Employment turnover and the public allocation of unemployment insurance

Employment turnover and the public allocation of unemployment insurance

Journal of Public Economics 73 (1999) 55–83 Employment turnover and the public allocation of unemployment insurance ´ John Hassler a , *, Jose´ V. Ro...

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Journal of Public Economics 73 (1999) 55–83

Employment turnover and the public allocation of unemployment insurance ´ John Hassler a , *, Jose´ V. Rodrıguez Mora a,b a

Institute for International Economic Studies, Stockholm University, S-106 91 Stockholm, Sweden b ´ Trıas ´ Fargas 25 – 27, Department of Economics, Universitat Pompeu Fabra, Ramon 08005 Barcelona, Spain Received 1 May 1998; received in revised form 1 November 1998; accepted 1 November 1998

Abstract Unemployment benefits are higher and turnover between unemployment and employment is lower in Europe than in the U.S. We model the political determination of the unemployment insurance to explain these differences. We show that saving and borrowing is a good substitute for unemployment insurance when turnover is high. With high turnover, the median voter thus prefers low unemployment insurance. With low turnover, generous unemployment insurance becomes more valuable. If the median voter cannot bind future voters, the voting cycle must, however, be long in order to support a high level of insurance. Endogenizing turnover produces the possibility of multiple political equilibria.  1999 Elsevier Science S.A. All rights reserved. Keywords: Employment turnover; Unemployment insurance; Voting JEL classification: D72; J6; J65

1. Introduction In this paper, we build a model of the political determination of the unemployment insurance level. The analysis will have many features in common with the influential paper by Wright (1986). Like Wright, we will disregard incentive effects *Corresponding author. Tel.: 146-8-162070. E-mail address: [email protected] (J. Hassler) 0047-2727 / 99 / $ – see front matter  1999 Elsevier Science S.A. All rights reserved. PII: S0047-2727( 99 )00004-3

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of unemployment insurance as well as other problems related to imperfect information. The level of the unemployment insurance will be determined by the preferences of a median voter. We will focus on a case where all individuals are ex-ante identical. There will, however, be two important generalizations relative to the previous literature. The first concerns the completeness of the capital market. The previously analyzed cases are polar opposites – no credit market at all, i.e. no saving and borrowing and a complete market including private unemployment insurance.1 We analyze the more realistic intermediate case, where individuals can borrow and save but not privately insure against the unemployment risk. The second major difference is that we analyze the consequences of relaxing the assumption that the median voter can fix the unemployment insurance forever. Instead, we assume that votes on the insurance level are taken periodically and that current voters cannot bind future voters. We will show that both these generalizations are not only realistic, but also fundamental to the conclusions. The empirical motivation for our analysis is that the generosity of the unemployment insurance varies substantially between countries. In particular, the unemployment insurance is much more generous in most European countries than in the U.S. This is reflected both in replacement ratios and in the period of time an unemployed is entitled to benefits.2 A second difference between the labor markets in Europe and the U.S. is that the average flows in and out of unemployment are much higher in the the U.S. than in Europe, i.e. employment turnover is higher in the U.S.3 Table 1 shows the flows into and out of unemployment in 1985 and 1993.4 The flows are expressed as the percentage of employed (unemployed) who became unemployed (employed) in an average month of the corresponding year. In Germany, for example, only 0.25% of the employed became unemployed in an average month in 1985. The figure for the U.S. was about 10 times higher. Similarly, 6.1% of the unemployed found employment each month while the percentage in the U.S. was almost seven 1

See, however, Flemming (1978) for a model of optimal unemployment insurance with capital markets. 2 OECD (1994) computes an index of the generosity of unemployment benefits in OECD countries. According to this index, the systems in Denmark and the Netherlands are the most generous, while the systems in the U.S. and Japan are the least generous. Due to multidimensional differences in the structures of the systems, any ranking can, of course, be ambiguous. However, it seems fairly clear that most European countries have a more generous unemployment insurance than the U.S. 3 We use the word turnover to denote the flow rates into and out of unemployment. This should be distinguished from job rotation which is typically used for flows on the labor market including flows of employed from one job to another. The level of job rotation defined in this way differs very little between the U.S. and Europe (see, for example, Bertola and Rogerson (1997) for a model of job-to-job rotation). In an analysis of unemployment insurance, the flows into and out of unemployment should be the main object of interest. 4 There is, of course, also substantial variation in the flow rates over age groups and industries. Disaggregating over age groups and industries, the higher turnover in the U.S. seems to prevail, however. See OECD (1995).

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Table 1 Flows in and out of unemployment 1985 Inflow a

Outflow a

1985

1993

1985

1993

Belgium Denmark Finland France Germany Italy Netherlands Spain Sweden U.K.

0.14 0.29 1.49 0.32 0.25 0.14 0.28 0.35 0.66 0.51

0.42 1.75 2.83 0.34 0.57 0.41 0.24 0.56 1.25 0.67

2.7 6.3 36.5 3.7 6.1 1.8 6.8 1.7 28.5 6.1

8.6 21.4 13.9 3.0 9.0 9.5 6.4 1.8 11.6 9.3

Average of above USA

0.44 2.45

0.90 2.06

10.0 41.4

8.45 37.4

a

As percent per month of source population. Source: OECD (1995).

times higher.5 The low flow out of employment means that the average risk of losing a job is low in Europe while high in the U.S. But then, why do we observe very comprehensive unemployment insurance systems in Europe and not in the U.S.? Our analysis will focus on three questions. First, does a low employment turnover mean that the employed would prefer a high unemployment insurance, financed by taxes on the employed? Second, if people vote sequentially on the level of unemployment benefits, can a high insurance level be sustained in a voting equilibrium? Third, can multiple equilibria exist if turnover and the unemployment insurance level are determined jointly? In answer to the first question, we first demonstrate the importance of recognizing that, in real life, individuals can self-insure, i.e. they can substitute saving and borrowing for unemployment insurance. To illustrate the importance of this, we start with a simple search model without saving. We show that such a model implies that the willingness of an employed person to pay for unemployment insurance increases in the flow rates into and out of unemployment. That is, large flows between unemployment and employment, as in the U.S., would tend to increase the level of insurance preferred by employed individuals. We will show that if we allow individuals to use a credit market to save or borrow, the relation between turnover and the preferences for unemployment insurance can be reversed. To understand this we must realize that, in the absence 5

Note, however, the large flows out of unemployment in Finland and Sweden, which might reflect the (unsustainably) low unemployment rates in these countries in 1985. For Sweden, a substantial part of the flow may have been into different unemployment programs where individuals are not considered as openly unemployed. By 1993, the outflow had fallen to 13.9 in Finland and to 11.6 in Sweden.

