Mandatory retirement savings in the presence of an informal labor market

This paper shows how mandating workers to save more for retirement can lead them to work informally and save less. Consider a worker who is more productive in the formal sector but works informally to avoid mandatory retirement contributions. Lowering the contribution rate (the share of wages mandated to be saved) will paradoxically increase her retirement savings. The reason for this is that working informally acts as borrowing against mandatory savings. The implicit cost of such borrowing, and hence the opportunity cost of working informally, rises as the contribution rate drops. This creates a substitution effect favoring formal work, driving the worker towards the formal sector. As her formal income increases, the base for her mandatory contributions rises, expanding her retirement savings. Therefore, the optimal contribution rate is no greater than the highest contribution rate under which the worker prefers to work exclusively in the formal sector.


Introduction
A key motivation for social security is the concern that people do not save enough for their retirement.This undersaving is explained by present bias, which is a behavioral bias that leads individuals to consume too much in the present relative to what their future selves would prefer (Feldstein 1985; Laibson 1998; İmrohoroglu et al 2003).The existence of this bias might lead one to conclude that people should be mandated to save a share of their labor income in their own retirement accounts.However, mandatory savings are ineffectual unless a borrowing constraint forbids agents from ending up with the same net savings as if there were no mandatory savings (Aaron 1966;  Blanchard and Fisher 1989).Nevertheless, since borrowing constraints appear to be pervasive, it is not unreasonable to think that mandatory savings are indeed an effective policy to increase retirement savings (Diamond 1977; Feldstein 2005; Kaplow 2010).In fact, starting with Chile in 1981, multiple countries have replaced their public payas-you-go pension systems with a fully funded system in which workers are mandated to save a share of their labor income in privately managed pension funds (Tapia and  Yermo 2007).
Unfortunately, the emerging economies that have implemented this type of reform have not seen their workers contributing to their retirement saving accounts as frequently as expected (Mesa-Lago et al. 2021).Instead, many workers still work in the informal sector, which consists of economic activities where no taxes or social security contributions are enforced.In Chile, where the reform was implemented in 1981, the share of the working-age population contributing to social security was 64% in 1979,  47% in 1990, and 65% in 2013.In Peru, where the reform was implemented in 1993, the share was 34% in 1992, 9% in 1995, and 28% in 2018. 1 Moreover, compliance with social security contributions is heterogeneous across the income distribution.In particular, the likelihood of contributing to social security increases with income.This is illustrated in Table 1 for the cases of Chile, Colombia, and Peru, three countries with mandatory retirement savings.In all three countries, the share of workers contributing to social security is the lowest at the bottom of the income distribution.After 40 years of implementing pension reforms based on mandatory savings, it is worth asking whether mandatory savings is an effective policy for increasing retirement consumption.The results I present suggest that at least some middle-income workers may have a good reason to be skeptical about the benefits of being mandated to save for their retirement, even if they do suffer from undersaving. 2 In this paper, I formally study the effects of mandating present-biased workers to save a fraction of their formal wages under the shadow of an informal labor market.As a motivating example, consider an agent who can secure a job in the formal sector for $10, 000 per year and face a contribution rate-the share of labor income required to be saved-of 20%.Alternatively, the agent can work informally for $9000 and face no mandatory savings.If there is no way to borrow against the $2000 mandated savings and the agent only cares about her present consumption, then she will choose the informal job despite the lower wage.From the perspective of such an impatient agent, taking the informal job is equivalent to remaining working formally but borrowing $1000 against the $2000 savings.As this example illustrates, working informally is equivalent to borrowing expensively.The previous example is extreme in the sense of assuming that the agent does not care at all about her future.I show in this paper how mandatory contributions to retirement saving accounts may still mislead workers with less extreme forms of present bias into working informally as a way to increase their present consumption at the expense of their retirement savings.This is because any present-biased worker with an active borrowing constraint perceives a fraction of the marginal contribution as a tax on her formal labor income.This is the case despite that every dollar contributed is saved in the worker's own individual retirement account.I show that a worker who is more productive in the formal sector will work informally when the effective tax is higher than the wage gap between the formal and informal sectors.If this is the case, then a lower contribution rate will reduce the effective tax and get the worker back into the formal sector.This will lead to higher retirement savings, despite the mandate to save a lower share of formal income.The analysis leads to the conclusion that the optimal contribution rate for a given worker is bounded by the highest rate that does not drive her to the informal sector.Unfortunately, this bound becomes lower as the present bias worsens.
Some standard assumptions regarding the preferences, present bias, and opportunity cost of working informally help to characterize the effect of a uniform contribution rate across the income distribution.To be effective, the uniform contribution rate has to be high enough to activate the borrowing constraints and to increase anyone's net savings.However, if this is the case, then at least some middle-income workers will be working informally.These workers will be better off with a lower contribution rate.
I illustrate the quantitative magnitude of the effects by carrying out a numerical exercise for Colombia, where around half the working population works informally.This exercise is based on Doligalski and Rojas (2023), who estimate both the expected formal and informal earnings for the Colombian working population.The numerical exercise suggests that (a) workers wealthier than 25.5% of the working population but poorer than the other 57.1% will be better off with a contribution rate lower than the current rate of 16%, and (b) a lower contribution rate will have small effects on their total labor earnings and their current consumption, but important effects on their formal labor earnings, savings, and retirement consumption.
Many studies have examined the role of social security in solving the undersaving problem generated by present bias.Examples include (Feldstein 1985; Laibson  1998; İmrohoroglu et al 2003; Amador et al. 2006; Cremer et al. 2008; Cremer 2009;  Andersen and Bhattacharya 2011; Bouchard St-Amant and Garon 2015), and Chu and  Cheng (2019).However, these studies do not account for the existence of an informal labor market as a mechanism to avoid social security contributions.Indeed, a different branch of the literature studies labor supply decisions in the presence of an informal sector.Examples in this branch include (Amaral and Quintin 2006; Galiani and  Weinschelbaum 2012; La Porta and Shleifer 2014; Ulyssea 2018), and Bobba et al.  (2021).However, few works have studied social security in a framework where households simultaneously decide how much to save and how much to work informally.Notable exceptions are Joubert 2015; Becerra 2017; McKiernan et al. 2020;  Ferreira and Parente 2020; Moreno 2021, and Tkhir (2021), who study the effect of social security reforms in the context of emerging economies.The reforms studied in these works involve intragenerational and intergenerational transfers that naturally distort both saving and labor supply decisions.In this paper, I take a step back and focus on a situation where there are no transfers other than those between present and future selves.This allows me to obtain analytical rather than numerical results and to pinpoint the effect of mandatory savings across the income distribution without the confounding effect of transfers between agents.More important, my paper conveys a more general message to any policymaker facing an informal labor market: If the contribution rate is high enough to drive someone who is more productive formally to work informally, then this person will be better off with a lower contribution rate.
My paper is also related to the empirical literature on the effect of pension wealth on voluntary savings.Examples include (Poterba et al. 1996; Gale 1998; Attanasio and  Rohwedder 2003; Chetty et al. 2014), and Lachowska and Myck (2018).The concern in this literature is how voluntary savings react to an additional dollar of pension wealth.Although the literature is well aware of the need to assess the effect of pension wealth on all assets and liabilities to properly measure any crowding out, it does not account for the reallocation of labor to the informal sector as an alternative way of borrowing.In this paper, mandatory savings do not crowd out voluntary savings when the borrowing constraint is active, but they may reallocate labor in a way equivalent to borrowing at an interest rate higher than the interest rate on savings.In this sense, this paper parallels the work of authors such as Valdés-Prieto (2004) and Andersen  and Bhattacharya (year), who study environments where agents are forced to save but can still borrow at an interest rate above the interest rate on savings.My paper additionally shows how the contribution rate affects the equivalent cost of borrowing, thereby translating into novel and unexpected ways in which the contribution rate affects the consumption path.Interestingly, my paper has some parallels with the work of Bilal and Rossi-Hansberg (2021), in which financially constrained agents "borrow" by moving physically to a location where they pay lower rent, foregoing the prospects of living in a more privileged location in exchange for smoothing their consumption path.
The remainder of this paper is organized as follows.Section 2 describes the model and its analytical results.Section 3 presents some numerical examples on the effects of decreasing the contribution rate using parameters for Colombia.Section 4 presents final remarks and suggests future paths of research.

