Bruno Decreuse
email: bruno.decreuse@univ-amu.fr

Guillaume Wilemme
email: guillaume.wilemme@univ-amu.fr.

Age discontinuity and nonemployment benet policy evaluation through the lens of job search theory *

Keywords: C41, H3, J21, J64 RDD, Age discontinuity, Nonstationary job search theory

A recent strand of papers use sharp regression discontinuity designs (RDD) based on age discontinuity to study the impacts of minimum income and unemployment insurance benet extension policies. This design challenges job search theory, which predicts that such RDD estimates are biased. Owing to market frictions, people below the age threshold account for future eligibility to the policy. This progressively aects their search outcomes as they get closer to entitlement.

Comparing them to eligible people leads to biased estimates because both groups of workers are actually treated. We provide a nonstationary job search model and quantify the theoretical biases on the datasets used in the literature. Our results suggest that the employment impact of minimum income policies are (signicantly) under-estimated, whereas the impacts of benet extensions on nonemployment duration are (not signicantly) over-estimated.

Introduction

Several papers use an age discontinuity to evaluate the impacts of nonemployment benets on job search outcomes. [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] on Canadian data and [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF][START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF] on French data examine the minimum income impact on employment. [START_REF] Lalive | How do Extended Benets aect Unemployment Duration? A Regression Discontinuity Approach[END_REF] on Austrian data, [START_REF] Caliendo | Benet Duration, Unemployment Duration and Job Match Quality: A Regression-Discontinuity Approach[END_REF] and Schmieder et al. (2012aSchmieder et al. ( ,b, 2016) ) on German data study the eects of unemployment benet extensions on unemployment duration and post-unemployment wages. In all these papers, the fate of people right after entitlement is compared to the fate of those right before it. This design challenges job search theory. On the ground that this theory oers a comprehensive view of labor market phenomena, RDD estimates based on age discontinuity should be biased.

In this paper, we highlight the roles played by market frictions and workers' expectations. Owing to frictions, workers below the threshold age respond to future labor market conditions in a continuous way. Age-conditional policies aect their current behavior, implying that people non-eligible to the policy are actually treated. From theoretical perspective, neglecting such eects leads to (signicantly) underestimate the minimum income impacts on Canadian and French data and (not signicantly) over-estimate the benet duration impacts in Austria and Germany (though this may still matter in the latter case).

We start by discussing the likely bias aecting RDD when using an age discontinuity to evaluate minimum income and benet extension policies (Section 2). We rst provide simplied arguments to understand the nature of the bias and guess the key factors behind its magnitude. We argue that search frictions imply that individuals are treated before entitlement and that the magnitude of pre-entitlement eect should be larger in case of minimum income policies than in case of benet extension policies. Then we examine the Local Average Treatment Eect (LATE) and decompose the RDD estimate expectation into the true eect and the potential bias due to the combination of frictions and rational expectations.

The bias should be negative in case of minimum income policies and positive in case of benet extension policies.

We then provide a job search model tailored to study age-conditional minimum income and unemployment benet potential duration (Section 3). This model features three key elements of job search theory: (i) frictions prevent workers from instantaneously nding jobs; (ii) search eort increases the arrival rate of job oers; (iii) individuals rationally anticipate policy eligibility. The typical job search problem is nonstationary: as time passes, individuals get closer to entitlement. This continuously aects the current value of future streams of income and, therefore, the time pattern of search eorts.

We use our model to quantify the impact of age-conditional minimum income policies on employment (Section 4). We calibrate its structural parameters on the Canadian data used by [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] and the French data used by [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF][START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF]. Well before entitlement, jobseekers are already impacted because they have a large chance to remain unemployed after the age threshold. Therefore they progressively reduce search eort. The comparison of individuals shortly before and after entitlement is misleading because they are similarly exposed to the policy. Our calibrations suggest that this eect quantitatively matters and theory predicts that the papers previously quoted underestimate the minimum income impact.

We also use our model to quantify the impact of age-conditional unemployment benet extensions on employment (Section 5). We calibrate its structural parameters on the Austrian data used by [START_REF] Lalive | How do Extended Benets aect Unemployment Duration? A Regression Discontinuity Approach[END_REF] and the German data used by [START_REF] Caliendo | Benet Duration, Unemployment Duration and Job Match Quality: A Regression-Discontinuity Approach[END_REF] and Schmieder et al. (2012aSchmieder et al. ( ,b, 2016)). Before the age threshold, job-seekers expect that they will have access to longer benets in case they nd a job and lose it afterwards. This provides early incentive to search. Therefore the comparison of individuals before and after the age threshold overestimates the benet duration impact on unemployment duration. However, our calibrations reveal that this eect is quantitatively small, especially in the Austrian case.

The pattern of transitions involved in the reasoning does not frequently occur, i.e., over a year, the probability of nding a job times the probability of losing it can be neglected.

Our paper applies search theory arguments to typical labor market issues where this theory is used. Therefore, it is not only food for thought for researchers specialized in public policy evaluation, but also a test of the theory with regard to its ability to reproduce actual labor market outcomes. Search theory provides a consistent framework to describe how fast people nd jobs and how much they get paid. This framework is based on the notion of search frictions: people cannot instantaneously match and need to spend time and resources to get matched. An alternative view is that people can nd jobs very rapidly and only those who cannot or do not want to nd jobs stay nonemployed. Under this alternative view, unobserved worker heterogeneity gives the false impression that everyone needs time. In the Walrasian limit of our model, people wait the very last second to modify their behavior; in such a case the RDD based on age discontinuity is unbiased.

This paper owes a lot to the literature on nonstationary search. Several papers focus on the potential duration of benets and show that the search eort increases and the reservation wage decreases as the exhaustion date gets closer (see, e.g., [START_REF] Mortensen | Unemployment Insurance and Job Search Decisions[END_REF][START_REF] Van Den Berg | Nonstationarity in Job Search Theory[END_REF][START_REF] Cockx | Imperfect Monitoring of Job Search: Structural Estimation and Policy Design[END_REF].

Eligibility eects are important and their consideration improves our understanding of job-nding rates by unemployment duration. In this paper, we model the benet duration as a Poisson process instead of a deterministic duration. We do so because there are additional eligibility criteria to UI benets so that the potential UI duration varies more across unemployed workers than expected. Having a random potential UI duration is a shortcut to account for such eects. The model can easily be modied to account for a xed potential duration. Closer to us, there is a growing literature on life-cycle eects in job search models. [START_REF] Chéron | Life-Cycle Equilibrium Unemployment[END_REF]; [START_REF] Hairault | Distance to Retirement and Older Workers' Employment: The Case for Delaying the Retirement Age[END_REF][START_REF] Hairault | Optimal Unemployment Insurance for Older Workers[END_REF][START_REF] Hairault | Equilibrium Unemployment and Retirement[END_REF] focus on the retirement age and its impact on the pattern of training and search investments decisions. [START_REF] Menzio | Directed Search over the Life Cycle[END_REF] analyze job-to-job transitions and the wage distribution during the life-cycle. The main lesson is that the retirement age aects workers well before retirement. This explains why older workers search less when unemployed and receive less training when employed. This also explains wage and employment patterns by age. We use insights from these models to quantify the theoretical bias of RDD based on an age discontinuity. Schmieder et al. (2012a[START_REF] Schmieder | The Eects of Extended Unemployment Insurance over the Business Cycle: Evidence from Regression Discontinuity Estimates over 20 Years[END_REF] provide nonstationary job search models to explain their results. Nonstationarity is not based on age eligibility to benets. Rather, it involves the eects of the potential duration of benets and the duration spent in unemployment. As time passes, the wage oer distribution worsens. Job-seekers reduce their search eort and receive lower wages. We do not examine wages the interested reader can modify our model to allow for duration-dependence in the wage oer distribution.

Instead, we complete their analysis by emphasizing the role of the age threshold. This allows us to show that their RDD has a very small bias despite workers are actually treated before entitlement.

There is already a literature on the evaluation of pre-treatment eects in the context of policy discontinuity with respect to time (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., [START_REF] Crépon | Information Shocks and the Empirical Evaluation of Training Programs during Unemployment Spells[END_REF][START_REF] Crépon | Information Shocks and the Empirical Evaluation of Training Programs during Unemployment Spells[END_REF][START_REF] Van Den Berg | Policy Discontinuity and Duration Outcomes[END_REF]. This literature is dominated by the 'No anticipation assumption' whereby individuals do not modify their behavior in expectation of the treatment date (see [START_REF] Abbring | The Nonparametric Identication of Treatment Eects in Duration Models[END_REF]. Though excluding the nonstationarity featured in this paper, this assumption is actually compatible with ex-ante policy eects, individuals acting in expectation of being treated at some future unknown date. The closest papers in spirit evaluate anticipated impacts through the individual disclosure of information on the treatment date (see, e.g., [START_REF] Black | Is the Threat of Reemployment Services More Eective Than the Services Themselves? Evidence from Random Assignment in the UI System[END_REF][START_REF] Giorgi | Long-Term Eects of a Mandatory Multistage Program: The New Deal for Young People in the UK[END_REF][START_REF] Crépon | Information Shocks and the Empirical Evaluation of Training Programs during Unemployment Spells[END_REF][START_REF] Van Den Berg | Policy Discontinuity and Duration Outcomes[END_REF], for participation to job search assistance or training programmes, and Lalive et al., 2005, for UI benet sanctions). This consists in comparing hazard rates for those who are notied some months or even weeks in advance to those who are not. These papers nd signicant eects of notication. Our paper complements this literature in three ways. First, we analyze a dierent economic issue, namely age-conditional nonemployment benets instead of active labor market programmes.

Second, we provide a nonstationary job search model that explicitly predicts the magnitude and direction of the RDD bias as a function of worker age. Third, we calibrate this model to quantify the theoretical biases in several studies.

Main intuitions

We rst provide a detailed intuition on the reason why matching frictions imply that workers below the age discontinuity are aected by age-conditional benets. We then discuss the potential bias that aects RDD estimates.

We are interested in the two types of labor market policies discussed in Introduction: age-conditional minimum income and age-conditional UI extensions. Both policies are likely to aect job search behavior.

Only the people above the threshold age A are eligible to such policies. However, people below A are also treated when 1. individuals can signicantly impact the outcome variable; 2. individuals anticipate that they will be treated later; 3. tomorrow "matters" for today's decision.

The rst condition implies there is a behavioral response to the policy. The second condition requires that individuals are forward-looking and suciently well-informed on the policy. The last condition depends on the magnitude of search frictions, which we now explain.

Heuristic argument

Search frictions and expectations imply that workers are treated by age-conditional benets before being entitled to them. We discuss the sign of this pre-treatment impact on workers' welfare as well as its likely magnitude according to worker age.

