Master Thesis
Cognitive Biases in Dynamic Job Search
Jonas Fluchtmann [201403265] Supervisor: Rune Vejlin
Aarhus Universitet The master thesis is submitted for the degree of Master of Science in Quantitative Economics (IMSQE) Submitted: August 11, 2016 (Deadline: August 14, 2016) This thesis may be made publicly available.
Abstract
In this thesis I apply dynamic job search models to hazard rates out of unemployment on the Danish labor market and estimate the structural parameters using a minimum-distance approach. To contrast from standard job search theory, which struggles to adequately predict hazard rates, I enrich the structural model with cognitive biases known from behavioral economic theory. These extensions are closely related to the irrational behavior economic agents often execute in real life situations and show a striking degree of increases in the fit to the empirical hazard rates. While these improvements have been demonstrated before, I combine different approaches that led to contradictory policy advices and therefore I am able to obtain new insights. Further, I present suggestive evidence that, given the estimated structural models, a policy change from the current one-step system to a front-loaded two-step policy with a decreased overall benefit level could lead to shorter unemployment durations, less governmental expenditure on the unemployment insurance system as well as increased welfare for the unemployed job seekers on the Danish labor market.
Contents 1 Introduction
1
2 Literature Review
3
2.1
Theory of Structural Job Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Cognitive Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2.1
Reference Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2.2
Present Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2.3
Overconfidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Applications in Structural Job Search Models . . . . . . . . . . . . . . . . . . . . . .
10
2.3
3 Theoretical Framework
12
3.1
Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.2
Reference Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.3
Present Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.4
Overconfidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.5
Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4 Data and Institutions
19
4.1
The Danish Unemployment Insurance System . . . . . . . . . . . . . . . . . . . . . .
19
4.2
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.3
Empirical Hazard Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
5 Solving the Model and Estimation
30
5.1
Solving the Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
5.2
Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5.3
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
5.4
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
6 Results
39
6.1
Benchmark without Saving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
6.2
Benchmark with Saving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
7 Robustness and Consistency Checks
49
7.1
Pre-Reform Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
7.2
Identity Matrix Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
7.3
AR Reference Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
7.4
Constant Belief Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
7.5
Alternative Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
7.6
Reservation Wages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
8 Implications
58
8.1
Unemployment Insurance Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
8.2
Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
8.3
Finding an Optimal Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
8.4
Nudging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
9 Discussion and Limitations
66
10 Conclusion
67
11 Bibliography
70
A Tables
78
B Figures
93
C Proofs
120
D Matlab Codes
122
1
Introduction
In academia and public alike, the scheme of unemployment insurance (UI) is an actively debated topic and the Danish system, which is characterized by relatively generous replacement rates, has been subject to several reforms over the last decades. These reforms mainly focused on cutting the length of benefit entitlement once individuals become unemployed and before the transition into social security with significantly lower transfers. The most recent reform in 2010 cut the entitlement length from four years by half. It was aimed at reducing the increased level of unemployment since the European financial crisis in 2008 and strongly influenced the path of the hazard rates out of unemployment (Hermansen, 2014). Generally, the hazard paths are characterized by strong exit rates right after the beginning of the unemployment spell that quickly decline, just to rise again in anticipation of the exhaustion of the unemployment insurance benefits. After this transition into social security transfers, the hazard rates shrink again to eventually settle at a somewhat constant level. While the evaluation of policy changes, prior to their legislation, often relies on structural models, it is striking that the standard model of job search theory is not able to sufficiently replicate the patterns present in the hazard rates. Thus, policy evaluation that employs the standard job search theory needs to be viewed with extreme caution. To overcome the problems of bad data fit, research in labor economics recently showed increased interest in the theories of behavioral economics, first established with the seminal prospect theory by Kahneman and Tversky (1979). Especially cognitive biases, which are well documented through a number of studies and experiments, like reference dependence, present bias and overconfidence, can possibly explain features of the hazard rates that the standard model cannot fit. Thus, a combination of job search theory with concepts of behavioral economics promises a more realistic modelling of job search efforts. This could eventually lead to more accurate policy evaluation. Further, the design of the unemployment insurance would possibly be able to exploit the behavioral factors that govern the search decisions of the unemployed job seeker. In this thesis I am building on the job search model by DellaVigna et al. (2016) who already included reference dependence and present bias in their estimations. These extensions have shown
1
to fit the patterns of the unemployment hazard rates on the Hungarian labor market very well and clearly outperform the standard formulation of job search theory. Further, I am invoking the concept of overconfidence in the subjective job finding probability, introduced by Spinnewijn (2015) in a similar model. The novel part of my estimations is not only the combination of both approaches, but I additionally alter the definition and formulation of the overconfidence to the assumption of a bias that vanishes over time, i.e. the unemployed job seekers learn about their true probability of finding employment over the course of the unemployment spell. I estimate the structural parameters of the model with the minimum-distance approach that minimizes the weighted distance between parametric moments, generated by the model, and the empirical counterparts, i.e. the hazard rates out of unemployment. The thesis aims at answering the question if features of behavioral economics can also explain the hazard rates out of unemployment on the Danish labor market as well as if the novel definition of overconfidence can increase the data fit. Further, the work tries to shed light on the effects of possible policy changes and the opportunities of exploiting the behavioral components of the unemployed agents decision making. The thesis therefore contributes to the literature on behavioral labor economics as well as the research on optimal unemployment insurance design. Following this introduction, the second section gives a literature review that introduces the theory of structural job search and cognitive biases. While the introduction to job search theory is rather basic, I put more focus on the behavioral extensions that are a part of the later model estimations, as they are the more novel part of the current research in this field. The section concludes with an overview of the main works that have already combined the theories of behavioral economics and structural job search and states the importance of combining different aspects. The third section introduces the theoretical framework of this thesis, i.e. the job search model that is used in the estimation process. I start with the primitive standard formulation of job search and successively introduce the behavioral model extensions that are presented in the prior literature review. Following this, the fourth section gives an overview of the Danish unemployment insurance system and its recent reform as well as the process of obtaining, cleaning and preparing the labor market data from the servers of Danmarks Statistik. The section concludes with the process of
2
estimating the hazard rates out of unemployment which form the empirical base for the structural model estimations. In section five I talk about the mechanisms that are applied to solve the theoretical framework, as well as the estimation technique used to obtain the structural parameters and follow up with a presentation of the results for the benchmark models in section six. I present consistency- and robustness checks in section seven to analyze the reliability of the model in terms of the fit to hazard rates obtained before the unemployment insurance reform as well as re-estimations with altered model specifications. Further in section eight, I talk about the implications that the results have and evaluate possible policy changes that aim to exploit the behavioral patterns active in job search. Additionally, I give a small introduction into a promising new field of research which could reduce unemployment spell lengths at low costs, based on the results obtained in this work. In section nine I summarize the limitations that were reached in the process of the thesis and discuss possible points of concern. Lastly, I conclude on the thesis and the found evidence as well as the proposed policy changes.
2
Literature Review
In this section I give an overview on the related job search literature and its history as an integral part of labor economics as well as cognitive biases as important psychological features that might enrich the underlying theories of job search. Both topics are central to this thesis and build the main theoretical framework. I conclude this section with an overview of literature that has already combined both fields.
2.1
Theory of Structural Job Search
The theory of job search was initially established in the early 1970s to contrast from standard labor supply theory with the seminal job search models of Mortensen et al. (1970) and McCall (1970) and has since become an important strain of general labor economics. The theory mainly aims to describe the transition from unemployment to employment. The general assumption behind this
3
theory is that informational frictions on the labor market influence the dynamics of job search for job seeking individuals and gives answers about the nature of involuntary unemployment which could not exist under the purely Walrasian markets as postulated in the standard labor supply theory. The job search theory was formulated as a dynamic problem under frictions, such as uncertainty and lack of information, where individuals receive exactly one wage offer per period, live forever and decide when to accept an offered wage ω, while any accepted job is held forever and the search itself is a costly process. The individuals stop the job search if a certain job offer ω is higher than their reservation wage that equals the expected marginal return from searching further. Therefore, the process is modeled as an optimal stopping problem and can often be reformulated in Bellmann equations which are solved by the use of dynamic programming techniques. The theory found wide ranges of application and variation resulting in a rich stream of publications over the following decades. Gronau (1971) for example relaxes the infinite lifetime assumptions and finds that the reservation wages tend to shrink in anticipation of the last period, as the individual faces trade-offs between wage-changes and unemployment. Burdett (1978) introduces on-the-job search to the search theory that can lead to wage increases at the same employer, even with the assumption of no human capital accumulation. Jovanovic (1979) works with the assumption that important characteristics of an employee-firm-match, like the future wage structure and productivity, are unknown and get slowly revealed over time by learning of both sides. This leads to turnovers if one of the sides does not see the match as a good choice in retrospective. The efforts of extending job search theory eventually led to a Nobel Memorial Prize in Economic Sciences in 2010 for the seminal contributions of Peter A. Diamond (1982a, 1982b), Dale T. Mortensen (1970, 1986) and Christopher A. Pissarides (1979, 1984, 2000), who included a matching function between job seekers and firm vacancies to job search models. By their combination of the individual and the firm side, the search process created certain external effects that the agents do not take into account as increased search effort decreases the employment prospects for other job seekers. However, it increases the prospects for the firm to fill a vacancy. Further, job search theory has concentrated on finding explanations for the wage dispersion that is found in the empirical data, even among workers with the same characteristics. Burdett and
4
Judd (1983) work with an equilibrium price setting model under market frictions where consumers sample different amounts of prices and the price distribution is common knowledge. The firms are then faced by uncertainty about the prices the consumers hold. Burdett and Judd show that it is optimal for firms to draw random from the price distribution, effectively leading to dispersion in the prices for the consumers. This setting can be transferred to job search where job seekers sample wage offers from firms. Albrecht and Axell (1984) introduce heterogeneity in the valuation of leisure time among the job seekers, leading to different reservation wages. The individuals with lower valuation of leisure then accept lower wages resulting in wage dispersion. In Burdett and Mortensen (1998) on-the-job search serves as an explanation of wage dispersion as firms may offer higher wages to employ workers from competitors.
2.2
Cognitive Biases
Cognitive biases are systematic errors in human behavior that move the individuals away from behaving purely rational in the sense of the homo oeconomics, the standard assumption of economic theory. These biases therefore lead to irrational judgments of market situations as well as to a distorted reality of the economic agents. Cognitive biases often occur due to limitations of human mental capacities as well as due to bounded rationality problems (Bless et al., 2004). The biases have significant implications on broad fields of application and often need to be taken into account when giving policy advice or similar, as shown by Kahneman and Tversky (1996). A broad introduction to several cognitive biases as well as laboratory and field evidence can be found in DellaVigna (2009). In the following subsections I briefly introduce the cognitive biases that seem to have important implications on the field of job search theory.
2.2.1
Reference Dependence
Reference dependence was first established in the seminal prospect theory by Kahneman and Tversky (1979) as a critique on expected utility theory for economic decision making under risk. Their ideas have since spread across several applications in economic theory and are regarded as one of the founding works in behavioral economics. Kahneman and Tversky (1979) claim that optimal choices 5
of economic agents are influenced by gains or losses in comparison to a reference point and provide a wide range of examples, also stating that individuals are more sensitive towards losses than gains (”loss aversion”). The examples include experiments where students and faculty members were subject to different lotteries as well as a reflection of the lotteries around 0. This reflection reversed the preference order of the individuals from risk seeking in the gain domain to risk aversion in the loss domain. This is contradictory to expected utility theory and a clear departure from the standard model which only puts its focus on the final expected outcome1 . Further developing on the theory of Kahneman and Tversky (1979), K¨oszegi and Rabin (2006) transfer the idea to economic models by adding an additional ”gain-loss”-term to the final utility of the individual. Whether something is considered a gain or a loss is evaluated according to a given reference point which can be formed by different means (status quo, certainty equivalent, goals, social comparison etc.), but is mostly assumed to be formed by the rational expectations about future outcomes. In this setup consumption utility at time t can be written as
ut (ct |rt ) = m(ct ) + n(ct |rt ) =
m(ct ) + η[m(ct ) − m(rt )]
if ct ≥ rt
m(ct ) + ηλ[m(ct ) − m(rt )] if ct < rt where ct and rt denotes consumption and the reference point respectively. The specification of m(·) and n(·) always depends on the given setting, i.e. the form of the utility function and the specification of the reference point. Generally, losses to the reference point are weighted stronger than gains, which is in line with the findings of the prospect theory. To achieve this loss aversion, an additional parameter λ ≥ 1 is added to the general gain-loss parameter η that captures the sensitivity towards deviations from the reference point. This puts an extra weight on the loss domain and increases the effect of negatively deviating from rt . In the light of job search the reference dependence can have an effect as job seekers might compare their current unemployment insurance benefits to their prior income and thus exercise more search effort right after the beginning 1 Further effects can be found in Kahneman and Tversky (1979) and include isolation- and certainty effects as well as non-linear probability weighting that overweights low probabilities and underweights high probabilities
6
of the unemployment spell2 . Several studies have examined evidence for reference dependence in the field, summarized in DellaVigna (2009). Kahneman et al. (1990) for example allocate mugs to a randomly selected group while also having a reference group not receiving a mug that is able to buy the mugs from the recipients. Both groups were then asked to state their willingness to pay (buyers) and their willingness to accept (sellers) with respect to the mug. According to standard theory one would expect that the average willingness to pay coincides with the average willingness to accept. Kahneman et al. however observe that the latter one is approximately twice as high as the former, indicating a deviation from standard decision making. This can be explained by the reference dependent utility, as the ones who received a mug form a reference point, the mug endowment, and losing their endowment feels worse than not receiving the mug.
2.2.2
Present Bias
Standard economic theory normally assumes that economic agents time discount rates are constant in every period, which in turn implies that the agents time preferences are the same at every point in time. Evidence however shows that individuals might be subject to time-inconsistent preferences. Thaler (1981) finds annual discount rates varying from 345 percent for a delayed benefit in one month to 19 percent for a delayed benefit in ten years. This suggests that an individuals discount rate is higher in the short run than in the long run and casts doubt on the constant discount rate assumption of the standard theory. More laboratory evidence is summarized in Loewenstein and Prelec (1992) and Frederick et al. (2002). Laibson (1997) and O’Donoghue and Rabin (1999) introduce the quasi-hyperbolic discounting to capture the time-inconsistency in discrete time by adding an additional discount factor β ∈ (0, 1] between the current and all subsequent periods. The discounted utility in a given period t thus becomes:
Ut = ut + β
∞ X
δ i ut+i
i=1 2 Note
that this would require a backwards looking reference point - a deviation from the rational expectation reference point.
7
Clearly β < 1 induces a steeper discounting in the closer future than in the periods further away while β = 1 would lead back to the standard model of time discounting. Looking at job search, the present bias can induce procrastination as unemployed job seekers might fall into behaviors that offer rewards in the present and costs in the future. Assume for example using time for leisure activities that generate positive utility in the present, but create costs in terms of foregone employment possibilities in the future instead of putting effort in the search for employment that only promises delayed benefits. There is plenty of evidence in the field to prove the quasi-hyperbolic discounting assumption, again summarized in DellaVigna (2009). One of those studies finds evidence on course homework in combination with deadlines (Ariely and Wertenbroch, 2002) where half of the students in an executive education class at the Massachusetts Institute of Technology were able to set their own binding deadlines for three compulsory class assignments. The students were also able to set no deadline at all, effectively moving a quasi-deadline to the end of the course. The other half of the class was subject to fixed and equally spaced deadlines over the semester and delayed hand-in of the assignments was punished with lower grading in both groups. The authors find that grades in the fixed deadline group were significantly higher than the grades in the group that chose their own deadlines. To isolate the effect of the deadline type (self chosen vs. externally set) on the grade outcome the authors also compare the group of equally spaced self chosen deadlines to the no-choice group, finding no significant difference. This implies that students are subject to the present bias induced by hyperbolic discounting and thus set suboptimal deadlines. As the students actually set costly (sub-)optimal deadlines Ariely and Wertenbroch show that the students are willing to pay to partly overcome the self control problems. Another study by DellaVigna and Malmendier (2006) shows that members of three American health clubs, who sign up for a monthly lump-sum contract that involves, in contrast to pay per visit, free entrance during the contract duration, pay on average $17 per visit while they would be able to just pay $10 per visit when not signing up for the contract. The authors conclude that this is due to overestimation of the future health club attendance, induced by time-inconsistent preferences.
8
2.2.3
Overconfidence
Besides the before-mentioned cognitive biases, overconfidence is also a well established concept. It assumes that people could possibly over- or underestimate their own ability or skills. Several studies find that people tend to overconfidence in hard tasks, but conversely to underconfidence in easy tasks (Burson et al. (2006), Moore and Cain (2007), Grieco and Hogarth (2009)). In the light of job search this can be thought of as overestimating the own ability of finding a job in the next week or month or more generally as overestimating the marginal job search efficiency of one extra unit of time spend on the search for employment. These effects are big for rather complicated tasks that involve more effort and tend to be very low or not prevalent for tasks that are easy to finish or accomplish (Buehler et al., 1994). Assuming a task that is hard enough to lead to overconfidence which creates an outcome with a certain probability, affected by effort, overconfidence can be formulated as the following: π(et ) ≤ π ˜ (et ), ∀ et ≥ 0 where π(et ) is the actual probability of the outcome realizations and π ˜ (et ) is the perceived but biased probability, due to overconfidence. Evidence in the field was found by Malmendier and Tate (2008) examining the management overconfidence of CEOs. The auhtors use CEOs who hold their own company options until expiration as a proxy for overconfidence. The long holding of the stock options is interpreted as estimating the future performance of their company too high and therefore as overconfidence. The overconfident CEOs are in fact significantly more likely to commence a merger with internal funding where they see external funding as overpriced. Malmendier and Tate also rule out insider information as a possible explanation for this and conclude that overconfidence leads to the bad performance of firms that commence mergers. Another field example for overconfidence can be found in the before-mentioned health-club study by DellaVigna and Malmendier (2006) where the individuals overestimate their self-control and thus pay too much for the lump-sum contracts.
