Documents de Travail du Centre d Economie de la Sorbonne

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1 Documents de Travail du Centre d Economie de la Sorbonne The three orlds of elfare caitalism revisited Sarah BROCKHOFF, Stéhane ROSSIGNOL, Emmanuelle TAUGOURDEAU Maison des Sciences Économiques, boulevard de L'Hôital, Paris Cedex 13 htt://centredeconomiesorbonne.univ-aris1.fr/bandeau-haut/documents-de-travail/ ISSN : X

2 The three orlds of elfare caitalism revisited Sarah Brockhoff Stéhane Rossignol Emmanuelle Taugourdeau March 7, 2012 Abstract We introduce a ne ay to model the Bismarckian social insurance system, stressing its cororatist dimension. Comaring the Beveridgean, Bismarckian and Liberal systems according to the majority voting rule, e sho that for a given distribution of risks inside society, the Liberal system ins if the inequality of income is lo, and the Beveridgean system ins if the inequality of income is high. Using a utilitarian criterion, the Beveridgean system alays dominates and the Bismarckian system is referred to the Liberal one. JEL Classification: D63; D72; H53 Keyords: social insurance systems olitical economy Bismarck Beveridge inequality redistribution Bielefeld University, Bielefeld Graduate School of Economics and Management, and University of Freiburg, Institute for Public Finance II, Bertoldstraße 17, Freiburg, Germany. sarah.brockhoff@econ.uni-freiburg.de. Financial suort from the German Academic Exchange Service is gratefully acknoledged. LED, University Paris 8 Saint-Denis, 2 rue de la Liberté, Saint-Denis Cedex, France. strossignol@gmail.com CNRS, Centre d Economie de la Sorbonne (CES, IDEP, ENS Cachan, 61 avenue du résident Wilson, Cachan Cedex, France. emmanuelle.taugourdeau@enscachan.fr. 1

3 1 Introduction This aer comares the three main systems of elfare caitalism Beveridgean, Bismarckian, Liberal as analyzed by Esing-Andersen (1990 from both a ositive and a normative ersective. To do this, e introduce a ne ay to model a Bismarckian tye of social insurance to account for the fact that Bismarckian systems are organized around grous of agents. We aim to focus on the redistributive design of these different regimes and comare the referred systems from both ersectives. The background for our considerations is the folloing: In many countries ith a Bismarckian system, such as Germany, Austria, France or Belgium, a variety of social rotection funds for illness, occuational injury, family or ension cover secific grous of eole. For instance, the set of French social insurance funds refers to rofessional grous such as railay and ublic transortation system emloyees, seamen, civil servants, agricultural orkers, entrereneurs, etc. For occuational injury, the German insurance system is similarly organized on a rofessional grou basis: Secific emloyer s mutual insurance associations cover the commercial, agricultural or the ublic sector as ell as railay orkers, firefighters and local authority emloyees etc. 1 There are other examles in Bismarckian countries here the formation of grous are a result of the agents choice. For instance, in Belgium or Germany, eole can choose from a (large range of health insurance funds. These funds are organized on the level of geograhic coverage, emloyers, craft guilds, etc. 2 The recognition of this organizational and strongly cororatist feature of the Bismarckian system goes back to the seminal ork of Esing-Andersen (1990: cororatism as tyically built around occuational grous seeking to uhold [...] status distinctions and used these as the organizational nexus for society and economy. (. 60. To be recise, Esing-Andersen (1990 clustered elfare states as conservative, social-democratic and liberal regime tyes. In line ith the established economic literature e retain for the first to systems the nomenclature of Bismarckian and Beveridgean systems. 1 German Social Security La, Book Nr. VII 2 In Germany, before the amendment to the Social Security La in 1996, eole had to insure themselves according to the selection criterion of the health insurance funds. Therefore, these funds covered only eole ho exactly matched their selection criterion, e.g. they lived in a secific geograhic region, they orked for a secific emloyer or in certain craft guilds etc. Noadays, eole can choose hich fund they ant to be insured in, cf. German Social Security La, Book Nr. V. Further source:.roseur.org 2

4 As ell as other dimensions, one imortant asect that distinguishes these systems is their degree of income redistribution. Firstly, Liberal systems are associated ith a very lo degree of income redistribution since they mainly encourage rivate insurance. Secondly, Beveridgean systems, based on the rinciles of universality, uniqueness and uniformity of benefits, are associated ith a high degree of income redistribution. This is due to roortional tax rates but flat benefits. Finally, Bismarckian systems are associated ith a loer degree of income redistribution. Most often, they have been modeled in the literature as a global insurance system, organized by the state, here individuals ay taxes roortional to their incomes, but indeendent of their risks, and receive benefits roortional to their income. The roblem ith this ay of modeling the Bismarckian system is that it ignores the cororatist attribute of such systems: If individuals are differentiated by income and by risk, then a ooling of individuals ith secific risks inside each fund leads to intra-grou horizontal redistribution in the Bismarckian system, i.e. it leads to redistribution from lo-risk to high-risk agents inside each fund. 3 As a consequence, each fund is characterized by its secific average risk. Transferred to the level of individual references, this imlies that individuals ho bear a high risk may benefit from a lo average grou risk. In a similar vein, the introduction of both income and risk heterogeneity of individuals leads to another kind of horizontal redistribution inside the Beveridgean systems: Here, redistribution is based on individual risk and on the distribution of risks inside the entire society. In our terminology, this tye is called global horizontal redistribution and it comlements the usual vertical income redistribution of the Beveridgean system. Again, transferred to the level of individual reference, oor and/or high-risk individuals benefit from the Beveridgean system. The Liberal system is characterized by neither horizontal nor vertical redistribution, since it consists of a rivate insurance mechanism, ith a contribution rate that is roortional to individual risk and income. In the folloing, e rovide a model hich accounts for all of these redistribution atterns by analyzing individual and aggregate references for the three systems. There are to strands of literature hich are related to our model. The first strand has determined both the tye and the size of social insurance or social security systems, resectively, and therefore refers to the exlicit 3 In our model, individuals can be thought of being differentiated ith resect to risk along a horizontal axis and ith resect to income along a vertical axis. The notion of horizontal refers to the redistribution of risk (i.e. from lo-risk to high-risk eole ith to asects: ithin the entire society, or ithin grous. Accordingly, vertical refers to redistribution of income (i.e. from rich to oor eole. See also Section

