AGE DISCONTINUITY AND NONEMPLOYMENT
BENEFIT POLICY EVALUATION THROUGH THE
LENS OF JOB SEARCH THEORY

Bruno Decreuse, Guillaume Wlemme

► To cite this version:

Bruno Decreuse, Guillaume Wlemme. AGE DISCONTINUITY AND NONEMPLOYMENT BENEFIT POLICY EVALUATION THROUGH THE LENS OF JOB SEARCH THEORY. International Economic Review, 2024, 10.1111/iere.12726. hal-04718007

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# AGE DISCONTINUITY AND NONEMPLOYMENT BENEFIT POLICY EVALUATION THROUGH THE LENS OF JOB SEARCH THEORY*

BY BRUNO DECREUSE AND GUILLAUME WILEMME

Aix Marseille University, CNRS, and AMSE, France; University of Leicester, U.K.

Recent papers use regression discontinuity designs (RDDs) based on age discontinuity to evaluate social assistance (SA) and unemployment insurance (UI) extension policies. Job search theory predicts that such designs generate biased estimates of the policy-relevant treatment effect. Owing to market frictions, people below the age threshold modify their search behavior in expectation of future eligibility. We use a job search model to quantify the biases on various datasets in the literature. The impacts of SA benefits on employment are underestimated, whereas those of UI extensions on nonemployment duration are overestimated. The article provides insights for RDD evaluations of age-discontinuous policies.

## 1. INTRODUCTION

Several papers use age discontinuity to evaluate the impacts of nonemployment benefits on job search outcomes. Lemieux and Milligan (2008) with Canadian data, and Bargain and Doorley (2011, 2017) with French data, examine the impact of social assistance (SA) benefits on employment. Lalive (2008) and Nekoei and Weber (2017) with Austrian data, Caliendo et al. (2013) and Schmieder et al. (2012a, 2012b, 2016) with German data study the effects of unemployment insurance (UI) extensions on unemployment duration and postunemployment wages. In all these papers, the fate of people right after entitlement is compared to the fate of those right before it.

Job search theory predicts an anticipation effect that blurs this quasi-experimental setting. Owing to frictions, workers below the age threshold respond to future labor market conditions in a continuous way. Age-conditional policies affect their current behavior, implying that people noneligible to the policy actually respond to it in expectation of future treatment. This anticipation behavior generates an interpretation bias when estimating the effect of the policy. Regression discontinuity designs (RDDs) rightly capture the entitlement effect, the effect of *becoming* entitled to extra benefits when reaching the age threshold, but not the effect of the policy, the effect of *being* entitled to extra benefits compared to a counterfactual without the policy. The so-called stable unit treatment value assumption (SUTVA) does not hold to estimate the policy effect.

This article proposes a method to estimate the effect of the policy and the interpretation bias. The method combines the insights from the RDD with the structure of a theoretical

<footnote>*Manuscript received February 2023; revised July 2024.</footnote>

<footnote>We received many suggestions from the participants to the following events and seminars: CREST, Le Mans/GAINS, University of Sheffield, SaM Essex, AMSE-Banque de France, TEPP Aussois, AFSE Orléans, SED St. Louis, AMSE summer school, and PET Strasbourg. We thank David Card, Nicolas Debarsy, Manon Garrouste, Rafael Lalive, Thomas Lemieux, and Pierre-Carl Michaud for their insightful discussions. This work was supported by French National Research Agency Grants ANR-17-EURE-0020 and ANR-17-CE26-0013, and by the Excellence Initiative of Aix-Marseille University - A*MIDEX. It used the SPECTRE High Performance Computing Facility at the University of Leicester. Please address correspondence to: Guillaume Wilemme, University of Leicester, University Road, Leicester LE17RH, United Kingdom. E-mail: guillaume.wilemme@le.ac.uk.</footnote>
model. RDDs emphasize that the age discontinuity in a policy design provides a source of identification to measure the (local) effect of the policy. We show that this source of identification is insufficient to estimate the policy's effect in the presence of search frictions (Section 2). The solution is to use the additional restrictions derived from a job search model. Doing so, we contribute to clarifying the presaging claim made by Lee and Lemieux (2010): "At this time, we are unable to provide any more specific guidelines for analyzing these age/time discontinuities since it seems that how one models expectations, information, and behavior in anticipation of sharp changes in regimes will be highly context-dependent." Our article illustrates the benefits of dialog between the structural and the quasi-experimental approaches in applied economics.

The method consists in estimating a model with search efforts and age-conditional policies. The typical job search problem is nonstationary. As time passes, individuals get closer to entitlement. These dynamics continuously affect the current value of future income streams and, therefore, the age pattern of search efforts. The behavior of the outcome variable around the age discontinuity, the focus of RDDs, defines moments to fit the model. The estimated model allows us to compare the counterfactual in which the policy is not age-conditional to the counterfactual without the policy. This way, we can compute the policy's effect and the interpretation bias of the RDD estimate. We cover different empirical implementations of this method using a generic search model. The objective is to gauge the RDD bias in practice and highlight the factors that contribute to the bias's magnitude. The overarching model is defined in Section 3.

We assess the sign and the magnitude of the interpretation bias in four cases. These cases correspond to two policies: an age-conditional increase in SA benefits and an age-conditional extension of the UI duration. We find that RDDs (significantly) underestimate the impacts of an SA increase in the Canadian and French cases and (not significantly) overestimate the impacts of a UI extension in the Austrian and German cases.

We first quantify the bias of RDDs that estimate the impact of age-conditional SA increases on employment (Section 4). We calibrate the model on the Canadian data used by Lemieux and Milligan (2008) and the French data used by Bargain and Doorley (2011, 2017). In both cases, we replicate the RDD estimate and compute the bias. Well before entitlement, job seekers are already impacted because they have a high chance of remaining unemployed after the age threshold. Therefore, they progressively reduce search effort. Theory predicts that the RDD underestimates the impact of an SA increase. In our calibration, the bias amounts to 20-40% of the impact. For future research implementing a similar RDD, we caution that the bias is large when the job-finding rate is low. We also recommend not using the employment rate as the outcome variable but the transition rate into employment.

We also quantify the bias of RDDs that estimate the impact of age-conditional UI extensions on the nonemployment duration (Section 5). We calibrate the model on the Austrian data used by Lalive (2008) and the German data used by Caliendo et al. (2013) and Schmieder et al. (2012a, 2012b, 2016). Before the age threshold, job seekers expect access to extended benefits if they find a job and lose it afterward. This expectation provides an early incentive to search. Therefore, comparing individuals before and after the age threshold overestimates the impact of UI extension on unemployment duration. Our calibrations reveal that this effect is quantitatively small in the Austrian case but matters more in the German case. In the German case, the bias amounts to 10-50% of the impact, depending on the specification. However, the impact remains within the confidence interval of the RDD estimate. Here, we caution that the bias is large when the duration of jobs is short, and the job-finding rate is low. The bias depends on the job seekers' expectations of obtaining UI benefits after the next job.

Our article provides food for thought for researchers specialized in public policy evaluation and a test of search theory's ability to replicate labor market outcomes. SA increases and UI extensions are typically interpreted using this theory. This statement is especially true for UI extensions. Schmieder et al. (2012a, 2016) rationalize their results with job search models abstracting from workers' age. We show how a calibrated model reproducing their quantitative
results also predicts the magnitude of the RDD bias. The statement is less true for the effects of SA increases. Lemieux and Milligan (2008) and Bargain and Doorley (2011, 2017) have in mind the nonfrictional labor supply model where employment effects come from participation choices. We challenge this view and, on the contrary, posit that employment effects, if any, derive from job search behavior. It should seem obvious to readers that employment effects involve both the participation and search margins.

This article owes a lot to the literature on nonstationary search. Several papers focus on the potential duration of benefits and show that the search effort increases and the reservation wage decreases as the exhaustion date gets closer (see, e.g., Mortensen, 1977; van den Berg, 1990; Cockx et al., 2018). The literature also highlights the effect of UI benefits on noneligible unemployed workers, called the entitlement effect (see, e.g., Mortensen, 1977; Holmlund, 1998; Krueger and Meyer, 2002). Eligibility effects are important, and their consideration improves our understanding of job-finding rates over the unemployment spell. Following earlier authors, we model the exhaustion of unemployment benefits as a Poisson process instead of fixing a deterministic maximum duration (see, e.g., Fredriksson and Holmlund, 2001). This assumption enables us to consider age as the only dimension of nonstationarity. We do so also because there are additional eligibility criteria for UI benefits, so the maximum UI duration varies a lot across unemployed workers. The model can easily be modified to account for a fixed maximum UI duration.

Closer to us, there is a literature on life-cycle effects in job search models. Chéron et al. (2011, 2013) and Hairault et al. (2010, 2012, 2015) focus on the retirement age and its impact on the pattern of training and search investment decisions. Menzio et al. (2016) analyze job-to-job transitions and the wage distribution over the life cycle. The main lesson is that the retirement age affects workers well before retirement. As a result, older workers search less for jobs when unemployed and receive less training when employed. This mechanism can explain wage and employment patterns by age. We use insights from these models to quantify the bias of RDDs in the context of age-conditional policies.

Schmieder et al. (2012a, 2016) provide nonstationary job search models to explain their results. In these models, nonstationarity does not derive from age eligibility for benefits. Instead, it involves the effects of the potential duration of benefits and the duration spent in unemployment. As time passes, the wage offer distribution worsens. Job seekers reduce their search effort and receive lower wages. We do not examine wages—the interested reader can modify our model to allow for duration-dependence in the wage offer distribution. Instead, we complete their analysis by emphasizing the role of the age threshold. We show that their RDD has a small bias.

There is already a literature on the pretreatment effects of policies with a time discontinuity (be it calendar time, age, or unemployment duration). These papers focus on training policies and their impacts on employment or hazard rates to employment (see, e.g., Crépon et al., 2009, 2018; van den Berg et al., 2020). This literature involves the “No anticipation assumption,” whereby individuals do not modify their behavior in expectation of the treatment date (see Abbring and van den Berg, 2003). Though excluding the nonstationarity featured in this article, this assumption is actually compatible with ex ante policy effects, individuals acting in expectation of being treated at some unknown date. Several works evaluate anticipated impacts through the individual disclosure of information on the treatment date (see, e.g., Black et al., 2003; De Giorgi, 2005; Crépon et al., 2018; van den Berg et al., 2020, for participation in job search assistance or training programs, and Lalive et al., 2005, for UI sanctions). Their approach compares hazard rates for those who are notified some months or even weeks in advance to those who are not. These papers find significant effects of notification. Our article complements this literature in three ways. First, we analyze a different economic issue: age-conditional nonemployment benefits instead of active labor market programs. Second, we provide a nonstationary job search model that explicitly predicts the magnitude and direction of expected effects as a function of worker age. Third, we calibrate this model to quantify the theoretical biases in four well-known studies.
Three papers are close in spirit. Gautier et al. (2018) combine a structural model with a quasi-experiment to quantify the impact of job seeker counseling, accounting for equilibrium effects. This article and ours highlight how standard tools like a difference-in-difference or an RDD produce biased estimates of the policy-relevant effect. Hagedorn et al. (2013) evaluate UI extensions during the Great Recession. By exploiting state border policy discontinuities and elements of search theory, they propose an estimation method to account for firms' expectations of policy changes. In comparison, our approach has to be more structural because workers' expectations cannot be inferred from the data in the studies we cover. Card et al. (2015) estimate the effect of unemployment benefits using the fact that these benefits are capped. They argue that workers cannot precisely anticipate the policy kink because it varies yearly based on wage inflation.<fnref n="1" /> This assumption is less likely to hold when the eligibility criterion is a fixed age threshold.