9
2.3
Applications in Structural Job Search Models
Recently behavioral economics also spread into the field of labor economics as well as into the subfield of job search. While the main interest of this thesis lies on structurally estimated models, it is to note that some other contributions included concepts of cognitive biases, focusing for example on the influence of backward-looking reference points on reservation wages (Koenig et al., 2016), labor supply reactions to wage changes (Doerrenberg et al., 2016) or wage rigidity induced by loss aversion in search and matching (Eliaz and Spiegler, 2013). Turning to the main subject of this thesis, several studies have since examined possible applications of reference dependence, present bias and biased beliefs in structural models of job search. The first notable applications can be found in DellaVigna and Paserman (2005) and Paserman (2008). Both studies apply hyperbolic discounting to dynamic models of job search. DellaVigna and Paserman (2005) empirically test a dynamic job search model using standard exponential discounting against the same one with hyperbolic discounting, in line with a present bias assumption. They find that the impatience is negatively correlated with the exit rate out of unemployment as well as with the search intensity of the job seekers. They find there is no effect on acceptance probabilities of wage offers and the reservation wage. The found evidence supports the assumptions that present bias and time inconsistency strongly influence the job search for unemployed agents and has clear implications for labor market policies. Paserman (2008) follows up on this research by quantitatively estimating the degree of the present bias in dynamic job search and predicts effects of altered UI benefit policies. The study estimates the present bias parameter β below 1, in line with the hyperbolic discounting assumption. The impact of a policy change that induces higher search-effort, and thus working against the present bias, is highly dependent on the underlying structure of the model and policy intervention. There are however interventions that may increase search effort and social welfare whilst decreasing the duration of the unemployment spell. Building up on the assumption of biased beliefs, Spinnewijn (2015) examines the influence of overconfidence in job search. The study finds empirically that unemployed job seekers strongly overestimate their probability to find a job. Spinnewijn finds that within a 1996/1998 sample of 1,487
10
job seekers in Michigan and Maryland, the individuals expect to remain unemployed for roughly 6.8 more weeks from the time of the examination. The actual continued unemployment duration of those respondents was approximately 3 times larger than their expectation and in total over 80 percent underestimated their remaining unemployment duration. This confirms the prior assumption that job seekers might overestimate their job finding probability and also suggests that individuals save too little to protect themselves against future income drops. Further, Spinnewijn estimates a structural model of job search including overconfidence to characterize optimal unemployment insurance policies when beliefs about the job search probability are biased. Here the unemployed job seeker receives an unemployment insurance benefit which is in turn financed by a tax subtracted from her pre-unemployment income. Spinnewijn argues that the handling of people who stay unemployed for a long time directly influences the incentives for short-term unemployed on both the savings and the search effort level, depending on the expected duration of unemployment for each individual. In Spinnewijns setting an overconfident individual prefers a lower benefit, leading to a lower tax as the job seeker expects to stay unemployed shorter than the actual realized duration. If in addition to the overconfidence the externality, induced by differences between actual and perceived returns to search, is small, then the optimal unemployment insurance policy is characterized by increasing benefits and taxes for people who remain in unemployment for a long time. This can be done with fairly low incentive trade-off costs for the short-term unemployed individuals as overconfident unemployed are less reactive to future incentives. Finally, DellaVigna et al. (2016) present a study involving a prototypical job search model with savings that adds reference dependence to differentiate from the standard model of structural job search. In contrast to most current research (see f.e. K¨oszegi and Rabin (2007)) DellaVigna et al. (2016) use a backwards looking reference point that accounts for the differences between the current benefit level and the prior wage income. The study presents a minimum distance estimation of the model parameters using hazard rates out of unemployment as empirical moments. The empirical base of this paper is collected from Hungarian unemployment data with the interesting characteristic of a change from a single-step to a two-step benefit system in 2005 and shows distinct differences in the behavior of the unemployed job seekers in both regimes. The estimated hazard rates, here
11
analogously defined as the search effort in the model, show a strong increase in the data fit compared to the standard model. As predictions show a unjustifiable strong degree of impatience, DellaVigna et al. extend the model to allow for present bias in terms of hyperbolic discounting. The results show a significant degree of present bias as well as a reference dependence that lasts a little longer than half a year. In contrast to the findings of Spinnewijn (2015) for overconfident job seekers, the conclusion of DellaVigna et al. points to a decreasing multi-step benefit system if the unemployed are reference dependent. As both recommendations point in different directions it is thus necessary to combine both models to find the optimal unemployment insurance policy under possible reference dependence, overconfidence and present bias.
3
Theoretical Framework
The main model of this thesis builds on the discrete time job search intensity model of DellaVigna et al. (2016). Their work is the first job search model adding reference dependence to yield more realistic predictions of search effort choices by unemployed individuals. They build on the frameworks developed by Card et al. (2007) and Lentz and Tranæs (2005). The following sections successively introduce the theoretical framework and all its extensions used in this thesis. By the end of this chapter I will have presented a fully flexible model with added reference dependence, present bias and overconfidence. It is straightforward to disable some of these extensions in the estimations by setting certain model specific parameters to 0 or 1, respectively.
3.1
Standard Model
In each of the periods the unemployed job seeker faces a decision of how much to consume of her income and how much to save for precautionary efforts that are aimed to smooth the consumption around the benefit exhaustion. In addition to that, she needs to choose the utility maximizing job search effort st ∈ [0, 1] which represents the probability of receiving a job offer in the upcoming period. Naturally the wage w connected with a job offer lies above the current unemployment benefit bt and once a job is found it is kept forever. The model is simplified by the assumption that the wage
12
offer is constant over all periods and the job seeker always accepts it, ruling out reservation wages3 . The search effort results in costs each period represented by the twice continuously differentiable cost function c(st ) with c0 (st ) > 0, c00 (st ) < 0, c(0) = 0 and c0 (0) = 0. During the unemployment the agent receives transfers bt in each of the periods. The unemployment insurance benefits are granted for the first T periods. After this, the eligibility of UI benefits is exhausted and the agent enters the social security system receiving the significantly lower social security transfers. While the individual is unemployed and seeks for a job, her utility is given by: h i E U VtU (At ) = max v(ct ) − c(st ) + δ st (At+1 )Vt+1|t+1 (At+1 ) + (1 − st (At+1 ))Vt+1 (At+1 ) st ,ct
(1)
subject to ct = At + yt −
At+1 1+R
(2)
U E where Vt+1 is utility of being unemployed in period t + 1 and Vt+1|t+1 is the value of being employed
in period t + 1 conditional on finding a job starting in period t + 1. The consumption utility is v(ct ) and the future period is discounted by δ. In every period the individual needs to make a consumption-savings decision that satisfies the budget constraint (2) where At represents the stock of savings and yt is the transfer income, both at period t. The agent is only allowed to save, imposing a no-borrowing constraint. The value function for being employed is given in the form of a Bellman equation: E VtE (At ) = max v(ct ) + δVt+1|t (At+1 ) ct
(3)
The interior solution4 of solving the value function of being unemployed for the optimal s∗t (At+1 ) implies: E U c0 (s∗t (At+1 )) = δ[Vt+1|t+1 (At+1 ) − Vt+1 (At+1 )]
(4)
Defining the inverse of the first derivative of the cost function as C(·) = c0−1 (·) gives the following 3I 4I
consider reservation wage decisions in the robustness checks later. additionally check for solutions at the boundaries of st in the model computations.
13
solution for s∗t (At+1 ): E U s∗t (At+1 ) = C(δ[Vt+1|t+1 (At+1 ) − Vt+1 (At+1 )])
(5)
After the benefits drop at T the search effort becomes stationary as there are no income related shocks anymore. The standard model is therefore solved from the point of the benefit drop T for every preceding period by backwards induction/dynamic programming.
3.2
Reference Dependence
With the first extension the agent also faces a reference-dependent gain-loss utility in the fashion of K¨ oszegi and Rabin (2006). This works in the way that the individual compares her consumption decision to her prior wage level for the first N periods after the unemployment spell. Note that the gain-loss utility also affects the periods after an individual changes from unemployment benefits to social security. The reference-dependent utility function is given as:
u(ct | rt ) =
v(ct ) + η[v(ct ) − v(rt )]
if ct ≥ rt
(6)
v(ct ) + ηλ[v(ct ) − v(rt )] if ct < rt where η is the weight on gain-loss utility, fixed at 1 as in most reference-dependence literature, and λ is the parameter specifying the loss-aversion which is assumed to be larger than 1. While being unemployed the reference point rt is always higher than or equal to the current benefit level and therefore the individual experiences a loss and weights the difference with ηλ. Note that if I set η to 0 I arrive back at the basic model without gain-loss utility. Contrary to the current research (K¨oszegi and Rabin, 2007), that assumes forward-looking reference points, DellaVigna et al. model the reference point as a backward-looking weighted average of the current income and the preceding N periods. Therefore it is defined as:
rt =
t X 1 yk N +1 k=t−N
14
(7)
It is a strong assumption that the unemployed job seeker compares her current consumption to her average past income, it is however necessary to make the model computable, as other variations, for example a forward looking reference point or one that considers actual consumption rather than income, increase the computational scope enormously and are by no means feasible in this thesis. I will however consider an AR(1)-variation of the reference point in the robustness checks. Analogously to the standard model, I can write down the value of being unemployed when the agent is subject to reference dependent utility: h i E U (At+1 ) + (1 − st (At+1 ))Vt+1 (At+1 ) VtU (At ) = max u(ct |rt ) − c(st ) + δ st (At+1 )Vt+1|t+1 st ,ct
(8)
subject to ct = At + yt −
At+1 1+R
(9)
Again I get the value function for being employed in the form of a Bellman equation:
E VtE (At ) = max u(ct |rt ) + δVt+1|t (At+1 ) ct
(10)
The interior solution of the optimal search effort does not change in this case and is still represented by equations (4) and (5). The model with reference dependence is solved by backwards induction from the point the optimal search effort is in steady state and therefore stationary, i.e. N periods after the last drop in transfer payments.
3.3
Present Bias
To allow for inconsistency in the discounting between different periods DellaVigna et al. (2016) introduce an additional discount factor β ∈ (0, 1] between the current period and the future. This builds up on the theory of hyperbolic discounting (sometimes β − δ discounting) by Laibson (1997) and O’Donoghue and Rabin (1999). Just like DellaVigna et al. I assume that the agent behaves naive according to her time preference, i.e. she assumes that she will only be faced by δ-discounting in the future and thus thinks she will follow a time consistent consumption and search effort path
15
while being faced with the additional discount factor β only in the current period. Reaching the next period however, she is again faced with the additional β and thus deviates from her initial consumption and search effort plans that just involved δ-discounting. The value function of being unemployed in this case can be written as: h i E U (At+1 ) + (1 − st (At+1 ))Vt+1 (At+1 ) (11) VtU,n (At ) = max u(ct |rt ) − c(st ) + βδ st (At+1 )Vt+1|t+1 st ,ct
U E where Vt+1 (At+1 ) and Vt+1|t+1 (At+1 ) are given by equations (8) and (10). The characterisation of
the optimal search effort then changes to:
E U c0 (s∗t (At+1 )) = βδ[Vt+1|t+1 (At+1 ) − Vt+1 (At+1 )]
3.4
(12)
Overconfidence
To allow the agent to be biased in her belief about her employment prospects I manipulate the agents beliefs about the job finding probability, inspired by Spinnewijn (2015). I however allow the agent to learn about her biased beliefs over time and update them in a way that it approaches the actual probability of finding a job. I formulate the perceived job finding probability as:
sˆt (st , At+1 ) = min st (At+1 ) · (π t + 1), 1 , π ∈ [0, 1)
(13)
where st (At+1 ) is the actual job finding probability and π is the parameter that determines the strength of the bias5 . For π ˆ = 0 I end up with the prior model without any bias in the beliefs. It is obvious that the biases diminishes over time and approaches st (At+1 ): lim sˆt (st , At+1 ) = lim min st (At+1 ) · (π t + 1), 1 ≈ st (At+1 )
t→∞
t→∞
5 Note that the magnitude of the biased belief is limited by twice the actual job finding probability. This is slightly inelegant and rules out stronger biases. Thus it might need to be re-formulated for extended works with this model. Nevertheless, the following results in section 6 and 7 never estimate π and the respective standard error with a magnitude that casts doubt on the used assumption.
16
The final value of being unemployed in the fully flexible model version thus becomes: h i E U (At+1 ) + (1 − sˆt (st , At+1 ))Vt+1 (At+1 ) VtU,n (At ) = max u(ct |rt ) − c(st ) + βδ sˆt (st , At+1 )Vt+1|t+1 st ,ct
(14) and the characterisation of the optimal search effort updates to:
c0 (s∗t (At+1 )) =
βδ(π t + 1)[V E
t+1|t+1 (At+1 )
U − Vt+1 (At+1 )]
U βδ[V E t+1|t+1 (At+1 ) − Vt+1 (At+1 )]
if st (At+1 )(π t + 1) ≤ 1
(15)
t
if st (At+1 )(π + 1) > 1
The assumption that the job seekers update their beliefs is contrary to Spinnewijn (2015) who finds in US data that learning about the true probability of finding employment presumably does not exist, which is in line with a laboratory experiment by Huffman et al. (2006). Danish survey data however shows a different picture. Here I make use of the Work, Unemployment, and Early Retirement-survey collected by the Centre for Comparative Welfare Studies at the University of Aalborg (2008). The survey data was collected with two (almost) identical questionnaires in 2006 as well as 2008 among 7,544 respondents6 . The goal was to gain insights about the labour supply and marginalization on the Danish labour market and the survey design allows interesting comparison between employed, unemployed and early retired individuals in Denmark. In the data I find 501 unemployed individuals that were searching actively for employment at the time of the surveys and who gave answers about their beliefs how much time will pass until new employment is found. While this is a small statistical population, the design of the survey questions gives some interesting insights into whether Danish unemployed job seekers are subject to learning about their subjective job finding probability or not. As the survey is unfortunately not linkable to register data I cannot gain further insights about the differences between expected and actual unemployment duration from the time of the questionnaire. However, table 2 in the appendix shows the job finding beliefs of job searchers by their time already spent in the current unemployment spell as well as average hours spent on job search in the month prior to the questionnaire and the average number of jobs 6 A small amount of questions differs between both survey rounds, they are however not relevant in the context of this thesis.
17
applied in the prior year. The pool of unemployed that were searching for a job applied for 39.98 jobs on average in the year prior to the survey and spent 25.4 hours in searching for jobs in the month prior to the questionnaire. Looking at the differences between the unemployed by their time already spent in the current unemployment spell, I find that the predicted unemployment duration basically increases over the unemployment spell length. This holds regardless the fact that the hours searched for employment are relatively stable in all groups and the number of jobs applied in the prior year often increases with the unemployment spell duration (until 4 years of unemployment). Even though the scope of the survey is fairly weak this gives the impression that Danish job seekers in fact update their beliefs over time and even become quite desperate. Following these findings I apply the proposed learning-bias in the model, but I will nevertheless estimate model parameters using a constant bias model like Spinnewijn (2015) in the robustness checks later.
3.5
Heterogeneity
One often observes decreasing hazard rates when analyzing the duration of unemployment of active job seekers (Eckstein and Van den Berg, 2007). When employing the simple standard model of section 3.1 the search efforts cannot shrink and thus fail to properly model any of the features found in most of the empirical data. It is however possible to introduce heterogeneity in the unemployment benefits or the search costs that enable the possibility of declining search efforts even in the standard model (Mortensen and Pissarides, 1999). In this thesis I will re-estimate the models of section 3.1 to section 3.4 with two different search cost types kl and kh as well the initial population share of the low cost type pl . While DellaVigna et al. (2016) supply three different cost types in their estimations, I mostly end up with identification issues when supplying more than two different types and thus stick to this method. Due to the heterogeneity I will have two different types of unemployed job seekers that have different probabilities of finding employment and thus some of them will leave the unemployment pool faster than others. Therefore, I have different search efforts for the two cost types and report the average search intensity in the population. In the long run however, all types converge to the same hazard rate (Eckstein and Van den Berg, 2007). By adding the heterogeneity I expect some 18
of the models to generate a closer model fit, especially when lacking the reference dependence that leads to shrinking search efforts after benefit drops.
4
Data and Institutions
In this section I introduce the Danish unemployment insurance system and its reform in 2010 to give some intuition about the regulations on the Danish labor market. Further, I turn to the process of obtaining the empirical data that is used in this thesis. This includes definitions and processes of preparing the data from its raw source on the servers of Danmarks Statistik as well as a short overview of the descriptives of the obtained sample. Lastly, I turn to the process of estimating the hazard rates out of unemployment that are used in the structural estimation of the search efforts.
4.1
The Danish Unemployment Insurance System
The danish unemployment insurance system is based on a voluntary scheme. Individuals who decide to join the insurance system have the right to receive unemployment benefits under special requirements once they become unemployed and register as unemployed at the Public Employment Service. To become insured one needs to be registered at one of several unemployment insurance funds ”arbejdsløshedskasse”, A-kasse in short. Once being registered the individuals need to pay a monthly fee which can be written off against the tax payments. To receive the benefits the unemployed must be a member of an A-kasse for a minimum amount of one year and have worked at least 1,924 hours in the preceding three years. This roughly represents one year of full-time work and to regain access to the benefits after their exhaustion an unemployed individual needs to fulfill this criteria again. For recent graduates who join an unemployment insurance fund within 2 weeks of their graduation the preceding education is regarded as employment and thus they are generally eligible to receive benefits right after finishing education. There is also a part time insurance available which amounts to two thirds of the full-time insurance described before. The unemployment insurance benefit amounts to up to 90% of the prior income, but is capped at 836 DKK per day for 5 days a week as per 2016, which represents 4,180 DKK per week for
19
recently employed and 686 DKK per day for 5 days a week (3,430 DKK per week) for recent graduates7 . Even though the unemployment insurance funds are private associations, the benefits are mostly financed by the Danish state. Therefore the amount and the duration of the benefits is subject to parliamentary legislation and the duration of the benefits as well as the requirements to receive it were subject to several reforms over the last decades. Until 1993 the time frame over which the benefits were payed was unbounded and it then got changed to 7 years. In the following years it was subject to more changes, from 7 to 5 years in 1995 as well as from 5 to 4 years in 1998. Since then the system remained mainly untouched until 2010. In the wake of the European economic and financial crisis, a reform of the unemployment insurance scheme was suggested with the aim to reduce the relatively high unemployment and the implementation was eventually legislated in June 2010 due in the following month. The main part was a cut in the maximum unemployment insurance benefit entitlement from 4 to 2 years as well as harmonising the requirements for regaining access to benefits after their exhaustion from having employment for 26 weeks in the preceding 3 years to the before-mentioned 52 weeks. After exhausting the right to receive benefits, i.e. after staying unemployed longer than the unemployment insurance period, the unemployed enter the social security system and receive the lower social assistance benefits ”kontanthjælp”. The individuals are however required to comply with regulations according to their household wealth before being able to receive social security payments. The social assistance benefits amount to roughly 80% of the unemployment insurance benefit for recipients with children and on the other hand to 60% for recipients without children. The social assistance is granted without a time constraint, but individuals are however subject to small deductions if they stay in the system for more than six months. The initial aim of significantly reducing the unemployment rate, subject to stark rises since 2008, through the unemployment insurance reform was not fulfilled and the reform itself was subject to strong opposition of the Danish trade unions who feared an erosion of the income security in Denmark (Madsen, 2013). To respond to these factors the government implemented ”acutemeasures” that delayed the benefit exhaustion for long-term unemployed who would lose their right 7 In 2008 this was 703 DKK per day for former employed, respectively 576 DKK for recent graduates, as well as 766 DKK and 628 DKK per day in 2011
20
Figure 1: Unemployment insurance benefits and the reform
Note: Visual description of the danish unemployment insurance reform in 2010 and the following acute-measures, extracted from Hermansen (2014). People entering unemployment insurance in the first half of 2008 were still eligible for 4 years of benefits while people entering in the second half got their unemployment insurance period extended to match 4 years, after which they were able to enter the special education benefit. The education benefit got extended with ”akutpakke 3” as well as the implementation of the ”arbejdsmarkedsydelse”. Both new benefit levels are matched to the level of social security and thus people entering unemployment insurance in the beginning of 2011 are the first ones who are subject to a 2 year unemployment insurance benefit period.
to benefits in late 2012 by another six months. In addition to that, a special education benefit was introduced from 2013 aimed at unemployed that exhaust their benefit duration and enter educational programmes to increase their job finding probability. This was followed by temporary ”arbejdsmarkedsydelse”. The level of the new education benefit and ”arbejdsmarkedsydelse” is however set to the same level as social security payments. The acute-measures moved the final implementation of the 2 year unemployment insurance benefit period to people who became unemployed since the beginning of 2011.