Age is continuous and indexed by a ≥ 0. Unemployed workers nd jobs at rate m and lose them at rate q. The wage is w per unit of time. We consider two dierent types of age-conditional benets. In both cases, unemployment income is b 0 before age A. In the rst case, all unemployed workers receive b 1 ≥ b 0 after age A. In the second case, only those who lost their job after age A receive b 1 , whereas the others receive b 0 . This case mimics benet extension policies where the benet level does not change but the potential benet duration increases after age A.

We start with the rst case where all unemployed receive b 1 after age A. We denote by S(a) the match surplus, i.e., the asset value dierential between employment and unemployment. This quantity is important because it indicates how much the job-seekers are willing to nd jobs. For all a ≤ A,

rS(a) = w -b 0 -(m + q)S(a) + S (a), (1)
rS(A) = w -b 1 -(m + q)S(A).
(2)

These arbitrage equations receive the following interpretation. The left-hand side is the nancial return to match surplus. The right-hand side is its labor market return. It is composed of the income dierential between employment and unemployment, w -b 0 , plus the loss of surplus induced by the transition rates m (while unemployed) and q (while employed), plus appreciation due to age as the worker becomes closer to entitlement to b 1 . This third term is zero after age A.

The pre-entitlement match surplus solving this problem is

S(a) = S 0 (1 -F (a)) + S 1 F (a), (3)
where

S i = (w -b i )/(r + m + q) is the stationary match surplus associated to benet level b i , i = 0, 1,
and

F (a) = exp[-(r + m + q)(A -a)] is the eective discount factor.
The match surplus at age a is a weighted mean of the two stationary match surpluses. The eective discount factor increases with age and decreases with the the sum of interest rate, r, and transitions rates, m and q. It is equal to one when a = A. Therefore the match surplus decreases with age from S 0 to S 1 .

Let ε(a) = d ln S(a)/d ln b 1 be the elasticity of match surplus with respect to age-conditional benet b

1 . When b 0 = b 1 , we have ε(a) = ε(A)F (a) < 0. (4)
The pre-entitlement match surplus negatively responds to an increase in post-entitlement benets. The magnitude of the pre-entitlement elasticity, ε(a), is proportional to the post-treatment one, ε(A). The factor of proportion is lower than one and increases with age to reach one at age A where the two elasticities coincide. It also decreases with the job-nding rate, reaching 0 in the Walrasian case where the job-nding rate tends to innity.

Next, we consider the second case where age-conditional benet b 1 is restricted to those who lost their job after age A. For all a ≤ A, rS(a) = w -b 0 -(m + q)S(a) + S (a),

(5)

rS(A) = w -b 0 -qΣ(A) -mS(A), (6) rΣ(A) = w -b 1 -(m + q)Σ(A). (7)
Workers after age A are not automatically entitled to higher benets. They need to experience an employment episode. This is why the computation of stationary match surplus for nonentitled workers, S(A), involves the stationary match surplus for those entitled, Σ(A). The transition towards entitlement occurs at rate m. Once entitled, workers will never be nonentitled again.

The pre-entitlement match surplus solving this problem is (9)

S(a) = S 1 + b 1 -b 0 r + m + q 1 + q r + m F (a) .
Unlike the previous elasticity ε(a), the elasticity e(a) is positive. An increase in age-conditional benets paid to job losers increases the value of matching for those nonentitled. This type of phenomenon is known in the search literature as the entitlement eect (see, e.g., Mortensen, 1973). The quantitative impact is proportional to the elasticity ε(a). This means that |e(a)| increases with age and decreases with the job-nding rate. Theoretically, the factor of proportion q/(r + m) can be lower or greater than one.

In practice, q << m so that |e(a)| < |ε(a)|. Therefore, the pre-entitlement impact of benets restricted to job losers after age A is likely much smaller than the impact of benets unconditionally paid to all unemployed after this age.

We summarize the main lessons taught by this simple model.

(i) Age-conditional benets aect the pre-entitlement match surplus. It follows that workers are actually treated before entitlement age.

(ii) The magnitude of pre-entitlement impact increases with worker age and search frictions. In the Walrasian case where m is arbitrarily large, the impact is zero.

(iii) Age-conditional benets paid to all unemployed after age A reduce the pre-entitlement match surplus.

(iv) Age-conditional benets restricted to job losers after age A increase the pre-entitlement match surplus.

(v) Given realistic values for m and q, benets paid to all unemployed after age A have a larger impact on pre-entitlement match surplus than those restricted to job losers after this age.

RDD bias

We now turn to a description of the RDD bias. This description involves the fact that search eorts increase with expected match surplus. This is a very intuitive idea that we will formally establish later.

Age-conditional minimum income and UI extensions are binary treatments that impact job search outcomes like the employment probability, unemployment duration or job-nding rate. To t with the previous discussion, we discuss the case of the hazard rate from unemployment to employment. Individuals are indexed by i and their age is a i . We neglect other forms of observable heterogeneity to simplify exposition.

Let D i denote treatment exposure with D i = 1 if treated and D i = 0 otherwise. We have D i = 1

when a i ≥ A.
E[Y 1 i -Y 0 i |a i = A] in a regression discontinuity design (RDD).
The key problem comes from those who are too young and supposedly non-treated but who already respond to the treatment in expectation of future entitlement. Thus we need three dierent potential outcome variables: Y 1 i for those with age a i ≥ A; Y 0 i the ideal outcome for those who do not expect treatment exposure; Ỹ 0 i for those not currently exposed, a i ≤ A, but expecting future entitlement.

The assignment variable is D i = 1(a i ≥ A). The observed outcome is

Y i = D i Y 1 i + (1 -D i ) Ỹ 0 i = Y 1 i if a i ≥ A Ỹ 0 i if a i < A , (10) which departs from the ideal outcome D i Y 1 i + (1 -D i )Y 0 i . Econometricians can observe Y 1
i after enti- tlement. However, they cannot observe Y 0 i before it. Instead, they observe Ỹ 0 i .

The two necessary conditions to estimate a LATE are (i) the treatment is "as good as random", i.e.,

Y 0 i , Y 1 i ⊥ ⊥ D i , and (ii) conditional expectations E[Y 0 i |a i = a] and E[Y 1 i |a i = a] are continuous in a.
Sticking to these assumptions, we can estimate the standard econometric model

Y i = β 0 + β 1 1(a i ≥ A) + δ(a i ) + i , ( 11
)
where δ is a continuous function. 1

The RDD estimator, β1 , is a consistent

estimator of E[Y 1 i -Ỹ 0 i |a i = A].
This expectation, that we call the RDD estimand, may dier from the LATE, which can be decomposed as follows:

E[Y 1 i -Y 0 i |a i = A] = E[Y 1 i -Ỹ 0 i |a i = A] RDD estimand + E[ Ỹ 0 i -Y 0 i |a i = A] Bias (12)
The RDD estimand coincides with the LATE when Ỹ 0 i is very close to Y 0 i around A. Otherwise, the RDD estimand features a positive bias with respect to the LATE when the behavioral response of individuals 1 We can of course add controls X i to estimate the expectation conditional on X i .

Age (a)

Job-nding rate (Y )

E[Y 1 i |a i = a] E[Y 0 i |a i = a] E[ Ỹ 0 i |a i = a] A LATE (a) Virtual age discontinuity Age (a)
Job-nding rate (Y )

A • • • • • • RDD estimate (b) Inferred age discontinuity
E[Y 1 i |a i = a] E[Y 0 i |a i = a] E[ Ỹ 0 i |a i = a]
A, i.e. E[Y 1 i -Y 0 i |a i = A] = 0.
In practice, econometricians do not observe the continuous curve depicted by the left-hand side gure.

Instead, individuals are observed at discrete time intervals as shown in the right-hand side gure. The linear regression provides the RDD estimate through extrapolation, choosing the degree of sophistication with the function δ. Here we show a linear spline model. 2 As evidenced by the right-hand side gure, the RDD underestimates the policy impact on the job-nding rate. Moreover the extrapolation procedure strengthens the bias.

Figure 2 focuses on age-conditional UI benet extensions. This corresponds to benets paid to those who lost their job after age A in the previous sub-section. To be eligible, nonemployed individuals need rst to nd a job and lose it after age A. As explained in the previous subsection, the pre-entitlement expected match surplus increases with age to reach a higher stationary level than for those who are entitled, i.e., those who remain unemployed after age A still need to nd a job to become eligible to the policy. This explains why the pre-entitlement hazard rate increases with age and why the comparison of those eligible right after age A to those right before this age involves a discontinuity in the hazard rate.

Here again, the RDD estimate signicantly diers from the LATE. The right-hand side gure shows that the RDD overestimates the policy impact.

The bias can be reduced by expanding the observation window around the discontinuity. If workers change their behavior one year before eligibility, we can simply drop all individuals who will be eligible in less than a year. This strategy of voluntarily losing observations in a window [A -ε, A] before the threshold has two limits. First, at individual level, the age pattern right before A -ε becomes more and more disconnected to the age pattern right after this interval as ε increases. Second, population heterogeneity between individuals at age A -ε and at age just above A also increases with ε. The design no longer responds to the criteria dening a randomized experiment, which is usually a strength of RDD [START_REF] Lee | Regression Discontinuity Designs in Economics[END_REF].

Expected eects of future treatment are already known in the literature on policy evaluation. However, identication involves the 'No expectation' assumption whereby workers do not modify their behavior in expectation of the treatment date (see [START_REF] Abbring | The Nonparametric Identication of Treatment Eects in Duration Models[END_REF]. This assumption forbids the type of patterns reported by Figures 1 and2. Nevertheless, within this framework, [START_REF] Crépon | Information Shocks and the Empirical Evaluation of Training Programs during Unemployment Spells[END_REF] and van den Berg et al. ( 2018) exploit heterogenous notication dates on future training to measure 2 The linear spline specication consists in estimating δ(a) as a linear function of a with a dierent slope below and above

A.

expected eects. This consists in comparing the behavior of those notied at a given date with the one of those unnotied at the same date.

3 Job search with age entitlement to nonemployment benets

We provide a continuous-time model with age-conditional benets and search eorts. This model is as simple as possible: workers are homogenous and risk-neutral, and there is no on-the-job search.

The time unit is the month. Benets respond to age in the two ways previously discussed by the literature. At age A, assistance benets suddenly increase from b min 0 to b max 0 . This corresponds to the Canadian case examined by [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] and the French one analyzed by [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF][START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF]. Moreover, workers dismissed after age A can receive UI benets for a longer period.