## 2. THE INTERPRETATION BIAS

This section defines the bias predicted by job search theory within a quasi-experimental framework. In doing so, we show that no model-free econometric strategy can circumvent the bias in most situations. We shed light on how econometric choices, such as the selection of bandwidth and the degree of the age polynomial, can influence the bias in our estimates.

Consider a nonemployment benefit policy, such as increasing SA benefits or extending the maximum UI duration. The policy corresponds to a binary treatment, depending on whether the individual is eligible for the policy. Let $T_i$ denote treatment exposure with $T_i = 1$ if individual $i$ is eligible and $T_i = 0$ otherwise. The treatment may impact a job search outcome $Y_i$, such as the employment probability, unemployment duration, or job-finding rate. The outcome is $Y_i^1$ when the individual is treated and $Y_i^0$ when untreated. Individuals have different ages $a_i$. To simplify the exposition, we abstract from other forms of observable heterogeneity that would affect the outcome variable.<fnref n="2" /> A policymaker is typically interested in the difference between $Y_i^1$ and $Y_i^0$, a treatment effect. The problem is that $Y_i^1$ and $Y_i^0$ are usually not observable simultaneously with empirical data.

The principle of the RDD is to use the age discontinuity in eligibility. If individuals are eligible for the policy only if they are above age $A$, then treatment exposure is $T_i = 1$ ($a_i \ge A$). The discontinuity in the outcome variable at age $A$ thus estimates the local average treatment effect (LATE) defined by $E[Y_i^1 - Y_i^0|a_i = A]$.

Job search theory predicts that those who are too young and supposedly untreated may already respond to the treatment in expectation of future entitlement. Therefore, there are three different potential outcome variables: $Y_i^1$ for those with age $a_i \ge A$; $Y_i^0$ the potential outcome for those who do not expect to be eligible; $\tilde{Y}_i^0$ for those not currently exposed, $a_i \le A$, but expecting future entitlement.<fnref n="3" />

The observed outcome is

$$ (1) \qquad Y_i = T_i Y_i^1 + (1 - T_i) \tilde{Y}_i^0 = \begin{cases} Y_i^1 & \text{if } a_i \ge A, \\ \tilde{Y}_i^0 & \text{if } a_i < A, \end{cases} $$

<footnote n="1">¹ In the appendix of the working paper version (2012), they show that a standard job search model also predicts a kink in the stationary wage density only if workers can anticipate the kink precisely.</footnote>

<footnote n="2">² In general, the outcome variable depends on the current age and the entry age into unemployment. In this section, $a_i$ corresponds to the current age, and we abstract from heterogeneity in the entry age into unemployment. Our analysis of the RDD bias extends to heterogeneity in the entry age and other forms of observable heterogeneity.</footnote>

<footnote n="3">³ In this section, we focus on outcomes that are not predetermined variables in search models. Predetermined variables, such as employment and unemployment rates, slowly respond to labor market changes according to theory. Not only $Y_i$ is different from the potential outcome $Y_i^0$ before the threshold, but $Y_i$ is also different from the potential outcome $Y_i^1$ after the threshold. See Subsection 4.4 in the case of an SA increase.</footnote>
<figure><img src="image_2.png" /></figure>

NOTES: The left-hand side graphs depict the job-finding rate as a function of age in the three possible cases: treated, $Y_i^1$ (in red), nontreated, $Y_i^0$ (in blue), treated in expectation, $\tilde{Y}_i^0$ (in black). The right-hand-side graphs illustrate the estimation with actual observations. The narrative of the RDD estimation says we can infer $\mathbb{E}[Y_i^0|a_i = a]$ for $a_i < a$ and $\mathbb{E}[Y_i^1|a_i = a]$ for $a_i > a$. What we do infer, however, is $\mathbb{E}[\tilde{Y}_i^0|a_i = a]$ for $a_i < a$.

<caption>FIGURE 1</caption>

ILLUSTRATION OF AN INTERPRETATION BIAS IN AN RDD

which departs from the desired outcome $T_iY_i^1 + (1 - T_i)Y_i^0$. With empirical data, $Y_i^1$ is observable after the age entitlement. However, it is not $Y_i^0$ that is observable before entitlement but $\tilde{Y}_i^0$.

An RDD aims at estimating the jump in outcome at the age threshold, which we call the RDD estimand. It is defined by

$$ \lim_{a \to A^+} \mathbb{E}[Y_i|a_i = a] - \lim_{a \to A^-} \mathbb{E}[Y_i|a_i = a]. $$

Under standard assumptions on continuity and independence of the outcome variable, the RDD estimand equals

$$ (2) \qquad \mathbb{E}[Y_i^1 - \tilde{Y}_i^0|a_i = A] = \underbrace{\mathbb{E}[Y_i^1 - Y_i^0|a_i = A]}_{\text{LATE}} + \underbrace{\mathbb{E}[Y_i^0 - \tilde{Y}_i^0|a_i = A]}_{\text{Bias}}. $$

The proof is in the online appendix. The RDD estimand coincides with the LATE when $\tilde{Y}_i^0$ is very close to $Y_i^0$ around A. Otherwise, the RDD estimand features a bias. When the behavioral response of individuals below A is qualitatively similar to the response of those above A, the bias has the opposite sign of the LATE. In that case, an RDD underestimates the magnitude of the effect. When the two responses are negatively correlated, the bias has the same sign as the LATE. In that case, an RDD overestimates the magnitude of the effect.

Figure 1(a) illustrates a case of positive bias with a negative LATE. Workers below the eligibility age already adjust their behavior; hence the observed outcome, in black, departs from the counterfactual outcome in a world without the policy, in blue. Depending on the policy and the outcome variable, the observed outcome can converge or diverge from the counterfactual outcome in a world with unconditional policy, in red. In terms of experiment, workers just above A correspond to a treatment group but workers just below A do not correspond to the relevant control group to evaluate the increase in benefits.
The bias we put forward complements more conventional sources of statistical bias (e.g., misspecification, selection issues, and measurement errors). It arises as a difference between the RDD estimand and the LATE of interest. In contrast, other statistical biases usually arise as a difference between (the expectation of) the RDD estimator and the RDD estimand. This particularity of the bias has two crucial implications. First, the bias is a matter of interpretation. The RDD approach estimates $E[Y_i^1 - \hat{Y}_i^0 | a_i = A]$, which is the effect of receiving specific benefits compared to not receiving benefits yet. If we redefine the LATE of interest as this effect, then there is no bias by construction. Second, the bias arises for any econometric method used to estimate the RDD estimand. In the article, we run OLS regressions with eligibility as an explanatory variable, following the four empirical evaluations. The bias would remain with more sophisticated methods such as local linear regressions (Hahn et al., 2001).

Figure 1(b) illustrates how the econometric specification matters. The outcome represented by a continuous curve in Figure 1(a) is not directly observable with low-frequency empirical data. Instead, individuals are observed at discrete time intervals as shown in Figure 1(b). Given the degree of the age polynomial, the linear regression estimates the jump at the threshold age by extrapolation. Here, we adopt a linear spline specification, as defined in Appendix A.1.

The choice of observation window is not innocuous. As we will explain, the job search theory predicts that the pre-entitlement response increases as individuals get closer to age A. Therefore, the gap between the potential outcome $Y_i^0$ and the observed outcome $\hat{Y}_i^0$ is the largest just below the age threshold. Reducing the observation window could magnify the RDD bias. We prove this result when potential outcomes have no trends in the online appendix.

An alternative to the standard RDD approach would be to ignore workers close to the threshold and compare workers way below A, whose anticipation response is negligible, with workers above A. This is the “donut-hole” approach (Cattaneo et al., 2019). This approach, however, requires that workers way below A face labor market conditions similar to those above age A. This assumption may be violated in practice because, for example, younger workers have less experience, or because the pool of unemployed workers becomes populated by less employable individuals over time. To summarize, there is a dilemma between comparing workers close to the age threshold on the one hand—with the issue that those just below the threshold may already react to the policy—and, on the other hand, comparing workers far away from the threshold—with the issue that these workers face too different labor market conditions.

Before turning to the theoretical model, we highlight the fact that the choice of the age polynomial is not innocuous. The pre-entitlement response of workers in the model generates an endogenous age pattern before the eligibility age. The age polynomial may capture this endogenous pattern, thereby amplifying the bias. In the online appendix, we formally prove this result in the case of age-conditional nonemployment benefits when the outcome variable $Y_i$ is the employment status of individual i.

### 3. JOB SEARCH MODEL WITH AGE-CONDITIONAL NONEMPLOYMENT BENEFITS

We focus on two nonemployment benefit policies: an increase in SA benefits and an extension of the duration of UI benefits. When these policies have an age requirement, they impact the job search behavior of the people below the age threshold when (i) finding a job takes time, (ii) individuals can significantly impact labor market outcomes, and (iii) individuals anticipate that they will be treated later. The first condition involves delays between search effort and search success. The second condition implies there is a behavioral response to the policy. The last condition requires that individuals are forward-looking and sufficiently well-informed about the policy.

We develop a parsimonious model with these three ingredients. Our objective is to emphasize the critical parameters that affect the interpretation bias. The model is general enough
to encapsulate the two policies. Our framework could serve as a backbone of a fully fledged model aiming to estimate precisely the treatment effect in a particular context.

3.1. *Theoretical Framework.* Workers are risk-neutral and differ in age *a*, which is continuous and measured in months. They enter the labor market at age $A$ as unemployed without UI benefits, and retire at age $\bar{A}$. We suppose that the retirement income is independent of employment history and, therefore, does not affect decisions in their working life. The discount rate is *r*. Workers search for jobs whereas unemployed but cannot search on the job. Employed workers receive a wage of *w*. Unemployed workers are either eligible for UI benefits or noneligible, in which case they receive SA benefits. The generosity of these benefits depends on the age-conditional policy in the two ways previously discussed in the literature. At the threshold age *A* in $[A, \bar{A}]$, SA benefits jump from $b_0$ to $b_1$. This policy change corresponds to the Canadian case studied by Lemieux and Milligan (2008) and the French case studied by Bargain and Doorley (2011, 2017). Moreover, workers dismissed after age *A* can receive UI benefits $b_{UI}$ for a longer duration than younger workers. We capture this as a lower rate of exhausting UI benefits. The corresponding Poisson rate drops from $\lambda_0$ before age *A* to $\lambda_1$ after this age. This policy change corresponds to the Austrian case studied by Lalive (2008) and the German case studied by Schmieder et al. (2012a, 2012b, 2016). Treating the exhaustion of UI benefits as random is a shortcut to abstract from other UI considerations. For instance, workers must be employed for a specific time to be eligible for benefits in practice, which would introduce another dimension of nonstationarity alongside age in the model.

We model the job search behavior of a typical nonemployed individual. Workers choose a search effort *s* that comes at cost *c*(*s*), with *c*(·) > 0, *c''*(·) > 0, and *c*(0) = *c*(0) = 0. In exchange for the cost, workers receive wage offers at rate *ms*. Each wage offer is a random draw from an age-invariant distribution. As we do not use wage data in Sections 4 and 5, we suppose that the lowest wage in the distribution is higher than the reservation wage. Therefore, changes in the job-finding rate are only due to changes in search effort.

