mmanuscript No. (will be inserted by the editor)
A DICHOTOMOUS ANALYSIS OFUNEMPLOYMENT WELFARE
Xingwei Hu
Received:
Abstract
In an economy which could not accommodate the full employmentof its labor force, it employs some labor but does not employ others. The bi-partition of the labor force is random, and we characterize it by an axiom ofequal employment opportunity. We value each employed individual by his orher marginal contribution to the production function; we also value each un-employed individual by the potential marginal contribution the person wouldmake if the market hired the individual. We then use the aggregate individualvalue to distribute the net production to the unemployment welfare and theemployment benefits. Using real-time balanced-budget rule as a constraint andpolicy stability as an objective, we derive a scientific formula which describes
Xingwei HuInternational Monetary Fund, 700 19th St NW, Washington, DC 20431, USATel.: +202-623-8317Fax: +202-589-8317E-mail: [email protected]: 0000-0001-8454-3974 a r X i v : . [ q -f i n . GN ] J un Xingwei Hu a fair, debt-free, and asymptotic risk-free tax rate for any given unemploy-ment rate and national spending level. The tax rate minimizes the asymptoticmean, variance, semi-variance, and mean absolute deviation of the underlyingposterior unemployment rate. The allocation rule stimulates employment andboosts productivity. Under some symmetry assumptions, we even find thatan unemployed person should enjoy equivalent employment benefits, and thetax rate goes with this welfare equality. The tool employed is the cooperativegame theory in which we assume many players. The players are randomly bi-partitioned, and the payoff varies with the partition. One could apply the fairdistribution rule and valuation approach to other profit-sharing or cost-sharingsituations with these characteristics. This framework is open to alternativeidentification strategies and other forms of equal opportunity axiom.
Keywords
Tax Rate · Unemployment Welfare · Equal Opportunity · FairDivision · Balanced Budget · Shapley Value
JEL Codes
C71 · D63 · E24 · E62 · H21 · J65
Acknowledgements
The author thanks Lloyd S. Shapley, Stephen Legrand, Bruce Moses,Alisha Abdelilah, and seminar participants at International Monetary Fund, Stony BrookUniversity, and Central South University for their suggestions. DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 3
The problem with which we are concerned relates to the following typi-cal situation: consider an economy which could not achieve full employmentof its labor force, and therefore some people are employed, and others arenot. As the employed receives wages and employment benefits (e.g., pension,health insurance, social security, education allowances, paid vacation), shouldthe unemployed receive some unemployment welfare? If YES, how much isfair? In a specific jurisdiction system, the term “unemployment welfare” heremay also mean “unemployment benefits,” “unemployment insurance,” or even“unemployment compensation.” In an advanced economy, the answer to thefirst question is likely YES. This paper answers the second question by justi-fying a fair share of unemployment welfare for the unemployed and deriving afair tax rate for the employed. Fair unemployment welfare and a fair tax rateare among the most fundamental topics of our society.The fair-division problem arises in various real-world settings. For a sim-ple motivating example, let us consider a k -out-of- n redundant system in en-gineering which has n identical components, any k of which being in goodcondition makes the system work properly. When valuing the importance ofeach component (either working or standby), one may intuitively claim thatthese components should be equally valued. A very similar situation occurs ina simple majority voting where not all the voters would support the proposalto vote; thus, the proposal could be passed or failed. Nevertheless, voters aresupposed to have the same voting power no matter what they support and Xingwei Hu for whom they vote. For another example, in the health insurance industry,not all of the policyholders are ill and use the insurance to cover their medicalexpenses. The question is how to fairly share the total medical cost amongboth the ill policyholders and the non-ill ones. In a labor market, we have asimilar, but more complicated situation: on the one hand, the market couldnot hire all of its labor force even though everyone in the market would liketo be employed; on the other hand, the participants in the market have het-erogeneous performance in the production. There are four common featuresin these examples: a coalition of players with cooperative nature, a randombipartition of the players, a payoff associated with the partition, and an ob-jective to share the payoff with all the players. This paper derives a solutionfor situations with these features. In the k -out-of- n redundant system and thesimple majority voting, we expect equality of outcome.We face a few challenges to deal with when fairly distributing the welfareand benefits, both generated by the employed. First of all, fairness may be anabstract