Unemployment Insurance Savings Accounts in a life-cycle model with job search

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1 Unemployment Insurance Savings Accounts in a life-cycle model with job search Sijmen Duineveld University of Augsburg October 19, 2015 Abstract The differences between a traditional Unemployment Insurance (UI) system and a system with individual Unemployment Insurance Savings Accounts (UISA) are analyzed in a life-cycle model with savings and job search. The disutility of working is calibrated on the elasticity of unemployment duration with respect to the level of unemployment benefits for the U.S. It is found that UISAs can provide more insurance and improve incentives at the same time as the optimal replacement rate with UISAs is much higher, while the unemployment rate is lower. Compared to an optimal traditional UI system, an optimal UISA system increases welfare with around 25% of the Net Present Value of all UI contributions. Keywords: Unemployment Insurance, Unemployment, Job Search, Moral Hazard, Life Cycle JEL Classification: J65, D91, H31, D82 1 Introduction This research investigates whether an Unemployment Insurance (UI) system with individual Unemployment Insurance Savings Accounts (UISA) can improve welfare compared to a traditional Unemployment Insurance system. The main difference between a traditional UI system and a UISA system is that in the latter the unemployment insurance contributions are added to a personal savings account instead of a public unemployment insurance fund. If one gets unemployed in the UISA system, the unemployment benefits are funded by withdrawals from the UISA. If the balance of the UISA is positive at retirement age, the savings are added to private wealth. If negative, they are bailed out. I am grateful to the participants of the Public Finance Research Seminar at the Ifo- Institute in Munich for their suggestions. 1

2 1.1 Theoretical advantage of UISA Research has shown that a large part of the welfare state does not redistribute income from high to low (lifetime) income households but merely redistributes income over the life-cycle. Bovenberg, et al. (2008) find that in Denmark as much as 75% of total taxes (for funding public transfers) are used only for redistribution over one s life-cycle instead of redistribution between different people. This means there is a weak actuarial link between the taxes one pays and the benefits one receives, so the labour market is distorted more than necessary. In addition to that benefits also create moral hazard that results in overuse of social institutions (Bovenberg, et al., 2008). Individual savings accounts can reduce both the distortion in the labour market and moral hazard at a relative low cost in terms of increased inequality in life-time income (Bovenberg, et al., 2008). Unemployment Insurance Savings Accounts can have two potential benefits, which are providing liquidity insurance and lifetime redistribution. Liquidity insurance With perfectly functioning capital markets everyone would be able to smooth his lifetime consumption, but many unemployed have limited liquidity (Chetty, 2008) and may face borrowing constraints. With UISAs the government provides credit with future contributions as collateral (Bovenberg, et al., 2008). This means that there is no distorting in the effort to find a job for those agents that expect a positive UISA balance at retirement (Feldstein, 2005), since these agent completely self-insure themselves with the government merely functioning as creditor. Lifetime redistribution With a UISA system redistribution towards the lifetime poor occurs via the bailing out of negative balances at retirement age. As a result it is difficult to improve labour market incentives for the poor by introducing UISAs. For example, in a UISA system with a lower limit on liability, someone at this lower limit has the same incentives as in a classic UI system: he has to pay UI contributions which act like a tax if he expects the balance on his UISA to remain negative. Similarly his withdrawals when unemployed are like normal unemployment benefits. So for agents that expect a negative balance on their UISA at retirement there is no improvement compared to the current UI system in the U.S. (Feldstein, 2005). The UISA can therefore not avoid the high marginal tax rate at low lifetime incomes as a byproduct of redistribution (Bovenberg, et al., 2008). The gains from a UISA system crucially depend on the correlation between income shocks over the life-cycle. If shocks are uncorrelated over time and individuals, meaning when income shocks are small compared to lifetime incomes, then there is ample scope for self-insurance (Bovenberg, et al., 2008). 2

3 1.2 Review of literature In order to analyze optimal Unemployment Insurance schemes in a realistic way the use of a life-cycle perspective and search intensity as a choice variable seem crucial. The life-cycle perspective is important because an environment with infinitely lived agents makes redistribution less relevant as (discounted) income varies less between agents, so less income has to be redistributed. In addition, liquidity insurance can be implemented more easily without creating moral hazard as agents have infinite time to repay their debts. In fact, models with infinitely-lived agents and an interest rate equal or higher than one divided by the discount rate have the odd feature that agents will accumulate infinite assets as a means of self-insurance (Chamberlain and Wilson, 2000). The importance of search effort as the most important choice variable is mentioned by Lentz (2009). He argues that empirical evidence from Devine and Kiefer (1991) shows that rejection or acceptance of job offers plays a small role in the observed unemployment hazard rate variation and indicates that search intensity is the most important driver for unemployment hazard. For these reasons the model in this paper features agents with both a finite life and job search. In addition, agents are able to accumulate assets. In the literature on optimal unemployment insurance combining these three aspects has only been done earlier by Heer (2006), who uses a very similar model, but does not evaluate Unemployment Insurance Savings Accounts. In terms of the analyzed institutions this paper is most close to the paper by Hopenhayn and Hatchondo (2011). Hopenhayn and Hatchondo (HH henceforth) use a life-cycle model with wealth accumulation to analyze different Unemployment Insurance systems, including UISA systems. In HH s model job offers arrive exogenously with a wage that depends only on the characteristics of the worker. This reduces moral hazard as public and private interests are more aligned. More precisely, the exogenous arrival of job offers has important implications, because the optimal UI scheme only has to make the employed slightly better off than the unemployed to induce agents to accept a job offer. In a model with endogenous search effort as analyzed in this paper incentives have to be stronger, because the intensity of job search depends on the difference between the employed and the unemployed in terms of expected utility. A further difference between HH and this paper is that HH do not optimize for a UI scheme in terms of contribution rates or replacement rates, although they review the effects of a limited number of different contribution and replacement rates. In this paper the differences between Unemployment Insurance Savings Accounts and traditional Unemployment Insurance are analyzed for optimal UI policies, in terms of optimal replacement rates for a given duration of UI benefits (either 6 months or indefinitely) as is more or less standard in the literature (see for example Wang & Williamson, 2002; Davidson & Woodsbury, 1998). The effects of the optimal policies in terms of welfare and unemployment rates will reviewed. This is the first paper that analyzes Unemployment Insurance Savings Ac- 3