5 distinction beteen Bismarckian and Beveridgean systems, see Casamatta et al. (2000b (for social insurance and Pestieau (1999 (for social security. They analyze the otimal size of the system (in terms of the tax rate ith the tye of system chosen at the constitutional stage. Their main result is that the degree of redistribution affects the olitical suort for the size of the system. 4 Rossignol and Taugourdeau (2004 study both the level of tax rate and the tye of system ithin a robabilistic model of electoral cometition. They roved that the chosen social insurance system is that hich minimizes the contribution rate for a high relative risk aversion, and that the reverse is true for a lo relative risk aversion. Moreover, Conde- Ruiz and Profeta (2007 rovide an OLG model of social security here the size and the tye of system is determined simultaneously, yet issue-by-issue. They find that the key determinant hich shaes their analytical result is income inequality: The Beveridgean system can be suorted by a coalition of lo and high income individuals. We comlement this first strand of literature in to ays. Firstly, the Bismarckian system is modeled as a cororatist one, hich enables us to distinguish it more clearly from the Liberal system. Secondly, the choice of the system is determined alternatively according to a ositive and a normative criterion, that e are able to comare. The second strand of literature this aer refers to analyzes the link beteen income inequality and the level of redistribution inside society. Indeed, in our model, the degree of inequality of income and the distribution of risk crucially affect the choice of an agent, hich affects the choice of the system for both ositive and normative criteria. The link beteen income inequality and redistribution has first been highlighted in the ell-knon Meltzer and Richard (1981 general equilibrium model of a labor economy here the share of redistributed income is determined by majority voting. 5 Their main finding is that if mean income rises relative to the income of the median voter, then redistribution increases. In other ords, a more unequal income distribution leads to more redistribution. In addition to the standard redistributive mechanism from rich to oor, insurance motives have also been introduced in the analysis of elfare olicies. For instance, Moene and Wallerstein (2001 sho that the redistributive and the insurance mechanisms ork in oosite directions in the sense that suort for social 4 See also Cremer et al. (2007 for the effect of myoic and non-myoic individuals on social security. 5 In addition, see Romer (1975 and Roberts (1977 on hose results Meltzer and Richard (1981 build uon. 4

6 insurance sending declines ith increased income inequality. 6 Finally, Kim (2007 extends the analysis of redistribution based on insurance motives by introducing a distribution of risks inside the society, here the level of risk deends on the agent s sector of activity. The main result of this model is that olitical demand for unemloyment insurance is clearly influenced by both the distribution of risks and income. As already indicated, our model rovides a comlete differentiation of individuals along three dimensions: income, individual risk and grou risk. This is a key oint of our analysis. In the revious literature, Casamatta et al. (2000b introduce heterogeneity of individuals by a one dimensional differentiation ith three discrete levels of income but the same robability of receiving income or relying on social benefits. Casamatta et al. (2000a and Conde-Ruiz and Profeta (2007 differentiate along to dimensions, namely age (orking young vs. retired old and the level of income (continuous in Casamatta et al., 2000a, discrete in Conde-Ruiz and Profeta, A related double differentiation of individuals ith regard to income and likelihood of illness is found in Gouveia (1997 ho analyzes the outcome of majority voting over the ublic rovision of a rivate good (in articular, health care. We concentrate on the case of insurance systems that cover unemloyment, occuational injury or health risks. Individuals earn a age income in the good state of the orld and receive insurance benefits in the bad state of the orld. Furthermore, they are members of a grou hich is characterized by a grou-secific risk distribution. This imlies that grous can be ranked according to the average risk of its members. We incororate into our analysis a Liberal insurance system reflecting an actuarial fair rivate insurance, a Beveridgean system involving redistribution for the entire society and a Bismarckian system comrising redistribution beteen high-risk and lorisk individuals ithin a grou. In a to stage model, first, the system of insurance is decided and second, the level of the tax rate is determined. The choice of the tax rate and the choice of the system are determined according to a ositive criterion, then comared to a normative one. In the folloing e sho that by majority voting, the Liberal system ins if the inequality of income is lo and the Beveridgean system ins 6 Moene and Wallerstein (2001 focus on the imact of income inequality on the suort of elfare sending hen elfare benefits are targeted toards the emloyed or the unemloyed. See also Iversen and Soskice (2001 for a similar model analyzing social olicy references hich deend on different tyes of skill investments reflecting unemloyment risks. Bénabou (2000 analyzes the imact of inequality and redistributive olicies that enhance efficiency ithin a stochastic groth model. 5

