On the optimal design of pension systems

Abstract

The present paper aims to quantify efficiency properties of flat and earnings-related pay-as-you-go financed social security systems of various institutional designs in order to identify an optimal pension design. Starting from a benchmark economy without social security, we introduce alternative pension systems and compare the costs arising from liquidity constraints as well as distortions of labor supply versus the benefits from insurance provision against income and lifespan uncertainty. Our findings suggest an optimal replacement rate of about 50 % of average earnings. In our model a single-tier earnings-related pension system yields the highest efficiency gains dominating flat benefits as well as two-tier systems of any form. We also show that the negative correlation between pension progressivity and pension generosity of real-world social security systems can be justified on efficiency grounds. Finally, our results indicate a positive impact of means-testing flat benefits against earnings-related benefits within multi-pillar pension systems.

Appendix: Computation algorithm

In the first step of our simulation analysis we compute the initial steady state of the model. The computation of an equilibrium follows the Gauss-Seidel method introduced by Auerbach and Kotlikoff (1987). Given our set of exogenous parameters and initial guesses for aggregate variables and the bequest distribution, we calculate factor prices, household’s decision rules and value functions.

In order to obtain the solution for the household’s optimization problem (1), the state space is discretized, i.e. we use the state vector ${z_j=(s,a_j,ep_j,\eta_j)\in {\mathcal{J}} \times {\mathcal{S}} \times {\mathcal{A}} \times {\mathcal{P}}\times {\mathcal{E}}}$ with discrete sets ${{\mathcal{J}}=\{1,\ldots, J\}, {\mathcal{S}}=\{1,\ldots,S\}, {\mathcal{A}}=\{a^1,\ldots, a^{n_A}\}, {\mathcal{P}}=\{ep^1,\ldots, ep^{n_P}\}}$ and ${{\mathcal{E}}=\{\eta^1,\ldots, \eta^{K}\}}$. For all possible states z _j we compute the optimal decisions from (1) over an equidistant multidimensional grid.

As we model both uncertainty and means-testing, (1) is not differentiable in every (c _j , ℓ _j ). Furthermore, V(z _j+1) is only known in a discrete set of points ${z_{j+1}\in \{j+1\}\times{\mathcal{S}} \times {\mathcal{A}} \times {\mathcal{P}} \times{\mathcal{E}},}$ resulting in our maximization problem not being analytically solvable. Hence, we use backward induction through the following maximization and interpolation algorithms to numerically find solutions for the household’s optimization problem:

First, we compute (1) in the maximum age of J for all possible states z _J . The probability of surviving to period J + 1 is zero for all households, there is no bequest motive and agents are not allowed to work after the mandatory retirement age—hence, V(z _J+1) = 0 and it is optimal for agents to consume all remaining assets.

Given all possible V(z _J ), we compute (1) for all $j=J-1,\ldots,1$. In some model specifications we’re modeling means-tested pensions, in which case V(z _j ) is not globally smooth and concave as the value function shows a considerable amount of local optima. Therefore it is essential to use a computation algorithm which is able to accurately find the global maximum of V(z _j ), i.e. which is robust against falsely selecting a local optimum. One method of dealing with this problem is a “brute force” approach, i.e. a discrete grid-search on a finely spaced grid with a large number of points. This method is robust against falsely selecting a local maximum as the global one, but is computationally very expensive.

To reduce computation time we consequently use Powell’s algorithm (Press et al. (2001), pp. 406ff.) to find optimal solutions for (1) while guarding against the selection of local optima by adopting a section-wise optimization routine over the grid: We split the multidimensional grid into p _a  × p _p sections, for each of which we find independent local optima of V(z _j ) using Powell’s algorithm. The local optimum with the highest value of V(z _j ) is identified as the global optimum of V(z _j ). Furthermore, we undertake careful robustness checks by varying the number of optimization sections as well as changing the resolution of the multidimensional grid.

Since Powell’s algorithm requires a continuous function, we interpolate V(z _j+1) using linear interpolation, i.e. we find a function sp _j+1 that satisfies the interpolation conditions

$$ sp_{j+1}\left(j+1,a_{j+1}^l,ep_{j+1}^m\right)=E\left[V(z_{j+1})\right] $$

for all grid points $l=1,\ldots,n_A$ and $m=1,\ldots,n_P$.

Given the optimal decisions of households, we compute their distribution and update the initial guesses by computing aggregate assets, labor supply and consumption. We also obtain the social security contribution rate which balances the budgets of the public pension system. The procedure is repeated iteratively until convergence is reached.

After obtaining the initial equilibrium we change the exogenous policy parameters and solve for the transition path between the initial and the final steady state. Assuming in the first guess that variables would remain constant along the transition even with alternative exogenous policy parameters, we update all individual and aggregate variables for each transition period until convergence is reached.