4 counts in combination with search effort and wealth accumulation in a life-cycle setting. Furthermore, this paper seems unique in the calibration of the disutility of working on the elasticity of the average unemployment duration with respect to the level of unemployment benefits. 2 Model The model is a life-cycle model where agents accumulate wealth as in Hopenhayn and Hatchondo (2011) and where agents also have to search (at a cost) for a new job when they are unemployed. Agents can accumulate wealth for precautionary motives and life-cycle purposes, but they can not borrow. However, the balance on the Unemployment Insurance Savings Account can be negative, but it is bailed out when negative at the first period of retirement. 2.1 Utility function Agents maximize utility: { T +D E 0 t=1 β t 1 [u (c t ) d t a t ] where E 0 is the expected value before the start of the first period, u (c t ) is the period t utility from consumption c t, T is the number of periods in working age, and D is the number of periods in retirement, β is the discount rate, d is the (fixed) discomfort of working and a is the job search intensity with the latter two being zero during retirement. The model differentiates between retired individuals and the working age population. In retirement a public pension is received. At working age an agent either works full-time with disutility d and gets paid the wage or is unemployed, receives unemployment benefits and decides on his job search intensity. 2.2 Retired individuals The retired individuals optimize (1) for t = [T + 1,..., T + D] subject to the following two constraints: } (1) Rk t + n c t + k t+1 (2) k t 0 (3) The first constraint is the budget constraint with private wealth in retirement denoted by k, the gross interest rate by R and the public pension by n. The second constraint is the borrowing constraint. If the balance of the Unemployment Insurance Savings Account is positive in the first period of retirement it is added to private wealth, k T +1. Writing the optimization problem as a Lagrangian yields: 4

5 L = E { T +D t=t +1 β t 1 [u (c t ) + δ t (Rk t + n c t k t+1 ) + γ t k t ] Combining the First Order Conditions (FOC) with respect to c t and k t+1 yields the Euler equation: } (4) u (c t ) = βru (c t+1 ) + γ t+1 (5) which implies consumption is constant if R = 1/β (as in this paper) and wealth is positive at the beginning of retirement (when the non-negativity constraint on private capital k does not bind), conditions which both hold in this paper. 2.3 Working age: employed individuals Employed individuals can only choose their consumption level and whether to quit their job or not. There is one difference between the models of a traditional UI system and a UISA system, and that is that in a UISA system the state vector also contains wealth in the UISA. The Bellman equation that workers optimize in periods t = 1,..., T is: V (k t, q e t ) = max c t,k t+1 u (c t ) d+ β max { (1 λ) V ( k t+1, q e t+1) + λv ( kt+1, q e t+1), V ( kt+1, q u1 t+1)} (6) where d is the disutility of working, λ the exogenous job loss probability, and V (k t, qt z ) is the value function for an individual with private wealth k t and qt z indicates the other state variables of an agent with status z with z = [e, u 1 ], where e means the agent is employed and u 1 means the agent is in his first period of unemployment. In a traditional UI system the vector q contains only the status, qt z = {z t }, but with UISAs this vector also contains s t, the savings in the UISA, so qt z = {s t, z t }. The maximum operator in the second line of (6) reflects endogenous quits. If the value of being unemployed is larger than the value of be employed, V ( ( k t+1, qt+1) u1 > V kt+1, qt+1) e, then the employee will quit his job at the end of the period. The constraints of the optimization are: Rk t + (1 τ µ) w c t + k t+1 (7) k t+1 0 (8) where τ is the tax rate and µ is the UI contribution rate, both as a percentage of the gross wage, which is normalized to 1. In a traditional UI system µw will be added to a public UI fund, while in a UISA system it will be added to the private UISA. Taxes are used to fund a budget surplus, but also any shortfall 5