The model
Consider a continuum of agents distributed in [0, 1] with full support.Agents live for two periods.They work in the first period and retire in the second.Hence, their first period of life is their working age, and their second is their retirement age.Leisure gives them no utility, and labor is their only endowment.The agent's location i ∈ [0, 1] defines her formal wage w f i and her informal wage w s i .The wage w f i (w s i ) represents her (in)formal earnings if she spends all her working age in the (in)formal sector.If an agent i spends a fraction z ∈ [0, 1] of her working age in the formal sector, then her formal earnings will be w f i z, her informal earnings will be w s i (1 − z), and her overall labor earnings before any taxes and contributions will be w Spending a share z of the working age working formally is equivalent to choosing a level of formal earnings y = w f i z, where y ∈ [0, w f i ].Therefore, the overall labor earnings before any taxes and contributions can be rewritten as For analytical convenience, I use y instead of z as the choice variable.
Agents are forced to save a fraction τ ∈ [0, 1) of their formal earnings y.Therefore, the disposable income in the first period of life is The mandatory savings are invested in a riskless asset that yields a gross return R > 0, which implies that if an agent i chooses a formal income y, then she receives Rτ y dollars in benefits for her retirement.Note that τ does not represent a tax rate: the benefits (Rτ y) discounted with a factor R −1 are equal to the contributions (τ y).Therefore, it is not straightforward to argue that workers will perceive the contributions as a tax and that they will have an incentive to avoid them. 3 worker may also voluntarily save x dollars at a gross return R, but she is subject to the borrowing constraint x ≥ 0. This constraint avoids the irrelevance of mandatory savings: If the agent could borrow at a rate R, then any mandatory savings would be offset by the equivalent borrowing, thereby implying that her net savings would be the same as if there were no mandatory savings.The gross return R is exogenous, as in a small open economy. 4n agent who consumes c 1 dollars in her working age and c 2 dollars in her retirement age has an experienced lifetime utility equal to u(c 1 ) + δu(c 2 ).I assume that the instantaneous utility function u is (1) increasing, (2) strictly concave, (3) twice differentiable, and (4) satisfies lim c→0 u (c) = ∞.The parameter δ ∈ (0, 1) is the subjective factor that the agent should be using to discount the future instantaneous utility u(c 2 ).Nonetheless, she actually uses a subjective factor δβ, where β ∈ (0, 1) represents her present-bias parameter.Her working-age decisions are driven by her decision lifetime utility, which is given by u(c 1 ) + δβu(c 2 ).5 I assume that there are no commitment devices the agent can use to maximize her experienced rather than her decision utility.
The solution for an agent who is more productive informally trivially involves working exclusively in the informal sector, regardless of the contribution rate.In contrast, an agent who is more productive formally may work informally to avoid mandatory savings.I focus for now on the case of an agent who is more productive in the formal sector (i.e., w f i > w s i ).The agent maximizes her decision utility subject to the feasibility constraints.Formally, she solves the following problem: subject to: The first-order conditions of Problem Eq. 1 are as follows: , with equality if ŷi < w f i . (3) The left-hand side of inequalities (Eqs. 2 and 3) is the effective pension tax rate (Feldstein and Liebman (2002)).Moreover, the right-hand side of inequality (Eq. 3) is the productivity gap between working in the formal and informal sectors.These two concepts are key since the worker decides where to work by comparing the effective pension tax rate with her productivity gap.The existence of an effective pension tax depends on the factor used to discount a dollar of retirement benefits.As usual, this factor is given by δβu ( ĉ2i ) u ( ĉ1i ) .If there are any voluntary savings left, then Eq. 2 implies that this factor equals R −1 .In this case, no mandatory contributions to the retirement savings accounts are perceived as a tax.However, with no voluntary savings left and an active borrowing constraint, benefits are discounted with a factor below R −1 .This implies a positive effective pension tax rate.
Note that the decisions regarding voluntary savings and the allocation of labor between the formal and informal sectors become intertwined.Since an active borrowing constraint is necessary for an effective pension tax, it is never optimal to work informally while having any voluntary savings left.This is formally stated in the following lemma: Lemma 1 The solution to Problem Eq. 1 when w f i > w s i has the following properties: Proof It follows from the first-order conditions of Problem Eq. 1.Another necessary condition for working informally is that 1 − − τ < 0. In this case, the agent can increase her take-home payment by working informally, despite that this comes at the expense of her retirement consumption.As such, working in the informal sector becomes an alternative way of borrowing.
Nonetheless, there is no reason to work informally as long as there are any voluntary savings left.In this case, the worker can still trade R dollars of retirement consumption for one dollar of working-age consumption.However, once the voluntary savings are depleted and the borrowing constraint is active, the worker might consider working informally as a way to increase her take-home payment and consume more in the present.By working informally, the worker can trade Rτ dollars of retirement con- − τ dollars of working-age consumption. 6Therefore, the borrowing cost when no voluntary savings are left is given by − 1 − This cost is greater than R, however.This shows that working informally is implicitly an expensive way of borrowing.Crucially, the borrowing cost is decreasing in the contribution rate τ .Paradoxically, "borrowing" becomes less expensive when the worker is compelled to save more.As a result, a higher contribution rate creates a substitution effect in favor of working-age consumption and against retirement consumption.This effect is the driver of the comparative statics shown further below.
The question that follows is under which condition the borrowing constraint is active.Naturally, this condition is that the contribution rate is above the saving rate that the present-biased self prefers the most.Let τ d i be the saving rate that agent i will choose to maximize her decision utility if there were no mandatory savings.I label this rate as the biased saving rate for short.The biased saving rate is characterized by the following Euler condition: As long as the contribution rate τ is below the biased saving rate τ d i , the net savings are always a share τ d i of formal income.However, once the contribution rate is above the biased saving rate, the borrowing constraint becomes active.In this case, one might imagine that net savings increase with a higher contribution rate.This is not necessarily the case, however, as suggested previously and shown further below.
An effective pension tax is necessary but not sufficient to drive the worker into the informal sector.Whether the agent works informally depends on her productivity gap.If the effective pension tax rate faced when working exclusively in the formal sector is not higher than the productivity gap, then there are no reasons to work informally.However, this is not the case if continuing to work exclusively in the formal sector implies an effective pension tax rate above the productivity gap.In this case, the agent will find it optimal-from the perspective of her decision utility-to work for at least some time informally.This allows her to increase her working-age consumption at the expense of her retirement consumption.This increases the marginal rate of substitution between future and present consumption up to the point where the effective pension tax rate falls back to equal the productivity gap.The following lemma formally characterizes this solution: Lemma 2 Define t i as the effective pension tax rate for agent i when she works exclusively in the formal sector and her borrowing constraint is active: (5) The solution to Problem Eq. 1 is as follows: and xi is the solution to the following equation: and ŷi is the solution to the following equation: Lemma 2 implies that there is an upper bound on the contribution rate that i tolerates before moving to the informal sector.I denote this bound as τ s i and label it the "savings-maximizing contribution rate" for reasons shown further below.The savings-maximizing contribution rate is the contribution rate at which the effective pension tax rate faced when working fully formally is equal to the productivity gap.It follows from Eq. 5 that the savings-maximizing contribution rate is defined by the following equation: Note that the savings-maximizing contribution rate τ s i is above the biased saving rate τ d i .This implies that the agent will still tolerate a contribution rate just above the saving rate preferred by her present-biased self before considering migrating to the informal sector.
It is worth highlighting that the worker may work informally as a way to smooth her consumption.If the contribution rate τ is above the savings-maximizing contribution rate τ s i , then the agent works informally.Although every dollar contributed is saved in the worker's savings account, the worker has an active borrowing constraint.Therefore, the factor she uses to discount a dollar of retirement consumption is lower than R −1 .Nonetheless, this factor is still higher than the factor she would use if she were working exclusively in the formal sector: In other words, working informally relaxes the borrowing constraint, allowing the worker to increase her decision utility by increasing her working-age consumption at the expense of her retirement consumption.Naturally, this decision may not be optimal from the perspective of her experienced utility.