7 if the inequality of income is high. Emloying a utilitarian criterion, the Beveridgean system dominates both the Bismarckian and Liberal systems but the Bismarckian system is referred to the Liberal one. This aer is organized as follos: Section 2 introduces the model. In Section 3 e analyze the airise references of individuals and determine the tye of elfare system chosen by majority voting. In Section 4 e analyze the outcome of a utilitarian social lanner and comare the results of both criteria. We conclude in Section 5. 2 The model The society is divided into grous hich are denoted by k = 1,..., M and there are N k members er Grou k. 7 There are N agents in the society ith N = M k=1 N k. An agent i of Grou k has an income i and a risk i to lose this income. A high level of i imlies that agent i is risky in terms of bad health or unemloyment, for instance. Each grou k is characterized by a secific distribution function of risk f k. To concentrate our analysis on the heterogeneity of the distribution of risk, e suose that the distribution function of income g is similar in each grou. Moreover, for the sake of the readability of our results, e assume that the distribution of incomes and risks are indeendent. Therefore, grous are heterogeneous ith resect to risks but homogeneous ith resect to income distribution. We no describe the distribution of income and risk in more detail. 2.1 Distributions of income and risk The distribution of income for each grou is reresented by the robability density function g defined on [ inf ; su ] ith average income = g ( d. The function g is ositively skeed such that median income m is loer than average income. Income levels can then be ranked as 0 inf m su. The distribution of risk deends on the grou k and imlies a grousecific risk robability density function f k defined on [0; 1]. This function f k is ositively skeed, as ell, and roduces a articular intra-grou average risk k, here k = 1 N k fk (d and N k = f k (d ith f k 0. Let f be the risk robability density function of the entire society, i.e. f = M k=1 f k. 7 These grous could be rofessional grous (e.g. the service sector, the agricultural sector, the industrial sector etc. or other tyes of grous. 6

8 We normalize N = 1 = f. The average risk in the entire society is = f(d. We assume that the intra-grou average risks are ranked as 1 < 2 < 3 <... < M (1 In addition, e ostulate that m,k = m for every k, i.e. the median risk of each grou m,k corresonds to the median risk in society m, even if the distribution of risk inside each grou is different. Ho can e justify these to assumtions? It is clear that there is a majority of lo-risk eole in each grou. It is reasonable to assume that the grous are mainly differentiated by the distribution of their highrisk members. This imlies that the grous have different average risks k (i.e. 1 <... < M, but aroximatively similar median risks m,k (i.e. m,k = m for every k. Finally, based on the ositive skeness of function f k, e no ostulate k, m < k hich imlies m <. In the folloing, e ill resent emirical justification for the relationshi beteen median risk and average risk. 2.2 Emirical evidence It is a ell-knon stylized fact that income distributions in many develoed countries exhibit ositive skeness, see, e.g. Neal and Rosen ( To establish the ositive skeness of the risk distribution e can refer to the same line of argument as before: We ant to sho that there is a majority of lo-risk and a minority of high-risk members in each grou. In our model, risk refers to the robability of having to rely on (social insurance benefits due to unemloyment or illness. We rovide for each of these risk factors an emirically observable roxy. For unemloyment e comare median and average duration of unemloyment using data from OECD countries for the years 2000 to We find that the roortion of countries here median duration of unemloyment is clearly smaller than average duration is substantial for the hole 8 Early contributions to the literature analyzing functional forms of earnings caacities are Staele (1942, Miller (1955, or Harrison ( For reasons of comarability across all OECD countries e chose unemloyment rates of male ork force. 7

9 time eriod, see Figure 1. Overall, less than 5% of total observations exhibit a reverse relationshi ith average risk loer than median risk % 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% median duration < average duration interval median = interval average median duration > average duration Figure 1: Median vs. average duration of unemloyment in OECD countries, roortions of countries relative to all OECD countries, , male ork force. Source: OECD Labour Market Statistics, on calculations. For illness our basic hyothesis is that eole affected by chronic health roblems or disability bear a higher risk of having to rely on insurance ayments. Since these eole constitute a minority in society, average risk ill be loer than median risk. Indeed, data from OECD (2010b shos that the self-assessed revalence of chronic health roblems or disability is loer than 15% on OECD average for the hole orking age oulation. Even for age grou 50 64, the roortion of eole ith self-assessed chronic health roblems or disability is loer than 25% on average and only for fe countries a little higher than 30%. Given a minority of eole bearing a high risk due to chronic health roblems and disabilities, the majority of eole has quite a lo risk. Moreover, if e consider health exenditures as a roxy of the health risk, then it clearly aears that mean health exenditures are consistently higher than median health exenditures (Jung and Tran, Data from OECD (2010a. Estimation of median and average duration of unemloyment and calculations of average duration of unemloyment are available from the authors uon request. 8

10 2.3 The three systems The agent i earns ith robability (1 i an income i hich is subject to a ayroll tax t, such that (1 t i is his net of tax income. With robability i the agent receives social insurance benefits b i hich, in the case of a Beveridgean system (BE, are identical for all agents b i = b BE. In the case of the Bismarckian system (BI, social insurance benefits b i are roortional to individual income but the coefficient of roortionality is identical for all agents inside the grou k, i.e. b i = b BI k ( i = c k i. Finally, in the case of a Liberal system (L, benefits that an agent receives in the bad state of the orld are actuarially comuted, based on both his risk i and the age i that he ould receive in the good state of the orld, i.e. b i = b L ( i, i. No redistribution occurs in this last system. Hence, under the Liberal system, the budget constraint for each agent i is given by (1 i t i = i b L ( i, i hich immediately imlies b L ( i, i = 1 i t i i Under the Bismarckian system, the budget constraint in Grou k is 1 N k ((1 t fk (g ( dd = 1 N k b BI k ( f k (g ( dd and since b BI k ( i = c k i it imlies 1 N k ((1 t fk (g ( dd = 1 N ck k f k (g ( dd thus c k = 1 k t, and finally k b BI k ( i = 1 k t i k Lastly, under the Beveridgean system, the social insurance budget constraint satisfies the identity ((1 tf(g ( dd = b BE f(g ( dd, hich imlies b BE = 1 t ith = f(d. The elfare function under the Beveridgean system, for an individual i of risk i if the tax rate is t, is no: ( 1 W BE (t, i, i = (1 i U((1 t i + i U t Analogously, the grou-secific elfare function for the Bismarckian system for a member i of Grou k, is ( 1 Wk BI (t, k i, i = (1 i U((1 t i + i U t i k 9