We capture this as a smaller rate of losing benet entitlement: the Poisson rate goes from λ max before age A to λ min after this age. This corresponds to the Austrian case studied by [START_REF] Lalive | How do Extended Benets aect Unemployment Duration? A Regression Discontinuity Approach[END_REF] and the German case highlighted by Schmieder et al. (2012aSchmieder et al. ( ,b, 2016)). The reason why we treat the loss of benet entitlement as a random event is because there are additional criteria to get entitled to UI. Therefore the max duration, in practice, varies across individuals of the same age and same date of job loss.

We model the job search behavior of a typical nonemployed individual. Before age A, the model is nonstationary because the worker's situation changes as the threshold age gets closer. Let S(a, w, b) be the match surplus for a worker of age a ≤ A, when the wage is w and the worker is entitled to benet b. In practice the benet can take three values: b U I , the regular UI benet, and b min 0 and b max 0 , the assistance benets when workers are, respectively, below and above age A. After age A, the model is stationary as the expected utility of a worker does not directly depend on age. However, unemployed workers who lose their job before age A experience the higher rate of losing benets λ max , whereas the others are submitted to the lower risk λ min . We denote by Σ(w, b, λ) the match surplus for a worker of age a > A when the wage is w, the worker is entitled to nonemployment benet b and the risk of losing UI benet is λ.

Workers make search eorts. The search eort s comes at cost c(s), with c (.) > 0, c (.) > 0 and c(0) = c (0) = 0. In exchange for the cost, workers receive wage oers at rate ms. Each wage oer is a random draw from the cumulative distribution function Φ. The corresponding probability density function is ϕ. In the remaining, we suppose that the lower bound of the support of the distribution e.g., the minimum wage is higher than the reservation wage. Employed workers lose their job at rate q.

We have

rS(a, w, b) = min s≥0 w -qS(a, w, b U I ) -b + c(s) -ms S(a, b) + λ max (S(a, w, b min 0 ) -S(a, w, b)) + S a (a, w, b) , (13) rΣ(w, b, λ) = min s≥0 w -qΣ(w, b U I , λ min ) -b + c(s) -ms Σ(b, λ) + λ(Σ(w, b max 0 , λ) -Σ(w, b, λ)) , (14)
S(A, w, b) = max{Σ(w, b, λ max ), Σ(w, b max 0 , λ max )}, (15)
where

S(a, b) = ∞ wmin S(a, w, b)ϕ(w)dw and Σ(b, λ) = ∞ wmin Σ(w, b, λ)ϕ(w)dw.
The rst two equations describe non-arbitrage equations before and after age A. We explain the rst one, the second one receiving a similar interpretation. The left-hand side is the nancial return on match surplus, whereas the right-hand side is its market return. Job seekers choose their search investment s. This comes at cost c(s) and gives access to the average match surplus S(a, b) with ow probability ms. The match surplus increases with the wage w and decreases with nonemployment income b. When hit by job destruction, workers join the pool of unemployed and receive UI benets. This is why the right-hand side is decreased by qS(a, w, b U I ). Workers receiving UI benets may also lose UI benet entitlement. Therefore the right-hand side is also decreased by λ max (S(a, w, b min 0 ) -S(a, w, b)). Lastly, the term S a (a, w, b) accounts for appreciation with age because assistance benets increase after A and the rate of losing UI entitlement goes down.

The purpose of unemployed workers is to minimize the match surplus, i.e., make their welfare as high as possible while unemployed. The rst-order conditions to the optimization problem give

c (s(a, b)) = m S(a, b), (16) c (σ(b, λ)) = m Σ(b, λ), ( 17
)
where the functions s and σ stand for the optimal search eort before and after age A. In both cases, the marginal cost of search is equal to its marginal gain, i.e., the contact rate by unit of search, m, times the average match surplus corresponding to their situation.

In the parameterized version of the model used for calibration, we suppose that the cost of eort is The dynamics of the dierent populations of unemployed is characterized in Appendix A. This is necessary because the typical outcome used to evaluate age-conditional minimum income policies is the employment rate -and not the job-nding rate as discussed in the previous section.

isoelastic with c(s) = s 1+α , α > 0. Let also B 1 = α(m/(1 + α)) 1+α α . Then stationary average match surpluses solve r Σ(b, λ) = w -b -q Σ(b U I , λ min ) -B 1 Σ(b, λ) 1+α α + λ( Σ(b max 0 , λ) -Σ(b, λ)),
, r S(a, b) = w -b -q S(a, b U I ) -B 1 S(a, b) 1+α α + λ max ( S(a, b min 0 ) -S(a, b)) + Sa (a, b), (19)
We now turn to the two cases of interest, i.e., age entitlement to assistance benets and age entitlement to extended UI benets.

Age entitlement to assistance benets

This section focuses on age entitlement to assistance benets. Our purpose is to illustrate the theoretical bias of the RDD used by [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] on Canadian data and [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF][START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF] on French data. We rst discuss how dierent groups of workers should react to an age-conditional increase in assistance benets. Then we calibrate our model on Canadian and French data.

Model

We

U I ) = w -b U I -q Σ(b U I ) -B 1 Σ(b U I ) 1+α α + λ( Σ(b max 0 ) -Σ(b U I )), (20) r Σ(b max 0 ) = w -b max 0 -q Σ(b max 0 ) -B 1 Σ(b max 0 ) 1+α α . ( 21
)
We rst nd Σ(b max 0

) and replace it in (20) to nd Σ(b U I ). ) decrease. Therefore these workers search less intensively in expectation of future entitlement. This eect grows as workers get closer to entitlement.

Before age A, r S(a, b U I ) = w -b U I -q S(a, b U I ) -B 1 S(a, b U I ) 1+α α + λ( S(a, b min 0 ) -S(a, b U I )) + Sa (a, b U I ),(22) r S(a, b min 0 ) = w -b min 0 -q S(a, b min 0 ) -B 1 S(a, b min 0 ) 1+α α + Sa (a, b min 0 ). (23
Our model predicts the behavioral response of workers before entitlement. The magnitude of this response governs the theoretical bias discussed in this paper. We now quantify it on French and Quebecois data.

Quebecois data

From 1969 to 1989, social assistance benets in Quebec were conditional on claimants' age with a discontinuity at 30. [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] use the age discontinuity to quantify the impact of assistance benets on the employment rate.

We parameterize our model to quantify the theoretical bias associated with this design. We use information available in [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] to x income parameters. We then set the elasticity of the marginal search cost, α, the job-nding rate per unit of search, m, and the job separation rate, q, to match the age pattern of the employment rate around 30. The rate of losing UI benets, λ, is deduced from the expected potential UI duration. We then reproduce the RDD estimations of [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] on the simulated employment rate.

We follow [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] and focus on drop-out men without dependent children. The annual interest rate is 5%. [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] provide income gures for 1986. The mean yearly wage was $13,924 and nonemployed workers received $185 per month of assistance benets below 30 and $507 after 30. By normalizing the mean wage to one, we obtain b min 0 = 0.159 and b max 0 = 0.437.

We get details on the UI scheme in 1986 from [START_REF] Lin | Employment Insurance in Canada: Policy Changes[END_REF]. The replacement rate was 60% and, therefore, we set b U I = 0.6.

The potential duration of unemployment was limited to 50 weeks and had two components: a work component based on weeks worked and a regional component increasing in unemployment rate. We set λ such that the model-based expectation of the potential UI duration, 1/λ, equalizes the expectation of the potential duration based on the 1986 rules linking the max benet duration to the previous employment duration. We only consider the last employment spell. If this spell is t, then the individual can receive UI benets for at most f (t), where f is dened from [START_REF] Lin | Employment Insurance in Canada: Policy Changes[END_REF]. Parameter λ solves

1 λ = ∞ 0 f (t).te -qt dt.
We simulate the pattern of unemployment by age to broadly match the empirical pattern. We start at age 25 and suppose that 68% of these workers are employed as reported by [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF]. None of the unemployed is initially entitled to UI benets. In practice we choose the parameter combination that minimizes the sum of squared deviations between each target and its simulated counterpart.

Intuitively, parameters α and m are identied by the employment rate and its change after entitlement.

The job separation rate, q, is identied by the speed of employment rate adjustment. The rate of losing UI benets, λ, is identied by the relationship linking 1/λ to the theoretical distribution of employment spells and its induced impact on the mean potential UI benet duration. Notes. Targeted moments are borrowed from [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF]. 1, second column, in [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF]. The simulated data are the dots depicted on Figure 3d. The LATE is the dierence between the two counterfactual employment rates at the age threshold.

The elasticity α of the marginal search cost is 0.54, which implies that the elasticity of the search eort to the match surplus is about two. The job separation rate, q, is 0.05, which corresponds to a mean employment duration of 20 months. This is remarkably short but not so surprising for the population of interest. We discuss alternative values in subsection 4.4. Lastly, the rate of losing UI benets, λ, is 0.11. scenarios where assistance benets do not change with age.

Figures 3a and3b reveal that workers respond to the policy change well before 30. In Figure 3a, match surpluses start to decrease at 28. Then they drop by 15-20% depending on the group of nonemployed workers. Workers gradually decrease search eort as a result. In Figure 3b, the job-nding rate of workers under social assistance decreases by 35%, whereas it decreases by 25% for workers under UI.

Figure 3c shows the implied changes in both stocks of unemployed workers. The percentage of workers under UI slightly decreases, whereas the percentage of workers under social assistance dramatically increases from 15% to 24%.

Figure 3d displays the age pattern of the employment rate. The black curve shows the simulated employment rate with age-conditional assistance benets. Dots show the mean employment rate over a full year, i.e., the dot at age a is the mean employment rate between a and a + 1.3

The two other curves depict the counterfactual scenarios where benets do not change with age. The employment rate starts decreasing at 28 and converges towards the red counterfactual employment rate.

The convergence process is almost completed at 32. Changes in job-nding rates immediately impact the ow of workers nding a job, but need time to aect the stock of employed workers.

Figure 3d visually depicts the quantitative impact of the policy as the spread between the two counterfactual scenarios. The eect of raising assistance benets by 170% is a decline in employment rate by 8 percentage points. Using the black curve in the neighborhood of the age discontinuity leads to a dierent computation. Comparing the employment rate at 29 to the one at 30 for instance underestimates the policy impact by 4 percentage points.

Shortening the age span increases the bias, whereas opening the age window reduces it. Of course this latter statement does not account for the risk of increasing heterogeneity between the two groups of workers, a major preoccupation of the RDD.

Table 3 shows the RDD estimates of [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] and our own estimates based on simulated data. We choose their specication with the employment rate at the Census age ("ERC"). Our simulated data are represented by the dots on Figure 3d. We provide several estimates, corresponding to two dierent polynomial specications for age. We also report the LATE, as dened by the dierence between the blue curve and the red one at age 30.

