Wor King Papers. Economics Working Papers. Business Cycle Dependent Unemployment Benefits with Wealth Heterogeneity and Precautionary Savings

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1 Wor King Papers Economics Working Papers Bsiness Cycle Dependent Unemployment Benefits with Wealth Heterogeneity and Precationary Savings Mark Strøm Kristoffersen As of Janary 2012 ECON-ASB working papers are inclded in Economics Working Papers

2 Bsiness Cycle Dependent Unemployment Bene ts with Wealth Heterogeneity and Precationary Savings Mark Strøm Kristo ersen y Agst 29, 2012 Abstract In the wake of the nancial and economic crisis the discssion abot social insrance and optimal stabilization policies has re-blossomed. This paper adds to the literatre by stdying the e ects of a bsiness cycle dependent level of nemployment bene ts in a model with labor market matching, wealth heterogeneity, precationary savings, and aggregate ctations in prodctivity. The reslts are ambigos: both procyclical and contercyclical nemployment bene ts can increase welfare relative to bsiness cycle invariant bene ts. Procyclical bene ts are bene cial de to contercyclicality of the distortionary e ect (on job creation) from providing nemployment insrance, whereas contercyclical bene ts facilitate consmption smoothing. JEL classi cation: E32, H3, J65. Keywords: Unemployment insrance, bsiness cycles, wealth heterogeneity, precationary savings. Comments from Torben M. Andersen, Mikkel N. Hermansen, Nicolas Petrosky-Nadea, Rne M. Vejlin as well as participants in seminars at Aarhs University, in the 2012 annal meeting of the Danish Econometric Society, and in the 6th Nordic Smmer Symposim in Macroeconomics are grateflly acknowledged. All remaining errors are, of corse, my own. y Aarhs University, Department of Economics and Bsiness, address: mkristo ersen@econ.a.dk. 1

3 1 Introdction In the wake of the nancial and economic crisis the discssion abot social insrance and optimal stabilization policies has re-blossomed. To let labor market policies, e.g. the nemployment insrance scheme, depend on the position of the bsiness cycle has been emphasized as a way to strengthen both social insrance and the (atomatic) stabilization of economic ctations, bt the literatre (both theoretical and empirical) on these sbjects is still of modest size. The optimal design of nemployment insrance (UI) has been stdied for several years, see Fredriksson & Holmlnd (2006) for a srvey of the literatre. However, only recently the literatre has started to investigate the e ects of UI across the bsiness cycle. The rsts to consider UI in a bsiness cycle context were Kiley (2003), Sánchez (2008), and Andersen & Svarer (2011b), who all sed static models not allowing for shifts between good and bad times. Recently, the literatre has been fast-growing, and bsiness cycle dependent UI is now being analyzed in dynamic models allowing for shifts between recessions and booms, see 1 Moyen & Stähler (2009), Andersen & Svarer (2010), Kroft & Notowidigdo (2010), Landais, Michaillat & Saez (2010), Mitman & Rabinovich (2011), Jng & Kester (2011), Ek (2012), and Schster (2012). However, the conclsions are open as some papers sggest that nemployment bene ts (both level and dration) shold be contercyclical, i.e., the UI system shold be more generos in bad times than in good times, whereas others sggest procyclical UI generosity. Importantly, all of the above do not allow for savings, and ths, they neither allow for wealth heterogeneity nor partial self-insrance in the form of precationary savings. 2 Acconting for precationary savings is potentially very important when stdying the insrance e ects of nemployment bene ts. The opportnity for individals to self-insre has important conseqences for optimal UI, as shown by Abdlkadiro¼gl, Krşç & Şahin (2002). They also show that UI schemes which are designed ignoring the possibility of partial selfinsrance via savings can actally be harmfl to the economy. Acconting for the heterogeneity of economic agents, e.g. wealth heterogeneity, has proven to be crcial when answering important economic qestions. As an example, the welfare costs of bsiness cycles are orders of magnitde larger in models with heterogeneos agents than originally sggested by Lcas (1987, 2003), see e.g. Storesletten, Telmer & Yaron (2001), and Krsell, Mkoyama, Şahin & Smith (2009). For srveys of the fast growing literatre with heterogeneos agents models see Heathcote, Storesletten & Violante (2009), and Gvenen 1 For considerations abot the practical implementation of bsiness cycle contingent nemployment insrance along with more thorogh reviews of this literatre (both the theoretical and the empirical) see Andersen & Svarer (2009, 2011a). 2 Landais et al. (2010) brie y consider self-insrance in the form of home prodction in their one-period model. 2