21
4.2
Data
The data used in this thesis mainly comes from the DREAM database of the Danish Ministry of Employment ”Beskæftigelsesministeriet”. The DREAM registry data contains information on weekly transfers for the whole Danish population and is collected by the Ministry of Employment, the Ministry of Education as well as SKAT and other population registers. Data is available from mid 1991 and the latest version accessible to me, at the time of collection, ran until September 2015. The received transfers in the data are distinguished by type and thus make it possible to calculate the unemployment duration in weeks for every person registered in Denmark as well as the transition into social security, transitions out of the labor force and several other states. From January 2008 and onwards the database also displays monthly information on employment. This makes it possible to focus on unemployed job seekers who entered unemployed insurance directly from employment and leave the unemployment insurance respectively social security system into employment. The post-reform regime with unemployment insurance benefits being transferred for two years is going to be the main empirical base used in the later estimations. The pre-reform regime with benefits being transferred for four full years will however be used to test the estimation outcome for its accuracy. In the following I present the definitions of states and the process of cleaning the data. These definitions are mostly based on Hermansen (2014) who provides an interesting analysis on the impact of the 2010 reform on the duration of the unemployment spells.
Unemployment: Individuals in unemployment insurance benefits. After the unemployment insurance exhaustion the ”job-parat” social security group, which includes individuals who enter the social security system but are still registered as active job seekers as well as the people who pass into the special-education benefit are also regarded as unemployed.
Self support: Individuals who do not receive any transfers and are not registered as employed in the respective month as well.
22
Employment: Individuals who do not receive any transfers in a given week and are also registered as employed in the respective month. Note that the employment information is just provided on a monthly basis and requires a paid ”AM”-contribution (compulsory labor market contribution in the Danish tax system).
Not in the labor force: Individuals who have no immediate prospect to return to work, i.e. people who are in sickness or education benefits (like SU and adult education) as well as people in social security that are not actively searching for a job (not ”job-parat”). Note that the special education benefit of the acute measures is not included here, but is regarded as unemployment when the job seeker transitions from benefit exhaustion into it.
Censored: Individuals who end their unemployment spell without leaving into employment or the ones who leave out of the labor force. As very few people stay in unemployment after a certain time I regard people who stay longer than this8 as censored. People who die during their spell, leave the country, enter maternity leave, job-rotation, a fleksjob or self support (see above) are also regarded as censored.
Just like Hermansen (2014) I allow for some special short-terms conditions in the transfer between the different states to cope with short term irregularities in the unemployment spell duration as well as conditions on when to regard something as an unemployment spell in the context of this thesis. I only regard a spell as unemployment if first, the unemployment duration lasts at least one week, second, the unemployed individual needs to have been in the employment category for four continuous weeks in the eight weeks prior to the spell as well as start the spell with receiving unemployment insurance benefits. I also regard individuals who start their spell in the social security category as censored. Third, the spell needs to end with a transition to the employment category. Here I allow for up to four weeks in the self-support category, but employment must at least be 8 I.e.
week 190 in post-reform regime as well as week 220 in the pre-reform regime
23
held for four continuous weeks. Spells that end with leaving the labor force or transitioning into self-support longer than four weeks are not regarded as unemployment spells and are censored. Further, I exclude individuals who start their unemployment spell below the age of 25 as they mostly have not enough work experience and prior income for the context of this analysis. I also exclude people who reach the age 50 during a spell as they are subject to different labor market regulations like early retirement and seniorjobs. Additionally, I just include the first spell for every person who enters unemployment as well as only persons who have the right for a full benefit period before entering the unemployment category. Due to the fact that interruptions in the unemployment spell occur rather frequently, I allow the individuals to temporary leave the unemployment category for up to three weeks and still regard the spell as uninterrupted. Of course the temporary leave out of the unemployment is not counted onto the unemployment duration. By restricting the sample to people who either enter unemployment in 2008 or 2011 I am able to follow the individuals in both regimes for a possible full unemployment insurance duration until the benefits are exhausted. See again figure 1, where it is shown that people who enter unemployment in 2008 are the last ones who are eligible for full four years of benefits and people who enter in 2011 are the first ones who are only eligible for full two years. One needs to however be aware that there are clear, crisis related, macroeconomic differences between the two samples. This is nevertheless the closest look at which I can compare both unemployment insurance regimes as spells in between both years were subject to several duration prolongings and stabilization policies as stated before. Table 1 in the appendix shows the basic descriptives for the pool of unemployed in both the preand the post-reform regime. It becomes obvious that both cohorts are not comparable in terms of the composition of characteristics. This is one of the reasons why I will not supply data for both regimes in the estimation process later described in section 5, i.e. I am just estimating the parameters on the post-reform cohort. Clearly, also the danish economy was subject to a recession and strong effects due to the European economic and financial crisis. This might have had some effects on the pools of unemployed. I am nevertheless confident that I can make use of the prereform data to roughly check the validity of the estimated model parameters as presented in section 6.
24
The structural estimation which is presented later needs certain values for the prior wage, the unemployment insurance benefit level, the social security level as well as the re-employment wage. As stated before, the unemployment insurance allows the unemployed to claim 90% of their prior income with an upper cap of 7,660 DKK for two weeks in 2011 as well as 7,030 DKK in 2008. Due to regulations of Statistics Denmark I am not able to export the exact median values for income data and thus I round it up or down to the 100 DKK level which then satisfies the rules. Using this I find that the median unemployed job seeker in fact claims 7,660 DKK when becoming unemployed in 2011 as well as a social security level of 4,600 DKK. The rounded median pre-unemployment income is 11,300 DKK for two weeks and the corresponding re-employment income 11,800 DKK. This is interesting as re-employment wages are typically lower than the pre-displacement wages (Quintini et al. (2013), Burda and Mertens (2001), Kuhn (2002)), but Denmark shows a relatively narrow wage structure due to the generous benefit system and a high level of organisation of employers and employees trough unions (Dahl et al., 2013). Even though Dahl et al. show that the level of centralized union-based wage bargaining decreased steadily over recent years, with increases in decentralized firm-level bargaining, the Danish wage distribution for full time employees shows one of the lowest dispersions in OECD comparison. The interdecile ratio between the 90 percent decile and the 10 percent decile amounts to 2.50 in 2010 while the OECD average is at 3.46 (OECD., 2016). Due to these facts it is possible that re-employment wages stay on the same level and might even increase slightly as on one hand the wage distribution is quite flat and the wage offers to job seekers are not likely to fall substantially below the prior wages and on the other hand the generous unemployment insurance system allows the unemployed to decline low wage offers and wait for a higher, acceptable wage offer without high opportunity costs. For the pre-reform cohort I however observe a slight decrease in the re-employment wages, possibly due to higher competition on the job market in the course of the crisis9 . 9 From 11,600 DKK to 11,200 DKK. The median job seeker again claims the full amount of unemployment insurance benefits.
25
4.3
Empirical Hazard Rates
The empirical base for the model estimation of this thesis are the non-parametric hazard rates out of unemployment. As mentioned earlier I use the post-reform data for the estimation and in turn use the pre-reform data to check the consistency of the model specifications. There are two main reasons for this: First, due to the length of the unemployment insurance benefit period in the pre-reform regime and no availability of employment and income data prior to 2008 I am not able to cover a large enough sample of unemployed job seekers after the benefit exhaustion. Therefore, I would not be able to export enough data from the servers of Statistics Denmark to include the highly interesting and important post benefit exhaustion dynamics in the estimation. This is however possible for the post-reform data to a certain extent. Second, due to the reform I can use the pre-reform data to check if the model actually works properly on the data. If I find a great model fit on the post-reform data, but the same parameters fail to give acceptable fit to the pre-reform data then I can clearly deny the assumption that the model represents the true data generating process. I use a simple, non-parametric approach to obtain the hazard rates out of unemployment. The DREAM data is obtained in discrete time on a weekly level and with the prior definitions I am able to obtain the weekly employment/unemployment status for every individual in the population. I aggregate the data to bi-weekly intervals for two main reasons: First, I am able to extract a larger scope of the data when aggregating it as Statistics Denmark has clear and strict rules for exporting data from their servers. Second, due to the aggregation I can speed up the model estimation process immensely as the main computationally intensive part is the non-linear solving of intertemporal decision problems involving a large asset grid for every interval point. By bisecting the amount of interval points I can decrease the time needed for the model computation by almost a half. Due to the discrete bi-weekly data collection I am not able to indicate the exact point in time a change in the labor market status occurred. However, there are still methods to estimate the hazard rates available. Following Jenkins (2005) I define the time an event occurred by T . It is not possible to observe T exactly because I can only observe the status at the discrete times ti .
26
Thus, if one observes a status change at ti it happened somewhere in between ti−1 and ti . The difference between ti−1 and ti in this setting naturally amounts to two weeks. By F (ti ) = P (T ≤ ti ) I follow Jenkins in denoting the cumulative distribution function of T and further the survival rate is defined as S(ti ) = 1 − F (ti ) = P (T > ti ), which is interpreted as the probability of staying in the initial state longer than ti . The interval hazard rate is then the probability of leaving the state conditional on still being in the state at ti . With a few steps of basic algebra I get to an expression in terms of the survival rates:
h(ti ) = P (ti < T ≤ ti+1 |T ≥ ti ) =
P (ti < T ≤ ti+1 ) S(ti ) − S(ti+1 ) S(ti+1 ) = =1− P (T ≥ ti ) S(ti ) S(ti )
(16)
Given the definition of the interval hazard rate I need an estimate for the interval survival function. To get these I use the common maximum-likelihood survival function estimate proposed by Kaplan and Meier (1958). This estimate needs an ordering of the state durations according to t1 < t2 < ... < tm , where tm is the end point of the accessible data. The amount of individuals who leave a state between ti−1 and ti is denoted by di and the amount of individuals who are in the state at ti−1 is denoted by ni , commonly referenced as the population at risk. Then I can write the Kaplan and Meier estimator by:
ˆ i) = S(t
Y
(1 −
i:t(i) 0
(20) to solve the maximization problem I am using dynamic programming tools that simplify the dynamic decision problem of asset and consumption choices by interpreting it as a series of smaller problems which are easier to deal with. The following definitions are based on Kuhn (2006) who gives a clear and concise introduction into numerical dynamic programming and its applications in economic problem settings. Starting from the latter formulation of the value function, let me further invoke Bellmann’s principle of optimality for decision problems that are similar in each period: ”An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.” (Bellman, 1957) This means that an optimal decision path chosen in the present needs to be optimal in all subsequent periods to be classified as an optimal policy. The concept is basically similar to the subgame perfect equilibrium in game theory, reached once the chosen decision path represents a Nash equilibrium in every subgame of the complete decision tree. Invoking this principle, the problem basically simplifies to a two-period decision problem where the optimal decision is optimal for all future
31
two-perioid decision problems. Starting from this point, I drop the time indices and write:
V E (A) = max v(c) + η {[1 − (1 − λ)H (w − c)] [v(c) − v(w)]} + δV E (A0 ) c>0
(21)
with A the current asset stock and A0 the asset stock in the coming period. In equation (10) the value function is already present in the recursive form of a Bellmann equation, where the value function V E (A) is the unknown. This will be used to solve the dynamic programming problem. Specifying a value for V E on the right hand side of the equation and maximizing specifies a new value for the left hand side. Thus, the Bellmann equation maps a function into another function which do not necessarily need to be the same. To illustrate this one can replace the max operator and define a new operator Ψ : V E → V E :
v(A) = Ψ(v(A0 ))
(22)
The operator Ψ(·), mapping from the space of bounded functions in the space of bounded functions, can be shown to be a contraction mapping with modulus δ. Let me introduce the theory behind contraction mappings (following Stokey et al. (1989) and Ok (2007)): Definition 5.0.1 Let X be any metric space. A self-map Ψ on X is said to be a contraction (or a contractive self-map) if there exists a real number 0 < δ < 1 such that
d(Ψ(x), Ψ(y)) ≤ δd(x, y) ∀x, y ∈ X.
(23)
This means that a differentiable real function on the metric space A ⊂ R, A 6= ∅ is a contraction, given that the derivative is strictly bounded by ±1 at any point on A. It ultimately leads to the Banach Fixed Point Theorem which will give us important features to solve the problem at hand: Theorem 5.1 Let X be a complete metric space. If Ψ : X → X is a contraction, then there exists a unique x∗ ∈ X such that Ψ(x∗ ) = x∗ . Of course I first need to verify that the Bellmann equation satisfies the conditions to be a contraction 32
on a complete space. This leads to the conclusion that the value function actually exists and it is therefore the single solution to the Bellman equation. Iterating on the Bellman operator converges to the value function from any initial guess. The proof can be found in the appendix and I continue with the algorithm features given that Ψ has a unique fixed point. The following value function iteration algorithm is commonly used in dynamic programming problems and employs the unique fixed point characteristic of a contraction mapping on a complete metric space, as explained before, and is converging to the true value function with a chosen convergence criterion > 0 (Kuhn, 2006).
Algorithm Outline
11
1. Choose a convergence criterion for |Vi (A) − Vi−1 (A)| < 2. Start with an initial guess for the value function V0 (A), e.g. V0 (A) = 0 3. For every A ∈ X evaluate for every c ∈ Γ(A) the current utility from the choice if the value function Vi (A) was the correct value function, i.e. calculate V˜i (A, c) = u(c|r = w) + δE [Vi (Ψ(c, A))] 4. Choose for every A: c∗ (A) = arg max u(c|r = w)+δE [Vi (Ψ(c, A))] and Vi+1 (A) = max V˜ (A, c) c∈Γ(A)
c∈Γ(A)
5. Check convergence. If the value function has not yet converged go back to step 3 and use Vi+1 (A) as guess for the value function, otherwise stop. Once convergence is achieved I get the true value of being employed in period t + N given a chosen asset grid X 12 . Since the numerical routine can not handle infinite dimensional objects for the possible states A directly, some discretization is needed. The asset grid represents the discretized state space in period t + N and I need to supply a fine grid to give the agent the choice between different asset and consumption paths in the preceding periods. Due to the no-borrowing constraint, 11 Just as in the proof for the Bellman operator being a strict contraction on a complete metric space, Γ(A) represents the feasible correspondence that consists out of all feasible states of c given a value for A. It therefore represents all achievable consumption/asset choices given an asset level A in the current period. 12 To speed up the process I run the algorithm four times without step 4 and just use the maximization in step 4 every fifth iteration. This method can achieve a significantly faster convergence which is beneficial as this step is one of the most time consuming parts of the estimation.
33
the minimum element of the grid is naturally 0 and the maximum element is defined by the asset stock in period t added to the income and its return over all periods between t and t + N . In the main estimations I use a grid that puts more weight on the lower end of the asset states to give E the representative agent more choices here. Given Vt+N |t (At ), which was found through the value
function iteration, I can find the optimal consumption path in the preceding periods given an initial asset level at t:
E (At ) = Vt|t
N −1 X
δ i−1 v(ct+i ) + η
i=0
N −1 X
δ i−1 {[1 − (1 − λ)H (rt+i − ct+i )] [v(ct+i ) − (rt+i )]} +
(24)
i=0
E + δ N Vt+N |t (At+N +1 )
At all times the intertemporal budget constraint between t and t + N needs to be satisfied. By repeatedly inserting I get from ct = At + w − N X i=1
At+1 1+R
to: N
ct+N − w X w ct+i−1 = A − + t (1 + R)i−1 (1 + R)T (1 + R)i−1 i=1
(25)
where At and ct+N are given through the asset grid in period t and the corresponding consumption grid in period t+N found with the value function iteration. For every grid point of the consumption state space at t + N I apply the fmincon routine, pre-implemented in MATLAB, to find the utility maximizing consumption and asset path given an initial asset level, subject to the budget constraint over all N periods. The fmincon routine is a nonlinear programming solver that finds minima of constrained nonlinear multivariable functions and works fine in this context. Additionally, I supply a positivity constraint for the consumption path. For every At I now choose the utility maximizing E consumption path and thus get Vt|t (At ) over the whole asset grid.
Given the now found values of being employed in the steady state I can solve for the steady state value of being unemployed VtU,n (At ) using the fsolve rootfinding-routine in MATLAB and then calculate the search effort with equation (15). Having found the value functions and search efforts over the whole grid in the steady state, I move one period backwards while initializing a new grid of feasible asset choices in the prior period. For every point on the new asset grid I choose the
34
asset level of the coming period that maximizes the current value of being unemployed and update the asset path. At this stage the reference point has changed and thus I need to recompute the value of being employed. Having found all value functions I can move one period backwards again. I repeat all these steps until I reach the first period, repeatedly updating the asset path for all grid points selecting the utility maximizing consumption choices. In the first period I choose the initial consumption-savings decision that maximizes the utility and the asset path determined by this action is the final output of the model, defining consumption and search levels over all periods.