Employed workers lose their jobs at a rate $q_a$, an exogenous function of age. This age trend allows the model to capture life-cycle patterns in the empirical studies. The job separation rate's age profile impacts the job-finding rate's age profile through the choice of search efforts.<fnref n="4" /> In the calibration of the model to the four policy cases, we parameterize an isoelastic cost function, $c(s) = s^{1+\alpha}$ and a linear job loss rate, $q_a = q_A + (a - A)q'_A$, with parameters $\alpha$, $q_A$, and $q'_A$.

Let $U_a(b, \lambda)$ be the asset value of unemployment at age *a* when entitled to benefits *b* that are exhausted at rate $\lambda$. At equilibrium, benefits can take the value $b_0$ or $b_{UI}$ before age *A*, and $b_1$ or $b_{UI}$ after age *A*. The UI exhaustion rate can only be $\lambda_0$ before *A* but either $\lambda_0$ or $\lambda_1$ after *A*. The two UI schemes coexist after *A* because the exhaustion rate is determined not by the current age but by the age at dismissal. Unemployed workers who lose their job before age *A* experience the higher exhaustion rate $\lambda_0$, whereas the others face the lower rate $\lambda_1$. Let $W_a(w)$ be the value of being employed with a wage *w* and $\bar{W}_a$ its expectation over the wage distribution. Denote the age derivative of any value $V_a$ by $\dot{V}_a$. The values solve the Bellman equations

$$ (3) \qquad rW_a(w) = w + q_a[U_a(b_{UI}, \lambda_{1[a \ge A]}) - W_a(w)] + \dot{W}_a(w), $$

$$ rU_a(b, \lambda) = \max_{s \ge 0} \left\{ \max\{b, b_{1[a \ge A]}\} - c(s) + ms[\bar{W}_a - U_a(b, \lambda)] \right\} $$

<footnote n="4">⁴ In the calibration, we use the same data as in the empirical studies, which prevents us from estimating more than one exogenous age trend. The advantage of an age trend in the job separation is that it generates age dynamics in the flows in and out of employment (see Créchet et al., 2023, for an analysis of life-cycle flows). Conversely, modeling an exogenous age trend in the baseline job-finding rate *m* would not affect job separation rates.</footnote>
$$ (4) \qquad \left. \begin{array}{l} +\lambda[U_a(b_{1[a\ge A]}, 0) - U_a(b, \lambda)] + \dot{U}_a(b, \lambda) \end{array} \right\} $$

with the termination conditions $W_{\bar{A}}(w) = U_{\bar{A}}(b, \lambda) = 0$ and the values being continuous at age A.

These two Bellman equations describe nonarbitrage conditions. The left-hand side is the financial return, whereas the right-hand side is the market return. Being employed yields the wage w until dismissal, which occurs at rate $q_a$. Dismissed workers then join the pool of unemployed and receive UI benefits. The exhaustion rate $\lambda_{1[a\ge A]}$ depends on whether workers have reached the age eligibility at the time of dismissal. When unemployed with current benefits b, workers can always opt for age-conditional SA benefits, $b_{1[a\ge A]}$. Incurring the cost c(s) gives access to the expected worker's surplus $S_a(b, \lambda) \equiv \dot{W}_a - U_a(b, \lambda)$ with flow probability ms. Workers receiving UI benefits may also exhaust these benefits at rate $\lambda$. This event corresponds to trading $U_a(b, \lambda)$ for $U_a(b_{1[a\ge A]}, 0)$. Finally, the terms $\dot{W}_a(w)$ and $\dot{U}_a(b, \lambda)$ account for appreciation or depreciation with age.

Workers choose the search effort that maximizes the value of unemployment. The first-order conditions to the optimization problem give

$$ (5) \qquad c'(s_a(b, \lambda)) = mS_a(b, \lambda), $$

where $s_a(b, \lambda)$ stands for the optimal search effort. The marginal cost of searching equals its marginal gain, that is, the contact rate by search unit, m, times the expected worker's surplus.

Substituting the optimal search effort, we can define the job-finding rate $\mathcal{J}(S_a(b, \lambda))$ and the net returns to search $\mathcal{R}(S_a(b, \lambda))$ as two increasing functions of the expected surplus.<fnref n="5" /> With $\bar{w}$ denoting the expected wage, the expected surplus solves

$$ (6) \qquad \begin{aligned} rS_a(b, \lambda) = \bar{w} - \max\{b, b_{1[a\ge A]}\} - q_a S_a(b_{UI}, \lambda_{1[a\ge A]}) - \mathcal{R}(S_a(b, \lambda)) \\ + \lambda[S_a(b_{1[a\ge A]}, 0) - S_a(b, \lambda)] + \dot{S}_a(b, \lambda), \end{aligned} $$

given the termination condition $S_{\bar{A}}(b, \lambda) = 0$ and continuity at age A.

We obtain a system of differential equations that only involves the surpluses in four different states of unemployment: $(b_0, 0)$, $(b_1, 0)$, $(b_{UI}, \lambda_0)$, and $(b_{UI}, \lambda_1)$. Once these equations are solved, we can determine the job-finding rates in each state. In online Appendix B, we characterize the populations in each state and the corresponding nonemployment durations. These steps are necessary to define the outcome variables used in the RDD regressions. Specifically, the evaluations of an SA increase focus on the employment rate, whereas the evaluations of a UI extension focus on the nonemployment duration.

Nonstationarity in age means that the surpluses solving (6) vary with age, or $\dot{S}_a(b, \lambda) \neq 0$. As a result, job-finding rates also vary with age. There are three sources of nonstationarity. The first is the horizon effect when $\bar{A}$ is finite. The second is the age gradient in job separations, when $q_a$ is not constant. The last one is the age-conditional policy.

The first two sources of nonstationarity are standard in life-cycle job search models. In a world with a finite horizon only (with $q_a$ constant and no age-conditional policy), surpluses decrease with age toward zero at retirement. In a world with only an age-varying job separation rate (with an infinite horizon and no age-conditional policy), an increasing profile in $q_a$ generates a decreasing trend in the surplus, as the duration of jobs is shorter when workers age. A decreasing profile of the surplus leads to a similar trend in search efforts and job-finding rates.

<footnote n="5">⁵ These functions are defined by $\mathcal{J}(S) = mc'^{-1}(mS)$ and $\mathcal{R}(S) = mc'^{-1}(mS)S - c(c'^{-1}(mS))$, with $\mathcal{J}'(S) = \frac{m^2}{c'(c'^{-1}(mS))} > 0$ and $\mathcal{R}'(S) = mc'^{-1}(mS) > 0$.</footnote>
The last source of nonstationarity derives from the age-conditional policy. In a world with age-conditional policy only (with an infinite horizon and a constant job separation rate), the surpluses are constant after age A because workers then do not anticipate any change in the environment. However, the surpluses are nonstationary before age A. Young workers’ situation continuously changes as they get closer to the age threshold.

The effect of the policy can be analyzed with the surplus equation (6). As expected, job seekers entitled to more generous benefits find jobs less rapidly, or $S_a(b_0, 0) \ge S_a(b_1, 0)$. Job seekers entitled to extended UI benefits also find jobs less rapidly, or $S_a(b_{UI}, \lambda_0) \ge S_a(b_{UI}, \lambda_1)$. We prove these results in online Appendix B.3.

**3.2. Interpretation Bias in the Model.** The situation with an age-conditional policy can be compared to two fictitious worlds. The first one is the world without the policy, in which nobody can obtain an SA increase or a UI extension. Let $\Psi_a(b, \lambda)$ be the expected surplus at age a in this world. The second one is the world with unconditional policy, in which anyone can benefit from the SA increase or the UI extension. Let $\Sigma_a(b, \lambda)$ be the expected surplus at age a in this world. Using the same reasoning as above, we obtain the dynamic equations

$$ (7) \qquad r\Psi_a(b, \lambda) = \bar{w} - \max\{b, b_0\} - q_a\Psi_a(b_{UI}, \lambda_0) - \mathcal{R}(\Psi_a(b, \lambda)) + \lambda[\Psi_a(b_0, 0) - \Psi_a(b, \lambda)] + \dot{\Psi}_a(b, \lambda), $$

$$ (8) \qquad r\Sigma_a(b, \lambda) = \bar{w} - \max\{b, b_1\} - q_a\Sigma_a(b_{UI}, \lambda_1) - \mathcal{R}(\Sigma_a(b, \lambda)) + \lambda[\Sigma_a(b_1, 0) - \Sigma_a(b, \lambda)] + \dot{\Sigma}_a(b, \lambda), $$

with the termination conditions $\Psi_{\bar{A}}(b, \lambda) = \Sigma_{\bar{A}}(b, \lambda) = 0$. The surpluses are defined exactly like in Equation (6) after substituting the age-conditional policy parameters $b_{1[a\ge A]}$ and $\lambda_{1[a\ge A]}$ by, respectively, $b_0$ and $\lambda_0$, and $b_1$ and $\lambda_1$. Note that the surpluses that solve these equations are stationary in an environment with infinite horizon and constant job separation rate.

Suppose we want to estimate the response of job-finding rates to the policy. Such a policy response can be measured by comparing $\mathcal{J}(\Sigma_a(b, \lambda))$ to $\mathcal{J}(\Psi_a(b, \lambda))$. However, only $\mathcal{J}(S_a(b, \lambda))$ can be recovered from observational data. The LATE, RDD estimand, and interpretation bias can be defined as explained in Section 2. We provide below a general intuition.

All workers above age A can benefit from the policy. Therefore, these workers behave as if they were in the world with unconditional policy. Indeed, for $a \ge A$, the surpluses $S_a$ and $\Sigma_a$ solve the same ordinary differential equations with the same boundary conditions at the retirement age $\bar{A}$. They are thus identical, and so are the job-finding rates, $\mathcal{J}(S_a(b, \lambda)) = \mathcal{J}(\Sigma_a(b, \lambda))$ for $a \ge A$.

Workers below age A do not behave as if they were in a world without the policy. For $a < A$, the surpluses $S_a$ and $\Psi_a$ solve the same equations but have different boundary conditions. Workers being forward-looking implies that the surpluses are continuous at age A, so that $\lim_{a \to A^-} S_a(b, \lambda) = S_A(b, \lambda) = \Sigma_A(b, \lambda) \ne \Psi_A(b, \lambda)$. Therefore, the surplus $S_a$ departs from the potential surplus $\Psi_a$. As a result, the observed job-finding rates do not correspond to the relevant counterfactual job-finding rates for $a < A$. To conclude, the RDD that compares workers just below and just above age A cannot recover the effect of interest.

The anticipation of the policy affects workers the most just below the age threshold. The effect may be negligible for young workers far enough from the threshold. One could show that the gap between $S_a$ and $\Psi_a$ asymptotically disappears as $a$ diminishes. That is the reason why the “donut-hole” approach may be appealing. However, owing to the age-varying job separation rate and the finite horizon, the surplus $\Psi_a$ at an age way below A may significantly depart from the surplus $\Psi_A$ at the age threshold. For instance, these younger workers may face a higher job loss rate than workers at A.
4. AGE-DEPENDENT SA

This section focuses on policies that raise SA benefits. The RDD relies on the strict age requirement that workers must meet to be eligible for increased benefits. We show that the bias is sizeable, ranging from 20% to 40% of the LATE, in two studies: Lemieux and Milligan (2008) with Canadian data and Bargain and Doorley (2011) with French data. In a sensitivity analysis, we draw general lessons to assess when the bias is large.