4 (2011). Several papers have stdied nemployment insrance in models with savings and heterogeneos agents, see Yong (2004), Pollak (2007), Reichling (2007), Lentz (2009), Krsell, Mkoyama & Şahin (2010), Vejlin (2011), and Mkoyama (2011). papers stdy nemployment insrance in a bsiness cycle context. In this sense, Costain & Reiter (2005) are more related to this paper. However, none of these They nd that procyclical social secrity contribtions are optimal, while nemployment bene ts shold be almost constant across states. However, their model is di erent from the model sed in this paper in some respects, for example their asset strctre is more simplistic, i.e., the interest rate is xed, there is no physical capital in the prodction process, and wages are independent of asset holdings. This paper stdies the e ects of a bsiness cycle dependent level of nemployment bene ts in a model with wealth heterogeneity and precationary savings. We se the model with aggregate ctations in prodctivity from Krsell et al. (2010), who only explicitly analyze UI in the steady state version of their model. Since labor spply and search e ort are both exogenos, UI does not case moral hazards, and therefore the optimal UI scheme does not trade o insrance and incentives to work/search, bt instead it trades o insrance and job creation. The model is basically a merger between two strands of the literatre: i) the Bewley-Hggett- Aiyagari model, 3 where risk-averse consmers face idiosyncratic earnings risks, against which they can only insre partially (throgh savings), and ii) the Diamond-Mortensen-Pissarides (DMP) search/matching model 4 of the labor market, where eqilibrim nemployment and vacancies are determined endogenosly. The model is calibrated to US data, and we nd that both procyclical and contercyclical nemployment bene ts can increase welfare relative to bsiness cycle invariant bene ts. Procyclical UI is bene cial becase the distortionary e ect of UI (on job creation) is contercyclical, whereas contercyclical UI is bene cial becase it facilitates consmption smoothing and raises mean consmption. It trns ot that there is a non-monotone relationship between these conteracting e ects. The largest welfare gain (in consmption eqivalent terms) is obtained by having procyclical nemployment bene ts when UI bene ts are conditioned on (crrent) prodctivity. This nding is robst to changing the calibration strategy, bt it trns ot that the chosen calibration strategy is crcial for the magnitde of the welfare gain obtained by shifting from constant UI across the bsiness cycle to procyclical UI bene ts. However, if UI bene ts are conditioned on either the nemployment level or lagged prodctivity instead and the pblic 3 See Bewley (n.d.), Hggett (1993), and Aiyagari (1994). 4 See Diamond (1981), Pissarides (1985), and Mortensen & Pissarides (1994). 3

5 bdget is allowed to work as a b er, the largest welfare gain is achieved from contercyclical UI generosity. The rest of the paper is strctred as follows: The model is presented in Section 2, and Section 3 explains how the model is solved nmerically. Section 4 considers the e ects of bsiness cycle dependent nemployment bene ts, whereas Section 5 contains varios robstness checks. Finally, Section 6 concldes. 2 Model We se the model with aggregate prodctivity shocks from Krsell et al. (2010) with some minor extensions. Time is discrete. Following Krsell & Smith (1998) it is assmed that aggregate prodctivity 5 z takes on two vales, z = g in good periods, and z = b in bad periods (with g > b > 0), and it follows a rst-order Markov process, where the probability of moving from state z to state z 0 is denoted zz 0 2 [0; 1]. 2.1 Matching The labor market has a Diamond-Mortensen-Pissarides search & matching strctre. Ths, nemployed workers and vacant jobs coexist, and existing matches between a worker and a rm are assmed to be separated at the exogenos rate 2 (0; 1). Unemployed workers and vacant jobs are randomly matched according to the aggregate matching fnction M (; v), which exhibits constant retrns-to-scale and is increasing in both argments. is the nmber of nemployed workers, and v is the nmber of vacancies. Ths, the job nding probability w is w = M (; v) = M 1; v = M (1; ) where the vacancy-nemployment ratio is de ned as v. The worker nding probability f is f = M (; v) v = M v ; 1 = M 1 ; 1. Hence, the law of motion for the nemployment rate is given by 0 = (1 w ) + (1 ) (1) 5 For notational convenience time sbscripts are left ot throghot the model description. 4

6 where a prime ( 0 ) denotes a next period variable. 2.2 Asset strctre It is assmed that no markets exist for insrance against idiosyncratic employment shocks. However, there exist two assets, capital k and eqity x, where capital is sed in the prodction process and the eqity is a claim for aggregate rm pro ts. The joint distribtion of assets and employment across consmers is denoted S, and the aggregate state in any given period is governed by (z; S). Next period s distribtion of assets is determined in this period since it depends only on the consmers asset accmlations and portfolio choice decisions. Likewise, next period s distribtion of the employment statses 6 (the fraction being employed and nemployed, respectively) is also determined in this period since it follows from the law of motion of aggregate nemployment in (1). Therefore, the joint distribtion of assets and employment across consmers in the next period is determined in this period, and we can write S 0 = (z; S). (2) Hence, the aggregate state in the next period is either (g; S 0 ) or (b; S 0 ). Let the consmer s state variable be a [1 + r (z; S) ] k + [p (z; S) + d (z; S)] x which is total asset holdings of the individal, and where is the depreciation rate of capital; r (z; S) is the interest rate; p (z; S) is the eqity price; d (z; S) is the dividend. Like Krsell et al. (2010) we implement the portfolio choice by considering two Arrow secrities, each paying one nit of the consmption good in a given state and nothing in the other state. This implementation is withot loss of generality since the two assets, aggregate capital and eqity, can be sed to create these secrities. transformation, see below. Investment rms carry ot this Let Q z 0 (z; S) denote the price of an Arrow secrity that provides one nit of the consmption good in the next period if and only if the next period aggregate prodctivity is z 0 when the 6 Note that only the aggregate distribtion of employment statses is determined this period. Next period s employment state is still ncertain at the individal level. 5