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of a credit market, the unemployment insurance has two functions – to smooth income and to insure against uncertain income shocks. Introducing a credit market means that individuals get another instrument at their disposal. However, the degree of substitutability between these two instruments depends on the rate of turnover. Low turnover means that income shocks associated with unemployment are more persistent than when turnover is high. Highly persistent shocks imply that uncertainty over long horizons is high, which strengthens the insurance motive. On the other hand, with low persistence the problem is rather to smooth income between short but frequent unemployment and employment periods, which can effectively be done by saving and borrowing. Consequently, saving and borrowing is a good substitute for unemployment insurance when turnover is high, but much worse when it is low.6 This establishes a motive for the employed to introduce generous unemployment benefits when employment turnover is low. In order to address the second question, we relax the assumption that the employed can permanently fix the insurance level. Instead, we assume that a vote is taken with regular intervals. The median voter can then only choose the level of insurance until the next vote, not forever. With low turnover, insurance is mainly important in the long run because, in the short run, the employed can be relatively sure of keeping her job. But if the periods between votes are short, the median voter can only influence the insurance level in the short run. During this period, she will most likely not benefit from an unemployment insurance. Since the median voter must nevertheless bear the cost of the insurance herself, she will thus vote for low insurance or no insurance at all. We show that the voting cycle may have to be very long for the sequential voting outcome to be close to the permanent insurance case. In a low turnover economy, everybody would benefit from a high insurance level. As we will see, however, the tension between the short-run good and the long-run best becomes more severe in a low turnover economy, so that the long-run best is more difficult to achieve. Boadway and Wildasin (1989) discuss pension systems and state that, if voters do not expect their own votes to affect future voting outcomes, young non-altruistic voters will not vote for high pensions financed by high taxes on labor, if the period until the next vote is short compared to the remaining period before their retirement. A similar mechanism is at work here. The focus of this paper is to analyze how different turnover rates affect preferences for insurance. This is a first step in an investigation of the joint determination of turnover and benefit levels. A priori, we expect causality to run in both directions. In addition to the mechanism discussed above, we expect that the rate of turnover is affected by the benefit level. High benefits may cause ‘better’ matches by facilitating longer search and thus lead to a larger number of relatively stable and long-term positions. Such a mechanism is developed in Marimon and Zilibotti (1997). They assume that the degree of specialization differs between 6

Gruber (1997) analyzes the consumption smoothing effect of unemployment insurance.

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Europe and the U.S. The educational system in the U.S. seems to provide less specialized skills than, for example, the German vocational high school programs. A higher degree of specialization makes the matching problem more difficult and can thus make turnover slower by increasing the length of both unemployment and employment spells. Furthermore, if unemployment insurance is generous, we should expect individuals to be willing to become more specialized, which would reduce turnover. In general, high unemployment insurance coverage is likely to affect the search behaviour of the agents, so that the turnover rate decreases and the unemployment level increases. In this paper, we extend the model to allow for an endogenous determination of turnover, while we abstract for the effects of unemployment insurance on the level of unemployment. This is done in the simplest possible way, by letting the unemployed make a once-and-for-all choice of the rate of turnover. In ongoing work (Hassler et al., 1998) we analyze, and explicitly model, the joint determination of turnover, unemployment and insurance in detail. We will see the most preferred rate of turnover of the unemployed increase in the level of unemployment benefits. Thus, high (low) unemployment benefits cause low (high) turnover which implies that the employed prefer high (low) benefits. This feedback mechanism can produce multiple steady states. One of these, which we label Europe, is characterized by generous unemployment benefits and low turnover, while a second, labelled U.S., is characterized by low benefit levels and high turnover. Stricter legal regulations on hiring and firing is another possible explanation for the low rates of turnover found in Europe. Such regulations are, of course, not determined by the unemployed themselves, but rather by the same political institutions that determine the level of unemployment benefits. In this case, our results could be interpreted as showing that high (low) insurance and low (high) turnover are complements and should be observed more often than the opposite combination. The paper is organized in the following way. In Section 2, we construct the basic model to be used in the paper. In Subsection 2.1 the individuals are not allowed to save and borrow. This assumption is relaxed in Subsection 2.2. In Section 3, we analyze the effects of introducing sequential voting. Section 4 endogenizes the rate of turnover and Section 5 concludes. Most proofs and technical details are included in a mathematical appendix available from the authors upon request.

2. Preferences for unemployment insurance

2.1. A flow model without savings Consider the following discrete time search model. Individuals receive a net income of w e when employed. When unemployed, they receive unemployment

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benefits, denoted w u . If the individual is employed, there is an exogenous probability q that she will become unemployed between the current and the next period. We call this probability the separation rate. Similarly, an unemployed individual may become employed the next period with the exogenous probability h, called the hiring rate. The only state variable for the individual is the employment status l that can assume values e and u, denoting employed and unemployed. The value function for the median voter must satisfy 1 V(e) 5 U(w e ) 1 ]]((1 2 q)V(e) 1 qV(u)), 11r

(1)

1 V(u) 5 U(w u ) 1 ]]((1 2 h)V(u) 1 hV(e)), 11r where U( ? ) is the per period utility function and r is the subjective discount rate. It then follows that the expected utility in the two states is given by (1 1 r)[(r 1 h)U(w e ) 1 qU(w u )] V(e) 5 ]]]]]]]]], r(r 1 h 1 q)

(2)

(1 1 r)[(r 1 q)U(w u ) 1 hU(w e )] V(u) 5 ]]]]]]]]]. r(r 1 h 1 q) After substituting the CARA utility function 2 e 2g c for U( ? ), we can use the value function for studying individual preferences over the level of an unemployment insurance financed by a non-negative pay-roll tax t.7 Income when employed is w e 5 w(1 2 t ) where w is the gross wage. Let d denote the steady state dependency ratio, i.e. the ratio of the number of unemployed to the number of employed. Note that d 5 q /h. Let us now make the following definition. Definition 1. The unemployment insurance system is actuarially fair if w u 5 wt /d. Note that if individuals have different separation and hiring rates, actuarial fairness for a particular individual should be calculated using her own separation and hiring rates. Let a denote the deviation from actuarial fairness in the system. We define a from the following relation: wu a 5 1 2 ]]. wt /d

(3)

A non-zero a, i.e. a deviation from actuarial fairness, can arise for several reasons. 7

The reasons for choosing the CARA utility function as the base case will become clear below. We will, however, also provide a numerical analysis of the case of constant relative risk aversion (CRRA).

