اندازه مطلوب امنیت اجتماعی در حضور بازار سهام

The paper develops a stylized overlapping generations economy with random production and a stock market. The impact of a Social Security system on production, asset markets, and consumer welfare is analyzed. It is shown that any reduction in the contribution rate fosters capital accumulation and increases asset prices, wages, and production output. Different welfare criteria are applied to determine the optimal size of Social Security. It is shown that there exists a unique contribution rate which is long-run optimal, socially optimal, and time-consistent in the sense that no generation has an incentive to change it.

Do stock market investments provide a better form of retirement provision than Social Security? This question plays an important role in the debate about the design and reform of the US and other Social Security systems. People who affirm it typically argue that empirical returns on stock market investments are in general higher than those earned by the Social Security system. As a consequence, they favor a privatized system with a reduced share of public pension benefits and a larger share of private investment for retirement in the stock market. Opponents of such a reform, however, argue that stock market investments are subjected to capital market risks. Hence, the proposed privatization would lead to an unfavorable increase of the risk to which retirement incomes are exposed. The latter argument stresses the importance to incorporate the role of risk and uncertainty when performing a theoretical study of Social Security. In addition, several types of interaction between the Social Security system and other economic variables are known to be important in the literature. In this regard, as first shown by Feldstein (1974) the presence of Social Security affects consumers’ savings behavior and crowds out private investment which harms the accumulation of capital. By the same mechanism, changes in the parameters of the Social Security system will in general affect (equilibrium) prices on asset markets and, therefore, stock market returns. This insight makes apparent that a simple static comparison of stock market and Social Security returns should be done with care and calls for a framework that incorporates the mutual interactions between Social Security, asset markets, and the production process. The existing studies in the literature account for these interactions in different ways and to different degrees of generality. A large body of the existing models takes the production process as exogenous and studies the efficiency properties of equilibria in a stochastic overlapping generations (OLG) framework. Examples of such pure exchange models may be found in Chattopadhyay and Gottardi (1999), Demange and Laroque (1999), or Peled (1984). The employed welfare concepts are usually those of ex-ante optimality (EAO) and conditional Pareto optimality (CPO); cf. Demange (2002) for a survey. These concepts generalize the traditional Pareto criterion to stochastic OLG economies. For a study of Social Security, however, the framework with an exogenous production process seems inappropriate as it does not incorporate the adverse effect on capital accumulation exhibited above. An endogenous description of capital accumulation and the production process in a stochastic OLG framework is developed in Wang (1993). His model was extended by Hauenschild (2002) to show that the adverse effect of Social Security on capital accumulation extends to a general class of stochastic OLG economies. The importance of this insight derives from the fact that the first welfare theorem need not hold in OLG economies and equilibria may be inefficient due to an overaccumulation of capital. In such a situation, the introduction of Social Security may lead to a welfare improvement by implementing a dynamically efficient allocation. A second potential welfare gain through the presence of Social Security is stressed and analyzed in Krüger and Kübler (2006) and Gottardi and Kübler (2008). They show that in the presence of incomplete financial markets, Social Security may lead to a welfare improvement due to an improved risk sharing between generations. Further recent studies of these risk-sharing effects may be found in Bohn (2009) and Olovsson (2010). In most of these models the notion of a stock market corresponds to a market for real capital which is built one for one from previous output. Abel (2003) generalizes this structure by assuming a non-linear capital adjustment technology to obtain a non-trivial capital price process. This modification allows him to analyze the impact of changes in the population structure on capital/stock prices and consumer welfare under different Social Security systems. Other assets such as bonds, etc. do not exist in his model. Their existence typically requires a multiperiod OLG structure as in Krüger and Kübler (2006) or Olovsson (2010). In this case, however, the derivation of analytical results and closed form solutions seems impossible. The present paper seeks to contribute to the discussion motivated above by studying the welfare effects and optimal size of Social Security in the presence of a stock market. The study complements the above approaches with respect to both the underlying framework as well as the employed welfare concepts. As for the framework, the assumptions made with respect to technologies and consumer preferences are similar to Abel (2003) sacrificing full generality to obtain closed form solutions and analytical results. The main difference to his and other existing studies is the assumption that capital is installed in (and owned by) firms and cannot be traded between generations directly. Instead, the notion of a stock market corresponds to a market where claims on the firms’ dividend payments are traded between generations. In addition, a bond market exist which is used by the firm to finance its capital investment and provides a riskless investment possibility to consumers. This structure was developed in Dechert and Yamamoto (1992) and Magill and Quinzii (2003) and is compatible with two-period lived consumers. Worth noting is the fact that stock market investments are essentially unproductive in this setting which stands in stark contrast to their role, e.g., in Abel (2003). The present paper reveals how crucial this difference is for the welfare effects of Social Security. With reference to the discussion of whether privatizing Social Security is favorable, the main goal of the paper is to determine the optimal size of the system corresponding to an optimal contribution level. For this purpose, the paper employs several welfare concepts including long-run optimality, social optimality, and time consistency. Unlike the aforementioned Pareto criteria, these concepts incorporate the trade-off between the interests of different generations and permit to identify a unique optimal allocation. Existing studies which also employ (versions of) these concepts may be found e.g., in Abel (2003), Bohn (2009), Demange and Laroque (2000) and Hillebrand (2011), who conducts a related study in a model with endogenous labor supply and no asset markets. The paper is organized as follows. Section 2 introduces the underlying model describing the behavior of consumers and the firm. Existence of equilibrium and the equilibrium effects of Social Security are studied in Sections 3 and 4. Sections 5, 6 and 7 employ the welfare concepts described above to determine the optimal size of Social Security. Section 8 draws some conclusions, proofs for all results are placed in the Appendix.