6 in a UISA system as a result of negative UISA balances that have to be bailed out. For the employed in a UISA system wealth in the savings account in the next period is given by: s t+1 = max (s t+1,min, R s s t + µw) (9) where R s is the interest rate that applies to the UISA, which will be set equal to the normal interest rate, R s = R, and s t+1,min is the minimum balance on an Unemployment Insurance Savings Account. This level is set such that UISA wealth at retirement, s T +1, will be zero if an agent stays employed until his retirement. In practice s t,min is the level at which bail outs start. The First Order Condition with respect to consumption is: u (c t ) = δ t (10) where δ t is the multiplier of the budget constraint (7). Since we allow for endogenous quits the optimization for capital in the next period has to be split in the two cases: (1) the agent keeps his job into the next period; (2) the agent quits at the end of the period. The FOCs for each case are, respectively : { β [1 λ] V ( ) k t+1, qt+1 e + λ V k t+1 ( kt+1, q u1 t+1 k t+1 )} + γ t = δ t (11) β V ( ) k t+1, q u1 t+1 + γ t = δ t (12) k t+1 where δ t is multiplier of the budget constraint (7) and γ t is multiplier on the non-negativity constraint of private capital (8). Using the FOC for each case it can be determined which case gives the highest utility. 2.4 Working age: unemployed individuals Unemployed individuals also have to decide on their consumption and savings, but they have one extra control variable, which is search effort. Search effort determines the chance of finding a job, but results in disutility. The Bellman equation for the unemployed is: V (k t, q u t ) = subject to: max u (c t ) a t + c t,a t,k t+1 β { p (a t ) V ( k t+1, q e t+1) + [1 p (at )] V ( k t+1, q un t+1)} (13) Rk t+1 + b z,t c t + k t+1 (14) k t+1 0 (15) a t 0 (16) 6

7 where a t is search effort, p (a t ) is the chance of finding a job given effort a t, b z,t are the benefits for the unemployed with status z, and u n in the superscript refers to the case where the agent is unemployed one more period. If the agent s UI benefits have not expired he will get UI benefits, b u,t, and otherwise he has to rely on Social Assistance, which is 0 throughout this paper. In case of a UISA system UI benefits are deducted from the UISA, meaning that all UI benefits are paid from the UISA. So for the unemployed next period s wealth in the UISA is given by: s t+1 = max (s t+1,min, R s s t b z,t ) (17) As mentioned earlier, agents are being bailed out when they reach s t+1,min. The FOCs for c t and k t+1 can be combined to yield: { u (c t ) = β p (a t ) V ( ) k t+1, qt+1 e + [1 p (a t )] V k t+1 ( kt+1, q un t+1 k t+1 )} + γ t (18) where γ t is the multiplier on the non-negativity constraint on private wealth (15). The FOC for a t is: 1 = β { p (a t ) V ( k t+1, q e t+1) p (a t ) V ( k t+1, q un t+1)} + ϕt (19) where ϕ t is the multiplier on the non-negativity constraint on search effort (16). Using the matching probability p (a t ) = 1 e rat this can be simplified to: { a t = max 0, 1 r log ( rβ [ V ( k t+1, qt+1) e ( V kt+1, qt+1)]) } un (20) where the multiplier on the non-negativity constraint for search effort is implemented through the max operator. The model is numerically solved backwards so at time t the values for V ( k t+1, q z t+1) are known. Search effort therefore only depends on k t+1 and the optimization problem can be reduced to a single variable (k t+1 ) problem. 3 First Best solution The First Best solution is the optimal solution from a social planner perspective when he can observe search effort. In this case the agent will enjoy full insurance and thus get a constant consumption stream until his death (Pavoni, 2007). The agents thus do not have any savings. The First Best solution consists of a constant consumption stream and an optimal search effort for unemployed agents, which will be age dependent. 7

8 The objective of the social planner is to maximize total expected welfare, which is the sum of the discounted utility from consumption, disutility of working for the employed and disutility from search effort for the unemployed: { T +D max E 0 t=1 β t 1 u ( c) d T β t 1 (1 Ψ t ) t=1 } T β t 1 Ψ t a t t=1 (21) where E 0 is the expected value before the start of the first period, Ψ t is the unemployment rate in period t and a t is the optimal search effort in period t and c is the maximum feasible constant consumption level. The optimization is subject to the total resource constraint: w T (1 Ψ t ) t=1 T +D 1 R t 1 c t=1 1 = G (22) Rt 1 where the first term is total gross labour income, the second term are the expenses on consumption and G is the targeted Net Present Value (NPV) of the budget surplus. The First Order Conditions with respect to search effort, a t (for every t), are: T j=t+1 [ d a j ] β j 1 Ψ j + φ w T j=t+1 1 Ψ j R j 1 = β t 1 Ψ t (23) where φ t is the multiplier on the resource constraint (22), or more accurately the marginal social benefit of an increase in resources and the terms Ψj in the summations are the marginal changes in future unemployment rates resulting from a change in current period s search effort. To solve these FOCs further both φ and Ψj have to be calculated. The marginal social benefit of an increase in the Net Present Value of total resources (or government/social planner budget),φ = V G, is calculated using the marginal discounted utility of consumption: φ = V T G = +D T +D u ( c) β t 1 / 1/R t 1 = u ( c) (24) t=1 t=1 The marginal changes in future unemployment levels, Ψj from the the law of motion for the unemployment rate:, can be calculated Ψ t+1 = λ (1 Ψ t ) + [1 p (a t )] Ψ t (25) where the first term are the employed that loose their job with probability λ and the second term are the unemployed that did not find a job, with probability 1 p (a t ). The changes in future unemployment levels are thus given by: 8