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One obvious reason is administration costs. Another reason is that differences in unemployment rates among individuals is less than fully reflected in different tax rates and / or benefits. Assume, for example, that the insurance system operates under a balanced budget and that all individuals face the same tax rate and the same unemployment benefits. Furthermore, assume that individuals belong to different types, denoted i, each with a particular dependency ratio d i . It is then straightforward to show that for individual i, a 5 1 2 d i /d,

(4)

where d is the aggregate dependency ratio, i.e. the economy-wide ratio of unemployed to employed. We will focus on the cases when 1 . a $ 0 under the maintained assumption that the median voter faces a long-run unemployment risk that is no larger than the aggregate unemployment rate. We assume that the median voter is employed and we will thus concentrate on her preferences over t.8 Since l is the only state variable, the preferred level of t is fully determined by the median voter’s employment status. To find that tax level we maximize V(e) with respect to t. The first-order condition for a maximum is

F

S

d 1 q(1 2 a) t 5 ]]] 1 1 ]ln ]]]] 12a1d gw (1 1 r)(dr 1 q)

DG

.

(5)

Note that with no deviation from actuarial fairness, the most preferred tax rate is

F

S

d 1 q ]] 1 1 ]ln ]]]] 11d gw (1 1 r)(dr 1 q)

DG.

(6)

The first term, d /(1 1 d), corresponds to full insurance. The term ln(q /((1 1 r)(dr 1 q))) is negative, provided that the discount rate is strictly positive, which then implies less than full insurance. This result was established by Wright (1986). Now, let us consider how the median voter’s preferences change with turnover. Result 1. The most preferred insurance level (strictly) increases in turnover when individuals do not have access to capital markets for borrowing and saving and discounting is (strictly) positive. Proof. The derivative of (5) with respect to the separation rate, holding q /h constant at d can be written

8

In this section, we analyze the welfare of the employed for different t, given that it is held constant forever. The value of a constant t that maximizes the value function of the currently employed may, however, not be attainable in a political equilibrium with finite voting cycles. We will return to this issue in Section 3.

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rd 2 ]]]]]]] . 0. g qw(1 2 a 1 d)(rd 1 q)

(7)

With strictly positive discounting, full insurance is suboptimal since increasing risk only has second-order negative effects when moving from full insurance. On the other hand, the reduced tax gives more money to spend today, while the loss comes in the future and is thus valued lower. This has positive first-order effects on the value function of the employed. This latter effect is stronger when separation rates are low, since the unemployment period is expected to be further into the future. This is the intuition behind Result 1.

2.2. Allowing savings Now assume that individuals can save and borrow but not privately insure against the unemployment risk. They receive interest on their financial assets, denoted A t , and the interest rate is, for simplicity, assumed to coincide with their subjective discount rate r. In addition to the employment status l, the amount of financial assets now also enters as an argument of the value function. The finite horizon value functions are given by

O(1 1 r)

A t 11 5 (1 1 r)(A t 1 w t 2 c t ),

5

T

Vt (A t , l t ) 5max Et cs

2s 1t

s 5t

U(c t ) s.t. A t given,

(8)

A T 11 $ 0,

with u(c) 5 2 e 2g c , w(1 2 t ),

5

w t 5 wt (1 2 a) ]]], d

if l t 5 e (employed), (9) if l t 5 u (unemployed),

e with probability (1 2 q),

5

l t 11 5

if l t 5 e,

u with probability q,

if l t 5 e,

u with probability (1 2 u),

if l t 5 u,

e with probability h,

if l t 5 u.

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It is straightforward to show that as the horizon T goes to infinity, the value-functions 9 converge to 11r V(A t , e) 5 2 ]] e 2g ([r / (11r)] A t 1c e ) , r 11r V(A t , u) 5 2 ]] e 2g ([r / (11r)] A t 1c u ) , r

(10)

where c e and c u are constants satisfying e 2g c e 5 (1 2 q) e 2g (r(w e 2c e )1c e ) 1 q e 2g (r(w e 2c e )1c u ) ,

(11)

e 2g c u 5 (1 2 h) e 2g (r(w u 2c u )1c u ) 1 h e 2g (r(w u 2c u )1c e ) , which ensures that the Bellman equations are satisfied. It can also be shown that consumption equals the annuity value of financial assets plus a constant that depends on the current employment status, i.e. r ]] A t 1 c e , 11r ct 5 r ]] A t 1 c u , 11r

5

if employed, (12) if unemployed.

In a mathematical appendix, available upon request from the authors, we show that there is always a unique solution to the maximization problem. Furthermore, we show that the constants in (12) satisfy wt (1 2 a) w(1 2 t ) . c e . c u . ]]], d 9

(13)

We have used discrete time for convenience, but the results would not change if we used continuous time. In such a case the value functions would be 1 V(A t , e) 5 2 ] e 2g ((r / A t )1c e ) , r 1 V(A t , u) 5 2 ] e 2g ((r / A t )1c u ) , r where c e and c u would be constants satisfying q g (w e 2 c e ) 5 ][e 2g (c u 2c e ) 2 1], r h g (w u 2 c u ) 5 ][e 2g (c e 2c u ) 2 1], r with properties similar to (11)

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which implies that the individual saves when employed and dissaves when unemployed. An important feature of the value functions is that wealth only enters through the multiplicative term e 2g [r / ( 11r)] A t . We can then define

H

V(l) ;V(A t ,l) e g [r / ( 11r)] A t , l 5

e,

if employed,

u,

if unemployed.