4.1. *Model.* We suppose $\lambda_0 = \lambda_1 = \lambda$. An increase in age-conditional SA benefits affects all workers, not only those above the age threshold receiving SA benefits but also those below this age who modify their job search behavior in expectation of future entitlement.

To assess the quantitative role of the perfect foresight assumption, we augment the model with uncertainty about the policy change. Suppose young workers believe that there is a probability $\xi$ that SA benefits increase from $b_0$ to $b_1$ at the age threshold. This assumption only modifies the continuation conditions at age $A$, as explained below.

We simply denote $S_a(b) \equiv S_a(b, \lambda)$ and expand Equation (6). Above the age threshold, for $a \ge A$, we obtain $S_a(b_1)$ and $S_a(b_{UI})$ by solving

$$ (9) \qquad rS_a(b_1) = \bar{w} - b_1 - q_a S_a(b_{UI}) - \mathcal{R}(S_a(b_1)) + \dot{S}_a(b_1), $$

$$ (10) \qquad rS_a(b_{UI}) = \bar{w} - b_{UI} - q_a S_a(b_{UI}) - \mathcal{R}(S_a(b_{UI})) + \lambda[S_a(b_1) - S_a(b_{UI})] + \dot{S}_a(b_{UI}), $$

with the terminal conditions, $S_{\bar{A}}(b_1) = S_{\bar{A}}(b_{UI}) = 0$.

Below the age threshold, for $a < A$, the surpluses $S_a(b_0)$ and $S_a(b_{UI})$ solve

$$ (11) \qquad rS_a(b_0) = \bar{w} - b_0 - q_a S_a(b_{UI}) - \mathcal{R}(S_a(b_0)) + \dot{S}_a(b_0), $$

$$ (12) \qquad rS_a(b_{UI}) = \bar{w} - b_{UI} - q_a S_a(b_{UI}) - \mathcal{R}(S_a(b_{UI})) + \lambda[S_a(b_0) - S_a(b)] + \dot{S}_a(b_{UI}), $$

with the new boundary conditions, $\lim_{a \to A^-} S_a(b_0) = \xi\Psi_A(b_1) + (1-\xi)S_A(b_1)$ and $\lim_{a \to A^-} S_a(b_{UI}) = \xi\Psi_A(b_{UI}) + (1-\xi)S_A(b_{UI})$. These conditions mean that workers are still uncertain about the policy even the day just before turning age $A$. Then, the day workers turn $A$, they learn that they are eligible for more generous assistance benefits, so the surpluses jump to $S_A(b_1)$ and $S_A(b_{UI})$. When $\xi = 0$, we obtain the dynamics with perfect foresight where the surpluses are continuous at age $A$. When $\xi = 1$, workers behave as if they were in a world without the policy below the threshold, or equivalently $S_a(b) = \Psi_a(b)$ for $a \le A$.

We study the impact of an increase in $b_1$. For workers under SA benefits and above the age threshold, $a \ge A$, the surplus $S_a(b_1)$ decreases as they receive increased nonemployment income. Therefore, these workers reduce search investment and the job-finding rate $\mathcal{J}(S_a(b_1))$ decreases. For workers entitled to UI benefits, the decline in surplus is lower because they first need to lose entitlement before getting the new level of assistance benefits. The qualitative impact on the search effort is the same as for nonentitled workers, and the job-finding rate $\mathcal{J}(S_a(b_{UI}))$ also decreases. The proofs are in online Appendix B.3.

For workers below the age threshold, $a < A$, the boundary conditions imply that the surpluses also reduce after an increase in $b_1$. The reason is that the value of nonemployment situations in the future rises, lowering the worker's surplus today: both $S_a(b_0)$ and $S_a(b_{UI})$ decrease. Therefore, these workers search less intensively in expectation of future entitlement. This effect grows as workers get closer to entitlement.

We now characterize the bias of the RDD. Focusing on unemployed workers with SA, the LATE on job-finding rates at age $A$ is the gap $\mathcal{J}(\Sigma_A(b_1)) - \mathcal{J}(\Psi_A(b_0))$. However, what is ideally observed is the RDD estimand $\mathcal{J}(S_A(b_1)) - \lim_{a \to A^-} \mathcal{J}(S_a(b_0))$. Unless workers do
not anticipate the policy with $\xi = 1$, the RDD estimand differs from the LATE. When workers perfectly anticipate the policy, or $\xi = 0$, the RDD estimand equals zero because there is no jump in the outcome variable at age $A$. At the limit, workers just below $A$ behave like they are already eligible.

In practice, RDD estimates may still be significantly different from zero due to observing individuals at discrete time intervals. Not observing individuals very near the threshold is particularly concerning, as this is where the pre-entitlement response is most pronounced. We will find nonzero RDD estimates despite a zero RDD estimand in the calibration results.

An SA increase reduces job-finding rates, implying that the LATE is negative. Since workers already search less intensively before reaching the age threshold, an RDD would underestimate the LATE in absolute terms. The magnitude of this response governs the theoretical bias discussed in this article. We now quantify it in the Canadian and French studies.

## 4.2. Calibration.
The model is calibrated by fitting the age patterns of the employment rate, which is the outcome of interest.

Lemieux and Milligan (2008) consider SA in Quebec, Canada, where benefits were conditional on claimants' age with a discontinuity at 30 between 1969 and 1989. Bargain and Doorley (2011) consider the French *Revenu Minimum d'Insertion*, an assistance income paid to individuals older than 25 between 1988 and 2009. The SA increase has the same magnitude in the two policy cases. SA benefits almost triple when workers reach the age threshold. We restrict our analysis to the same populations of interest as in these two studies. These are male high-school dropouts without dependent children in Canada and single male high-school dropouts in France.

The annual interest rate is fixed at 5%, and the mean wage is normalized to 1. Workers enter the labor market as unemployed without UI benefits at age 18 and retire at 65. We assume perfect foresight of the policy in the benchmark calibration, $\xi = 0$. We obtain the benefit incomes $b_0$, $b_1$, and $b_{UI}$ from the two studies. In the Canadian and French UI schemes, workers are entitled to a maximum UI duration $P(t)$, which depends on their employment history $t$. We set $\lambda$ such that the expected maximum UI duration in the model, $1/\lambda$, equals the expectation of $P(t)$ over the last job's duration:

$$ (13) \qquad \frac{1}{\lambda} = \int_{0}^{\infty} P(t) q_A e^{-q_A t} dt, $$

where $q_A e^{-q_A t}$ is the probability density function of job duration with a separation rate of $q_A$.<fnref n="6" /> Online Appendix C provides additional calibration details, including the policy rules $P(t)$.

We then estimate the elasticity of the marginal search cost $\alpha$, the job-finding rate per search unit $m$, the job separation rate $q_A$, and the age gradient $q'_A$ to match the age pattern of the employment rate around the age threshold. As a sensitivity analysis, we also calibrate the model while fixing $q_A$ at different values in online Appendix E.1, and while fixing $\xi$ at nonzero values in online Appendix E.2.

We provide an identification argument below. Abstract first abstract from any age variation in job separations. The elasticity parameter $\alpha$ is identified by the magnitude of the employment drop before and after the age threshold. A large gap between the employment rates before and after implies that workers have strongly responded to the policy, so the elasticity of cost to search efforts must be low. The baseline job-finding rate, $m$, and the job separation rate, $q_A$, are identified by the average employment rate and the speed of adjustment. A high employment rate means that $m/q_A$ must be high. The faster the employment rate adjusts around the threshold, the more dynamic the labor market is, which means that both $m$ and $q$ are higher.

<footnote n="6">⁶ To avoid numerical complexities, we abstract from the age variation in the job loss rate in the formula, which is ex-post consistent with a low magnitude of $q'_A$. We focus only on the last employment period.</footnote>
<figure><img src="image_3.png" /></figure>

NOTES: Employment rates in the data are from Lemieux and Milligan (2008) and Bargain and Doorley (2011). In the model, the employment rate of the age group *a* is computed as the integral of the employment rate from *a* to *a* + 1. The shaded areas show untargeted employment rates.

<caption>FIGURE 2
MODEL FIT FOR THE AGE-CONDITIONAL SA INCREASE</caption>

The age gradient of the job separation rate is parametrically identified from the age trend far enough after the age threshold, when the effect of the age-conditional policy has dissipated. The employment rate converges to the counterfactual employment rate corresponding to a world with unconditional policy. This convergence occurs within the observation window for plausible values of $\alpha$, $m$, and $q_A$. Therefore, the age variation in the employment rate far from the threshold is not due to the age-conditional policy but to the two other life-cycle elements: the horizon effect, which is captured by the fixed parameter $\bar{A}$, and the age-varying job separation rate, which is captured by the free parameter $q'_A$. This age variation for middle-aged workers thus identifies the age gradient $q'_A$.$^7$

The empirical moments we match are employment rates by age groups in years provided by the two studies. The theoretical moments are the employment rates averaged per age group. Consistently with our identification argument, we incorporate more data points on the right to identify the age gradient in job separations. We target employment rates between the ages of 27 and 35 in Canada. For France, the empirical employment rate before 25 strongly increases with age in the data (see Figure 2). Our model cannot satisfactorily explain these dynamics, which are likely due to a staggered entry into the labor market. We therefore reduce the window compared with the Canadian case and target employment rates between the ages of 24 and 30. The parameters minimize the sum of squared deviations between each target and its simulated counterpart.

The parameters for Canada and France are reported in Table 1, and the model fit is plotted in Figure 2. The model captures well the targeted employment levels before and after the threshold age, and the long-term trend. It replicates untargeted employment rates, except for France below age 23 where the model overshoots the employment rate.

Turning to the estimated parameters, the elasticities $\alpha$ of the marginal search cost imply that the elasticities of the search effort to the worker's surplus $(1/\alpha)$ are about 1.5 for Canada and 4 for France. This difference indicates that workers in the French case respond more to a policy change than in the Canadian case. In both cases, the job separation rates are high, im-

<footnote>$^7$ The parametric assumption on the job loss rate is essential to the identification argument. With a more flexible or nonparametric form, it would not be possible to identify separately the magnitude of the employment drop or the speed of adjustment from the age gradient in job separations. In other words, it is always possible to find a shape for $q_a$ to generate any pattern in the employment rate. Note the linear parameterization cannot generate the nonlinear decreasing age patterns of separations in the data (see, e.g., Menzio et al., 2016; Créchet et al., 2023).</footnote>
<table><caption>TABLE 1
MODEL PARAMETERS FOR THE AGE-CONDITIONAL SA INCREASE</caption><thead><tr><th rowspan="2">Policy Case</th><th colspan="9">Parameters</th></tr><tr><th>A</th><th>b<sub>0</sub></th><th>b<sub>1</sub></th><th>b<sub>UI</sub></th><th>α</th><th>m</th><th>q<sub>A</sub></th><th>q'<sub>A</sub></th><th>λ</th></tr></thead><tbody><tr><td>Canada</td><td>30</td><td>0.159</td><td>0.437</td><td>0.6</td><td>0.69</td><td>0.156</td><td>0.051</td><td>-5.51e<sup>-5</sup></td><td>0.11</td></tr><tr><td>France</td><td>25</td><td>0.23</td><td>0.602</td><td>0.68</td><td>0.241</td><td>0.276</td><td>0.0683</td><td>-7.16e<sup>-5</sup></td><td>0.0648</td></tr></tbody></table>

NOTE: In both policy cases, we set $\underline{A} = 18$, $\bar{A} = 65$, $r = 0.0048$, $\bar{w} = 1$, and $\xi = 0$.

plying that jobs last about 20 months in Canada and 15 months in France. These large rates are consistent with the fact that the population of interest faces precarity in the labor market. The French youth were (and still are) disproportionately exposed to short-term contracts, especially less educated workers. These job separation rates translate to UI exhaustion rates at 0.11 and 0.065, corresponding to an expected maximum UI duration of nine months in the Canadian case and 15 in the French one. Finally, we find an age profile of job separation rates that is decreasing in Canada and increasing in France, mapping the different patterns in the two studies.

