systems with the functions of entertainment and leisure. Also these models are compared and analyzed. In the basis, the author pays more attention to establish social security system with bequest by using AK model, in order to establish mathematic model, some preparing works are done. As follows:This model is supposed that each agent lives for three periods. Each middle-aged agent has exactly one child and each generation has a unit mass. In middle age, everyone receives abequest Bt from his old aged parent, divides one unit of time between leisure Zt and labor Lt,and allocates labor earnings plus his received bequest between middle-ageconsumption Ct and saving St for old age consumption Dt+1.In old age, agents live inretirement and allocate income plus social security benefits between consumption and bequests to children. There is a social security program that taxes labor income at a flat rateτand provides an annuity benefit Tt to each retiree. The agent's budget constraint is given bywhere Wt refers to the wage rate and Rt the interest factor.The preferences are assumed to be altruistic toward children:where Vt stands for the welfare of a middle-aged agent and Vt+1 the welfare of his child. The parameter a measures altruism toward the child,βthe subjective discount factor,σtheinverse of elasticity of intertemporal substitution, andψthe taste for leisure.The production function is given bywhere Y is output, K capital,(?) the average capital per worker, A the productivity parameter, andθthe share parameter of capital. In equilibrium,Kt =(?)t,production factorsare paid by their marginal products:Wt=(1-θ)AKtLt-θ and Rt=θALt1-θ.A social security program may be funded or unfunded, and may or may not link an individual's amount of social security benefits to his own contribution, permitting four typesof programs. The capital market equilibrium in this closed economy requires Kt+1= St withunfunded social security, and Kt+1= St+τWtLt with funded social security.With those preparing works mentioned above, the study begins to establish model for every type of social security system.Unfunded social security with earnings-independent benefitsIn this case, the amount of social security benefits is equal for all retirees: where the upper bar stands for the average value.In order to ease of notation, defineΓX= Xt/Yt for Xt=Bt,Ct,Dt,or St as the ratioof variable X to output Yt.The solution for the proportional allocations of output and time is implicitly given as follows:How does labor respond to a tax rate change? The answer was given by Zhang Jie in her published paper in 2003. Here, the influence needs to be summarized in the thesis.From (7) to (9), the conclusion is drawn as follows:elasticity of intertemporal substitution is less than one,σ≥1.At this time,σ>(?) iscorrect. So whenσ≥1,(?)<0; for the condition ofσ≤(?),(?)is ambiguous.Now, the influence of tax rate change on bequest will be discussed. Partial derivative toτand sorting out on the sides of (8),Whenσ>ψL/(1+ψL) andψL>1-L,(?)>0 is correct, that is to say,ΓB(theratio of bequest to output) increases with the increasing of tax rate. To specialize the condition ofσ,whenσ≥1 andψL>1-L,obviously we can know thatσ>ψL/(1+ψL) andψL>1-L,so (?)>0 is still in effect. However, the welfare impact of a rise in the social security tax rate is ambiguous, as the decline in consumption reduces welfare but the rise in leisure raises welfare (to be dealt with in simulation later). We sum up these results below:Proposition 1. Ifσ>ψL/(1+ψL),a rise in the tax rate for unfunded social security with earnings-independent benefits reduces labor supply, per capita output, consumption, and the growth rate of capital, The impact on welfare is ambiguous; ifσ>ψL/(1+ψL)andψL>1-L,a rise of tax rate increasesΓB.Unfunded social security with earnings-dependent benefitsThe social security budget balance in this case is given asThe parameterφlinks a retiree's past labor income to the amount of his social security benefits.The implicit solution for (ΓB,ΓS,L,R) is given by (7), (9), the interest rate equation, andThe last term in the bracket [?] in the RHS of (14),τ(αR)1/σ/R,captures the relationshipbetween the amount of social security benefits expected by a worker and his wage income..Zhang Jie in her published paper in 2003 discussed the impact of labor supply on tax rate of social security. The general idea will be repeated here.Form (7), (9), (14), we can get equation as follows:where F=-τα1/σR(1/σ)/σ(1-L)(1-θ)[1+(α/β)-1/σZ-ψ(1/σ)/σ][σ-(1-θ)(1-L)(1-σ)]+τα1/σR(1/σ)/σψ(1/σ)L(1-L)(α/β)-1/σZ-ψ(1/σ)/σ.Under the adequate condition, that isσ≥1,we can deduce F<0.Thus (?)<0 as in the previous case. But forσ≥1,the magnitude of (?) is smaller in this case than in the previous case. So the distortions on output andgrowth are smaller too in this case than in the previous one.Now, let's discuss the impact of tax rate change on bequest. We can get the following equation from partial derivative toτon the sides of (14),So forσ≥1 andψL > (1-L)[1-(αR)1/σ/R],(?)>0.To specialize the condition tobeσ≥1 andψL>1-L,we can get obviouslyσ≥landψL>(1-L)[1-(αR)1/σ/R].In thiscase,(?)>0 is still in effect. But the magnitude of (?) is bigger in this case than in the previous case.Proposition 2:Ifσ≥1,a rise in the tax rate for unfunded social security with earnings-dependent benefits reduces labor supply, per capita output, and the growth rate, The impact on welfare is ambiguous. Forσ≥1,the sizes of the tax effects on labor supply, output and growth rate are smaller with earnings-dependent than with earnings-independent socialsecurity benefits. Forσ≥1 andψL>1-L,a rise in the tax rate increasesΓB.And theimpact of tax rate change onΓB in unfunded social security with earnings-dependentbenefits is bigger than in unfunded social security with earnings-independent benefits.Funded social securityThe social security budget balance requiresThe capital market clears when Kt+1=St+τWtLt as aforementioned.The solution for the proportional allocations of output and time is given below: Let (?)S=ΓS+τ(1-θ),(?)B=ΓB+τ(1-θ). Substitute (?) forΓi (i=B,S) in (18)-(20). The(7)-(9) will be get from it. Thus, the impact of tax rate change on saving rates, labor supply, output level, growth rates and bequest are all equal across the two cases, implying also the same welfare.Proposition 3: When benefits are equal for all old-aged agents, funded social security is the same as unfunded social security in terms of aggregate saving, labor supply, output, growth rate of capital, bequest, and welfare.This observation follows directly from proposition1, 2, 3:Corollary: For all old-aged agents, the impact of funded social security and unfunded social security with earnings-independent labor supply, output, and growth rate of capital is bigger than that in unfunded social security with earnings-independent, while the impact of the previous one on bequest is smaller than the latter. In all, the previous one has larger distortions than the latter. |
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