Comparative statics
The effect of the contribution rate (τ ) on the level of mandatory savings (τ ŷi ) depends on whether the agent works formally for the entirety of her working age (y = w f i ) or for at least some time informally (y < w f i ).Even if the borrowing constraint is active, the agent may still tolerate a higher contribution rate before working any time informally.Nonetheless, if the agent is already working informally, then her mandatory savings fall as the contribution rate increases.This happens because a higher contribution rate (τ ) is insufficient to offset a lower formal income ( ŷi ), meaning that the retirement savings (τ ŷi ) are lower.This is formally shown in the following proposition: Proposition 1 Consider the case where w f i > w s i .Define i as the elasticity of formal income ŷi with respect to the contribution rate τ : The effect of the contribution rate (τ ) on the level of mandatory savings (τ ŷi ) is intimately related to the effect of the contribution rate on retirement consumption ( ĉ2i ).This is because they are one and the same when the borrowing constraint is active.As previously shown, an agent working informally is implicitly borrowing each dollar of present consumption at a cost of − 1 − Rτ dollars of future consumption.Paradoxically, a higher contribution rate makes borrowing cheaper, at least for those who are already working informally.However, the only way to borrow against retirement consumption when there are no voluntary savings left is by reducing the level of mandatory savings (τ ŷi ).The only way to achieve this is by decreasing formal income ( ŷi ) in a way that offsets the higher contribution rate (τ ).
The effect of the contribution rate on present and future consumption might be the complete opposite of what a mechanical analysis that ignores the behavioral responses would suggest.Suppose that the contribution rate increases from τ to τ , as illustrated by Fig. 1.If there were no opportunities to work informally and the borrowing constraint were active, then the consumption bundle would shift from Working-age consumption would fall, and retirement consumption would increase.However, if there is a chance of working informally and the contribution rate τ is above the savings-maximizing contribution rate τ s i , then the Fig. 1 Effect of the contribution rate on present and future consumption effect on retirement consumption is actually the opposite.As the contribution rate increases from τ to τ , the consumption bundle shifts from ( ĉ1i , ĉ2i ) to ( ĉ 1i , ĉ 2i ).
Retirement consumption falls from ĉ2i to ĉ 2i because of both income and substitution effects.Even more surprising, nothing rules out an increase in present consumption from ĉ1i to ĉ 1i .This is because the income and substitution effects go in opposite directions.These results are formally shown in the following proposition: Proposition 2 Consider the case where which has an ambiguous sign.
Proof It follows from Lemma 1 and Proposition 1.
The effect of the contribution rate on welfare-as defined not by the decision utility but by the experienced utility-is not straightforward because of the ambiguous effect of the contribution rate on present consumption.Nevertheless, it can be shown that agents who are more productive formally but who are working partially in the informal sector will be better off with a lower contribution rate.To show this, define τ e i as the saving rate that maximizes the experienced utility if migrating to the informal sector were impossible.I label this rate the ideal saving rate.The ideal saving rate τ e i is defined by the following equation: Whether a worker can be forced to save what is best for her depends on the inequality between the ideal saving rate and the savings-maximizing contribution rate.This inequality depends on the productivity gap.If the productivity gap is large enough, then the savings-maximizing contribution rate is above the ideal saving rate.In this case, the best that can be done for the experienced utility of an agent is to set the contribution rate equal to the ideal saving rate.The present-biased agent will perceive the contributions as a tax, but the productivity gap will deter her from working informally.However, if the productivity gap is not large enough, then the savingsmaximizing contribution rate is below the ideal saving rate.In this case, the best that can be done for an agent's experienced utility is to set the contribution rate equal to her savings-maximizing contribution rate.If the contribution rate were higher than the savings-maximizing contribution rate, the worker would work informally, and both her retirement savings and her welfare would be lower.In other words, the savingsmaximizing contribution rate of agent i is a bound on her optimal contribution rate.
The following proposition formalizes this point: Proposition 3 Define W i := u( ĉ1i ) + δu( ĉ2i ) as the indirect experienced utility of an agent i for whom In summary, the contribution rate begins to have detrimental effects once it reaches a level at which the effective tax rate of being exclusively formal exceeds the productivity gap.These effects are as follows: (1) the worker begins to work informally, (2) her savings, retirement consumption, and welfare decrease as the contribution rate increases, and (3) her working-age consumption might actually increase with the contribution rate.In other words, well-intentioned efforts to help a present-biased worker with her undersaving problem begin to backfire if these efforts are overly demanding.