4.3. *Bias in the Studies.* Figures 3(a) and (b) depict the job-finding rates predicted by the calibrated model. The continuous lines correspond to the situation where SA benefits increase at the threshold age. The dashed and dotted lines correspond to alternative scenarios where SA benefits do not change with age. When SA benefits are age-conditional, workers respond two years before becoming eligible in the Canadian simulation and one year before in the French one. The job-finding rate of workers receiving SA benefits drops, respectively, by 25% and 60%.<fnref n="8" /> Workers above the age threshold behave as if they were in a world with unconditional access to higher SA benefits. The graphs of the corresponding surpluses and populations by state of unemployment are provided in the online appendix.

Figures 3(c) and (d) display the age pattern of the employment rate. As for the job-finding rates, the employment rate starts decreasing two years before the age threshold in Canada and one year before in France. Changes in job-finding rates immediately impact the *flow* of workers entering employment. However, the employment rate is a *stock* (or predetermined variable) and, therefore, takes time to adjust. That is why the employment rate still departs from the ideal counterfactual employment rate after the age threshold. After the age threshold, it takes two years for Canada and one year for France to eliminate this gap with the counterfactual employment rate.

The LATE on the employment rate is the spread between the two counterfactuals at the threshold age A. Raising assistance benefits reduces the employment rate by 7 percentage points in Canada and 8.5 percentage points in France. Using the black curve in the neighborhood of the age threshold leads to a different computation. For instance, comparing the employment rate at age 29.5 to the one at 30.5 in the Canadian case gives a rough RDD estimate of 4 percentage points. For France, the comparison between ages 24.5 and 25.5 leads to an estimate of 6 percentage points. This simple comparison underestimates the policy effect by 30% and 40% for each case.

The bias remains with more sophisticated RDD estimators. To prove that statement, we use the model to generate simulated data sets to run the same regressions as in the two studies. Our simulated data points correspond to the aggregated employment rates plotted on Figure 2.<fnref n="9" /> Table 2 reports the RDD estimates for different specifications. Appendix A.1 defines these specifications.

<footnote n="8">⁸ Job-finding rates are quite large for French workers under SA. This result derives from the high job loss rate chosen by our calibration procedure. Short employment durations must be balanced by short nonemployment durations to match a given employment rate.</footnote>

<footnote n="9">⁹ Note the discrete approximation of the running variable, here age, generates an additional bias in the RDD estimates (see Dong, 2015).</footnote>
<figure><img src="image_4.png" /></figure>

NOTES: The black continuous curves correspond to the outcomes when the SA increase is conditional on age. The blue dashed, and the red dotted curves correspond to the counterfactual outcomes when there is no SA increase and when the SA increase is unconditional. SA benefits are always $b_0$ in the first counterfactual and always $b_1$ in the second. In panels (a) and (b), the curves marked with dots and stars correspond to unemployed workers under social assistance (SA) and unemployment insurance (UI).

<caption>FIGURE 3
SIMULATIONS OF AN AGE-CONDITIONAL SA INCREASE</caption>

The estimated effects in the data and from the model are very close. They are the closest in the cubic and linear spline specifications. The only time the model does not match the empirical RDD estimate is in the linear and quadratic case for France, where Bargain and Doorley (2011) report positive estimates. However, Bargain and Doorley (2011) discard these estimates that are driven by the employment rate pattern at early ages (Figure 2).

Comparing the RDD estimates from the data to the model-predicted LATE, we find that the RDD bias ranges from 20% to 40% in both cases. Although these are sizeable magnitudes, the RDD biases are not always statistically significant at 95% confidence. Note, however, that the bias is significant in the linear spline specification, which is the preferred specification in many RDD studies.

There is a subtle difference between the RDD estimates from the data and the model. The estimate decreases with the polynomial order when using the model but increases with the data. With the model, increasing the polynomial order captures the declining employment pattern due to the pre-entitlement effect, leading to smaller estimates. The two studies do not find this result because the age polynomial captures additional employment trends and potential noise before and after the age threshold.
<table><caption>TABLE 2
RDD ESTIMATES FOR THE AGE-CONDITIONAL SA INCREASE</caption><thead><tr><th rowspan="2"></th><th colspan="2">Canada</th><th colspan="2">France</th></tr><tr><th>Data</th><th>Model</th><th>Data</th><th>Model</th></tr></thead><tbody><tr><td>RDD estimates</td><td></td><td></td><td></td><td></td></tr><tr><td>Linear</td><td>-0.041<br>(0.012)</td><td>-0.052</td><td>0.027<br>(0.018)</td><td>-0.078</td></tr><tr><td>Quadratic</td><td>-0.051<br>(0.012)</td><td>-0.045</td><td>0.067<br>(0.013)</td><td>-0.068</td></tr><tr><td>Cubic</td><td>-0.048<br>(0.014)</td><td>-0.043</td><td>-0.065<br>(0.014)</td><td>-0.068</td></tr><tr><td>Linear spline</td><td>-0.049<br>(0.011)</td><td>-0.047</td><td>-0.049<br>(0.010)</td><td>-0.069</td></tr><tr><td>Quadratic spline</td><td>-0.056<br>(0.018)</td><td>-0.026</td><td>-0.067<br>(0.017)</td><td>-0.052</td></tr><tr><td>LATE</td><td></td><td>-0.069</td><td></td><td>-0.085</td></tr></tbody></table>

NOTE: The outcome variable is the employment rate. The “Data” estimates and standard errors are taken from table 1, second column, in Lemieux and Milligan (2008) and table 1, first column, in Bargain and Doorley (2011). The “Model” estimates are obtained from simulated employment rates, aggregated by age group, depicted in Figure 2. The regressions are run with the same window as in the two studies, 25–39 for Canada and 20–35 for France. The LATE is the difference between the two counterfactual employment rates at the age threshold.

These results are not driven by a large calibrated value of the job separation rate. When calibrating the model with alternative values of $q_A$, we find that the magnitude of the bias decreases with the job loss rate (online appendix). A higher job loss rate leads to setting a higher job-finding rate to keep the employment rate constant. Therefore, this is akin to reducing search frictions and the corresponding need to anticipate the age discontinuity well in advance. By sticking to a large job loss rate, we report conservative values for the bias.

### 4.4. When is the Bias Large?
Our sensitivity analysis identifies the factors responsible for a sizeable bias and provides practical insights for applied researchers. The effect of the job-finding and job separation rates on the bias is represented in Figure 4 whereas the sensitivity to other key parameters is reported in Table A.2.

*Job-finding and job separation rates.* To assess the role of search frictions in generating a sizeable bias, we compute the RDD bias and the LATE after shifting the baseline job-finding rate *m* and the job separation rate at the age threshold $q_A$.

The RDD bias is the largest when it is hard to find a job, both in magnitude and relative to the LATE. For instance, if the baseline job-finding rate were twice lower, the RDD bias would amount to 50% of the LATE in Canada and 35% in France. This finding is consistent with intuition. The longer it takes to find a job, the earlier workers respond to the policy. A change in job separation rate affects not only the level of employment rates before and after the threshold but also the speed of adjustment around the threshold. Numerically, a higher job separation rate implies a higher RDD bias, but the LATE is also higher. The bias as a share of the LATE remains constant after shifting the job separation rate.

*Age gradient and horizon effect.* Employment rates in Canada and France do not follow the same patterns after the age threshold. These life-cycle trends have no quantitative impact on the size of the bias. For instance, removing the age variation in job separation rates, by setting $q'_A = 0$, does not markedly shift the bias or the LATE.

The Canadian policy occurs at a later age than the French one. Therefore, the horizon effect may affect the two cases differently. We find no impact of the horizon effect in Canada and France. For illustration, the retirement age should be around 40 in Canada to observe a meaningful change in the bias. In that case, the bias would increase because the employment rate will not have time to converge to the counterfactual rate after the age threshold.
<figure><img src="image_5.png" /><figcaption>(a) RDD bias, Canada</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b) RDD bias, France</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(c) LATE, Canada</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(d) LATE, France</figcaption></figure>

NOTES: The RDD bias and the LATE (in absolute value) on the employment rate are represented as functions of pa- rameters *m* and *q*<sub>A</sub> in the base-2 log scale. The RDD bias is computed with a linear spline specification. In the Cana- dian case, *m* = 0.0156 and *q*<sub>A</sub> = 0.051. In the French case, *m* = 0.276 and *q*<sub>A</sub> = 0.0683. The cross at the center marks the location of the calibrated parameters.

<caption>FIGURE 4
BIAS AND LATE AS A FUNCTION OF *m* AND *q*<sub>A</sub>, AGE-CONDITIONAL SA INCREASE</caption>

*Perfect foresight.* Workers’ anticipation of the SA increase triggers a response before the age threshold. What happens when workers do not have perfect foresight? We compute the RDD estimate predicted by the model when we increase the uncertainty parameter ξ. As expected, the bias decreases as ξ increases. The decrease, however, is small. At the limit when workers do not know about the policy before reaching age A, when ξ = 1, the bias still amounts to 20% and 12% of the LATE.<fnref n="10" />

The bias remains when young workers are entirely ignorant about the SA increase only be-
cause the employment rate is predetermined. When ξ = 1, the employment rate overlaps with
the counterfactual rate in a world without the policy, below age A. However, from age A,
the employment rate smoothly decreases to converge to the counterfactual rate with uncon-
ditional policy. This continuous decrease, instead of a jump, is the source of the bias.

Note that the uncertainty captured by ξ differs from a take-up rate. When workers reach age A, they all obtain SA benefits if they are unemployed without UI. A less-than-100% take-up rate in the model would likely produce a similar effect on the bias, but it would also affect the LATE. The lower the take-up rate, the lower the LATE.

<footnote n="10">¹⁰ Alternatively, we calibrate the model while fixing different values of ξ in online Appendix E.2. The versions with different values of ξ predict biases of the same magnitude as in the benchmark calibration.</footnote>
*The choice of outcome variable.* Both studies take the employment rate as the outcome variable. The sluggish dynamics of the employment rate contribute to the bias, even when workers do not expect the SA increase.

Alternatively, one could use the average job-finding rate as the outcome variable. The job-finding rates of unemployed workers under SA and UI are state variables and, therefore, overlap with the counterfactual rates right from age A. Although the *average* job-finding rate of unemployed workers is predetermined (because the share of workers in each state is predetermined), it could depart less from the relevant counterfactual than the employment rate.

We propose to compute the policy's effect on the employment rate using job-finding rates as the regression outcome. We first estimate an RDD on the average job-finding rate. Then, we use an approximation formula to compute the predicted effect on the employment rate from these RDD estimates.<fnref n="11" /> Although this method does not reduce the RDD bias compared to the benchmark, it performs better when workers are uncertain about the policy.