The estimated eects, whether on actual or simulated data, are always lower than the LATE. The spread amounts to between 2 to 4 percentage points. Though the two sets of estimates are actually close, there is a subtle dierence between them. The impact decreases with the polynomial order when using simulated data, whereas it increases when using actual data. With simulated data, increasing the polynomial order captures the declining employment pattern due to pre-entitlement eect. The estimated impact at the age discontinuity is correspondingly reduced. [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] do not obtain this property because the age polynomial captures a dierential employment trend before and after 30 at work on real data. This dierential trend is not predicted by our model and likely not related to minimum income.

French data

From 1988 to 2009, the French revenu minimum d'insertion (RMI) was an assistance benet paid to individuals older than 25. [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF] use this discontinuity to evaluate the impact of assistance benets on the employment rate. We proceed similarly to the Quebecois case.

We apply the model to the population of single male high-school dropouts. The annual interest rate is 5%. [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF] provide relevant income gures for 1999. Unemployed workers below 25 receive on average 206 euros per month of housing benets. After 25, the sum of housing benets and RMI is 539 euros. Moreover, the ratio of this income to the median wage income is about 66%. We suppose that the ratio of assistance benets to the mean wage is the same.

We use details about the French UI scheme in 1999 from [START_REF] Pommier | L'Indemnisation du Chômage en 1999 et[END_REF]. The replacement rate was 68%, which leads to b U I = 0.68. We use the French policy rules to dene λ as a function of q like in the Quebecois case.

We simulate the pattern of employment by age. We start at age 20 and suppose that 64% of the corresponding workers are employed, the employment rate reported by [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF].

None of the unemployed is initially entitled to UI benets.

The employment rate before 25 strongly increases with age in the data. Our model cannot well explain these dynamics. We therefore reduce the window around the entitlement age (compared with the Quebecois case) and consider only three employment rates. The model perfectly matches these three moments as there are three free parameters. As in the Canadian case, parameters m and α are identied with the level of employment rate and its change, whereas parameter q is identied by the adjustment speed.

Table 4 displays the dierent parameters and key model outcomes. The elasticity α is smaller than in the Canadian case, 0.32 against 0.54. This indicates that the search eort is more responsive to changes in match surplus in the French case (the elasticity is 3) than in the Canadian one (the elasticity is 2).

As in the Canadian case, the job separation rate is large at 0.0737, implying that jobs last 14 months on average. This comes as a surprise given the French separation rate is usually low by international standard. However, the average rate hides strong disparities between young workers and the rest of the crowd. The French youth were (and still are) disproportionately exposed to short-term contracts and associated precarity. This phenomenon was particularly striking for the low skilled, here composing the population of interest. Anyway, we discuss alternative values for q in subsection 4.4. Notes. Targeted moments are borrowed from [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF].

Figures 4a and4b show the simulated patterns of match surpluses and job-nding rates by age. Each gure displays four curves. The solid curves correspond to nonemployed workers under UI (blue curve) and social assistance (orange curve) when assistance benets are conditional on age. The dashed curves correspond to the counterfactual scenario where assistance benets do not increase at 25. depicts the resulting patterns for the two stocks of nonemployed workers.

The match surplus decreases by 7 to 25% between 24 and 25. The corresponding job-nding rates decrease from 50% at 24 to 20% at 25 for workers under social assistance and from 19% to 15% for workers under UI. The stock of workers under social assistance starts increasing after 24 to reach 0.10 at 25 instead of 0.03 in the counterfactual scenario.

Job-nding rates are quite large for low-skilled French workers, especially for those under social assistance. This result is induced by the high job loss rate chosen by our calibration procedure. To match a given employment rate, short employment durations must balanced by short nonemployment durations.

Figure 4d displays the simulated employment rate and compares it to two counterfactual scenarios.

When assistance benets remain at b min 0 , the blue curve shows that the employment rate converges The policy impact is the dierence between the two counterfactual scenarios. This gives 7 percentage points at age 25. By comparing employment rates at 24 and 25, the estimated impact is lower by 2 percentage point. Here again, shortening the age window further reduces the estimated impact, whereas expanding the window increases the estimated impact.

In Table 6, we simulate a counterfactual dataset from the theoretical model and apply the same linear regressions as [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF]. They aggregate the employment rate by age group using years and quarters for workers between 20 and 35. They also play with the age window for robustness purpose. We reproduce their estimates in the two columns entitled "Simulated". We also report the LATE, here again dened as the dierence between the two counterfactual scenarios at age 25.

Despite the rst line of Table 6 displays a positive impact on actual data, [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF] see their estimates in the range "between -0.069 and -0.049". We obtain simulated coecients of the same magnitude. The LATE is above the upper bound of the estimates, implying a theoretical bias of about 1 to 2 percentage points.

Discussion

We show that the RDD bias should be large on both datasets. In theory, the RDD underestimates the actual impact of nonemployment benets by 2 to 4 percentage points in the Canadian case. This amounts to between 25 and 50% of the estimated impact. In the French case, gures are slightly lower: 1 to 2 percentage points, which represents between 15 and 33% of the estimated impact.

Such magnitude of the theoretical bias depends on two key properties: (i) our frictional model is a good approximation of the data generating process and (ii) its parameters have been correctly set.

Our model emphasizes the role of search frictions. This leaves aside alternative determinants of employment basd on participation choices. Of course, participation decisions are themselves impacted by search frictions, i.e., workers are less willing to participate when it is hard to nd jobs. However considering such choices in the context where they dier across workers would reduce the role played by search frictions, thereby lowering the bias we report here.

The parameters of the model have been set to match various targets. However, the result is not entirely convincing because in both cases the job loss rate is remarkably large. One may wonder what the bias would be in alternative parameterizations where the job loss rate is smaller.

In Appendix D, we consider such alternative parameterizations for both cases. We show that the magnitude of the bias decreases with the job loss rate. Intutitively, considering a higher job loss rate leads to set a larger job-nding 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 actually report conservative values for the bias.

UI benet extension

We now focus on age-conditional extensions of UI benets. [START_REF] Lalive | How do Extended Benets aect Unemployment Duration? A Regression Discontinuity Approach[END_REF] on Austrian data and Schmieder et al. (2012aand Schmieder et al. ( ,b, 2016) ) on German data study experimental settings where the potential duration of benets is conditional on age at dismissal. They associate their ndings to search theory, but do not discuss the non-stationarity induced by age.

Model

We suppose λ min < λ max and b min 0 = b max 0 = b 0 . Like the previous section, we argue that a decrease in λ max aects all workers: not only those receiving extended benets, but also those below the age threshold who take care of future employment gains in terms of benet entitlement.

We simulate the pattern of unemployment by age from age 46 and focus on a particular outcome of interest, the average unemployment duration. As in the Quebecois and French cases, we are left with three parameters: the elasticity of the marginal search cost, α, the contact rate by unit of search, m, and the job loss rate, q. Parameters α and m are calibrated to match the average unemployment durations before and after the reform. In practice, we choose the average unemployment duration for workers between 49.75 and 50 and for workers between 50 and 50.25.

As the average unemployment duration is discontinuous at the threshold, the job loss rate cannot be calibrated as in the Quebecois and French cases. Here we use external information on job destruction.

Winter-Ebmer ( 2003) study the same reform and nd that between 25 and 22% of employed workers lose their job within a year. We accordingly set q to 0.02. In the Appendix, we report simulation results with alternative values for q. 14.1

Note. The dependent variable is the expected nonemployment duration (in weeks) at the age at start of the spell. The empirical estimates are taken from Table 2 in [START_REF] Lalive | How do Extended Benets aect Unemployment Duration? A Regression Discontinuity Approach[END_REF]. The LATE is the dierence between the two counterfactual nonemployment duration at the age threshold in gure 5d.

We nd a low elasticity of the marginal cost, α, compared with the Canadian and French cases of the previous section, 0.02 against 0.54 and 0.32. This indicates a very large behavioral response to changes in match surplus: the corresponding elasticity is about 50. Notes for gures 5a and 5b. The blue and orange continuous curve respectively correspond to the case of an unemployed worker receiving UI benets for a short period and an unemployed worker receiving the social assistance. The green curve represents the case of a worker who is eligible to longer UI benet. The dashed curves represent the counterfactual scenario without policy change at the age threshold. Notes for gures 5c and 5d. The black curve is the predicted expected unemployment duration. The blue and red curves correspond to the counterfactual durations when eligibility to UI does not depend on age. People lose UI entitlement at rate λmax in blue and λ min in red.

workers could get extended benets after 48. The dashed curves show the counterfactual scenario where the potential UI benet duration does not vary with age.

The main lesson displayed by Figures 5a and 5b is that pre-entitlement eects are small. Unemployed workers increase their search eort to account for the possibility of extended UI after job loss. But this increase is quantitatively small and does not start before age 49.5, i.e., 6 months before the threshold age. This small behavioral response of non-eligible workers goes along with a large impact of the policy on eligible workers. Their job-nding rate is half the one of non-eligible ones under UI.

Figures 5c and 5d depict, respectively, the age patterns of the expected UI duration and expected nonemployment duration. The UI duration diers from the nonemployment duration when the worker exhausts her benets. In each case, the black line shows the simulated duration with UI extension after 50. The dashed lines display the two counterfactual scenarios when benets are extended since 46 (red curve) or never extended (blue curve). The true policy impact is measured by the dierence between the red curve and the blue one; the RDD exploits the change in the black curve around age 50.

Lalive focuses on the expected unemployment duration. Figure 5d shows that the red dashed line is very close to the black line before 50, whereas the blue dashed line is very close to the black line after 50. Workers increase search eort before turning 50 and the expected unemployment duration slightly declines. Therefore comparing the expected unemployment durations of workers dismissed before and after 50 overestimates the policy impact. But the bias can hardly be seen.

Table 9 replicates the RDD based on age discontinuity in [START_REF] Lalive | How do Extended Benets aect Unemployment Duration? A Regression Discontinuity Approach[END_REF]. He runs regressions at individual level. We run similar regressions at age cell level and monthly frequency. We assume the worker age is uniformly distributed and compute the average unemployment duration for each age group.

The results conrm that the bias is very small. The estimated impact is about 14 weeks of additional unemployment and we reproduce it by applying the same strategy on the simulated data. It is slightly larger than the LATE computed as the dierence between the red dashed curve and the blue one in Figure 5d.

Considering alternative values for the job loss rate does not invalidate the results reported here. In the Appendix, we numerically show that the bias increases with the job loss rate. It amounts to between 1.1 (estimate on simulated data) and 1.8 weeks (exact LATE computed from the counterfactual trajectories of the average unemployment duration) when the job loss rate reaches 0.1, a very large value.

German data

From 1987 to 1999, German workers who started their unemployment spell before 42 were entitled to 12 months at best of UI benets. If their unemployment spell started between 42 and 44, they were entitled to 18 months. Schmieder et al. (2012a) use this age discontinuity to quantify the impact of benet extensions on average nonemployment and benet durations.

