تامین اجتماعی به عنوان تعادل مارکوف در مدل OLG

This paper studies the characteristics of intergenerational transfers in a standard overlapping generations model with short lived governments that care about the welfare of young generations only. A number of authors have shown that simple intergenerational games, in which in each period the current young generation plays as a dictator, are able to deliver political equilibria with social security even if the underlying competitive equilibrium is not dynamically inefficient. These authors have either derived pure steady state results or have relied on subgame-perfectness. This paper extends these results deriving Markov subgame perfect equilibria (i.e. that depend only upon the period t state variable, which is the stock of capital). Non-Markov subgame perfect equilibria assume agents know all the past history of the game; they cannot predict when the social security system will emerge and whether or not it will eventually emerge; they prescribe that generations that never deviated may be punished. Markov equilibria, placing more restrictions on the structure of the game, are able to deliver solutions that do not suffer from these drawbacks. As the paper shows, however, Markov strategies may produce unstable dynamics.

Intergenerational redistribution is a very important issue in the political debate: on the one hand pension systems can be manipulated for political purposes; on the other, it is not clear how a transfer scheme should be designed to be optimal and thus more secure from political pressures, considering that it should be flexible enough to adjust to exogenous (such as population or technological) shocks. In this context, it is very important to understand the widespread role of intergenerational transfers and why pay-as-you-go (PAYGO) social security systems developed and became stable institutions of modern societies. In particular, the existence of PAYGO schemes seems puzzling, given that successive generations cannot subscribe any commitment to pay old age benefits to older generations. There are many explanations in the literature of why PAYGO social security have been introduced and then expanded. Some relate to the incompleteness of private annuity markets and adverse selection, others to the fact that PAYGO systems can help to overcome inefficient intergenerational credit markets. The classical solution to the puzzle is that, if the economy is on a dynamically inefficient path (such that the interest rate falls below the economy’s growth rate), then the introduction of a PAYGO social security scheme is Pareto improving since it reduces the capital deepening. However, even in this case, a PAYGO social security scheme is a dynamically inconsistent agreement between successive generations. Young generations would be better off discontinuing the PAYGO scheme and setting up a new one. Hence the question arises of why PAYGO schemes survive. Recently a growing literature has analyzed the role of PAYGO pension schemes in the context of majority voting in overlapping generations models. Azariadis and Galasso (1997), Galasso (1998), Cooley and Soares (1998) show that equilibria exist in which social security emerges through a majority voting mechanism. Boldrin and Rustichini (2000) show that “a PAYGO public pension plan is a subgame perfect equilibrium of a majority voting game in an OLG model with production and capital accumulation when the growth rate of total labor productivity and the initial stock of capital satisfy a certain set of restrictions.” These contributions show that a PAYGO system can be supported by a strategy which stipulates that a punishment, in the form of no old age transfer, must be inflicted to all generations that in the past have modified its rules. That is, a generation which votes to deviate from the prescribed level of social security payroll tax would be punished and would not receive any old age benefit. A generation which decides not to inflict the punishmentwould not receive any old age benefit as well (in this case, it would contribute to the system but would not receive any benefit back). These equilibria have some limitations. In the first place, both steady states (Cooley and Soares, 1998) and dynamic (Boldrin and Rustichini, 2000) subgame perfect equilibria (SPE) assume agents know the complete past history of the game and use this information when deciding their actions. Secondly, they have not sharp implications: as Boldrin and Rustichini (2000) show, although the social security system can be sustained in equilibrium, the model cannot predict when it will emerge and whether or not it will eventually emerge. Third, they prescribe that the PAYGO is never started again if one generation de viated in the past; if having the PAYGO is beneficial to the generation that starts it and to the following ones, these would get punished by the strategy even if they never deviated. In this paper I show that to support the social security system it is not necessary to condition the transfer to the elderly to their past behavior. The system can be supported by a simple negative relation between the level of saving of each generation and the amount of old age transfers that the same generation receive: if a generation decides to reduce the contribution to the system below the equilibrium level, it will increase its saving and will reach old age with a higher level of the capital stock (in the form, for example, of high private pension accounts); if the same generation would receive a reduced old age transfer as a result of the higher level of capital stock owned, it could be better off not reducing the contribution to the system in the first place. This simple rule in fact implements a punishment type strategy as in Boldrin and Rustichini (2000) that says: “check the capital stock; if it is too high, that means the current elderly did not transfer the amount of resources as they should have, hence lower the transfer to them.” The paper shows that this rule has a Markovian representation. Thus, the Markovian equilibrium requires simply to check the level of the capital stock and to punish agents in proportion of the amount of their deviation from the equilibrium path. The model I present shares many of the characteristics of the one of Boldrin and Rustichini (2000): I assume as well that the median voter is a member of the young cohort, that young cohorts (or the governments representing the young) have to decide upon the payroll tax that the young have to pay to the old as a fraction of their wage, that this payroll tax can change with the capital stock and that saving and consumption are determined by individual maximization. The main difference is that I focus on Markov subgame perfect equilibrium (SPE). Focusing on Markov strategies I am able to restrict the set of potential equilibria, which in standard SPE are usually many. In fact, since the model entails a game among generations, the strategy chosen by the current government depends on the course of action it expects for the future (retirement benefits for the current young will in fact depend on the payroll tax that future young will decide to pay). In this set up, there may be many policy trajectories that are self-fulfilling. Markov equilibria, assuming that strategies depend only on the value of the state variable (which is the capital stock) put strong constraints on the set of admissible forecasts. Since the economy is stationary, it is reasonable to restrict the attention to policies that depend solely on the capital stock (and are thus independent of the period in which the government is in power), that is to policies of the type τ(kt ). In fact there is no reason why a government in power at time t1, when the capital stock is k1, should behave differently from a government that is in power at time t2, when the capital stock is k2, if k1 = k2.1 Strategies that depend solely on the value of the prevailing state variable are usually referred to as Markov strategies.Markovian equilibria in principle do not suffer from the mentioned limitations of more general subgame perfect ones. First, agents need to know only the current value of the state variable, which in the model is the stock of capital, and not all the past history of the game. Second, since the equilibrium strategy will set a relation between the payroll tax that youngworkers decide to pay and the stock of capital which is owned by the elderly, for each level of the capital stock there will be an equilibrium tax. Finally, the equilibrium implies that the level of the payroll tax changes continuously as the economy grows (or shrinks); if a generation deviates from the equilibrium path, this will affect the capital stock and the future course of payroll taxes, but it does not imply that the PAYGO will collapse forever. I will show that theMarkovian equilibrium adds on agents’ reasons to support a PAYGO system. As in Boldrin and Rustichini (2000), the PAYGO can arise either due to the classical efficiency-enhancing role of PAYGO social security systems or to the fact that the transfer of resources from the young to the old reduces the level of capital; in this latter case, if the return per unit of capital has an elasticity greater than one, a reduction in saving increases the total return more than proportionally, thereby increasing lifetime income of the young generation (the “saving monopolist” argument). In the Markovian solution, a generation may want to increase the transfers to the elderly also as this would secure, through a reduction in its level of saving, an increase in old age benefits supplied by future generations. This is a continuous version of the punishment type strategy as in Boldrin and Rustichini (2000). It is different, since Boldrin and Rustichini (2000) assume agents either decide to pay the given payroll tax (set by the previous generation) or not to pay at all. It is worth noting that the punishment strategy needed to support the PAYGO as a political equilibrium actually leads to a higher steady state level of capital than in the model where payroll taxes are exogenously given. That is in equilibrium, since pension benefits are a decreasing function of the capital stock, a higher level of saving than in the competitive model is necessary in order to achieve consumption smoothing. Against all these pluses the paper shows that Markov strategies, when simple logarithmic utility and Cobb–Douglas production functions are assumed, may produce unstable dynamics. The paper is organized as follows. Section 2 describes the model and Section 3 the equilibrium concept. Section 4 present the solution assuming a logarithmic utility function. In Section 5, I consider a more general utility function (CRRA) and I use numerical simulation techniques to approximate the policy function and to analyze the dynamic properties of the equilibrium. Section 6 concludes.

