Social Security: Long-Term Financing and Reform

Definition

The term “social security: long-term financing and reform” is used to denote the analysis and study of policies oriented to guarantee the long-run sustainability of social security systems. Focusing on pension systems, the principle of the social security system is to guarantee benefits when retired that are financed by contributions during working ages. The sustainability of a social security system is guaranteed if benefits paid out are covered by contributions to the system. Alternatively, sustainability of a social security system may be less stringently defined by requiring that the debt per worker of the social security system remains constant over time. Based on future projections of liabilities and revenues of social security programs, different reform scenarios can be developed. Parametric reforms of social security systems are targeted toward changes in either benefits, or contributions, or retirement ages. Structural reforms of social security systems denote the...

Appendix

Basic Sustainability Equation

The following equation represents the basic equation for the evolution of debt/fund of the pension system at time t:

$$ \frac{\mathrm{dD}(t)}{\mathrm{dt}}= rD(t)-T(t)+N\left(0,t\right){\int}_R^{\omega }{e}^{- na}l(a)b\left(a,t\right) da-\tau (t)N\left(0,t\right){\int}_E^R{e}^{- na}l(a){y}_l\left(a,t\right) da, $$

(A.1)

where D(t) is the total debt at time t, r is the interest rate, T(t) denotes the total taxes levied to finance the pension system, N(0, t) denotes the babies born at time t, R is the age at retirement, ω is the maximum lifespan (See “Maximum Lifespan”), n is the population growth rate, l(a) is the probability of surviving to age a, b(a, t) is the pension benefit at age a at time t, τ(t) is the social contribution rate, E is the length of schooling, and y _l(a, t) is the labor income at age a at time t.

By expressing Eq. A.1 in terms of the debt per potential worker at time t – i.e., $d(t)=\frac{D(t)}{L(t)}$, where L(t) is the total potential working age population – the social security budget constraint becomes:

$$ \frac{\partial d(t)}{\partial t}=\left(r-n\right)d(t)-\frac{T(t)}{L(t)}+\frac{\int_R^{\omega }{e}^{- na}l(a)b\left(a,t\right) da}{\int_E^R{e}^{- na}l(a) da}-\tau (t)\frac{\int_E^R{e}^{- na}l(a){y}_l\left(a,t\right) da}{\int_E^R{e}^{- na}l(a) da}. $$

(A.2)

Defining the average pension benefit at time t, b(t), and the average labor income at time t, y _l (t), as

$$ {\displaystyle \begin{array}{l}b(t)=\frac{\int_R^{\omega }{e}^{- na}l(a)b\left(a,t\right) da}{\int_R^{\omega }{e}^{- na}l(a) da},\\ {}{y}_l(t)=\frac{\int_E^R{e}^{- na}l(a){y}_l\left(a,t\right) da}{\int_E^R{e}^{- na}l(a) da},\end{array}} $$

respectively, we have

$$ \frac{\partial d(t)}{\partial t}=\left(r-n\right)d(t)-t(t)+b(t)\frac{\int_R^{\omega }{e}^{- na}l(a) da}{\int_E^R{e}^{- na}l(a) da}-\tau (t)\ {y}_l(t). $$

(A.3)

Defining the old-age dependency ratio, α, as

$$ \alpha =\frac{\int_R^{\omega }{e}^{- na}l(a) da}{\int_E^R{e}^{- na}l(a) da}. $$

(A.4)

Substituting Eq. A.4 in Eq. A.3 gives

$$ \frac{\partial d(t)}{\partial t}=\left(r-n\right)d(t)-t(t)+b(t)\alpha (t)-\tau (t)\ {y}_l(t). $$

(A.5)

Finally, assuming the debt per worker remains unchanged, d(t) = d, we obtain Eq. 1

$$ \left(r-n\right)d+ b\alpha =\tau\ {y}_l+t. $$

Maximum Pension Benefit

First, we relate the average pension to the initial pension benefit. Assuming a mature pension system in which labor income increases at a rate g, then the pension benefit at age a can be expressed in terms of the initial pension benefit as b(a) = b(R)e ^{−g(a − R)}. Using the “mean value theorem for definitive integrals,” the average pension benefit can be expressed in terms of the pension benefit of new retirees as follows:

$$ b=\frac{\int_R^{\omega }{e}^{- na}l(a)b(a) da}{\int_R^{\omega }{e}^{- na}l(a) da}=b(R){\int}_R^{\omega }{e}^{-g\left(a-R\right)}\frac{e^{- na}l(a)}{\int_R^{\omega }{e}^{- nu}l(u) du\ }\ da=b(R){e}^{-g\left(\overline{A}-R\right)}, $$

(A.6)

where $\overline{A}$ is the average age of pensioners. For notational simplicity, we define the difference $\overline{A}-R$ as A _R, or the average number of years lived as a pensioner.

