The positive impact of unemployment benefits on flows into social security disabiliby insurance

Cole Pearce, 11668741 Masters of Economics

Monetary Policy and Banking Specialization

The Positive Impact of Unemployment Benefits on Flows into Social

Security Disability Insurance

Supervisor: Marcelo Zouain Pedroni August 15, 2017

Statement of Originality

This document is written by Cole Pearce who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of

Abstract

I use panel-data from all US states and the District of Columbia from 1984 to 2016 to determine if increasing unemployment benefits reduces the net new beneficiaries of the United States Social Security Disability Insurance (SSDI) program. Instead, I find that a one percent increase in unemployment benefits increases the annual net flow into SSDI by 0.034% of the working age population, or approximately 77 thousand additional beneficiaries nationally. A calibrated model returns a decrease in net flow, as the insurance value of unemployment benefits encourages workers to seek the most highly-valued earnings, wages. I conclude with suggestions for methodological improvements that could solve this puzzle.

Table of Contents:

1. Introduction 4 2. Literature Review 6 3. Empirical Results a. Regressions 9 b. Data Description 11 c. Results 12

d. Awards per Application 14 e. Potential Explanations 15

4. Modeling the Effect of Unemployment Benefits a. Main Assumptions 17

b. Application Clearing 19

c. Equilibrium 20

d. The Elasticity of Net Flow into Disability 21

e. Decomposition 22

f. Testing the Model 24

5. Conclusion 27

6. Bibliography 29

Appendix 1: Robustness Check 31

1. Introduction

The number of beneficiaries of United States Social Security Disability Insurance (SSDI) increased from 1.66% of the working age population in 1984, when the program assumed its current form, to 4.14% in December 2016. Beneficiaries qualify for the program if they have met previous work requirements and a mental or physical disability would prevent them from working for at least one year or would result in death. Beneficiaries exit the program if they become employed. For workers whose conditions are sufficiently serious to qualify but can still perform some labor, this creates a choice: participate in the labor market’s cycle of employment and unemployment or draw a disability benefit, potentially until retirement.

Figure A. Increase in the number of SSDI Beneficiaries since 1984

Source: Social Security Administration

Admitting those who are capable of work reduces labor force participation and incurs significant public cost in terms of both in-cash benefits and Medicare insurance coverage. Still, there is little public or political support for tightening the SSDI screening procedure. Policymakers should consider alternative methods of encouraging marginal applicants into the labor market. My research question is whether increasing unemployment benefits would lower net inflows into SSDI.

It would be reasonable to expect that higher unemployment benefits reduce the net flow of beneficiaries into SSDI. With higher unemployment benefits, being laid off is less costly, which increases the value of labor market participation relative to an SSDI award. Unemployment benefits may also provide the means for credit-constrained workers to engage in retraining or to search for higher-paying jobs.

I run three OLS regressions with state and time fixed effects on annual panel-data of unemployment benefits on the net inflow of SSDI beneficiaries from 1987 to 2016 for the fifty states and the District of Columbia. I find a 1% increase in unemployment benefits increases the annual net flow of beneficiaries into SSDI by 0.034% of the working age population. If this increase occurred across the United States, about 77 thousand additional people would become beneficiaries of SSDI each year.

Fig B. Unemployment Benefits and Net Inflow into Disability per Working Age Capita

Source: Author’s Calculations

I create a simple model to examine this surprising result intuitively. In the model, unemployed workers allocate search effort between searching for employment and applying for a disability program. Depending on the relative value of discounted wages and disability benefits, the elasticity of net flow into the disability program to unemployment benefits could be either positive or negative. When calibrated, the model suggests that the elasticity of net flow into SSDI to unemployment benefits is negative, as higher unemployment benefits incentivize workers to enter the more highly-valued state, employment.

To conclude, I suggest paths for future research that could resolve this gap between empirics and intuition. Further empirical research could use discrete changes in legislation, such as shifts in states’ maximum unemployment benefit duration following 2010, or, for researchers with access to restricted Social Security Administration data, decompose net flows into applications, awards, and outflows. Additionally, the model could be improved by taking into account the limited duration of unemployment benefits or by introducing heterogenous wages or screening rates.

2. Literature Review

I use Benitez-Silva, Disney, and Jimenez-Martin’s (2010) division of disability into two types, work disability and health disability. In turn, this division is based on Nagi’s (1961) definition of disability as an inability to perform socially-assigned roles.

Health disability refers to medical conditions or their symptoms that inherently limit an individual’s capabilities. The incidence of health disability changes slowly over time as medical technology develops and different illnesses or conditions become more or less common. (Benitez-Silva, Disney, Jimenez-Martin, 2010) For the purposes of this paper, I define health disability as conditions that deny an individual gainful employment, regardless of their employer’s ability to adapt the work environment, the individual’s desire to continue working, or the individual’s willingness to change occupations.

Work disability refers to society’s or an individual’s belief that the individual is incapable of being employed. Work disability can be affected by cultural norms and labor market conditions, so its incidence changes more quickly than health disability’s. (Ibid, 2010) I define work disability as a condition that may prevent gainful employment, depending on the willingness of employers to accommodate the condition and an individual’s willingness to continue working or change occupations if necessary.

This paper is driven by Autor and Duggan (2003)’s finding that the 1984 reforms of Social Security Disability Insurance (SSDI) made it far easier for the work-disabled to be awarded benefits. Previously, an applicant’s medical diagnosis would need to meet or exceed one of a defined list of conditions. Following the 1984 legislation, greater weight was attached to applicant’s self-reported pain or discomfort, and if applicants’ medical condition was not severe, they could still be awarded benefits if they were elderly or had little education. The rigor of continuing eligibility checks also fell, lowering the risk a beneficiary would be forced to leave the program. Duggan and Imberman (2009) suggest that the 1984 reforms account for one-half of the increase in male beneficiaries and one-third of the increase in female beneficiaries and that increases in unemployment cause SSDI applications or bring them forward in time.

Further motivation comes from Autor and Duggan (2006) and Autor (2015), which examine the negative impact on SSDI’s growth on the labor market and the fiscal health of the Social Security system. Despite improvements in medical technology and legally-mandated improvements to workplace accommodation for the disabled, the gap in employment rates between middle-aged workers with and without a self-reported disability grew by ten percentage points between 1988 and 2008. SSDI’s share of annual Social Security outlays rose from

Furthermore, Autor (2015) suggests there is limited scope for direct reforms to SSDI. Directly removing beneficiaries from the program, such as attempts in the early 1980s or the 1996 removal of substance-abusers, were unsuccessful, as most removed beneficiaries successfully appealed or reapplied with a different condition. The Ticket to Work program, which offers partial disability benefits to SSDI beneficiaries who return to the labor market, has had limited uptake. Kostol and Mogdan (2014) show that a Norwegian policy similar to the proposed Benefits Offset reform, in which beneficiaries lose only $1 of benefits for every $2 they earn above the earnings threshold, boosts labor force participation by 8.5 percent. However, Norway has roughly twice the disability beneficiaries per capita of the US, so gains would likely be smaller.