7 crrent state is (z; S). Then, the asset prices mst satisfy the following no-arbitrage conditions 1 = Q g (z; S) [1 + r (g; S 0 )] + Q b (z; S) [1 + r (b; S 0 )] (3) p (z; S) = Q g (z; S) [p (g; S 0 ) + d (g; S 0 )] + Q b (z; S) [p (b; S 0 ) + d (b; S 0 )]. (4) since we can perfectly track the retrns on capital and eqity by investing in the two Arrow secrities. 2.3 Consmers There is a continm of consmers with mass 1, and these are either employed (1 ) or nemployed (). The consmers face an exogenos borrowing constraint at a and are heterogeneos with respect to employment stats and asset holdings. Labor spply and search e ort are both exogenos Unemployed consmers Let a 0 z 0 denote the consmer s demand for an Arrow secrity that pays ot one nit of the consmption good in the next period if and only if the next period s aggregate prodctivity trns ot to be z 0. The nemployed worker s optimization problem is U (a; z; S) = max c;a 0 ga;a 0 b a (c) + zg (1 w (z; S)) U a 0 g; g; S 0 + w (z; S) W a 0 g; g; S 0 + zb [(1 w (z; S)) U (a 0 b; b; S 0 ) + w (z; S) W (a 0 b; b; S 0 )]g sbject to c + Q g (z; S) a 0 g + Q b (z; S) a 0 b = a + h T and S 0 = (z; S) where () is an increasing and strictly concave instantaneos tility fnction; c is the consmption level; 2 (0; 1) is the discont factor; w (z; S) is the job nding probability de ned above; h is nemployment bene ts before tax; T is a lmp-sm tax paid by all consmers; U (a; z; S) is the vale of being nemployed with asset holding a, and W (a; z; S) is the vale of being employed taking the wage determination into accont. Let the decision rle, i.e., the optimal soltion to the optimization problem, for a 0 z 0 z0 (a; z; S). be 6

8 2.3.2 Employed consmers The employed worker s optimization problem is fw (w; a; z; S) = max c;a 0 g a;a0 b a (c) + zg U a 0 g ; g; S 0 + (1 ) W a 0 g; g; S 0 + zb [U (a 0 b; b; S 0 ) + (1 ) W (a 0 b; b; S 0 )]g sbject to c + Q g (z; S) a 0 g + Q b (z; S) a 0 b = a + w T and S 0 = (z; S) where w is the wage. Hence, W f (w; a; z; S) is the vale of being employed given the wage w, and W (a; z; S) is the vale of being employed when taking the wage determination into accont. Denoting the wage fnction, i.e., the otcome of the wage determination, as w =! (a; z; S) the relationship is W (a; z; S) W f (! (a; z; S) ; a; z; S). Let the decision rle for a 0 z 0 be ee z0 (w; a; z; S) for a given wage, and de ne e z (a; z; S) 0 ee z0 (! (a; z; S) ; a; z; S). 2.4 Firms Assme a one- rm-one-job strctre. To nd a vacant worker the rm posts a vacancy. The vale of a vacancy, V (z; S), is V (z; S) = +Q g (z; S) (1 f (z; S)) V (g; S) + f (z; S) +Q b (z; S) (1 f (z; S)) V (b; S 0 ) + f (z; S) J g (a; z; S) ; g; S 0 f (a; S) da J ( b (a; z; S) ; b; S 0 ) f (a; S) da where is the vacancy cost; f (z; S) is the worker nding probability de ned above; f (a; S) is the poplation of nemployed workers with asset holdings a, and ths, f (a; S) = is the density fnction of the nemployed workers over a; J (a; z; S) is the vale of a lled job taking the wage determination into accont, and hence, the integrals show the expected vale of matching with an nemployed worker (given a ftre state) taking the wage determination and the individal decision rles into accont. The rm disconts ftre vales by the Arrow secrity prices since 7

9 these are the rates at which the non-constrained consmers discont ftre states. 7 free entry of rms which implies that rms post vacancies v (z; S) ntil V (z; S) = 0. There is A matched rm rents capital from the consmers at a rental rate of r (z; S) and pays the worker a wage w. The vale of a lled job given the wage w, e J (w; a; z; S), is therefore h ej (w; a; z; S) = e (w; z; S) + Q g (z; S) h +Q b (z; S) V (b; S 0 ) + (1 V (g; S 0 ) + (1 ) J i ) J e e g (w; a; z; S) ; g; S 0 i e e b (w; a; z; S) ; b; S 0 where the instantaneos pro t is de ned as e (w; z; S) max k fzf (k) r (z; S) k wg, and zf (k) is the prodction fnction, which is increasing and strictly concave in the capital inpt k. Again, we can de ne J (a; z; S) e J (! (a; z; S) ; a; z; S). The rst-order condition implies r (z; S) = zf 0 (k). Symmetry implies that in eqilibrim each rm has the same capital stock, and the capital stock per job is ~ k = ^k=(1 aggregate capital stock. Therefore, the eqilibrim pro t can be written as (a; z; S) zf ~k ); where ^k is the r (z; S) ~ k! (a; z; S) : (5) The dividend is then calclated as the total pro ts mins the total vacancy costs, that is d (z; S) = (a; z; S) f e (a; S) da v (6) i.e., aggregated pro ts over all rms (=all jobs) since f e (a; S) is the poplation of employed workers with asset holdings a. 2.5 Wages When an nemployed worker and a vacant job get matched, the wage is determined throgh Nash Bargaining. Ths, the wage in a match inclding a worker with asset holdings a solves 1 max fw (w; a; z; S) U (a; z; S) ej (w; a; z; S) V (z; S) w where 2 (0; 1) is the bargaining power of the worker. The soltion is described by w =! (a; z; S). 7 In the nmerical soltion of the model it trns ot that very few (if any) consmers have a binding borrowing constraint. 8

10 The bargained wage depends on the asset holdings of the worker, becase these a ect both the worker s otside option in the crrent period and the chosen asset holdings next period (contingent on the ftre aggregate state) Investment rms We envision competitive investment rms who sell contingency claims to consmers by rearranging capital and eqity. Asset market clearing reqires e z (a; z; S) f 0 e (a; S) da+ z 0 (a; z; S) f (a; S) da = (1 + r (z 0 ; S 0 )) ^k 0 +p (z 0 ; S 0 )+d (z 0 ; S 0 ) for each z 0, together with the no-arbitrage conditions (3) and (4). (7) 2.7 Government The government provides (partial) nemployment insrance as it pays ot nemployment bene ts to the nemployed workers. This is nanced via a lmp-sm tax levied on all consmers. Assme (for now) that the pblic bdget has to balance each period, that is T = h (8) i.e., the tax revene eqals total pblic expenditres. 3 Solving the model This section explains how the model is solved nmerically. Frthermore, it brie y discsses, how the economy behaves in the benchmark of constant UI bene ts across the bsiness cycle. 3.1 Comptation In Appendix A it is shown that the resorce balance condition (goods-market eqilibrim condition) ^c + h^k0 (1 ) ^k i = zf ~k (1 ) v (9) 8 Note that the wage is reset every period, also for pre-existing matches. Ths, it is implicitly assmed that rms cannot commit to ftre wages. 9