We apply the model to the population of households with children. The annual interest rate is 5%.

Income in the dierent states is borrowed from Schmieder et al. (2012a). The replacement rate for UI benet is 68%, which corresponds to the average for individuals with children. We set the replacement rate for social assistance at 35%, the average for married individuals.

We simulate the pattern of unemployment by age starting from age 48. Schmieder et al. (2012a) provide the average UI duration before and after the threshold. We use these moments to calibrate the rates at which workers lose UI entitlement, λ max and λ min . Like in the Austrian case, α and m are calibrated to match the average nonemployment durations. We do not have external information on the job loss rate, q. Instead, we calibrate q using information about cumulated nonemployment spells in the ve years after the start of the rst nonemployment spell. From Schmieder et al. (2012b), we set the average number of days of nonemployment over this period to 907 and convert it in months.

4 4 The theoretical distribution of the number of nonemployment days can be recovered from the theoretical distribution With ve empirical targets and ve free parameters, we exactly match the moments. Note. The dependent variables are the UI benet duration and nonemployment duration. The empirical estimates are taken from Table 2 in Schmieder et al. (2012a). The LATE is the dierence between the red curve and the blue one, as displayed by Figures 6c and6d.

Parameters λ max and λ min are slightly above the values matching the potential UI durations before and after 42. This implies that, on average, workers are entitled to UI benets for (slightly) shorter durations than indicated above. We interpret this nding as resulting from additional eligibility conditions that typically reduce the potential UI duration.

Figure 6a shows the average match surpluses in three cases. The green curve depicts X1 . It is constant and expected by workers who get dismissed after age A, thereby receiving b 1 . The blue curve depicts X0 (a), which is of interest for workers dismissed before age A. It gradually increases until A is reached and then stays constant at X0 . Lastly, the orange curve shows the counterfactual average match surplus when the benet level is b 0 irrespective of age. It is initially very close to X0 (a) and the spread gradually increases, reecting the possibility for workers to get access to b 1 in case they accept a job oer.

Figure 6b shows the resulting job-nding rates. The job-nding rate linearly increases with the match surplus. The dierence between the actual case and the counterfactual one increases with age but remains of nonemployment and employment spells. In practice, we use a Monte-Carlo method to compute the expected number of nonemployment days. Like the Austrian case, the RDD overestimates the policy impact. The quantitative eect is more pronounced in the German case: the impact on the expected unemployment duration is overestimated by 0.2 month, about a fourth of the reported impact.

Table 12 reproduces the two regressions in Schmieder et al. (2012a). As in the Austrian case, we mimic their regressions performed on individual data by aggregating workers by age cell in our simulations. Age is in days. The rst set of regressions uses the actual UI duration as the dependent variable and the second one uses the nonemployment duration. We report the results with a linear spline and dierent bandwidths.

The regressions on simulated data reproduce quite well the results on actual data. The estimated impact is around 0.8 months of nonemployment and overestimates the actual impact by 0.2 months. The estimated impact on the UI duration is both stronger and closer to the actual one.

Discussion

We here discuss why we nd a small RDD bias in case of UI benet extensions. We develop the intuitions exposed in subsection 2.1. Namely, benets paid after the threshold age have small impacts on preentitlement match surplus for realistic values of the job loss rate, q, and the contact rate per unit of search, m.

The RDD bias arises because unemployed workers before age A have pre-entitlement incentive to nd a job in order to receive longer benets in case they lose their job after entitlement. Therefore the bias increases with the likelihood of a particular transition pattern, i.e., nding a job before entitlement and losing it after entitlement. This likelihood is small when compared with alternative transition patterns.

It follows that expectation eects do not impact much the pre-entitlement job search behavior.

Figures 7a and 7b illustrate this reasoning. They depict the bias as a function of m and q. The bias is the job-nding rate dierential between unemployed workers receiving UI benets just before entitlement and counterfactual workers in a world without policy change at age threshold. For the baseline parameterizations, indicated by crosses, the bias is the gap at threshold age between the dashed and continuous blue curves in Figures 5b and6b.

Parameter m has a non-monotonic impact. The dierential is small when m is low why would (a) Austrian case (b) German case

Figure 7: Theoretical bias as a function of and q Notes. This gure depicts the dierence in job-nding rates between workers under UI just before the age threshold and counterfactual workers who are not confronted to such a policy change. The job-nding rate dierential is represented as a function of parameters m (left axis) and q (right axis) in base-10 log scale. In the Austrian case, m = 0.0687 and q = 0.020. In the German case, m = 0.109 and q = 0.0259.

workers search if they had no chance of nding a job? and when m is large workers can simply wait the very last minute to take a job. Conditional on the job-nding rate, the dierential is maximized for intermediate values of m.

Parameter q increases the dierential. When q is very small, UI benets do not matter and the dierential is low. As q increases, the pre-entitlement match surplus becomes more impacted by postentitlement UI benets. This aects the pre-entitlement behavioral response of the job-seekers, thereby increasing the RDD bias.

Figures 7a and7b reveal that the magnitude of the bias can be very large, but only for parameterizations involving large values of m and q. In particular, the job loss rate must be much higher than empirically plausible values. This is especially true in the Austrian case where q needs to be above 0.2 to see anything on the job-nding rate dierential. This amounts to ten times the value used for the simulations.

It follows that the dierential must be small for reasonable values of m and q. Therefore it makes sense to compare unemployed workers before and after the threshold age. Though both workers are actually treated, the behavioral response of the supposedly nontreated can be neglected.

Conclusion

Job search theory is a consistent framework to describe labor market phenomena. We use it to discuss the potential bias of popular RDD to evaluate minimum income and UI benet extension policies. The design involves an age discontinuity in treatment exposure, i.e., only those above a specic age have access to minimum income or extended benets. Search frictions imply that this design is biased: individuals should progressively modify their search behavior before entitlement. We calibrate a nonstationary job search model on the various datasets used by the literature. According to our calibrations, the impact of age-conditional minimum income is under-estimated and this quantitatively matters, whereas the impact of age-conditional UI benet extensions is over-estimated but the bias can be neglected. These results do not prove that the designs are biased per se. The biases are conditional on the fact that job search theory is an accurate description of actual labor markets. The simple models that we use certainly overstate the magnitude of search frictions because they do not feature worker heterogeneity or participation choices. Therefore a rst task consists in enriching 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 ndings.

However, the parameters of this model also predict the magnitude of the RDD bias.

A The dierent populations of unemployed by age

The size of the labor force is normalized to one. At age a, the mass of unemployed workers receiving social assistance is u 0 (a). Similarly, u short U I (a) and u long U I (a) are the masses of unemployed receiving UI benets for, respectively, a regular period and an extended one.

When a ≤ A,

du 0 da (a) = λ max u short U I (a) -ms(a, b min 0 )u 0 (a) (30) du short U I da (a) = q[1 -u 0 (a) -u short U I (a)] -[ms(a, b U I ) + λ max ]u short U I (a) (31)
The inow of workers receiving social assistance now has two components: those who lose UI entitlement at rate λ max and those who lose it at rate λ min . There is no inow of workers in the short-period entitlement regime. Workers who lose their job at rate q get access to extended benets.

f U I (B|a) =    exp - B 0 (λ max + ms(a + t, b U I ))dt .(λ max + ms(a + B, b U I )) if a < A, exp [-(λ min + mσ(b U I , λ min ))B] .(λ min + mσ(b U I , λ min )) if a ≥ A. . ( 36
) with s(a + t, b U I )) = s(a + t, b U I ) if a + t < A, or s(a + t, b U I ) = σ(b U I , λ max ) if a + t > A.
The pdf is the product of two terms. The rst term with the exponential is the survival probability, i.e., the probability of receiving UI benets for at least duration B. The second term is the hazard rate of losing UI benets at a + B.

The expected benet duration is E

(B|a) = ∞ 0 Bf U I (B|a)dB. For a ≥ A, this conditional expectation is E(B|a) = (λ min + mσ(b U I , λ min )) -1 if a ≥ A. ( 37
)
We now characterize the age-conditional pdf of non-employment duration D. If a < A,

f N E (D|a) = f U I (D|a). ms(a + D, b U I ) λ max + ms(a + D, b U I ) + D 0 f U I (B|a). λ max λ max + ms(a + B, b U I )
. exp - Notes for gures 8b and 8b. The black curve is the predicted employment rate when unemployd workers get extended UI benets after 50. The blue and red curves correspond to the counterfactual employment rates in absence of age discontinuity.

People lose UI eligibility at rate λmax when the curve is blue and λ min when red.

D Robustness with respect to the job destruction rate

Each of the calibrations exposed in Sections 4 and 5 involve a particular assumption to set the job loss rate, q. Tables 13 to 16 report the results for alternative job loss rates. We nd i) the absolute value of the LATE decreases with q,

ii) the estimate on simulated data tends to increase with q. Note. The job-nding rates correspond to workers nonentitled to unemployment insurance. The estimates corresponds to the estimated eects from simulated data with a linear spline specication. The estimate on actual data comes from [START_REF] Lemieux | Incentive Eects of Social Assistance: A Regression Discontinuity Approach[END_REF] reported in Table 3. Note. The job-nding rates correspond to workers nonentitled to unemployment insurance. The estimates corresponds to the estimated eects from simulated data with a linear spline specication in years. The estimate on actual data comes from [START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF] reported in Table 6. Note. The job-nding rates correspond to workers receiving unemployment insurance. The estimates corresponds to the estimated eects from simulated data with a linear spline specication in years. The estimate on actual data comes from [START_REF] Lalive | How do Extended Benets aect Unemployment Duration? A Regression Discontinuity Approach[END_REF] reported in Table 9.

  S(A) = S 1 + (b 1 -b 0 )/(r + m) ≥ S 1 , the inequality being strict when b 1 > b 0 . These workers are still nonentitled to b 1 after age A because they have not experienced a job episode lasting after this age. Let e(a) = d ln S(a)/d ln b 1 . For b 0 = b 1 , we have e(a) = -ε(a) q r + m > 0.

Figure 1 :

 1 Figure 1: RDD bias in case of age-conditional minimum income

  b = b U I , b max 0 and λ = λ min , λ max . The worker starts with S(0, b min 0 ). Then, the dynamics of the nonstationary match surplus is, for all a < A and b = b U I , b min 0

  with the terminal conditions S(A, b) = max{ Σ(b, λ max ), Σ(b max 0 , λ max )}. The search eorts are s(a, b) = B 2 S(a, b) 1/α and σ(b, λ) = B 2 Σ(b, λ) 1/α , where B 2 = (m/(1 + α)) 1/α . The model is stationary after age A. Job seekers dier in two dimensions: whether they are entitled to UI benets or not, and, for the entitled, their risk of losing entitlement. That b max 0 < b U I implies Σ(b max 0 , λ) > Σ(b U I , λ) for λ = λ min , λ max . Therefore σ(b max 0 , λ) > σ(b U I , λ) and the nonentitled nd jobs more rapidly than the entitled at given rate of losing UI benets. That λ min < λ max implies that Σ(b, λ max ) > Σ(b, λ min ) for b = b max 0 , b U I . Therefore σ(b, λ max ) > σ(b, λ min ) and those who lose their jobs after age A nd jobs less rapidly than those who lose their jobs before age A.