Effect of present bias on the optimal contribution rate
Interestingly, the optimal contribution rate for agent i-given by min{τ e i , τ s i }-may fall as present bias worsens.Recall that the higher the present bias is, the lower the parameter β.The parameter β does not affect the ideal saving rate τ e i , but it does affect the savings-maximizing contribution rate τ s i .In particular, the highest contribution rate that i tolerates before moving to the informal sector declines as present bias worsens. 7If the present bias of agent i is bad enough to drive the savings-maximizing contribution rate τ s i below the ideal saving rate τ e i , then the optimal contribution rate for agent i decreases as her present bias worsens.Intuitively, the retirement benefits are discounted more severely as present bias worsens, increasing the temptation to begin working informally.This leads to the necessity of decreasing the contribution rate to deter the agent from working informally.Eventually, if the agent becomes fully myopic (β = 0), then the best that can be done for her experienced utility is to set her contribution rate equal to her productivity gap, which is the savings-maximizing contribution rate under full myopia. 8igure 2 illustrates the effect of present bias on the savings-maximizing contribution rate (τ s i ), the ideal saving rate (τ e i ), and therefore the optimal contribution rate (min{τ e i , τ s i }).If the present bias is not severe (β is close to one), then the optimal contribution rate is the ideal saving rate τ e i .However, if the present bias is severe (β is close to zero), then the optimal contribution rate is the savings-maximizing contribution rate τ s i .Figure 2 also presents the relationship between present bias and the biased saving rate τ d i .Naturally, the gap between the ideal saving rate τ e i and the biased saving rate τ d i closes as the present bias fades.Furthermore, the savings-maximizing contribution rate τ s i is always above the biased saving rate τ d i , as previously stated.However, the gap between the savings-maximizing contribution rate and the biased saving rate crucially depends on the productivity gap 1 − . If earnings in the formal and informal sectors become more similar, then the gap between the savings-maximizing rate and the biased saving rate narrows.As this happens, it becomes more difficult to force the agent to save more.Eventually, if there is no productivity gap, then the savingsmaximizing contribution rate and the biased saving rate coincide.These insights are important for the results shown further below.

