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The social security system has been 100 years of history since its establishment. Atthe basic livelihood needs of the members of society, it plays an important role. And thesystem of pay-as-you-go social security has been relatively perfect. The large majority ofthe world’s countries have adopted the social security system . Because China has enteredthe aging society since the end of the last century, the problem of the old age pension hasbeen attracted to the people’s attention. We use a mathematical model to simulate the pay-as-you-go social security system approximately in the real society. And the conclusions areimportant significant.In the first chapter, the paper firstly introduces the basic knowledge and the backgroundof the social security system.In the second chapter we use the mathematical model to simulate the pay-as-you-gosocial security system. The utility function is given byBecause of statistics of the average retirement age and the average life expectancy in ourcountry and other western countries, we use the form ofα>ε.Production function is given by the Cobb-Douglas production functionwhere Y is aggregate output, K aggregate physical capital, lt the input of labor per middle-aged agent, L the amount of aggregate middle-aged agents and Ht the human capital per middle-aged agent. A middle-aged agent devotes vnt hours to rearing children, htnt to edu-cating children, and the rest of time (1 ? vnt ? htnt) to working.The human capital of a child Ht, is given byAnd , qt is the amount of goods invested in the child, ht the e?ectiveunits of the parent’s time invested in the children, xt grandpa(or grandma) investing theamount of goods in child. And Hˉt is the average human capital.The consumption functions are respectively given by :τis the ?at rate, s the saving rate, r the interest rate, (1 ? vnt ? htnt)wtHt the wage income.And wt is the wage rate per unit of e?ective labor, T the social security benefit per retiree. Amiddle-aged agent receives a bequest bt, from his parents at the beginning of period t, andleaves a bequest, bt+1, to each child in period t + 1. Here grandpa(or grandma) gives thegoods xt+1ntnt+1 to grandsons.In the pay-as-you-go social security system,We divided the pensions into three types: social security with purely universal benefits; socialsecurity with purely earnings-dependent benefits and social security with mixed benefits.Our ultimate goal is to know that fertility, human capital and so on are how to a?ect the welfare of society. The maximize utility function is giving byaccording to are constants, all grow at the same rate g under the steay-state balancedgrowth conditions.The first-order conditions with respect to each variable, SoSo we can give the following propositions .Proposition 1., increasing the tax-rate or the earnings-dependent of social security benefits has no e?ect on the saving rate.The effects of the benefit formula on fertility, human capital investments and growth aremore complicated. To better illustrate that, we look at three cases:1.Old age pensions with purely earnings-dependent benefits, whenφ= 0. Proposition 2 , i.e., increasing the tax-rate has a positive e?ect onγb.Proposition 3 If , increasing the tax-rate has apositive e?ect on n.Proposition 4 If , increasing thetax-rate has a negative e?ect on (1 ? vn ? hn).Proposition 5 i.e., increasing the tax-rate has a negative e?ect on h;, i.e., increasing the tax-rate has a negative e?ect on h; i.e., increasingthe tax-rate has a negative e?ect onγx.Proposition 6 Under some conditions , i.e., increasing the tax-rate has anegative effect on g .1.Old age pensions with purely earnings-dependent benefits whenφ=ατ.Proposition 7 Ifα→1,βε>γ, , i.e., ifα→1,βε>γ, increasing thetax-rate has a positive effect onγb.Proposition 8 Ifρ>ρmin时, , i.e., increasing the tax-rate has a negativee?ect on n.Proposition 9 Ifρ>ρmin, , i.e., ifρ>ρmin, increasing thetax-rate has a positive effect on (1 ? vn ? hn).Proposition 10 , i.e., increasing the tax-rate has a positive e?ect on h;ifα→1, , i.e., ifα→1, increasing the tax-rate has a positive e?ect onγq;, i.e., increasing the tax-rate has a positive e?ect onγx.Proposition 11 Under some conditions , i.e., under some conditions in-creasing the tax-rate has a positive e?ect on g .3.Old age pensions with mixed benefits, when 0 <φ<ατ.It is a internationally recognized standard that the economic output and growth rate canmeasure social welfare having the progress or not. Then when the pensions are dependentwith the personal wages. Reducing population, increasing the human capital will enhancethe growth rate and improve social welfare.
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