11 is fl lled, where ^c is aggregate private consmption. That is, aggregate private consmption pls investments eqal aggregate otpt (net of aggregate vacancy costs), where we can de ne ^y zf ~k (1 ) v. Frthermore, for a complete de nition of the recrsive competitive eqilibrim see Krsell et al. (2010). Their appendix also contains a detailed description on how to solve the model nmerically, and therefore this section only smmarizes the method sed. Using the idea of Krsell & Smith (1998), consmers are assmed to have bonded rational perceptions of the evoltions of key economic variables, i.e., we apply the method of "approximate aggregation". 9 Hence, consmers perceive the next period aggregate capital stock, ^k 0, as a (log-linear) fnction of z; ^k;. The same is tre for this period s, p, d, and Q g (or Q b, see below). Krsell et al. (2010) show that these simple prediction rles are highly accrate with R 2 s above 0:999, and with very small forecasting and prediction errors. The nmerical soltion 10 of the model proceeds as follows (sing the z = g case for illstration): 1) Gess on the law of motion for aggregate capital, i.e., ^k 0 as a (log-linear) fnction of z; ^k;. 2) Gess on coe cients of the prediction rles for, p, d and Q g as (log-linear) fnctions of z; ^k;. 3) Calclate 0 from the law of motion in (1). 4) Calclate Q b sing the no-arbitrage condition in (3) and the rst three steps. 5) Perform the individal maximization and determine the wages from the Nash bargaining. 6) Simlate the economy for many periods 11 sing the reslts from the previos steps, and pdate the forecasting and prediction rles sing the data from the simlation. Iterate ntil the forecasting and prediction rles converge (gain s cient accracy). The reslting forecasting and prediction rles for the standard case are presented in Appendix B. 3.2 Calibration We apply the calibration of Krsell et al. (2010) who calibrate the model to t US data. A period is chosen to be six weeks. The prodction fnction is zf (k) = zk. The parameters = 0:3, = 0:01, and = 0:995 are chosen sing three calibration targets: a capital share of 0:3, an investment-otpt ratio of 0:2, and an annal rate of retrn on capital of 0:04. Also, the borrowing constraint is chosen as a = 0. The tility fnction is (c) = log (c). 9 Withot this assmption, consmers needed to know the law of motion for the entire distribtion of agents, which is an in nite-dimensional object. "Approximate aggregation" assmes that a nite set of moments is s cient for forecasting ftre economic variables. 10 We se 60 grid points in the a direction for the vale fnctions, 15 points in the a direction for the wage fnction, 4 points in both the ^k and the direction. We interpolate between grid points sing cbic splines in the a direction and linear interpolation in the other directions. 11 The economy is simlated for 2000 periods, and we disregard the rst 500 periods. 10

12 Following Cooley & Prescott (1995) the prodctivity levels are chosen to be g = 1:02 and b = 0:98 yielding an nconditional mean prodctivity of 1. Following Krsell & Smith (1999) and Krsell et al. (2009) the average dration of each boom (or recession) is set to two years, i.e., 16 periods in this model, which implies bb = gg = 0:9375 with 12 bg = 1 bb and gb = 1 gg. In the standard case, the matching parameters are calibrated following Shimer (2005). In the benchmark with constant nemployment bene ts across the bsiness cycle we set h = 0:99, which trns ot to be approximately 40% of the average wage. The separation rate is = 0:05. The matching fnction is M (; v) = v 1. Aiming for = 1 in eqilibrim pins down = 0:6. Frthermore, = 0:5315 is chosen sch that = 1 satis es the free-entry condition. Finally, = = 0:72, again following Shimer (2005). 3.3 Benchmark This section brie y discsses the benchmark case of invariant UI bene ts across the bsiness cycle. Table 1 smmarizes the means and ctations of the key economic variables across the bsiness cycle. Table 1: Smmary statistics for the benchmark of invariant UI bene ts z v ^k w h mean g +2.00% 0.63% +2.36% +2.99% +0.43% +2.01% 0.00% T RR p d r ^y ^c mean g 0.63% 2.04% +2.06% % +1.74% +2.16% +0.44% Note: mean denotes the nconditional mean, i.e., 0:5(x g+x b ), where x z is the average of x across periods with z = z: g is the percentage deviation of the average across good states from the nconditional mean. Ths, per de nition b = g, and only g is shown. w is the average wage; RR h T w T is the average replacement ratio; ^y is the aggregate otpt (net of vacancy costs); ^c is the aggregate consmption. From Table 1 it is seen that the benchmark economy behaves as expected in several aspects. Vacancy creation, and thereby also the v- ratio, is procyclical. Hence, the job nding rate is procyclical, which leads to a contercyclical nemployment rate. Ths, there is a clear negative relationship between aggregate prodctivity and nemployment, and the correlation is corr (z; ) = 0: The nconditional probability of being in a bad state is ths Pr (z = b) = 1 gg 2 bb gg = gb Hamilton (1994, p. 683), and similarly Pr (z = g) = bg gb+bg = 1 2. gb+bg = 1 2, cf. 11