Two previous papers have examined if federal unemployment benefit extensions during recessions reduce SSDI applications and awards. Mueller, Rothstein and von Wachter (2016) do not find a significant impact of Emergency Unemployment Compensation (EUC) unemployment benefit extensions on SSDI awards. Using Current Population Survey microdata, they conclude this is likely due to the lack of recent labor force participation of most awardees, which limits access to unemployment benefits. However, they find slightly more than a quarter of awardees were recently attached to the labor force, and determine the average member of this group would qualify for unemployment benefits. Even in this group though, unemployment benefit uptake is only ten percent. This suggests a limited magnitude for the impact of unemployment benefits on disability inflows. By contrast, Rutledge (2013) uses job-loss, disability application, and earnings microdata from the Survey of Income and Program Participation Gold Standard File and finds that workers who received an unemployment benefit extension are 58% less likely to apply for SSDI than those who did not. This effect held across age groups and education levels. My own research differs in that I examine the overall generosity of unemployment benefits rather than unemployment benefit duration and that I use state-level data from 1986-2016 rather than solely during the Great Recession.

Instead of unemployment benefits, Low and Pistaferri’s (2014) estimated life-cycle model with heterogeneous productivity and health examines the impact of welfare programs such as food stamps on SSDI applications and awards. They conclude that food stamps serve as substitutes for SSDI for non-disabled or partially-disabled applicants. However, more generous food stamps increased flows of highly-productive severely-disabled workers into the program, as the welfare program smoothed their consumption during the lengthy application process.

Previous works have also investigated the prevalence of work disability within SSDI, which serves as the maximum that any change in labor market factors, such as unemployment benefits, would be able to lower inflows. Maestas, Mullen & Strand (2013) and French & Song (2014) use variation in initial examination worker or appellate judge strictness to determine the proportion

of applicants that could either be accepted or rejected. For these marginal applicants, about a quarter of the total, being awarded disability benefits lowers future employment on average by 28 or 26 percent. Maestas, Mullen and Strand emphasize the heterogeneity within this group, finding that awards did not lower future employment for the most impaired marginal applicants but lowered it by 50 percent for the least impaired.

3. Empirical Results

In the United States, federal law determines the general structure of unemployment compensation programs, with each state determining the specifics of its own program. Workers become eligible to receive unemployment benefits after earning a predetermined amount or working for a percentage of the previous 12 months. They receive these benefits if laid off without personal fault. States determine many characteristics of unemployment benefits: the percentage of previous wages workers receive, the maximum or minimum amount, and the maximum duration a worker can receive benefits.

I use this variation in unemployment compensation programs across states and over time to determine the impact of an increase in unemployment benefits on the net flow into SSDI.

3a. Regressions

I first regress the logarithm of unemployment benefits on the net flow into SSDI, once without state or time fixed effects, once with state fixed effects, and once with state and time fixed effects.

Regression 1:ΔD_it= α_i+ α_t+ β₀+ β_{1 it}b + μ_it

where α_iand α_tare state and time fixed effects respectively, β₀ the constant, b_it the natural logarithm of unemployment benefits, and μ_it the error term.

In the United States, workers receive unemployment benefits only if they lose their job unwillingly and with no personal fault. In recessions, workers are less likely to quit and are more likely to be laid off, therefore unemployment benefits per unemployed worker increase during recessions. Likewise, during low points in the business cycle, the labor market becomes less attractive relative to disability benefits, likely increasing SSDI inflows. To account for this potential bias, I control for growth in real per capita state GDP to remove business cycle effects. Regression 2: ΔD_it = α_i+ α_t+ β₀+ β_{1 it}b + β₂Δgdp_it+ μ_it

where gdpΔ _it is the year-on-year percentage point change in a state’s real per capita gross domestic product.

Finally, I control for other relevant macroeconomic variables, such as the percentages of the working age population who were unemployed, employed, or beneficiaries of SSDI in the

previous year, real wages, and the percentage of SSDI applicants who were awarded benefits. I detail briefly why each control is included, and its expected sign.

Within the United States, workers on unemployment insurance receive benefits equal to a portion of their previous wage, up to a maximum level of weekly benefits set by their state government. As a state’s average wages increase, unemployment benefits should rise, provided they are not bounded by the state’s maximum weekly benefit level. At the same time, a higher average wage makes labor market participation more attractive for the marginally disabled, reducing inflows into SSDI.

Policymakers may favor more generous unemployment insurance in times or states where a greater percentage of the workforce is unemployed. Accordingly, I control for the percent of the working age population that was employed or unemployed in the previous year. As SSDI rejects applicants with gainful employment, inflows should fall as a greater percentage of the workforce becomes employed.

Another factor could be existing stocks of disability beneficiaries. As I assume that Americans’ health was constant from 1984-2016, a larger stock of beneficiaries means that a state should have fewer workers with disabilities sufficient to qualify for SSDI benefits. Additionally, a larger stock of beneficiaries suggests higher outflows, as more awardees leave the program due to reaching retirement age or dying. If states with lower unemployment benefits have larger stocks of SSDI beneficiaries, the coefficient for unemployment benefits could be biased upward. One final possibility is that unemployment benefits reduce the number of marginal disabled applying, but that the screening procedure becomes more lenient as fewer work-disabled apply, causing larger net inflows. I control for the national awards to applications ratio, except when time fixed effects render this unnecessary.

Regression 3: ΔD_it= α_i+ α_t+ β₀+ β_{1 it}b + β_{2 it}w + β₃Δgdp_it+ β_{4 t}g + β₅D_i,t−1+ β_{6 i,t−1}E + β₇U_i,t−1+ μ_it

where wit is the logarithm of average annual wages, g_t the national awards-to-applications ratio, D_i,t−1the percentage of the working age population who were disabled worker beneficiaries in the previous year, Ei,t−1the percentage of the working age population who were employed the previous year, and Ui,t−1the percentage of the working age population who were unemployed the previous year.

3b. Data Description

Net Flows into SSDI

I obtain annual data for the number of SSDI disabled worker beneficiaries by state from Table 5.J8 of the Social Security Supplement of the following year with three exceptions. The 1986 data comes from the 1988 Supplement, while the 1984 and 1985 data come from tables 130 and 133 of the 1986 Supplement respectively. Unfortunately, the number of awards and exits is restricted at the state level, so I obtain the net change in disabled worker beneficiaries by taking the difference of the number of beneficiaries in December of each year and the previous year. As states’ populations are different and experienced different rates of growth from 1984 to 2016, I normalize net changes in disabled worker beneficiaries by dividing them by annual intercensal estimates of each state’s working age population by the US Census Bureau.

Average Unemployment Benefits

To obtain the amount of benefits the average unemployed person would receive in a given year, I take the total annual unemployment insurance compensation in each state from the Bureau of Economic Analysis, then divide by the number of unemployed in that state and year. 1

Macroeconomic Controls

I obtain state-level per capita real GDP growth from the Bureau of Economic Analysis. However, state-level GDP growth data is only available from 1987, restricting the dataset by three years for regressions 2 and 3.

I use state-level average annual private wage data from the Bureau of Labor Statistics Quarterly Census of Employment and Wages.

I use the Bureau of Labor Statistics Annual Averages of Unemployment and wage and salary employment data from the Bureau of Economic Analysis to determine the number of employed and unemployed in each state annually. I then divide this by annual intercensal estimates of states’ working age population to render them comparable across states.