Now, consider the pension benefit of new retirees cannot exceed the average disposable income, after financing the pension system, of current workers; that is,

$$ b(R)\le \left(1-\tau \right)\ {y}_l-t. $$

(A.7)

Multiplying both sides of Eq. 1 by minus one, adding in both sides of the equality the average labor income, and using Eq. A.7 give

$$ b(R)\le {y}_l-\left(r-n\right)d- b\alpha =\left(1-\tau \right){y}_l-t. $$

(A.8)

Using Eq. A.6 and dividing both sides of A.7 by the average labor income gives

$$ \frac{b(R)}{y_l}\le \frac{1}{1+{e}^{-g{A}_R}\alpha }-\frac{r-n}{1+{e}^{-g{A}_R}\alpha}\frac{d}{y_l}, $$

(A.9)

which coincides with Eq. 3.

Sustainability Factor

Assuming d = 0, a balanced pension system satisfies

$$ b\alpha =\tau\ {y}_l. $$

(A.10)

Substituting Eq. A.6 in Eq. A.10 gives

$$ b(R){e}^{-g{A}_R}\alpha =\tau\ {y}_l\Rightarrow \phi := \frac{b(R)}{y_l}=\frac{\tau }{\alpha }\ {e}^{g{A}_R}. $$

(A.11)

Taking logarithms in Eq. A.11 and differentiating it with respect to time gives

$$ \frac{1}{\phi}\frac{d\phi}{d t}=\frac{1}{\tau}\frac{d\tau}{d t}-\frac{1}{\alpha}\frac{d\alpha}{d t}+\frac{d\left(g{A}_R\right)}{d t}. $$

(A.12)

Sustainable Internal Rate of Return of a Pension System

Let us assume a stable population that grows at a rate n and an economy whose wages increase annually at a rate g. In a mature and balanced pension system, all contributions equal all benefits claimed:

$$ {\int}_E^R{e}^{- na}l(a)\tau {y}_l(a) da={\int}_R^{\omega }{e}^{- na}l(a)b(a) da. $$

(A.13)

Using the equality b(a) = b(R)e ^{−g(a − R)} in Eq. A.13 gives

$$ {\displaystyle \begin{array}{ll}{\int}_E^R{e}^{- na}l(a)\tau {y}_l(a) da=& b(R)\\ {}& \times {e}^{gR}{\int}_R^{\omega }{e}^{-\left(n+g\right)a}l(a) da.\end{array}} $$

(A.14)

Let us denote i as the internal rate of return of the system. Thus, the present value, discounted at a rate i, of the stream of social contributions equals to the present value, discounted at rate i, of the stream of benefits received:

$$ {e}^{\left(i-g\right)R}{\int}_E^R{e}^{-\left(i-g\right)a}l(a)\tau {y}_l(a) da=b(R){e}^{iR}{\int}_R^{\omega }{e}^{- ia}l(a) da. $$

(A.15)

From Eq. A.15 the initial benefit at age R is

$$ b(R)=\tau {e}^{- gR}\frac{\int_E^R{e}^{-\left(i-g\right)a}l(a){y}_l(a) da}{\int_R^{\omega }{e}^{- ia}l(a) da}. $$

(A.16)

Substituting Eq. A.16 in Eq. A.14, and rearranging terms, it can be studied whether the benefits claimed exceed the contributions collected

$$ \frac{\int_R^{\omega }{e}^{-\left(n+g\right)a}l(a) da}{\int_R^{\omega }{e}^{- ia}l(a) da}{\int}_E^R{e}^{-\left(i-g\right)a}l(a)\tau {y}_l(a) da-{\int}_E^R{e}^{- na}l(a)\tau {y}_l(a) da\gtreqless 0. $$

(A.17)

Taking ${\int}_E^R{e}^{- na}l(a)\tau {y}_l(a) da$ as common factor gives

$$ {\displaystyle \begin{array}{l}{\int}_E^R{e}^{- na}l(a)\tau {y}_l(a)\\ {}\begin{array}{l}\\ {}\times da\left(\frac{\int_R^{\omega }{e}^{-\left(n+g\right)a}l(a) da}{\int_R^{\omega }{e}^{- ia}l(a) da}\frac{\int_E^R{e}^{-\left(i-g\right)a}l(a){y}_l(a) da}{\int_E^R{e}^{- na}l(a){y}_l(a) da}-1\right)\end{array}\\ {}\times \gtreqless 0.\end{array}} $$

(A.18)

Notice the sign of Eq. A.18 depends on the sign of the term in parenthesis. After some manipulation of Eq. A.18, we have

$$ \left({\int}_R^{\omega }{e}^{-\left(n+g-i\right)a}\frac{e^{- ia}l(a)}{\int_R^{\omega }{e}^{- iu}l(u) du} da{\int}_E^R{e}^{-\left(i-g-n\right)a}\frac{e^{- na}l(a){y}_l(a)}{\int_E^R{e}^{- nu}l(u){y}_l(u) du} da-1\right), $$

which using the “mean value theorem for definitive integrals” gives

$$ \left({e}^{\left(i-n-g\right)\left({W}_R-{W}_L\right)}-1\right), $$

(A.19)

where W _R and W _L are the average age of pensioners and the average age of workers, respectively. Provided W _R > W _L, the above equation implies that the pension system generates a deficit (>0) or surplus (<0) depending on the value of the internal rate of return of the pension system (i) as follows:

$$ \left\{\begin{array}{cc}>0& i>n+g,\\ {}=0& i=n+g,\\ {}<0& i<n+g.\end{array}\right. $$

(A.20)