Effects across the income distribution
A natural starting point for mandatory savings-and the most frequently implemented policy-is to have a uniform contribution rate τ for all workers in the economy: regardless of the wage, all workers are required to save the same share of their wages.This makes perfect sense if everyone suffers from the same degree of present bias, preferences are homothetic, and there is no opportunity to work informally.However, if there is an informal sector and the opportunity cost of working formally is heterogeneous across the income distribution, then a uniform contribution rate will have heterogeneous effects across the population.Is the uniform rate of mandatory savings helping the poor?The wealthy?Neither?
The answer to this question depends on how the population is distributed between formal and informal activities.I assume that both w f i and w s i are continuous and increasing in i, and thus, agents are sorted in [0, 1] according to their productivity.I assume that the productivity gap increases with i, meaning that the opportunity cost of working informally is higher for wealthier people.The assumption is consistent with the stylized facts of Table 1 and is also corroborated in the data by authors such as Almeida and Carneiro (2012) and Doligalski and Rojas (2023).Intuitively, it is less costly to transition from being a formal construction worker to an informal street vendor than to transition from being a chief financial officer to an accountant at a small, informal firm.
Assumption 1 also implies that if w f 0 ≤ w s 0 and w f 1 > w s 1 , then there is a unique agent a ∈ [0, 1) such that w f a = w s a .This partitions the set of agents [0, 1] into two subsets: (i) the set [0, a] of agents whose informal productivity is at least equal to their formal productivity, and (ii) the set (a, 1] of agents who are more productive in the formal sector. 9Crucially, note that the productivity gap 1 − The solution for agents in [0, a] trivially involves working exclusively in the informal sector.However, it is not necessarily the case that all agents in (a, 1] will work exclusively in the formal sector.This is due to the incentive to avoid at least part of the contributions.For ease of exposition, I assume that the productivity gap for workers at the top of the income distribution is always sufficiently high to not offer them any incentives to work informally. I further assume that preferences are represented by a CRRA utility function.In the context of this paper, this assumption implies that the biased saving rate τ d i , the ideal saving rate τ e i , and the effective tax rate t i are the same for everyone (i.e., independent of i).I drop the subindex i from these variables henceforward.
I also assume that the present-bias parameter β is the same for everyone.Nonetheless, there is evidence that present bias worsens as labor productivity decreases. 10That being said, all results presented below can be easily accommodated for this case.This is because all the results rely on a savings-maximizing contribution rate τ s i that increases with i-that is, poorer workers are less tolerant of mandatory savings.This is the case if Assumption 1 holds and the present-bias parameter increases with i. Nonetheless, I show the results for the case of homogeneous present bias for ease of exposition.
Assumptions 1 and 2 allow to identify the allocation of agents in [0, 1] between formal and informal activities as a function of the uniform contribution rate τ .Naturally, all workers in [0, a) work exclusively in the informal sector, regardless of the contribution rate.For agents in (a, 1], the solution depends on whether the contribution rate activates the borrowing constraint, as stated in the following Lemma: Lemma 3 Under assumptions 1 and 2, the following holds: 9 Naturally, if w f 0 = w s 0 , then a = 0 and so no worker is more productive informally. 10See Moser and Olea de Souza e Silva (2019) and Hosseini and Shourideh (2019), for example.
Figure 3 illustrates the results of Lemma 3 for the case where τ > τ d , which is the case where the contribution has an actual effect on anything relevant.Recall that agents are sorted from poorest to richest in [0, 1].Agents will be full-time formal, sometimes formal or full-time informal depending on (i) their productivity gap and (ii) the effective pension tax when working full-time formally.First, agents in [0, a] will be exclusively in the informal sector, as previously stated.Second, individuals in (a, b τ ) will not work all their working age in the formal sector.This is because if they were working exclusively in the formal sector, then the effective pension tax would be higher than their productivity gap.Third, the mass of individuals in [b τ , 1] will see no advantage in working anytime in the informal sector, despite an effective pension tax.In short, the richest (poorest) workers will be fully (in)formal, in line with their productivity.Nonetheless, middle-income workers will be partially informal, even though they are more productive in the formal sector.
As stated in Lemma 3, a sufficiently high uniform contribution rate τ implies that some workers will be working informally despite their higher productivity in the formal sector.Proposition 3 implies that these agents will be better off with a lower contribution rate.This is formally stated in the following proposition: Proposition 4 Under Assumptions 1 and 2, the following holds: Proof It follows from Proposition 3 and Lemma 3.
It is worth emphasizing the result in Proposition 4. If a uniform contribution rate is high enough to be helping anyone, then it is necessarily too high for some others.Helping some people to have a better retirement comes at the cost of displacing some The effect of present bias on informality for a given contribution rate can be derived from Lemma 3. A contribution rate τ greater than the biased saving rate τ d implies that everyone in (a, 1] has an active borrowing constraint.As a consequence, if these workers were working exclusively in the formal sector, they would face the effective tax rate t defined in Eq. 5. Note from Eq. 5 that the effective tax rate t increases as β decreases.This means that the effective tax rate t increases as the present bias worsens.Intuitively, a worsening in present bias lowers the perceived benefits of any retirement savings that cannot be offset by traditional borrowing.Recall from Lemma 3 that if τ > τ d , then there is an agent b τ ∈ (a, 1) such that everyone in [0, b τ ) will at least sometimes work informally.Furthermore, b τ increases if t increases, as shown in Fig. 3. Therefore, for a given contribution rate τ greater than τ d , a worsening in present bias increases the share of people working at least sometimes informally. 12onsider the effects of implementing a uniform contribution rate equal to the ideal saving rate τ e .This is the saving rate chosen by workers if they were maximizing their experienced rather than their decision utility.Therefore, it is not implausible to conceive of a naive policymaker who implements a uniform contribution rate τ e to address present bias.Nevertheless, this policy ignores the temptation to work informally.Given the lack of commitment devices, the naive policy drives at least some otherwise formal workers into informality.It follows from Eqs. 5 and 10 that the effective pension tax rate for anyone working formally full-time will be (1 − β) τ e .As a consequence, all workers with a productivity gap between 0 and (1 − β) τ e will be working at least some time informally.These workers will be better off with a contribution rate lower than τ e .
The drawbacks of implementing a uniform contribution rate equal to the ideal saving rate τ e increase as the problem of present bias worsens.The workers at the top of the income distribution who stay working exclusively in the formal sector will be achieving the highest feasible experienced utility.Nonetheless, this set shrinks as the present bias worsens.As this happens, more workers who are more productive formally are found working at least sometimes informally.Since these workers are better off with a lower contribution rate, the case for lowering the contribution rate becomes stronger.
As the previous paragraph suggests, it is never optimal for a social planner facing present bias to set a uniform contribution rate equal to the ideal saving rate τ e .This is formally stated in the following proposition:

Proposition 5 Let f i be the density function on individual types. Consider a social planner with Pareto weights λ i , an increasing continuous transformation of experi-enced utilities G(•) and the following social welfare function:
Suppose that Assumptions 1 and 2 hold and that β < 1.
then the uniform contribution rate τ that maximizes the social welfare function ( 11) is below the ideal saving rate τ e .
Proof See Appendix A.

