1 Unemployment Insurance

Transcription

1 1 Unemployment Insurance 1.1 Introduction Unemployment Insurance (UI) is a federal program that is adminstered by the states in which taxes are used to pay for bene ts to workers laid o by rms. UI started in 1935 with the Social Security Act and the goal, as it is today, was to provide partial wage replacement to involuntary unemployment. Questions: What is the optimal level and length of UI bene ts?

2 The tradeo is again insurance vs. distortion. On the one hand, consumers are better o because he/she is smoothing consumption using UI bene ts. But UI bene ts distorts behavior because they can cause workers not to try hard to nd a job, and they can also crowd out other forms of self insurance.

3 1.2 Baily (1978): Some Aspects of Optimal Unemployment Insurance The idea of the Baily model is to derive a simple and intuitive formula for the optimal replacement rate based on parameters that can be estimated empirically (as well as comparing the insurance vs. distortion tradeo ). Among the potential distortions on workers side focused on in this paper are: 1. extended durations (less search); 2. indirect e ects on savings, spousal work (crowd out of self insurance).

4 The main bene t of UI is consumption smoothing. The main goal is to compare the bene ts with the costs of the program, and gure out how the optimal level of UI bene t should be determined.

5 1.3 Model There are two periods. In period 1, the individual receives income y from her period 1 job. The tax rate on all incomes is t; and the income tax is used to nnance the UI system. In period 1, individual decides how much to consume and how much to save. Saving amount in period 1 is denoted by s: In the end of period 1, the individual is laid o with probability 1 ; which is assumed to be exogenous. (with probablity ; the worker is retained on her job). In period 2, if the worker was laid o at the end of period 1, she will choose work search intensity c;

6 A laid o worker may be able to nd a job in some fraction of the second period. The fraction of period 2 that workers spends working is a function of the intensity of job search c (in terms of resources spent, not the amount of time) and the reservation wage y n : (c; y n ) 2 (0; 1) n < 0: That is, the fraction of time that the individual will be working in second period is increasing in search intensity and will be larger if the reservation wage is lower. When a worker laid o in the end of the rst period is not working (which happens in a fraction 1 of the time) in the second period, she receives UI bene t in the amount of b;

7 The government must balance its budget in that its income tax must equal to its UI bene t payouts.

8 Comments: 1. Consumption is the only argument for the utility function. Leisure is ignored. 2. Bene ts do not a ect the probability of laying o workers. This will be true if there is perfect experience rating: the rm s payroll taxes are adjusted to make the rm indi erent between layo rates. 3. This is a partial equilibrium model.

9 1.4 Preliminary Analysis Total revenue: T R = yt + yt + (1 ) y n t Balanced budget: yt + yt + (1 ) y n t = (1 ) (1 ) b b = yt + yt + (1 ) (1 ) (1 ) y nt = (t) t The important thing to recognize that the replacement rate is endogenous to the tax rate because changes depending on the tax rate.

10 Second period income if the worker is laid o in the end of the rst period: y l = (1 ) (b c) + (1 t) y n Note that the UI bene t income is not taxed.

11 1.5 First Best Solution: Social Planner s Problem If the social planner can direct control workers behavior, the social planner will be solving max V = U [y (1 t) s] + U [y (1 t) + s] + (1 ) U [y l + s] fs;y n ;c;tg = U [y (1 t) s] + U [y (1 t) + s] 2 y+y+(1 )(c;yn )y n (1 )(1 (c;y n )) + (1 ) U 4 (1 (c; y n)) t + (c; y n ) (1 t) y n + s = U [y (1 t) s] + U [y (1 t) + s] " (1 + ) y + (1 ) U (1 ) t c + (c; y n) (c + y n ) + s # c 3 5 How to characterize the solution to this problem?

12 Note that since in this problem, the social planner is assumed to be able to choose all four variables, s; y n ; c; t; we just need to take rst order conditions with these four variables separately. FOC w.r.t. c (c; y n ) (y n + c) = 1 (c; y n ) LHS is the marginal return from more intensive search and the RHS is the marginal cost of more intensive search. FOC w.r.t. y n (c; y n n (y n + c) = (c; y n ) (2)

13 FOC w.r.t. s : U 0 [y (1 t) s] + U 0 [y (1 t) + s] + (1 ) U 0 (y l + s) = 0: (3) Finally, taking the derivative w.r.t. t : U 0 [y (1 t) s] + U 0 [y (1 t) + s] (1 + ) U 0 [y l + s] = 0: (4) Note that, summing over the last two optimality conditions, we get: U 0 [y (1 t) + s] = U 0 [y l + s] That is, the optimal solution must be that: the level of consumption in the second period should be independent on whehther or not one was laid o in the rst period.

