Actuarial Fairness Evaluation

Most discussions of social security programs evaluate this policy using measures based on the concept of actuarial fairness. The approach commonly used is to construct tax and benefit streams and compare the tax cost of the coverage to the level of benefits by computing either the present value of net benefits (PVNB) or the internal rate of return. The expected PVNB is computed using a market real interest rate. The internal rate of return is the rate of discount that equates the expected present value of benefits with the expected present value of taxes. So, these welfare measures compare the return on funds contributed to the system with the returns that would have been earned had the funds been invested in alternative applications such as bonds or the stock market.

The PVNB for an age i, type j agent at time t is given by

where

This measure corresponds to the difference between the present value of lifetime resources of the agent with social security and without social security when everything else remains constant.

Impact of Social Security on the Lifetime Utility of Agents

To evaluate the impact of the social security policy on the well-being of individuals, we need to compare the utility of individuals in the economy with social security to their utility in the economy without social security.

I assume that the economy is in a steady state without social security and I compute the path for the economy when a social security system is implemented. Unless otherwise specified, I assume that a social security system described by a constant tax rate on labor income,

, is introduced. The revenues generated are equally distributed as benefits among the retirees. I then compare the utility of agents in the economy without the system with the utility they attain when the system is implemented.

To assess how different individuals fare under a social security system, I construct an equivalent variation measure (EVM). I compute the level of transfers required to make each individual able to attain the same level of lifetime utility in the economy without social security as in the economy with the social security system. The transfer is optimally allocated by the individual to consumption and savings and allows her to attain the reference level of utility.

The compensation to be given to a type j, age i agent endowed with a level of assets a_i,j in an economy without social security and an aggregate state described by A, is x_i,j such that

, where

is the lifetime utility level of that agent in the environment we want to evaluate. The measure of welfare is then

, where r is the real interest rate in the reference steady state.

Preferences

I set the coefficient of risk aversion, ρ, to 2 and choose the value for the discount factor,

, so that the equilibrium interest rate is approximately 6%. I take the coefficient of consumption in the utility function,

, so that, on average, agents in the labor force allocate around 31% of their time to market activities.

Technology

Following Cooley and Prescott (1995), I set the share of labor in the production function to 0.6 and I set the depreciation rate so that the steady-state annual investment/capital ratio for this economy is 0.076.

The age-specific endowments of efficiency units for the economies with heterogeneous agents are taken from Hansen (1993).

Finally, I set the exogenous growth rate to be 1.83%.

Demographic Process

Social security

The social security tax rate is set exogenously to 12.8%. The total social security payroll tax including hospital and disability insurances (HI and DI) is 14.7%; the OASI tax rate is about 9.9%, HI payroll tax is about 2.9%, and DI tax is about 1.9%. Whereas HI can be modeled as a transfer to agents of age 65 and over, DI benefits would be a transfer to agents below age 65. So, I set the social security tax rate in the model to 12.8% [see Altig et al. (2001)].

I use this procedure to calibrate several economies first with intragenerational homogeneity and then with intragenerational heterogeneity: the benchmark economy where all agents live for T periods, an economy where social security benefits are proportional to retirees' contributions, one with lifetime uncertainty, and another where agents cannot borrow against future income.

The parameter choices for these economies are summarized in Table 1.

Benchmark Economy—No Lifetime Uncertainty

Intragenerational homogeneity

Figure 1 shows the evolution of equilibrium capital stock K, the effective labor supply L, the interest rate r, and the wage rate w along the transition path resulting from introducing a social security system (

%

) in a steady state without social security (

).

Transition dynamics.

Social security is an economywide policy (as opposed to an individual choice) that affects the behavior of all individuals in the economy. It distorts their savings and labor supply decisions and affects the aggregate level of capital and the supply of labor in the economy. Older generations have a larger marginal propensity to consume than younger ones. That is, older individuals consume a higher share of any additional income than do younger ones. The introduction of social security implies a redistribution of resources away from the young and toward older generations, which leads to an increase in the aggregate consumption level, a decrease in savings, and a corresponding decrease in the accumulation of physical capital. Furthermore, the increase in the taxation of labor income and in retirement benefits lowers the incentives of individuals to work and save. This leads to a decline of the capital stock and the effective labor supply, affecting the interest rate and wages, as can be seen in Figure 1. Hence, social security also changes individuals' labor income and the returns to their savings.