*General lessons.* To conclude, there are two takeaways for researchers implementing an age-based RDD to evaluate SA benefits. First, the bias is likely to be large when the job-finding rate is low. Second, using a stock variable like the employment rate as the outcome variable is not recommended. Regressions that use the age-conditional job-finding rate as the outcome generate less biased estimates of the policy-relevant effect, especially when workers do not anticipate the age-based policy.

## 5. AGE-DEPENDENT UI BENEFITS

This section focuses on policies that extend the duration of maximum UI benefits. The RDD is based on the requirement that workers receive extended benefits only if they lose their job after a certain age. The model is calibrated to two studies: Lalive (2008) with Austrian data and Schmieder et al. (2012a) with German data. We show that the bias is negligible in the Austrian case, and sizeable but not statistically significant in the German case. In a sensitivity analysis, we draw general lessons to assess when the bias is large.

5.1. *Model.* We suppose $b_0 = b_1$. Like in the previous section, we argue that an age-conditional UI extension in the form of a decrease in $\lambda_1$ affects all workers. It does not only affect those receiving extended benefits but also those below the age threshold who care about future employment gains in terms of benefit entitlement.

We expand the surplus equation (6) in this context. The model is augmented with a UI reciprocity probability. Employed workers who lose their jobs have a probability $\chi$ to be entitled to UI benefits, with $0 < \chi \le 1$.

Consider first workers above age A. For $a \ge A$, we have

$$ (14) \qquad rS_a(b_0, 0) = \bar{w} - b_0 - q_a[\chi S_a(b_{UI}, \lambda_1) + (1-\chi)S_a(b_0, 0)] - \mathcal{R}(S_a(b_0, 0)) + \hat{S}_a(b_0, 0), $$

$$ (15) \qquad rS_a(b_{UI}, \lambda_0) = \bar{w} - b_{UI} - q_a[\chi S_a(b_{UI}, \lambda_1) + (1-\chi)S_a(b_0, 0)] - \mathcal{R}(S_a(b_{UI}, \lambda_0)) + \lambda_0[S_a(b_0, 0) - S_a(b_{UI}, \lambda_0)] + \hat{S}_a(b_{UI}, \lambda_0), $$

$$ rS_a(b_{UI}, \lambda_1) = \bar{w} - b_{UI} - q_a[\chi S_a(b_{UI}, \lambda_1) + (1-\chi)S_a(b_0, 0)] $$

<footnote n="11">¹¹ The employment rate just above age A is approximated by $\frac{JFR}{JFR+q_A}$, where the average job-finding rate just above age A is JFR. The employment rate just below age A is approximated by $\frac{JFR-RDD}{JFR-RDD+q_A}$, where the RDD estimate on the average job-finding rate is RDD. The estimated effect is the difference between these two quantities.</footnote>
$$ (16) \qquad -\mathcal{R}(S_a(b_{UI}, \lambda_1)) + \lambda_1[S_a(b_0, 0) - S_a(b_{UI}, \lambda_1)] + \dot{S}_a(b_{UI}, \lambda_1), $$

with the terminal conditions $S_{\bar{A}}(b_0, 0) = S_{\bar{A}}(b_{UI}, \lambda_0) = S_{\bar{A}}(b_{UI}, \lambda_1) = 0$.

For younger workers, with $a < A$,

$$ (17) \qquad rS_a(b_0, 0) = \bar{w} - b_0 - q_a[\chi S_a(b_{UI}, \lambda_0) + (1-\chi)S_a(b_0, 0)] - \mathcal{R}(S_a(b_0, 0)) + \dot{S}_a(b_0, 0), $$

$$ (18) \qquad rS_a(b_{UI}, \lambda_0) = \bar{w} - b_{UI} - q_a[\chi S_a(b_{UI}, \lambda_0) + (1-\chi)S_a(b_0, 0)] - \mathcal{R}(S_a(b_{UI}, \lambda_0)) + \lambda_0[S_a(b_0, 0) - S_a(b_{UI}, \lambda_0)] + \dot{S}_a(b_{UI}, \lambda_0), $$

with the continuity conditions, $\lim_{a \to A^-} S_a(b_0, 0) = S_A(b_0, 0)$ and $\lim_{a \to A^-} S_a(b_{UI}, \lambda_0) = S_A(b_{UI}, \lambda_0)$.

We study the impact of a decrease in $\lambda_1$, leaving the proofs in online Appendix B.3. For workers above age A entitled to UI benefits, the surplus $S_a(b_{UI}, \lambda_1)$ decreases as they receive extended benefits. These workers reduce their search investment and find jobs at a lower rate. The effect is the opposite for workers not currently entitled to these UI benefits. The surpluses $S_a(b_0, 0)$ and $S_a(b_{UI}, \lambda_0)$ increase because the prospect of extended UI benefits raises the surplus. These workers search more intensively as a result. This is the entitlement effect described by Mortensen (1977).

For workers below age A, the continuity conditions imply that the surpluses also increase after a decrease in $\lambda_1$. Owing to the entitlement effect, the surpluses $S_a(b_0, 0)$ and $S_a(b_{UI}, \lambda_0)$ increase. This increase is stronger as workers get closer to the age threshold. Therefore, search efforts and job-finding rates increase with age below the threshold.

We now characterize the bias of the RDD. Focusing on unemployed workers with UI benefits, the LATE on job-finding rates at age A is the gap $\mathcal{J}(\Sigma_A(b_{UI}, \lambda_1)) - \mathcal{J}(\Psi_A(b_{UI}, \lambda_0))$. However, what is ideally observed is the RDD estimand $\mathcal{J}(S_A(b_{UI}, \lambda_1)) - \mathcal{J}(S_A(b_{UI}, \lambda_0))$. Unemployed workers just below A behave like unemployed workers who would get extended benefits after their next job. The RDD estimand equals the entitlement effect.

A UI extension reduces job-finding rates, implying that the LATE on nonemployment duration is positive. Since workers start searching more intensively before reaching the age threshold, an RDD overestimates the LATE in absolute terms.

5.2. Calibration. The model is calibrated for each study to fit the age patterns of the nonemployment duration, which is the outcome of interest.

Lalive (2008) considers the maximum UI duration in Austria, which was extended from 39 to 209 weeks for workers that become unemployed after age 50, between 1988 and 1993. Schmieder et al. (2012a) consider the maximum UI duration in Germany. From 1987 to 1999, workers who started their unemployment spell before age 42 were entitled to a maximum UI duration of 12 months. If their unemployment spell started between 42 and 44, they were entitled to 18 months. There is another UI extension at age 44, but we abstract from it in our analysis.<fnref n="12" /> The policy change is drastically different between the two cases. For Austria, the maximum UI duration was multiplied by more than 5, whereas it increased by 50% for Germany. We restrict our analysis to the same populations of interest as in these two studies. These are workers who have experienced continuous employment and are therefore eligible for the maximum UI duration, males in Austria and both genders together in Germany.<fnref n="13" />

<footnote n="12">¹² Ideally, one would simulate the model with both age thresholds. Since we find a quantitatively small effect with only one threshold at 42, we presume the simulation with two thresholds would provide similar results.</footnote>

<footnote n="13">¹³ Lalive (2008) also considers women but evaluates the effect on early retirement.</footnote>
The annual interest rate is fixed at 5%, and the mean wage is normalized to 1. In both policy cases, the mandatory retirement age is fixed at 65. The income parameters are obtained from the two studies. The details are in online Appendix C. As workers are eligible for the maximum UI duration, we set $\lambda_0$ and $\lambda_1$ so that the mean potential durations of benefits are 39 and 209 weeks in Austria and 12 and 18 months in Germany. We use the share of unemployed workers who receive UI benefits in the data, at 59% in Austria (Lalive, 2008) and 70% in Germany (Trenkle, 2023), to determine the UI recipiency probability.

As in the calibration of the previous policy, we are left with four parameters: the elasticity of the marginal search cost $\alpha$, the contact rate by search unit $m$, and the job loss rate $q_A$, and the age gradient $q'_A$. Our targeted moments consist of the average nonemployment duration at different ages around the threshold.

The identification argument is similar to Section 4. The elasticity parameter $\alpha$ is identified by the magnitude of the nonemployment duration jump at the threshold. A large jump implies that workers have strongly responded to the policy, so the elasticity of cost to search efforts must be low. The baseline job-finding rate $m$ is identified by the level of the nonemployment duration. A long duration means that $m$ must be low. The age gradient $q'_A$ is identified by the age patterns after the threshold age. For instance, an increasing pattern in nonemployment duration would be captured by a positive value for $q'_A$, so that the surplus and search efforts would decrease with age.

Unlike in the case of an age-conditional SA increase, the age pattern does not provide information to identify the job separation rate at the threshold $q_A$. The reason is that the nonemployment duration is discontinuous at $A$. Losing a job just before or just after the threshold assigns workers to two different UI benefit schemes. These workers choose a different search effort. Here, we use external information to calibrate the job destruction rate. Winter-Ebmer (2003) studies the same policy as Lalive (2008) and find that between 22% and 25% of employed workers lose their job within a year. We accordingly set $q_A$ to 0.02 to match this fact.<fnref n="14" /> We take the same value for Germany in the benchmark calibration.

The job separation rate may be a critical parameter to generate an anticipation effect before the age threshold. Accounting for the job loss rate, workers who find a job before age $A$ expect to keep the job sufficiently long to pass the threshold and enter the more generous UI system. As a sensitivity analysis, we calibrate the model with alternative values for $q_A$ in the online appendix. We also propose an extended calibration using richer information available for Germany. More precisely, the job separation rate $q_A$ is estimated using information on employment trajectories after the start of the first nonemployment spell. The UI exhaustion rates $\lambda_0$ and $\lambda_1$ can also be estimated using information on UI duration, that is, the duration for which workers receive UI benefits.

The empirical moments we match are nonemployment durations by age groups for workers initially entitled to UI benefits. The age groups correspond to quarters between ages 48 and 52 for Austria and four-month periods between ages 41 and 43 for Germany. In the latter case, we consider a smaller age window to avoid capturing age dynamics around age 44 that may be affected by the second UI extension. Nonemployment durations in Austria and Germany differ in levels. Below the age threshold, workers spend about three months as nonemployed in Austria compared to 15.5 months in Germany. This difference is substantial, given that the maximum UI duration is 9 and 12 months, respectively. After the age threshold, the average nonemployment duration jumps to seven months in Austria and 16.5 months in Germany.

The parameters for Austria and Germany are reported in Table 3. The model fit is plotted in Figure 5. The model captures well the level, the jump and the age pattern of nonemployment durations within the calibration window. The fit of untargeted nonemployment durations is also good at younger ages. For Austria, the model overshoots the duration of workers above 52. The gap with the data for older workers may be due to early retirement schemes that we do not model.