11 and the elfare function of an agent i under the Liberal system is: ( 1 W L i (t, i, i = (1 i U((1 t i + i U t i We aim to determine the referred system according to to alternative criteria, i.e. a ositive one, majority voting and a normative one, utilitarian criterion. In both cases, the timing of decisions is as follos: In the first stage, the elfare system is chosen. In the second stage, the level of the tax rate is chosen, according to the studied criterion. We ill solve these games by backard induction. For the sake of simlicity e secify the utility function to be U(x = ln x. 3 Majority voting 3.1 Choice of tax rate Maximizing the level of the elfare of a given agent i ith resect to the tax rate t i yields the same referred tax rate under the three systems: t i = i (2 The referred tax rate does not deend on income. Moreover, since agents are differentiated by their risk i, their references are single eaked ith resect to the tax rate. As a result, according to the majority rule, the tax rates that are chosen in both the Beveridgean and Bismarckian systems are those referred by the median voter, i.e.: t BE = t m = m t BI = t m,k = m,k Since all grous have aroximately similar median risks m,k (i.e. m,k = m for any k, the tax rate chosen by majority voting corresonds to the choice of the society s median agent and is the same in both the Beveridgean and Bismarckian systems i t BE = t BI = t m = m In the Liberal system the choice of tax rate is made indeendently by each agent and corresonds to his ersonal level of risk 11 t L = t i = i 11 For the sake of simlicity e refer to the term tax rate also ith regard to the Liberal system. Contribution rate ould be a more recise term. 10

12 Incororating the chosen tax rates in the elfare functions gives: ( 1 W BE (t m, i, i = (1 i ln((1 m i + i ln m ( 1 Wk BI k (t m, i, i = (1 i ln((1 m i + i ln m i ( k 1 W L (t i i, i, i = (1 i ln((1 i i + i ln i i i (3 (4 = ln ((1 i i (5 3.2 Individual references on the system Before determining the system that ould be chosen by majority voting, e need to study individual references for the systems using the tax rates e have just determined. We focus on a airise comarison of the three systems to have a comlete ranking of the systems for each agent Bismarck or Beveridge? We start by comaring the Beveridgean and Bismarckian systems ith the tax rates obtained by majority voting. Proosition 1. Agent i of Grou k refers a Beveridgean system to a Bismarckian one iff i < r k, here r k = 1 k 1 is an increasing function of k k, and thus of k. This agent refers the Bismarckian system iff i > r k. Proof. From Equations (3 and (4 hich is equivalent to W BE (t m, i, i > Wk BI (t m, i, i ( ( 1 1 i ln k m > i ln m i 1 > 1 k i k i.e. equivalent to i < r k, here r k is clearly an increasing function of k, and k is an increasing function of k according to Inequality (1. k 11

13 Note that the coefficient r k is a measure of the average risk k in Grou k, relative to the average risk of society,. If the average risk in Grou k coincides ith the society s average risk, then r k is equal to 1. If the average risk in Grou k is loer (higher than the society s average risk, then r k is strictly smaller (larger than 1. Since r k is an increasing function of k, e can rite: r 1 < r 2 <... < r j < 1 < r j+1 <... < r M With the usual ay of modeling the Bismarckian system, there is only one grou, i.e. M = 1. In this case, 1 = so that r 1 = 1. It immediately imlies that an agent of income i refers a Beveridgean system if i < and a Bismarckian one if i >. It is a articular case of our Proosition 1. With the more realistic ay of modeling the Bismarckian system that e adot here, the Bismarckian system is articularly interesting for agents ho belong to lo-risk grous, i.e. to Grou k ith k lo. Agents ho bear a high risk benefit from a grou ith a lo mean risk because of the intra-grou horizontal redistribution. Both the Beveridgean and the Bismarckian systems imly horizontal redistribution (i.e. from lo-risk to high-risk agents, but the only system ith vertical redistribution (i.e. from rich to oor agents is the Beveridgean one. Thus, oor agents refer Beveridge to Bismarck. An individual i refers a Beveridgean system if his income i is such that i < r k, as shon in Figure 2. In each grou, there may be a roortion of agents ho refer the Beveridgean system and another that refer the Bismarckian one. The roortion of agents ho refer the Beveridgean system increases ith the average risk of the grou. As a consequence, according to the ranking of the k, the roortion of agents ho refer a Beveridgean system is the loest in Grou 1 and the highest in Grou M. Note that the individual choice of the system only deends on the grou the individual belongs to, and on his individual income i, but not on his individual risk i. Finally, every agent of Grou k refers a Beveridgean system if su < r k hich is more likely to be true for high k (that is for a high average risk, hereas every agent refers a Bismarckian system if inf > r k hich is more likely to be true for lo k (that is for a lo average risk. 12