13 De to the balanced bdget reqirement, the lmp-sm tax is contercyclical as UI expenditres are higher dring recessions where more workers are nemployed. The average wage is near-proportional to z, and wages are procyclical. This makes the average replacement ratio contercyclical, i.e., the income of an nemployed worker relative to the (average) income of an employed worker is higher in bad times than in good times. The dividend is highly procyclical becase pro ts are very volatile. Finally, aggregate otpt (net of vacancy costs) is procyclical, while aggregate consmption is only slightly procyclical, i.e., consmers are to a large extent able to smooth consmption ot over the bsiness cycle. Table 1 also reveals that the ctations in, v, and over the bsiness cycle are very small. This is a well-known reslt from e.g. Shimer (2005). As a robstness check, in Section 5.2 we se a di erent calibration strategy inspired by Hagedorn & Manovskii (2008), which delivers mch more reasonable ctations in these key bsiness cycle variables. Finally, the Gini coe cient 13 for wealth in the benchmark economy is 0:3153 on average across the bsiness cycle, and it is 0:03% higher in good states. 4 Bsiness cycle dependent nemployment bene ts This section analyzes the e ects of allowing nemployment bene ts to depend on the position of the bsiness cycle. Similar to Costain & Reiter (2005), we assme that the government is interested in the welfare conseqences of a UI scheme where the level of nemployment bene ts depends linearly on aggregate prodctivity, that is 14 h = h + z z z z (10) where z is the average aggregate prodctivity across the bsiness cycle; h is the bene t level when aggregate prodctivity eqals its average; z is the policy choice variable, and it determines the degree of bsiness cycle dependence. A positive (negative) z implies pro(conter)- cyclical bene ts, while z = 0 implies a bsiness cycle independent UI level (the benchmark). P n i=1 f(ai)( i+ i 1) 13 Nmerically, the Gini coe cient for wealth is calclated as G = 1, where i n P i j=1 f (a j) a j, 0 = 0, f (a) is the discrete probability density fnction of asset holdings, and n is the nmber of grid points in the asset distribtion, see e.g. X (2004). 14 This UI scheme does not allow nemployment bene ts to depend on the whole history of shocks bt only on the crrent period shock, which has a practical implementation appeal more than a prely theoretical appeal. Frthermore, note that choosing h as in section 3.2 implies that the average level of nemployment bene ts is na ected compared to Krsell et al. (2010). 12

14 4.1 Welfare measre Consider the welfare conseqences of changing z from the benchmark of invariant nemployment bene ts ( z = 0). In order to be able to calclate the expected welfare gain for each individal, we will make the experiment of moving an individal along with its asset level and employment stats from the benchmark economy to an economy with a di erent z. As is typical in this literatre the welfare conseqences of sch experiments can be fond following Lcas (1987). The welfare gain,, can be calclated from 15;16 " 1 # " X X 1 # E 0 t log ((1 + ) c t ) = E 0 t log c EXP t t=0 where c t is the individal s consmption nder the benchmark case (bsiness cycle invariant bene ts), and c EXP t is the individal s consmption nder the experiment (holding xed the individal s asset level and employment stats in the initial period). Ths, measres the consmption eqivalent, i.e., by how mch shold the individal be permanently compensated in terms of consmption if not moving to the "new economy". Hence, > 0 ( < 0) means that the individal gains (loses) from the experiment. One advantage of the present welfare measre is that one can calclate the expected welfare gain for each individal in the benchmark economy, i.e., it is possible to distingish between poor and rich workers. A potentially important drawback of this welfare measre is the discrepancy between actal aggregate asset holdings in the economy and the asset holdings aggregated over all the individals moved to the new economy (one by one). The same applies for actal aggregate employment verss employment aggregated over the moved individals. Therefore, as a robstness check, one can apply the standard tilitarian welfare measre. t=0 Hence, we calclate the mean welfare for di erent z s and compare this to the mean welfare for z = 0, where welfare in a given period is de ned as W (a; z; S) f e (a; S) da + U (a; z; S) f (a; S) da i.e., we se the actal distribtion of asset holdings and employment statses, and not the benchmark distribtion. When solving the model nmerically, it has always been the case that the two approaches provide eqivalent conclsion, i.e., all reslts regarding the optimality of 15 In practice, is fond by rearranging to = exp V EXP V (1 ) 1, where V EXP hx 1 i hx 1 i E 0 t=0 t log c EXP t and V E 0 t=0 t log (c t ). In the tables we present the welfare gain in percentage, i.e., This welfare measre is also sed by e.g. Krsell et al. (2010) and Mkoyama (2011) in models withot aggregate shocks, and by Costain & Reiter (2005) and Krsell et al. (2009) in models with aggregate shocks. 13