but vague concept. We believe that “fairness” is bound with the equal-ity of employment opportunity, not with the equality of outcome, nor with theequality of productivity. Furthermore, we also believe that everyone in the la-bor market could contribute in some way; but the opportunities are limited.Thus, unemployment is not a fault of the unemployed, nor a flaw of the labormarket, but a self-adjustment mechanism toward the efficiency of the market.Secondly, we attempt to apply a taxation policy to the labor market which op-erates in an ever-changing economy and with ever-changing productivity. For DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 5 it to be useful, the tax and division rule should be not only fair to all people butalso be able to cope with the uncertainty and sustainability in generating anddistributing the net production. Ideally, it should balance the account of valuegenerated and that of value paid in each employment contingency. Thirdly, ina perfect world, a fair tax rate should depend only on observed data, to avoidany excessively political bargaining and costly strategic voting. One major is-sue, however, is the non-observability of the heterogeneous-agent productionfunction in all employment scenarios at all the time. Another data issue isthe desynchronization between the unemployment rate and the tax rate; theformer is high-frequency data while the latter has a lower frequency. Often ina yearly time-frame, policymakers determine the tax rate after observing mostmonthly unemployment rates.Vast literature (e.g., Kornhauser 1995; Fleurbaey and Maniquet 2006) fromvarious aspects has studied the fairness in taxation and unemployment pay-ments. In particular, Shapley (1953) proposes an influential axiom of fairnessto develop a fair-division method, called the Shapley value, which is widelyused in distributing employment compensation and welfare (see, for exam-ple, Moulin 2004; Devicienti 2010; Giorgi and Guandalini 2018; Krawczyk andPlatkowski 2018). Beneath the pillars of the Shapley value and the Shapley ax-iom, however, are two underlying assumptions: players’ unanimous participa-tion in the production, and distributor’s complete information about the pro-duction function. Recently, Hu (2002, 2006, 2018) relax the unanimity assump-tion and generalize the Shapley value, using some non-informative probability
Xingwei Hu distributions for the dichotomy or bipartition of the players. In particular, Hu(2018) proposes using a Beta-Binomial distribution to address the equality ofopportunity. This current research also capitalizes on the Beta-Binomial dis-tributions. Furthermore, we do not assume the complete information aboutthe production function. We only need its value at one observation which doesoccur.The advantage of our approach is twofold. On the one hand, we provide agame-theoretical micro-foundation for a fair-division solution to distribute theunemployment welfare and employment benefits. One can apply the solutionconcept to many similar situations without substantive alternations, and onemay also extend the framework using other identification schemes, rather thanthe tax policy stability or unemployment rate minimization, detailed in Section4. On the other hand, the fair tax rate we provide is simple enough to be usedin practice. It relies only on the unemployment rate and a reserved portionof production, which is not for personal use. The total unemployment welfaredepends only on the tax rate and the observed production. We attempt toimmunize our solution from any unnecessary randomness, hypotheticals, am-biguity, and latency. These include, but are not limited to, the competitiveand cooperative features of the labor market, endogenous employment searchbehavior, non-linear schedule of tax rates, the exact sizes of the labor marketand time-varying unemployment population. With this simplicity in hand, acertain level of abstraction is necessary and any application of the fair solutionshould accommodate to the concrete reality.
DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 7
We organize the remainder of the paper as follows. Section 2 applies theframework of dichotomous valuation (or simply, “D-value”) in Hu (2002, 2006,2018) to value each person in the labor market, assuming equal employmentopportunity. The two sides of D-value depend on two unknown parametersand are aggregated separately for the unemployed and the employed labor.Next, Section 3 formulates a set of fair divisions of the net production, usingthe aggregate components of D-value. In this section, we base the set of fairtax rates on two accounting identities for a balanced budget. In Section 4, bymaximizing the stability of the tax rate or minimizing the expected posteriorunemployment rate, we identify a specific relation between the fair tax rateand the unemployment rate. The particular solution is robust under a fewother criteria. Section 5 lists three other applications (namely, voting power,health insurance, and highway toll) of the distribution rule, and Section 6suggests several ways to extend this framework. The account is self-contained,and the proofs are in the Appendix.