9 Ψ t+1 = p (a t ) Ψ t (26) Ψ j = [ 1 p ( a ) ] Ψ j 1 j 1 λ for j > t + 1 (27) where the first equation is the marginal change in unemployment in the next period, and the the second equation the marginal change in unemployment in later periods. With the expressions for φ and Ψj equation (23) can be solved using fixed point iteration. First, by making a initial guess for c and a t for all t and then recalculating a t working backwards in time (given c), and finally calculating the maximum possible c. These steps are then repeated until convergence. In this process the analytical solution for a t is calculated as follows (working backwards, starting at T 1). First we will write the terms Ψj in (23) in vector notation, simplify the notation using γ j = [ 1 p ( ) ] a j λ and substituting forward we obtain: Ψ t+1 Ψ t+2. Ψ T = p (a t ) Ψ t [1 p ( a t+1 ) λ ] Ψt+1. [ 1 p ( a T 1 ) λ ] ΨT 1 = p (a t ) Ψ t 1 γ t+1. γ T 1 γ t+2 γ t+1 The vectors in the sums in (23) are rewritten as Γ = [ ] d a t+1 β t [ ] d a t+2 β t+1 X 1 =. and X 2 = φw [d a 1 T ] βt notation for (23): which has the solution: 1 R t 1 R t R T 1 1 γ t+1. γ T 1 γ t+2 γ t+1 (28), to yield the simplified p (a t ) Ψ t (X 1Γ + X 2Γ) = β t 1 Ψ t (29) a t = max { 0, 1 r log [ 1 r β t 1 (X 1 + X 2 ) Γ ]} (30) 9

10 4 Calibration The model is calibrated using U.S. data. The unemployment benefits are fixed at 50% of the net wage as in Wang & Williamson (2002) with a maximum duration of 6 months. If someone is unemployed for longer he doesn t get any benefits. The public pension is fixed at 50% of the net wage as in Conesa & Krueger (1999). Pay-roll taxes (including unemployment insurance premiums) are set at 15% of the gross wage. The utility of consumption is of the CARA-type, u (c) = exp( αc) α with α = 2 as is standard in the literature. The time period in the model is one month. The model is calibrated such that it matches the average duration and average unemployment rate of U.S. data over the period Over this period the average duration of unemployment is 3.31 months and the average unemployment rate 6.00% (CPS data). In addition, the model is calibrated to match the elasticity of the average unemployment duration with respect to the level of unemployment benefits, which is set at 0.5. This is approximately the middle of the range reported by Krueger & Meyer (2002) for the U.S.. The three parameters that are calibrated for are the job loss probability, λ, the search efficiency in the matching function, r, and the disutility of working, d. The job loss probability is not taken from the data as unemployment spells of less than one month are not possible in the model and hence should be ignored. The results from the calibration are a job loss probability of 1.66% per month. This is significantly lower than the 5.32% reported by Shimer (2005), but that number includes many very short term unemployment spells, which are ignored in this model except for their influence on the average duration. Parameter values are listed in table 1. The interest rate is set at R = 1/β. The Compensating Variation of the disutility of working is about 1/3 of the net wage, which intuitively seems realistic. 5 Optimal UI policies The optimal policies in this research consist of constant contribution rates for a given UI benefit duration, with 6 months and indefinite UI benefit durations being analyzed. Given the long computation times the optimization of policies is limited to two variables: the (constant) replacement and the UI contribution rate in a UISA variant. The use of constant replacement rates allows for a comparison of the results with others, since others have also reported optimal constant replacement rates for either 6 months or indefinite UI benefit durations. The choice to analyze only constant contribution rates, however, does not imply that it is believed that constant replacement rates are optimal. In fact, Hopenhayn & Niccolini (1997) found that the optimal replacement rate declines with unemployment duration. In addition they found that it is optimal to impose a tax on reemployed individuals that is increasing with the duration of the last unemployment spell. Although Shimer and Werning (2008) find that constant benefits and a constant reemployment tax (both independent on the 10