14 BE System BI System su r M r 3 r 2 r 1 inf M-1 Figure 2: Individual references: BE or BI? Bismarck or Liberal? No e comare the Bismarckian and Liberal systems. An agent i of Grou k refers a Bismarckian system if W BI k (t m, i, i > W L (t i, i, i hich leads to the folloing roosition: Proosition 2. Agent i of Grou k refers a Bismarckian system to a Liberal one iff i > k, here k only deends on the grou of the agent, and is an increasing function of k. This agent refers the Liberal system iff i < k M k Proof. See Aendix A. Note that the choice beteen L and BI does not deend on the income earned by the agent in the good state of the orld, because in both systems there is no vertical redistribution. A Bismarckian system imlies intra-grou horizontal redistribution (i.e. from lo-risk to high-risk agents in oosition to the Liberal system. As 13

15 L System BI System 1 ˆ M ˆ 3 ˆ 2 ˆ M-1 Figure 3: Individual references: L or BI? a result, high-risk eole refer the Bismarckian system to the Liberal one. For a given agent of risk i, the Bismarckian system is more interesting if the other agents of the grou are lo risk. If k is lo (i.e. k lo, Grou k is a very lo-risk grou. It is then interesting to have a Bismarckian system for an agent of this grou, as it aears in Figure Beveridge or Liberal? No e comare the Beveridgean and Liberal systems. An agent i refers the Beveridgean system iff M k W BE (t m, i, i > W L (t i, i, i hich leads to the folloing roosition: Proosition 3. Agent i refers a Beveridgean system to a Liberal one iff i < ŵ( i here ŵ( i is an increasing function of i, ith ŵ(0 = 0 and ŵ(1 = + This agent refers the Liberal system iff i > ŵ( i 14

16 Proof. See Aendix B. Figure 4 resents the artition of the oulation beteen those ho refer a Liberal system and those ho refer a Beveridgean system. The reference deends both on the income and the risk suorted by the agent. The curve reresenting the function ŵ characterizes the boundary beteen both regimes. Therefore, a combination of income and risk on the boundary makes the agent indifferent to both regimes. An agent i of income i and risk i refers a Beveridgean system against a Liberal one iff i < ŵ( i, here ŵ is an increasing function of i. Agents ith a sufficiently high income and relatively lo risk refer the Liberal system. Agents ith a sufficiently lo income and relatively high risk refer the Beveridgean system because they benefit from vertical and horizontal redistributions. ˆ L > BE ˆ ŵ( i L < BE 1 Figure 4: Individual references: L or BE? A Beveridgean system imlies both global horizontal and vertical (i.e. from rich to oor agents redistribution. Both high-risk and/or oor agents have an incentive to choose a Beveridgean system to benefit from redistribution. Conversely, lo-risk and/or rich agents benefit more from suorting their referred tax rate. In addition, these agents do not benefit from redistribution. Therefore, they are in favor of a Liberal system. 15

17 3.2.4 Summary of individual references Overall, there are three tyes of redistribution mechanisms hich essentially determine individual references. They are summarized in Table 1. Redistribution mechanism Vertical redistribution Global horizontal redistribution Intra-grou horizontal redistribution Effective in BE BE BI Table 1: Summary of redistribution mechanisms Figure 5 gives an overvie of the artition functions for individual references for a given Grou k. Firstly, the Beveridgean system is clearly referred by agents ho are characterized by a combination of very lo income and very high risk. Hoever, Beveridge is also referred by oor agents ho suort a relatively small risk if the income effect of a high vertical redistribution dominates. BI System L System BE System BI = L BI > L > BE L =BE L > BI > BE k+1 BI > BE > L L > BE > BI BE > L > BI k +1 BE > BI > L BE = BI Figure 5: Overvie of artition functions for individual references 16

18 Secondly, the Liberal system is referred by agents ith a lo risk, from the quite rich to the very rich agents because lo-risk agents are against horizontal redistribution and rich agents are against vertical redistribution. Hoever, a Liberal system is also referred by oor agents ho have a very lo risk: If this lo-risk effect dominates the income effect from vertical redistribution inside the Beveridgean system, then these agents also refer Liberal to Beveridge. The additional advantage of a Liberal system is that the tax rate is not chosen by a decision-maker, but is the one referred by the agent. Thirdly, agents are in favor of a Bismarckian system if they are sufficiently rich and have a level of risk beyond a certain threshold since the Bismarckian system features intra-grou horizontal redistribution but no vertical redistribution. The imact of a higher grou risk on the artition sace of individual references can be seen by considering references of Grou k+1. Comared to Grou k, the indifference curves of agents in this grou ill move for BE = BI uards and for BI = L to the right. This is indicated in Figure 5. As a consequence, the sace here the Bismarckian system is referred becomes smaller because a Bismarckian system is more favorable for loer grou risk. 3.3 Choice of the system under majority voting Before studying the choice of a utilitarian lanner e first focus on a simle ositive decision rule: majority voting. We restrict our analysis to airise comarisons of the choice of the systems. Note that there is not necessarily unanimity ithin a grou regarding a referred system Bismarck or Beveridge? In the folloing e define as the oor those agents hose income is loer than the average income ( i < and as the rich those agents ho have a higher-than-average income ( i >. We study the imact of a mean reserving sread (hereafter referred to as MPS. 12 This means that rich eole become richer, oor eole become oorer, but the average income remains unchanged. Recall that the indicator of risk of Grou k relative to society s risk, r k, is ranked as follos: r 1 <... < r j < 1 < r j+1 <... < r M 12 Note that an MPS is related to second-order stochastic dominance, this is ell defined in Mas-Colell et al. (1995, chater 6.D. 17