Before our formal discussions, let us introduce a few notations. For a generaleconomy, we assume that its labor force consists of the employed labor and theunemployed labor, ignoring any part-time labor. Let N = { , , · · · , n } denotethe set of people in the labor force, indexed as 1 , , ..., n , and let S ⊆ N denotethe random subset of the employed labor in N . For any subset T of N , let | T | denote its cardinality; for notational simplicity, we often use n for | N | , t for Xingwei Hu | T | , and s for | S | . Let us also write the employment rate as ω = sn , which is oneminus the unemployment rate. Besides, we employ the vinculum (overbar) innaming a singleton set; for example, “ i ” is for the singleton set { i } . Also, weuse “ \ ” for set subtraction, “ ∪ ” for set union, and β ( · , · ) for a two-parameterBeta function. The Appendix defines ∆ , ∆ , · · · , ∆ as shorthand notations.2.1 Equality of Employment OpportunityEqual employment opportunity is widely acknowledged and is the start-ing point or axiom for us to study fairness and welfare. In the United States,for example, equal employment opportunity has been enacted to prohibit fed-eral contractors from discriminating against employees by race, sex, creed,religion, color, or national origin since President Lyndon Johnson signed Ex-ecutive Order 11246 in 1965. In the literature , there are abundant qualitativedescriptions, informal or formal, about the equal opportunity (e.g., Friedmanand Friedman 1990; Roemer 1998; Rawls 1999). In this section, we introducea quantitative and probabilistic version of equal opportunity whereby the em-ployment opportunity is assumed equitable for all persons in the labor force.We assume three layers of uncertainty for the random subset S while main-taining the equality of employment opportunity. In the first layer, the employ- Besides the justification from equality of opportunity, unemployment welfare could alsocome from other considerations (e.g., Sandmo 1998, Tzannatos and Roddis 1998, Vodopivec2004). These include, but are not limited to, social protection, insurance of income flow,poverty prevention, the efficiency of the labor market, and political considerations. In thispaper, we focus solely on equal employment opportunity. DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 9 ment size | S | follows a binomial distribution with parameters ( n, p ). Whenindependence is assumed, p is the probability of any given person being em-ployed. In the second layer, the unknown parameter p has a prior Beta distri-bution with hyperparameters ( θ, ρ ), where θ > ρ > p ∈ (0 , | S | = s is p θ − (1 − p ) ρ − β ( θ, ρ ) ns p s (1 − p ) n − s = n ! s !( n − s )! p θ + s − (1 − p ) ρ + n − s − β ( θ, ρ ) . (1)Eq.(1) implies the following marginal probability density for | S | = s : P ( | S | = s ) = Z n ! s !( n − s )! p θ + s − (1 − p ) ρ + n − s − β ( θ,ρ ) d p = n ! s !( n − s )! β ( θ + s,ρ + n − s ) β ( θ,ρ ) , s = 0 , , · · · , n. (2)In the third layer, for any given employment size s , all subsets of size s havethe same probability of being S . As there are n ! s !( n − s )! subsets of size s in N ,the probability for the employment scenario S = T is P ( S = T ) = P ( | S | = s ) n ! s !( n − s )! = β ( θ + s,ρ + n − s ) β ( θ,ρ ) , if t = s ;0 , otherwise . (3)Clearly, the equality of employment opportunity is implied in the assumedtriple-layered uncertainty. Furthermore, equal opportunity is also assumed forall coalitions of the same size.By Eq.(1) and Eq.(2), the posterior density function of p given | S | = s is n ! s !( n − s )! p θ + s − (1 − p ) ρ + n − s − β ( θ,ρ ) n ! s !( n − s )! β ( θ + s,ρ + n − s ) β ( θ,ρ ) = p θ + s − (1 − p ) ρ + n − s − β ( θ + s, ρ + n − s ) . Thus, the posterior employment rate follows a Beta distribution with param-eters ( θ + s, ρ + n − s ). In the following, let us use ˜ p n,ω to denote the posterioremployment rate given the observance of | S | = nω . In contrast, p is the unob-servable prior employment rate, and ω is the observable employment rate.2.2 Value of the Employed and the UnemployedFor any T ⊆ N , we use a heterogeneous-agent production function v ( T )to measure the aggregate productivity when S = T . We assume the net-profitproduction function v ( T ) excludes the labor cost which compensates the timeand efforts devoted by the employed labor in producing v ( T ). To isolate theadded value by the labor force alone, we also assume that v ( T ) excludes thecost of consumed physical and financial resources. Both the labor cost andthe resource cost are exempt from taxation in a firm. Thus, without loss ofgenerality, we may assume that v ( ∅ ) = 0 for the empty set ∅ . However, v ( T )does not necessarily increase with T nor with its size | T | . To retain its labor orto minimize its labor turnover, a firm would share part of its net profit withits employees. To keep things simple, we use the term “employment benefits”to denote the employees’ profit-sharing part in v ( T ), in contrast to the term“unemployment welfare.”Let us formally introduce two components of the D-value. For any i ∈ N ,to analyze its marginal effect on the value generating process, we consider twojointly exhaustive and mutually exclusive events: DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 11 – Event 1 : i ∈ S . Then, i ’s marginal effect is v ( S ) − v ( S \ i ), called marginalgain , in that he or she contributes v ( S ) − v ( S \ i ) to the production, due tohis or her existence in S . The expected marginal gain is γ i [ v ] def= E (cid:2) v ( S ) − v ( S \ i ) (cid:3) . (4)In the above, “def” is for definition and “ E ” for expectation under theprobability distribution specified by Eq.