Ageing-driven pension reforms

Abstract

This paper stems from the observation that there are two worldwide trends, pension reform and population ageing, and asks whether the two may be related. Exploring the cases of pension reform in different countries, we find that, although they are very different, the cases share a common characteristic: they shift risks away from workers towards those who are retired. Furthermore, population ageing, by increasing the weight of the elderly relative to working generations, raises the price of intergenerational risk sharing. Combining these findings, we argue and show formally that pension reform can be seen as a welfare-best response to population ageing.

Appendices

Appendix A: The case of a PAYG scheme

This appendix contains the details of the maximization of the social welfare function in the case of a PAYG scheme.

Equation (14) in Section 4.3 expresses social welfare in the case of a PAYG scheme, which we repeat here for convenience:

$$\begin{array}{@{}rcl@{}} W_{p}&=& \!E \left[ n_{y} \left( \frac{\left( w\!-\left( \frac{E(\mu )}{1+E(r)}\right)\beta \omega w \!-\left( \phi_{p} \mu + (1-\phi_{p}) E(\mu)\right)(1\,-\,\beta)\omega w \right)^{1-\gamma}}{1-\gamma} \!\right)\right. \\&& \left.+\frac{n_{o}}{1+{\Delta}}\!\left( \frac{\left( \left( \frac{E(\mu )}{1+E(r)}\right)\beta \omega w \left( \frac{1+r}{\mu}\right) \,+\,\left( \frac{\phi_{p} \mu + (1-\phi_{p}) E(\mu)}{\mu} \right) (1\,-\,\beta) \omega w \right)^{1-\gamma}}{1-\gamma} \right) \!\right]\\ \end{array} $$

(24)

The first derivative with respect to the policy parametre ϕ _p reads as follows:

$$\begin{array}{@{}rcl@{}} \frac{\partial W_{p}}{\partial \phi_{p}} &=& E \left[ -n_{y} \left( w-\left( \frac{E(\mu )}{1+E(r)}\right)\beta \omega w -\left( \phi_{p} \mu + (1-\phi_{p}) E(\mu)\right)(1-\beta)\omega w \right)^{-\gamma} \right.\\ &&(\mu-E(\mu))(1-\beta)\omega w \\&&+\frac{n_{o}}{1+{\Delta}} \left( \left( \frac{E(\mu )}{1+E(r)}\right)\beta \omega w \left( \frac{1+r}{\mu}\right) \,+\,\left( \frac{\phi_{p} \mu + (1-\phi_{p}) E(\mu)}{\mu} \right) (1\!-\beta) \omega w \right)^{-\gamma} \\&&\left.\left( \frac{\mu-E(\mu)}{\mu} \right) (1-\beta)\omega w \right] \end{array} $$

(25)

The expression of the second derivative of W _p with respect to ϕ _p is derived in a similar way:

$$\begin{array}{@{}rcl@{}} \frac{\partial^{2} W_{p}}{(\partial \phi_{p})^{2}}\!\! &=&\! \!E \left[ \!-\gamma n_{y} \left( w-\left( \frac{E(\mu )}{1+E(r)}\right)\beta \omega w \!-\left( \phi_{p} \mu + (1-\phi_{p}) E(\mu)\right)(1-\beta)\omega w \right)^{-\gamma-1}\right.\\ &&\left( (\mu-E(\mu))(1-\beta)\omega w \right)^{2} \\&&- \gamma \frac{n_{o}}{1\,+\,{\Delta}} \!\left( \!\left( \frac{E(\mu )}{1+E(r)}\right)\!\beta \omega w\left( \frac{\!1+r}{\mu}\right) \!\,+\,\left( \frac{\!\phi_{p} \mu + (1-\phi_{p}) E(\mu)}{\mu} \right) (1\,-\,\beta) \omega w \right)^{\!-\gamma-1}\\ &&\left.\left( \left( \frac{\mu-E(\mu)}{\mu} \right) (1-\beta)\omega w \right)^{2} \right] \end{array} $$

(26)

This expression shows that the second derivative of W _p with respect to ϕ _p is unambiguously negative.

Unfortunately, the first-order condition is nonlinear in ϕ _p and it is not possible to directly derive an analytical expression for ϕ _p . We proceed by writing ∂ W _p /∂ ϕ _p as a function of μ and r:

$$ \frac{\partial W_{p}}{\partial \phi_{p}}=0 \quad \rightarrow \quad E({\Omega}_{p}(\mu,r))=0 $$

(27)

Now, we elaborate the second-order Taylor approximation of Ω _p around the point (E(μ), E(r)). This produces the following equation,

$$\begin{array}{@{}rcl@{}} &&E \left( {\Omega}_{p}(E(\mu),E(r))+ {{\Omega}_{p,\mu}}(E(\mu),E(r))(\mu-E(\mu))\right. \\&&+{{\Omega}_{p,r}}(E(\mu),E(r))(r-E(r)) \\&&+\frac{1}{2}\left[{{\Omega}_{p,\mu \mu}}(E(\mu),E(r))(\mu-E(\mu))^{2}+{{\Omega}_{p,r r}}(E(\mu),E(r))(r-E(r))^{2} \right. \\&&\left.\left.+ 2 {{\Omega}_{p,\mu r}}(E(\mu),E(r))(\mu-E(\mu))(r-E(r))\right] \right)=0 \end{array} $$

(28)

where Ω _p,x refers to the derivative of Ω _p to x and Ω _{p,x y} refers to the derivative of Ω _p,x to y.