11 Table 1: Parameters Description (symbol) Value Net wage UI contributions (µ) Tax (τ) Net UI benefits (b) Net Social Assistence 0 Net public pension (n) Maximum UI duration in months 6 Working age (T in years) 40 Retirement (D in years) 15 Population Discount factor (β) Risk aversion (α) 2.00 Job loss probability (λ in %) 1.66 Matching efficiency (r) 1.65 Disutility working (d) Compensating Variation of disutility a a C.V. is calculated here as ĉ is average consumption d u (ĉ) for simplicity, where length of the unemployment spell) are very close to optimal their results depend crucially on the exogenous job arrival rate. With exogenous job arrivals - that differ only in the wage they offer - the incentives to accept a job offer do not have to be as strong as with job search as private and public interest are much more aligned, since both benefit from a higher wage. Thus in their model moral hazard is not as strong as with search effort, where the private agent has to exert costly search effort to find a job. So despite strong indications that in models with job search UI benefits should decline with the length of the duration of the unemployment spell this research only optimizes constant replacement rates for given UI durations, except for a UISA system where also the UI contribution rate is optimized for. To compare the effects of traditional UI systems with systems based on Unemployment Insurance Savings Account the results are given for four different UI systems. These are (1) Benchmark, the UI system in the U.S.; (2) Traditional UI, which is the optimal traditional UI scheme; (3) UISA-Fixed, which is the optimal UISA scheme with the same contribution rate as in the optimal Traditional UI scheme; (4) UISA, which is the optimal UISA scheme where also the contribution rate is optimized for. For the last 3 systems the results are reported for a system with benefits for a maximum of 6 months and for indefinite benefits. The Benchmark is the UI system in the U.S. with a maximum of 6 months UI benefits with the level of benefits set at 50% of the net wage. It should be noted that in the UISA schemes the contributions will continue 11

12 Table 2: Optimal policies with max. 6 months UI benefits Benchm. Trad. UI UISA-F UISA Net wage Net UI benefits UI contributions Unemployment rate (%) Compensating Variation as % NPV UI in Trad. UI a a The Net Present Value of UI contributions in Traditional UI scheme is indefinitely, unlike in Hopenhayn & Hatchondo (2011) where UI contributions to the UISA stop once a maximum balance on the UISA is reached. For all schemes the Net Present Value of the government budget (including any shortfall from bail outs in the UISA case) is the same as in the Benchmark case. In the optimal Traditional UI scheme the contribution rate is set such that the Net Present Value of the UI budget is 0. In the UISA models the government has to bail-out balances that are negative at retirement, so tax revenues have to be higher in order to keep the government budget at the same level as in a traditional UI setup. In the first period all agents are unemployed and have no wealth as in Hopenhayn & Hatchondo (2011). 5.1 Maximum 6 months UI benefits The main results for optimal UI schemes with a maximum duration of 6 months of benefits are listed in table 2. The Compensating Variations are calculated relative to the variant with the highest utility, which is the UISA scheme, where the UI contribution rate is also optimized for (last column). The Compensating Variation is calculated as the extra wealth needed to get the same expected utility at the beginning of period 1 for an unemployed agent with no wealth (the status of all agents in the model at the beginning of period 1). The Compensating Variations seem small compared to the monthly Gross Wage, which is 1. However, unemployment is low at just 6% in the Benchmark case, which means UI contributions constitute only a small part of total wage income. As can be seen in table 2 UI contributions are just 2.71% of the Gross Wage. Therefore, the welfare gains are also computed as a percentage of the average Net Present Value of all UI contributions in the optimal traditional UI scheme, which is 3.72 here. From the improvement in welfare in the optimal Traditional UI it can be concluded that unemployment benefits are too generous in the Benchmark case, since benefits are about 1/4 smaller in the optimal Traditional UI system, while the welfare is about 25% higher in the latter. With these lower benefits the welfare improves mostly as a result of stronger incentives to search for a job, which increases total income and consumption, and results in a drop of the 12

13 unemployment rate from 6.00% to 5.30%. When the Optimal Traditional UI is transformed into the UISA-F system, which has the same UI contribution rate (but a slightly higher tax), the Compensating Variation is reduced from roughly 23% to about 4.5% of the NPV of UI contributions in the optimal Traditional UI case. It can thus be concluded that a UISA system can improve welfare even when net wages and benefits are (almost) the same. This results from the better incentives in a UISA system as the private costs of unemployment are higher, because UI benefits are drawn from the UISA balance. So if the agent expects to end his working life with a positive UISA balance then every period of unemployment directly reduces his expected pension. Similarly, every period in employment directly increases his wealth via the UISA scheme. In short, private and public interests are more aligned for those expectation a positive UISA balance at the end of their working life. In this UISA-F variant about 63.5% of the agents enter retirement with a positive balance. The private costs of unemployment also increase in a UISA system, because of the interest rate applied to UISA balances. The interest costs (income) on a negative (positive) UISA balance increase the costs (benefits) of being unemployed (employed) in comparison to a traditional UI scheme. The interest increases private costs of UI and therefore reduces the redistributional effects of unemployment insurance. To bail out negative balances the total tax revenue has to be increased in the UISA case. However, the unemployment rate is lower with UISA so the tax increase is minimal at only 0.1% of the Gross Wage. The resulting slightly lower net wage has a negligible effect on incentives. The total effect on search incentives with a UISA is quite strong as the unemployment rate is only 4.76% compared to 5.30% for the optimal Traditional UI system. When the UI contributions are also optimized in a UISA scheme the welfare increases, but only by about 4.5% of the NPV of UI contributions. The increase in welfare is the effect of more insurance as well as better incentives. The better insurance is the result of net benefits that are more than two times higher. The better search incentives are mostly the result of UI contribution rates that are 3.6 times higher. This ensures that agents are much more likely to end their working life with a positive balance on their UISA with only about 3.3% of the agents entering retirement with a negative balance. This means that more than 96% of the agents self-insure completely. The better incentives from that are reflected in an unemployment rate of just 4.60%, despite 5% lower net wages. The unemployment rate is quite close to the First Best unemployment rate of 4.23% (not reported). 5.2 Indefinite UI benefits When unemployment benefits last indefinitely (until retirement) instead of 6 months the results are very similar as can be seen in table 3. The Benchmark is still the current UI system with maximum 6 months of benefits, while the other variants have indefinite UI benefits. Interesting is that welfare is higher with 13