(3). – Event 2 : i S . Then, i ’s marginal effect is v ( S ∪ i ) − v ( S ) in that S faces a marginal loss v ( S ∪ i ) − v ( S ), due to i ’s absence from the employed laborforce S . In other words, the person would increase the production by v ( S ∪ i ) − v ( S ) if the market included him or her in S . The expected marginalloss is then λ i [ v ] def= E (cid:2) v ( S ∪ i ) − v ( S ) (cid:3) . (5)We let γ i [ v ] be the employment benefits i receives when he or she is em-ployed, and let λ i [ v ] be the unemployment welfare i receives when he or sheis unemployed. Note that, even if i always remains employed, both S and S \ i change daily, if not hourly; thus i ’s marginal gain is not a constant. Similarly,even if i remains unemployed for a while, S , S ∪ i , and i ’s marginal loss are notconstant. To account for this uncertainty, we have already taken expectationsin Eq.(4) and Eq.(5) when defining γ i [ v ] and λ i [ v ].A few key points worth mentioning to help us understand the profit-sharingstrategy. First, in addition to receiving employment benefits, the employed la-bor also receives the reimbursement for labor cost, which compensates its human capital usage in generating v ( S ). Human capital also accumulates inprior employment and pre-employment education. Labor, physical, and finan-cial costs are not part of the generated value v ( S ). The unemployed labor,however, only receives unemployment welfare. Secondly, if i ∈ S , then v ( S \ i )is not observable while we observe v ( S ). Similarly, when j S , we cannotobserve both v ( S ∪ j ) and v ( S ) simultaneously. Thus, we need to transformthe aggregate marginals into observable forms, such as those in Theorem 1.Thirdly, the aggregate employment benefits P i ∈ S (cid:2) v ( S ) − v ( S \ i ) (cid:3) is not neces-sarily equal to v ( S ), the value collectively generated by S . Thus, we distributesome of the surpluses v ( S ) − P i ∈ S (cid:2) v ( S ) − v ( S \ i ) (cid:3) to the unemployed labor N \ S .The distribution is not through personal giving but government taxation andunemployment payment systems. This distribution channel also appeals to usfor the aggregate benefits and aggregate welfare at the national level, as statedin Theorem 1. Theorem 1
The aggregate components of the D-value are P i ∈ N γ i [ v ] = E (cid:20) P i ∈ S (cid:0) v ( S ) − v ( S \ i ) (cid:1)(cid:21) = nβ ( θ + n,ρ ) β ( θ,ρ ) v ( N ) + P T ⊆ N : T = N t ( θ + ρ − − nθρ + n − t − β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T ) , P i ∈ N λ i [ v ] = E " P i ∈ N \ S (cid:0) v ( S ∪ i ) − v ( S ) (cid:1) = P T ⊆ N : T = ∅ t ( θ + ρ − − n ( θ − θ + t − β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T ) − nβ ( θ,ρ + n ) β ( θ,ρ ) v ( ∅ ) . (6) DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 13
By Eq.(3), the expected production is E [ v ( S )] = X T ⊆ N β ( θ + t, ρ + n − t ) β ( θ, ρ ) v ( T ) . (7)Out of the 2 n employment scenarios for S , we observe only one. Let us considerthis particular scenario at S = T , which occurs with probability β ( θ + t,ρ + n − t ) β ( θ,ρ ) and which generates the value v ( T ). After the production, we face a challengeof fairly dividing v ( T ) between T and N \ T , whom both have an entitlementto v ( T ). Our division rule should fully respect the entitlement claim: each em-ployed person receives his or her expected marginal gain, and each unemployedperson receives his or her expected marginal loss. As a contrast, the Shapleyvalue distributes v ( N ) to all players in N ; Hu(2018) formulates solutions todivide E [ v ( S )] and v ( N ) − E [ v ( S )]. Besides γ i [ v ] and λ i [ v ], we should reservea portion of v ( T ) for the common good of the economy and society.3.1 A Real-Time Balanced Budget RuleAs noted above, we purposely divide the net production v ( T ) into threecomponents. The first one is for employment benefits. We compare the coeffi-cients of v ( T ) in Eq.(7) and that of E (cid:20) P i ∈ S (cid:0) v ( S ) − v ( S \ i ) (cid:1)(cid:21) in Eq.(6), ignoringthe probability density for the scenario; the employed labor T should retain t ( θ + ρ − − nθρ + n − t − v ( T ) as their employment benefits. The rest, h − t ( θ + ρ − − nθρ + n − t − i v ( T ),pays to the government as “tax” . Besides, we assume both λ i [ v ] and γ i [ v ] are In practice, not all unemployment welfare comes from the taxation system. In the UnitedStates, for example, it is funded by a compulsory governmental insurance system, which4 Xingwei Hu tax-exempt in order to avoid any double taxation. Thus, we define the tax rate τ as τ def = 1 − s ( θ + ρ − − nθρ + n − s − , s = 1 , · · · , n − . (8)For the time being, the rate relies on the labor market size n and the em-ployment rate ω = | S | n , but not on the production v ( S ). In this definition, weexclude two extreme but unlikely cases at ω = 0 and ω = 1. Secondly, we com-pare the coefficients of v ( T ) in Eq.(7) and that of E " P i ∈ N \ S (cid:0) v ( S ∪ i ) − v ( S ) (cid:1) in Eq.