19 Proosition 4. (i If the inequality of income is lo, i.e. here, if r j < inf < su < r j+1, then in Grous 1, 2,..., j, there is unanimity in favor of the Bismarckian system, and in Grous j + 1,..., M, there is unanimity in favor of the Beveridgean system. (ii A Mean Preserving Sread (MPS of the distribution of incomes imlies a loer olitical suort for the Bismarckian system in Grous 1,..., j, and a loer olitical suort for the Beveridgean system in Grous j + 1,..., M. (iii With a sufficiently large MPS, there is a majority in favor of the Beveridgean system. Proof. See Aendix C. The decisive factor hich determines the tye of system is income inequality. An interretation of this Proosition is as follos: (i If the inequality of income is lo, the imact of vertical redistribution can be neglected. In this case, comaring BI and BE systems means that to different tyes of horizontal redistribution are comared: intra-grou horizontal redistribution in the BI system and global horizontal redistribution in the BE system. The intra-grou horizontal redistribution is more favorable for grous 1,..., j because for them k < holds true. Therefore, each member in these grous refers a BI system. The reverse is true for grous j + 1,..., M: They benefit more from global horizontal redistribution since < k. Consequently, all agents in these grous refer a BE system. (ii Let us consider Grous 1,..., j after an MPS. The rich agents of these grous ill still refer the BI system. The same is true for the rather oor eole, because they benefit from their lo intra-grou risk k in the BI system. The oorest eole become even oorer ith the MPS, so that they finally refer BE because it allos vertical redistribution. For agents ho belong to Grous j + 1,..., M, again, the reverse is true. With an MPS, the oor agents of these grous ill still refer BE, and also the rather rich eole. The richest agents become even richer ith the MPS, so that they finally refer BI because it does not imly vertical redistribution. Therefore, if a majority of eole belongs to Grous 1,..., j, then BI is adoted by majority voting. BE is adoted if the reverse is true. (iii In case of a very large inequality of income (i.e. very large MPS, the effect of vertical redistribution dominates horizontal redistribution: eole 18

20 ith an income i loer than almost all refer BE. Then, there is a majority for BE since m <. This effect is in line ith the result from Meltzer and Richard (1981 hich states that hen the share of redistributed income is determined by majority voting, a more unequal income distribution leads to more redistribution Bismarck or Liberal? We no focus on the choice beteen a Bismarckian and a Liberal system. The inequality of income has no imact on the olitical suort of the Liberal system against the Bismarckian one, because in both systems the social benefit is roortional to the income, i.e. there is no vertical redistribution. Recall e have assumed that the median voter is the same in each grou, i.e. m,k = m for every k. Proosition 5. According to the majority voting criterion a Liberal system is adoted against a Bismarckian one. Proof. We can set ( 1 h k ( m = Wk BI (t m, m, i W L (t k i, m, i = m ln k m 1 m Since m,k = m for every k, and m < k for every k, then h k ( m < 0. From Proosition 2 and its roof, e are then able to state that for all agents i ith i m, that reresent at least 50% of the voters of each grou, the Liberal system is referred. The main advantage of the Liberal system is that agents can choose their individually referred tax rate. The main advantage of the Bismarckian system is intra-grou horizontal redistribution; hoever, this advantage only alies to a minority of eole. As a result, the Liberal system is referred by a majority of agents Beveridge or Liberal? Let us no study the majority choice beteen the Beveridgean and Liberal systems. According to Proosition 3, e kno that an agent of income i and risk i refers the Beveridgean system against the Liberal one iff i < ŵ( i. 19

21 The function ŵ deends on m,, and. In the folloing roosition, e sho that the olitical suort for BE (against L deends on the level of inequality of income in the society. Proosition 6. (i If the inequality of income is lo, here more recisely, if inf > η, here η only deends on the distribution of risks, and 0 < η < 1, then a majority of agents refers a Liberal system to a Beveridgean one. (ii A Mean Preserving Sread (MPS of the distribution of income imlies a higher olitical suort for the Beveridgean system. (iii With a sufficiently large MPS, there is a majority in favor of the Beveridgean system. Proof. See Aendix D. Again, there is a recise interretation of this roosition. (i If the inequality of income is lo, vertical redistribution does not matter. In this case, BE vs. L means global horizontal redistribution vs. no redistribution at all. Global horizontal redistribution is in favor of high-risk agents hich are a minority. In turn, there is a majority for the L system. (ii The higher the inequality of income, the stronger the vertical redistribution. Poor agents are in favor of vertical redistribution. Since they constitute more than 50% of the oulation, the suort for BE increases. (iii With a sufficiently large inequality of income, the effect of vertical redistribution dominates, so oor agents ill be in favor of BE. 4 Utilitarian criterion In this section, e focus on a normative criterion, analyzing the choice of a utilitarian social lanner. 4.1 Choice of tax rate The Liberal system is characterized by total liberty of choice for any individual. Similarly to the majority voting analysis, the individually referred tax rate is alied, i.e. t L = t i = i Under the Beveridgean system, the utilitarian lanner determines the common otimal tax rate, t BE u, by maximizing the average of the individual 20