The annual national award to application ratio is available from the Social Security Administration, and is used as a control variable in regressions without time fixed effects.

1 _{This measure of unemployment benefits includes information on the percentage of the unemployed who qualify for} UI benefits. A regression to check robustness that only includes the average value of benefits for unemployed who qualify produces similar results. See Appendix 1.

3c. Results

Table 1: Impact of Unemployment Benefits on Net SSDI Inflows Dependent Variable: Annual Net SSDI Inflows per Working Age Capita

No Fixed Effects State Fixed Effects State and Time Fixed Effects Observations 1632 1632 1632 Unemployment Benefits (Regression 1) Log UI Benefits 0.0009*** (9.15e-05) 0.0026*** (0.0001) 0.0003* (0.0001) R2 0.049 0.257 0.001

Controlling for State Real GDP Growth (Regression 2) Observations 1530 1530 1530 Log UI Benefits 0.0008*** (9.59e-05) 0.0026*** (0.0001) 0.0002 (0.0001) State per capita GDP

Growth (0.0008) 0.0001 0.0022** (0.0007) (0.0006) 0.0002 R2 0.047 0.256 0.001 Observations 1530 1530 1530 Controlling for Macroeconomic Conditions (Regression 3) Log UI Benefits 0.0012*** (9.67e-05) 0.0028*** (0.0001) 0.0004** (0.0001) Log Wage -0.0023*** (0.0003) (0.0006) 0.0003 -0.0016** (0.0007) State per capita GDP

Growth -0.0013* (0.0007) -0.0003 (0.0007) 8.89e-05 (0.0006) National Award per

Application Ratio 4.83e-05*** (3.76e-06) 2.98e-05*** (4.21e-06) - Lagged Disability as Percent of Working Age Population 0.0002*** (1.11e-05) -6.94e-05*** (2.19e-05) -8.23e-05*** (2.21e-05) Lagged Employment as Percent of Working Age Population -7.76e-07 (1.41e-06) 5.20e-06 (7.77e-06) -9.45e-06* (5.62e-06) Lagged Unemployment as Percent of Working Age Population 0.0001*** (1.51e-05) 0.0001*** (2.08e-05) 6.01e-05*** (1.99e-05) R2 0.289 0.343 0.034

Contrary to my hypothesis, an increase in unemployment benefits increases net flow into disability for all regressions, with and without fixed effects. This change is statistically significant at the 5% level in all cases except for regressions 1 and 2 with state and time fixed effects.

Using the R-squared of Regression 1 without state and time fixed effects, variation in unemployment benefits accounts for only 5% of variation in net inflow into disability. This small R-squared is expected, as according to Mueller, Rothstein, and Von Wachter (2016), only ten percent of eligible awardees receive unemployment benefits during their application process.

3d. Awards per Application

The national award to application ratio was a statistically significant control in previous regressions. Restrictions on state-level data mean that I cannot eliminate the possibility that unemployment benefits increase the number of awards by drawing away the most capable work-disabled, thereby reducing the scrutiny of the screening process on others.

Intuitively, if Americans’ health has remained roughly constant from 1984 to 2016, the distribution of the severity of disability should remain constant across time. Workers with the most severe disabilities have the greatest incentive to apply, so as the number of applications rises, the average severity of an applicant’s disability declines. If workers with more severe injuries are more likely to be awarded benefits, as the annual number of applications rises, it is reasonable to expect that a lower percentage of applications will be awarded benefits.

I use national-level data to observe the correlation between applications and awards more generally. In _{​Figure C, annual awards appear to rise with an increase in applications, albeit far} less than one for one.

Figure C. ​National Awards and Applications per Working Age capita

Source: Author’s Calculations

To determine how the number of applicants affects the likelihood of receiving an SSDI award more precisely, I regress each year’s national award to application ratio on the number of applications per working age capita. I also run a second regression controlling for the percentage

of the working age population who are unemployed and for national per capita GDP growth, in case the screening procedure becomes more lenient during economic downturns.

Regression 4: g = α + β₁Ω+ μ

Regression 5: g = + α β Ω₁ + β₂U + β₃ΔGDP + μ

Table 2: Test for Declining Award to Applications Ratio Dependent Variable: National Award to Application Ratio

Regression (4) (5) Observations: 33 33 Constant 56.331*** (2.626) 58.913*** (4.723) Applications per Working Age capita

(Percentage Points)

-16.050*** (2.653)

-16.944*** (3.561) National Unemployed per Working Age

capita (Percentage Points)

-0.166 ( 0.769)

Real National GDP Growth per capita -0.336

(0.493)

Adjusted R2 0.527 0.503

I reject the null hypothesis that the percentage of the working age population applying does not impact the ratio of awards to applications. If the number of applications per working age capita increases by one percentage point, the likelihood of an applicant becoming a beneficiary falls by 16%.

Even at its 2010 peak, fewer than 1.5% of the national working age population applied for SSDI. At this level, while more applicants lower the award-to-application ratio, increasing the number of applicants still increases the total number of awards. Despite the heterogeneity of application rates between states, it is therefore unlikely that higher unemployment benefits raise net inflows into SSDI by keeping marginal workers in the labor force.

3e. Potential Explanations

Why then does an increase in unemployment benefits increase net flows into disability?

One possible explanation lies in the ‘experience-rating’ of unemployment insurance, employers pay additional taxes depending on the the ratio of unemployment benefits paid to former workers

to currently paid wages. If states increase the former by virtue of raising the maximum weekly benefit or maximum duration of benefits, firms have a greater incentive to push laid-off workers to apply for SSDI rather than draw benefits. However, it seems unlikely that former employers have sufficient leverage to shepherd laid-off workers through the lengthy application process if it is against workers’ interests. (_{​Autor, 2015)}

Likewise, states have a fiscal incentive to shift workers from state-funded unemployment insurance programs to the federally-funded SSDI program. Still, states have limited control over workers’ decision to apply. (Ibid., 2015)

Alternatively, Low and Pistaferri’s (2014) results suggest that increasing the generosity of welfare programs such as food stamps could increase if some workers prefer disability to employment, but were unwilling to go through a period of low consumption during the application process. Food stamps would allow them to smooth consumption, moving marginal workers into SSDI. However, unemployment benefits differ from food stamps by increasing the value of future employment, as a portion of those earnings become available if laid off. Welfare benefits do not depend on past work, so they lack this effect.

While this is the most compelling explanation, it runs counter to Rutledge’s (2013) empirical work, which shows that SSDI applications peak following the expiration of unemployment benefits. This suggests that workers see unemployment benefits as a substitute for disability benefits, rather than a method to smooth consumption during the application process.

I cannot explain why higher unemployment benefits increase the net inflow to SSDI. To formally show why I believe this result to be counterintuitive, I present my logic in a simple model in the following chapter.

4. Modeling the Effect of Unemployment Benefits

I create a simple model to formalize my intuition. The model builds upon the observed relationship between applications and the awards to applications ratio to create a clearing-mechanism for the application process. Workers’ demand for disability benefits is determined by labor market factors such as unemployment benefits and wages as well as the realized award to application ratio. Government officials’ willingness to supply new disability benefits is declining in the realized award to application ratio.