15 pro- and contercyclical UI bene ts, respectively, are sstained when applying this alternative, tilitarian welfare measre, and therefore only the former welfare measre will be presented below. 4.2 Welfare conseqences Table 2 shows the welfare conseqences 17 of moving the agents along with their employment statses and asset holdings from the benchmark economy with constant nemployment bene ts to an economy with bsiness cycle dependent nemployment bene ts and slope parameter z. The table shows that both procyclical and contercyclical UI bene ts can increase welfare relative to constant UI bene ts across the bsiness cycle. The mean welfare gain is largest in case of procyclical UI bene ts (with z = 1:76). The nemployed and the employed experience almost the same welfare gains on average. Actally, even the poorest nemployed who face a binding borrowing constraint prefer procyclical bene ts. Most of the employed gain in both good and bad periods, whereas the nemployed gain more in good periods than they lose in bad periods. On average, every consmer in the economy gain when moving from the benchmark of constant UI bene ts to procyclical UI generosity, which is somewhat srprising. On the other hand, if the cyclicality of bene ts is too strong, e.g. z = 10, both pro- and contercyclical UI bene ts are harmfl to the agents. Consmers facing a binding borrowing constraint will still gain in periods where bene ts (and ths consmption) are raised, bt they will lose mch more in periods where bene ts are lowered de to diminishing marginal tility. The maximm attainable mean welfare gain is 0:002% of consmption, which is small compared to the welfare gains fond in other stdies. Mitman & Rabinovich (2011) nd that the optimal UI scheme, which overall implies procyclicality of both bene t level and dration, yields a mean welfare gain of 0:67% of consmption compared to the crrent US system. Ek (2012) considers both di erentiated taxes and bene t levels, and she nds that taxes shold be procyclical whereas bene ts shold be contercyclical, which yields a mean welfare gain of 0:01% of consmption compared to the optimal niform system. However, both these stdies ignore savings, and therefore they are likely to overrate the welfare gains from bsiness cycle dependent nemployment bene ts since self-insrance is not possible. In a model allowing for savings, Costain & Reiter (2005) calclates the welfare costs of bsiness cycles to be 0:269% of consmption by comparing their static model, i.e., withot 17 We present the average across good periods, g, and bad periods, b, along with the nconditional mean 0:5 g + b. In contrast, Krsell et al. (2009) choose to pick a random good period and a random bad period. However, the welfare gains are almost constant conditional on aggregate prodctivity, and therefore the two methods are (almost) eqivalent. 14

16 aggregate ctations, to their dynamic benchmark. They nd that the optimal policy with strongly procyclical taxes and slightly procyclical bene ts eliminates arond 70% of the welfare costs of bsiness cycles, i.e., it implies a mean welfare gain of 0:185% of consmption compared to the dynamic benchmark. In contrast to this paper, Costain & Reiter (2005) do not consider aggregate assets, e.g. physical capital and eqity, and ths, implicitly they do not allow for insrance against aggregate shocks. However, they are able to calclate the welfare costs of bsiness cycles in the case where "aggregate insrance" is attained. Using this, the welfare gain from the optimal policy described above is 0:049% of consmption relative to the case with "aggregate insrance". This seems to be the relevant nmber for comparison with this paper, and their nmber is mch larger for two reasons: i) the calibration strategy, cf. Section 5.2, and ii) Costain & Reiter (2005) consider two policy variables, the cyclicality of bene ts and the cyclicality of the pblic de cit, against only one, the cyclicality of bene ts, in this paper where the pblic bdget (for now) is reqired to balance each period. The welfare gains sggested by both Costain & Reiter (2005) and Ek (2012) primarily stem from allowing taxes to vary over the bsiness cycle, since they nd that di erentiated taxes over the bsiness cycle reslt in mch larger welfare gains than di erentiated UI bene ts. As this paper focses only on the latter policy variable, one wold therefore expect mch smaller welfare gains from the optimal policy. Table 2: Welfare conseqences of di erent degrees of cyclicality in UI bene ts z z Welfare gains (in %) Fraction gaining (in %) overall nempl empl C ec overall nempl empl 10:00 g :00 b :00 mean :00 g :00 b :00 mean :00 g :00 b :00 mean :00 g :00 b :00 mean :76 g :76 b :76 mean :00 g :00 b :00 mean :00 g :00 b :00 mean Note: The table shows the welfare gains from moving the agents from an economy with constant UI bene ts to an economy with cyclically dependent bene ts and slope parameter z. C and ec denote the nemployed and employed, respectively, with a binding borrowing constraint. g is a good state, b is a bad state, mean denotes the nconditional mean, i.e., 0:5( g + b ). : Ronded to %. z = 1:76 is the level that maximizes the mean welfare gain. 15

17 4.3 E ects on key economic variables Table 3 shows how the key economic variables are a ected by changing the cyclicality in UI bene ts, z. The table reveals some of the conteracting e ects working in favor of procyclical and contercyclical UI bene ts, respectively. From Table 2 we already know that the relative importance of these conteracting e ects is non-monotone in z. Table 3: Averages of key economic variables z z (%) v ^k w h T RR p d ^y ^c 1:00 g :00 b :00 mean :00 g :00 b :00 mean :76 g :76 b :76 mean :00 g :00 b :00 mean Note: The table shows the averages across good and bad periods, respectively. mean denotes the nconditional mean, i.e., 0:5(x g+x b ), where x z is the average of x across periods with z = z: w is the average wage; RR h T is the average replacement ratio. The aggregate otpt is ^y zf ~k (1 ) v. z = 1:76 is the level that maximizes the mean welfare w T gain. Compared to the benchmark ( z = 0) procyclical nemployment bene ts lead to stabilization of most of the economic variables. Ths, nemployment, vacancies, and the v- ratio are all stabilized with procyclical UI generosity. When bene ts are raised in booms, the wage will increase, and therefore rms will post fewer vacancies. This leads to a decrease in the v- ratio, which lowers the job nding probability, and therefore the nemployment level will increase. Similarly, vacancy creation will be higher in recessions where bene ts are lowered, which makes job nding easier and nemployment decreases. Frthermore, procyclical bene ts stabilize the price of eqity, pro ts, and aggregate otpt (de ned as the right-hand side of (9)), whereas wages are destabilized. The aggregate otpt is stabilized becase the nmber of prodcers (= the employment level) is stabilized, and becase the aggregate vacancy costs are stabilized. In the benchmark, the average replacement ratio is contercyclical de to procyclical wages. However, if the procyclicality of UI bene ts is s ciently strong, the average replacement ratio will be procyclical, which is the case for the optimal degree of procyclicality ( z = 1:76), in which case UI bene ts are 3:56% higher in good times than on average across the bsiness cycle, and the (average) replacement ratio is 1:50% higher. For contercyclical UI bene ts the exact opposite happens. Most variables are destabilized, e.g. nemployment, bt wages are stabilized since they increase dring recessions de to a better otside option of the worker, and they decrease dring booms where the otside option of the worker is worsened. Hence, agents need to save less for bad times and more for good 16