(14)

Clearly, we can use V(l) to find individual preferences over different values of t. Since V(l) is independent of A t , preferences over different values of t are also independent of A t . Although our economy will consist of individuals with different asset levels, the characterization of political preferences is simple.10 Now let us consider how the most preferred tax and insurance rates vary with employment turnover. The following two results show that this cannot be a monotonic relation. Result 2. If turnover is zero, the value of unemployment insurance maximizing the utility of the employed workers is zero. Proof. This is obvious, since the net wage decreases with the level of insurance in all future states of the world if turnover is zero. Result 3. As turnover with a given level of unemployment increases to infinity, the value functions become independent of the employment status of the individual. A proof of the previous result can be sketched in the following way. As turnover goes to infinity, the conditional probabilities P(l t 1 s 5 eul t 5 e) and P(l t 1 s 5 eul t 5 u) for s . 0 converge to the unconditional probability 1 /(1 1 d). Since this means that future income becomes independent of the current state, the value functions must coincide.11 Corollary 2. As turnover with a given level of unemployment increases to infinity, consumption becomes independent of the state, and the value of insurance goes to zero. Proof. If the value functions are independent of states, so is also consumption. Then marginal utility is independent of the state and there is thus no risk to insure. The intuition for the result that the need for insurance vanishes as turnover 10 This is, of course, an example of the well-known result that CARA utility implies that preferences over lotteries with fixed absolute sizes are independent of wealth. 11 A formal proof is available in a mathematical appendix available upon request from the authors.

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becomes high is the following. If turnover is high, an individual will experience a large number of unemployment and employment spells over a given period of time. The law or large numbers then assures that the time spent in unemployment over that period becomes certain as turnover goes to infinity. Consumption smoothing can then be perfectly achieved with saving and borrowing – no unemployment insurance is needed. It should be noted that this is certainly not the case when the individual cannot save and borrow. In that case, consumption in the two states differs by assumption when unemployment insurance is imperfect and the value of insurance is always positive. We now know that the tax level preferred by an employed voter is zero both when the turnover is zero and when it is very high. Obviously, there are intermediate parameter values for which the most preferred tax rate is positive. The following result will shed light on the relation between the most preferred tax rate and intermediate rates of turnover. Result 4. Let d denote the tax level corresponding to full insurance and let D denote the ratio of the utility in the two states, i.e. D ; expsg (c e 2 c u )d, at the employed median voter’ s most preferred value of t. Then: (i) The most preferred tax rate is unique and satisfies t , d. (ii) If the most preferred tax is strictly positive, it satisfies 11r

ln( D ) 1 lns 1 2 q 1 q Dd 2 lns s 1 2 hd D 1 hd t 5 d 2 d ]]]]]]]]]] , g rw

where D is the unique solution to s 1 2 hd D d (1 1 r) 5 d ]]] 1s1 2 dd s 1 2 hd D 1 h

q D ]]] 12q1q D

.

(iii) If ds1 1 rd . 1, the non-negativity constraint on t will be binding so the most preferred tax and insurance rates are zero. (iv) Both employed and unemployed have single-peaked preferences over t. (v) If the most preferred tax rate is strictly positive, the unemployed would prefer higher insurance than the employed. Proof. In a mathematical appendix available upon request from the authors, where also necessary and sufficient conditions for the most preferred tax rate to be positive are derived.

2.3. Numerical illustrations Let us now consider some numerical examples to illustrate the analysis. We use the parameter values from Table 2. In the upper panel of Fig. 1 we plot the expected utility of an employed individual for different values of the replacement ratio, i.e. the ratio of unemployment benefits to net wages, when a, that is the

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Table 2 Parameters Period r g w h low h high d qlow qhigh a

1 month 3% per year 2 1 1 / 18 per month 1 / 6 per month 10% h low / 10 h high / 10 0 and 3%

deviation from actuarial fairness, is zero.12 In the high turnover case, the hiring rate h is such that the expected length of the unemployment period is 6 months. In the low turnover case, the duration is 18 months. The separation rates are proportional to the hiring rates, so that the dependency ratio is 10% in both cases. We see that the values of the replacement ratios maximizing utility of the employed approximately coincide. The maximum is achieved at an unemployment compensation of 69 and 67% of the net wage for low and high turnover, respectively. The value functions are single-peaked in the figure, as was shown to be a general property of the value functions. This makes it possible to identify a median voter in the case of heterogeneity with respect to turnover and actuarial fairness. There is a crucial difference between the two curves in Fig. 1 – the curve in the high turnover case is much flatter. This is an illustration of Result 3, where we showed that risk disappears as turnover increases towards infinity. With an actuarially fair insurance system, the value functions then become horizontal. The relatively low degree of curvature implies that the value of the insurance (i.e., the utility loss of moving from optimal to no unemployment insurance) is smaller in the high turnover case. This implies that a small deviation from actuarial fairness affects the preferred insurance level more the higher the turnover is. This is illustrated in the lower panel of the figure, where we have calculated the expected utility for the two cases, now setting a equal to 3%. In the high turnover case, we get a considerable reduction in the most preferred unemployment compensation, which is now as low as 3%. With a slightly larger deviation from actuarial fairness, the most preferred level falls to zero. The effect is much smaller when turnover is low. The most preferred unemployment compensation falls to 48%, when a is set to 3%.13 Why is insurance more important in the low turnover case? An intuitive 12

As seen from the expression for the value functions, V(e) and V(u) are monotone transformations of c e and c u , so we plot the latter as functions of the replacement ratio. 13 Similarly, a positive deviation from actuarial fairness (a , 0) will have larger positive effects on the most preferred unemployment compensation.

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Fig. 1. Value functions of employed.

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explanation is that low turnover means that the income shock associated with a job loss is more persistent than with high turnover. It is well known that saving and borrowing is a good substitute for insurance, when shocks have low persistence. It should also be noted that the variance of the length of unemployment spells equals 1 /h 2 and thus increases in the expected length of the unemployment period. As noted above, an individual expecting to see many spells of unemployment / employment within a given horizon faces less uncertainty than an individual with few but longer spells of unemployment. With high turnover, the individual’s problem is largely to translate her variable income into a smooth consumption stream. This can be done by the insurance system, but also almost equally well by saving and borrowing on the credit market. A small deviation from actuarial fairness makes the capital market preferable. With low turnover on the labor market, the opposite is true. Insurance is then important, since an unusually long unemployment spell, with large effects on lifetime utility, is much more likely to occur. For example, the probability that someone who is currently unemployed must wait more than 3 years before becoming employed is 13.5% in the low turnover case but only 0.2% in the high turnover case. The probabilities of more than 5 years of unemployment are 3.6 and 0.005%, respectively. In the upper panel of Fig. 2 we plot the most preferred replacement ratio for employed workers against the hiring rate, for different values of the deviation from actuarial fairness. The other parameters are given in Table 2 and the separation rate q is set to h / 10, so that the dependency ratio and thus unemployment is kept constant, regardless of the rate of turnover. The lower panel shows the same relation, but now with the rate of turnover measured by the expected duration of the unemployment period. In Fig. 2 we see that for all levels of deviations from actuarial fairness, the most preferred tax level increases very steeply in turnover when the latter is low. The highest optimal tax rates are achieved for low rates of turnover, and these maxima are achieved at lower rates of turnover the higher is the deviation from actuarial fairness. For the four examined values of deviations from actuarial fairness (0, 0.03, 0.10, and 0.30), the maximum replacement ratio occurs at turnover rates corresponding to an unemployment duration of 18, 80, 143 and 250 months, respectively. As turnover increases from these low rates, the most preferred tax level decreases monotonically.14