14 Table 3: Optimal policies with indefinite UI benefits Benchm. (6 m.) Trad. UI UISA-F UISA Net wage Net UI benefits UI contributions Unemployment rate (%) Compensating Variation as % NPV UI in Trad. UI a a The Net Present Value of UI contributions in Traditional UI scheme is indefinitely lasting unemployment benefits despite lower benefits, which Wang & Williamson (2002) also found for traditional UI. The main difference between indefinite benefits and a maximum of 6 months of benefits is the optimal level of UI benefits, with benefits levels being about 30% lower for most variants with indefinite duration. In all other aspects the results for benefits that last indefinitely are very similar to those for benefits that last 6 months. Overall it can be concluded that UISA systems can improve welfare compared to traditional UI system wit a welfare difference of about 25% of the NPV of UI contributions in the optimal Traditional UI case. The robustness of this result will be discussed in Section 8. 6 Policy functions To get a better insight in the results as discussed in the previous section the policy functions for the different models will be reviewed in this section. The policy functions are for the setup with unemployment benefits of infinite duration. The label UI refers to the optimal Traditional UI scheme and the label UISA to the optimal UISA scheme, where also the UI contribution rate is optimized for. The first policy function that is analyzed is search effort over the life-cycle at average wealth levels as shown in figure 1. Since unemployment benefits last indefinitely and the replacement rate does not depend on the duration of the spell unemployed individuals only differ in their wealth level. In the first periods search effort is highest for the UI case, because many agents have low consumption levels as they have no or very little wealth and therefore depend almost completely on UI benefits, which are low in the optimal Traditional UI scheme. In the UISA case search effort at the start of working age is lower, because agents are better insured and thus have less incentives to search for a job. In fact, the search effort in the UISA case is very close to search effort in the First Best case. Later in life search effort in the Traditional UI case drops dramatically, and is much lower than with the other schemes. The reason that search effort is 14

15 0.5 Figure 1: Search effort over the life-cycle at average wealth levels UI UISA 1st best Age significantly higher in the UISA case is that it has much better incentives. As mentioned before, even though the replacement rate is higher, a large part of the UI insurance is self-insurance as about 98.5% of the agents end their working life with a positive UISA balance, meaning these agents self-insure completely. This induces agents to search harder, although less hard than in the First Best case. In figure 1 it is also clearly visible that search effort drops close to the retirement age of 65, even in the First Best case. The reason being that the benefits of finding a job decline as the job will last only till retirement. Figure 2 shows the level of search effort for given levels of private wealth at the age of 30. For the UISA variant the policy functions are shown for three different levels of UISA balances: S min = 12.01, the minimum that occurred in the simulations, S = 0 and the average S = The First Best is shown as a horizontal line as agents do not have any wealth. In the figure a negative relationship between search effort and private wealth is clearly visible. At higher wealth levels the marginal utility of consumption is lower, and thus also the difference in expected utility between being employed and being unemployed is smaller. The smaller difference in utility between being employed and unemployed reduces search effort (Lentz, 2009). 15

16 Figure 2: Search effort at age UI UISA(S min = 12.01) UISA(S=0) UISA(Avg. S = 0.99) 1st best Private savings (K) Search effort does depend quite strongly on both private wealth, k, and wealth in the savings account, s. Especially in the UI case we see that search effort depends very much on the wealth level as these agents have higher marginal utilities when they are unemployed, since UI pay outs are low. In the case of the UISA scheme search effort is much lower for agents that do no expect to end their working life with a positive balance on their UISA (dashed line). They have little incentives to search for a job, because the replacement rate is high. However, the agents that expect to end their working life with a positive balance on their UISA (S = 0 and S = 0.99) search harder for a job when they are unemployed. In fact their search effort is quite close to the First Best case. Figure 3 with search effort at the age of 60 looks quite different from the one at age 30. In the UISA case search effort is again low for the agents with a large negative UISA balance, and their search effort declines further with wealth as the marginal utility of consumption decreases. This becomes especially pronounced at wealth levels of about 40 at which point they expect their future search effort to drop to zero when they do not find a job soon, and so reduce search effort even more. This sharp drop in search effort is also observed for agents with a UISA 16

17 Figure 3: Search effort at age UI UISA(S min = 2.92) UISA(S=0) UISA(Avg. S = 27.31) 1st best Private savings (K) balance of 0. These agents, however, search much harder for a job, especially at low private wealth levels as they expect to end their working life with a positive USIA balance if they find a job soon and their marginal utility is high. Agents in the UISA scheme with a large positive UISA balance let their search effort depend more linearly on private wealth as they expect to keep searching for a job until very shortly before retirement (see figure 1) and thus expect to find a job. Their search effort is lower than in the UI case although they self-insure themselves completely, since their UISA balances are high enough to withdraw UI benefits until retirement. The explanation is that in the UI case the benefits are lower and thus the marginal utility of finding a job and increasing consumption is higher. 7 Discussion and comparison of results The main objective of this research is comparing a traditional UI system with a UISA system. An indication of the validity of the results are the optimal replacement rates in comparison to optimal replacement rates found by others. 17