Notice also the immediate response of the factor prices to the implementation of the social security system due to the decrease in the labor supply. There is an instant increase in wages, which benefits the young, and a decline in the interest rate, which is detrimental to old workers and retirees who get a higher share of their income from capital. Following the initial period, capital decreases while labor supply has an initial surge and drops again. This generates a drop in the wage rate and a rise in the interest rate while the economy converges to a steady state with a lower capital level.

How the generations value social security

The values, as a fraction of per capita output, of the equivalent variation measure (EVM) and of the PVNB for each type of agents as a function of age are shown in Figure 2.

Benchmark calibration—PVNB and welfare measure.

An important attribute of the equilibrium is that both the PVNB and the EVM are negative for the youngest generations (below 40–45 years old) and positive for the remaining ones, peaking for the youngest retirees.

However, the welfare loss of the young generated by the social security system is lower than what is implied by the PVNB while the welfare gains of the middle-aged are higher than those implied by the actuarially fair measure. This disparity becomes clear in Figure 4 where the difference between these two measures is represented, ranging from around 70% of the steady-state output per capita to about −10% of output.

In a small open economy, where factor prices remain unchanged, both measures are slightly different (see Figure 3). The difference between the two measures follows the evolution of the present value of agents' lifetime labor income, and it represents the welfare impact of the distortionary tax levied on labor income. Besides the income effect of social security, measured in this case by the actuarially fair measure, the introduction of the social security tax rate distorts the price of leisure relative to consumption.

Benchmark calibration—small open economy.

The significance of the impact of the tax distortion on welfare also shows that linking social security contributions to benefits can have some impact on welfare because it might reduce the distortionary effect of social security taxation. Furthermore, because the distortionary effect increases more than proportionally with the tax rate, the mismeasurement of actuarially fair measures will increase with the predicted future increase in the social security tax rate.

In a closed economy, the negative distortionary effect of taxation is more than offset by the impact of social security on factor prices (see Figure 4).

Benchmark calibration—closed economy.

Older agents are affected by the initial decrease in interest rates, wherease younger retirees have time to benefit from the future increase in interest rates. The middle-aged, with most of their working life behind them and with significant assets, benefit from the initial increase in wages and the future increase in interest rates without being significantly affected by future decreases in wage rates. Younger agents are hurt by the future decrease in wages, and their lower income does not allow them to accumulate many assets and take advantage of the future increase in interest rates.

The size of the disparity between the two measures illustrates that the general equilibrium effect of social security on the utility of agents, through its impact on the levels of capital, effective labor supply, and on factor prices is important.

In Table 2, I compare the bias of the actuarial fair measure for short-run and long-run evaluations of the impact of social security on an age-1 individual. For the small open economy, there is no difference, but for the closed economy, the difference is significant. The bias more than quadruples from the short run to the long run. The general equilibrium effects of social security are much more pronounced in the long run, as we can see in Figure 1. The equilibrium levels of capital as well as the labor supply decrease dramatically with the introduction of a pay-as-you-go social security system. The real rate of return on private savings increases while the wage rate decreases.

More importantly, when the objective of the analysis is to evaluate the impact of social security on an individual, comparisons across steady states can be even more misleading than the use of actuarial fair measures. We need to compare the utility of agents when there is no social security to the one they would get if social security were to be implemented.

Intragenerational heterogeneity

Because the distortionary tax and the general equilibrium effects might affect heterogeneous individuals diversely, I study, in this section, the sensitivity of the bias of the actuarially fair measures to heterogeneity. I focus on two sources of intragenerational heterogeneity, allowing agents to differ in their efficiency levels,

, in one experiment, and in their discount factors,

, in another experiment.

Efficiency

I start by differentiating agents according to their efficiency levels,

. I use the same calibrations as Altig et al. (2001), ordering 12 different agent types from the less efficient and poorer (type 1) to the more efficient and richer (type 12).

The graphs (Figures 5 and 6) show the bias of the PVNB relative to the welfare measure across ages for different types of agents. It has the same configuration across ages as in the economy with homogeneous agents. It is clear that this bias varies across types and is related to the differences in agents' “labor effectiveness.”