14 Plugging the above equality into the FOC w.r.t. s or t above, we get: U 0 [y (1 t) + s] = U 0 [y (1 t) s] ; thus, the optimal solution must have s = 0: No private savings!

15 That is, the rst best solution will involve full insurance. The idea of the proof is simply an argument of perturbation: if there is no full insurance, then a small change in the tax rate (while keeping the behavior of the consumers xed, as we assumed that the social planner can do in this rst best scenario) to smooth consumption in the two states of the world in the second period can lead to an improvement of consumer welfare. Why should the optimal saving be zero? The reason for this is that saving, as an instrument to smooth consumption, works well when future incomes streams are not stochastic. In the current setting, because layo at the end of rst period is stochastic, second period income stream is stochastic: it could be y (1 t), or y l : A certain level of private saving could not equate the marginal utility of consumption in the two states (layo, no layo ). Buying insurance is more e cient and can be thought of as a contingent saving.

16 1.6 Second Best Solution Now we characterize the second best solution, assuming that workers set their individual search and job acceptance behavior taking the prevailing UI bene t and tax rates as given (i.e., workers ignore the e ect of their own behavior on the overall UI system, which is valid under an assumption that the worker is just a representative worker). Then the optimal bene t and tax rates are set by the government, given the behavioral responses of the workers to the UI system. Workers maximize expected utility by choosing s; y n ; c; and taking as given b and t : max fs;y n ;cg V = U [y (1 t) s] + U [y (1 t) + s] + (1 ) U [(1 (c; y n )) (b c) + (c; y n ) (1 t) y n + s]

17 FOC w.r.t c and y n are: c : y n (c; y n ) (y n (1 t) + c b) = 1 (c; y n (c; y n ) [y n (1 t) + c b] = (c; y n ) (1 t) n Notice the di erences between the FOC for c in the second best and rst problems in the calculation of marginal bene t and marginal costs. (the di erence is between private and social marginal e ects.) Comparing (1) with (5), the marginal bene t for search intensity n (y n + c) for the rst best, but [y n (1 t) + c b] for the second best, while the marginal cost is the same. Thus we can conclude that the second best seach intensity is going to be lower than the rst best. (This is one of the distortions);

18 The di erence between the rst and second best level of y n is somewhat more ambiguous without further assumptions on U and (; ) : Under certain conditions (see Bailey s paper for more details), one can < < 0: Thus, higher tax rates (which implies higher UI bene ts) will lead to lower search intensity, higher reservation wage and lower fraction of time working in the seond period. This is the disincentive e ects of the UI program.

19 The government s problem is to choose the tax rate t to max ftg V [s (t) ; c (t) ; y n (t) ; ; t] The rst order condition @c + Using the envelope theorem, we know that the rst three terms are zero. Thus, the rst order condition for + = 0

20 Take these derivatives. To make progress on understanding the optimal tax rate (and the optimal bene t), we can do a Taylor series approximation of the above rst order condition around the consumption level if laid o in the second period, i.e., around c l = y l + s. Denote the consumption in period 1 as c e = y (1 t) + s:and let c = c e c l be the drop of consumption caused by layo. There exists some 2 (c l ; c e ) such that c " U 00 () U 0 (y l + s) # = [(1 ) y n + (1 + ) y] 2y (1 ) ( t) d dt (1 ) t 2y dy n dt Interpretation:

21 the LHS measures the cost to workers of the income risk from the probability of layo that they face. If UI taxes and bene ts are increased, this reduces the variability of workers income and consumption, which reducing the welfare consequences of the income risk. The rst term in RHS is a measure of the distortion of search behavior (recall that the term d=dt captures the e ect of t on both c and y n ), the second term adjusts the distortion from the search behavior to re ect that the payroll tax itself is distortionary. We can ignore this last term. Under the assumption that y n y; we have [(1 ) y n + (1 + ) y] 2y 1

22 Also using the fact that b = t, we can approximate where Eb u bene t. t 1! d dt b 1 d (1 ) db = E u b is the elasticity of unemployment duration with respect to UI Finally, if c e and c l are close, c " U 00 () U 0 (y l + s) # c c e " cl U 00 (c l ) U 0 (c l ) # = c c e R (c l ) where R (c l ) represents the relative risk aversion in the layo state. Hence, the optimal bene t condition is approximately: c c e R (c l ) E u b :

23 Interpretation of the Bailey condition: The optimal unemployment insurance bene t level is set when the proportional drop in consumption resulting from layo, times the degree of relative risk aversion of workers (evaluated at the level of consumption when laid o ) is equal to the elasticity of the duration of unemployment with respect to balanced budget increase in UI bene ts and taxes. 1.7 Calibration In order to obtain some quantitative sense of what the optimality condition implies for the optimal replacement ratio, Bailey rst showed that c=c e is related to the replacement ratio.