The model is capable of producing both positive and negative elasticities of net flow into disability to unemployment benefits, depending on whether the award to applications ratio is above or below its maximizing level and whether the value of wages is greater than the value of disability benefits.

In contrast to the empirical results, the calibrated model predicts a negative elasticity of net inflow into disability to unemployment benefits.

4a. Main Assumptions

The economy exists in discrete time over an infinite time horizon, and is populated by a continuum of risk-neutral workers who seek to maximize the present value of earnings. I normalize the population to one.

Workers may be employed, unemployed, or a beneficiary of the disability program. _{​E​, ​U​, and ​D} denote the share of workers in each of these states. Workers discount future earnings by r > 0 and have a retirement hazard rate, _​k​.

Retiring workers are replaced with new workers, who have a health disability with exogenous probability _{​i. ​Health-disabled workers immediately enter the disability program. New workers} without a health disability are employed with exogenous probability _{​a​, while the remainder are} unemployed.

Wages, unemployment benefits and disability benefits are exogenously determined by firms and governments. Wages and disability benefits are each higher than unemployment benefits.

When employed, workers receive an exogenous wage, _{​w​. However, they have an exogenous risk,} , of losing their job and becoming unemployed each period. Their condition is represented by

λ

[1]: V_E = 1

1+r ( + λw VU + ( − λ − k1 )V ) E

When in the disability program, workers receive a disability benefit until retirement. They cannot reenter the labor market. Their condition is represented by the following Bellman equation:

[2]: V_D = _1+r1 ( + ( − kd 1 )V ) _D

To improve readability, I define ︿was the present value of discounted wages and λ︿ as workers’ concern regarding the possibility of job loss. Similarly, d︿ is the present value of discounted disability benefits. w︿ ≡ w , and r+k+λ λ ︿ , ≡ λ r+k+λ d ︿ ≡ d r+k

The values of employment and the disability program can then be written as: [3] V_E = w︿+ λ︿V_U

[4] V_D = d︿

When unemployed, workers receive an unemployment benefit, _{​b​. All unemployed workers have} one measure of effort to expend each period. They can choose to spend it applying to the disability program or searching for work. Ω_D is the percentage of effort devoted to applying to disability, while 1-Ω_Dis the percentage devoted to searching for work. The likelihood that workers receive a job or are awarded disability benefits scales with the search effort they put in. For each measure of effort a worker puts into job search, they have an exogenous likelihood, _​a​, of becoming employed. However, the likelihood of being awarded disability benefits, _{​g​, is} endogenously determined based on Ω .

Modeling the results of regressions 4 and 5, officials of the disability program tighten their screening procedure as they handle more applications. The award-to-application ratio, _{​g​, then} falls as more unemployed apply and rises as fewer apply. With x as an positive exogenous constant, officials are willing to offer Ω_S new disability benefits each period:

[5] Ω_S = 1 − x g

quantity demanded increasing in the award-to-application ratio. Government officials’ willingness to supply disability benefits falls as the award to application ratio rises. The applications process clears when the two are equal.

Having set up the application and job search process, the Bellman equation for unemployment is: [6] V_U = 1

1+r ( + (1b − ΩD)aVE + ΩDgVD+ ( − (11 − ΩD)a− ΩDg− k)V )U

4b. Application Clearing

The value of unemployment is determined by the choice between applying for disability or searching for a job. The unemployed allocate their one measure of effort so that the value of unemployment is maximized.

Depending on the award to application ratio, _{​g​, the unemployed would typically choose to} allocate all or none of their effort toward applying for disability. However, this would cause the number of applications to exceed or fall short of the applications government officials will allow at that award to application ratio. In those scenarios, the application process fails to clear.

A unique awards to applications ratio, g*, _{​makes unemployed workers indifferent between} searching for jobs and applying for disability. At g*, unemployed workers can set their application effort equal to government officials’ application demand in order to clear the application process. The application process is depicted in Figure D below.

Figure D. ​Supply and Demand for Disability Benefits

4c. Equilibrium

Given wages, unemployment benefits, and disability benefits as well as rates of job loss, hiring, retirement, health disability and future discounting ( , a, k, i, r)λ , the equilibrium is the series of values for the award-to-application ratio (_{​g​), supply of disability benefits (} Ω_S), demand for disability benefits (Ω_D), value of unemployment ( V_U),value of employment ( V_E), and the shares of workers in employment, unemployment, and the disability program ( _{​E​, ​U​, ​D​ )}

such that [1]-[6] are satisfied, supply and demand for disability benefits (Ω_S, Ω )_D are equal, and the demand for disability benefits and the award-to-application ratio maximize the value of unemployment, [6].

I list the unique equilibrium values below, see Appendix 2 for a mathematical proof. The equilibrium value of g is given by:

[7] _​

g

*

₌

a(1−λ)(b−w) ︿

b−d + a (w−d) λ+r+k

At the equilibrium award to application ratio, government officials supply the equilibrium amount of awards, which is also the equilibrium demand for disability benefits:

[8] Ω_S* _{= Ω}

D* = 1 − x a(1−λ)(b−w)

︿

b−d +λ+r+ka (w−d) Then, the equilibrium values of unemployment and employment are: [9] _​

V

* U

=

b+(1−Ω )aw + Ω g d* ︿ * *︿ r+k+(1−Ω )a(1−λ)+Ω g* ︿ _{* *} [10] _​

V

* E

= w

︿

+ λ

︿ b+(1−Ω )aw + Ω g d* ︿ * * ︿ r+k+(1−Ω )a(1−λ)+Ω g* ︿ _{* *}

Lastly, the equilibrium shares of workers in employment, unemployment or the disability program are:

[11]

E

*

₌

k+λ(1−_{k+(1−Ω )a+Ω g}(1−Ω )a_** _{* *}) (1−i)ak(1+_{k+(1−Ω )a+Ω g}(1−Ω )(1−a)_** _{* *})

[12]

U

*

=

(1−i)k(1−a(1− ))

λ λ+k

k+Ω g +(1−Ω )a(1−* * * λ )

[13]

D

*

=

Ω g

* * (1−i)(1−a(1− )) λ λ+k k+Ω g +(1−Ω )a(1−* * * λ ) λ+k

+ i

4d. The Elasticity of Equilibrium Share of Workers in the Disability Program to Unemployment Benefits

Having found the equilibrium values of application demand and the award-to-application ratio, I calculate the elasticity of net flow into disability to unemployment benefits.

An increase in unemployment benefits shifts the level at which unemployed workers are indifferent between applying for disability and searching for work.

[14] dg_db*

=

(b−d+ a (w−d)) λ+r+k 2

a(1−λ)(1+︿ a )(w−d) λ+r+k

If wages are higher than disability benefits, fewer workers apply for disability benefits when unemployment benefits are increased. This increases the award-to-application ratio. If disability benefits are higher than wages, more workers will apply when unemployment benefits increase, driving down the award-to-application ratio.

I then calculate how a change in unemployment benefits impacts the equilibrium share of workers in the disability program. I find that an increase in unemployment benefits could either increase or decrease the share of disability beneficiaries, depending on whether an increase in unemployment benefits raises or lowers the award-to-application ratio and whether the award-to-application is above or below a threshold set by parameters.