22 elfares. Since U BE = W BE (t BE u,, f(g(dd [ ( 1 = (1 ln((1 t BE u + ln ( 1 = (1 ln(1 t BE u + ln tbe u ] tbe u f(g(dd + (1 ln + ln here ln stands for the mean of ln i. Then max U BE imlies t BE t BE u =. u Under the Bismarckian system, the utilitarian lanner determines the otimal tax rate of Grou k, t BI u,k, by maximizing the average of the individual elfares of Grou k. 13 Since U BI,k = 1 Wk BI N (tbi u,k,, f k(g(dd k = 1 [ (1 ln (( 1 t BI N k u,k = (1 k [ ln ( 1 t BI u,k + ln ] + k ln + ln ( 1 k k ( 1 k then maximizing U BI,k yields the otimal tax rate t BI u,k = k. 4.2 Choice of the system k ] t BI u,k f k (g(dd + k ln t BI u,k + kln The Beveridgean system yields the folloing social elfare: ( 1 UBE = (1 ln(1 t BE u + ln tbe u + (1 ln + ln ( 1 = (1 ln(1 + ln + (1 ln + ln = ln(1 + (1 ln + ln (6 The Bismarckian system roduces the social elfare: M UBI = N k UBI,k k=1 13 Generally, in Bismarckian countries, tax rates differ across funds. 21

23 here UBI,k = (1 k [ ln (1 k + ln ] ( 1 k + k ln + k ln k + k ln = ln (1 k + ln k thus U BI = M N k [ln (1 k ] + ln (7 k=1 The Liberal system gives the social elfare: UL = W L (t ; ; f(g ( dd = ln ((1 f(g ( dd = ln + ln (1 (8 The comarison of the different elfare functions leads to the folloing roosition. Proosition 7. A utilitarian lanner has the folloing references: BE > BI > L Proof. Comaring (6 and (7 gives [ M ] UBE UBI = ln(1 + (1 ln + ln N k [ln (1 k ] + ln = [ ln(1 k=1 ] M N k [ln (1 k ] + [ ln ln ] > 0 k=1 due to the Jensen inequality. Comaring (7 and (8 gives [ M ] UBI UL = N k [ln (1 k ] + ln = k=1 M N k [ln (1 k ] ln (1 > 0 k=1 UBI U L is ositive due to the Jensen inequality. [ ] ln + ln (1 22

24 Under the utilitarian criterion, the Beveridgean system is alays referred even if the distribution of risk is strongly asymmetric in favor of lo-risk agents (say, 85% of agents have a risk loer than the average risk. If e comare BE and BI using a utilitarian criterion, to effects are clearly in favor of BE: the vertical redistribution that benefits oor eole more than it hurts rich eole, the global horizontal redistribution that benefits agents belonging to high-risk grous more than it can hurt agents belonging to lo-risk grous. Note that even if there is no inequality of income, BE is still referred because of this second effect. Similarly, if e comare BI and L, the intra-grou horizontal redistribution is only at ork in BI so that it is referred by the utilitarian olicymaker. 4.3 Do the ositive results meet the normative recommendations? In order to comare the results obtained under both criteria, e need to evaluate the imact of every redistribution mechanism (vertical and horizontal either ith our ositive or normative criterion. A utilitarian lanner is in favor of vertical redistribution since it benefits lo income agents more than it hurts high income agents. This argues for BE rather than for BI or L. Similarly, the majority voting rule suorts any vertical redistribution because high income agents are a minority in the society. Again, this argues for BE. On the one hand, a utilitarian lanner is also in favor of any horizontal redistribution since it benefits high-risk agents more than it hurts lo-risk agents. This argues for BE or BI rather than L. On the other hand, the majority voting rule does not suort any horizontal redistribution because high-risk agents constitute a minority in the society. This argues for L. In addition, the utilitarian lanner gives riority to global horizontal redistribution comared to intra-grou horizontal redistribution since the first one is a broader tye of redistribution. This argues for BE against BI. The intra-grou horizontal redistribution is referred under a majority voting rule if and only if there is a majority of agents in good grous, i.e. Grou k such that k <. This last case argues for BI. As a consequence of all these redistribution effects, the utilitarian lanner refers a more redistributive system, i.e. BE. The referred system under a 23

25 majority voting rule deends on the relative sizes of income inequality and risk inequality as ell as on the roortion of agents belonging to good grous. More recisely: the loer the income inequality, the higher the olitical suort for L, the higher the income inequality, the higher the olitical suort for BE, so that the majority voting choice corresonds to the utilitarian choice if the inequality of income is sufficiently high. Moreover, the higher the roortion of agents belonging to good grous, the higher the olitical suort for BI. More recisely, hen comaring BE and BI, a majority voting rule leads to adot BI if the inequality of income is lo, hich is in oosition to the utilitarian choice. Finally, hen only BI and L are comared, the utilitarian criterion leads to refer BI since it allos horizontal redistribution. This is in oosition to the result ith majority voting, since ith m < only a minority of the society benefits from horizontal redistribution. 5 Conclusion We have studied the three main tyes of elfare caitalism ithin a simle economic model hich incororates secific grous. In articular, e have introduced a more accurate ay to model the cororatist Bismarckian system, taking into account the fact that this system allos intra-grou horizontal redistribution, as outlined by Esing-Andersen (1990. For the choice of the elfare system using the majority voting rule, e have shon the influence of the inequality of income, distribution of risk and the grou structure. Under a utilitarian criterion, the Beveridgean system is alays referred. Moreover, the Bismarckian is alays referred to a Liberal one. The utilitarian reference for the Beveridgean system may exlain the evolution of the Bismarckian countries toards mixed systems incororating an increasing art of Beveridgean characteristics. This aer offers reliminary results hich allos us to state that the main results concerning the choice of the elfare system are crucially modified under the ne ay of modeling the Bismarckian system. This is a first ste in a research rogram that should be encomassed by the develoment of ne studies incororating this ne ay of modeling. 24