(6); the unemployed labor N \ T should claim t ( θ + ρ − − n ( θ − θ + t − v ( T ) fromthe tax revenue τ v ( T ) as its unemployment welfare. Thirdly, we assume that areserved proportion, δv ( T ), is not individually and not directly distributed tothe labor force N . As a consequence, the tax rate τ includes both the reserveratio δ and the proportion to the unemployment welfare, i.e., τ ≡ δ + s ( θ + ρ − − n ( θ − θ + s − . (9)The reserve δv ( T ) is meant to serve the general public’s interests and tohave broad appeal, rather than the individual needs. More specifically, δv ( T )includes, but is not limited to, the payments to the population who are notin the labor force, to the corporate equity earnings not used for employmentbenefits, to the public administration and national defense, to the public wel-fare, to the tax deficit from the past, to the future development, and so on. By manages the collection and payment accounts. However, the contribution to and distributionfrom the insurance system are de facto of a type of payroll tax. In Australia, unemploymentwelfare, as part of social security benefits, is funded through the taxation system. In reality, the two cases have zero probability to occur, even though the probabilitymodel (3) gives a very slight chance. DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 15 admitting the corporate equity earnings to the reserve δv ( T ), we have deliber-ately ignored firm-specific re-distribution processes of the net corporate earn-ings, and have purposely avoided the associated corporate income taxation.In practice, government expenditure and corporate earnings are two majorcomponents of the reserve δv ( T ). When the government has other sources oftax revenue, a pro rata share of its expenditure will come from δv ( T ). Besides,the government could implement a countercyclical fiscal policy by adjustingits spending level in δv ( T ) such that it counteracts the ratio of the corporateearnings to the production. Indeed, by Eq.(9), τ automatically reacts posi-tively to the change of the corporate earning ratio, other things remaining thesame.Put in another way, the real-time balanced budget rule implied in Eq.(8)and Eq.(9) forbids any borrowing between different labor market scenarios orforbids any inter-temporal borrowing. Thus, this sustainable taxation policymeets the needs of the present market scenario without compromising the abil-ity of future market scenarios to meet their own needs. In practice, however, itis challenging, if not impossible, to enforce or enact the balanced budget ruleat the labor market scenario level or on a real-time basis. In the United States,for example, the employment rate ω changes daily and is recorded monthlyby the Bureau of Labor Statistics; as a policy variable, the tax rate τ changesyearly. Striving for the balanced budget rule to the highest degree, one could including that in all levels of government but excluding all unemployment welfare pay-ments.6 Xingwei Hu minimize the variance of the employment rate ω within a yearly time frame.In a perfect situation, the employment rate closely follows a degenerate prob-ability distribution and remains almost constant within the year. We discussthe variance minimization in Section 4. In contrast to the exhaustive distri-bution of the net production in the national accounts, a household could stillmaximize its utility through inter-temporal borrowing, saving, lending, andconsuming.3.2 Sets of Feasible SolutionsAlthough the terms “tax rate” and “tax rule” have been used loosely andinterchangeably about τ , we must distinguish between them to have an accu-rate discussion. As a tax rule or tax policy, τ is a function of ( n, ω, δ ), subjectto the equality of employment opportunity and the balance of budget as spec-ified in Eq.(8) and Eq.(9). In contrast, as a tax rate, τ is merely the value ofthe function at a specific ( n, ω, δ ) .For a given triple of ( n, ω, δ ), there are three indeterminates ( θ, ρ, τ ) in thesystem of two equations Eq.(8) and Eq.(9). Let Ω n,ω,δ denote the set of allfeasible combinations of ( θ, ρ, τ ) which satisfy both Eq.(8) and Eq.(9): Ω n,ω,δ def = ( θ, ρ, τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ = 1 − nω ( θ + ρ − − nθρ + n − nω − ,τ = δ + nω ( θ + ρ − − n ( θ − θ + nω − , ≤ τ ≤ , θ > , ρ > . . From now on, a “fair tax rate” could mean a solution of τ in Ω n,ω,δ ; or it couldbe a limit of tax rates which satisfy Eq.(8) and Eq.(9). DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 17
In general, for a reasonable δ , a finite n , and a ω ∈ (0 , Ω n,ω,δ . In this case, we need one more restriction to solve( θ, ρ, τ ) uniquely. To do so, one could capitalize on the statistical relationbetween ω and ( θ, ρ ): the prior mean and mode of ω are θθ + ρ and θ − θ + ρ − ,respectively (e.g., Johnson et al. 1995, Chapter 21). Alongside this direction,for example, we could set θθ + ρ (or θ − θ + ρ − ) to be the historical average (ormode, respectively) of ω in the previous year. Alternatively, we could set θθ + ρ or θ − θ + ρ − to be a target employment rate or the natural employment rate.However, this type of identification schemes requires additional input (e.g.,historical average, target employment rate, or natural employment rate).Furthermore, to derive a unique solution from Ω n,ω,δ or its boundary, weshould base our deviation on observed data only. One concern in the input( n, ω, δ ) is the size of the labor market n . Even though the size n is not randomin our model of equal employment opportunity, it is likely a time-varying latentvariable in that there is no clear cut between entry to and exit from the labormarket; and many depressed unemployed persons may not actively seek newpositions. In practice, n changes daily whereas τ most likely changes yearly.No matter how it changes and how much latent it is, however, we are confidentthat n is a large number in the general economy. Thus, in contrast to Eq.(8)and Eq.(9), we seek a fair tax rule which is valid for all large n ’s but is notspecific to a particular n . As a result, the tax rule we are targeting should notinvolve n , and we may write it as τ ( ω, δ ) : (0 , × (0 , → (0 , To visualize the solution set Ω n,ω,δ , let us write ( θ, ρ ) in terms of ( n, δ, τ, ω )using Lemma 1 in the Appendix. Figure 1(a-b) plots the feasible solution setsfor n = 10000, δ = .