[15] dD_db*

= ( − i

1

)(1

− a −

(1

_λ+kλ

))

_{(k+g (1−xg )+axg (1−* * * λ ))} λ+k 2 ((1 − 2xg )k − ax g (1 − * 2 * 2 λ ))

λ+k db dg*

Lastly, I use this to write the elasticity of equilibrium share of workers in the disability program to unemployment benefits. [16]

_​ dD_db*_Db*

= ( − i

1

)(1

− a −

(1

_λ+kλ

))

_{(k+g (1−xg )+axg (1−* * *} λ ₎₎ λ+k 2 ((1 − 2xg )k − ax g (1 − * 2 * 2 λ )) λ+k db dg* _b D* 4e. Decomposition

I now describe the model’s results. An increase in unemployment benefits can increase or decrease the award-to-application ratio. These changes in the award-to-application ratio affect the equilibrium share of workers in the disability program by adjusting the percentage of the unemployed entering into the disability program each period and by enlarging or shrinking the equilibrium share of the unemployed.

As disability increases in relative attractiveness, the award to application ratio falls

The awards to applications ratio, g*, is realized at the level at which unemployed workers are indifferent between applying to disability and searching for jobs. When deciding between these options, workers consider both the value of disability benefits relative to wages and the likelihood that they would be awarded a disability benefit against the likelihood they would find a job. If a change in labor market factors makes disability more attractive, the awards to applications ratio must fall to maintain parity. Likewise, if disability becomes less attractive, the awards to applications ratio must rise to maintain indifference.

Recall that as the awards to applications ratio rises, there are fewer applicants. In this way, the model uses the attractiveness of disability relative to the labor market to determine applicant flows in line with intuition.

Increasing unemployment benefits could make disability application more or less attractive

How does the realized awards to applications rate move with b? In dg_db*

=

_(b−d+ a _(w−d)) ,

λ+r+k 2

a(1−λ)(1+︿ a )(w−d) λ+r+k

the critical component is w − d . As the denominator is squared and both components of are always positive, determines whether an increase in

(1

)(1

)

a − λ

︿

+

a

λ+r+k w − d

unemployment benefits increases or decreases the awards to application rate.

Higher unemployment benefits encourage workers to seek the most highly paid option

Intuitively, the dependence of dg_db* on w − d makes sense. Higher unemployment benefits make failure to find a job or receive disability benefits less costly. Accordingly, workers place less weight on their likelihood to succeed in the job search or disability application process, and more on the returns of each path. If wages are higher than disability benefits, workers will increasingly prefer to apply for jobs. The award-to-application ratio must rise in response to restore indifference. The reverse is true if workers value disability benefits above wages.

One award-to-application rate maximizes the percentage of unemployed entering disability

The award-to-application ratio rises as more of the unemployed apply, and falls as more unemployed do. Because of the quadratic nature of g (Ω)Ω * , the percentage of the unemployed entering the disability program each period is maximized when the award-to-application ratio is

equal to 1 . Increases in the award-to-application ratio above result in fewer awards as the

2x 2 2x1

decreasing number of applicants outweighs applicants’ enhanced likelihood of being awarded disability benefits. Decreases below 1 likewise result in fewer awards, as the increase in

2x

applicants is outweighed by applicants’ diminishing likelihood of success. The opposite relations hold for changes that move the award-to-application ratio closer to 1 .

2x

Increases in equilibrium application effort also increase application eligibility

The term −ax g (1 − 2 * )

​

reflects how the equilibrium share of unemployed workers

2 _λ

λ+k

(k+g (1−xg )+axg (1−* * * _λ+kλ ))2

changes following a change in the award to application ratio.

Intuitively, additional application effort not only changes an unemployed worker’s chance of gaining disability benefits, it also reduces the worker’s chance of finding a job. This increases the equilibrium share of unemployed workers relative to employed workers, so that more workers will apply for disability in any period. Likewise, a decrease in application effort increases workers’ chance of finding a job, meaning that there are fewer unemployed to apply for the disability program each period.

2 _{This effect can be isolated by treating the equilibrium share of unemployed workers as a constant, then} differentiating [13].

4f. Testing the Model

I use comparative statics to determine whether the model predicts a positive or negative elasticity of the equilibrium share of workers in the disability program _{​to unemployment benefits in} conditions that resemble the US economy from 1984 to 2016.

Parameters Calibrated Value Target

λ

(exogenous risk of job loss)

0.09 The annual likelihood to be laid off or discharged, discounted by the likelihood to have obtained another job within 3 months 3

k

(retirement hazard rate)

0.02 With an even population distribution, 1/49 of the 15-64 year old population retires each

year. I round this to 1/50.

i

(percentage of population with health-disabilities)

0.015, 0.02 0.03

I choose a range of possible i, as the true incidence of health disability is unknown. For the lowest, I use the approximate percentage of the working age population that were SSDI beneficiaries in 1984. The 4 others are more lenient versions of this base.

a

(job-finding rate with full search effort)

0.75 As a base, I use Hall and Schulhofer-Wohl’s 60% likelihood of a worker who was laid off months ago having a job in 12 months time. As the modeled job finding rate will fall based on application effort, I divide the base by 0.8 to account for the average number of applications per unemployed person from

1984-2016, ~.2.

x

(parameter for the strictness of work-disability screening)

1.775 I use the awards to applications regression coefficient if the constant

was standardized to one.

r

(discount rate)

.04 The average net real interest rate from 1984 to 2016

Three parameters, _{​w, d, ​and ​b remain free. Following Mueller, Rothstein, and Von Wachter, I} calibrate wages at two-thirds of the national average for 1984-2016, or about 25 thousand dollars. For _{​b​, I use the national annual average for 1984-2016, or about 5 thousand.}

The value of the disability benefit is the sum of the average annual in-cash benefit, the average annual value of Medicare for disabled beneficiaries, and the value of the additional leisure from

3 _{I combine JOLTS data with the analysis and results of Hall and Schulhofer-Wohl (2015). The typical exogenous} job-loss rate in the literature is calibrated monthly using JOLTS. To calibrate on an annual basis, I need to find the likelihood of losing one’s job without regaining it in the intervening months.

4 _{Prior to 1984, the Reagan administration sharply tightened SSDI award screening and increased scrutiny of} continuing disability reviews on existing beneficiaries. I assume only the health-disabled remained in the program

neither working nor searching for work. Social Security Administration data suggests an average annual real in-cash benefit of $11,500 from 1984-2016, and Gelber, Moore, and Strand’s (2017) estimate for the value of Medicare health insurance is approximately $6,500 in 2009 dollars. I then set the leisure utility of disability in order to target the average, maximum, and minimum award-to-application ratios from 1984-2016, 41%, 52% and 32% respectively. This suggests an average value of _{​d​ of 24.1 thousand dollars, a maximum ​d​ of 24.8, and a minimum ​d​ of 23.6.} Then, to determine whether the model suggests a positive or negative elasticity of equilibrium share of disability beneficiaries to unemployment benefits, I compare the equilibrium share of disability beneficiaries under the base parameters and when unemployment benefits are increased by one thousand dollars.