2.4. Constant relative risk aversion The CARA utility function is convenient, but not necessarily the best way of representing preferences. The main intuition behind our results, that saving and borrowing provide a good (bad) substitute for insurance if turnover is high (low), 14

We have not been able to prove that the relation between turnover and most preferred taxes is single-peaked, nor have we found any combinations of parameters for which this is not the case.

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Fig. 2. Preferred replacement ratio of employed for different deviations from actuarial fairness and turnover rates.

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should also apply to other utility functions. The purpose of this section is to extend the numerical illustrations to the case of constant relative risk aversion, to get an indication of whether this is the case. We thus need to calculate the value functions of the problems given by Eqs. (8) and (9) with U(c) replaced by c 12g 9 2 1 lim ]]]. g 9→g 1 2 g 9

(15)

By iterating on the Bellman equation, we find numerical solutions to the value functions.15 Of course, assets will now affect preferences and, unlike in the previous section, we will not be able to provide theoretical result. Nevertheless, the numerical solutions may give some indications of the generality of our results. In Fig. 3 we plot value functions for the employed against the replacement ratio. We set g 5 1, a 5 0 and the values in Table 2 for the remaining parameters. The

Fig. 3. Value functions of employed with CRRA utility and actuarially fair insurance. ] This is done by the following steps. (1) Fix a sequence of asset levels, A [A ; h 2 10, 2 9, . . . ,40j. 0 0 (2) Guess the associated values of V (A,e) and V (A,u). (3) Fit cubic splines to V 0 in order to make V 0 ] defined for all A [ R. (4) For A [A, find V 1 (A,l) from V 1 (A,l) 5 max c hU(c) 1 (1 1 r)21 EV 0 ((A 1 vl 2 c)(1 1 r),l9)j. (5) Iterate from (3) until sufficient convergence is achieved. 15

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Fig. 4. Value functions of employed with CRRA utility.

four panels are drawn for assets equal to 210, 0, 12 and 36 times the monthly gross wage. The figure shows that the most preferred replacement ratio falls as assets increase. This is, of course, due to the fact that higher levels of assets make the individual less averse to the unemployment risk. More importantly, we find that the curves are much flatter for the high turnover case, exactly as with constant absolute risk aversion. We thus expect a small deviation from actuarial fairness to have a substantially larger effect on the most preferred replacement ratio, when turnover is high. This is shown in Fig. 4, where we set a to 3 and 10% and show the value functions for assets equal to 12. As we see, the decrease in actuarial fairness reduces the most preferred replacement ratio from 58 to 30 and 9% in the high turnover case. In the low turnover case, on the other hand, the fall is from 70 to 57 and 38%. It is also clear that the increase in welfare associated with the introduction of an unemployment insurance system is substantially larger when turnover is low. Our conclusion is thus that our findings in the previous subsections are not idiosyncratic to the CARA utility function.

3. Sequential voting In the previous section, we assumed that the median voter could fix the tax and benefit level forever. Now, let us consider the case when taxes can be fixed only for some finite number s $ 1 of periods. We assume that the tax rate is set one period before it begins to apply, so that the voter does not know her employment status when the tax rate begins to apply. The tax rate that is determined at t thus applies to t 1 1 . . . t 1 s. In addition to assets and employment status, the value functions now clearly depend both on the tax rate determined in the current period and on the tax rates that will apply thereafter. Let V(A t ,e,t,t e ,s) denote the value function for the employed median voter who at time t expects taxes to be set to t e

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from t 1 s 1 1 and herself sets the tax rate to t for t 1 1 to t 1 s.16 It can be shown that 17 11r V(A t ,e, t, t e , s) 5 2 e 2g [r / (11r)] A t ]] e 2g c e,s , r

(16)

11r V(A t , u, t, t e , s) 5 2 e 2g [r / (11r)] A t ]] e 2g c u,s , r where c e,s and c u,s are functions of the parameters of the problem, in particular of t, t e and s. Consumption is given by r ]]A t 1 c e,s , 11r ct 5 r ]]A t 1 c u,s , 11r

5

if employed, (17) if unemployed.

From (16) we see that the value function is proportional to e 2g [r / (11r)] A t , so preferences over t are independent of wealth. As before, it is then easy to identify the median voter as any of the employed. Furthermore, this implies that no strategic motive is involved in voting. Changing the tax rate for t 1 1 only affects the asset distribution in the future. Since assets are irrelevant for preferences over tax rates, the current median voter cannot affect future votes, if we restrict the attention to Markov strategies. She thus only has to consider what is her preferred tax rate until the next vote. This implies that the median voter solves max V(A t , e, t, t e , s). t

(18)

We may now look for a time-consistent dynamic voting equilibrium. Here, this means that if the median voter, who is employed, expects a tax rate of t * from the next voting period and onwards, she votes for the same tax rate now and thus sets t 5 t * for the coming s periods. We thus require that a time-consistent dynamic voting equilibrium for the tax rate t * satisfies

t * 5 arg max V(A t , e, t, t * , s). t

(19)

The immediate question is then if the most preferred long-run tax rate can be sustained in a dynamic voting equilibrium. The following result shows that when s 5 1 the answer is a clear ‘no’.18 16

The tax rate in the current period is also set to t e , although it is straightforward to change this assumption. We could compute the value function for all possible sequences of tax rates, although this may require some tedious calculations. Preferences over tax rates are independent of wealth for all such sequences. 17 Proof in the mathematical appendix available upon request from the authors. 18 Intuition and the numerical exercises below suggest that this result should extend to any finite s.