It is easy to see that this reduces to the following condition:

$$\begin{array}{@{}rcl@{}} &&{\Omega}_{p}(E(\mu),E(r)) + \frac{1}{2}\left[{{\Omega}_{p,\mu \mu}}(E(\mu),E(r))Var(\mu)\right.\\ &&\left.+ {{\Omega}_{p,r r}}(E(\mu),E(r))Var(r) + 2 {{\Omega}_{p,\mu r}}(E(\mu),E(r))Cov(\mu,r) \right] =0 \end{array} $$

(29)

Elaborating this condition gives us the expression for ϕ _p in the main text (Eq. (15)).

Appendix B: The case of a funded scheme

This appendix contains the details of the maximization of the social welfare function in the case of a funded scheme.

Equation (22) in Section 4.3 expresses social welfare in the case of a funded scheme, which we repeat here for convenience:

$$\begin{array}{@{}rcl@{}} W_{f}&=& E \left[ n_{y} \left( \frac{\left( w - \left( \frac{E(\mu )}{1+E(r)} \right)\omega w + \mu \phi_{f} \left( \frac{E(\mu)}{1+E(r)} \frac{1+r}{\mu} -1 \right) (1-\beta) \omega w \right)^{1-\gamma}}{1-\gamma} \right)\right.\\ &&\left.+\frac{n_{o}}{1+{\Delta}}\left( \frac{\left.\left( \frac{E(\mu )}{1+E(r)} \frac{1+r}{\mu} \right) \omega w - \phi_{f} \left( \frac{E(\mu )}{1+E(r)} \frac{1+r}{\mu} -1 \right) (1-\beta) \omega w \right)^{1-\gamma}}{1-\gamma} \right) \right]\\ \end{array} $$

(30)

The derivative of W _f with respect to ϕ _f reads as follows:

$$\begin{array}{@{}rcl@{}} \frac{\partial W_{f}}{\partial \phi_{f}} &=& E \left[ n_{y} \!\left( w - \left( \frac{E(\mu )}{1+E(r)} \right)\omega w + \phi_{f} \left( \frac{E(\mu)}{1+E(r)} (1+r)- \mu \right) (1-\beta) \omega w \right)^{-\gamma}\right. \\ &&\left( \frac{E(\mu)}{1+E(r)} (1+r) -\mu \right) (1-\beta)\omega w \\ &&- \frac{n_{o}}{1+{\Delta}} \left( \left( \frac{E(\mu)}{1+E(r)} \frac{(1+r)}{\mu} \right) \omega w - \phi_{f} \left( \frac{E(\mu)}{1+E(r)} \frac{(1+r)}{\mu} -1 \right) (1- \beta) \omega w \right)^{-\gamma} \\ &&\left.\left( \frac{E(\mu)}{1+E(r)} \frac{(1+r)}{\mu} -1 \right) (1-\beta)\omega w \right] \end{array} $$

(31)

The expression of the second derivative of W _f with respect to ϕ _f is derived in a similar way:

$$\begin{array}{@{}rcl@{}} \frac{\partial^{2} W_{f}}{(\partial \phi_{f})^{2}}\! &=&\! \!E \left[ -\gamma n_{y} \left( w \!- \left( \frac{E(\mu )}{1+E(r)} \right)\omega w + \phi_{f} \left( \frac{E(\mu)}{1+E(r)} (1+r)- \mu \right) (1\!-\beta) \omega w \right)^{-\gamma-1}\right. \\ &&\left( \frac{E(\mu)}{1+E(r)} (1+r)- \mu \right)^{2} \left( (1-\beta)\omega w \right)^{2} \\ &&\!\!- \frac{\gamma n_{o}}{1+{\Delta}} \left( \left( \frac{E(\mu )}{1+E(r)} \frac{1+r}{\mu} \right) \omega w \,-\, \phi_{f} \left( \frac{E(\mu)}{1+E(r)} \frac{(1+r)}{\mu} -1 \right) (1\!-\beta) \omega w \right)^{-\gamma-1} \\ &&\left.\left( \frac{E(\mu)}{1+E(r)} \frac{(1+r)}{\mu} -1 \right)^{2} \left( (1-\beta)\omega w \right)^{2} \right] \end{array} $$

(32)

This expression shows that the second derivative of W _f with respect to ϕ _f is unambiguously negative.