5.2
Calibration
To solve the model and to estimate the parameters of interest I need to establish some further assumptions. Following DellaVigna et al. (2016) I define the search cost function, that depends on the effort input, in power form such that c(st ) =
ks1+γ t 1+γ .
This way γ represents the inverse search
effort elasticity with respect to the individuals net valuation of employment. This can be seen by equation (4) which DellaVigna et al. define as υ. The elasticity of s∗ with respect to υ then becomes ηs,υ =
ds dυ
· υs = γ1 . Further, I also implement the utility as a simple log function, v(b) = ln(b) which
brings concavity of the utility. Lastly, I assume a fixed two-week discount factor δ = 0.995 just as DellaVigna et al. do it in most cases. This is briefly discussed in Paserman (2008) as a way to circumvent identification problems of the exponential discounting parameter and it is a common practice in studies that are not explicitly interested in δ. The bi-weekly pre-unemployment wage is set to the rounded median of the individuals who became unemployed in 2011 and claimed benefits, that is 11,300 DKK and the corresponding reemployment wage is 11,800 DKK. The claimed unemployment insurance benefit is 7,660 DKK and the rounded social security level for the median claimant is 4,600 DKK. In the benchmark estimations I follow DellaVigna et al., assuming that the unemployed job seeker starts the unemployment spell with zero assets from the period before, that is she would need to build up a buffer against the possible income shock when the benefits are exhausted by saving up from her benefits and not dissave from prior employment savings. Additionally, the return on savings R is set to zero to isolate the effect of saving to buffer against the income shock. 35
5.3
Estimation
As using the maximum likelihood method for the estimation in the context of this thesis is computationally not feasible, I employ the minimum-distance method, as described in Cameron and Trivedi (2005), which is also used by DellaVigna et al. (2016). Minimum-distance estimation is a mathematical technique that allows to estimate model parameters in settings were a maximum likelihood method is not practicable due to problems in obtaining the likelihood functions or difficulties in their evaluation. Clearly this is the case in the model of this thesis, just like in many other models of optimal dynamic behaviour and discrete choice. The method replaces the likelihood function with the distance between empirical and parametric moments and minimizes this objective function. The parametric moments are the search intensities for every period obtained from the model, which was descried in section 3, while the empirical moments are the hazard rates out of unemployment obtained in section 4. As the search intensity represents the objective probability of leaving unemployment in the following period, the empirical hazard rates are the corresponding moments to these under the assumption that the model embodies the true data generating process. With this assumption holding, the unemployed job seekers choose their search effort such that it maximizes equation (14). The vector of model parameters is then estimated by:
θˆ = arg min (m(θ) − m) ˆ 0 W (m(θ) − m) ˆ
(26)
θ
where θˆ is the minimum-distance estimate of the underlying parameters of the model, i.e. λ, the magnitude of the loss aversion, N , the adjustment speed of the reference point in biweekly intervals, β, the discounting parameter of the hyperbolic-discounting assumption, π, the parameter defining the bias in the beliefs about the subjective job finding probability, as well as k and γ, the parameters defining the search costs and their elasticity towards gains from entering employment. In some of the estimations I assume heterogeneity in the search costs which leads to two types of agents with population shares pl for the lower cost type kl as well as 1 − pl for the higher cost type kh . Even though I do not present it here, I generally found severe identification problems when including more cost types. This might be due to the fact that the loss aversion as well as
36
the overconfidence already explain large parts of the non-linearities in the models. The moments generated by these parameters are m(θ) and the empirical moments found in the underlying data are m. ˆ The estimator is weighted by a weighting matrix W which is recommended to be a diagonal matrix with the inverted variance of each supplied moment to decrease the influence of moments that have a high variance. I obtain the variance through formula (18). As a robustness check I employ a simple identity matrix for the weighting later. The minimum-distance estimator (26) achieves normality with variance
ˆ = (G ˆ 0 W G) ˆ −1 · (G ˆ 0 W · V ar(m) ˆ · (G ˆ 0 W G) ˆ −1 V (θ) ˆ · W G)
where G =
∂m(θ) ∂θ 0 |θˆ
(27)
(Cameron and Trivedi, 2005).
In the estimation of the parameters I supply 79 bi-weekly moments from the empirical data. As it is not guaranteed that I can assume a steady-state search effort from the last period supplied, and the restrictions of Statistics Denmark do not allow me to export more moments, I will only supply 79 model generated moments to the estimator. It is however possible that N is large enough that the steady state is achieved after the last supplied moment. Therefore, I calculate the model on the whole range until the steady state, but cut the data at period 79 in the estimation. To find the minimum of the objective function (26) I use the MultiStart class of the Global Optimization Toolbox in MATLAB, as conventional solvers tend to find local solutions and computation times turn out to be extremely large, especially with fine grids. The MultiStart class runs fmincon routines from a range of different starting values and obtains the global solution to the problem as the minimum of all local optima. Additionally, it offers parallel computing options that run the estimations with different starting values on multiple processor cores, effectively separating sub-tasks of the computations to more processor cores in order to decrease overall running time. I start the MultiStart estimation for the hand-to-mouth models with a scattered grid of 2000 starting points13 and upper bounds set to 1 for the parameters β and π as they can’t exceed this boundary by definition. Additionally, I set lower bounds equalling 0 for all estimated parameters as 13 In most cases I find the global optimum with a much smaller amount of starting points, but with 2000 points I can be relatively sure to find the right solution.
37
all are weakly positive by definition14 . For the estimations including the consumption and savings decisions for the job seekers I limit the starting points to 50 as the computation times increase immensely15 and the feasibility of larger amounts of starting points faces a trade-off with the size of the asset grid, which is set to 160 points16 . Unfortunately I can not guarantee that I will obtain the exact global solution with this procedure, but the available time for this thesis restricts the use of larger amounts of starting points. In this case I would be bound to significantly reduce the asset grid size which could lead to distortions of the outcome. I am aware of the limitations my procedure of obtaining the solutions has, but I am nevertheless confident that it can yield interesting conclusions that can set the ground for further projects.
5.4
Identification
Due to the chosen estimation strategy all parameters are identified jointly in the minimum-distance process. Just like in DellaVigna et al. (2016) and Paserman (2008) it is however possible to roughly classify the central sources of identification of the parameters. The discounting parameter β is identified by the strength of the peaks around the changes in income as well as the general level of the search efforts. The decision to save enables the more patient job seekers however to smooth the consumption around the benefit drop with less intense search effort increases. The inverse of the search cost elasticity towards utility gains of becoming employed, i.e. the search cost curvature γ, is identified by the sharpness of the search effort reactions towards changes in the income, such as the initial unemployment spell or the transition into social security at week 104. When looking at the standard model one can clearly observe that the heterogeneity in the search costs enables shrinking search efforts after the benefit exhaustion compared to the model without heterogeneity. Thus the search costs ki are identified by the non-linearities in the path of the hazard rates as well as by the overall level of the search efforts while scaling them up or down. Further, the loss aversion 14 Depending on the stability of the estimation I sometimes supply upper and lower boundaries for parameters that represent an economical reasonable interval. 15 The estimations were running for several weeks due to the high dimensionality induced by the consumption-saving decisions and I was not able to find a more efficient way of computing. 16 To give the agent more choices at the lower end of the asset space three quarters of the points are equally spaced increments from the minimum to a quarter of the maximum asset level in every period. The last quarter of grid points are equally spaced increments from the latter to the maximum asset level.
38
parameter λ is identified by the magnitude of the spike around the income shocks as well as the dynamics leading up to it. The length of this effect, i.e. the time until the search effort becomes constant after the transition into social security identifies the adjustment speed of the reference point N . Lastly, the overconfidence parameter π is identified relative differences between the initial search efforts to the later level, especially around the income shocks.
6
Results
This section presents the estimation results explained in the prior section. First I have a look at basic hand-to-mouth estimates that do not allow to accumulate savings. I present different model variations sequentially turning model extensions on after presenting a standard model in the beginning, but I am adding present bias to all model extensions17 . Following this I introduce heterogeneity in the search costs to the estimations and present the results. At the end I repeat the prior steps while allowing the unemployed individual to save up income to secure against the sharp income drop once the UI benefits are exhausted. I put more focus on the different parameter estimates in the first two subsections to give some intuition about the model behavior and focus more on the changes in the last two subsections that involve consumption-saving decisions. The parameter estimates can be found in the appendix as well as the corresponding plots of hazard rates and parametric search efforts18 . For the plots of the search efforts when allowing for saving I additionally supply plots of the path of the asset stock as well as the different value functions in the utility to go a bit into detail here. The tables report standard errors in parenthesis that are calculated using (27), where I estimate the Jacobian matrix to obtain G =
∂m(θ) ∂θ 0
|θˆ with a
finite difference approach provided in the Adaptive Robust Numerical Differentiation toolbox for MATLAB. Further, I provide the goodness of fit measure (GOF) that is simply the value of the objective function at the optimal solution. 17 Adding present bias did always either not change the model predictions or improve the fit to the empirical data. Presenting all models with and without present bias would have been to much in the scope of this thesis. 18 For the plots of the models that allow for endogenous saving I increase the grid size to 400 to smooth the plots.
39
6.1
Benchmark without Saving
First, I present the significantly simpler model estimations that assume hand-to-mouth consumption and therefore ignore the possibility to use savings as a buffer against the income drop after the UI benefit exhaustion. All results can be found in table 3 as well as figure 5. I first estimate the standard model without heterogeneity which simply fits the equations of section 3.1 to the data and thus estimates only the search cost elasticity γ as well as the cost parameter k. As expected, the primitive standard model (1) can not explain any of the patterns and dynamics of the empirical data. It is not able to get hold of the initial shrinking search effort right after the peak of the unemployment spell start. In fact, the model even shows increasing search intensity from the beginning to the point of unemployment insurance exhaustion. This strongly contradicts the patterns of the data until approximately week 70. The search intensity reaches its maximum at week 104 when the transfers drop to social assistance and from there on the steady state is reached and the search effort stays at a stable level high above the actual hazard rates. In addition to the poor data fit, I observe parameters that fall out of any justifiable range. The search cost elasticity γ is estimated as 7.38 with immense search costs and standard errors. Further, I observe a wide range of local optima that do not differ strongly from the global estimates in terms of data fit even with highly varying parameters. All this indicates that a primitive standard model without savings and heterogeneity is not able to model the search intensity of unemployed job seekers. Next, I estimate a model that adds reference dependence to the standard version (2) just as in section 3.2. Therefore, I set η to 1 and add the loss aversion parameter λ and the duration until the reference point has adapted N to the estimation. Further, I include the present bias parameter β between the current and all future periods, just as presented in section 3.3. The data fit shows strong improvements for all patterns of the empirical data. The initial dynamics are captured reasonably well and also the increasing search effort from week 70 until the unemployment insurance benefit exhaustion is present while the spike at week 104 is almost as distinctive as in the empirical data. After the transfer drop I however observe a too strong and quick drop in the search efforts that even fall marginally close to 0 from week 150 onward. This way the model clearly
40
undershoots the dynamics after entering the social security. This fact can possibly be attributed to the weighting matrix W which emphasizes the initial dynamics much stronger, due to higher identification of these moments. As very few people are still left in unemployment after the transfer drop, the variance of the empirical hazard rate increases strongly from that point and thus the influence of the difference between empirical and parametric moments shrinks in the estimation process. I will present estimations with the identity matrix weighting in the robustness checks later to check if they lead to improvements. The parameter estimates of this model versions fall into more reasonable ranges, but some are still relatively high. I find a search cost curvature γ of 0.14 that implies more realistic reactions to income changes relative to the prior model as well as search costs k at 1,117. Further, I find a reference dependence duration N of 43.07, equalling 527 days approximately as well as extremely high loss aversion λ = 37.66. This clearly falls out of usual literature estimates (Tversky and Kahneman, 1991, 1992). I expect the loss aversion λ to be this high to capture the initial dynamics after the unemployment spell as they show a strong decrease in search efforts with a low variance due to a high population still in unemployment. This comes at the cost of undershooting the search intensity after the drop in transfers, induced by the over-pronounced λ. Additionally, I find the hyperbolic discounting parameter β to be 1 and thus leading no present bias. Next, I turn to the model including biased beliefs (3) as introduced in section 3.4, but without reference dependence. The parametric estimates of the search effort show a really close fit to the initial dynamics of the empirical hazard rate data, nevertheless they clearly fail to capture the distinctive peak at week 104 as well as the following patterns. Just like in the standard model, the parametric moments reach their steady state at the time of the benefit exhaustion and thus overshoot the actual search effort. It is however to note that this is still an improvement to the heavy drop of the reference dependence model according to the goodness of fit measure which on the other hand depends as well on the weighting matrix W . The parameter estimates show a higher cost curvature with γ = 0.26 and a search cost parameter k = 194. The parameter inducing the biased belief π is estimated at 0.92 and thus turns out to be quite strong. This implies that the bias is still 0.13 after one year, but it basically vanished after 2
41
years at the benefit drop with 0.01 and therefore the subjective perceptions about the job finding probability almost adapted to the true probability at this time. Again, I do not find present bias in this model with β = 1. Finally, I estimate the full model (4) as shown in section 3.4. The fit of the model is strong, especially for the first time after the benefits drop. Compared to the prior models I am able to catch the initial high hazards and the following dynamics with sharp drop much more closely. It is to mention that the spike is not captured as closely as in the simple reference dependent model. Turning to the parameter estimates, I find a reference dependence parameter λ = 3.79 which is really close to usual literature estimates as well as a duration N = 10.26 and thus approximately 144 days. I still find a pretty low cost curvature γ = 0.15 as well as a cost parameter of k = 143. Further, I estimate a present bias parameter β = 0.87 which is slightly to high to be in line with Laibson (1997), who finds β to be somewhere between 0.51 and 0.82, but however at the upper end of Paserman (2008), who estimates it to be between 0.40 and 0.89 based on the hyperbolic discounting job-search model by DellaVigna and Paserman (2005). Paserman however found on NSLY data that the degree of present bias decreases with increasing wage levels. The generally high wage structure in Denmark therefore might be an explanation of β being relatively high. I find an overconfidence parameter of π = 0.95 implying a strong bias in the perception of the job finding probability that diminishes very slow over time, i.e. the bias is still 0.26 after one year. At the time of the benefit exhaustion it shrunk to 0.08 and the representative agent almost adapted the true job finding probability in his perceptions. The confidence intervals around the parameters of interest are relatively tight in this specification, unlike in the reference dependence model.
Heterogeneity In this part I present results that add unobserved heterogeneity in the search costs to the model. This way I enable a possible shrinking search effort after the benefit exhaustion in week 104 even for variants like the standard or overconfidence model that arrive in a steady state at this point with homogeneous search costs. Table 4 reports the results of the estimation and the corresponding plots can be found in figure 6.
42
Even though the standard model (5) in this setting still performs relatively bad, I observe big improvements compared to the variant without unobserved heterogeneity. The parametric search efforts now show a shrinking dynamic after the unemployment spell begins, while the spike at week 104 is however not present. The curvature is estimated at a high γ = 5.46, while I estimate the population share of the low cost type pl at 0.36. I find search cost parameters with kl = 40, 389 and an extremely high kh . Turning to the model including reference dependence (6), I observe strong improvements in the model fit compared to the version without unobserved heterogeneity. The initial patterns right after the unemployment spell are captured well and also the peak shows a strong fit. The loss aversion λ = 1.64 shows a more reasonable estimate while the duration slightly shrinks to N = 15.36, 215 days respectively. The two cost types are found with pl = 0.77 and are associated with cost parameters kl = 49 and kh = 186, as well as γ = 0.08 that is lower than in all other models. While almost all parameters fell into reasonable ranges, I estimate the present bias β at 0.98 which is relatively high. The standard errors at the parameters of interest are much smaller than in the homogeneous search cost version and thus possibly indicate a better identification. When just adding the biased beliefs and present bias to the standard model (7), I again observe a worse data fit than in the reference dependence model with present bias. It however shows improvements to the same model without heterogeneity as the search efforts decline after the transition into social security while I even observe a slight spike at week 104. I estimate a cost curvature γ = 0.37 that is higher than most estimates before, excluding the standard model. The bias paramter π is estimated at 0.86. In this specification I observe two types with pl = 0.88 and cost parameters kl = 42 and kh = 454. I find a present bias paramter β = 0.70 that falls into the ranges of Laibson (1997) and Paserman (2008). The full model (8) shows interesting patterns concerning the reference dependence. The adjustment speed of the reference point shrinks to N = 1.61, 23 days respectively. This is a very short duration in comparison to prior estimates, but the heterogeneity-induced non-linearity of the model, combined with the overconfidence, captures most of the model dynamics and patterns. This however leads to a really interesting behavior after the unemployment insurance benefit exhaus-
43
tion with a much steeper drop in the search efforts, further induced by a slightly high λ = 5.57. This matches the empirical data quite well and the fit on the following and preceding data is also relatively strong. The present bias parameter β = 0.63 shows the lowest estimate in all model variants so far and fits rather good into the literature estimates of Laibson (1997) and Paserman (2008). Again, the initial data patterns are matched strongly due to the overconfidence π = 0.88 and the cost curvature is higher than in most of the single type models, but I nevertheless obtain a comparably high standard error on the cost parameters. Concluding on this section, I can say that only the full model (4 + 8) as well as the reference dependence model including unobserved heterogeneity (6) achieve acceptable model fit. Nevertheless, I observed that the overconfidence model with heterogeneous job seekers (7) captures most of the data dynamics very good as well, while only the benefit exhaustion peak at week 104 is not really distinct. It is however to note that the reference dependence model estimates the discounting parameter β at unusually high rate according to prior literature. It appears that the full model, which also shows a slight improvement in terms of the model fit19 , estimates the parameters at a more reasonable level. While the estimate for the present bias β falls into a range that was found before by Paserman (2008), I find a very short reference dependence adjustment speed in the 2-type version that induces short influence of the loss aversion. Most of the non-linearities are captured by the overconfidence and heterogeneity among workers.
6.2
Benchmark with Saving
After having presented the results using models that assume simple hand-to-mouth consumption, I now allow endogenous saving that enables the possibility of smoothing the consumption around the points of the benefit exhaustion. While this will likely lead to a less sharp spike in the search efforts at week 108, it allows for a more realistic behavior. Even though I found a substantial degree of present bias in the full model, individuals should have a natural interest in transferring parts of their income prior to the benefit shock to future periods to feel less ”pain” induced by the fall in 19 Note that this can also be induced by the increased degrees of freedom when including the additional bias parameter. Naturally, adding a new parameter to a model increases its flexibility and thus might be able to come closer to the real data even at unrealistic estimates.