1, and any ω ∈ (0 , θ and ρ . From Figure 1(c-d), we observe thatboth θ and ρ drop sharply on the other side of the straight line. As a matter offact, any point on the other side of the straight line does not represent a fairsolution, owing to the positivity requirement of θ and ρ . Actually, the straightline has a tax rule τ ( ω, δ ) = 1 − ω + δω . This linear relation between τ and ω can be seen from Lemma 1: on the one hand, when ∆ ≡ ω + τ − δω − >
0, both θ = n ω∆ + n∆ + ∆ n∆ + ∆ and ρ = n (1 − ω ) ∆ + n∆ + ∆ n∆ + ∆ increase to + ∞ as n → ∞ ,due to the positivity of ∆ ≡ δω − ω − τ + 2 >
0; on the other hand, when ∆ <
0, both θ and ρ decrease to −∞ as n → ∞ . Thus, the singular line isexpressed as ∆ = 0, or τ ( ω, δ ) = 1 − ω + δω . Furthermore, for any fair tax ruleon the side of ∆ >
0, the posterior employment rate ˜ p n,ω has a degeneratedlimit distribution as n → ∞ . This is claimed in Theorem 2. The next sectionanswers a related question: which tax rule makes the distribution convergencefastest. The answer happens to be the solution on the singular line. Theorem 2
For any fair tax rule τ ( ω, δ ) ∈ (1 − ω + δω, , as n → ∞ , ˜ p n,ω converges in distribution to the degenerate probability distribution with massat ω . DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 19(a) (b)(c) (d)
Fig. 1: Solutions of Eq.(8) and Eq.(9) for n = 10000 , δ = . , ω ∈ (0 , In this section, we derive the limit fair tax rule τ ( ω, δ ) = 1 − ω + δω froma few different angles. At a minimum, a good tax rule should not discourageemployment incentives and productivity, such as detailed in Theorem 5 and7. On the other end, we expect a good rule to be robust and optimal undermultiple criteria. We study five criteria in Theorem 3, 4, 5, 6, and 8, any ofwhich uniquely identifies the solution. They either minimize the employmentmarket risk or maximize the employment expectation, under a given marketcapacity and a budget balance constraint. First, we should heavily exploit the observed market behavior. While anefficient labor market stimulates productivity v ( S ), a higher employment ratedoes not imply higher productivity and vice versa — the production function v ( S ) does not necessarily increase with the employment size | S | . Acemogluand Shimer (2000) find that a moderate level of unemployment could boostproductivity by improving the quality of jobs. Indeed, it fosters peer pressurein producing v ( S ), allows workers to move on from declining firms, and enablesrising companies and the economy as a whole to optimally respond to externalshocks. Therefore, our tax rule would not merely target a higher expectation ofemployment rate, but grounds its assumption in the observed market behavior,and lets the market itself respond with a higher employment rate, to themaximum extent permitted by the budget rule and the market capacity.Secondly, from a statistical viewpoint, our tax rule relies on a realizationof p , not on the uncertainty of the realized value. A natural way to factor outthis uncertainty is to study the posterior rate ˜ p n,ω , where both s and ω areno longer random. As Theorem 2 indicates, ω is informative and indicativein revealing the central tendency of the posterior labor market, delineated by˜ p n,ω . Asymptotic dispersion of ˜ p n,ω and the market’s response to the tax rule τ ( ω, δ ) are among other essential ingredients in the complete profile of ˜ p n,ω .Thus, we set optimal criteria to minimize dispersion measures or maximizethe expected market response. DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 21 ω , τ migrates the risk fromthe unemployment rate to the tax rate. With the absence of other exogenousshocks, indeed, stability in the tax rate is equivalent to that in the unemploy-ment rate. The same argument is also valid when the exogenous shocks areorthogonal to that in ω .When we use the variance of ˜ p n,ω to measure its instability, Theorem 3states that the tax rule τ ( ω, δ ) = 1 − ω + δω minimizes the asymptotic vari-ance of ˜ p n,ω . Furthermore, it is also the limit of variance-minimizing tax rulesfor finite labor markets. We add the restriction “ θ, ρ ≥ n ” in the finite labormarkets to ensure the positivity of the hyperparameters ( θ, ρ ). Theorem 3 As n → ∞ , the tax rule τ ( ω, δ ) = 1 − ω + δω minimizes the limitvariance of √ n ˜ p n,ω , i.e., argmin τ lim n →∞ VAR ( √ n ˜ p n,ω ) = 1 − ω + δω. Besides, lim n →∞ argmin τ VAR (cid:0) √ n ˜ p n,ω (cid:12)(cid:12) θ, ρ ≥ n (cid:1) = 1 − ω + δω. Furthermore, both the min-imum limit variance and the limit minimum variance are zero.