Similar to the case of a PAYG scheme, we proceed by writing ∂ W _f /∂ ϕ _f as a function of μ and r:

$$ \frac{\partial W_{f}}{\partial \phi_{f}}=0 \quad \rightarrow \quad E({\Omega}_{f}(\mu,r))=0 $$

(33)

The second-order Taylor approximation of the function Ω _f (μ,r) around the point (E(μ),E(r)) implies the following expression that is linear in ϕ _f :

$$\begin{array}{@{}rcl@{}} {\Omega}_{f}(E(\mu),E(r)) + \frac{1}{2}&&\left[{{\Omega}_{f,\mu \mu}}(E(\mu),E(r))Var(\mu) \right. \\ &&+ {{\Omega}_{f,r r}}(E(\mu),E(r))Var(r) \\&&\left.+ 2 {{\Omega}_{f,\mu r}}(E(\mu),E(r))Cov(\mu,r) \right] = 0 \end{array} $$

(34)

Elaborating this condition yields the closed-form expression for ϕ _f in the main text (see Eq. (23)).

Appendix C: The case of endogenous private saving

Here, we assume that the household chooses his private savings such as to maximize his intertemporal utility function. This means we have to elaborate the following first-order condition: $c_{y}^{-\gamma }=(1/(1+\delta ))E((1+r)c_{o,+1}^{-\gamma })$. In our case with stochastic capital income and stochastic non-capital income, an analytical solution for the condition does not exist. We therefore elaborate a first-order Taylor approximation for the term of which the expectation is included in the first-order condition with respect to the two risk factors in the model. Including this approximation in the first-order condition allows us to derive an explicit solution for private savings:

$$ a=\frac{w-p-\left( \frac{1+E(r_{+1})}{1+\delta}\right)^{-1/\gamma}\omega w}{1+\left( \frac{1+E(r_{+1})}{1+\delta}\right)^{-1/\gamma}\left( \frac{1+E(r_{+1})}{E(\lambda_{+1})}\right)} $$

(35)

This equation looks intuitive: a higher after-premium wage income increases savings, a high pension benefit due to the PAYG scheme lowers private saving and the intertemporal elasticity of substitution, 1/γ, drives the impact of the rate of return upon savings. Variance and covariance terms do not appear in Eq. (35); the first-order Taylor approximation removes precautionary savings from the model. Also intuitive is that a higher expected survival ratio implies higher private savings and that this effect is smaller the more generous are pension benefits.

Applying the same procedure as in the main text, we now derive the following expression for ϕ _p in the case of a PAYG scheme:

$$ \phi_{p} = \frac{D_{1}+D_{2}(1-\tilde{Cov}(\mu,r))}{D_{1}+\frac{D_{3}}{D_{4}}} $$

(36)

where

$$\begin{array}{@{}rcl@{}} D_{1}&=&\frac{\omega}{E(\mu)} \end{array} $$

(37)

$$\begin{array}{@{}rcl@{}} D_{2}&=&\frac{\alpha_{-1}}{E(\mu)}D_{\mu} \end{array} $$

(38)

$$\begin{array}{@{}rcl@{}} D_{3}&=&\frac{D_{\delta}^{1-\gamma} D_{\mu}^{2}}{(1+D_{\delta}D_{\mu})^{1-{\gamma}}}\omega (D_{\mu}(1-E(\mu)\omega)+ \omega)^{-\gamma-1} \end{array} $$

(39)

$$\begin{array}{@{}rcl@{}} D_{4}&=&\frac{1}{1+{\Delta}}(D_{\mu} \alpha_{-1}+ \omega)^{-\gamma-1} \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} D_{\delta}&=& \left( \frac{1+E(r)}{1+\delta} \right)^{-1/\gamma} \end{array} $$

(41)

$$\begin{array}{@{}rcl@{}} D_{\mu}&=&\frac{1+E(r)}{E(\mu)} \end{array} $$

(42)

$$\begin{array}{@{}rcl@{}} \alpha&=&a/w \end{array} $$

(43)

Equation (36) is more complex than its counterpart in the main text, Eq. (15). As a result, we cannot sign ∂ ϕ _p /∂ E(μ) and cannot derive analytically that an increase in E(μ) will always decrease ϕ _p . Therefore, we resort to numerical simulation. We run 100,000 simulations that combine different values of E(μ), α ₋₁, E(r), $\tilde {Cov}(\mu ,r)$ and γ. These simulations confirm the earlier result: except for the cases in which ϕ _p is zero, all cases imply a negative relationship between E(μ) and ϕ _p . Hence, we conclude that our results are not driven by our choice not to derive private savings from optimizing behaviour.