44
income. This is especially magnified due to the loss aversion of the reference dependence. The hand-to-mouth models in table 3 and 4 showed a particularly strong data fit for the full model (4), even in the absence of heterogeneous agents. In this section I present the same models with just one type of agents while now allowing for the before-mentioned endogenous saving. The results can be found in table 5 and figure 7. Figure 8 plots the value of being employed and the value of being unemployed in all respective settings, while figure 9 displays the path of the asset stock of the unemployed job seeker. Again, I observe a bad data fit for the standard model (9). The search efforts are increasing slightly from the beginning of the unemployment spell to the point of the benefit exhaustion which is in stark contrast to the actual empirical data and the search efforts stay constant from week 104 onwards, just like before. On the parameter side, I observe really high search costs combined with a high search cost curvature γ. Allowing for endogenous saving results in a strong asset accumulation from week 90 until the point of benefit exhaustion. From this point onward the job seeker depletes the savings until week 120 to keep the consumption on a higher level and thus she enables a smoother transition into social security. The asset stock is essentially zero at all other times. Turning to the reference dependence model (10), I again observe a slight over- and undershooting of the empirical hazard rates, especially between weeks 30 to 80 as well as after the benefit exhaustion peak. The spike at this point is however, just as expected, much less sharp and relatively smooth and thus with a worse fit than in the hand-to-mouth model. Like in the model without saving (3) I observe an extremely high loss aversion λ = 41.18 that is hardly justifiable by the underlying theory. Further, I observe the same lack of present bias due to β = 1 and a search cost curvature γ that differs just slightly. In comparison with the standard model (9) the agent saves much less of the income to buffer against the income shock and also just saves around this point in time. At the highest point of asset accumulation the agent roughly saves half as much as in the standard model. Due to the reference dependence, there are also larger impacts on the shape of the value of employment and unemployment over time. While the standard model shows a mostly constant value of employment
45
(with an exception around the benefit exhaustion), the reference dependence model has an increasing valuation of both the value of being employed as well as the value of being unemployed. This is induced by the adjusting reference point, i.e. as long as the reference point decreases the agent feels less and less losses each period and thus the valuation of unemployment as well as employment increases until the reference point is adjusted. The value of unemployment shows a strong decrease around the benefit exhaustion due to the transfer of income into the future and the anticipation of the income shock, but nevertheless increases afterwards due to before-mentioned reasons. The overconfidence model (11) shows almost identical search efforts than before with a slightly smoother pattern around week 104. Again, it is not able to capture the search efforts after the transition into social security due to steady state search efforts from the point all assets are depleted. The patterns of the asset accumulation are roughly similar to the standard model (9) with a slightly higher stock at the exhaustion of the unemployment insurance benefits. Also the paths of the values of being unemployed and employed follow the same pattern, with an exception of a concave shape of the former, induced by a much lower search cost curvature γ = 0.24. The rest of the parameters shows roughly the same magnitude, again with no present bias β = 1. Further turning to the full model (12), I see a small decrease in the model fit around the benefit exhaustion peak with a less sharp and generally smooth spike, but the model still performs well and better than other models with endogenous saving and homogeneous agents. It is also the only model variant in this setting that is able to properly capture the data patterns after the unemployed transition into social security. The parameter estimates just differ slightly from the ones where I didn’t allow for saving (4), with an exception of the overconfidence parameter π that is estimated at the same point. Interestingly, the present bias parameter β increases to 0.93 and thus falls out of the literature estimates of Laibson (1997) and Paserman (2008). Just as in the reference dependence model (3), I only observe a short period of asset accumulation to buffer against the income drop that stays on roughly the same level as in the latter model. Due to the faster adjustment induced by a lower N I observe a constant value of employment once the reference point reached the unemployment benefit level, but the general pattern is similar to the value of employment in the reference dependence model. The value of being unemployed shows a
46
different pattern however. Just from the beginning of the unemployment spell it decreases at an almost constant rate until week 104, with a slight exception of a lower decrease while the reference point is not yet adjusted. This is induced by the overconfidence and therefore higher valuation of the future20 . From the benefit exhaustion until the reference point reached the point of the social security transfers, I observe an increasing value of being unemployed, just as in the reference dependence model. Afterwards the valuation reaches a constant steady state rate. Heterogeneity Following the practice as in the hand-to-mouth estimations, I now allow for heterogeneity in the search costs, namely again two different types. As the prior estimations showed good improvements even for models that did not add reference dependence in the specifications, I expect a significantly better model fit than in the preceding section with homogeneous search costs. The results can be found in table 6 and figure 10. Figures 11 and 12 display the paths of the value functions as well as the asset stock. Again, I observe strong increases in the model fit for the standard model (13). It captures the data patterns well until roughly week 80 when the hazard rates increase in anticipation of the benefit exhaustion. The steady state search effort is reached slightly later than in the hand-to-mouth model induced by saving efforts for the same consumption smoothing goals as in the preceding section. The saving itself is however slightly higher than before, but exhibits roughly the same patterns. Due to the much lower search cost curvature γ = 1.52 I now observe a convex path of the value of being unemployed while the value of being employed shows no notable differences. For the high cost type I obtain a very high standard error on the search costs kh , possibly induced by the still bad data fit after the benefit exhaustion where the search efforts are constant and higher than the empirical hazard rates. The reference dependence model (14) provides clear improvements after the benefit exhaustion in comparison to the standard model and the specification without heterogeneity (10), but has problems in capturing the peak at week 104. I observe a much too smooth behavior around this 20 The agent thinks she will be more likely to get a job than she actually is and thus the expected (yet biased) value of the future in the current value of unemployment is higher than without overconfidence.
47
time, possibly induced by increases in the total amount of saved assets. The initial patterns after the unemployment spell start display a very good data fit however. Allowing for heterogeneity changes the path of the value of being unemployed right at the beginning of unemployment, as the value is now slightly decreasing for the first few weeks. Other patterns remain roughly the same, also for the value of being employed. Just as the general behavior when allowing for heterogeneity in the search costs showed before, I observe a higher search cost curvature and still a much too high λ. Interestingly, I observe a model fit for the overconfidence model (15) that is practically as good as the reference dependence specification (14) and all data dynamics are captured well with the exception of a too small spike around week 104. Similarly to the hand-to-mouth model (7), I get decreasing search efforts from after this period which comes much closer to the data than without allowing for heterogeneity. The value of being unemployed in this specification is almost decreasing linear, slightly different than before. I again observe the highest asset accumulation in this variant that already starts at week 66 and thus the agent is very much forward-looking. I also observe present bias in this version with β = 0.90 as well as the same overconfidence bias as before with π = 0.82. Lastly, I turn to the full model (16) that, as expected, shows the best data fit in this section, but also lacks to display the distinctively sharp spike at the transition into social security transfers. Nevertheless, all other data patterns are captured extremely close. The loss aversion λ = 4.63 as well as the present bias β = 0.7 fall into reasonable ranges that are very well justifiable according to the underlying theory. While the saving of the agent increases in this specification, it is still much lower than in the standard (13) or overconfidence model (15) and more comparable to the pattern in the reference dependence specification (16). The most notable change in the path of the value of being unemployed is a change to a concave pattern rather than the almost linear decrease up until week 104 before. This section showed that allowing for endogenous saving, even though it is more realistic, leads to a less strong data fit to the empirical data among all model specifications. Again I saw the best fit in the full model version with and without heterogeneity (12 + 16) and the reference dependence
48
(14) as well as the overconfidence specification (15), both with heterogeneous search costs, achieved an acceptable fit that almost reaches the behaviour of the full model (16). I nevertheless observe much higher standard errors overall in this section than in the one where saving was not included and generally the parameter estimates become slightly less reasonable (with an exception for the full model).
7
Robustness and Consistency Checks
In the following subsections I consider the robustness of the main models to alternative variations of model specifications as well as consistency checks with the pre-reform data. While some of the used model specifications appear reasonable concerning the fit to the empirical data, it is left to check if it could lead to improvements when changing some of the assumptions. Due to time restrictions in this thesis and the extreme computation lengths of the model estimations that include endogenous saving, I only report model variations with hand-to-mouth consumers, ruling out consumptionsavings decisions. Note that the conclusions do not necessarily need to carry trough to the savings models and therefore only lead to suggestive results.
7.1
Pre-Reform Regime
As shown before, I found a strong fit to the empirical data when using the full specified model with and without heterogeneity heterogeneity as well as the reference dependence and overconfidence models with heterogeneity. The structure of the available data and the unemployment insurance reform of 2010, explained in the earlier sections, makes it especially interesting to check whether the model estimates lead to consistent model behaviour, i.e. to check whether the estimated search efforts would also lead to a reasonable model fit for the pre-reform data. If this would lead to search efforts that are far off from the empirical data I could conclude that there surely must be something else going on in the behaviour of the job seekers that I did not capture in my model variations. First, I have a look at the overconfidence model with heterogeneity (7) which however was not able to sufficiently capture the dynamics of the benefit exhaustion. The results of the application
49
to the pre-reform data can be found in figure 13 in the appendix. The model does not exhibit a strong fit to the empirical hazard rates and is again not able to capture the peak at the benefit exhaustion which here happens at week 208, i.e. after 4 years. While there is arguably some degree of fit to the initial hazard rates after the begin of the unemployment spell, this is not strong enough to consider this model fitting properly to the data. When allowing for endogenous saving, which can be found in figure 14, the patterns of the model (15) do not change notably. Turning to the reference dependence model with heterogeneity (6) which performed strong on the post-reform data, I find a strikingly good data fit. The model is able to capture the initial shrinking hazard rates well and also comes close to the following dynamics. Additionally, I observe that the patterns leading up to the benefit exhaustion peak are captured in a satisfactory manner. It is however to note that, even though I do not observe the data further, the parametric and empirical hazard rates seem to drift apart after the unemployment insurance benefits are exhausted. I expect that I would observe a bad data fit from this point if more data would be at hand. When allowing for asset accumulation (14) I observe a worse fit to the pre-reform data. The search-efforts practically overshoot the hazard rates from week 150 onward, induced by saving efforts and therefore smoothed consumption paths. This also leads to a spike that is much too flat in comparison with the empirical data. Further looking at the simple full model (4) which performed quite good before, I find pre-reform search efforts that capture the patterns of the data very well. While there is a slight undershooting right at the unemployment spell start, I get a fit as close as with reference dependence solely (6) and the shrinking search effort after week 208 even suggests a better fit after the benefit exhaustion if I would observe more data. The model with endogenous saving (12) exhibits a slightly better data fit in the initial patterns, but somewhat overshoots the empirical data after week 180, also leading to a too smooth spike. This is nevertheless better than in the reference dependence model (14) and the trend, albeit no further observable data, points in the right direction. The estimations showed that including heterogeneity to the full model (8) increases the data fit further. Nevertheless this does not carry trough to the pre-reform data. The data fit is highly comparable to the overconfidence model (7), obviously induced by the quick adjustment of the
50
reference point. Further, the peak at the benefit exhaustion is basically non-existent and thus the full model with heterogeneity does very poor when taking the pre-reform data into consideration. When allowing for endogenous saving (16) however, I observe strong increases in the model behavior especially along the path until the transition into social security. The peak at this point itself is captured relatively bad albeit coming much closer to the data than in the hand-to-mouth version (8). I conclude on this section that the full model without heterogeneity (4) as well as the reference dependence model with heterogeneity (6) are the only hand-to-mouth variants that show consistent behaviour between pre- and post-reform data. It becomes striking that the complete full model with heterogeneity (8) lacks a way to capture the benefit exhaustion peak. While all models in the endogenous saving case showed flaws I again observed the best data fit in the full model without heterogeneity (12). I am nevertheless aware that there also might be other things going on between the years 2008 and 2011 that alter the search efforts and thus could lead to differences between the parametric and the empirical data.
7.2
Identity Matrix Weighting
In this subsection I re-estimate the models using another weighting matrix in the objective function (26), namely the identity matrix W = IT . As the optimal weighting, using the diagonal of the inverse of the data variance-covariance matrix, puts more weight on better identified moments I ran into small problems with some of the models estimated earlier. This was mostly due to very low influence of the dynamics after the benefit exhaustion peak as the population in unemployment became relatively small at this point. With the new estimation method used in this section I put equal weights on all moments over the whole time frame available and thus hope to find a better fit at the problematic points. Generally, I do not observe improvements of the overall model fit as well as the goodness of fit (which makes sense as the goodness of fit is measured on the optimal W ) as it can be seen in figures 15 and 16. Especially for the reference dependence (22) and the full model with and without heterogeneity (20 + 24) as well as for the overconfidence model with heterogeneity (23) I observe 51
no noteworthy changes in the model behavior. The overconfidence model (19) however shows some changes after the point of the benefit exhaustion. Instead of observing completely flat search efforts from that point onward I now observe decreasing search intensity where also the curvature changes to a concave shape contrary to all prior estimations. This change leads to a more realistic behaviour of the model, even though the fit over the whole range decreases. Interestingly, the identity matrix weighting slightly affects the simple reference dependence model (18). The search efforts tend to marginally reach zero a bit slower while the initial peak after the unemployment spell is somewhat underestimated compared to the good fit of the benchmark model, but the general dynamics do not change. Looking at the parameter estimates in tables 7 and 8, I observe an increase of λ for the reference dependence model (18) as well as an increase of N for the full model (25). Generally, I do not observe further notable changes, except for increased standard errors for some of the parameters. It is however to note that the reference dependent model with heterogeneity (22) achieves the best fit on the data when calculating the GOF measure of the benchmark section.
7.3
AR Reference Dependence
DellaVigna et al. additionally introduce an alternative reference point specification defined as an AR(1) process: rt = ρrt−1 + (1 − ρ)bt = (1 − ρ)
∞ X
ρi bt−i
(28)
i=1
By doing this, the added reference dependence becomes smoother around the times of the adjustment compared to equation (7). The AR parameter ρ now captures the speed of the adjustment and by restricting it to ρ ∈ [0, 1) the reference dependence is guaranteed to disappear asymptotically and not as abrupt as in the standard formulation after N periods. The dynamics of the AR(1) reference point specification in comparison with the benchmark version, i.e. an income average over N periods can be seen in the following figure with ρ = 0.9 and N = 40 as well was w = 6500 and b = 4450.
52
Figure 4: Comparison of the two different reference point specifications
It is obvious that the AR(1) specification shows a much smoother behaviour around the time time of the reference point adjustment in the benchmark model. Table 9 and figure 17 now show the estimation results using this specification in an 3-type heterogeneity model. The model fit for the reference dependence version with heterogeneity (26) as well as both full model versions (27+28) is strong and comparable for all three specifications. It captures all distinct dynamics of the empirical hazard rates very well, even though I also do not obtain the benefit exhaustion peak as strong as in the data. Looking at the reference dependence model without heterogeneous agents (25), I observe a bad data fit that overshoot the hazard rates after week 75 strongly. The auto-correlation parameter ρ is estimated at 0.94 for the reference dependence specification with heterogeneity as well as 0.91 and 0.92 for the full model versions. To compare it with the benchmark reference dependence duration I can estimate the ”half-life” of the AR(1) reference dependence with (Choi et al., 2004): H(ρ) =
ln(0.5) ln(ρ)
The half-life here indicates the time until the influence of the income in a given period reduced to the half of its initial value and thus the half life in the benchmark is reached after
N 2.
This
happens after approximately 108, 72 as well as 11 days (for models 6, 4 and 8). The half-life of the reference dependence in the AR(1) model variant is however circa 125, 122 and 160 days (for models 26, 27 and 28) and therefore generally longer then in the benchmark models. The overconfidence
53
parameter π is increasing in comparison to the benchmark model estimates, while β is estimates at 1 for all three variations. Perhaps more interesting is the increase of the loss aversion parameter λ for all variations. This becomes however quite high in comparison to the usual literature estimates of the parameter. Further, I observe relatively high standard errors for the loss aversion parameter for the full model with heterogeneity. This casts doubt on the proper identification of the model specification with an AR(1) reference point.
7.4
Constant Belief Bias
Spinnewijn (2015) argues in his paper that learning most likely does not exist among overconfident unemployed job seekers in the United States. Even though I found light evidence that Danish agents might learn about their actual probability of finding a job over the curse of the unemployment spell, I nevertheless estimate a ”constant” overconfidence model in which the job seekers do not update their perceived job finding probability. It then becomes:
sˆt (st , At+1 ) = min {st (At+1 ) · π ¯ , 1}
(29)
and thus the overconfidence effect does not change over time. Table 9 and figure 18 present the estimation results of this model (29), again also employing the 2-type heterogeneity model (30). The specifications perform slightly worse than the benchmark setting, mainly due to the less sharp search effort increase in anticipation of the transition into social security. I also observe a slight undershooting after the benefit drop, which is however still acceptable in the model that allows for heterogeneity. The model with homogeneous search costs undershoots the patterns more severe. The hyperbolic discounting parameter β is relatively low in both specifications, but still in line with Laibson (1997) and Paserman (2008). The loss aversion λ is quite far apart in both models and shows high standard errors that cast doubt on the identification. I also find strong differences in the adjustment speed. While π ¯ was not restricted to be on the unit interval I find it almost exactly at 1.00 in the 2-type version while it falls just slightly lower in the basic full model. Concluding, I can say that even though a constant overconfidence increases the model fit of the
54
reference dependence model (2) substantially, it cannot reach the accuracy of the full model with overconfidence that vanishes over time (4).
7.5
Alternative Samples
In this subsection I re-estimate the full model, which performed strong in the benchmark estimations, using different samples of the unemployed. As the re-employment wage of an employed individual normally depends on skills, it makes intuitive sense to compare sub-samples of individuals divided by their income level. An individual with a relatively high wage might be more skilled than an individual with a lower wage and thus possibly leave the unemployment with higher speed. There might also be differences between several parameters among both samples as I looked at the median unemployed in the benchmark analysis. I divide the population into two groups, i.e. the group below the 33rd percentile as the low income group as well as the group above the 66th percentile as the high income group. Interestingly, I do not find big differences between both groups in terms of the dynamics of the hazard rates which can be seen in figure 19. There is however different behaviour around the benefit exhaustion peak. While the highest hazard rate in the high income sample appears at week 104, I find the peak maximum for the low income sample at week 106 and thus one period later. I also find the initial hazard rates and the peak itself slightly higher in the high income sample. The estimated search efforts in both samples show an acceptable model fit, while the model on the low income sample shows a much sharper and quicker income drop after the unemployment insurance benefits are exhausted. Looking at the parameter estimates, presented in table 9, I find clear differences that lead to the prior found disparity in the estimated model dynamics. Interestingly, I find a rather low π = 0.67 and a long adjustment speed N = 31.18 on the higher income population (32) in contrast π = 0.91 and λ = 1.24 on the low income sample (31). Rather surprisingly I find a much higher search cost curvature for the high income individuals with γ = 1.09. This implies less sharp reactions around income changes and leads to the relatively flat search efforts after the benefit exhaustion. These characteristics carry trough to the model when allowing for heterogeneity, albeit showing a 55
different magnitude. Perhaps most interesting is a present bias parameter β = 0.34 for the high income population as well as relatively low overconfidence. While the found differences on both samples are rather interesting, I need to question the results due to the lack of sufficient data after the benefit exhaustion peak. I suspect that the results would differ if I could observe a longer span of the data and thus use more moments in the estimation. Therefore, I restrict myself from further interpreting these results, while I still highlight the possible differences especially concerning the peak at the transition into social security. I also observe high standard errors, especially for the loss aversion λ as well as for the present bias parameter β, possibly induced by the relatively few moments that could be supplied in the estimations.