Heterogeneous labor efficiency—small open economy.

Heterogeneous labor efficiency—closed economy.

In the small open economy (Figure 5), the bias is generated by the distortionary tax on labor income and as expected agents with higher effective labor supply are more hurt by tax distortions.

In the closed economy (Figure 6), more efficient agents benefit more from the initial increase of the wage rate and are more hurt by decreases in future wage rates. Because they also accumulate more assets, they gain more from future increases in interest rates while being more affected by the current decrease in the interest rate.

So, the bias associated with the PVNB measure depends not only on the age of individuals but also on the type. Consequently, the relative impact of social security on different types of individuals is also mismeasured when using an actuarially fair approach.

In Table 3, I compare the bias of the actuarially fair measure for short-run and long-run evaluations. Again, in a small open economy, there is no distinction, but in a closed economy the difference is significant. In fact, in a closed economy, less effective age-1 agents benefit from social security by more than the PVNB measure indicates, but all agents born in the steady state of the economy with social security are much worse off than what the actuarially fair measure shows.

Discount factor

Carroll and Samwick (1997) show that much of the variation in wealth accumulation over the life cycle can be related to the values of agents' discount rates. Krusell and Smith (1998) establish that heterogeneity in the discount factor can significantly improve the ability of their model to match the U.S. wealth distribution. Following these results, I differentiate agents according to their discount factors,

, and analyze the bias of actuarially fair measures, given this source of heterogeneity. As in Samwick (1998), I assume that there are seven types of agents in each cohort. I then set the values of the parameters so that the model matches the targeted values described in the calibration section while the percent deviation from the mean of the discount factors and the population shares of each type of agents are consistent with the distribution of households' rates of time preference estimated by Samwick. Keep in mind that the aim of the experiment is to construct a model with some illustrative value and not an attempt to match the earnings or wealth inequality observed in the U.S. economy. I set the average discount factor to 0.971 and the coefficient of consumption in the utility function to 0.3; the values for β and

are described in Table 4.

The bias of the actuarially fair measure for different types of agents across ages is represented in Figures 7 and 8. For many agent types, the bias has a similar configuration across ages as in the economy with homogeneous agents. But, unlike in the previous case, the bias does not seem to change monotonically with the differences in the discount factor and it is clearer that the PVNB does not capture the intragenerational impact of social security. In the small open economy (Figure 5), the bias is once again related to the distortionary effect of social security. This effect increases with the effective labor supply. Agents with a lower discount factor care more about current leisure and supply less labor in the earlier periods of their lives while accumulating negative levels of assets in order to finance consumption; hence, the impact of tax distortions on their utility is relatively small. However, as they get older, they work more to finance consumption and pay back their loans. The distortionary effect of social security is then higher relative to that for other agents from the same cohort. Conversely, younger agents with higher

allocate more of their time to work and are more affected by the tax distortion. As they grow old, they can finance consumption with their accumulated wealth and decrease their supply of labor to enjoy more leisure. They are then less affected by the distortionary tax.

Heterogeneous discount factors—small open economy.

Heterogeneous discount factors—closed economy.

However, the distortionary effect of social security seems also to be low for the younger agents with the highest levels of

. Whereas young agents' labor supply increases with the discount factor, the elasticity of labor supply decreases. For the higher levels of

, the labor supply elasticity is very low and labor supply becomes less responsive. Even though agents allocate most of their time to labor, the tax rate becomes less distortionary.

As in the preceding section, in the closed economy (Figure 8), the tax distortion becomes almost irrelevant. Agents with a lower discount rate borrow and do not work in their early years. Because the rate of interest increases and the wage rate decreases later in the transition, when they supply more labor and are in debt, they are worse off than what is indicated by the PVNB measure. Older agents enjoy the initial increase in wages when they are supplying more labor, but they also have accumulated debt and are going to face significantly higher interest payments. Agents with a higher discount rate supply more labor initially, benefiting from the early increase in the wage rate and save more, accumulating more wealth. Hence, they gain from the future increase in the interest rate.