<footnote n="14">¹⁴ With $q = 0.02$, the share of workers losing their job within a year is $1 - \exp(-12 \times 0.02) \approx 21\%$.</footnote>
<table><caption>TABLE 3
MODEL PARAMETERS FOR THE AGE-CONDITIONAL UI EXTENSION</caption><thead><tr><th rowspan="2">Policy Case</th><th colspan="9">Parameters</th></tr><tr><th>A</th><th>b<sub>0</sub></th><th>b<sub>UI</sub></th><th>χ</th><th>α</th><th>m</th><th>q'<sub>A</sub></th><th>λ<sub>0</sub></th><th>λ<sub>1</sub></th></tr></thead><tbody><tr><td>Austria</td><td>50</td><td>0.42</td><td>0.6</td><td>0.59</td><td>0.027</td><td>0.0749</td><td>3.87e<sup>-5</sup></td><td>0.111</td><td>0.0207</td></tr><tr><td>Germany</td><td>42</td><td>0.35</td><td>0.68</td><td>0.7</td><td>0.99</td><td>0.111</td><td>2.13e<sup>-5</sup></td><td>0.0833</td><td>0.0556</td></tr></tbody></table>

NOTE: In both policy cases, we set $\underline{A} = 18$, $\bar{A} = 65$, $r = 0.0048$, $\bar{w} = 1$, and $q_A = 0.02$.

<figure><img src="image_9.png" /></figure>

NOTES: Nonemployment durations are aggregated by quarter for Austria (Lalive, 2008) and four-month period for Germany (Schmieder et al., 2012a). In the model, we approximate these aggregated durations as explained in online Appendix C. The shaded areas show untargeted nonemployment durations.

<caption>FIGURE 5
MODEL FIT FOR THE AGE-CONDITIONAL UI EXTENSION</caption>

Turning to the estimated parameters, the job separation rate increases as workers age in both cases. We find a low elasticity of the marginal cost, $\alpha$, at 0.03 for Austria compared with 0.99 for Germany. This elasticity indicates a very strong behavioral response to changes in workers' surplus in the Austrian case. For Austria, the short nonemployment durations before the age threshold suggest that workers quickly find a job. Most nonemployed workers do not even exhaust their benefits, unlike German workers. The effect of a UI extension on the returns to search is therefore small.<fnref n="15" /> However, the response of job seekers doubles the average nonemployment duration. Only a low elasticity can rationalize that strong response.

5.3. *Bias in the Studies.* Figures 6(a) and (b) depict the job-finding rates predicted by the calibrated model. The continuous lines correspond to the situation where workers can receive UI benefits longer after an age threshold. The dashed and dotted lines correspond to alternative scenarios where the UI duration does not change with age. When the UI duration is age-conditional, workers receiving SA or UI benefits search slightly more before the age threshold than those in the counterfactual without UI extension. After the age threshold, workers can now receive UI benefits for longer. These workers search less intensively than workers entitled to short-time UI benefits. The corresponding surpluses and populations of unemployed by age are available in the online appendix.

These search behaviors translate into short nonemployment durations for those who are below the threshold in comparison to workers above it (Figures 6(c) and (d)). Nonemployment durations below the threshold are shorter than in the counterfactual without the policy. This

<footnote n="15">¹⁵ The expected worker's surplus only decreases from 14.2 to 14.</footnote>
<figure><img src="image_10.png" /></figure>

NOTES: The black continuous curves correspond to the outcomes when the UI extension is conditional on age. The blue dashed, and the red dotted curves correspond to the counterfactual outcomes when there is no UI extension and when the UI extension is unconditional. In the first counterfactual, unemployed workers are either under SA (dots) or short-time UI with $\lambda_0$ (stars). In the second one, they are either under SA or long-time UI with $\lambda_1$ (triangles). Nonemployment durations are computed for workers claiming UI at the start of the unemployment spell, averaged by age.

<caption>FIGURE 6
SIMULATIONS OF AN AGE-CONDITIONAL UI EXTENSION</caption>

difference is the RDD bias, as it implies a gap between the LATE and the jump in nonemployment durations. The RDD bias is almost imperceptible for Austria.

To illustrate the role of different specifications in capturing the RDD bias, we generate simulated data sets to estimate the same regressions as in the two studies. Both studies run regressions at the individual level. We run similar regressions on average nonemployment durations by age groups in months instead of at the individual level. The studies consider different specifications. Lalive (2008) tries several age polynomials whereas Schmieder et al. (2012a) adjusts the bandwidth around the threshold. Unless stated otherwise, the regressions use average durations between ages 46 and 54 for Austria with an eight-year window (±4) and between 40 and 44 for Germany with a four-year window (±2). Table 4 reports the RDD estimates, the estimand, and the LATE.

For both studies, the RDD estimand predicted by the model is within the range of estimates obtained from the data. The model produces RDD estimates that are also very close to their empirical counterparts. Consistently with Figure 6, the theoretical bias, measured by the difference between the RDD estimand and the LATE, amounts to less than 2% of the LATE for
<table><caption>TABLE 4
RDD ESTIMATES FOR THE AGE-CONDITIONAL UI EXTENSION</caption><thead><tr><th rowspan="2"></th><th colspan="2">Austria</th><th colspan="2">Germany</th></tr><tr><th>Data</th><th>Model</th><th>Data</th><th>Model</th></tr></thead><tbody><tr><td><em>RDD estimates</em></td><td></td><td></td><td></td><td></td></tr><tr><td>No trend</td><td>3.37<br>(0.25)</td><td>4.58</td><td></td><td>0.87</td></tr><tr><td>Linear spline</td><td>3.41<br>(0.45)</td><td>2.52</td><td></td><td>0.82</td></tr><tr><td>Cubic spline</td><td>2.57<br>(0.8)</td><td>2.76</td><td></td><td>0.82</td></tr><tr><td>±2</td><td></td><td>2.71</td><td>0.78<br>(0.09)</td><td>0.82</td></tr><tr><td>±1</td><td></td><td>2.75</td><td>0.92<br>(0.12)</td><td>0.82</td></tr><tr><td>±0.5</td><td></td><td>2.76</td><td>1.04<br>(0.17)</td><td>0.82</td></tr><tr><td>±0.2</td><td></td><td>2.76</td><td>0.79<br>(0.27)</td><td>0.82</td></tr><tr><td>RDD estimand</td><td></td><td>2.76</td><td></td><td>0.82</td></tr><tr><td>LATE</td><td></td><td>2.72</td><td></td><td>0.7</td></tr></tbody></table>

NOTE: The outcome variable is the nonemployment duration for workers initially claiming UI benefits. The “Data” estimates and standard errors are taken from table 2, the three first columns, in Lalive (2008) and table 2 in Schmieder et al. (2012a). The estimates and standard errors for Austria are converted into months. The “Model” estimates are obtained from simulated data built from the nonemployment duration on Figures 6(c) and (d). The RDD estimand is the jump at age A of the durations with age-conditional policy. The LATE is the difference between the two counterfactual nonemployment durations at the age threshold. The specifications with varying bandwidths use a linear spline polynomial.

Austria and about 17% for Germany. Even for Germany, the LATE remains within the confidence interval of two standard errors in all the specifications. However, the bias reaches 50% of the LATE in the ±0.5 specification.

When calibrating the model with different values of $q_A$, we find that the magnitude of the bias increases with the job loss rate (online appendix). The bias is not statistically significant for Austria but is significant for Germany when $q_A$ is higher than 0.05. A higher job loss rate implies that future potential UI benefits matter more in the expected surplus. This effect increases the gap between the job-finding rates of unemployed workers entitled to short UI duration and those entitled to long UI duration, which is the entitlement effect.

In the extended calibration of the German case, we find a bias of a larger magnitude at 35% of the LATE. This result comes from a job separation rate of 0.04, a larger value than in the benchmark calibration.

5.4. *When is the Bias Large?* Our sensitivity analysis identifies the factors responsible for a sizeable bias and provides practical insights for applied researchers. The effect of the job-finding and job separation rates on the bias is represented in Figure 7 whereas the sensitivity to other key parameters is reported in Table A.3.

*Job-finding and job separation rates.* The RDD bias arises because unemployed workers below age A have a pre-entitlement incentive to find a job to receive extended benefits when they lose their job after the threshold. Consequently, the bias increases with the likelihood of a particular transition pattern, that is, finding a job before entitlement and losing it after entitlement. This likelihood is small compared to alternative transition patterns for realistic values of the contact rate per search unit *m* and the job separation rate $q_A$. It follows that anticipation effects do not strongly impact job search behaviors before the age threshold.
<figure><img src="image_11.png" /></figure>

NOTES: The RDD bias and the LATE on the nonemployment duration are represented as functions of parameters *m* and *q*<sub>A</sub>, with a linear scale for Austria and a base-2 log scale for Germany. The RDD bias is computed with a linear spline specification with a four-year window. In the Austrian case, *m* = 0.0749 and *q*<sub>A</sub> = 0.02. In the German case, *m* = 0.111 and *q*<sub>A</sub> = 0.02. The cross at the center marks the location of the calibrated parameters.

<caption>FIGURE 7
BIAS AND LATE AS A FUNCTION OF *m* AND *q*<sub>A</sub>, AGE-CONDITIONAL UI EXTENSION</caption>

The RDD bias and the LATE are represented in Figure 7 for alternative parameters of the job-finding and job separation rates. There is a stark contrast between the two cases due to the different search elasticity α. The low elasticity implies that a small change of parameters in the Austrian case leads to a strong shift of the LATE. However, both cases generate similar qualitative patterns. The LATE is higher when jobs are shorter in duration, and to some extent, when it is harder to find jobs. These elements contribute to making employment less attractive and discourage search efforts. When these elements are combined, the RDD bias is the highest, in absolute terms and relative to the LATE. For Austria, when the job separation rate at the threshold is 10% higher and the baseline job-finding rate is 10% lower, the LATE skyrockets to 21 months whereas the RDD bias does not reach one month (about 3% of the LATE). For Germany, the RDD bias reaches 1.5 months, three times the LATE, when the job separation rate is twice as high and the baseline job-finding rate twice as low.

This analysis shows that the RDD bias for age-conditional UI extensions can be significant and sizeable, especially when job-finding rates are low and job separation rates are high.
***Age gradient and horizon effect.*** The age pattern of job separation rates is stronger and workers are closer to the retirement age in Austria, compared to Germany. These differences may explain differences in bias.

The age profile in job separation turns out to affect the LATE only for Austria. Removing the age gradient reduces the LATE by 16% in that case. The intuition is that, when $q'_{A} = 0$, finding a job is more rewarding because workers expect to stay longer in employment when they are above A, compared to the case when $q'_{A} > 0$. However, the RDD estimand follows the same pattern, so that the RDD bias does not markedly change.

The horizon effect does not matter quantitatively in both cases. The mandatory retirement age should be below 60 in Austria or below 55 in Germany to observe a change in the LATE.

***UI recipiency.*** Workers below the age threshold respond to their expectation to obtain benefits after losing their next job, which depends on the UI recipiency rate. We compute the RDD estimate for different probabilities for $\chi$. As expected, the lower the probability of receiving UI benefits, the lower the RDD bias. Conversely, if everyone claimed UI benefits after losing their jobs, the RDD bias would be at 27% of the LATE in the German case.

***General lessons.*** The first lesson is that the RDD bias depends on the frictions’ strength in the labor market. It is likely to be large when workers stay unemployed for a long time and do not obtain stable jobs. The second lesson is that anticipating future eligibility to UI benefits is critical. When workers do not easily obtain UI benefits after the next job, the entitlement effect is small, and so is the bias.

## 6. CONCLUSION

Job search theory is a consistent framework to describe labor market phenomena. We use it to discuss the potential bias of RDDs evaluating SA increases and UI extensions. The design involves an age discontinuity in treatment exposure, that is, only those above a specific age can access more generous or extended benefits. Search frictions imply that this design is biased because individuals should progressively modify their search behavior before entitlement. This anticipation behavior implies that the SUTVA does not hold.