We provide a few comments to help clarify any potential misunderstandingsabout the theorem. First, a stable √ n ˜ p n,ω is achieved at n = ∞ where the limitvariance is zero. However, ˜ p n,ω is still exposed to exogenous shocks, such asthose studied in Pissarides (1992) and Blanchard (2000). Secondly, it is worthemphasizing that 1 − ω + δω is the limit tax rule as n → ∞ . For a large but finite n , a small positive number could be added to 1 − ω + δω to ensure thepositivity of θ and ρ . That small positive number, however, is negligible;thus, we can practically use the rule τ ( ω, δ ) = 1 − ω + δω without any addition.Moreover, a higher-order approximation τ = 1 − ω + δω + ω (1 − ω )(1 − δ ) n couldbe an excellent alternative to τ = 1 − ω + δω . Thirdly, with a zero or nearzero variance in the unemployment rate, labor mobility means one layoff andone new hire should almost coincide to ensure the total employment size | S | remains nearly constant. It also means that the sizes of employment and labormarket change proportionally so that their ratio remains unchanged. Lastly,though the posterior distribution is skewed, the tax rule minimizes both theoverall risk and the one-sided risk of √ n ˜ p n,ω , as stated in Theorem 4. Inparticular, a policymaker’s concern is on the downside risk only. Theorem 4 As n → ∞ , the tax rule τ ( ω, δ ) = 1 − ω + δω minimizes both thelimit lower semivariance and the limit upper semivariance of √ n ˜ p n,ω . See details in the proof of Theorem 3. See the proof of Theorem 3 for details. DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 23 sures other than the posterior variance or semivariance. Meanwhile, it helpsmitigate income inequality.The rule τ ( ω, δ ) = 1 − ω + δω is an effective taxation strategy to maximallyboost the employment size without breaking the opportunity equality and thebudget balance. For an economic policymaker, one primary concern is theforward-looking employment profile ˜ p n,ω . By Theorem 2, the mean of ˜ p n,ω converges to ω as n → ∞ for any fair rule τ ( ω, δ ) ∈ (1 − ω + δω, n is finitely large, they respond adversely to an increasing τ (cf Theorem 5).Consequently, to maximize the posterior mean, we should minimize the taxrate τ while still maintaining the conditions τ ( ω, δ ) ∈ (1 − ω + δω, θ > ρ >
0. Thus, the limit of fair tax rates which maximize the mean of ˜ p n,ω wouldbe 1 − ω + δω . As a remark, the condition ω > . Theorem 5
For any ω ∈ ( . , and a finitely large n , the mean of ˜ p n,ω reactsnegatively to an increasing tax rate τ ∈ (1 − ω + δω, . As a consequence, lim n →∞ argmax τ MEAN (cid:18) ˜ p n,ω (cid:12)(cid:12) θ, ρ ≥ n (cid:19) = 1 − ω + δω. Furthermore, the tax rule also minimizes the mean absolute deviation(thereafter, MAD) from the mean as n → ∞ . For a Beta distribution, es-pecially with large parameters, MAD is a more robust measure of statisticaldispersion than the variance. The MAD around the mean for the posterior˜ p n,ω is (e.g., Gupta and Nadarajah 2004, page 37): E [ | ˜ p n,ω − E (˜ p n,ω ) | ] = 2( θ + s ) θ + s ( ρ + n − s ) ρ + n − s β ( θ + s, ρ + n − s )( θ + ρ + n ) θ + ρ + n . (10) In the next theorem, we identify the same tax rule by minimizing the asymp-totic MAD.