How much Social Security do we need when there is a stock market? To answer this question, the present paper developed a stochastic overlapping generations model where consumers share aggregate production risk by trading in risky shares and riskless bonds. Social Security was modeled as a static pay-as-you-go system which redistributes resources between young and old consumers. Within this framework, the interactions between Social Security, asset markets, and the production sector were analyzed. It was shown that an increase in the size of the system affects young consumers’ investment behavior which in turn has an adverse effect on asset prices, capital accumulation, and production output. To determine the optimal size of the system, different normative concepts were applied to measure consumer welfare at different contribution rates. Based on these concepts, a unique contribution rate was identified which was long run optimal, socially optimal, and time consistent. This contribution rate determines the optimal size of Social Security in the presence of a stock market. The answer to the question raised in the introduction whether or not privatizing Social Security is favorable therefore depends on whether the optimal contribution rate determined in (33) is positive. For empirically reasonable parameterizations (such as View the MathML sourceα≈13), however, one would expect this value to be either negative or close to zero. If this is the case, a retirement system based on private investment which levies a lump-sum tax on capital income is favorable and would improve the existing Social Security system according to the employed welfare concepts. Interestingly, the empirically likely case of a contribution rate larger than the optimal value, i.e., View the MathML sourceτ>τopt, implies by virtue of (34) that the returns generated by private investment are indeed larger than those earned by the Social Security system but also display a higher variance. This seems in line with the perceived trade-off between higher returns and higher risk due to the privatization of Social Security as discussed in the introduction. As the results of this paper were obtained for a particular setting that permitted the derivation of analytical results, it is natural to ask whether the findings carry over to scenarios with more general preferences, technologies and noise processes. In particular, it would be important to know under what conditions the welfare criteria employed in this paper continue to identify the same optimal contribution level. In this regard, Hillebrand (2011) shows that this result breaks down if labor supply is endogenous. Since it seems quite difficult to obtain analytical results in a more general setting, one will most likely have to use numerical simulations as in Krüger and Kübler (2006). The structure of the model and the employed welfare concepts, however, should be amendable to such extensions and the generalization of the results marks a challenge for future research.