7.6
Reservation Wages
DellaVigna et al. (2016) note the relatively small impact of reservation wages on the dynamics of search efforts in line with other research (Card et al. (2007), Krueger and Mueller (2014), Schmieder et al. (2015)). I nevertheless follow their approach in still estimating a model where the agents accept or decline obtained wage offers to give intuition about possible implications in this setting, especially with the added overconfidence. Further, Koenig et al. (2016) note the importance of backwardlooking reference points to replicate theoretical predictions of a canonical job search model. As a full structural model with saving and consumption decisions was not feasible on time I focus on the computationally simpler model with hand-to-mouth consumption. This specification achieved strong fit in the benchmark estimations as well as a reference dependence variant. The set-up of the model is basically the same as in the benchmark case, but now the agents draw possible wage offers wt+1 from a stationary log-normal distribution with characteristics close to the actual data and need to decide weather to accept an offer or to search further in hope to obtain a better wage. While Flinn and Heckman (1982) note the importance of the functional form of the wage offer distribution, I settle with stationary log-normal distribution as the results here are only meant to be suggestive. The value of being unemployed can then be formulated as:
56
VtU,n
Z = u(yt |rt ) − c(st ) + βδ sˆt (st )
∞ E U max{Vt+1|t+1 (wt+1 ), Vt+1 }dwt+1
+ (1 −
U sˆt (st ))Vt+1
(30)
0
For simplicity I follow DellaVigna et al. and get rid of any gain or loss utility once a job is found. It is straight-forward to show21 that the agent forms an reservation wage equal to:
u
∗ wres,t+1 = eVt+1 (1−δ)
(31)
∗ and thus only accepts wage offers wt+1 ≥ wres,t+1 .
For the estimation I supply additional moments, namely the average log-re-employment wage at every point of exit from the pool of unemployed. Unfortunately the coverage of the wage income data is lacking for some individuals and thus I am not able to export and further use a data scope as broad as before and need to settle with a few less moments. The results of this estimation can be seen in figure 20 and 21 as well as table 10 and are performed for the reference dependence (35 + 37) as well as the full model (36 + 38) with and without heterogeneous agents. Generally, the model fit for the hazard rates becomes worse, especially the hazard rates after the peak of the benefit exhaustion appear to be captured really weak in both variations. Interestingly, I find a really strong fit for the re-employment wages in the full model without heterogeneity (36). Up until week 104 the model captures all dynamics almost perfectly on both the wages and the hazard rates. After this point I am subject to really volatile re-employment wages due to the smaller pool of people still in unemployment, but the parametric re-employment wages still fit quite good over the average of these moments. Both reference dependence models fail to capture the patterns on the re-employment wages and are generally far off from the empirical data. Especially the initial re-employment wages are increasing while the data shows decreases. This is due to the fact that an unemployed job seeker accepts lower wages to limit the negative impact of the reference dependence until the reference 21 The
E reservation wage is characterised by Vt+1|t+1 (wres,t+1 ) = rearranged (remember simple log-utility) to the presented solution.
57
v(wres,t+1 ) 1−δ
U = Vt+1 which just needs to be
point slowly adjusts to the benefit level. The overconfidence in the simple full model (36) seems to work against this and thus provides much more realistic parametric moments that come at the cost of a loss aversion parameter λ that is practically equal to the fixed η. This implies that losses are weighted the same as gains by the job seeker which impacts the search effort behaviour after the benefit exhaustion as I observe almost flat search efforts. For the full model with heterogeneity (38) I observe a high loss aversion λ combined with a very quick adjustment of the reference point. This induces a closer data fit for the hazard rates with a really steep drop after the benefit exhaustion. On the re-employment wages this shows up as a quick and steep increase in the wages while the prior data fit is still very strong. In both models that include heterogeneity it is however striking that the population share of the low type is estimated at 0.99 and thus practically no heterogeneity exists. This becomes even more obvious when taking the extreme standard errors on the search costs for the high type kh into account. In conclusion, even though I found strikingly good fit to the log wages in the full models and even a good hazard rate fit for the full model with heterogeneity, I perceive the high standard errors and almost non-existent high cost type as problematic, possibly hinting to some identification problems of the model including reservation wages.
8 8.1
Implications Unemployment Insurance Policies
Following Paserman (2008), I can use the earlier estimated models to actually evaluate hypothetical changes in the unemployment insurance system as the search effort was linked to the hazard rates out of unemployment. This becomes especially interesting when considering the reference dependence and a hypothetical multi-step unemployment insurance system under which the actual paid out benefits drop more often than just after 104 weeks when the unemployed job seeker enters social security. I can analyse the job seekers welfare, the government expenditure as well as the speed of reemployment induced by different policies and thus I am able to give simple policy recommendations based on the results. As Paserman points out, one needs to be aware that the presented model is 58
not a general equilibrium model. The models lack for example the government side that finances the unemployment insurance transfers through taxes. Policy analysis on partial equlibria, just like in the case of this thesis, therefore needs to be viewed with caution. There are three main policies of interest that I am evaluating. The first one simply cuts the benefit level b over the whole range of unemployment insurance entitlement by a certain amount so that the unemployed job seekers receive less benefits in every period. Similar policies that involve cuts or increases of the benefit level have been studied widely (Carling et al. (2001), Eugster (2015), Lalive et al. (2006)) and generally point to positive effects of benefit cuts on the unemployment duration, i.e. a lower (higher) benefit level decreases (increases) the unemployment duration. Second I evaluate a reduction of the unemployment insurance entitlement length, i.e. a cut of the duration the benefits b are paid to the unemployed job seeker before she transitions into social security. The shortening and extension of benefit entitlement was also subject to prior examination (Van Ours and Vodopivec (2006), Card and Levine (2000)) and the evidence leads to roughly same conclusions as benefit cuts. Third, I evaluate a relatively new idea of re-designing the unemployment insurance system. Just like DellaVigna et al. (2016) propose, consider now an unemployment insurance system that pays benefits b1 for T1 periods until the unemployed individuals transition to benefits b2 from period T1 until T . From T onwards the individuals who are still unemployed are subject to social security transfers b. Clearly the danish system can be represented by b1 = b2 = b and T equal to 2 years, respectively 4 years for the pre-reform regime. Assuming now that a hypothetical change from a one-step to a two-step system (”front-loading”) would occur in the unemployment insurance. This would change the system from b1 = b2 to b1 > b2 in a way such that the total amount of benefits payed for an unemployed job seeker within T periods is exactly the same as in the current setting, i.e.:
b1 T1 + b2 (T − T1 ) = bT
(32)
The new step after T1 here could possibly exploit the reference dependence of the job seekers to
59
induce higher search efforts around the new benefit drop due to the same dynamics present in the empirical data when transitioning into social security. The conclusion will however highly depend on the magnitude of loss aversion λ as well as the length until the reference point adapts N .
8.2
Evaluation
For the analysis I use the benchmark model estimations that generated the closest fit to the actual post-reform data as well as to the pre-reform hazard rates to make sure it is sufficiently representing the actual behaviour on the job market. I do not make use of the model versions that I examined in the robustness checks and model variations in section 7 as they either did not significantly improve the data fit and increase the computational intensity. Additionally, a fairly big amount of the alternative models exhibited less reasonable parameter estimates as well as high standard errors, casting doubt on their proper identification. The here analyzed models include the full model with and without heterogeneity as well as the reference dependence and overconfidence models with heterogeneity in the search costs. For all models I am going to analyze the hand-to-mouth variant and the version that allows for endogenous saving. Due to the dynamic inconsistency induced by the present bias, a definition of the job seekers welfare is hard to choose as plans and actions differ over the time paths. I follow Paserman (2008) here who evaluates the utility of the long-run selves of the job seekers, just like in O’Donoghue and Rabin (2001). This means that I evaluate the decision paths of the present biased agent using normal exponential discounting. Further, I evaluate the mean expected duration in unemployment under all policy regimes. The mean expected duration statistic is also called the expectation of life and defined as the integral of the Kaplan and Meier (1958) estimate: Z µ ˆ= 0
∞
ˆ S(t)dt =
m X ˆ (i) ) (t(i) − t(i−1) )S(t
(33)
i=1
Lastly, I present the government expenditure which I evaluate after four years. To calculate it, I project the parametric survival rates for four years (at which point almost all unemployed would have left into employment) and multiply the respective point estimate of the survival rate by the
60
monetary transfer for every period and sum it afterwards. I present several hypothetical policy changes that might alter the search efforts and thus the speed of reemployment with possible changes in the government expenditure as explained earlier. First, I evaluate the current system to have a benchmark for further analysis and comparisons and then consider a possible cut of the benefit level b by either 10, 20 or 30%. Following this I evaluate a cut of the entitlement length by also 10, 20 or 30%. Next I will turn to the front-loading of the benefit path by introducing one step after exactly one year. This will be done by increasing b1 by either 10, 20 or 30% while b2 is cut in order to satisfy equation (32). Table 11 shows the effects of cutting the benefit level b by either 10, 20 or 30% in comparison to the benchmark policy. Figures 22 and 23 plot the policy changes of 10% as well as 30% benefit cuts against the initial search effort paths. Generally, the policy changes lead to higher initial search efforts that increase with the intensity of reducing the benefits. Especially the reference dependence model without saving seems to be very reactive to the benefit cuts right at the beginning of the unemployment spell start. The policy further induces a stronger decline in the search efforts that eventually lead to a smaller peak at the point of the benefit exhaustion. The magnitude of the difference between the benchmark spike and the one under the different benefit cut regimes depends highly on the model specification as differences in the full model versions as well as in the overconfidence specification are rather small for the 10% cut, but already lead to larger disparities in the reference dependence variants. The 30% cut results in almost flat patterns at the point of transition into social security. Due to the increased search efforts right at the start of the unemployment spell I observe strong reductions of the expected unemployment spell length for the average unemployed job seeker as well as corresponding reductions in the expected government expenditure. Obviously, the magnitude of the spell length reduction as well as the reduction of the expenditure depends on the intensity of the benefit cut and generally increases it, with an exception of the reference dependence model with saving. While the improvements in duration and expenditure seem favourable for the government side, they lead to reduction of the perceived utility for the job seeking unemployed. Now I turn to the effects of decreasing the benefit entitlement length as presented in 12. The
61
changes of a decrease in the duration of benefit transfers are plotted in 24 and 25 for a 10% as well as 30% entitlement length decrease. The policy changes move the spike to earlier periods and additionally lead to increases over the whole range until the individuals move into social security transfers. All of the three policies show decreases of the expected spell lengths for the average unemployed job seeker that amount to roughly a third to a quarter of the reductions in the benefit cut policy. The effects on the government expenditure are rather small and only reduce the amount spend in four years in a few of the model specifications. While the perceived utility of the unemployed job seekers decreases over all policies and model specifications, I observe only very small effects on this level, with an exception of the reference dependence model that allows for savings. Table 13 shows the effects of front-loading the benefit path by an increase of either 10, 20 or 30% in comparison to the benchmark policy. Figures 26 and 27 plot the policy changes of 10% as well as 30% front-loading against the initial patterns. As expected, the policy changes lead to an additional earlier spike in the search intensity at the point of the first transition from the higher benefits b1 to the lower level b2 . The magnitude of this peak obviously depends on the degree of front-loading, but I observe this pattern in all model specifications. Due to the smaller drop at the point of the unemployment insurance benefit exhaustion, induced by a lower b2 , I observe a smaller spike here. The decrease of the search efforts for a 10% front-loading is rather small for all models. Especially for the full model with heterogeneity in the search costs, the full model with savings as well as both overconfidence model specifications, I just observe a marginal difference here. Additionally, the policy changes induce smaller search effort right at the beginning of the unemployment spell. After roughly 20 to 30 weeks the search intensity becomes higher than in the benchmark case in anticipation of the first income shock. Lastly, I observe an immense first spike in the reference dependence model without asset accumulation. While the initial search efforts are by far the lowest under all specifications they eventually reach an intensity at the first transfer shock almost three times as high as the initial peak once the individual transitions into social security. Turning to the effects on the average unemployed job seeker, as presented in table 13, I observe varying results for different models on the three analyzed levels. The higher benefits in the first year induce an increase in the perceived utility that becomes stronger the higher the level of b1 is set.
62
This can mainly be explained by the normal discounting δ, but also partly by the degree of present bias β that further decreases the influence of future income streams in the utility. Interestingly, I observe clear differences between the hand to mouth models compared to the versions that allow for endogenous savings in terms of the expected spell lengths of the average unemployed. While in the hand-to-mouth case only the full model without heterogeneity decreases the spell length, I find spell length decreases for all models and policies in the models that allow for endogenous saving, with an exception for the overconfidence specification. The government expenditure increases nevertheless over all model specifications and front-loading intensities.
8.3
Finding an Optimal Policy
Unfortunately none of the evaluated policies were able to yield beneficial effects on all of the three analyzed levels, i.e. decreases in the average unemployment spell length, decreases in the government expenditure as well as increases in the perceived utility of the job seeker. I nevertheless found strong decreases of the average spell length and government expenditure for (almost) all model specifications when cutting the benefits as well as strong utility increases when changing to a front-loaded benefit system. It therefore might be promising to combine both policies. In the following I evaluate a policy change to a front-loaded benefit system with an added cut in the overall benefit level. I cut the benefits b by either 10 or 20% to bcut .22 Again, the front-loading is characterized by b1 > b2 , where bcut increases by either 10, 20 or 30% to b1 . Further, b2 is again set to satisfy:
b1 T1 + b2 (T − T1 ) = bcut T
(34)
But can this policy lead to a desirable outcome? It turns out it possibly can. Table 14 and 15 show the effects of this policy change while figures 28 to 31 again plot the altered search efforts. Compared to the simple front-loading policy I observe increases in the initial search efforts induced by the lower benefit level and thus stronger effect of the of the loss aversion. Further, I observe a 22 I do not evaluate a 30% cut as before. Combined with the front-loading it mostly leads to a benefit level b that 2 would fall below the level of social security and is therefore not desirable.
63
smaller second peak at week 104 when the unemployed transition into social security. These effects become stronger in the model specifications that allow for saving while the first peak, the point of changing from b1 to b2 , becomes stronger. Turning to the effects of the policy change, I find desirable outcomes for all 30% front-loading policies when cutting the benefits overall by 10% and additionally for the 20% front-loading in some of the specifications. While the spell lengths and the government expenditure decrease, I also find relatively strong positive effects on the perceived utility of the job seekers. For the 20% benefit cut I only observe utility increases for in the full model without heterogeneity. As I found overall positive effects of the 10% benefit cut and 30% front-loading policy for all model specifications, it seems possible to exploit the cognitive biases that the unemployed have been showed to be subject to in the prior estimations. This policy appears to be more desirable than the sole effect of cutting the benefit level or the introduction of a front-loaded system as it might be able to balance the often conflicting interests of the providing government and the receiving individuals. Nevertheless, one needs to be aware of the fact that the utility increase and spell length decrease is detected for the average unemployed job seeker. Due to the front-loading of the benefits, the individuals who are already at risk of long-term employment might not prefer this policy23 .
8.4
Nudging
While policy changes in the unemployment insurance system are a straight-forward and actively debated way to influence the behavior of the unemployed job seekers, there might be a more subtle method to intervene that I am shortly introduce here. There is no way to make proper use of these ideas in the scope of this thesis and it is just meant as a suggestive overview of a growing field of research that is increasingly trying to exploit the phenomena also found in this thesis. Recently scientific interest has risen in low-cost interventions that provide information about job prospects and the efficiency of early search as well as advice on what types of jobs to search for (Altmann et al. (2015), Belot et al. (2015)). These studies build up on the idea of ”nudging” (Thaler 23 In further analysis I could compare the welfare for both heterogeneity types instead of only taking the average into account.
64
and Sunstein, 2003) which argues that certain indirect methods can influence economic individuals subconscious decision making to achieve desired outcomes and choices without enforcing it through commandments and prohibitions or altering economic incentives (like a change in the benefit policy). Therefore it might be possible to further exploit the cognitive biases that affect job search through small but targeted and well developed interventions. Naturally these interventions come at very low costs and are thus highly interesting for the extremely expensive unemployment insurance system in Denmark and other welfare states. The aforementioned study by Altmann et al. (2015) makes use of the provision of information to unemployed job seekers in form of a brochure that contains detailed descriptions of the importance of early search as well as information about the state of the economy and how to search for a job effectively. Altmann et al. evaluate the effect of the brochure in a large scale field experiment among unemployed job seekers in Germany and find positive, yet insignificant, effects on re-employment and the related income earned in the following year. However, they find significant and relatively strong positive effects for the group of individuals at the risk of long-term unemployment. The recent study by Belot et al. (2015) uses a smaller scaled experiment where unemployed job seekers in Scotland were able to search for jobs in the laboratory for several weeks with a specifically designed web interface. The treatment group was subject to a widening of the possible vacancy pool with other relevant occupations, not directly related to the prior job. This increased the number of job interviews for the treatment group significantly, especially for the group that was searching for a relatively narrow set of possible employments before. The two studies show that it could be possible to make use of nudging to lead to positive effects in job search. It is however left to study this further with well developed field experiments. There could for example also be the possibility to work against the problems induced by present bias. Babcock et al. (2012) notes that the time inconsistency induced by it could let job seeking individuals procrastinate by giving in into their short-run incentives (leisure vs. efforts of search). The long-run incentives are tied to finding a job which possibly lies in the distant future and thus the individuals are subject to a severe self-control problem that lets them often decide against their long-run interest. It could therefore be possible to use frequent low-cost reminders via SMS or email
65
that make the benefit of increasing the search effort more apparent. It might also be possible to use small benefits and/or fines, also on a high-frequency basis, that directly influence the utility of the agents and thus might trigger higher search for employment.