24 Roughly, consider a worker who experiences an eighteen week spell of unemployment during one year. Suppose, as the data seems to suggest that a typical unemployed worker has about a week s wage in his savings. A typical freshly unemployed worker worked about 34 weeks in the year of his unemployment. He, therefore, receives as income 34 weeks of wages, 17 weeks of UI bene ts and also consumes saving equal to 1 week s wages. (He receives only 17 weeks of bene ts in line with the standard practice of commencing payment of UI only after a one week delay.) The relation between c=c e, the drop in consumption, and b=y; the ratio of weekly UI bene t to weekly after-tax wage income is then calculated as

25 follows: c l = b y + 1 or, c e = 51 thus, c c e = 51 b y = 3 : b y c c e One also needs some estimate for the risk aversion and the elasticity of unemployment spells with respect to bene t. Show Table 2 Here.

26 2 Meyer (1990, Econometrica): Unemployment Insurance and Unemployment Spells This paper estimates the e ects of the level and length of unemployment insurance (UI) bene ts on unemployment durations. Such parameter estimates, as we saw in Bailey s paper, are crucial for guring out the optimal level of UI bene ts. Meyer s ndings can be summarized as follows: The coe cient on the UI bene t level is precisely estimated and implies that a 10 percent increase in bene ts is associated with an 8.8 percent decrease in the hazard.

27 The coe cient suggests a relatively large disincentive e ect of UI.

28 3 Gruber (1993, AER): The Consumption Smoothing Bene ts of Unemployment Insurance While Bailey s paper used in his calibration a huristic relationship between c=c e and the replacement ratio b=y; Gruber sets out to estimate this relationship using data. He estimates regressions of the form where: C i = + 1 X i + 2 UI i + " i C is the change in long consumption when the individual becomes unemployed;

29 X is a vector of individual characteristics which may a ect the consumption change; UI measures the replacement rate (ratio of bene ts to wages) for which an individual is eligible. A consumption smoothing e ect of UI would be represented by a nding of 2 > 0: A discussion about using UI bene t eligibility instead of actual UI bene ts received. There are two considerations: 1. Receipt of UI, and the amount of UI received, is endogenous. (Take up of UI is only 67 percent among those eligible for bene ts.)

30 If factors that determine UI take up are correlated with consumption changes from job loss, estitating the above eq. using actual UI bene ts received would not be valid for predicting the response to future changes in UI negerosity. 2. The data on actual UI receipts are very noisy. He nds that 2 = :265: Gruber then performed a calculation of the optimal UI bene ts, using his own estimate of the relationship between c=c e and b=y; using the estimate of Eb u from Meyer (1990) and various values of risk aversion parameters.

31 4 Shavell and Weiss (1979) Shavell and Weiss s seminal paper attempts to study the optimal time path of UI bene ts in a dynamic model, which against tries to gure out the optimal trade o between insurance and distortions, but in a dynamic setting. 4.1 Model Consider a worker who is already unemployed. He is risk averse with a concave utility function u () ;

32 During each period, the unemployed worker rst collects UI bene ts b t and then either nds a job or does not. [The probability of nding a job p t may depend on his job search e fort e t.] If he nds a job in period t; it is assumed that he is rst paid in period t + 1 and for convenience, it is assumed that he works at that job forever. If the worker receives a wage o er, let w t denote the wage o er received in period t and let f (w t ; e t ) denote the probability density of a wage o er given e ort. The worker also needs to decide whether to accept a wage o er he received. Let wt denote the reservation wage in period t;

33 Let r be the one period discount rate (assumed to be equal to the interest rate). Finally suppose that the government has B 1 budget in the UI fund per unemployment worker. Thus, the sequence of UI bene ts fb t g 1 t=1 must satisfy B 1 = b 1 + (1 p 1) b r = 1X t=1 b t Q tj=1 1 pj (1 + r) t 1 : + (1 p 1) (1 p 2 ) b 3 (1 + r) 2 + :::

34 More generally, let B t = 1X k=t b k Q k 1 j=t 1 pj (1 + r) k t : Note: Because of the assumption of the constant UI budget assumption made in this paper, the focus of this paper di ers from that of Bailey. Baily was interested in how the level of B should be optimally determined, here, the question is, given B; how it should be spread over time.