18 7.1 Discussion of methodology A crucial assumption in this research is that agents start their working age life unemployment and without any wealth as in Hopenhayn & Hatchondo (2011). Although unemployment is more common early in working life (Stiglitz & Yun, 2005) some objections can be raised against letting all agents start unemployed. For example, in the U.S. one is only eligible for UI if one has been employed before. However, as mentioned above, the prime interest of this research is to compare a traditional UI system with a UISA system, while using a life-cycle perspective allowing for wealth accumulation and realistic interest rates. As the public pension is limited to 50% of the net wage agents save for retirement, resulting in very high wealth accumulation. Agents can utilize this wealth for consumption smoothing in case of unemployment, which is not realistic as retirement savings cannot be used for this purpose in reality. Hence, in this model agents can self-insure to a very high extend through their retirement savings and results in unrealistically low optimal replacement rates, if agents start their working life employed (see table 8). To counter balance this abundance of liquid funds it is assumed that agents start their working life unemployed and without wealth. One may argue that this is not realistic at all, but it should be realized that other researches have also increased the dependence on UI in artificial ways. For example, most researchers limit savings to ensure more dependence on UI. Savings have been limited by assuming there is no wealth accumulation at all, allowing the government to set consumption directly (Davidson & Woodsbury, 1997; Hopenhayn & Nicolini, 1997 and 2009), or by restricting wealth accumulation in some other way, like Wang & Williamson (2002) who use a zero interest rate on private savings. In addition, most other authors do not use a life-cycle approach and thus abstract from retirement savings, with the only exceptions in table 4 being Heer (2006) and Hopenhayn & Hatchondo (2011). Summarizing, letting all agents start unemployed and without wealth it is justified by the observation that retirement risk is concentrated early in careers (Stiglitz & Yun, 2005). Even though there are arguments against doing that, the optimal replacement rates found in this research are similar to those others have found as can be seen in table 4. Thus the comparison between a traditional UI system and a UISA system is at least based on replacement rates that are in line with those of others. 7.2 Comparison optimal replacement rates From table 4 it is clear that the optimal replacement rates for traditional UI found in this research are relatively low compared to those found by others. One reason that replacement rates are relatively low in this research is that agents have full access to their retirement savings (as mentioned in subsection 7.1), which are substantial (on average about 50 times the gross wage at retirement). These savings provide agents with liquidity to self-insure and smooth consumption over the life-cycle. This makes the benefits of UI smaller compared 18

19 Table 4: Optimal replacement rates in literature Opt. Repl. rate Duration Type This research months Traditional UI This research 0.24 Traditional UI This research months UISA This research 0.54 UISA Davidson & Woodsbury months Traditional UI Davidson & Woodsbury Traditional UI Wang & Williamson months Traditional UI Wang & Williamson 0.24 Traditional UI Hansen & Imrohoroglu Traditional UI Heer 0.7 a Traditional UI Hopenhayn & Hatchondo 0.3 b 6 months UISA a for Germany. b from discrete choice between 0.3, 0.6 and 0.9. to models where agents have no savings as in Davidson and Woodsbury (1997) or less savings as in Wang and Williamson (2002). Another reason that replacement rates might be lower in this research is that agents have a finite life, which tends to make insurance more costly. The reason is that when agents approach their retirement age the benefits of searching for a job declines as they will only benefit a relatively short time from finding a job. So even in the First Best case search intensity is reduced the closer one gets to retirement (see figure 1). Thus with finite lives higher replacement rates reduce search effort especially before retirement. The papers by Davidson & Woodsbury (1997), Wang & Williamson (2002) and Hansen & Imrohoroglu (1992) abstract from this retirement issue that comes with the life-cycle approach. Summarizing, agents are unemployed and without wealth at the start of their life-cycle, which should result in high replacement rates. Despite that, the optimal replacement rates for traditional UI found here are relatively low compared to others, because this research uses search effort and the life-cycle approach. It is unclear to what extend this affects the welfare gains from implementing a UISA system. However, the optimal replacement rates for a traditional UI system versus a UISA system differ by at least a factor two, which indicates strong differences in incentives between the two systems. Although the optimal replacement rate for a UISA system as found by Hopenhayn & Hatchondo seems much smaller than the optimal replacement rate found here, it has to be noted that their optimal replacement rate is almost the same (0.3) as in this research when the UI contribution rate is fixed (at 1.7%, see table 2). However, it is hard to say what would be the outcome if Hopenhayn & Hatchondo (2011) would have truly optimized their policies instead of discretely varying a couple of key parameters and choosing the best 19