9
Discussion and Limitations
In this section I discuss the limitations and constraints of this thesis to put the following conclusion into proper context. Obviously, any economic model is always an abstraction from reality and thus can never perfectly predict the behavior of economic agents in the real world. The same goes for the postulated model of this thesis, but I found a reasonably strong fit for some of the model specifications when computing the search efforts on the data obtained before the reform of the unemployment insurance system in 2010. Thus, I assume that a part of the estimated structural models predict the behavior of the unemployed job seeker justifiably well to give the suggestive policy evaluation in section 8. I nevertheless need to take the fact into account that all policies are evaluated on a partial equilibrium model and results therefore need to be regarded with proper caution. For a general critique of the estimation process, I need to note the restrictions I applied, mostly for the model specifications that allow for endogenous savings. Clearly, it would have been optimal to increase the size of the asset grid, giving the agent more options, as well as the amount of starting points for the minimization of the objective function (26). The latter is problematic as the process generally led to several local minima and I cannot surely guarantee that I obtain the single global solution. Both restrictions needed to be imposed due to the time constraints of this thesis and the extremely demanding model computation, induced by the high dimensionality of the consumption-saving decisions on the whole asset grid. The computations were running for several weeks and just got finished right in time to include them in the final project. For a more serious and sophisticated analysis of the models, I need to re-estimate the endogenous saving specifications on a higher set of starting points as well as a larger asset grid. There might be a possibility to speed up the computation process itself, but as of now I was not able to find a way to seriously
66
decrease the computation time. Further, even though I presented several robustness checks, there is still room for more and advanced changes in the model specifications24 , especially as all of them were restricted to the hand-to-mouth models. The alternative specifications presented in this thesis typically led to a decrease of the model fit and higher standard errors, but mostly exhibit comparable model patterns. Emphasizing on the overconfidence, I found a significantly better data fit with the novel definition of a bias that vanishes over time although the evidence for this behavior was relatively weak and contradictory to the findings of Spinnewijn (2015) and Huffman et al. (2006). Also, due to the formulation chosen here, the overconfident job seeker can never have a bias in his subjective job finding probability that is larger than twice as high as the actual probability of finding a job. Even though I never estimated a bias parameter close enough to π = 1, this is inelegant and urges to possibly re-formulate the overconfidence for further research.
10
Conclusion
In this thesis I found evidence that job search models, enriched with extensions closely related to cognitive biases, can explain patterns of the Danish unemployment hazard rates in a strikingly good manner, while the standard model of job search is exhibiting strong difficulties in the fit to the empirical data. The hazard rates are characterized by decreasing intensities from the beginning of the unemployment spell and a pronounced spike at the point of transition into the lower social security payments. The models were estimated using a minimum-distance approach and support the findings of DellaVigna et al. (2016) who introduced reference dependence to job search and applied a similar econometric strategy in estimating the model parameters. The results show a significant impact of reference dependence and present bias on the path of search efforts for the unemployed job seekers. While a model with reference dependence and present bias is already able to replicate the most distinct data features, it is a novel model extension, building on the idea of overconfidence in job 24 E.g. a change of the specification of the utility function, different types of heterogeneity, estimation of δ, positive asset stock at the unemployment spell start and others.
67
search by Spinnewijn (2015), that I introduced in this thesis which leads not only to a closer data fit on the whole range of hazard rates, but also results in parameters that are closer to usual literature estimates. The model even shows a surprisingly strong fit in absence of heterogeneity, unlike all other specifications. In contrast to Spinnewijns’ constant bias of the confidence, I introduced a ”learning model” of the bias that vanishes over time as the unemployed job seekers learn more about their actual probability of finding a job. The results hold no matter if I allow for endogenous saving or not, but it is to note that, due to the immense computational intensity of the models with consumption-saving decisions, I needed to restrict the econometric approach by searching on a much smaller amount of possible starting points than in the simpler hand-to-mouth version. Therefore I cannot guarantee that I find the global minimum of the objective function that was supplied in the estimation process. The results with endogenous-saving thus need to be viewed with slight caution and leave the possibility for re-estimation if more time and resources are available. In the last part of this thesis I evaluated possible policy changes with the earlier estimated structural models. The found evidence opens the door for a possible re-structuring of the unemployment insurance by exploiting the distinct features of cognitive biases. While usual policy propositions, like a cut of the benefit level or the entitlement period, lead to unemployment spell length and government expenditure decreases they also decrease the perceived utility of the unemployed job seeker. A new idea of front-loading the benefit path, proposed by DellaVigna et al. to exploit the loss aversion of reference dependence, however leads to strong utility increases over most of the tested model specifications while the government expenditure also rises. As all of the mentioned policy proposals did not lead to desirable effects for the unemployed and the government alike, I combined the front-loading of the benefit path with a general decrease of the benefit level. It turned out that this combination leads to strong welfare increases as well as decreases in the unemployment spell length and the government expenditure over all model specifications. Future research could dig deeper into the effects of the policies and aim to find an optimal unemployment insurance, given the influence of cognitive biases among unemployed. Besides this, nudging with small and targeted interventions which exploit behavioral patterns, that showed to
68
be active in job search, presents a promising field for extended research, especially considering the low-costs and therefore feasibility of large-scale field experiments, just like in Altmann et al. (2015).
69
11
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Appendix
A
Tables
Table 1: Descriptive statistics of the population used in the structural estimation: Means, standard deviation (in parenthesis) and t-statistic
UI benefit regime
Number of Spells Female Dane Western Nonwestern Age Start Disposable Income in DKK* Compulsory education** Vocational education** Gymnasium education** Short academic education** Medium academic education** Long academic education** Missing education
Pre-reform
Post-reform
30,700
57,023
0.495 (0.50) 0.885 (0.32) 0.040 (0.20) 0.076 (0.26) 36.886 (7.28) 182,169 (115,087) 0.102 (0.30) 0.505 (0.50) 0.037 (0.19) 0.051 (0.22) 0.134 (0.34) 0.105 (0.31) 0.067 (0.25)
0.456 (0.49) 0.891 (0.31) 0.047 (0.21) 0.062 (0.24) 37.302 (7.50) 198,494 (78,494) 0.083 (0.28) 0.553 (0.50) 0.035 (0.18) 0.047 (0.21) 0.135 (0.34) 0.091 (0.29) 0.057 (0.23)
T-stat
10.89 -2.84 -4.8 7.52 -7.91 -24.77 9.67 -13.72 1.31 2.74 -0.58 6.58 6.05
*Income in 2007 and 2010 respectively; **I had only access to highest attended education level, the figures do not represent actual finishing of the education line
78
Table 2: Statistics of the ’Work, Unemployment, and Early Retirement’-survey, belief about duration until a new job is found by months already in the current unemployment spell
Months since last paid job Belief: Months until new job
120
Hours searche last month
24 Never
32.6 60.0 15.8 5.0
31.4 12.3 10.0 2.0 15.7
34.5 24.3 25.0 45.0 2.0 15.4
35.9 34.2 16.9 8.0 26.8
32.3 24.5 26.7 16.8 80.0 12.2
32.4 26.0 38.7 54.4 20.0 10.1
28.8 38.1 48.7 10.0 0.0 12.3
10.5 2.3 18.8 4.0 3.6
Jobs applied last year
24 Never
28.3 32.0 23.3 0.0
49.8 43.0 62.5 3.3 44.3
53.6 26.2 45.7 32.5 4.0 48.1
71.6 50.5 38.3 21.7 24.4
33.0 81.5 38.0 44.5 50.0 38.9
70.8 28.0 39.2 138.6 7.0 36.4
22.7 36.1 30.1 17.5 0.0 31.9
25.3 23.7 17.5 18.5 9.1
Share of beliefs
24 Never
67.6% 17.6% 11.8% 0.0% 0.0% 2.9%
64.5% 16.1% 3.2% 4.8% 0.0% 11.3%
45.7% 30.9% 7.4% 2.5% 2.5% 11.1%
31.4% 29.4% 17.6% 2.9% 0.0% 18.6%
25.7% 18.9% 16.2% 8.1% 1.4% 29.7%
30.6% 6.1% 12.2% 10.2% 4.1% 36.7%
26.0% 15.6% 9.1% 2.6% 1.3% 45.5%
13.6% 13.6% 18.2% 0.0% 9.1% 45.5%
Hours searched last month
28.3
14.3
24.4
24.3
32.1
30.3
23.0
7.8
Jobs applied last year
20.9
40.6
35.0
41.3
47.7
53.3
23.1
18.8
34
62
81
102
74
49
77
22
Number of answers
79
Benchmark models Table 3: Benchmark, Structural Estimation of Model Variations with Hand-to-Mouth Consumers and no Heterogeneity
Standard
Ref. Dep
Overconf.
Full
(1)
(2)
(3)
(4)
Utility Parameters Discount factor β
-
Loss aversion λ
-
1.00 (0.023)∗ -
Adjustment speed N
-
1.00 (0.012)∗ 37.66 (17.971) 43.07 (1.376)
0.87 (0.012) 3.79 (0.170) 10.26 (0.410)
Confidence Parameters Overconfidence π
-
-
0.92 (0.001)
0.95 (0.002)
7.38 (1.237) 17955 x 107 (70157 x 107 )
0.14 (0.014) 1116.50 (547.842)
0.26 (0.012) 194.11 (19.746)
0.15 (0.010) 143.23 (10.543)
Model Fit Used moments
79
79
79
79
Estimated parameters
2
6
5
7
10792.00
831.00
719.51
258.15
Search Cost Parameters Curvature γ Search costs k
GOF ∗
-
: Standard errors need to be viewed with caution as the parameter estimates lie at the upper boundary of the parameter space and thus might take greater values if not restricted.
80
Table 4: Benchmark, Structural Estimation of Model Variations with Hand-to-Mouth Consumers and 2-Type Heterogeneity
Standard
Ref. Dep
Overconf.
Full
(5)
(6)
(7)
(8)
Utility Parameters Discount factor β
–
Loss aversion λ
-
0.70 (0.011) -
Adjustment speed N
-
0.98 (0.011) 1.64 (0.027) 15.36 (0.592)
0.63 (0.090) 5.57 (1.690) 1.61 (0.363)
Confidence Parameters Overconfidence π
-
-
0.86 (0.037)
0.88 (0.009)
5.46 (2.667) 0.36 (0.010) 40389 (16646) 7.07 x 109 (6.01 x 109 )
0.08 (0.003) 0.77 (0.007) 48.80 (3.704) 185.78 (6.411)
0.37 (0.013) 0.88 (0.037) 42.01 (17.909) 453.62 (321.385)
0.31 (0.017) 0.84 (0.077) 30.72 (11.640) 201.85 (138.467)
Model Fit Used moments
79
79
79
79
Estimated parameters
5
7
6
9
572.59
269.06
326.05
166.66
Search Cost Parameters Curvature γ Population share low type pl Search costs low type kl Search costs high type kh
GOF
81
-
Estimates with endogenous saving Table 5: Benchmark, Structural Estimation of Model Variations with Endogenous Saving and no Heterogeneity
Standard
Ref. Dep
Overconf.
Full
(9)
(10)
(11)
(12)
Utility Parameters Discount factor β
-
Loss aversion λ
-
1.00 (0.017)* -
Adjustment speed N
-
1.00 (0.011)* 41.18 (16.230) 38.00 (0.942)
0.93 (0.002) 4.74 (0.098) 10.00 (0.113)
Confidence Parameters Overconfidence π
-
-
0.92 (0.001)
0.95 (0.000)
6.96 (1.122) 4825 x 107 (7362 x 106 )
0.13 (0.010) 1073.90 (449.049)
0.24 (0.010) 185.90 (14.762)
0.13 (0.005) 166.35 (3.836)
Model Fit Used moments
79
79
79
79
Estimated parameters
2
6
5
7
10827.00
1263.10
811.63
289.37
Search Cost Parameters Curvature γ Search costs k
GOF ∗
-
: Standard errors need to be viewed with caution as the parameter estimates lie at the upper boundary of the parameter space and thus might take greater values if not restricted.
82
Table 6: Benchmark, Structural Estimation of Model Variations with Endogenous Saving and 2Type Heterogeneity
Standard
Ref. Dep
Overconf.
Full
(13)
(14)
(15)
(16)
Utility Parameters Discount factor β
-
Loss aversion λ
-
0.90 (0.077) -
Adjustment speed N
-
1.00 (0.110)* 26.87 (26.233) 48.02 (17.488)
0.70 (0.128) 4.63 (1.180) 22.88 (1.382)
Confidence Parameters Overconfidence π
-
-
0.86 (0.017)
0.17 (0.077)
1.52 (0.370) 0.40 (0.010) 63.02 (34.899) 7565.90 (9421.900)
0.43 (0.041) 0.31 (0.044) 351.87 (153.640) 1863.50 (718.090)
0.35 (0.015) 0.86 (0.043) 69.14 (32.994) 540.01 (259.601)
0.72 (0.227) 0.74 (0.029) 83.86 (39.571) 1008.80 (740.869)
Model Fit Used moments
79
79
79
79
Estimated parameters
5
7
6
9
638.94
340.90
361.38
232.7596
Search Cost Parameters Curvature γ Population share low type pl Search costs low type kl Search costs high type kh
GOF ∗
-
: Standard errors need to be viewed with caution as the parameter estimates lie at the upper boundary of the parameter space and thus might take greater values if not restricted.
83
Robustness Table 7: Structural Estimation of Model Variations with Hand-to-Mouth Consumers and no Heterogeneity, Identity Matrix Weighting
Standard
Ref. Dep
Overconf.
Full
(17)
(18)
(19)
(20)
Utility Parameters Discount factor β
-
Loss aversion λ
-
0.79 (0.013) -
Adjustment speed N
-
1.00 (0.032)∗ 49.80 (57.488) 51.00 (3.631)
0.84 (0.011) 4.41 (0.190) 7.27 (0.228)
Confidence Parameters Overconfidence π
-
-
0.94 (0.001)
0.95 (0.002)
6.63 (1.533) 1.80 x 1011 (9.52 x 1011 )
0.18 (0.064) 1755.40 (1942.700)
0.12 (0.004) 75.44 (5.080)
0.14 (0.008) 118.67 (7.604)
Model Fit Used moments
79
79
79
79
Estimated parameters
2
6
5
7
0.016 15397.00
0.005 2169.00
0.004 1545.30
0.001 350.492
Search Cost Parameters Curvature γ Search costs k
GOF∗∗
∗
-
: Standard errors need to be viewed with caution as the parameter estimates lie at the upper boundary of the parameter space and thus might take greater values if not restricted. ∗∗ : While the GOF is not directly comparable due to other weighting, I also present the goodness of fit measure using the optimal weighting matrix at the bottom. Note that this will always be worse than in the benchmark models due to the structure of the objective function.
84
Table 8: Structural Estimation of Model Variations with Hand-to-Mouth Consumers and 2-Type Heterogeneity, Identity Matrix Weighting
Standard
Ref. Dep
Overconf.
Full
(21)
(22)
(23)
(24)
Utility Parameters Discount factor β
–
Loss aversion λ
-
0.69 (0.112) -
Adjustment speed N
-
0.98 (0.010) 1.69 (0.027) 15.35 (0.561)
0.85 (0.016) 4.48 (0.571) 7.41 (0.265)
Confidence Parameters Overconfidence π
-
-
0.85 (0.011)
0.95 (0.004)
0.08 (0.000) 0.75 (0.004) 44.53 (0.262) 178.46 (1.547)
0.08 (0.003) 0.78 (0.007) 48.81 (3.451) 185.78 (6.011)
0.38 (0.014) 0.85 (0.045) 38.64 (15.901) 306.13 (186.468)
0.14 (0.035) 0.91 (0.077) 122.10 (243.655) 153.52 (363.512)
Model Fit Used moments
79
79
79
79
Estimated parameters
5
7
6
9
0.005 2705.70
0.001 275.49
0.002 338.41
0.001 346.63
Search Cost Parameters Curvature γ Population share low type pl Search costs low type kl Search costs high type kh
GOF∗
∗
-
: While the GOF measure is not directly comparable due to other weighting, I also present the goodness of fit measure using the optimal weighting matrix at the bottom. Note that this will always be worse than in the benchmark models due to the structure of the objective function.
85
Table 9: Structural Estimation of Model Variations with Hand-to-Mouth Consumers, Different Specifications Ref. Dep
Ref. Dep.
Full
Full
Full
Full
Full
Full
Full
Full
AR(1)
AR(1) 2-type
AR(1)
AR(1) 2-type
const. bias
const. bias 2-type
low wage
high wage
low wage 2-type
high wage 2-type
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
0.001 (0.005) 86.40 (41.761) 0.93 (0.002) -
1.00 (0.002)* 22.33 (0.116) 0.94 (0.001) -
1.00 (0.006)* 6.36 (0.085) 0.91 (0.003) -
1.00 (0.028)* 36.54 (23.090) 0.92 (0.004) -
0.61 (0.021) 23.31 (27.041) -
0.53 (0.026) 5.01 (4.268) -
1.00 (0.043)* 9.32 (9.601) -
1.00 (0.226)* 13.52 (9.307) -
1.00 (0.692)* 10.82 (20.079) -
0.34 (1.238) 3.81 (4.595) -
0.67 (0.034)
0.39 (0.015)
0.11 (0.008)
0.32 (0.145)
30.00 (0.865) 0.18 (0.042)
1.61 (1.111) 0.13 (0.059)
1.24 (1.002) 0.27 (0.025)
31.18 (1.807) 1.09 (0.107)
6.02 (1.377) 0.63 (0.682)
28.75 (4.507) 1.34 (0.385)
Confidence Parameters Overconfidence π
-
-
-
-
-
0.96 (0.015) -
-
Constant overconfidence π ¯
0.98 (0.002) -
0.97 (0.115)
1.00 (0.003)
0.91 (0.004) -
0.67 (0.043) -
0.50 (0.198) -
0.29 (0.154) -
Model fit Used moments
79
79
79
79
79
79
59
57
59
57
Estimated parameters
5
7
6
8
6
8
6
6
8
8
2166.40
274.67
253.27
174.42
419.00
210.59
113.36
126.92
75.39
115.05
Utility Parameters Discount factor β Loss aversion λ AR(1) parameter ρ Adjustment speed N
86 Curvature γ
GOF ∗
: Standard errors need to be viewed with caution as the parameter estimates lie at the upper boundary of the parameter space and thus might take greater values if not restricted.