In Table 5, we can see that, in the closed economy, the results from the preceding section are reversed. In this case, the three wealthiest agent types benefit from social security by more than indicated by the actuarially fair measures. The remaining agents would be much worse off than what is indicated by the actuarially fair measure. Furthermore, whereas the PVNB decreases with the agent type, the reverse happens for the welfare measure, which becomes positive for the wealthiest agents, indicating that they would be better off if they would be born in the steady state of the economy with social security.

Intragenerational homogeneity

The U.S. social security system has two components: one proportional to past contributions and another independent of past contributions. These components have distinct implications in terms of the redistributional impact of a social security system. Whereas the latter implies some redistribution from high to low earners, with high earners contributing significantly more than low earners and receiving the same benefits, the first does not imply any direct intergenerational redistribution.

Even though the comparison between these components tends to orbit around the redistributional impact of social security, a considerable role has also been given to the reduction in the distortionary effect of social security taxation provided when benefits are linked to past contributions. Auerbach and Kotlikoff (1987) show that there might be large efficiency gains from establishing a link between an agent's contributions and benefits. In this section, I analyze the impact of linking individual earnings to benefits on the bias of the actuarially fair measures by assuming that all social security benefits are proportional to the average labor income of the retiree:

and

. I then compare the results with the ones obtained for the benchmark economy.²

I compute the steady-state value of the replacement rate,

, such that the social security budget is balanced for a tax rate of 12.8%. I then compute the transition path resulting from implementing the social security system described by that replacement rate in a steady state without social security. I further assume that along the transition the social security tax rate is set to maintain a balanced government budget.

In Figure 9, we can see the bias of the actuarially fair measure in a small open economy. As expected, the introduction of a social security system in which benefits depend on earnings decreases the distortionary impact of social security taxation on the supply of labor relatively to the benchmark case. Furthermore, the coefficient of consumption in the utility function required to target the average time allocated to labor observed for the U.S. economy in this economy is lower than in the benchmark case. Agents with a lower coefficient of consumption in the utility function care more about leisure and have a higher labor supply elasticity. Their labor decisions respond more to the social security tax rate but they also supply less labor. The overall impact of tax distortions on utility is relatively small.

Earnings-dependent benefits—small open economy.

However, the amplitude of the bias in the closed economy is significantly higher (see Figure 10) than in the benchmark economy. We notice that the bias peaks for middle-aged agents at a much higher level and reaches more negative values for some of the current retirees but it is also less negative for younger agents.

Earnings-dependent benefits—closed economy.

The link between labor income and social security benefits increases the incentives to supply labor. This effect is partially offset by the decrease in the coefficient of consumption. A lower level of this parameter implies a higher elasticity of labor supply and hence a higher response of labor supply. The higher decrease in labor supply in response to the introduction of social security leads to a significantly higher rise in initial wages and drop in interest rates.

Furthermore, the earnings dependency of benefits also results in a bigger decrease in savings. Because the impact on capital is relatively higher than the one on labor supply, the future decrease of wages and increase of interest rates are also higher than in the benchmark case. The middle-aged have more to gain from the initial increase in wages and the future increase in interest rates. Younger agents benefit more from the initial increase of the wage rate, and are more affected by future decreases in wage rates. As agents accumulate more wealth than in the benchmark case, they gain from increases in future interest rates, while being more affected by the current decrease of the interest rate. Hence, the stronger response of the price factors to the introduction of social security results in a higher bias of the actuarially fair measures.

In Table 6, the results from the benchmark economy are confirmed. The difference between the short and the long run in the small open economy case is a result of the higher tax rates needed to finance social security benefits in the short run.

Intragenerational heterogeneity

I now look at the bias of the actuarially fair measures when agents differ according to their efficiency levels,

.

Even though a system with earnings-dependent benefits has different redistributional implications than a universal system, these differences are captured by the actuarially fair measures. As can be seen by comparing Figures 11 and 12 with Figures 5 and 4 (in the Benchmark section) the intragenerational bias of the actuarially fair measures is hardly affected by this aspect of the system.

Earnings-dependent benefits—heterogeneous labor efficiency, small open economy.

Earnings-dependent benefits—heterogeneous labor efficiency, closed economy.