Numerical examples
Some simple calculations suggest that a uniform contribution rate may indeed generate welfare losses for a significant share of the working populations of at least some of the emerging economies that have implemented this policy.Consider the case of Colombia, where approximately half the working population works informally.Table 1 shows the similarities between Colombia and Latin America regarding the share of informal workers by quintile of the income distribution.This makes Colombia an ideal candidate for a case study.The current contribution rate in Colombia is 16%.
If this rate corresponds to the naive policy τ e , then the effective pension tax rate for anyone always working formally is (1 − β) × 16%.Imai et al. (2021) report an average of 0.97 for the yearly present-bias parameter across the studies they consider in their metastudy.An estimate of 0.254 for the parameter β can then be obtained by taking this average and assuming that the working age lasts for 45 years.This estimate suggests an effective pension tax rate of 11.9%.As a consequence, anyone who is not at least 11.9% more productive formally will work at least some time informally.According to the Mincerian estimates by Doligalski and Rojas (2023), the bottom 25.5% of the earnings distribution is more productive informally, while the top 57.1% is at least 11.9% more productive formally.Therefore, 17.4% of the working population would have more savings, more retirement consumption, and more welfare if their contribution rate were lower than the current 16%.
The magnitude of the effect of reducing the contribution rate depends on the rest of the parameters of the model.These parameters are the return on savings R, the inverse of the elasticity of substitution σ , and the subjective factor δ. Table 2  parameter values used to construct the counterfactuals.The real net return in the last 5 years for the best-performing retirement portfolio in Colombia has been 4.4% per year, according to Parra et al. (2022).Compounding this return for 45 years yields an estimate of R of 6.94.The inverse of the elasticity of substitution σ is taken from Granda and Hamann (2015).Finally, the subjective factor δ is calibrated using Eq. 10 to match an ideal saving rate τ e of 16%.Under these parameters, every present-biased agent i ∈ [0, 1] would prefer a saving rate τ d equal to 9.6%.This is the saving rate that maximizes the decision rather than the experienced utility.Since there is a uniform contribution rate τ e of 16%, every worker in (a, 1] has an active borrowing constraint.Admittedly, it is not straightforward to identify the current contribution rate of 16% with the ideal saving rate.In Colombia, the contribution rate steadily increased from 8% in 1994 to 16% in 2006, with no further attempts to increase it ever since.However, the law that increased the contribution rate to 16% justified this increment by stating that "in the world, the average contribution rate ranges from 18% to 20% of income."13This hints that policymakers may still have thought that the ideal saving rate was higher than 16%.14

Policy experiment I: contribution rates conditioned on type
If the uniform contribution rate corresponds to the ideal saving rate τ e , then all agents in (a, b τ e ) will be better off with a lower contribution rate.Consider the following question.For each type i ∈ (a, b τ e ), what is the effect of reducing the contribution rate from τ e to the savings-maximizing contribution rate τ s i ?The objective of this policy experiment is to quantify the maximum welfare gains these workers can achieve by moving away from the current status quo without the need for any transfers between agents.
In the case of Colombia, the set (a, b τ e ) corresponds to the 17.4% of the working population who would be better off with a lower contribution rate.The results of reducing their contribution rate from 16% to τ s i are presented in Table 3.Each row represents a worker's type.Each type roughly represents between 2.4 and 2.5% of the working population.Column (1) presents the potential formal wage per year (as measured in 2013 US PPP dollars) for each type.Column (2) presents the position of each type of worker in the wage distribution, namely, the share of the working population with a potential formal wage less than or equal to the corresponding value in Column (1).Column (3) presents the productivity gaps between the formal and informal sectors for each type of worker.Note that the productivity gap increases as the potential formal wage increases, as formulated in Assumption 1.The information for these first three columns is again based on the Mincerian estimates of Doligalski  and Rojas (2023).15The contribution rates that maximize the experienced utility of each type of worker in (a, b τ e ) are presented in Column (4) of Table 3.These rates are the savingsmaximizing contribution rates found by solving the equation 1 − The savings-maximizing contribution rates increase with potential earnings: They start at 9.6% (the biased saving rate) and increase to 16% (the ideal saving rate).Column (5) presents the formal earnings with a rate τ s i relative to formal earnings with a rate τ e .Since the contribution rate is lower, formal earnings are higher.This increase is significant and reaches more than 50% for some workers.
The effect on retirement savings and retirement consumption is presented in Column (6) of Table 3.Despite the lower contribution rate, retirement savings and retirement consumption are higher.This is the case by construction since τ s i is the savingsmaximizing contribution rate for agent i.The growth in retirement savings and in retirement consumption almost reaches 10% for some workers, which represents a significant improvement in the well-being they enjoy at retirement.
A significantly higher formal income does not translate into a significantly higher total income.This is shown in Column (7) of Table 3, which indicates that total labor earnings do not increase by more than 2% for any type of worker.This is not surprising: ultimately, the productivity gaps between the formal and informal sectors are relatively low for this set of workers.Therefore, most of the welfare improvement is not coming from higher output but from postponing consumption.This is because postponing consumption becomes less painful as the contribution rate falls.Workers do not have to sacrifice as much consumption today to increase their retirement consumption.This is the force leading to higher retirement savings despite the mandate to save a lower share of formal income.
Working-age consumption increases with a lower contribution rate, as shown in Column (8) of Table 3.However, the effect is very marginal.Why is working-age of a model akin to a Roy model, following the methodology outlined in literature such as Heckman and  Honore (1990); Pratap and Quintin (2006)), and French and Taber (2011).
consumption not increasing significantly, given the lower contribution rates?This is because a lower contribution rate generates a substitution effect in favor of future consumption.This partially offsets the direct positive effect of a lower contribution rate on take-home payment, leading to modest gains in present consumption.Nonetheless, the effect on present consumption remains positive.Lowering the contribution rate increases both working-age and retirement consumption.
To quantify the effect on welfare (as measured by the experienced utility and not by the decision utility), I estimate the additional retirement-age consumption equivalent to changing the contribution rate from τ e to τ s i for each i ∈ (a, b τ e ).Formally, I seek the equivalent variation ev 2i τ s i that solves the following equation: The equivalent variations are presented in Fig. 4.They trace the change in retirement consumption found in Column (6) of Table 3.However, since lowering the contribution rate not only increases retirement consumption but also increases working-age consumption, the equivalent variations in Fig. 4 are above the additional retirement consumption in Column (6) of Table 3.For some workers, decreasing the contribution rate from 16 to 12.3% is equivalent to increasing their retirement-age consumption by 13.5%.Note that the contribution rate for every type above b τ e remains at 16%, leading to no effect on their welfare.