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Result 5. Define the most preferred long-run tax rate for the employed as t * ; arg maxt V(e,t ). If 1 2 h . q and T * . 0, T* cannot be sustained in a voting equilibrium where the median voter sets the tax rate for one period at a time. Proof. See Appendix A, Section A.1. We would intuitively expect the insurance motive for the employed to increase over time. The following result shows that this is correct.19 Result 6. Let t * . 0 denote the tax rate in a dynamic voting equilibrium with one-period voting cycles. If 1 2 h . q, the current median voter would like the tax rate two periods ahead to be strictly larger than t. Proof. See Appendix A, Section A.2. The assumption 1 2 h . q implies that the probability of being employed at some future date is always higher for a currently employed person than for an unemployed person. Given this assumption, it is straightforward to extend the proposition in the previous result to periods further in the future. Given Result 6, we expect the median voter to become more favorable to high taxes and unemployment insurance during the next voting cycle as voting cycles become longer. Making the voting cycle longer may thus be a way of making high unemployment insurance feasible in a voting equilibrium. How the length of the voting cycle affects insurance preferences may depend on the turnover rate, since the difference between long-run and short-run interests of the employed depends on turnover. To analyze this dependence, consider the following experiment. Set t e to the values preferred by an employed agent in the low and high turnover economies, if it were to be fixed forever. As in the previous section, these tax rates correspond to replacement ratios of 69 and 67% when insurance is actuarially fair. Let us now consider whether an employed person would prefer taxes to be set to zero during the coming s periods (months) rather than being kept at the long-run optimum for all periods. This will clearly depend on s, and we expect that, for large enough s, t e may be preferred to zero. In Fig. 5 we plot the temptation to deviate from the most preferred long-run insurance, represented by the increase in consumption generated (c e,s 2 c e ) at the time of the deviation, against s, expressed in years for the two cases. Three features should be noted here: In both turnover cases, the median voters prefer zero insurance during the next voting period, even if the voting cycle is rather long. We actually need voting

19

Shavell and Weiss (1979) provide a similar result.

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Fig. 5. Consumption increase in percent of gross wage employed persons if insurance is zero for the next voting cycle.

cycles exceeding 20 years for the median voter to prefer the most preferred long-run insurance over zero. Second, the temptation to deviate first increases in the length of the voting cycle, then decreases. Third, the initial increase in the temptation to deviate is stronger when turnover is low. When the voting cycle is around 5 years, the value of deviating from the long-run optimum is several times higher under low turnover than under high turnover. The temptation to deviate from a high insurance may thus be particularly strong when turnover is low. The previous results indicate that increasing the voting cycle may be an ineffective way of sustaining political support for generous unemployment benefits. This particularly seems to be the case when turnover is low and the long-run most preferred benefit level is high. The strength of the committment device needed in order to support a generous benefit system may thus increase as

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turnover decreases. In other words, the tension between the long-run and short-run interests of the median voter is stronger when turnover is low. In this case, increasing the length of the voting cycle may, in fact, increase the difficulty in sustaining the long-run most preferred level. When the level of benefits is reduced from the long-run most preferred level, the temptation to set them to zero naturally decreases. Thus, even if the long-run most preferred level cannot be sustained with finite voting cycles, lower levels might. Using numerical methods, it is straightforward to find the solutions to (19) for different lengths of the voting cycle. Voting equilibria for different lengths of the voting cycle are plotted in Fig. 6. The unemployment duration is, as above, 18 months in the low turnover case and 6 months in the high turnover case. As we see, the voting cycle must be very long to generate voting equilibria with high replacement ratios – even a 15-year voting cycle is not sufficiently long to support particularly high replacement ratios. We also see that in exact accordance with the results in the previous section, the sensitivity of the chosen level of insurance to the degree of actuarial fairness is much larger when turnover is high.

Fig. 6. Voting equilibrium replacement ratios for different voting cycles.

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4. Multiple stationary states The previous sections investigated the effects of turnover rates on the public allocation of unemployment insurance. In other words, we have a causality going from market structure to political outcome. It seems obvious that the reverse causality also exists, i.e. political decisions shape the environment of the individuals and, thus, affect their behaviour. Naturally, both effects affect each other, and in the economy both market structure and political outcomes are jointly determined. To analyze such a dual causality, we will here present a simple extension of our model that allows the turnover rate to be endogenous. The intuition behind the mechanism that creates a causality from the level of benefits to turnover is straightforward: if the level of unemployment benefits is high, agents will increase their aspiration levels and, on average, search longer before finding a job they can accept. The matching will thus be better and we expect the match to be more productive and last longer. Such a ‘picky’ search strategy implies low turnover, which will be too risky if unemployment benefits are low. In this manner, generous unemployment benefits produce low turnover, and, as we saw in Section 2.2, low turnover will be associated with a preference for generous unemployment benefits. It is outside the scope of this paper to provide an explicit formalization of how benefits affect turnover. Instead, we simply assume that labor market entrants can choose what turnover they will face in the future. Their choice also affects the wage they get when becoming employed. We denote the relation between wages and hiring rates w(h). We also assume the separation rate to be lower for good matches, i.e. it falls as the hiring rate decreases. In particular, we assume that the separation rate is proportional to the hiring rate, q(h) 5 dh. Now, let us define the function H(t ) ; arg max V(u; h, q(h), w(h), t ), h

(20)

where H(t ) gives the choice of a constant hiring rate maximizing the expected utility of the unemployed. It can be considered a stationary state optimal response function. We require the choice of h to be permanent. An economic interpretation of the assumption of a permanent turnover is the following. When entering the labor market, agents make a decision regarding the degree of specialization of their human capital. The unemployed search for a job matching their individual characteristics. The closer the match, the higher is the productivity of the match and thus the wage they receive, as in, for example, Marimon and Zilibotti (1997). We can think of the derivative of the wage with respect to the distance between the type of job and the type of individual as the degree of individual specialization. A highly specialized individual can get a high wage, but only if finding a job closely matching her particular type of skills. The degree of specialization can be assumed to be chosen at an early age and be too costly to change later in life. A more

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specialized individual can benefit more from being ‘picky’ when looking for a job since she thereby can get a higher wage and a more long-term job. She would thus be more willing to pay the cost of being ‘picky’, i.e., a longer search time. We consider our assumption of a choice of constant labor rotation to be a short-cut for a richer model along these lines.20 Now, let us turn to the employed. We assume that they use a political mechanism T, determining the tax rate that maximizes their expected future utility. This most preferred tax rate depends on their previous turnover choice. We can then define T(h) ; arg max V(e; h, q(h), w(h), t ). t

(21)