Intragenerational homogeneity

In this section, I study the value of social security as insurance against risk in an economy with incomplete asset markets. As in Hubbard (1987), uncertainty appears in the model as a positive probability of agents dying in any given period along their lives. I assume that, upon an agent's death, the government fully taxes her bequests and distributes the revenue equally among all surviving agents. Although social security plays an important insurance role against other types of risk, this straightforward way of introducing lifetime uncertainty allows me to show how the presence of risk can affect the evaluation of social security by individuals and increase the margin of error of actuarially fair measures, while avoiding intragenerational heterogeneity as a by-product. Therefore, even though this type of uncertainty could be at least partially insured against through private annuity markets, I assume that there is a market failure in the private provision of annuities. I do not make the absence of annuity markets endogenous to the model, referring the reader to Rothschild and Stiglitz (1976) and Eckstein et al. (1985) for possible explanations. When individuals face uncertain lifetimes and cannot resort to private annuities to insure for the possibility of living longer than expected, they resort to precautionary savings. Because of the inability to purchase annuities, agents have to save more than enough to finance their expected lifetime consumption if they want to avoid very low levels of consumption when they live longer than expected. Hence, even though individuals also discount the future more heavily, this type of uncertainty tends to increase capital accumulation.

As Hubbard and Judd (1987) show, when individuals face uncertainty over the lengths of their lives and there is a market failure in the private provision of annuities, there are large welfare gains associated with the public provision of annuities through social security. Therefore, an evaluation of social security that looks only at the net present value of transfers associated with this policy is erroneous because it does not account for the insurance value of social security.³

Uncertain lifetime—small open economy.

Notice that for the youngest generations, the bias of the actuarially fair measure is negative. On the one hand, they receive significantly fewer transfers because of the reduction in involuntary bequests. On the other hand, they have a very low probability of dying, and they heavily discount future periods when the insurance role of social security is more important. Therefore, they do not positively value the insurance scheme provided by the social security system.

In addition, in the closed economy (Figure 14), social security has the general equilibrium effects described in the previous sections and it will further decrease capital because individuals do not need to save as much to finance consumption if, after retirement, they live longer than expected.

Uncertain lifetime—closed economy.

For both the small open economy and the closed economy, the difference in the bias of the actuarially fair measure between the short run and the long run is significant (see Table 7). In the open economy the difference is because agents born before the implementation of social security have accumulated a level of assets different from the levels amassed had they been born under the new policy. This implies that the levels of accidental bequests received by young agents in the short run are different from the long-run ones.

Intragenerational heterogeneity

In this section, I study the bias of the actuarially fair measures across heterogeneous agents by allowing agents to differ in their efficiency levels,

.

There is some difference in the bias of the actuarially fair measure across types. While losses from the decrease in accidental bequests are identical for all agents, the insurance value of social security is lower for high-earnings agents. Their contributions are much higher than average but their annuities are the same (Figure 15).

Uncertain lifetime and heterogeneous labor efficiency—small open economy.

In the closed economy (Figure 16), the difference is mitigated by the benefits of a higher return to savings (increase in interest rate).

Uncertain lifetime and heterogeneous labor efficiency—closed economy.

In Table 8, I compare the bias of the actuarial fair measure for short-run and long-run evaluations. Again, for both the small open economy and the closed economy, the difference is significant and it is higher for the poorer agents who benefit from higher levels of redistribution through involuntary bequests in the short run.

However, to better evaluate the redistributive impact of social security and the bias of the PVNB measure in an environment with uncertainty, we should also consider that different types of individuals incur different levels of risk and have access to different instruments to insure against risk. For instance, typically, low-income agents face higher probabilities of death or health problems than high-earnings ones. Wealthier agents also have access to a wider variety of financial instruments that allow them to better insure against risk. The redistributive impact of social security would change in an environment with such features and this would also not be captured by the PVNB measure.

Intragenerational homogeneity

Implicit to actuarially fair measures is the assumption that agents can reallocate resources across their lives optimally. Consequently, if individuals' ability to do so is impeded, it results in another source of misevaluation of the welfare implications of a social security policy that reallocates resources across agents lives.

To study this type of bias, I analyze the value of social security in an economy with borrowing constraints. In this model economy, individuals cannot anticipate future incomes—labor income or social security benefits—for current consumption by accumulating negative net assets. That is,

.