18 times. For most agents the former dominates, and therefore consmption increases on average. Frthermore, this leads to a stabilization of aggregate consmption. Table 3 also shows that mean nemployment, which can be thoght of as the strctral nemployment level, is lowered when moving from the benchmark economy ( z = 0) to an economy with (optimal) procyclical bene ts ( z = 1:76), becase nemployment drops more dring recessions than it increases dring booms. Also, strctral nemployment is higher with contercyclical bene ts, becase nemployment increases more dring recessions than it decreases dring booms. Hence, the distortionary e ects of UI (on job creation) is contercyclical, at least for small absolte vales of z, and therefore procyclical UI bene ts can be welfare improving. This interpretation is in line with Andersen & Svarer (2011b) who nd that bene ts shold be lowest in the state with most distortions. On the other hand, if the procyclicality of UI bene ts is too strong (e.g. z = 5), strctral nemployment increases compared to the benchmark, and in fact, the nemployment rate will be higher dring booms than dring recessions. To sm p, procyclical UI generosity can be bene cial de to the contercyclical natre of UI distortions, whereas contercyclical UI generosity can be bene cial as it facilitates consmption smoothing and raises mean consmption. Table 1 showed that most consmers are able to smooth consmption fairly well, even withot contercyclical UI, and therefore procyclical UI generosity dominates contercyclical UI. Finally, Figre 1 shows a sample path for the nemployment rate in the benchmark case of constant UI and in case of (optimal) procyclical UI. It con rms that nemployment is stabilized in case of procyclical UI since nemployment increases (relative to the benchmark) in good times, where nemployment is low, and decreases in bad times, where nemployment is high. 5 Robstness In this section varios robstness checks are carried ot. 5.1 Alternative pblic bdget reqirement In the analysis above the pblic bdget was not allowed to act as a b er. However, an important argment in favor of UI is that it works as an atomatic stabilizer. Therefore, this section allows the pblic bdget to balance only on average To be more precise, the tax now solves T = 0:5 h g + h b, where hz is the average of h in periods with z = z, and 0:5 is the nconditional probability of each state. Ths, the tax is now constant over time. 17

19 Figre 1: Sample path for nemployment, constant UI verss procyclical UI time Benchmark Pro cyclical UI Note: The gre shows sample paths for the nemployment rate,, in the benchmark case of constant UI across the bsiness cycle (dotted line) and in the case of procyclical UI (fll line; z = 1:76). The reslts are presented in Tables 4 and 5 in Appendix C.1. They show that allowing the pblic bdget to balance only on average does not change the qalitative reslts, i.e., both procyclical and contercyclical UI bene ts can increase welfare, and procyclical nemployment bene ts ( z = 1:75) still yield the largest welfare gains. However, the maximm attainable mean welfare gain across individals is almost 50% lower than when the pblic bdget always balances. The reason is that nder a balance bdget reqirement the tax increase (decrease) conteracts the increase (decrease) in nemployment bene ts and, ths, wages. With procyclical UI bene ts, consmers no longer gain in bad periods from a low tax or s er in good periods from a higher tax the former trns ot to be dominant. For the same reasons the welfare gains from contercyclical bene ts are larger when the pblic bdget only balances over time. Hence, as is well-known in the literatre, reqiring the pblic bdget to balance each period works in favor of procyclical UI generosity, see e.g. Andersen & Svarer (2011b). The optimal procyclical UI scheme implies that the pblic bdget de cit is also procyclical (0:0025 on average across good states and 0:0025 across bad states), i.e., UI expenditres are higher in good states than in bad states, since the increase in the bene t level dominates the decrease in the nmber of nemployed workers when moving from a bad to a good state. 18

20 5.1.1 Unemployment dependent UI bene ts In practice, it seems easier to implement a scheme where UI bene ts are conditioned on the nemployment level. Ths, the level of nemployment bene ts is determined from h = h + where is the mean nemployment level. Note that the interpretation of is now a bit more complicated 19 since changing will also change. As opposed to the interpretation of z, a positive (negative) implies conter(pro)-cyclical bene ts. The reslts are presented in Tables 6 and 7 (bdget balance) and Tables 8 and 9 (bdget balances only on average) in Appendix C.2. contercyclical bene ts can increase welfare. Again, we nd that both procyclical and In the case of bdget balance, procyclical bene ts are optimal, as the welfare gain is maximized for = 5:00 with a mean welfare gain of 0:0101% of consmption. However, even thogh UI bene ts are 1:79% higher across good times than on average across the bsiness cycle, the (average) replacement ratio is 0:21% lower. Hence, the procyclicality of the UI bene ts is not strong enogh to revert the contercyclicality of the (average) replacement ratio de to procyclical wages. Frthermore, when the pblic bdget is allowed to work as a b er, the welfare gain is maximized in case of contercyclical bene ts ( = 1:20) with a mean welfare gain of 0:0085% of consmption. This UI scheme implies that the pblic bdget de cit is contercyclical ( 0:0013 on average across good states and 0:0013 across bad states), UI bene ts are 0:94% lower across good times than on average across the bsiness cycle and the (average) replacement ratio is 3:09% lower. Ths, conditioning UI on nemployment instead of prodctivity actally alters the choice between pro- and contercyclical UI. Overall we nd that nemployment dependent UI yields mch higher welfare gains than UI conditioned on prodctivity. This is not srprising since the latter (in or model) restricts UI bene ts to jmp between two levels only, whereas the former allows UI bene ts to di er between, say, mild and deep recessions. However, combined with the fact that it is easier, in practice, to condition UI on the nemployment rate, it is somewhat srprising that the approach of conditioning UI directly on nemployment, to the athor s knowledge, is new to literatre. 19 To solve this, cold be replaced by the nemployment level in steady state. However, this speci cation wold imply that in general the average h is no longer h, and therefore it wold be a di erent kind of experiment. 19