This paper has considered a standard overlapping generations model where individuals live and consume for two periods. They work during the first period and are retired in the second one. In this context I tried to understand why selfish young generations may be willing to pay a retirement income to their parents by means of a pay-as-you-go (PAYGO) social security system. In fact, unless the economy without the PAYGO scheme is on a dynamically inefficient path, young workers should be better off discontinuing the PAYGO scheme and investing the payroll tax in productive capital. Even when the economywithout the PAYGO scheme is on a dynamically inefficient path, young workers should be better off discontinuing the PAYGO scheme and setting up a new one. A number of authors have shown that simple intergenerational games in which in each period the current young generation plays as a dictator are able to deliver political equilibria with social security even if the underlying competitive equilibrium is dynamically efficient. This paper extends these results to equilibria that are Markov (i.e. they depend only upon the period t state variable, which is the stock of capital). The intuition is similar to the one developed in previous contributions. The equilibriumstrategy, which implies a reduction in payroll taxes as the capital stock increases, is simply the implementation of a punishment strategy in a Markovian context. It is worth noting that this strategy actually leads to a higher steady state level of capital than in the model where taxes are exogenously given: in equilibrium, since pension benefits are a decreasing function of the capital stock, a higher level of saving than in the competitive model is necessary in order to achieve consumption smoothing.