In a life-cycle economy, consumption-smoothing individuals wish to borrow during the early periods of their lives when their income is insufficient to cover their desired consumption. In the presence of borrowing constraints, they are not able to borrow from future earnings and they consume the totality of their after-tax labor income, below their optimal consumption levels. Since social security is financed through a tax on labor income, social security taxes depress consumption when the liquidity constraints are binding and make it more likely that they become binding. An increase in social security benefits and therefore on taxation might make agents worse off even if it increases their lifetime resources as asset market imperfections limit the extent to which agents can allocate their resources optimally over their lifetimes. Actuarially fair measures cannot account for the implications of the liquidity constraints faced by individuals.

This is evident in a small open economy. In Figure 17, we clearly see the bias of the actuarially fair measure becoming very large for agents for whom the borrowing constraint is active. Whereas in the absence of these constraints the bias was almost flat for young agents (see Figure 3), now it is initially very high with the actuarially fair measure grossly overestimating the cost of social security for these agents.

Borrowing constraints—small open economy.

This result might seem surprising because we could think that, to compute the welfare implications of this policy, we would have to subtract the utility cost of taxing constrained agents from the net present value of benefits. The welfare measure would therefore be lower than the actuarially fair measure. Because

, a borrowing-constrained agent's valuation of current resources relative to future resources is different from the market real interest rate and the difference increases exponentially with the number of periods an agent is constrained. That is, the discount rate used to compute the actuarially fair measures is not an accurate measure of a constrained agent's evaluation of future resources. When an agent cannot borrow, she values current resources much more than resources that are only made available to her in the future. In the case of social security, she values highly the negative impact of current taxation and she strongly discounts future payments and benefits. So, to correctly measure the value of social security for an agent, much more weight has to be given to the impact of the social security taxes that she has to pay in the initial periods of her life than to future transfers.

Furthermore, because the agent cannot borrow, her initial consumption is very low and her marginal utility very high and any small transfer has a significant impact on her utility. Hence, even though social security can have a much higher impact on the utility of a liquidity-constrained individual than on the utility of an individual that is not constrained, the absolute value of the current welfare measure can be much lower for the former than for the latter. Furthermore, the absolute value of this welfare measure is much lower than the actuarially fair measure for agents for whom the borrowing constraint is active. Consequently, the use of the current welfare measure to evaluate the impact of social security across agents can be misleading.

Therefore, I construct a measure that circumvents this problem and allows a straightforward comparison of the welfare impact of social security on heterogeneous individuals. As before, I compute the current value of the additional amount of goods to be given to an individual to make her able to attain the same level of lifetime utility in the economy with social security as in the economy without the social security system. Now, however, I delay the transfer to a period in time when the individual is no longer borrowing constrained; moreover, I also delay any negative transfers that make the constraint binding. I then compute the present value of the transfer using the equilibrium real interest rate as the discount rate. Hence forth, I call the resulting measure the “discounted” welfare measure.

Figure 18 shows the bias of the actuarially fair measure relative to the “discounted” welfare measure. We observe that the bias is very large for agents for whom the borrowing constraint is active. Whereas this bias was flat for young agents in the absence of liquidity constraints, now it is very high, with the actuarially fair measure grossly underestimating the cost of social security for these agents. The increase in the bias (in absolute value) is due to the high cost of social security taxation for borrowing-constrained individuals. To make those individuals as well off as with social security, we need to give them an amount equal to the net present value of benefits, and take away from them an amount that accounts for the negative impact of having to pay social security taxes when they are liquidity constrained.

Borrowing constraints—small open economy.

In a closed economy, the bias curve (see Figure 19) is not significantly different from the one in the benchmark economy (see Figure 4). Even though the negative impact of social security taxation on borrowing-constrained individuals is still present, it is mitigated by the impact of social security on factor prices. Upon the introduction of a pay-as-you-go social security system, agents' lifetime income decreases, future wages decrease while current real wage rate increases dramatically (see Figure 1). The decrease in lifetime income implies a higher decrease in current consumption. The increase in current wages and the decrease in future wages means that labor income profiles are smoother and the need to reallocate resources intertemporally by borrowing is lower. Hence, the borrowing constraint is less binding than in an open economy. Whereas before the implementation of social security, agents only started accumulating assets after age 29, upon the implementation of the system they start at age 26 with a significant reduction in the burden of the constraints, as can be seen in Figure 20, where I represent the lifetime asset accumulation of an agent in the steady state of the economy without social security and along the path of the economy once social security is implemented. Therefore, the impact of social security on borrowing-constrained agents is significantly less dramatic than in the previous case when factor prices were unchanged, and it can even be positive. The effect of the change in factor prices on borrowing-constrained individuals seems to be overlooked in the literature. For instance, Hubbard and Judd (1987), describe in some detail the importance of borrowing constraints in evaluating the welfare impact of social security but they do not mention this feature, which is also not uncovered by Conesa and Krueger (1999) where it combines with the effects of lifetime uncertainty and intragenerational heterogeneity.