Policy experiment II: a lower uniform contribution rate
Unfortunately, the contribution rate schedule suggested by Column (4) of Table 3 is not implementable when productivity gaps are private information.In particular, some workers might be tempted to work at least some time informally to face the same contribution rate as workers with slightly lower productivity.A possibility to   Assume that the Pareto weights are all equal and that function G in Eq. 11 is linear.If this is the case, then the computation for the Colombian economy yields that τ = 14.1%. 16This rate is below the current contribution rate of 16%, as expected from Proposition 5.The quantitative results of lowering the contribution rate from 16 to 14.1% are presented in Table 4. Unsurprisingly, formal income increases or remains unchanged for everyone, as shown in Column (5).Column (6) shows that the retirement consumption for most of the 17.4% in the middle of the distribution increases.However, retirement consumption falls by 11.6% for all workers in the top 57.1% of the income distribution.This is because these workers were already working exclusively in the formal sector but now face a lower contribution rate.Since they would prefer to save only 9.6% of their income, they do not respond to the lower contribution by increasing their voluntary savings.Total income increases by no more than 2% for no worker, as shown in Column (7).Finally, working-age consumption increases for everyone working for at least some time formally, as shown in Column (8).
The variations in retirement consumption equivalent to decreasing the uniform contribution rate from 16 to 14.1% are presented in Fig. 5.For the 17.4% of the workers in the middle of the income distribution, the equivalent variations are positive, although not as high as if every type of worker had an idiosyncratic contribution rate τ s i .However, for workers at the top 57.1% of the income distribution, lowering the contribution rate is equivalent to decreasing their retirement consumption by 2.1% (Fig. 5 presents the equivalent variations for some but not all of the types that face a negative equivalent variation).

Policy experiment III: nonuniform contribution rates under unobservable types
The social planner is not necessarily constrained to choose a uniform contribution rate.If nonuniform contribution rates are available but types are not observable, what should the social planner do?Bear in mind that the main result from Sect. 2 is that if a worker who is more productive formally is actually working informally, then she will be better off with a lower contribution rate.This raises a question: If the contribution rate does not have to be uniform, how could a social planner increase the retirement savings of workers who are more productive formally without leading them to work informally?Specifically, what is the best way of achieving the highest experienced utility for each profile under the constraints that (a) types are not observed, (b) agents work in the sector where they are more productive, and (c) there are no transfers between agents? 17et T (y) be the contribution to the individual retirement account when the formal earnings are y.In this setting, the average contribution rate T (y)/y may differ from the marginal contribution rate T (y).By construction, agents in [0, a] are not more productive formally and cannot be forced to save anything other than what their present-biased selves prefer.Nevertheless, a benevolent social planner would like to require each worker in (a, 1] to save at least what their present-biased selves would voluntarily save but no more than what their future selves would find optimal: However, if the social planner does not want workers in (a, 1] to work informally, then the effective marginal tax rate cannot be above the productivity gap: Equation ( 14) corresponds to the first-order condition for the worker's problem when there is a nonlinear policy T , when the borrowing constraint is active, and when the policy is such that it is optimal to work formally full-time. 18quations 13 and 14 suggest the following algorithm to increase savings without misleading anyone who is more productive formally to be working informally: 1. Start with type a, the pivotal worker who is equally productive in the formal and informal sectors.Equate her contributions to the voluntary savings preferred by her present-biased self since it is not possible to force her to save more than that: 2. As long as T (w f i ) < τ e w f i , increase the contribution for i > a up to the point where the marginal effective tax rate equals the productivity gap: The previous algorithm generates a contribution function T that partitions the set of agents [0, 1] into three subsets: (1) the set [0, a] of workers who are not more productive formally, (2) the set [a, d) of agents more productive formally but with a productivity gap not large enough to implement an average contribution rate τ e , and (3) the set [d, 1] of agents more productive formally and with a productivity gap large enough to implement the ideal saving rate τ e as their average contribution rate.
Consider applying the previous algorithm to the Colombian economy. 19The resulting average contribution rates are presented in Fig. 6 under the label . For .Incentive compatibility requires the following second-order condition: This inequality is guaranteed by the assumption that 1 − is increasing in i (Assumption 1).
19 Appendix B presents the algorithm for the case where there are discrete instead of continuous types.

Fig. 6
Average contribution rates contrast, the optimal idiosyncratic rates, if types were observable, are presented under the label min{τ s i , τ e }.If types are not observable, then the average contribution rate has to be flatter, meaning that T (w (a, d).This implies that b τ e < d, meaning that more workers have to face an average contribution rate below the ideal saving rate τ e when types are not observable.This gives them no incentives to pretend to earn a lower formal wage to face a lower contribution rate.
For better or worse, the status quo for the Colombian economy does not feature the absence of mandatory savings but presents a uniform contribution rate of 16%.What would be the effect of replacing this uniform contribution rate with the nonlinear policy T in Fig. 6?The results are presented in Table 5.The first three columns are the same as in Table 3. Column (4) in Table 5 presents the average contribution rate for each type of worker implemented when types are not observable.By construction, everyone who can obtain formal earnings higher than $5024 per year will work formally fulltime under policy schedule T .Therefore, the results in Column (5) represent the increase in formal labor income when everyone who is more productive formally abandons the informal sector.The effect on retirement consumption is presented in Column ( 6).Everyone with a potential formal wage between $5024 and $5902 has more retirement consumption, while everyone with a potential formal wage between $5902 and $8939 has less retirement consumption.However, this last set of workers is not necessarily worse off.This is because everyone with wages between $5024 and $8939 has higher working-age consumption, as shown in Column (8).Meanwhile, everyone with earnings above $8939 is unaffected since their average contribution rate remains at 16%.
To disentangle the effect on welfare, I estimate the additional retirement-age consumption under regime τ e equivalent to enjoying the consumption bundle achieved  under regime T .The equivalent variations are presented in Fig. 7. On one hand, everyone with potential earnings between $6778 and $8939 (14.5% of the working population) is worse off.This is not surprising since their average contribution rate has fallen while their formal labor income remains the same.On the other hand, Fig. 7 Equivalent variations: nonuniform rates with unobserved types everyone with potential earnings between $5024 and $6778 (17.4% of the working population) is better off.Some of them would have lower retirement savings, but this is compensated by enjoying more consumption in their working age.This shows that retirement savings are not always an appropriate metric for welfare, even in the presence of present bias.In summary, by replacing the uniform contribution rate of 16% with contribution function T in Column (4) of Table 5, the following results are obtained: (1) those who work fully informally and who correspond to the bottom 25.5% of the income distribution are not affected, (2) the next 17.4% begin to work fully formally and are actually better off, (3) the next 14.5% are worse off since they were already working fully formally but now face a lower contribution rate, and (4) the top 42.6% of the income distribution is unaffected since their average contribution rate remains at 16%. 20he institutional details of the Colombian social security system are in fact notoriously more complicated, but they suggest a status quo where formality is even more disadvantageous to workers than what is suggested by the previous numerical exercises.Among other things, (1) workers may opt for a parallel pay-as-you-go pension program, but this is only convenient for the top 5% of earners (Montenegro et al.  2018).Additionally, (2) although the contribution rate is 16%, only 11.5% of wages go to the individual's retirement account.The rest corresponds to fees and insurance (3%) and a transfer to a public fund designed to guarantee a minimum pension benefit (1.5%).However, to receive this benefit at retirement, workers have to contribute for at least 23 years.This requirement is rarely met by workers in the bottom half of the income distribution (Villar and Forero 2018).Furthermore, (3) formal jobs are subject to additional taxes and contributions that increase the benefits of working informally.Finally, (4) workers who would be more productive in the formal economy but do not reach the legal minimum wage are actually excluded altogether from the formal labor market, as they cannot contribute to social security when their earnings are below the legal minimum monthly wage.Since the minimum wage per year was $5862 in 2013 US PPP dollars, Table 3 suggests that 7.5% of the working population is excluded from the formal labor market but would be more productive working formally. 21t is worth emphasizing how institutional features can worsen the negative effects of mandatory savings.In countries like Colombia where unemployment and involuntary informality exist, the main mechanism highlighted by the paper can be exacerbated.Workers may see less reason to participate in formal employment and make contributions since they face the risk of not fulfilling the 23-years-of-contributions requirement for pension eligibility.Furthermore, those who do not meet the requirement end up subsidizing the minimum pension benefit of those who do. 22This creates stronger incentives to avoid contributions and engage in informal work for workers with a higher risk of falling into unemployment or involuntary informality.