21 5.1.2 UI bene ts depending on lagged prodctivity In reality, statistics on the position of the bsiness cycle always become available with a certain lag. Therefore, this section stdies the conseqences of conditioning UI bene ts on lagged prodctivity, instead of crrent prodctivity, i.e., h = h + z 1 z 1 z 1 z 1 where z 1 is previos period s prodctivity, and z 1 is mean lagged prodctivity. 20 the model we need to inclde z 1 as an additional state variable. To solve The reslts are presented in Tables 10 and 11 (bdget balance) and Tables 12 and 13 (bdget balances only on average) in Appendix C.3. Not srprisingly, both procyclical and contercyclical bene ts can increase welfare. With bdget balance, procyclical UI bene ts are optimal, as the welfare gain is maximized for z 1 = 1:25 with a mean welfare gain of 0:0012% of consmption. Here, bene ts are 2:21% higher across good times than on average across the bsiness cycle, bt the (average) replacement ratio is only 0:17% higher. In this case, there is no negative vale of z 1 delivering a positive welfare gain compared to invariant bene ts. Again we nd that when the pblic bdget is allowed to work as a b er, the welfare gain is maximized in case of contercyclical bene ts ( z 1 = 4:00) with a mean welfare gain of 0:0014% of consmption. contercyclical ( This UI scheme implies that the pblic bdget de cit is 0:0066 on average across good states and 0:0066 across bad states), UI bene ts are 7:06% lower across good times than on average across the bsiness cycle and the (average) replacement ratio is 9:57% lower. Hence, conditioning UI bene ts on lagged prodctivity reslts in the same qalitative conclsions as nemployment dependent UI. The reason is that the distribtion of asset holdings and employment statses is determined in the previos period, i.e., it depends on prodctivity in the previos period, cf. (1) and (2), and therefore lagged prodctivity matters more for the stats of the economic agents than crrent prodctivity. Recall that in the benchmark the correlation between crrent prodctivity and nemployment is corr (z; ) = 0:82, whereas the correlation between lagged prodctivity and nemployment is corr (z 1 ; ) = 0: The mean lagged prodctivity is calclated as z 1 = 0:5 z g zb where z g 1 is the average of lagged prodctivity across good periods (high crrent level prodctivity), and similar for z b 1 across bad periods. Again, this ensres that mean nemployment bene ts are still h. 20

22 5.2 Alternative calibration Like Krsell et al. (2010) we also consider a di erent calibration strategy inspired by Hagedorn & Manovskii (2008), who showed that the Diamond-Mortensen-Pissarides model is consistent with key bsiness cycle facts, in particlar it matches the empirically observed volatility of nemployment, vacancies, and the ratio between the two at bsiness cycle freqencies, if sing an alternative calibration strategy than Shimer (2005). In particlar, the vale of being nemployed is now mch closer to the vale of being employed since h will be mch larger. Frthermore, the bargaining power will be mch smaller, e ectively implying that wages are less responsive to prodctivity shocks. The vacancy cost is set to = 2:165 which is approximately 60% of the average labor prodctivity in the model. Following Hagedorn & Manovskii (2008) the bargaining power is set v to = 0:052. The matching fnction is M (; v) =. With = 0:7 and a job- nding ( l +v l ) 1=l rate of 0:592 as calibration targets we get l = 2:2. Finally, h = 2:29 implies that = 0:7 satis es the free-entry condition. Therefore, the replacement ratio trns ot to be very high, approximately 95%, and the interpretation of the vale of non-market activities h mst now cover mch more than jst UI, e.g. home prodction and self employment. 21 The smmary statistics for the benchmark model is fond in Table 14 in Appendix C.4. It shows that nemployment volatility is now mch higher, whereas wages are fairly rigid. The reslts are presented in Tables 15 and 16 (bdget balance) and Tables 17 and 18 (bdget balances only on average) in Appendix C.4. They show that sing an alternative calibration strategy does not change the overall conclsion: nemployment bene ts shold be procyclical when conditioning on prodctivity. Bt the welfare gains are mch larger since nemployment is mch more volatile with the Hagedorn-Manovskii calibration, whereas the wages are fairly rigid. In fact, the welfare gains are almost two orders of magnitde larger than with the standard calibration. Again we nd that reqiring the pblic bdget to balance each period works in favor of procyclical UI since the welfare gains from procyclical UI are higher in Table 15 than in Table 17. Frthermore, contercyclical UI bene ts are no longer welfare improving. The reason is that wages are mch less volatile compared to the standard calibration, and therefore there are only small gains from stabilizing wages (and ths consmption). These gains are dominated by the very strong contercyclicality of UI distortions. 21 In particlar, h = h UI +h non-ui, where only h UI is tax nanced via (8). We let h non-ui = 1:30 (= 2:29 0:99), and h UI is determined from (10) with h UI = 0:99. Alternatively (bt less realistic), it cold be assmed that all income from non-market mst be nanced via taxes. It trns ot that this experiment delivers exactly the same qalitative reslts. 21