Table 3. Model Comparative Statics

5

d w b D Δ D Average _​g 24.1 25 5 0.428 24.1 25 6 0.417 -.011 Minimum _​g 24.8 25 5 0.517 24.8 25 6 0.516 -.001 Maximum g 23.6 25 5 0.211 23.6 25 6 0.139 -.072

For the range of award-to-application ratios experienced between 1984 and 2016, the model

predicts that the equilibrium share of workers in the disability program will fall. This result is the opposite of the empirical findings.

Additionally, the calibrated model suggests that unemployment benefits decrease the inflow into disability most when workers value wages more highly than disability benefits. This reiterates that, by making failure to leave unemployment less painful, unemployment benefits encourage pursuit of the most highly paid state. When the difference between wages and disability benefits is smaller, the magnitude of this effect is reduced.

5 _{Each was tested with health disability, ​i, ​of .015, .02, and .03. However, as the effect of ​i ​on the change in} equilibrium D was small, I list only the comparative statics for i = .02

Finally, the model greatly overpredicts the equilibrium share of the population in the disability program. Partially, this is due the lack of other types of nonemployment, which likewise causes the model to overpredict the share of employed and unemployed. Another cause is that SSDI beneficiaries are older and in poorer health than the average member of the working age population. As such, their retirement rate (including the possibility of death) is likely higher than unemployed or employed workers. Because of these factors, the magnitude of the calibrated model’s estimate for the effect of an increase in unemployment benefits is likely inaccurate. However, neither of these factors should affect the sign of the change in equilibrium share of disability beneficiaries.

5. Conclusion

The Social Security Disability Insurance (SSDI) program provides necessary support to workers whose conditions keep them from gainful employment. However, its lenient screening procedure also awards benefits to workers who would be able to work in certain fields or with greater workplace accommodation. By drawing in these marginal workers, SSDI reduces labor force participation and incurs higher costs. With little political will to change the structure of the SSDI program, policymakers should consider alternative methods to keep marginal workers attached to the labor force. Unemployment benefits are more easily adjusted.

It is reasonable to expect that higher unemployment benefits encourage marginal workers to continue searching for jobs rather than applying for SSDI benefits. Unemployment benefits directly add to the value of participating in the labor market, both now and in the future, and could also allow credit-constrained workers to engage in retraining programs or otherwise seek to connect with a healthier industry if their old one is doing poorly. If their job search is initially unsuccessful, benefits tied to search incentivize further effort.

However, I find that, empirically, unemployment benefits increase net inflow into disability. If the average unemployment benefit provided in the US were to rise by one percent, 77 thousand more beneficiaries would enter the program annually.

To help understand this puzzling result, I develop a search model to formalize my intuition. The resulting model is capable of generating a positive or negative elasticity of net flow into the disability program to unemployment benefits. However, the calibrated model predicts a decrease in net inflows into SSDI in response to an increase in unemployment benefits.

As a logical exercise, the model succeeds at its goals, but has weaknesses when calibrated. It greatly overpredicts the equilibrium share of workers receiving disability benefits, and its application process does not clear if there is a large difference in the values of wages and disability benefits. A heterogenous health status, with screening leniency and the value of disability benefits falling with health, could resolve both of these issues. Another valuable change would be to assume workers have constant relative risk aversion rather than risk neutral preferences. This would provide a more realistic method of evaluating how the insurance value of unemployment benefits impacts net flow into disability, as well as changing how workers value consistent disability benefits relative to labor market participation. The impact of unemployment benefits could be realistically moderated by including other types of nonemployment, in which actors face a similar choice between looking for a job or applying for a disability, but do not receive unemployment benefits. Other nonemployment also allows a

more meaningful interpretation of the duration of unemployment benefits, as unemployed workers could exhaust their benefits by being unemployed for a number of periods.

Empirically, improvements are also possible. For researchers with access to restricted SSDI state-level award, application, and outflow data, the impact of unemployment benefits on net flow into disability could be decomposed into its constituent effects. As an alternative to dollar amounts, research could use discrete state legislative changes to unemployment compensation programs as an independent variable. This method could be used to isolate the effects of changes to maximum and minimum benefits, maximum benefit duration, or the percentage of wages paid in unemployment benefits to better inform future policy choices.

Perhaps these methodological improvements will be sufficient to resolve the puzzle: why does an increase in unemployment benefits increase net flows into SSDI? Until economists are able to reconcile this gap between intuition and data, we will be unable to explain the optimal use of unemployment benefits in relation to disability insurance.

6. Bibliography

Autor, D.H. (2015). "_{​The unsustainable rise of the disability rolls in the United States: causes, consequences and} policy options​," ​Chapters​,in: Social Policies in an Age of Austerity, chapter 5, pages 107-136 Edward Elgar Publishing.

Autor, D. H., & Duggan, M. (2006). The Growth in the Social Security Disability Rolls: A Fiscal Crisis Unfolding.

Journal of Economic Perspectives,​​20​ (3), 71-96.

Autor, D. and Duggan, M. (2003). The Rise in the Disability Rolls and the Decline in Unemployment. _​The

Quarterly Journal of Economics​, 118(1), pp.157-206.

Benítez-Silva, H., Disney, R. and Jiménez-Martín, S. (2010). Disability, capacity for work and the business cycle: an international perspective. _{​Economic Policy}​, 25(63), pp.483-536.

Isaacs, K.P. (2018). ​Unemployment Insurance: Consequences of Changes in State Unemployment Compensation Laws​.

French, E. and Song, J. (2014). The Effect of Disability Insurance Receipt on Labor Supply. _{​American Economic}

Journal: Economic Policy​, 6(2), pp.291-337.

Gelber, A., Moore, T. and Strand, A. (2017). The Effect of Disability Insurance Payments on Beneficiaries' Earnings. _{​American Economic Journal: Economic Policy}​, 9(3), pp.229-261.

Hall, R. and Schulhofer-Wohl, S. (2018). Measuring Job-Finding Rates and Matching Efficiency with Heterogeneous Job-Seekers. _{​American Economic Journal: Macroeconomics}​, 10(1), pp.1-32.

Kitao, S., 2014. A life-cycle model of unemployment and disability insurance. ​Journal of Monetary Economics ​, 68, pp.1–18

Kostøl, A. and Mogstad, M. (2014). How Financial Incentives Induce Disability Insurance Recipients to Return to Work. ​American Economic Review​, 104(2), pp.624-655.

Liebman, J. (2015). Understanding the Increase in Disability Insurance Benefit Receipt in the United States. _​Journal

of Economic Perspectives​, 29(2), pp.123-150.

Low, H. and Pistaferri, L. (2015). Disability Insurance and the Dynamics of the Incentive Insurance Trade-Off.

American Economic Review​, 105(10), pp.2986-3029.

Maestas, N., Mullen, K. and Strand, A. (2013). Does Disability Insurance Receipt Discourage Work? Using Examiner Assignment to Estimate Causal Effects of SSDI Receipt. ​American Economic Review​, 103(5), pp.1797-1829.

Michaud, A., Nelson, J. & Wiczer, D., 2016. Vocational Considerations and trends in Social Security Disability. ​The

Michaud, A. & Wiczer, D., The Disability Option: Labor Market Dynamics with Macroeconomic and Health Risks.