Final remarks
This paper presented a model intended to examine the conjecture that mandatory savings help to overcome the undersaving problem generated by present bias in the context of an informal economy.The model shows that mandating saving a uniform fraction of labor income ultimately partitions the working population into full-time formal workers, full-time informal workers, and workers who are sometimes formal and sometimes informal.For the latter, a lower contribution rate will actually lead to higher levels of savings and welfare.Some estimates for Colombia suggest that 17.4% of the working population-those wealthier than 25.5% of the population but poorer than the other 57.1%-will be better off with a lower contribution rate.A lower rate will have no significant effect on their current consumption or their total labor income but will significantly increase their formal income, their savings, and their retirement consumption.
Proposing to decrease the contribution rate of low-earning workers may raise the eyebrows of staunch defenders of forced savings, especially in the context of an aging population.However, this policy proposal by no means implies that a lower contribution rate will lead to sufficient retirement savings.Instead, it suggests that the labor income of individuals prone to informality is not the appropriate source of funding for retirement savings.Since the opportunity cost of working informally is comparatively higher for top earners, an alternative is taxing labor income at the top of the income distribution and subsidizing the retirement funds at the bottom.Nevertheless, the design of such a policy would require solving a screening problem across the working population, since formal income is self-reported and the productivity gap is not observable to social planners. 23Furthermore, the screening problem will have to account for the substitution between consumption and leisure, which adds another margin of reaction against mandatory savings.24Saving through spending-where a share of a bill becomes an automatic contribution to a retirement saving accountis another alternative to funding social security, as recently implemented in Mexico and discussed in Chile. 25Overall, future research should seek better mechanisms to finance retirement savings in the presence of an informal labor market.26the share of the working-age working formally.The solution will most likely feature bunching at the value of z where the minimum-time requirement is fulfilled.

Proof of Proposition 1
The results for the case where τ ≤ τ d i are the same as if there were no borrowing constraints.The results for the case where τ d i < τ ≤ τ s i are the same as if there were an active borrowing constraint but no chance of working informally.For the case where τ > τ s i , Eq. 7 implies that Q.E.D.

Proof of Proposition 3
The proof is presented for the case where τ > τ s i .Note that u(c 1 ) + δu(c 2 ) is not the agent's objective function.Therefore, the envelope theorem does not apply.From the definition of W , it follows that First, consider the case where δu ( ĉ2i ) u ( ĉ1i ) R ≤ 1.Since 1 − Second, consider the case where δu ( ĉ2i ) u ( ĉ1i ) R > 1. Introducing 1 − u ( ĉ1i ) R τ = 0 into Eq.19 implies that: 0. This last inequality and Proposition 1 imply that dW i dτ < 0. Q.E.D.

Proof of Lemma 3
Fix the contribution at a uniform rate τ for all i ∈ [0, 1].Assumptions 1 and 2 imply that 1 −

Proof of Proposition 5
Let g i denote the social marginal welfare weight for type i ∈ [0, 1]: Suppose that the uniform contribution rate τ that maximizes the social welfare function is equal to τ e .Given Assumptions 1 and 2, it follows from Lemma 3 that the population can be partitioned into [0, a], (a, b τ e ), and [b τ e , 1].Since β < 1, then a < b τ e .If τ e were the uniform contribution rate that maximizes the social welfare function, then the first-order condition for the problem of the social planner would be as follows: Thus, the left-hand side of Eq.21 is negative, leading to a contradiction.

Fig. 2
Fig. 2 Effect of present bias on various rates

Fig. 3
Fig. 3 Allocation of the working population between formality and informality

Fig. 4
Fig. 4 Equivalent variations: conditioning the contribution rate on worker's type

Fig. 5
Fig. 5 Equivalent variations: lowering the uniform contribution rate certain d ∈ (a, 1], set T (w f i ) = τ e w f i for all i ∈ [d, 1].
informal workers are always unaffected by the contribution rate and sodW i dτ = 0 for i ∈ [0, a].ii) Proposition 4 implies that if the contribution rate is τ e , then dW i dτ < 0 for i ∈ (a, b τ e ).iii)By Assumption 2 and the definition of τ e in Eq. 10, dW i dτ = 0 for i ∈ [b τ e , 1]. iv) The social marginal weight g i is positive for all i ∈ (a, b τ e ) since λ i f i > 0 for i ∈ (a, 1] and G(•) is an increasing function.

Table 2
Parameter values

Table 3
Policy experiment I: a contribution rate conditioned on type

Table 4
Policy experiment II: a lower uniform contribution rate

Table 5
Policy experiment III: a nonuniform contribution rate under unobservable types w