We calibrate a nonstationary job search model on the various data sets used in the literature. According to our calibrations, the impact of age-conditional SA increases is underestimated in absolute terms, whereas the impact of age-conditional UI extensions is overestimated in absolute terms. As a takeaway, the bias is large in the presence of strong search frictions, when the job-finding rate is low and the job separation rate is high. We also warn against the use of stock variables like the employment rate as the outcome. Finally, the magnitude of the bias critically depends on the ability of workers to anticipate their future search strategy. Recent literature shows that unemployed workers do not behave as rational agents, thus weakening the bias (see, e.g., Spinnewijn, 2015; DellaVigna et al., 2022).

The biases are conditional on the fact that job search theory accurately describes actual labor markets. The simple model we use certainly overstates the magnitude of search frictions because they do not feature worker heterogeneity or participation choices. Therefore, the first task is to enrich this model to make more reliable counterfactual predictions. Nevertheless, labor market researchers should be careful while interpreting the results of an RDD based on age discontinuity. A job search model can indeed rationalize their findings. However, the same model also predicts the magnitude of the RDD bias.

Research can pursue two directions. Our model can be extended to study alternative outcomes. Caliendo et al. (2013), Schmieder et al. (2016), and Nekoei and Weber (2017) examine the impacts of UI extensions on postunemployment wages and job match quality. We must consider the role of reservation wages and introduce duration-dependence in the wage offer distribution. Our model can also be extended to consider equilibrium effects, which would require a labor demand block.
**DATA AVAILABILITY STATEMENT** The data and code that support the findings of this study are openly available at https://doi.org/10.3886/E207842V1, or on the GitHub page https://github.com/gwilemme/RDD_age_disc.

## APPENDIX A

**A.1. Regression Discontinuity Design (RDD) Specifications.** In several instances of the article, we estimate the policy impact with an RDD based on an age discontinuity (see Tables 2 and 4). The RDD estimator $\hat{\beta}_1$ is based on the econometric model

$$ (A.1) \qquad Y_i = \beta_0 + \beta_1 \mathbf{1}\{a_i \ge A\} + \delta(a_i) + \epsilon_i, $$

where the function $\delta$ is continuous in age.

The specifications differ in the polynomial form of $\delta$. The polynomial form can be the same before and after the age threshold, or can be different. In the former case, function $\delta$ is simply a polynomial, the degree to which we have to choose its degree. In the latter case, function $\delta$ is a spline, with different polynomials before and after the age threshold.

In the polynomial case, the derivative of function $\delta$ is also continuous. The continuity forces the dynamics of the outcome of interest to be locally similar before and after the age threshold. In the spline case, the derivative of function $\delta$ is generally discontinuous. The RDD allows for potentially different local dynamics before and after the threshold.

Though we present several regression results in each case, we follow the other researchers and insist on the linear spline specification.

Table A.1 provides the list of specifications used in the article. Function $\delta$ does not feature a constant term because $\beta_0$ already accounts for it. Importantly, $\delta$ is continuous at A in all specifications and nondifferentiable at A in the spline specifications.

<table><caption>TABLE A.1
PARAMETERIZATIONS OF THE AGE TREND</caption><thead><tr><th></th><th>Specification for δ(a)</th></tr></thead><tbody><tr><td>No trend</td><td>0</td></tr><tr><td>Linear</td><td>δ<sub>1</sub>a</td></tr><tr><td>Quadratic</td><td>δ<sub>1</sub>a + δ<sub>2</sub>a<sup>2</sup></td></tr><tr><td>Cubic</td><td>δ<sub>1</sub>a + δ<sub>2</sub>a<sup>2</sup> + δ<sub>3</sub>a<sup>3</sup></td></tr><tr><td>Linear spline</td><td>δ<sub>1</sub>a + δ'<sub>1</sub>(a - A)1{a ≥ A}</td></tr><tr><td>Quadratic spline</td><td>δ<sub>1</sub>a + δ<sub>2</sub>a<sup>2</sup> + δ'<sub>1</sub>(a - A)1{a ≥ A} + δ'<sub>2</sub>(a - A)<sup>2</sup>1{a ≥ A}</td></tr><tr><td>Cubic spline</td><td>δ<sub>1</sub>a + δ<sub>2</sub>a<sup>2</sup> + δ<sub>3</sub>a<sup>3</sup> + δ'<sub>1</sub>(a - A)1{a ≥ A} + δ'<sub>2</sub>(a - A)<sup>2</sup>1{a ≥ A} + δ'<sub>3</sub>(a - A)<sup>3</sup>1{a ≥ A}</td></tr></tbody></table>

<table><caption>TABLE A.2
SENSITIVITY ANALYSIS, AGE-CONDITIONAL SA INCREASE</caption><thead><tr><th rowspan="2"></th><th colspan="3">Canada</th><th colspan="3">France</th></tr><tr><th>Estimate</th><th>LATE</th><th>Bias (%)</th><th>Estimate</th><th>LATE</th><th>Bias (%)</th></tr></thead><tbody><tr><td>Benchmark calibration</td><td>-0.047</td><td>-0.069</td><td>31.9</td><td>-0.069</td><td>-0.085</td><td>19</td></tr><tr><td><em>Other age effects</em></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>No trend, q'<sub>A</sub> = 0</td><td>-0.047</td><td>-0.07</td><td>32.7</td><td>-0.07</td><td>-0.085</td><td>17.8</td></tr><tr><td>Infinite horizon</td><td>"</td><td>-0.069</td><td>31.9</td><td>-0.069</td><td>"</td><td>19</td></tr><tr><td>Retirement at Ā = 45</td><td>"</td><td>"</td><td>"</td><td>"</td><td>"</td><td>"</td></tr><tr><td>Retirement at Ā = 40</td><td>-0.027</td><td>"</td><td>60.9</td><td>"</td><td>"</td><td>"</td></tr></tbody></table>

(Continued)
<table><caption>TABLE A.2
(CONTINUED)</caption><thead><tr><th rowspan="2"></th><th colspan="3">Canada</th><th colspan="3">France</th></tr><tr><th>Estimate</th><th>LATE</th><th>Bias (%)</th><th>Estimate</th><th>LATE</th><th>Bias (%)</th></tr></thead><tbody><tr><td><em>Uncertainty</em></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ξ = 0.25</td><td>-0.05</td><td>"</td><td>28.6</td><td>-0.071</td><td>"</td><td>17</td></tr><tr><td>ξ = 0.5</td><td>-0.052</td><td>"</td><td>25.4</td><td>-0.072</td><td>"</td><td>15.1</td></tr><tr><td>ξ = 0.75</td><td>-0.054</td><td>"</td><td>22.5</td><td>-0.073</td><td>"</td><td>13.6</td></tr><tr><td>ξ = 1</td><td>-0.056</td><td>"</td><td>19.7</td><td>-0.075</td><td>"</td><td>12.2</td></tr><tr><td><em>Job-finding rate as regression outcome</em></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>No uncertainty</td><td>-0.039</td><td>"</td><td>44.2</td><td>-0.067</td><td>"</td><td>21.2</td></tr><tr><td>ξ = 0.25</td><td>-0.048</td><td>"</td><td>30.8</td><td>-0.073</td><td>"</td><td>13.5</td></tr><tr><td>ξ = 0.5</td><td>-0.057</td><td>"</td><td>18.4</td><td>-0.079</td><td>"</td><td>6.5</td></tr><tr><td>ξ = 0.75</td><td>-0.065</td><td>"</td><td>6.8</td><td>-0.085</td><td>"</td><td>0.03</td></tr><tr><td>ξ = 1</td><td>-0.072</td><td>"</td><td>-4</td><td>-0.09</td><td>"</td><td>-6</td></tr></tbody></table>

NOTE: Each row consists of a version of the model that departs from the calibrated model. We report the RDD estimate, the LATE, and the RDD bias relative to the policy effect on the employment rate. We use the linear spline specification. When the average job-finding rate is used as the outcome variable instead of the employment rate, we use a back-of-the-envelope approximation to find the estimated effect on the employment rate.

<table><caption>TABLE A.3
SENSITIVITY ANALYSIS, AGE-CONDITIONAL UI EXTENSION</caption><thead><tr><th rowspan="2"></th><th colspan="3">Austria</th><th colspan="3">Germany</th></tr><tr><th>Estimate</th><th>LATE</th><th>Bias (%)</th><th>Estimate</th><th>LATE</th><th>Bias (%)</th></tr></thead><tbody><tr><td>Benchmark calibration</td><td>2.71</td><td>2.72</td><td>-0.3</td><td>0.82</td><td>0.7</td><td>18.1</td></tr><tr><td><em>Other age effects</em></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>No trend, q'<sub>A</sub> = 0</td><td>2.31</td><td>2.29</td><td>1.2</td><td>0.81</td><td>0.69</td><td>17.5</td></tr><tr><td>Infinite horizon</td><td>2.71</td><td>2.72</td><td>0.82</td><td>0.82</td><td>0.7</td><td>18.1</td></tr><tr><td>Retirement at A̅ = 60</td><td>2.68</td><td>"</td><td>-1.4</td><td>"</td><td>"</td><td>"</td></tr><tr><td>Retirement at A̅ = 55</td><td>0.74</td><td>0.98</td><td>-23.9</td><td>"</td><td>"</td><td>18.4</td></tr><tr><td>Retirement at A̅ = 50</td><td></td><td></td><td></td><td>0.83</td><td>0.65</td><td>27.3</td></tr><tr><td><em>UI recipiency</em></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>χ = 0.25</td><td>2.99</td><td>3.04</td><td>-1.9</td><td>0.88</td><td>0.83</td><td>6</td></tr><tr><td>χ = 0.5</td><td>2.78</td><td>2.79</td><td>-0.6</td><td>0.85</td><td>0.75</td><td>12.5</td></tr><tr><td>χ = 0.75</td><td>2.59</td><td>2.59</td><td>0.2</td><td>0.82</td><td>0.68</td><td>19.5</td></tr><tr><td>χ = 1</td><td>2.43</td><td>2.41</td><td>0.8</td><td>0.79</td><td>0.62</td><td>27.2</td></tr></tbody></table>

NOTE: Each row consists of a version of the model that departs from the calibrated model. We report the RDD estimate, the LATE, and the RDD bias relative to the policy effect on nonemployment duration. We use the linear spline specification with a four-year window.

The RDD specifications also differ in the choice of bandwidths, that is, the window of observations around the age threshold. For instance, a bandwidth $\pm h$ means that we retain observations such that $a \in [A - h, A + h]$.

## A.2. Sensitivity Analysis.

## SUPPORTING INFORMATION

Additional supporting information may be found online in the Supporting Information section at the end of the article.

Figure A.1: Simulations of an age-conditional SA increase, additional figures

Figure A.2: Simulations of an age-conditional UI extension, additional figures

Figure A.3: Simulations of an age-conditional UI extension, additional figures
Table A.1: Calibration with different values of $q_A$, Canada

Table A.2: Calibration with different values of $q_A$, France

Table A.3: Calibration with different values of $q_A$, Austria

Table A.4: Calibration with different values of $q_A$, Germany

Table A.5: Calibration with different values of $\xi$, Canada

Table A.6: Calibration with different values of $\xi$, France

Table A.7: Model parameters for the extended calibration

Figure A.4: Extended calibration for Germany

Table A.8: RDD estimates for the extended calibration

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