Both h 5 H(t ) and t 5 T(h) define mappings between the unemployment insurance rate (determined by t ) and the rate of turnover (given by h). We expect H(t ) to define a negative relation between unemployment insurance and turnover, as in Marimon and Zilibotti (1997). Furthermore, a high tax and insurance rate reduces risk, which also makes the more risky strategy of high specialization more attractive. This is the mechanism explored in Section 2.2. Now, consider T(h). This function is almost identical to the functions depicted in Fig. 2, the only difference being that the wage may now depend on h. We should thus expect T(h) to be decreasing over relevant ranges of h. In Fig. 7 we have plotted an example of the mappings defined by H(t ) and T(h).21 The horizontal axis represents the rate of turnover, indexed by the expected length of an unemployment spell (1 /h), and the vertical axis represents the insurance rate, indexed by the replacement ratio, i.e. unemployment benefits over net wages (t (1 2 a) /(1 2 t )d). We see that the curves in Fig. 7 intersect at two points. One corresponds to a stationary state with high turnover and low insurance, the expected length of an unemployment period is 6 months and unemployment benefits are 4% of the net wage. We call this the the U.S. stationary state. The other intersection occurs at a low turnover rate and a high level of unemployment insurance. The expected unemployment period here is 35 months and the insurance gives benefits of 60% of the net wage. We call this the European stationary state. We have thus established that our model can generate multiple stationary states.

20 In another paper (Hassler et al., 1998) we model this in a more realistic way. See also Marimon and Zilibotti (1997). 21 The wage function w(h) is assumed to be w 5 1 1 w 1 (1 2 e 2w 2 / h ). This function gives a wage of unity as expected unemployment duration approaches zero. As duration increases, the wage increases at a decreasing rate so that the wage approaches 1 1 w 1 as duration goes to infinity. The curvature of the wage function is determined by w 2 . The parameters used for plotting Fig. 7 are w 1 5 0.09, w 2 5 2 0.035, a 5 0.03, r50.03 / year and g 5 2.

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Fig. 7. The unemployed’s choice of turnover and the employed’s choice of unemployment insurance.

5. Concluding discussion Before making some concluding remarks, let us summarize our main results. First, the rates of flows in and out of unemployment are important determinants of preferences over the level of unemployment insurance. Low turnover means that the income shock associated with becoming unemployed is highly persistent. In this case, self-insurance via a capital market is a bad substitute for unemployment insurance. It is then in the long-run interest of the employed (as well as, of course, of the unemployed) to have an unemployment insurance with high replacement ratios, even when the system is inefficient or actuarially unfair for other reasons. Second, there is a tension between the long-run and short-run interest of the employed which prevents the long-run good from being achieved in a sequential voting equilibrium. The median voter prefers an insurance system with replacement rates which are low today but increasing over time. This is not feasible in a sequential voting equilibrium without a possibility to bind future voters. The tension between the long- and the short-run interest depends on the turnover rate. If unemployment insurance is introduced, the expected utility of employed individuals increases more the lower is the turnover rate. However, the lower the

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turnover rate the stronger is the immediate temptation to reduce the level of benefits and taxes. Third, if we allow the turnover rate to be endogenous, we observe causality running in both directions, which can produce multiple steady state equilibria. This may help us understand the widely different behaviour of the American and European labor markets. High (low) unemployment insurance incurs low (high) turnover, and low (high) turnover incurs high (low) unemployment insurance. We have shown that our first result above critically depends on the assumption that people can save and borrow. In our model, we made the rather unrealistic assumption that everybody has access to a perfect capital market. We must thus consider the potential consequences of borrowing limits and other capital market imperfections. Such assumptions complicate the model considerably. In particular, preferences over the insurance level become dependent on current assets, thus making it more difficult to find the median voter. What is probably more important is that if preferences depend on assets, the current median voter can change the behavior of the future median voter by affecting the wealth distribution via the insurance system. Finding a dynamic voting equilibrium is then, at best, difficult (Krusell et al., 1997, show examples where this is possible, however). Our second finding was that the short-run and long-run interests of the employed diverge. This implies that it may be difficult to sustain an insurance system with high replacement ratios, in particular when turnover is low, even though this is in the long-run interest of the employed (and, of course, the unemployed). The observation that unemployment insurance is generous and turnover is low in Europe was the starting point of this paper. The first finding provides us with a motive – low turnover is likely to make the employed want a high level of unemployment benefits. The second finding shows that the means for this may not exist if the level of unemployment insurance and its associated institutions can be changed without frictions. We do think, however, that our second finding points in an interesting direction. It shows that the employed in Europe have more to gain from building institutions that facilitate long-run arrangements of the unemployment insurance than the employed in the U.S. have. Similarly, the stronger tension between the short-run interest of the employed and the unemployed may also make it more important for the employed to try to build institutions where the unemployed have considerable political influence. We think that universal unemployment insurance coverage, unions and political labor parties are all means which can contribute to achieving such long-run social contracts. Such institutions may thus be more likely to develop in low turnover economies.22 The finding that the deviation between the long-run and the short-run interest is 22

´ See Esteban and Sakovics (1993) for a model where a small cost of changing institutions can prevent the breakdown of an intergenerational transfer scheme, in spite of short-run temptation to do so.

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larger when turnover is low is likely to be a general result. This could be applied to other forms of social security, like pensions, and health insurance. In general, we expect that stronger commitment devices are necessary if the current status of individuals is highly, but not perfectly, persistent. In this case, an increase in the length of the voting cycle may be an unproductive way of overcoming the political time inconsistency. In this paper, we have not explicitly modelled individual heterogeneity with respect to separation and hiring rates. We have showed that preferences are single peaked, so a majority voting mechanism would result in the median voters preferences determining the insurance rate. Our finding that preferences are non-monotonic in turnover implies, however, that the median voter is not immediately identified as the voter with median rate of turnover. Employed individuals with very low turnover and individuals with high turnover may share a low interest in unemployment insurance. The non-monotonicity also means that changes in the distribution of turnover rates may have substantial effects on politically chosen unemployment insurance rates, even if they do not affect the mean and median turnover rates. For example, say that there is a primary sector with very low turnover, for instance a public sector with lifelong employment. In addition, assume that there is a secondary sector with very high turnover, for example a non-regulated private sector. Now, assume that the lifelong employment contracts in the primary sector are terminated and that job-protection legislation is introduced in the secondary sector. This would then increase the turnover in the primary sector and decrease it in the secondary sector. Both these factors would lead to an increase in the number of individuals preferring high unemployment insurance and the median voter is likely to change, thereby incurring a political choice of higher unemployment insurance. Allowing ourselves to be speculative, we think that this sketches a development that may have occurred in many European countries in, say, the last two or three decades. A last issue concerns time limits on unemployment benefits. We have shown that employed individuals have a strong motive to insure against long spells of unemployment. Short spells can more easily be handled through self-insurance. This implies that an unemployment insurance covering only a limited (short) period of unemployment appears misguided from a purely insurance based point of view. This may be an explanation for the strong opposition in many European countries against imposing a time limit on unemployment benefits. This statement, as most of the analysis in this paper, is positive. Normative conclusions about the level of insurance must certainly take many factors that have been outside the scope of this paper into account. An important example of such factors is the moral hazard problem, due to imperfect control of the individual’s search intensity. The low rates of turnover found in Europe may cause a severe conflict between the goals of providing good insurance against bad luck on the labor market and constructing an unemployment insurance that mitigates various moral hazard problems.