20 Table 5: Parameters sensitivity analysis Stnd. (z 1 = u) α = 5.00 ɛ = 0.8 Empl. (z 1 = e) UI contr. (µ) Tax (τ) Risk aversion (α) Job loss prob. (λ in %) Matching efficiency (r) Disutility working (d) C.V. disutility a a C.V. is calculated here as d for simplicity, where ĉ is average consumption u (ĉ) result. 8 Sensitivity analysis The robustness of the results is checked by three changes: (1) a higher risk aversion parameter; (2) increasing the elasticity of the average duration of unemployment with respect to the level UI benefits (3) all agents start their working life employed instead of unemployed. In each case the model is calibrated anew, meaning that the exogenous job destruction rate, matching efficiency and disutility of working are calibrated for. In the first sensitivity analysis the risk aversion is set at 5.0 instead of 2.0 and in the second sensitivity analysis the mentioned elasticity is set at 0.8 instead of 0.5. The changes in the calibrated parameters are listed in table 5. The results for the optimal policies in case of the higher risk aversion are listed in table 6. Compared to the results with the standard risk aversion the UI benefits increased a little, but the unemployment rate increased significantly. The primary cause are the significant parameter changes in the calibration. As listed in table 5 the higher risk aversion resulted in a matching efficiency that was 7 times higher while the disutility of working is less than 1/8 of the value with the standard risk aversion. The higher risk aversion results in a shift in the trade-off between equity and efficiency in favor of equity. With indefinite benefits and a traditional UI system the unemployment rate increases almost a full percentage point as the relatively high replacement rate reduces search effort more than is made up for by the higher matching probability. However, with a UISA system the UI benefits go up further but result in a much lower unemployment rate. The welfare differences also increased quite a bit, which is expected as the higher risk aversion magnifies any welfare differences. When the elasticity of the average unemployment duration with respect to the level of UI benefits is increased from 0.5 to 0.8 the results change very little as seen in table 7. The calibration for this higher elasticity results in a higher disutility of working and a lower job destruction rate and a lower job finding 20

21 Table 6: Optimal policies with high risk aversion, α = 5.0 (indefinite benefits) Benchm. (6 m.) Trad. UI UISA-F UISA Net wage Net UI benefits UI contributions Unemployment rate (%) Compensating Variation as % NPV UI in Trad. UI a a The Net Present Value of UI contributions in Traditional UI scheme is Table 7: Optimal policies with high elasticity of unemployment duration w.r.t. level UI benefits, ɛ = 0.8 (indefinite benefits) Benchm. (6 m.) Trad. UI UISA-F UISA Net wage Net UI benefits UI contributions Unemployment rate (%) Compensating Variation as % NPV UI in Trad. UI a a The Net Present Value of UI contributions in Traditional UI scheme is probability, but the parameter changes are small and thus have little influence on optimal UI benefits, welfare or unemployment rates. Finally the results when all agents start their working life employed are listed in table 8. The most striking feature is that the optimal replacement rates are much lower, with a maximum of less than 15% of the net wage. As explained earlier these low replacement rates are the result of high retirement savings that facilitate self-insurance. However, when the welfare loss of the optimal Traditional UI system is compared with the optimal UISA system then the relative welfare loss is still quite high at about 13% of UI contributions, although the contribution rate is very low. As with the other variants the optimal UISA can provide much more insurance (replacement rates are almost 3 times higher) at little costs as the unemployment rate does not change much. 9 Conclusion The results of this paper suggest that Unemployment Insurance Savings Accounts can improve welfare compared to a traditional Unemployment Insurance system. These results were found using a rather simple setup for the UISA system with only optimization of a constant replacement rate and a constant 21

22 Table 8: Optimal policies when agents start employed (indefinite benefits) Benchm. (6 m.) Trad. UI UISA-F UISA Net wage Net UI benefits UI contributions Unemployment rate (%) Compensating Variation as % NPV UI in Trad. UI a a The Net Present Value of UI contributions in Traditional UI scheme is UI contribution rates in case of a UISA system. In addition, only a simple limit on liability (non-negativity at retirement) was used for the UISA system. This leaves leaves room for more improvement of a UISA system, for example by imposing a maximum on UI contributions as in Hopenhayn and Hatchondo (2011) or other limits on liability. The improvements in welfare using a UISA system compared to a traditional UI system come from two effects. The first effect are the better incentives, which increase search effort for jobs and results in lower unemployment rates. The second effect results from more insurance as the optimal UISA system has higher replacement rates. The improvement in welfare from a optimal traditional UI system to a UISA system were substantial, and around 25% of UI contributions in the former system, with optimal replacement rates within the range of optimal replacement rates found by others. Although these results depend on agents entering working age unemployed, the improvement in welfare is still 13% of UI contributions when agents enter working age employed, albeit that optimal replacements rates in this case are much lower. Overall it can be concluded that a UISA system can improve welfare substantially through better incentives and more insurance. References [1] Beauchemin, K., & Tasci, M. (2008). Diagnosing labor market search models: a multiple-shock approach (No. 0813). [2] Bovenberg, A. L., Hansen, M. I., & Sørensen, P. B. (2008). Individual savings accounts for social insurance: rationale and alternative designs. International Tax and Public Finance, 15(1), [3] Chamberlain, G., & Wilson, C. A. (2000). Optimal intertemporal consumption under uncertainty. Review of Economic Dynamics, 3(3),

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