Working Paper​.

Mitman, K. and Rabinovich, S. (2014). Unemployment Benefit Extensions Caused Jobless Recoveries!?. _​SSRN

Electronic Journal​.

Mueller, A., Rothstein, J. and von Wachter, T. (2016). Unemployment Insurance and Disability Insurance in the Great Recession. _{​Journal of Labor Economics}​, 34(S1), pp.S445-S475.

Rutledge, M. (2013). The Impact of Unemployment Insurance Extensions on Disability Insurance Application and Allowance Rates. _{​SSRN Electronic Journal}​.

Yin, N. (2015). Sicker and Poorer. ​Health Services Research and Managerial Epidemiology​, 2, p.233339281557158.

Appendix 1: Robustness Check

Regressions 1-3 are affected by changes in the portion of the unemployed who are ineligible for unemployment benefits. Notably, a person only receives unemployment benefits in the US if they are laid off for no fault of their own. If as David Autor (2013) writes, “workers are

disproportionately likely to apply for SSDI benefits when they involuntarily lose work—even if their job loss is unrelated to their health”, the first regression could be biased.

To see if this is the case, I regress annual net inflow into SSDI on the logarithm of the product of the average duration of state unemployment benefits and that state’s average weekly benefit, using data from Department of Labor Employment and Training Administration. I include in the 6

duration additional weeks of unemployment benefits available through the federal Emergency Unemployment Compensation programs. These programs provided additional weeks of

unemployment benefits during recessions, at beneficiaries’ previous level of weekly benefits. If EUC benefit extensions expired before September, I halved the increase in duration for the final year.

I use the same functional forms as regressions 1-3, only with replaced by the logarithm of theb_it alternative measure.

Regression 6: ΔD_it= α_i+ α_t+ β₀+ β_{1 it}b + μ_it

Regression 7: ΔD_it= α_i+ α_t+ β₀+ β_{1 it}b + β₂Δgdp_it+ μ_it

Regression 8: ΔD_it= α_i+ α_t+ β₀+ β_{1 it}b + β_{2 it}w + β₃Δgdp_it+ β_{4 t}g + β₅D_i,t−1+ β_{6 i,t−1}E + β₇U_i,t−1+ μ_it

Using this measure, unemployment benefits continue to have a positive impact on net inflow to SSDI. The size of the effect grows when controlling for macroeconomic variables, although the coefficient of unemployment benefits is only significant at the 10% level when controlling for macroeconomic variables with time and state fixed effects.

6 _{Data quality is an issue here. Numerous data entries were left blank. I replaced blank entries with the average of the} values immediately before and after the gap

Table 3

Dependent Variable: Annual Net SSDI Inflows per Working Age Capita Observations: 1530

OLS State Fixed Effects State and Time Fixed Effects Unemployment

Benefit

Log Average Benefit 0.0009***

(9.95e-05) 0.0014*** (0.0001) (0.0002) 0.0002

R2 0.045 0.108 0.001

Controlling for Real GDP Growth

Log Average Benefit 0.0008***

(0.0001) 0.0013*** (0.0001) (0.0003) 0.0002 Real per capita GDP

Growth 0.0003 (0.0008) 0.0015* (0.0008) 0.0002 (0.0006) R2 0.039 0.101 0.000 Controlling for Macroeconomic Variables

Log Average Benefit 0.0013*** (0.0001)

0.0021*** (0.0001)

0.0005* (0.0003) Log Wages -2.23e-03***

(0.0003)

0.0007 (0.0006)

-0.0017** (0.0007) Real per capita GDP

Growth -0.0015** (0.0007) (0.0007) -0.0007 -2.00e-05 (0.0005) Award to Applications 5.67e-05***

(3.80e-06) 3.69e-05*** (4.51e-06) - Lagged D 0.0001*** (1.12e-05) -0.0001*** ( 2.35e-05) -8.22e-05*** (2.23e-05) Lagged U 0.0001*** (1.60e-05) 6.76e-05 (2.3934e-05) 5.27e-05*** (1.99e-05) Lagged E -8.32e-07

(1.43e-06) 1.84e-05** (8.29e-06) -1.01e-05* (5.83e-06)

Appendix 2: Modeling Proof

In this Appendix, I show the equilibrium objects of the model as well as the elasticity of the equilibrium share of workers in the disability program to unemployment benefits.

Equilibrium Objects

First, I wish to find _{​g​*, the award to application ratio at which unemployed workers are} indifferent between searching for a job and applying for disability. This is the only award to application ratio at which applications submitted and processed can be equal, allowing the application process to clear.

I restate the Bellman equations that define the value of employment, disability and unemployment. [1]: V_E = 1 1+r ( + λw VU + ( − λ − k1 )V ) E [2]: V_D = _1+r1 ( + ( − kd 1 )V ) _D [3] V_U = 1 1+r ( + (1b − Ω)aVE+ ΩgVD+ ( − (11 − Ω)a− Ωg− k)V ) U

Rewrite [1] and [2] as:

V_E = w+λV_r+k+λU V_D = d r+k Define w︿ ︿. λ, and as︿d , w ︿ ≡ w r+k+λ λ ︿ , d ≡ λ r+k+λ ︿ ≡ d r+k Therefore, [4] V_E = w︿+ λ︿V_U [5] V_D = d︿

V_U = 1 1+r ( + (1b − Ω)a(w V ) gd 1 a g )V ) ︿ + λ︿ _U + Ω ︿+ ( − (1− Ω) − Ω − k _U Sort by V_U → aw gd (r+ k + ( − Ω1 )a(1− λ︿)+ Ωg)V_U = b + (1− Ω) ︿+ Ω ︿ [6]_​

u

V =

b+(1−Ω)aw + Ωgd︿ ︿ r+k+(1−Ω)a(1−λ)+Ωg︿

At the unique awards to applications ratio, _​g_{​*, the unemployed are indifferent between searching} for work or applying for disability. Then, at _{​g​*, any change in} Ω does not change the value of unemployment.

→

dΩ dV_U

= 0 :

(r+k+(1−Ω)a(1−λ)+Ωg)︿ 2 (−aw+gd)(r+k+(1−Ω)a(1−λ)+Ωg) − (−a(1−λ)+g)(b+(1−Ω)aw +Ωgd)︿ ︿ ︿ ︿ ︿ ︿ → − w d)(r 1 )a(1 ) g) (− (1 ) )(b 1 )aw gd) 0 = ( a︿+ g︿ + k + ( − Ω − λ︿ + Ω − a − λ︿ + g + ( − Ω ︿ + Ω ︿ w(r ) w(1 )(1 ) wΩg d da(1 )(1 ) dΩ 0 = − a︿ + k − a2︿ _{− Ω − λ}︿ _{− a}︿ _{+ g + g}︿ _{− Ω − λ}︿ _{+ g}2︿ → − a(1 ) w(1 )(1 ) daΩ(1 ) g w(1 )g dΩ) − ( b − λ︿ − a2︿ _{− Ω − λ}︿ _{− g}︿ _{− λ}︿ _{+ b + a}︿ _{− Ω + g}2︿ w(r ) (d ) da(1 ) a(1 ) aw 0 = − a︿ + k + g − b + g︿ − λ︿ + b − λ︿ − g ︿ Sort by g b ( w (1 )))g (b(1 ) (r )) ( − d + a ︿− d︿ − λ︿ = a − λ︿ − w︿ + k Isolate g to find _​g​*

g

*

₌

a(b(1−λ) − w(r+k)) ︿ _︿ b−d +a(w − d(1−λ))︿ ︿ ︿ Simplify using 1−λ_r+k︿ ₌ 1 λ+r+k

[7] g

*

₌

a(1−λ)(b−w) ︿ b−d + a (w−d) λ+r+k

I can use _{​g​* to find the supply of new disability benefits. The formula for the supply of new} disability benefits is:

[8] Ω_S = 1 − x g Substituting [7] into [8]:

Ω

_S

= 1 − x

a(1−λ)(b−w) ︿ b−d + a (w−d) λ+r+k

To clear the application process, the supply of new disability benefits must equal demand.