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Acknowledgements We thank Antonio Cabrales, Assar Lindbeck, Torsten Persson, Kjetil Storesletten and Fabrizio Zilibotti for helpful comments and discussions. We are also grateful to the editor of this journal and the two anonymous referees for giving ¨ important suggestions. Christina Lonnblad provided invaluable editorial assistance. John Hassler thanks the people at the Minneapolis Federal Reserve Bank for their hospitality. A.1. Temptation to deviate with one period voting cycles First, note that

S

≠V(A t ,e,t1 ,t,1) 1 ]]]] 5 ]] (1 2 q)(2w)U 9(c(A t 11 ,e)) ≠t1 11r

D

12a 1 qw ]]U 9(c(A t11 ,u)) . d

(A.1)

Now we want to show that (A.1) is strictly negative. First, since we assume that t * . 0, it is an interior maximum to V(A t ,e,t ), so it satisfies the first order condition 0

5

≠V(A t , e, t ) ]]] ≠t ≠V(A t 11 , e, t ) 1 5 ]] (1 2 q) (2w)U 9(c(A t 11 , e)) 1 ]]]] 11r ≠t ≠V(A t 11 , u, t ) 12a 1 q w ]]U 9(c(A t 11 , u)) 1 ]]]] . d ≠t

F

S

S

DG

D (A.2)

Now, since also ≠V(A t 11 ,e,t ) / ≠t 5 0, (A.1) and (A.2) implies that ≠V(A t ,e,t1 ,t,1) q ≠V(A t11 ,u,t ) ]]]] 5 2 ]] ]]]]. ≠t1 11r ≠t

(A.3)

What then remains is to show that ≠V(A t 11 ,u,t ) / ≠t is strictly positive. For this purpose, we first note that it follows from (A.2) that

S

D

(A.4)

DG.

(A.5)

≠V(A t 11 ,u,t ) q 12a U 9(c(A t 11 ,e)) 5 ]]] w ]]U 9(c(A t 11 ,u)) 1 ]]]] . d ≠t (1 2 q)w For an unemployed person, we have

FS S

D

≠V(A t ,u,t ) ≠V(A t 11 ,e,t ) 1 ]]] 5 ]] h (2w)U 9(c(A t 11 ,e)) 1 ]]]] ≠t 11r ≠t ≠V(A t 11 ,u,t ) 12a 1 (1 2 h) w ]]U 9(c(A t11 ,u)) 1 ]]]] d ≠t

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Using (A.4) in (A.5) yields

FS S

≠V(A t ,u,t ) ≠V(A t11 ,u,t ) 1 q 12a ]]] 5 ]] h ]] w ]]U 9(c(A t 11 ,u)) 1 ]]]] ≠t 11r q21 d ≠t

S

≠V(A t 11 ,u,t ) 12a 1 (1 2 h) w ]]U 9(c(A t11 ,u)) 1 ]]]] d ≠t

F

DD

DG G

≠V(A t 11 ,u,t ) 12q2h 12a 5 ]]]] w ]]U 9(c(A t 11 ,u)) 1 ]]]] . d ≠t s1 1 rd(1 2 q) This implies that

O

≠V(A t ,u,t ) 12a ` s ]]] 5 w ]] k U 9(c(A t 1s ,u)) . 0, ≠t d s 51 where k ; (1 2 q 2 h) / [(1 1 r)(1 2 q)] and U 9(c(A t 1s ,u)) is conditional on the median voter being unemployed from t 1 1 to at least t 1 s. A.2. Increasing preferred tax rate The derivative of V(A t ,e,t1 ,t,1) with respect to the tax rate in period t 1 2 and evaluated at t1 5 t 5 t * is given by

S

≠V(A t 11 ,e,t1 ,t,1) ≠V(A t 11 ,u,t1 ,t,1) 1 ]] (1 2 q)]]]]] 1 q ]]]]] 11r ≠t1 ≠t1

D

≠V(A t 11 , u, t1 , t, 1) 1 5 ]] q ]]]]]], 11r ≠t1

(A.6)

since ≠V(A t 11 ,e,t1 ,t,1) / ≠t1 5 0 by assumption. The derivatives ≠V(A t 11 ,e,t1 ,t,1) / ≠t1 and ≠V(A t 11 ,u,t1 ,t,1) / ≠t1 are given respectively by

S w ]]S 2hU 9(c(A 11r

D

w 12a ]] 2(12q)U 9(c(A t 11 1r(w e 2c e ),e))1q ]]U 9(c(A t11 1r(w e 2c e ), u)) , 11r d t 11

D

12a 1r(w u 2c u ),e))1(12h)q ]] U 9(c(A t 11 1r(w u 2c u ), u)) d (A.7)

Dividing by [w /(1 1 r)]g e 2g [r / ( 11r)] A t 11 , this can be simplified to

S S2h e

D D.

12a e g r(w e 2c e ) 2 (1 2 q) e 2g c e 1 q ]] e 2g c u 5 0, d e g r(w u 2c u )

2g c e

12a 1 (1 2 h)q ]] e 2g c u d

(A.8)

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Now, we know from Results 4 and 5 that c e . c u . Together with the assumption that 1 2 q . h, we have

S

D S

D

121 12a 0 5 2 (1 2 q)e 2g c e 1 q ]] e 2g c u , 2 h e 2g c e 1 (1 2 h)]] e 2g c u , d d which completes the proof.

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