26 Aendix A Proof of Proosition 2 According to (4 and (5, Wk BI(t m, i, i > W L (t i, i, i is equivalent to ( 1 k (1 i ln (1 m + i ln m > ln (1 i (9 k We set ( 1 k h k ( i = (1 i ln (1 m + i ln m ln (1 i ( k 1 k m = ln (1 m + i ln ln (1 i 1 m We have and h k (0 = ln (1 m < 0 and h k (1 + h k ( i = h k ( i = k ( 1 1 k + ln 1 i k 1 (1 i 2 > 0 m 1 m h k is a convex function on [0, 1, ith h k (0 < 0 and lim 1 h k ( = + so that there is clearly a unique k such that h k ( k = 0. Note that k deends only on k and m. ( In addition, h k ( k = ln (1 m + k ln 1 k k 1 m ln (1 k = 0 ( 1 According to the imlicit function theorem, k = hk k k h k > 0, k and k is an increasing function of k, so that k is an increasing function of k. m ; B Proof of Proosition 3 According to (3 and (5 e have H ( i, i = W BE (t m, i, i W L (t i, i, i ( 1 = (1 i ln [(1 m i ] + i ln ( 1 = ln (1 m ln (1 i + i ln 25 m m 1 m i ln [(1 i i ]

27 ( Moreover, H ( i, i = 0 ln 1 m 1 m ( i = 1 i ln 1 i ( 1 i 1 m 1/i 1 m i.e. H ( i, i = 0 1 m 1 m i = Then, an agent i refers BE to L iff H ( i, i 0, i.e. iff i ŵ( i, here ŵ( i = 1 m 1 m ( 1 m 1 i 1/i Let us sho that ŵ( i is an increasing function of i, ith ŵ(0 = 0 and lim 1 ŵ( = + ( ŵ( i = C ex (a( i, here C = 1 m 1 m > 0 and a( i = 1 i ln 1 m 1 i Clearly, e can rite that a(0 = lim i 0 a( i =, and a(1 = lim i 1 a( i = +, thus ŵ(0 = 0 and ŵ(1 = + We just have to sho that a( i is an increasing function of i. a( i = 1 i ln (1 m 1 i ln(1 i a ( i = 1 2 i ln (1 m i ln(1 i + 1 i (1 i 2 i a ( i = ln (1 m + ln(1 i + (1 i 1 = 1 ln (1 m + ln(1 i + (1 i Let us sho that 2 i a ( i 0 for any i [0; 1]. We set b( i = 2 i a ( i = 1 ln (1 m + ln(1 i + 1 (1 i here b ( i = 1 1 i + 1 (1 i 2 = i (1 i 2 > 0 and b(0 = ln (1 m > 0 Thus b( i > 0 on i [0; 1], so that 2 i a ( i > 0 and a( i is an increasing function of i. i C Proof of Proosition 4 (i According to Proosition 1, an agent i of Grou k refers BE to BI iff i < r k here r 1 <... < r j < 1 < r j+1 <... < r M. By assumtion, r j < inf, then for any agent i of Grou k ith k j, e have i inf > r j r k. Thus, there is unanimity in favor of BI in Grous 1, 2,..., j. Similarly, by assumtion, su < r j+1, then for any agent i of Grou k ith k j + 1, e have i su < r j+1 r k. Thus there is unanimity in favor of BE in Grous j + 1,..., M. 26

28 (ii Imact of an MPS. For k j, the agent i refers BE iff i < r k, here r k < 1. With an MPS, the roortion of eole ith i < r k increases, so that the olitical suort for BE increases. For k j + 1, the agent i refers BI iff i > r k, here r k > 1. With an MPS, the roortion of eole ith i > r k increases, so that the olitical suort for BI increases. (iii Imact of a sufficiently large MPS. For k j, ith a large MPS, the roortion of eole of income i [r k ; ] becomes very small, so that the roortion in favor of BE becomes arbitrarily near that of eole of income i <. For k j + 1, ith a large MPS, the roortion of eole of income i [; r k ] becomes very small, so that the roortion in favor of BI becomes arbitrarily near that of eole of income i >. Finally, hatever the grou, if the MPS is sufficiently large, then the roortion of eole in favor of BE is arbitrarily close to the roortion of eole of income i <. Since the median income m. is loer than, e can conclude that ith a sufficiently large MPS, there is a majority of eole in favor of BE against BI. D Proof of Proosition 6 (i For any agent i of risk i and income i : H ( i, i = W BE (t m, i, i W L (t i, i, i ( ( 1 m = ln (1 m ln (1 i + i ln + i ln 1 m i ( = h( i + i ln here h( ( i = ln (1 m ln (1 i + i ln 1 i m 1 m For an individual of income and risk i, h( i is the difference of elfares under BE and L. h ( i = 1 (1 i > 0, and h(0 = ln(1 2 m < 0, and ( h(m = m ln 1 m 1 m < 0 because m <. 27

29 h is a convex function ith h(0 < 0 and h(m < 0, thus h( i < 0 for all i m. h is a continuous function, ] then max 0 m h( < 0. Setting η = ex [max 0 m h(, e have then 0 < η < 1. By assumtion, η < inf, thus max ( 0 m h( < ln inf For every i, ith i m, e have H ( i, i = h( ( i + i ln i max 0 m h( + ln ( inf < 0 Then, any agent i such that i m refers L to BE, i.e. a majority of eole are in favor of L. (ii An MPS imlies that ln ( increases for a majority of eole because m <, thus it increases the olitical suort for the Beveridgean system. (iii ( With a sufficiently ( large MPS of the distribution of income, e have i ln i i ln m > h( for 50% of the oulation. Then, clearly H ( i, i > 0 for a majority of eole. 28

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