  I and λ min = λ max = λ. An increase in b max 0 aects all workers: not only those above the age threshold actually receiving b max 0 but also those below this age who modify their job search behavior in expectation of future entitlement. Consider rst workers above age A. We simply denote Σ(b) ≡ Σ(b, λ). We solve r Σ(b

.

  ) with the terminal conditions S(A, b U I ) = Σ(b U I ) and S(A, b min 0 ) = Σ(b max 0 ). Here also, we rst integrate (23) to nd the function S(a, b min 0 ) and then replace it in (22) to nd the function S(a, b U I ). Suppose there is an increase in b max 0 For workers above age A nonentitled to UI benets, the match surplus Σ(b max 0 ) decreases as they receive higher nonemployment income. Therefore these workers reduce search investment σ(b max 0 ) ≡ σ(b max 0 , λ) and the job-nding rate decreases. For workers entitled to UI benets, the decline in match surplus is lower because they rst need to lose entitlement before getting the new level of assistance benets. The qualitative impact on the search eort σ(b U I ) ≡ σ(b U I , λ) is the same as for the nonentitled and the job-nding rate also decreases. However the quantitative impact is smaller. For workers below age A, the terminal conditions S(A, b U I ) = Σ(b U I ) and S(A, b min 0 ) = Σ(b max 0 ) imply that the pre-entitlement match surpluses decrease at all ages, just like in Section 2. The value of nonemployment situations in the future rises and this is instantaneously reected by lower match surpluses, i.e., both S(A, b U I ) and S(A, b max 0

Figures

  Figures 3a to 3cdepict the outcomes of the calibrated model. Each gure displays four curves. The continuous lines correspond to nonemployed workers under UI (blue curve) and social assistance (orange curve) when assistance benets are conditional on age. The dashed lines correspond to alternative

Figure 4 :

 4 Figure 4: Simulation with French data Notes for Figures 4a, 4b and 4c. Continuous curves correspond to unemployed workers under UI benets (blue curve) and social assistance (orange curve) confronted to the age-conditional policy. Dashed curves represent the counterfactual scenarios without policy change at the age threshold. Notes for Figure 4d. The black curve is the predicted employment rate when assistance benets are conditional on age. The blue and red curves correspond to the counterfactual employment rates when assistance benets do not depend on age. Nonemployment income is b min 0 when the curve is blue and b max 0 when red.

Figure

  Figure5adisplays the age pattern of match surpluses, whereas Figure5bshows the corresponding job-nding rates. In each case there are ve curves. The solid curves in blue and orange are associated to unemployed workers under regular UI (blue curve) and assistance benets (orange curve) when job losers have access to extended UI benets after 50. The solid green curve corresponds to workers eligible to extended benets. In practice, this can only happen after 50, but we also show what would happen if

Figure 6 :

 6 Figure 6: Simulation with German dataNotes for gures 6a and 6b. The continuous curves correspond to unemployed workers receiving unextended UI benets (blue curve) and under social assistance (orange curve). The dashed curves represent the counterfactual scenario without policy change at the age threshold. The green curve correspond to unemployed workers receiving extended UI benets. Notes for gures 6c and 6d. The black curve is the predicted expected unemployment duration. The blue and red curves correspond to the counterfactual durations when eligibility to UI does not depend on age. People lose UI entitlement at rate λmax in blue and λ min in red.

  benets lose entitlement and enter the social assistance regime at rate λ max . Meanwhile, workers under social assistance nd jobs at rate ms(a, b min 0 ). In the second dierential equation, the inow of workers receiving benets is composed of those who lose their job at rate q. The outow is made of workers who nd jobs at rate ms(a, b U I ) or lose benet entitlement at rate λ max . When a ≥ A, du 0 da (a) = λ max u short U I (a) + λ min u long U I (a) -mσ(b max 0 , λ min )u 0 (a) = -[mσ(b U I , λ max ) + λ max ]u short U I (a) = q[1 -u 0 (a) -u short U I (a) -u long U I (a)] -[mσ(b U I , λ min ) + λ min ]u long U I (a)

  + t, b 0 )dt .ms(a + D, b 0 )dB (38) We distinguish two situations: either the worker nds a job before losing UI benets, D = B, or after benet exhaustion, D > B. The pdf f N E is the sum of two terms. The rst term corresponds to D = B. This is the pdf of the benet duration f U I weighted by the proportion of exits to employment among those who stop receiving benets. The second term corresponds to D > B. This is an integral over all possible values of B between 0 and D. The quantity f U I (B|a) λmax λmax+ms(a+B,b U I ) is the pdf of receiving UI benets for duration B multiplied by the proportion of exits to assistance benets among those who stop receiving UI benets. The exponential term is the probability of remaining unemployed between duration B and D. The term ms(a + D, b 0 ) is the job-nding rate for nonentitled workers who entered unemployment at age a and are still unemployed at age D. If a ≥ A, f N E (D|a) = e -(λmin+mσ(b U I ,λmin))D .mσ(b U I , λ min ) + D 0 e -(λmin+mσ(b U I ,λmin))B .λ min .e -mσ(b0,λmin)(D-B) .mσ(b 0 , λ min )dB

0

  + mσ(b U I , λ min ))(mσ(b 0 , λ min ) -mσ(b U I , λ min )) mσ(b 0 , λ min ) -(λ + mσ(b U I , λ min )) e -(λmin+mσ(b U I ,λmin))D λ min mσ(b 0 , λ min ) mσ(b 0 , λ min ) -(λ min + mσ(b U I , λ min )) e -mσ(b0,λmin)D (40) The expected nonemployment duration is E(D|a) = ∞ Df N E (D|a)dD. This expectation has a simple form when age is above the threshold. For a ≥ A, E(D|a) = mσ(b 0 , λ min ) -mσ(b U I , λ min ) (λ min + mσ(b U I , λ min ))(mσ(b 0 , λ min ) -λ min -mσ(b U I , λ min )) -λ min mσ(b 0 , λ min )(mσ(b 0 , λ min ) -λ min -mσ(b U I , λ min )) (41) C Employment with Austrian and German data (a) Population of nonemployed workers, Austrian case (b) Employment rate, Austrian case (c) Population of nonemployed workers, German case (d) Employment rate, German case

Figure 8 :

 8 Figure 8: Employment dynamics with Austrian and German dataNotes for gures 8a and 8c. The continuous curves depict the stocks of unemployed under UI (blue curve) and social assistance (orange curve) when unemployed above 50 receive extended UI benets. The dashed curves represent the counterfactual scenario without benet extension. Notes for gures 8b and 8b. The black curve is the predicted employment rate when unemployd workers get extended UI benets after 50. The blue and red curves correspond to the counterfactual employment rates in absence of age discontinuity. People lose UI eligibility at rate λmax when the curve is blue and λ min when red.

  A is qualitatively similar to the response of those above A. The bias is negative when the two responses are negatively correlated.Figures1 and 2illustrate the bias in the two dierent situations of interest. Figure1focuses on age-conditional minimum income. This corresponds to benets paid to all unemployed after age A in the previous sub-section. Therefore the pre-entitlement expected match surplus decreases with age and converges to the post-entitlement expected match surplus. As individuals make search eort on the basis of expected match surplus, search eorts gradually decrease with age well before entitlement. Slightly before the threshold age A, individuals are almost fully treated and behave as their older peers. Continuity implies that the RDD estimand is 0 in

	Job-nding
	rate (Y )
	• • •
	RDD estimand	RDD estimate
	LATE
		• • •
	A	A	Age (a)
	(a) Virtual age discontinuity	(b) Inferred age discontinuity
	Figure 2: RDD bias in case of age-conditional UI benet extension
	Note. The left-hand side graphs depict the job-nding rate as a function of age in the three possible cases: treated, Y 1 i (in
	red), non-treated, Y 0 i (in blue), treated in expectation, Ỹ 0 i (in black). The right-hand side graphs illustrate the estimation
	with actual observations. The narrative of the RDD estimation says we can infer E[Y 0 i |a i = a] for a i < a and E[Y 1 i |a i = a]
	for a i > a. What we do infer, however, is E[ Ỹ 0 i |a i = a] for a i < a.

below

Table 1

 1 displays the calibrated parameters, whereas Table2shows the moments that this calibration matches.

Table 1 :

 1 Parameters in the Quebecois case

Table 2 :

 2 Moments to t the values of α, m and q, Quebecois case

	Moments	Target	Model
	Employment rate between 27 and 28	66.4%	67.0%
	Employment rate between 28 and 29	66.2%	66.7%
	Employment rate between 29 and 30	66.0%	64.8%
	Employment rate between 30 and 31	61.1%	60.7%
	Employment rate between 31 and 32	59.8%	59.1%
	Employment rate between 31 and 32	57.6%	58.8%

Table 3 :

 3 Regression discontinuity estimates, Quebecois case Note. The dependent variable is the employment rate. The empirical estimates are taken from Table

		Estimated eect on the employment rate
	Polynomial specication for age	actual data	simulated data
	Linear	-0.041	-0.064
	Quadratic	-0.051	-0.047
	Cubic	-0.048	-0.047
	Linear spline	-0.049	-0.048
	Quadratic spline	-0.056	-0.037
	LATE	X	-0.082

Table 4 :

 4 Parameters in the French case

Table 5 :

 5 Moments to t the values of α, m and q, French case

	Moments	Target	Model
	Employment rate between 24 and 25	72.1%	72.1%
	Employment rate between 25 and 26	66.8%	66.8%
	Employment rate between 26 and 27	65.8%	65.8%

Table 6 :

 6 Regression discontinuity estimates, French case Note. The dependent variable is the employment rate. The empirical estimates are taken from Table1in[START_REF] Bargain | The Eect of Social Benets on Youth Employment: Combining Regression Discontinuity and a Behavioral Model[END_REF]. The simulated data are the dots depicted by Figure3d. The LATE is the dierence between the two counterfactual employment rates at the age discontinuity on Figure4d.