Borrowing constraints—closed economy.

Borrowing constraints—closed economy.

As usual, the bias more than doubles from the short run to the long run in the closed economy (see Table 9). Furthermore, for the small open economy there is a dramatic increase in the bias relative to the benchmark economy case (see Table 2)—the size of the bias doubles—whereas for the closed economy, the difference is significant but not as dramatic, which reinforces the findings from the short-run analysis.

Intragenerational heterogeneity

Once again, I differentiate agents according to their efficiency levels,

, to see how the bias differs across different types of agents.

The graphs (Figures 21 and 24) show the bias of the PVNB relative to the welfare measure for different types of agents across ages. It is again clear that this bias differs across types.

Borrowing constraints and heterogeneous labor efficiency—small open economy.

In the small open economy (Figure 21), for high-earnings agents, the age-profile of the actuarially fair measure bias is very similar to the one computed for the economy without borrowing constraints. However, agents with lower efficiency levels are relatively more hurt by the social security system than the PVNB measure indicates because their welfare levels are also affected by the impossibility of borrowing against future income. This not only occurs in the initial periods of their lives, but, as we can see in Figure 21, it becomes more important for the poorer middle-aged who would like to borrow against future retirement benefits, namely agents of types 1, 2, and 3. The actuarially fair measure grossly misestimates the impact of social security on these agents' well-being. Even though social security redistributes resources from the wealthy to the poor, as captured by the PVNB measure, it taxes the poor when they work and, because of the borrowing constraint, cannot anticipate their future benefits to attain higher levels of consumption prior to retirement.

In Figure 22, I show the asset accumulation of agents who are 50 years old when social security is implemented. I represent the asset accumulation profiles for the poorest types in the borrowing-constrained economy and in the unconstrained one (the benchmark economy). We can see that the poorest agents would like to borrow against their future income when they are young. When social security is implemented, they are 50 years old and start consuming out of their assets. They dissave and wish to borrow against their social security benefits.

Borrowing constraints and heterogeneous labor efficiency—small open economy.

In Figure 23, I show the asset accumulation of agents who are born in the implementation period. Again, I represent the asset accumulation profiles for the poorest types in the borrowing-constrained economy and in the unconstrained one (the benchmark economy). We can see that the two poorest agent types would like to be in debt for the duration of their lives, borrowing against their labor income and social security payments. Other, less poor agents wish to borrow against their labor income initially and, by the time they are middle-aged, have a positive net asset position. They would then dissave and borrow against their social security benefits. The actual response of the liquidity-constrained agents to the implementation of social security is clearly different from what would be optimal, and this is not accounted for by the actuarially fair measures.

Borrowing constraints and heterogeneous labor efficiency—small open economy.

In the closed economy (Figure 24) as in the case with intragenerational homogeneity, the biases are very similar to the ones in the benchmark economy. The only significant differences are for the poorest workers in the economy who would like to borrow against their social security benefits and, as in the previous cases, the actuarially fair measure does not take into account that they cannot do so.

Borrowing constraints and heterogeneous labor efficiency—closed economy.

In Table 10, I compare the bias of the actuarially fair measure for short-run and long-run evaluations. Again, for the small open economy, there is no change in the bias from the short run to the long run, but for the closed economy the difference is significant. Furthermore, while some of the less effective agents born in the implementation period benefit from social security (the ones that benefit from the redistribution of resources but are not heavily penalized by the borrowing constraint), agents born in the steady state of the economy without social security are better off than the ones born in the steady state of the economy with social security, independently of their types.