35 4.2 The Optimal Time Sequence of Bene ts assuming tha the unemployed have no wealth and cannot borrow In this case, an individual who is unemployed at the beginning of period t receives and consumes b t (his only source of income), and obtain utility u (b t ) ; Case No. 1: Suppose that Searc E ort does not a ect the wage o er probabilities. In this case, e t = 0 for all t: Further suppose that the wage of any job which might be found to be w; a constant.

36 In this special case (which is not supposed to be realistic, but rather will be serving as a benchmark), the only role of UI is to provide insurance to the worker before the worker nd a job (which is uncertain as to when will that happen). Intuitively, this would mean that the UI bene ts must be constant over time at a level that will satisfy the budget constraint.

37 Formally, max EU fb t g 1 1 t=1 " = p 1 u (b 1 ) + u (w) r = s:t: B 1 = +p 2 (1 p 1 ) 1X t=1 1X t=1 " # 2 ty 1 3 4p t 1 pj 5 4 j=1 b t Q tj=1 1 (1 + r) t 1 u (b 1 ) + u (b 2) 1 + r + u (w) r (1 + r) 2 u b j pj tx j=1 # + ::: (1 + r) j 1 + u (w) r (1 + r) t 1 3 5

38 Take derivative with respect to b k for all k = 1; ::: we have thus Q kj=1 1 thus b k is a constant. pj (1 + r) k 1 u0 (b k ) = u 0 (b k ) = Q kj=1 1 (1 + r) k 1 pj Proposition 1: suppose that unemployed individuals (i) has no wealth, cannot borrow and (ii) cannot in uence the probability of getting a job each period, then: UI bene ts should be the same from one period to the next (i.e., b 1 = b 2 = :::):

39 Case 2: Suppose that unemployed individuals do in uence the probability of getting a job by their choice of e ort e and the reservation wage w In this case, an individual who is unemployed at the beginning of period t and receives b t enjoys u (b t ) e t because search e ort involves disutility. Note that it is assumed that the marginal utilities of e ort and consumption are taken as independent. The probability of getting a job as a function of e ort e t and reservation wage wt is p t = p (w t ; e t ) = Z 1 w t f (w t ; e t ) dw t :

40 Let u t be the expected utility, discounted to t + 1; given that an individual gets a job in period t; u t = u (w t ; e t ) = Z 1 w t (1 + r) u (w t) f (w t ; e t ) dw t : r p t Conditional upon being unemployed at period t; an individual s expected utility, discounted to t is 02 1X ky kx u 31 b j e j E t 4p k 1 pj 5 4 (1 + r) j t + u k (1 + r) k t+1 5A : k=t j=t j=t According to the principle of optimality, w t ; e t must be chosen in each

41 period to solve E t = max fw t ;e tg ( u (b t ) e t + p (w t ; e t) u (w t ; e t) + (1 p (w t ; e t)) E t r From this problem we know that dp t de t+1 < 0: Thus anything which increases the utility of being unemployed in t + 1 increases the probability of that event happening. ) Using the same technique as that in the last proposition (but with much more analysis), we can show: Proposition 2: Suppose that the unemployed individual (i) has no wealth, cannot borrow, and (ii) can in uence the probability of getting a job each

42 period by their choice of reservation wage and a level of e ort devoted to job search. Then If the government does not monitor this choice on an individual basis, UI bene ts should decline from period to period, and although remaining positive, approach zero in the limit. (i.e. b 1 > b 2 > ::: > b t > 0 for all t and lim t!1 b t = 0; if the government does monitor this choice, then UI bene ts should be constant for period to period. The idea behind the rst result is as follows. Suppose that b t = b t+1 and consider a small actuarially fair reduction in b t+1 and increase in b t : Since initially u 0 (b t ) = u 0 (b t+1 ) ; the rst order approximation of the direct e ect of this change on expected utility will be zero. But this change will

43 lower E t+1 and thus increase the probability of nding a job in period t; which saves costs. Show Table 1: Optimal Time Sequence of Unemployment Insurance Bene ts that will achieve the same expected utility as the current UI system. The savings under the optimal time sequence of UI bene ts are about 16 percent of the current UI cost.

44 4.3 The Optimal Time Sequence of Bene ts Assuming that the Unemployed Begin with Positive Wealth or Can Borrow They also consider the case that individuals start with initial wealth z 0 ; or can borrow against their future income. For the analysis, they assume that individual do not in uence the probability of getting a job. They show that:

45 Proposition 3: Suppose that unemployed individuals (i) have wealth or can borrow and may save or dissave; (ii) cannot in uence the probability of getting a job. Then UI bene ts should at rst be zero, and then should rise to a constant value (i.e., 0 = b 1 = ::: = b T < b T +1 b = bt +2 = :::