	Estimated eect on the employment rate
	Empirical	Simulated

rapidly to 73%. When assistance benets are (unexpectedly) set to b max 0 at age 20, the red curve shows that employment rate converges to 66%. The simulated employment rate initially sticks to the blue curve, separates from it at age 24 and decreases to reach the red curve at age 26. By age 25, the employment rate is 69%. The speed of adjustment is twice faster than in Canada, a phenomenon mostly due to the high job separation rate.

Table 7 :

 7 Parameters in the Austrian case

	A	w	b 0	b U I	r	α	m	q	λ max	λ min
	50 1 0.28 0.6 0.0048 0.0195 0.0687 0.02 0.111 0.0207

Table 8 :

 8 Moments to t the values of α and m, Austrian case

	Moments	Target	Model
	Average nonemployment duration between 49.75 and 50	15.2 weeks	15.2 weeks
	Average nonemployment duration between 50 and 50.25	29.6 weeks	29.6 weeks

Note. Targeted moments are borrowed from

[START_REF] Lalive | How do Extended Benets aect Unemployment Duration? A Regression Discontinuity Approach[END_REF]

. The age is at start of unemployment spell.

Table 9 :

 9 Regression discontinuity estimates, Austrian case

		Estimated eect on nonemployment duration (in weeks)
	Polynomial specication for age	Empirical	Simulated
	No trend	14.6	14.1
	Linear spline	14.8	14.2
	Cubic spline	11.15	14.5
	LATE	X

Table 10 :

 10 Parameters in the German case

	A	w	b 0	b U I	r	α	m	q	λ max	λ min
	42 1 0.35 0.68 0.0048 0.739 0.109 0.0259 0.0941 0.0663

Table 11 :

 11 Moments used to t the values of α, m, q, λ max and λ min , German case Targeted moments are borrowed fromSchmieder et al. (2012a) andSchmieder et al. (2012b). We dene ages just before and just after 42 at 41.99 and 42.01. The age is at start of unemployment spell.

	Moments	Target	Model
	Average nonemployment duration before 42	15.6 months	15.6 months
	Average nonemployment duration after 42	16.4 months	16.4 months
	Average cumulated nonemployment duration after 42	29.8 months	29.8 months
	Average nonemployment duration before 42	6.7 months	6.7 months
	Average nonemployment duration after 42	8.5 months	8.5 months
	Note.

Table 12 :

 12 Regression discontinuity estimates, German case

			Estimated eect on duration (in months)
			Empirical	Simulated
	Bandwidth in years:	UI duration	Nonemployment duration	UI duration	Nonemployment
	±2 around 42	1.78	0.78	1.78	0.833
	±1	1.82	0.92	1.78	0.828
	±0.5	1.73	1.04	1.78	0.823
	±0.2	1.65	0.79	1.78	0.821
	LATE	X	X	1.74	0.6

B

  Computing the age-conditional average nonemployment and benet durationsLet B be the UI benet duration and D the non-employment duration. After age A, the Poisson rate of quitting benet entitlement is λ min + mσ(b U I , λ min ), i.e., rate of losing entitlement plus job-nding rate.Before age A, the rate of quitting benet entitlement is age-dependent.

	The probability density function (pdf ) of UI benet duration B, conditional on age at the start of the
	unemployment spell a, is

Table 13 :

 13 Robustness, Quebecois case

	Fixed parameters	q	0.005	0.01	0.025	0.05	0.1
	Calibrated parameters	α	0.133	0.298	0.526	0.541	0.44
		m	0.0163	0.0293	0.0736	0.148	0.292
	Employment rate (%)	27-28	67.5	67.5	67.4	67	66.6
		28-29	66.1	66.2	66.4	66.7	66.6
		29-30	64.2	64.2	64.4	64.8	65.4
		30-31	61.9	61.7	61.2	60.7	60.1
		31-32	59.8	59.6	59.3	59.1	59.2
		32-33	57.8	58	58.4	58.8	59.2
	Rate of loosing benets	λ	0.0888	0.0908	0.0975	0.11	0.139
	Job-nding rate at 27 Job-nding rate after 30	ms(27, b min 0 mσ(b max 0 )	) 0.00899 0.0205 0.0568 0.125 0.00328 0.0111 0.0354 0.078	0.294 0.174
	Estimate on actual data		-0.049	-0.049	-0.049	-0.049	-0.049
	Estimate on simulated data		-0.028	-0.037	-0.044	-0.048	-0.053
	LATE		-0.19	-0.14	-0.097	-0.082	-0.074

Table 14 :

 14 Robustness, French case

	Fixed parameters	q	0.005	0.01	0.025	0.05	0.1
	Calibrated parameters	α	0.106	0.262	0.355	0.330	0.332
		m	0.0217	0.0408	0.106	0.214	0.43
	Employment rate (%)	24-25	71.4	71.5	71.8	72	72.1
		25-26	68.1	68	67.6	67.1	66.6
		26-27	65.1	65.1	65.3	65.5	65.9
	Rate of loosing benets	λ	0.0351	0.0369	0.0429	0.0546	0.086
	Job-nding rate at 22 Job-nding rate after 25	ms(22, b min 0 mσ(b max 0 )	) 0.0224 0.0429 0.126 0.00249 0.0139 0.0502	0.286 0.115	0.587 0.242
	Estimate on actual data		-0.049	-0.049	-0.049	-0.049	-0.049
	Estimate on simulated data		-0.076	-0.089	-0.073	-0.056	-0.054
	LATE		-0.33	-0.2	-0.11	-0.085	-0.071

Table 15 :

 15 Robustness, Austrian caseJob-nding rate after 50, with λ max mσ(b U I , λ max )Job-nding rate after 50, with λ min mσ(b U I , λ min )

	Fixed parameters	q	0.005	0.01	0.025	0.05	0.1
	Calibrated parameters	α	0.00784	0.0118	0.0234	0.042	0.077
		m	0.0269	0.0408	0.0826	0.152	0.282
	Job-nding rate at 49, with λ max	ms(49, b U I )	0.218	0.217	0.212	0.205	0.195
			0.22	0.22	0.221	0.222	0.224
			0.13	0.13	0.13	0.13	0.13
	Estimate on actual data		14.8	14.8	14.8	14.8	14.8
	Estimate on simulated data		14.4	14.3	14.2	14	13.7
	LATE		14.3	14.2	14	13.6	13

Table 16 :

 16 Robustness, German caseJob-nding rate after 42, with λ max mσ(b U I , λ max ) 0.055 0.0551 0.0553 0.0556 0.0561 Job-nding rate after 42, with λ min mσ(b U I , λ min ) 0.0517 0.0518 0.0519 0.0521 0.0524 The job-nding rates correspond to workers receiving unemployment insurance. The estimates corresponds to the estimated eects from simulated data with a 2-year bandwidth. The estimate on actual data comes fromSchmieder et al. (2012a) reported in Table12.

	Fixed parameters	q	0.005	0.01	0.025	0.05	0.1
	Calibrated parameters	α	0.361	0.48	0.775	1.18	1.86
		m	0.0618	0.0787	0.108	0.129	0.137
		λ max	0.0943	0.0942	0.094	0.0937	0.0932
		λ min	0.066	0.0659	0.0657	0.0655	0.0652
	Job-nding rate at 40, with λ max	ms(40, b U I )	0.0548	0.0547	0.0545	0.0542	0.054
	UI duration
	Estimate on actual data		1.78	1.78	1.78	1.78	1.78
	Estimate on simulated data		1.80	1.80	1.80	1.80	1.80
	LATE		1.79	1.78	1.76	1.73	1.70
	Nonemployment duration
	Estimate on actual data		0.78	0.78	0.78	0.78	0.78
	Estimate on simulated data		0.80	0.81	0.81	0.82	0.82
	LATE		0.75	0.70	0.59	0.48	0.36
	Note.

The discrete approximation of the running variable, here age, generates a bias in the RDD estimates (see[START_REF] Dong | Regression Discontinuity Applications with Rounding Errors in the Running Variable[END_REF]. Dots are below the continuous employment curve because the curve is decreasing. For instance at 29, the dot corresponds to the average employment rate of workers between 29 and 30 and is lower than the employment rate at 29.

We can solve it by blocks. We solve the rst two equations in Σ(b U I , λ min ) and Σ(b 0 , λ min ) and then replace these values in the last two equations. The pre-entitlement match surpluses S(a, b 0 ) and S(a, b U I ) increase with age as workers get closer to entitlement. Therefore search eorts and the resulting job-nding rates also increase with age prior to entitlement age.

The calibration procedure exposed hereafter involves the average nonemployment duration and the average benet duration. We do so because they are outcomes used to evaluate the impact of benet extension policies. Both durations can be computed from the trajectories of the various job-nding rates.

Appendix B shows the details of the computation.

Austrian data

We now focus on the Austrian data used by [START_REF] Lalive | How do Extended Benets aect Unemployment Duration? A Regression Discontinuity Approach[END_REF]. From 1988 to 1993, the maximum UI duration after 50 was extended in some regions in Austria, from 39 weeks to 209 weeks. We follow Lalive and calibrate the model on males between 46 and 54 who have experienced continuous employment -women are aected by the early retirement policy in Austria. We then reproduce the RDD estimation on the simulated average nonemployment duration.

The annual interest rate is 5%. Here again, we normalize the average wage to one. [START_REF] Lalive | How do Extended Benets aect Unemployment Duration? A Regression Discontinuity Approach[END_REF] mentions a gross replacement rate of UI benets at 0.4 for the median wage earner without family allowance. We set the replacement rate of UI benets to 0.6. AWe also set assistance benets to 70% of UI benets, the ratio of median assistance to median UI benets in the data.

Lalive only selects workers who are eligible to the maximum UI duration. Therefore we set λ max and λ min so that the mean potential durations of benets are, respectively, 39 and 209 weeks.


G Molas

Marc Bocquet

J Buckley

J P Colonna

L Masarotto

H Grampeix

F Martin

V Vidal

A Toffoli

P Brianceau

L Vermande

P Scheiblin

M Gély

A M Papon

G Auvert

L Perniola

C Licitra

T Veyron

N Rochat

C Bongiorno

S Lombardo

B De Salvo

S Deleonibus

Thorough investigation of Si-nanocrystal memories with high-k interpoly dielectrics for sub-45nm node Flash NAND applications

HAL is

Introduction

The large success of mobile equipment is leading to a dramatic increase of the market for NAND Flash, key devices for mass data storage [1]. Discrete trap memories, such as TANOS [2] and Si-nc memories [3,4] are one of the most suitable candidates to push integration density further beyond the 45nm node, because of their good scalability, robustness against SILC and low floating gate (FG) to floating gate coupling. In particular, Si-nc memories offer the potential of better data-retention at high temperature (the stored electrons being trapped in the Si-nc energy conduction band, rather than in temperature-activated deep traps of nitride), as well as mitigated lateral charge migration (the Si-ncs being isolated from one to the other by silicon dioxide rather than nitride). However, up to now, Si-nc memories have always shown poor FN program/erase characteristics, due to the small Si-nc/control gate coupling. In order to overcome this issue, engineering of the tunnel dielectric and/or IPD [5,6] is necessary. In particular, in this paper, we present for the 1 st time to our knowledge an exhaustive experimental and theoretical study of Si-nc memories where the conventional HTO or oxide/nitride/oxide (ONO) interpoly dielectric is replaced by HfO2, Al2O3 or HfAlO based IPD. Excellent performance and clear correlations between the device electrical results and the IPD material properties are shown. We also suggest how to solve another issue of Si-nc devices, which makes them poorly interesting for multi-level NAND applications, i.e. the relatively small programming window. As a possible solution we propose the addition of a thin SiN layer over the Si-ncs, which allows for a significant increase of the memory window. Our approach is widely validated through indepth analysis of Si-nc memories with HfO2, Al2O3 or HfAlO IPD, based on several material results, electrical data on memory transistors, physical modelling and TCAD three-dimensional simulations.