Reservation Wages Table 10: Structural Estimation of Model Variations with Hand-to-Mouth Consumers and Reservation Wages
(36)
Ref. Dep 2-type (37)
Full 2-type (38)
0.87 (0.013) 2.89 (0.190) 29.53 (0.646)
0.69 (0.058) 1.01 (1.392) 68.55 (22.142)
0.87 (0.017) 4.49 (0.425) 26.85 (0.592)
0.72 (0.022) 21.13 (1.594) 2.00 (0.026)
-
0.89 (0.003)
-
0.90 (0.002)
0.11 (0.007) 98.86 (7.017) -
0.28 (0.039) 143.26 (74.403) -
0.13 (0.008) -
0.21 (0.006) -
Search costs low type kl
-
-
Search costs high type kh
-
-
0.99 (0.120)* 113.19 (11.821) 1440.07 (72360333)
0.99 (0.488)* 117.03 (10.866) 695.62 (1146573)
148
148
148
148
5
6
7
8
4470.50
1516.30
4467.30
1332.60
Utility Parameters Discount factor β Loss aversion λ Adjustment speed N
Confidence Parameters Overconfidence π Search Cost Parameters Curvature γ Search costs k Population share low type pl
Model Fit Used moments Estimated parameters GOF
Ref. Dep
Full
(35)
∗
: Standard errors need to be viewed with caution as the parameter estimates lie at the upper boundary of the parameter space and thus might take greater values if not restricted.
87
Policy Evaluation Table 11: Effects of Benefit Cut Policies on the Average Unemployed Job Seeker Compared to the Benchmark Case
Hand to mouth
Full 1-type
Full 2-type
Ref. Dep. 2-type
Overconf. 2-type
Endogenous saving
Spell length
Governm. Expend.
Perceived Utility
Spell length
Governm. Expend.
Perceived Utility
Benchmark
692
171,900
1808.0
695
172,530
1800.8
-10% -20% -30%
-73 -146 -219
-12,440 -25,300 -38,590
-1.2 -2.5 -3.7
-68 -138 -209
-11,450 -23,560 -36,380
-1.5 -2.9 -4.3
Benchmark
693
172,150
1845.2
696
172,620
1834.1
-10% -20% -30%
-68 -128 -183
-11,720 -21,680 -30,540
-1.1 -2.2 -3.4
-51 -96 -134
-8,450 -15,800 -22,320
-3.5 -7.2 -11.2
Benchmark
690
171,670
1804.9
699
173,370
1506.6
-10% -20% -30%
-113 -197 -266
-25,270 -44,200 -59,840
-2.0 -3.9 -5.7
-10 -9 3
-220 1,020 3,500
-29.0 -58.7 -89.2
Benchmark
696
172,560
1845.4
693
171,870
1838.4
-10% -20% -30%
-67 -125 -174
-11,530 -20,820 -28,500
-0.8 -1.5 -2.2
-62 -117 -166
-10,190 -18,740 -26,060
-1.0 -2.0 -3.0
Note: The benefit level b is cut over the whole entitlement period T . I compare cuts by either 10, 20 or 30% to the benchmark policy. Average spell length in days, government expenditure after 4 years in DKK and agents utility evaluated with exponential discounting.
88
Table 12: Effects of Entitlement Length Cut Policies on the Average Unemployed Job Seeker Compared to the Benchmark Case
Hand to mouth
Full 1-type
Full 2-type
Ref. Dep. 2-type
Overconf. 2-type
Endogenous saving
Spell length
Governm. Expend.
Perceived Utility
Spell length
Governm. Expend.
Perceived Utility
Benchmark
692
171,900
1808.0
695
172,530
1800.8
-10% -20% -30%
-27 -53 -81
-110 130 700
-0.01 -0.02 -0.04
-27 -54 -81
-150 -90 630
-0.1 -0.1 -0.1
Benchmark
693
172,150
1845.2
696
172,620
1834.1
-10% -20% -30%
-18 -38 -60
1,660 3,220 4,710
-0.1 -0.2 -0.3
-15 -31 -48
2,120 4,210 6,280
-0.4 -0.8 -1.4
Benchmark
690
171,670
1804.9
699
173,370
1506.6
-10% -20% -30%
-26 -54 -84
-190 -500 -980
-0.4 -0.7 -1.1
-24 -46 -67
-190 170 1,100
-4.7 -9.8 -15.6
Benchmark
696
172,560
1845.4
693
171,870
1838.4
-10% -20% -30%
-15 -31 -50
2,450 4,680 6,720
-0.2 -0.3 -0.4
-17 -35 -55
1,840 3,550 5,090
-0.1 -0.3 -0.5
Note: The benefit entitlement period T is cut by either 10, 20 or 30% to the benchmark policy. The unemployed job seeker thus enters the social security system faster, but benefits themselves are not affected. Average spell length in days, government expenditure after 4 years in DKK and agents utility evaluated with exponential discounting.
89
Table 13: Effects of Twp-Step Front-Loading Policies on the Average Unemployed Job Seeker Compared to the Benchmark Case
Hand to mouth
Full 1-type
Full 2-type
Spell length
Governm. Expend.
Perceived Utility
Spell length
Governm. Expend.
Perceived Utility
Benchmark
692
171,900
1808.0
695
172,530
1800.8
+10% +20% +30%
-3 -15 -34
5,940 11,230 16,500
1.2 2.3 3.4
-5 -20 -41
5,410 9,940 14,630
1.2 2.4 3.6
Benchmark
693
172,150
1845.2
696
172,620
1834.1
5 5 0
7,960 15,880 24,090
0.8 1.5 2.1
-7 -8 -4
4,970 12,990 23,680
2.0 3.9 5.6
Benchmark
690
171,670
1804.9
699
173,370
1506.6
+10% +20% +30%
18 28 31
12,590 26,070 40,370
1.0 2.0 3.0
-13 -26 -37
2,960 7,680 14,380
13.4 25.7 37.1
Benchmark
696
172,560
1845.4
693
171,870
1838.4
+10% +20% +30%
11 18 21
9,510 19,020 28,570
0.1 0.5 0.9
2 4 6
7,180 15,110 23,430
0.8 1.2 1.6
+10% +20% +30% Ref. Dep. 2-type
Overconf. 2-type
Endogenous saving
Note: The former benefit structure where b was payed over T periods is changed. I introduced two new benefits where b1 > b2 are payed out for T1 = 52 weeks, T2 = (T − T1 ) = 52 weeks respectively and b1 is either raised 10, 20 or 30% higher than b. The second step benefit b2 is set to satisfy b1 T1 + b2 (T − T1 ) = bT . Average spell length in days, government expenditure after 4 years in DKK and agents utility evaluated with exponential discounting.
90
Table 14: Effects of a 10% Benefit Cut Policy Combined with Two-Step Front-Loading on the Average Unemployed Job Seeker Compared to the Benchmark Case
Hand to mouth
Full 1-type
Full 2-type
Ref. Dep. 2-type
Overconf. 2-type
Endogenous saving
Spell length
Governm. Expend.
Perceived Utility
Spell length
Governm. Expend.
Perceived Utility
Benchmark
692
171,900
1808.0
695
172,530
1800.8
+10% +20% +30%
-71 -77 -90
-25,330 -19,550 -13,840
-0.2 0.9 1.9
-68 -77 -92
-24,750 -19,700 -14,430
-0.3 0.9 2.0
Benchmark
693
172,150
1845.2
696
172,620
1834.1
+10% +20% +30%
-61 -58 -58
-23,390 -15,840 -8,070
-0.3 0.3 1.0
-53 -52 -46
-22,390 -15,210 -6,120
-1.6 0.2 1.8
Benchmark
690
171,670
1804.9
699
173,370
1506.6
+10% +20% +30%
-88 -69 -57
-30,320 -17,180 -3,160
-1.1 -0.2 0.6
-20 -31 -42
-16,790 -12,930 -7,850
-15.2 -2.9 8.6
Benchmark
696
172,560
1845.4
693
171,870
1838.4
+10% +20% +30%
-53 -42 -35
-21,730 -12,640 -3,410
0.0 0.3 0.5
-56 -52 -48
-22,830 -15,520 -7,880
-0.5 -0.2 0.2
Note: The former benefit structure where b was payed over T periods is changed. First, b is cut by 10%. Then I introduce two new benefits where b1 > b2 are payed out for T1 = 52 weeks, T2 = (T − T1 ) = 52 weeks respectively and b1 is either raised 10, 20 or 30% higher than the new b. The second step benefit b2 is set to satisfy b1 T1 + b2 (T − T1 ) = bT . Average spell length in days, government expenditure after 4 years in DKK and agents utility evaluated with exponential discounting.
91
Table 15: Effects of a 20% Benefit Cut Policy Combined with Two-Step Front-Loading on the Average Unemployed Job Seeker Compared to the Benchmark Case
Hand to mouth
Full 1-type
Full 2-type
Ref. Dep. 2-type
Overconf. 2-type
Endogenous saving
Spell length
Governm. Expend.
Perceived Utility
Spell length
Governm. Expend.
Perceived Utility
Benchmark
692
171,900
1808.0
695
172,530
1800.8
+10% +20% +30%
-140 -142 -148
-54,130 -48,290 -42,510
-1.5 -0.6 0.3
-134 -137 -152
-52,860 -47,570 -43,220
-1.8 -0.7 0.3
Benchmark
693
172,150
1845.2
696
172,620
1834.1
+10% +20% +30%
-120 -116 -114
-50,430 -43,680 -36,760
-1.5 -0.8 -0.2
-95 -92 -92
-46,200 -40,180 -33,590
-5.4 -3.7 -2.3
Benchmark
690
171,670
1804.9
699
173,370
1506.6
+10% +20% +30%
-180 -164 -153
-64,430 -54,920 -44,020
-3.1 -2.4 -1.7
-16 -26 -53
-32,480 -29,530 -28,620
-44.7 -32.2 -22.0
Benchmark
696
172,560
1845.4
693
171,870
1838.4
+10% +20% +30%
-111 -101 -93
-48,770 -40,990 -33,050
-0.8 -0.5 -0.3
-110 -105 -100
-48,750 -42,280 -35,530
-1.6 -1.2 -0.9
Note: The former benefit structure where b was payed over T periods is changed. First, b is cut by 20%. Then I introduce two new benefits where b1 > b2 are payed out for T1 = 52 weeks, T2 = (T − T1 ) = 52 weeks respectively and b1 is either raised 10, 20 or 30% higher than the new b. The second step benefit b2 is set to satisfy b1 T1 + b2 (T − T1 ) = bT . Average spell length in days, government expenditure after 4 years in DKK and agents utility evaluated with exponential discounting.
92
B
Figures
Benchmark Estimates with Hand-to-Mouth Consumers No Heterogeneity
Figure 5: Benchmark, estimated model variations with hand-to-mouth consumers and no heterogeneity (red) and Empirical Hazard Rates (blue)
93
Heterogeneity
Figure 6: Benchmark, estimated model variations with hand-to-mouth consumers and two-type heterogeneity (red) and empirical hazard rates (blue)
94
Benchmark Estimates with Endogenous Saving No Heterogeneity
Figure 7: Benchmark, estimated model variations with endogenous saving and no heterogeneity (red) and Empirical Hazard Rates (blue)
95
Figure 8: Value of being employed (VE , blue) and value of being unemployed (VU , red) for model variations with endogenous saving and no heterogeneity
96
Figure 9: Path of the asset stock for model variations with endogenous saving and no heterogeneity
97
Heterogeneity
Figure 10: Benchmark, estimated model variations with endogenous saving and two-type heterogeneity (red) and empirical hazard rates (blue)
98
Figure 11: Value of being employed (VE , blue) and value of being unemployed (VU , red) for model variations with endogenous saving and two-type heterogeneity
99
Figure 12: Path of the asset stock for model variations with endogenous saving and two-type heterogeneity
100
Robustness checks Pre-reform regime
Figure 13: Estimated model variations with hand-to-mouth consumers (red) and empirical hazard rates for the pre-reform regime (blue)
101
Figure 14: Estimated model variations with endogenous saving (red) and empirical hazard rates for the pre-reform regime (blue)
102
Identity Matrix Weighting
Figure 15: Estimated model variations with hand-to-mouth consumers and no heterogeneity, estimated with identity matrix weighting, (red) and empirical hazard rates (blue)
103
Figure 16: Estimated model variations with hand-to-mouth consumers and two-type heterogeneity, estimated with identity matrix weighting, (red) and empirical hazard rates (blue)
104
AR Reference Dependence
Figure 17: Estimated model variations with hand-to-mouth consumers, estimated with AR(1) reference dependence, (red) and empirical hazard rates (blue)
105
Constant Overconfidence
Figure 18: Estimated model variations with hand-to-mouth consumers, estimated with constant overconfidence, (red) and empirical hazard rates (blue)
106
Alternative Samples
Figure 19: Estimated model variations with hand-to-mouth consumers, estimated on high- and low income samples, (red) and empirical hazard rates (blue)
107
Reservation Wage
Figure 20: Estimated model variations with hand-to-mouth consumers and no heterogeneity, estimated with reservation wage choices, (red), empirical hazard rates and log re-employment wages (blue)
108
Figure 21: Estimated model variations with hand-to-mouth consumers and two-type heterogeneity, estimated with reservation wage choices, (red), empirical hazard rates and log re-employment wages (blue)
109
Policy Evaluation Benefit Cuts
Figure 22: Search efforts under different benefit cut schemes, Benchmark model in blue, 10% cuts in dashed red line (–) and 30% cuts in dotted red line (:)
110
Figure 23: Search efforts under different benefit cut schemes, Benchmark model in blue, 10% cuts in dashed red line (–) and 30% cuts in dotted red line (:)
111
Entitlement Period Cuts
Figure 24: Search efforts under different entitlement period cut schemes, Benchmark model in blue, 10% cuts in dashed red line (–) and 30% cuts in dotted red line (:)
112
Figure 25: Search efforts under different entitlement period cut schemes, Benchmark model in blue, 10% cuts in dashed red line (–) and 30% cuts in dotted red line (:)
113
Two-Step Front-Loading
Figure 26: Search efforts under different two-step front-loading schemes, Benchmark model in blue, 10% front-loading in dashed red line (–) and 30% front-loading in dotted red line (:)
114
Figure 27: Search efforts under different two-step front-loading schemes, Benchmark model in blue, 10% front-loading in dashed red line (–) and 30% front-loading in dotted red line (:)
115
Combining Benefit Cuts and Two-Step Front-Loading
Figure 28: Search efforts under different two-step front-loading schemes with a 10% benefit cut, Benchmark model in blue, 10% front-loading in dashed red line (–) and 30% front-loading in dotted red line (:)
116
Figure 29: Search efforts under different two-step front-loading schemes with a 10% benefit cut, Benchmark model in blue, 10% front-loading in dashed red line (–) and 30% front-loading in dotted red line (:)
117
Figure 30: Search efforts under different two-step front-loading schemes with a 20% benefit cut, Benchmark model in blue, 10% front-loading in dashed red line (–) and 30% front-loading in dotted red line (:)
118
Figure 31: Search efforts under different two-step front-loading schemes with a 20% benefit cut, Benchmark model in blue, 10% front-loading in dashed red line (–) and 30% front-loading in dotted red line (:)
119
C
Proofs
Bellman operator is a strict contraction Aim of this section is to proof that the Bellman equation in fact satisfies the conditions to have a unique fixed point as theorem 5.1 constitutes. The following proofs and steps build on Stokey et al. (1989) and Ok (2007)25 . Let us start from the point where I defined the new operator Ψ : V → V that maps from the space of bounded functions into the space of bounded functions. I then have a problem of the form
Ψ(V )(x) = sup u(x, y) + δV (y) x,y
s.t. y ∈ Γ(x)
where Γ(x) is the correspondence for y given the state x, i.e. the set of feasible states for y for each x ∈ X. I assume that X is a convex subset of Rn and Γ : X → X is nonempty, compact-valued and continuous. The correspondence in this setting is given by A0 ∈ Γ(A) such that A0 = (1+R)[A+w−c] and satisfies this. I further assume that u : X × X → R is bounded such that ∃ u ¯ with u(x, y) < u ¯ when x ∈ X and y ∈ Γ(x). Making use of a logarithmic utility in the model setting I satisfy the assumptions as u(A0 ) = ln(A0 ) is clearly bounded from above given the state A and the correspondence A0 ∈ Γ(A). Now define the metric space (CB(X), d∞ ) by the space of bounded and continuous functions CB(X) ≡ {V : X → R continuous and bounded} with the usual sup norm kV k = sup |V (x)| and x∈X
metric distance d∞ (V, W ) = kV − W k = sup |V (x) − W (x)|. This is a complete metric space (see x∈X
Ok, 2007) and thus gives us the first condition of theorem 5.1 to proof a unique fixed point of Ψ. I now need to show that Ψ has a unique fixed point. This requires to show that Ψ is a contraction by theorem 5.1, but like in many cases it is not trivial to constitute this property. I can however invoke Blackwell’s sufficient conditions that, once satisfied, ensure that a given mapping 25 As well as major parts of Pablo Kurlats lecture notes on the Bellmann equation: http://web.stanford.edu/ pkurlat/teaching/ ~
120
is a contractive self-map: Theorem C.1 Let X ⊆ Rl and CB(X) be the space of bounded and continuous functions V : X → R, with the sup norm kV k = sup |V (x)|. If now Ψ : V → V satisfies: x∈X
a. [Monotonicity]: V, W ∈ B(X) and V (x) ≤ W (x) for all x ∈ X ⇒ (ΨV )(x) ≤ (ΨW )(x) for all x ∈ X; b. [Discounting]: There exists some δ ∈ (0, 1) such that [Ψ(V + a)](x) ≤ (ΨV )(x) + δa for all V ∈ B(X), a ≥ 0, x ∈ X.
Then Ψ is a contraction with modulus δ. I know that the conditions hold in this problem because for any x:
u(x, y) + δV (y) ≤ u(x, y) + δW (y) sup u(x, y) + δV (y) ≤ sup u(x, y) + δW (y) y∈Γ(x)
y∈Γ(x)
Ψ(V )(y) ≤ Ψ(W )(y)
and thus the monotonicity condition a. is satisfied. Further:
Ψ(V + a)(x) = sup u(x, y) + δ[V (y) + a] y∈Γ(x)
= Ψ(V )(x) + δa
and I also have the discounting condition satisfied.
121
D
Matlab Codes
The Matlab codes and programmes for the computation and estimation of the main models of this thesis can be found either under https://github.com/JonasFluchtmann/MatlabCodes or by request via
[email protected].
122