→ Ω_S = Ω_D = Ω [9]

Ω

*

= 1 − x

a(1−λ)(b−w) ︿ b−d + a (w−d) λ+r+k

Substitute [7] and [9] into [6] to find the equilibrium value of unemployment.

[10] _​

V

*

U

=

b+(1−Ω )aw + Ω g d* ︿ * *︿

r+k+(1−Ω )a(1−λ)+Ω g* ︿ _{* *}

Then substitute [10] into [4] to find the equilibrium value of employment: [11] _​

V

*_E

= w

︿

+ λ

︿ b+(1−Ω )aw + Ω g d*

︿ _{* *}︿

r+k+(1−Ω )a(1−λ)+Ω g* ︿ _{* *}

Lastly, I use [7] and [9] in order to determine the equilibrium shares of employed, unemployed, and disabled workers (_{​E​, ​U​, ​D​).}

[12] E = ( − λ − k1 )E+ ( − Ω1 *)aU + ( − i1 )ak

[13] U = ( − k − ( − Ω1 1 *)a_{− Ω}* *g )U_{+ λ + ( − i}E 1 )(1_{− a} )k [14] D = ( − k1 )D+ Ω* *g U _{+ k} i

I begin with the equilibrium share of employed workers, _​E​. Rewrite [13] as:

U = _{k+(1−Ω )a+Ω g}λE+(1−i)(1−a)k* * *

Then substitute this into [12]:

1 )E 1 )a 1 )ak

E = ( − λ − k + ( − Ω* λE+(1−i)(1−a)k

k+(1−Ω )a+Ω g* * * + ( − i

E = λ+k−_{k+(1−Ω )a+Ω g}(1−Ω )aλ** * * (1−Ω )a* (1−i)(1−a)k +(1−i)ak k+(1−Ω )a+Ω g* * * → [15]_​

E =

k+λ(1−_{k+(1−Ω )a+Ω g}(1−Ω )a_** _{* *}) (1−i)ak(1+_{k+(1−Ω )a+Ω g}(1−Ω )(1−a)_** _{* *})

I continue with the equilibrium share of unemployed workers, _​U​. Rewrite [12] as:

E = (1−Ω )aU+(1−i)ak* _λ+k

Then, substitute this into [13]:

1 1 )a g )U 1 )(1 )k

U = ( − k − ( − Ω* _{− Ω}* * _{+ λ}

λ+k (1−Ω )aU+(1−i)ak*

+ ( − i − a

Solve for _​U​:

U = λ λ+k +(1−i)(1−a)k (1−i)ak

k+(1−Ω )a+Ω g −* * * λ+k λ(1−Ω )a* →

[16]

U =

(1−i)k(1+λ+kaλ−a)

k+Ω g +(1−Ω )a(1−* * * λ ) λ+k

The final equilibrium object is the share of workers in the disability program, _​D​.

Substitute [16] into [14]:

1 )D g i

D = ( − k + Ω* * (1−i)k(1+λ+kaλ−a)

k+Ω g +(1−Ω )a(1−* * * λ )

λ+k + k →

[17]_​

D = Ω

* *

g

(1−i)(1+λ+kaλ−a)

k+Ω g +(1−Ω )a(1−* * * λ ) λ+k

+ i

The Elasticity of the Equilibrium Share of Disability Beneficiaries to Unemployment Benefits

Having found the equilibrium award-to-application ratio, disability benefit supply and demand, and the equilibrium share of the population in the disability program, I calculate the elasticity of equilibrium share of disability beneficiaries to unemployment benefits.

Taking the derivative of [17] with respect to b: db

dD*

=

(k+Ω g +(1−Ω )a(1−* * * λ ))2

(1−i)(1+aλ−a)( (k+Ω g +(1−Ω )a(1− ))−Ω g ( −a(1− ) )) λ+k db

dΩ g* * _{* * * λ}

λ+k * * db

dΩ g* * _λ

[18] dD_db*

=

(k+Ω g +(1−Ω )a(1−* * * λ )) λ+k 2

(1−i)(1+aλ−a)( (k+(1−Ω )a(1− ))+ Ω g a(1− )) λ+k db

dΩ g* * _{* λ}

λ+k dΩ*db * * λ+kλ

Next, I calculate dΩ_db* and dΩ g_db* * in terms of dg_db*

[19]_​ dΩ_db*

= − x

dg_db*

= g

Ω* * _g*_{− xg}*2

[20] dΩ g_db* *

=

dg_db*

− 2

x

dg_db*

g

*

Using [7], take the derivative of g*with respect to b

→

db dg*

=

(b−d+aw−da(1−λ))︿ ︿ ︿ 2 a(1−λ)(b−d+︿ a (w−d))−a(1−λ)(b−w) λ+r+k ︿ [21] dg_db*

=

(b−d+ a (w−d)) λ+r+k 2 a(1−λ)(1+︿ a )(w−d) λ+r+k

Using [9], [19], and [20], simplify [18] db

dD*

=

(k+Ω g +axg (1−* * * λ )) λ+k 2

(1−i)(1+aλ −a)( (k+axg (1− ))− Ω axg (1− )) λ+k db dΩ g* * _{* λ} λ+k db dg* * _{* λ} λ+k

_→

db dD*

=

_{(k+Ω g +axg (1−* * *} λ ₎₎ λ+k 2 (1−i)(1−a(1− λ ))( k+axg (1− )( − Ω )) λ+k db dΩ g* * _{* λ} λ+k db dΩ g* * db dg* *

→

db dD*

=

(k+g (1−xg )+axg (1−* * * λ )) λ+k 2 (1−i)(1−a(1− λ ))(( −2x g )k+axg (1− )( −2x g − +x g )) λ+k db dg* db dg* * * λ λ+k db dg* db dg* * db dg* db dg* *

→

db dD*

=

(k+g (1−xg )+axg (1−* * * λ )) λ+k 2 (1−i)(1−a(1− λ ))(( −2x g )k−ax g (1− ) ) λ+k db dg* db dg* * 2 * 2 λ λ+k db dg*

1

)(1

(1

))

db dD*

= ( − i

− a −

_λ+kλ _{(k+g (1−xg )+axg (1−* * * λ ))} λ+k 2 ((1 − 2xg )k − ax g (1 − * 2 * 2 λ )) λ+k db dg*