I. Device fabrication

Si-nc memories (samples S1 to S4) -Si-ncs were deposited by CVD (~6nm, d=9E11cm -2 ) on a 4nm thick thermally grown tunnel oxide. Si-ncs were passivated by a 750°C NH3 nitridation. Then different IPD stacks were deposited, composed of an 8nm high-k layer (HfO2, Al2O3 or HfAlO with ~30% of Hf) sandwiched between two 4nm-thick HTOs (S1 to S3, Fig. 1). Their respective EOT are 10nm, 10.5nm and 11.2nm. Poly-Si was used as a control gate. On some wafers, the interpoly HTO top oxide was skipped to reduce the EOT of the stack (S4, Fig. 1). In this case, a TiN control gate was deposited. The gate length was defined by electron beam lithography, down to 90nm. Si-nc/ SiN memories (sample S5 and S6) -In this case, Si-ncs were deposited by CVD (~8nm, d=9E11cm -2 ) and capped in-situ by a 2nm SiN layer. HfAlO based 3-layer (S5) and 2-layer (S6) IPD were integrated. Fig. 1 presents Si-ncs SEM plan views and TEM cross sections of the memory devices. Fig. 2 shows the infrared spectra of HfAlO layers, demonstrating the progressive shift of the AlO binding energy as %Hf increases. The optical bandgap of HfAlO layers (Fig. 2) increases linearly with %Al.

II. Results and discussion

1. IPD comparison -Fig. 3 shows the IV characteristics of high-k based 3-layer IPD capacitors. One can observe that at high fields, the insulation capabilities increase with the Hf content of the high-k [7]. Moreover, lower leakage currents are measured at high voltages on high-k based stacks with respect to HTO. Program erase (p/e) characteristics in FN-FN regime of the 3-layer IPD Si-nc memories are presented in Figs. 45. These results demonstrate the feasibility of NAND applications for Si-nc memories with high-k IPD thanks to the improved coupling ratio and low IPD leakage current at high voltages (Fig. 3). It also appears that the memory window is the largest with an HfO2 based IPD (fig. 6). Indeed, as the IPD EOT decreases, the coupling ratio increases, leading to higher Vth. In fig. 7 is plotted the required program voltage Vp for a 2.5V Vth in 1ms for the different IPD stacks, and it appears that Vp is reduced as the IPD EOT decreases, confirming our former hypothesis. Cycling characteristics are presented in Fig. 8. With HfO2 IPD, a constant memory window of ~3V is maintained even after 10 5 cycles, with a ~500mV Vth drift. The higher Vth drift of Al2O3 based IPD devices may be explained by the higher electric field in the Al2O3 layer (due to higher dielectric constant) for a given p/e condition. HfAlO offers a good compromise between HfO2 and Al2O3 in terms of memory window and low Vth drift. Retention characteristics of the different memory devices at RT and 125°C are plotted in Figs.9-10. Since the tunnel oxide thickness was the same for the different samples, we can conclude that the retention time is governed by the leakage current through the IPD stack. We observe that the charge loss is the highest for the HfO2 IPD memory while the Al2O3 IPD exhibits the best retention time and the slowest charge decay rate. This is consistent with the more elevated barrier height of Al2O3, which is relevant especially at low applied electric fields. Finally, the charge life time was extracted from retention measurements (Fig. 11), showing no temperature activation up to 125°C, and thus demonstrating the good high temperature behaviour of Si-nc memories.

Si-nc/SiN floating gate memories

Fig. 12 plots the program erase characteristics of Si-ncs/SiN double layer floating gate memories. With HfAlO three-layer IPDs, large memory windows of nearly 6V for 1ms programming time can be achieved. A reduction of p/e voltages by 2V is possible by using a 2-layer IPD rather than a three-layer one. Fig. 13.a shows the feasibility of 2-bits/cell operation in HC programming mode. The 2bits are clearly distinguished, with a 25% Vth separation between both of them, demonstrating the discrete character of the Si-nc/SiN floating gate. Retention characteristics at room temperature (fig. 13.b) show the same charge decay than Si-nc memories. However, a stronger activation is measured at 125°C compared to Si-nc memories due to the contribution of SiN traps. Good endurance performance (fig. 14) is also achieved. The memory window narrowing (not observed in Si-nc memories), can be explained by charges trapped in the SiN layer, which are more difficult to remove.

III. Modelling and numerical simulations

The physical model of Si-nc devices with high-k IPDs is illustrated in Fig. 15. The writing current through the tunnel oxide was calculated using the WKB formalism. Leakage currents through the IPD are given by experimental data (Fig. 3) [7]. The trapped charge in Si-ncs is calculated by the balance of the filling and emptying currents through the tunnel and interpoly dielectrics [8], respectively Jtun and JIPD. With a HTO IPD, JIPD~Jtun and FN programming is not efficient. On the other hand with high-k IPD JIPD<<Jtun, and the memory window is essentially governed by Jtun, controlled by the memory coupling ratio. Indeed as the IPD EOT decreases, the injected charge in the FG increases (due to higher coupling ratio) leading to a wider Vth, which is in agreement with our experimental results. The endurance measurements (Fig. 8) show that the IPD has a strong influence, meaning that trapping in the high-k layer rather than in the tunnel dielectric (as usually considered for cycling [9]) should be involved. We have therefore developed a cycling model (Fig. 16) that takes this trapping into account and have arbitrarily assumed it as happening along a plane in the middle of the high-k film. We show here the result of the cycling model in the case of S1 (IPD with HfO2). The trapping in the HfO2 is considered by using an empirical power law (used in [10] for silica; see modelling details in Fig. 16) that leads to good correspondence with the experimental results. TCAD 3D simulations (Fig. 17) using Synopsys tools were performed to evaluate the FG to FG coupling for future generations of ultra dense NAND memories [1]. A strong reduction (~2 decades) of the FG cross coupling in the 32nm node with Si-ncs is obtained compared to the case of a poly-Si FG, due to reduced coupling areas. This leads to a lower parasitic Vth of the disturbed cell for a given FG charge in the master cell. Note that for such scaled Si-nc memory devices where a very small number of Si-ncs per cell is involved, the use of self assembled FGs [11] will reduce the dispersion of the memory characteristics.

Conclusions

We successfully demonstrated the FN operation of Si-nc memories by integrating high-k IPDs, due to the improved coupling ratio and reduced IPD leakage currents. HfAlO based IPD combines the good quality of HfO2 and Al2O3 dielectrics, allowing 1ms programming, good endurance and long retention. Moreover Vth was enhanced to 6V by using a double Si-ncs/SiN FG with an HfAlO IPD. Finally physical modelling and TCAD simulation allow to explain the advantages of Si-ncs and high-k IPD on the memory characteristics.

III -Modeling and simulations

dot IPD FG th R C Q V    IPD tun FG IPD tun G IPD FG C C Q C C V C V       FG sub V , E  : transparency with WKB approximation   FG sub V , E  : transparency with WKB approximation     dE T k E E exp 1 ln . V , E h T k m q 4 V J B Fsub 0 FG sub 3 B Si 3 FG tun                          FG G IPD FG tun FG V V J V J dt dQ   

Fig. 1 (

 1 Fig.1 (a-b) TEM cross sections of 3-layers IPD memories. (c-d) SEM plan views of deposited Si-ncs (c) and Si-ncs/SiN (d) layers. Inset: TEM cross section of the double Si-nc/SiN layer. Bottom: schematics of the 3layer and 2-layer IPD Si-nc and Si-nc/SiN memory devices studied in this work.

Fig. 2 Fig. 3 Fig. 4 Fig. 5

 2345 Fig.2 Infrared spectra of 6nm ALD deposited HfAlO films of various Hf:Al compositions. Inset: measured optical bandgap (ellispometry) vs the Hf fraction of HfAlO alloys.

Fig. 6

 6 Fig.6 Comparison of Vth for a program voltage of 18V for memory devices with various 3-layer IPDs. Control devices with ONO and HTO IPD are also shown Fig.7 Program voltage required for a Vth of 2.5V with 1ms pulses, vs IPDs EOT. Different 3-layer and 2layer IPD are shown.

Fig. 8

 8 Fig.8 Endurance characteristics of 3-layer IPD memories (S1 to S3), with various O/high-k/O IPDs. Program: VP=18V/1ms, erase: VE=-15V/1ms.

Fig. 9

 9 Fig.9Normalized retention curves at RT of the programmed state (Vth0=2.5V) for devices S1 to S3.

Fig. 10

 10 Fig.10 Retention characteristics at 125°C of 3-layer IPD memories (S1 to S3), with various O / high-k / O IPD stacks (S1 to S3).

Fig. 11 Fig. 12

 1112 Fig.11 Life time vs measurement temperature for samples S1 to S3.

Fig. 13

 13 Fig.13 (a) Id(Vg) in forward (Vd=1.5V) and reverse (Vd=-1.5V) modes for S5. (b) Normalized retention curves of the programmed states of S5 (Vth=3.5V).

Fig. 14

 14 Fig.14 Endurance characteristics of Si-nc/SiN memories with 3-layer (S5) and 2-layer (S6) IPDs.

Fig. 15

 15 Fig.15 Simulated program/erase characteristics. Lines: simulations (Si-ncs area coverage [8]: Rdot=50%), symbols: data. In high-k IPD memories, as JIPD<<Jtun, the memory window is essentially controlled by the IPD EOT.

  Fig.17TCAD 3D simulation of the FG/FG cross coupling in a poly-Si FG memory and in a Si-nc memory. Vth is analytically calculated from the simulated FG potentials.

  Fig.16Simulated endurance characteristics for S1, assuming a progressive trapping in the middle of the IPD stack during cycling, modelled by an empirical law[9]. =-0.2, =6.35.10 -7 .