Theorem 6 As n → ∞ , the tax rule τ ( ω, δ ) = 1 − ω + δω minimizes the MADof n ˜ p n,ω around the mean, i.e. lim n →∞ argmin τ E (cid:20) n | ˜ p n,ω − E (˜ p n,ω ) | (cid:12)(cid:12)(cid:12)(cid:12) θ, ρ ≥ n (cid:21) =1 − ω + δω. Besides, argmin τ lim n →∞ E (cid:20) n | ˜ p n,ω − E (˜ p n,ω ) | (cid:12)(cid:12)(cid:12)(cid:12) θ, ρ ≥ n (cid:21) = 1 − ω + δω. i, j ∈ N with i = j , we say i uniformly outperforms j in v if – v ( T ∪ i ) − v ( T ) ≥ v ( T ∪ j ) − v ( T ) for any T ⊆ N \ i \ j ; and – v ( T ) − v ( T \ i ) ≥ v ( T ) − v ( T \ j ) for any T ⊆ N with i, j ∈ T .With these two inequality conditions, i has higher marginal productivity than j in all comparable employment contingencies − either both employed or bothunemployed. As productivity is highly valued in Eq.(4) and Eq.(5), j shouldreceive fewer employment benefits and less unemployment welfare than i does.This is formally claimed in Theorem 7. Besides, the theorem does not requirethe Beta-Binomial distribution, as long as i and j alone have the same chanceof being employed; other players in the labor market may have unequal em-ployment opportunities. The theorem is valid for all fair tax rules, includingthe special one τ ( ω, δ ) = 1 − ω + δω . DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 25
Theorem 7 If i ∈ N uniformly outperforms j ∈ N in v and they have equalemployment opportunity, then γ i [ v ] ≥ γ j [ v ] and λ i [ v ] ≥ λ j [ v ] . We say i, j ∈ N are symmetric in the production function v if they uniformlyoutperform each other. By Theorem 7, λ i [ v ] = λ j [ v ] and γ i [ v ] = γ j [ v ] if i and j are symmetric and they have equal employment opportunity. In other words,they should receive the same amount of unemployment welfare if both areunemployed; they should also receive the same amount of employment benefitsif both are employed.Without any further analysis of or any prior knowledge about the produc-tion function, symmetry among the unemployed (or the employed) could bea reasonable a priori assumption to distribute the unemployment welfare (oremployment benefits, respectively). For example, the Cobb-Douglas produc-tion function is symmetric among all employed persons in the labor market.Besides, if we assume symmetry among the employed labor and also assumesymmetry among the unemployed labor, then the tax rule τ ( ω, δ ) = 1 − ω + δω eliminates the income inequality, when the income is either employment ben-efits or unemployment welfare. Theorem 8 affirms this equality of outcome,but an employed individual and an unemployed one may not be symmetric in v . Furthermore, the theorem does not restrict the size of n and the specificprobability distribution for the equal employment opportunity. In the k -out-of- n redundant system mentioned in Section 1, accordingly, the n components(either working or standby) are equally important if they have equal qual- ity. Section 6 offers a few alternative probability distributions for the equalemployment opportunity. Theorem 8
Assume equal employment opportunity in N . If all employed indi-viduals are symmetric in v and all unemployed individuals are also symmetricin v , then τ ( ω, δ ) = 1 − ω + δω if and only if an unemployed person’s unem-ployment welfare equals an employee’s employment benefits. v ( S ). For example, let the labor cost for i ∈ S be the minimum wagerequirement from all j ∈ N \ S , who either are symmetric with i or uniformlyoutperforms i in v . That is, some j from N \ S can do i ’s work without compro-mising the production v . The minimum wage is called reservation wage , belowwhich j is unwilling to work. At this minimum market replacement cost, S can switch i with someone else from N \ S without sacrificing its net profit.In fact, j does not need to be symmetric with or outperform i as long as theprobability of v ( S ∪ j \ i ) ≥ v ( S ∪ i \ j ) is significantly large and the person hasa willingness to accept (e.g., Horowitz and McConnell 2003). The probabilityof such S could be difficult for job interviewers to approximate and estimate.At the other end, in order to avoid creating an undue disincentive to work, j ’s unemployment welfare must be less than the reservation wage plus em-ployment benefit. Moreover, to ensure every one of N is in the labor market, DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 27 the welfare is necessary to bind to the incentive-compatibility constraint. Thisincentive requirement places a lower bound for the labor cost.
In an abstract sense, the above account is a fair-division solution for thefollowing game-theoretic setting: there are a large number of players; the play-ers are randomly divided into two groups; the payoff comes from one group.A wide range of applications falls into this type of games. In this section, weanalyze three applications other than labor markets to show how to use theformula derived in the last three sections.5.1 Voting PowerIn a voting game (e.g., Shapley 1962), v : 2 N → { , } is a monotonicallyincreasing set function. Let S denote the random subset of voters who vote forthe proposal. The proposal passes when v ( S ) = 1; otherwise, it blocks when v ( S ) = 0. However, v should not mean “production” or alike.No matter the outcome, Hu (2006) describes γ i [ v ] as i ’s probability ofturning a blocked result to a passed one, and λ i [ v ] the probability of turninga passed result to being blocked. Thus, the sum of λ i [ v ] and γ i [ v ] quantifies i ’s power in the game.The ratio δ plays a role in some circumstances. Let us consider, for example,10% of the voters approve just a proposal before a referendum voting on the proposal, and assume the number of other voters’ support ballots follows aBeta-Binomial distribution.Many voting games are symmetric. In these cases, the equality of outcomebecomes an egalitarian allocation of power.5.2 Health InsuranceHealth insurance has two types of policyholders: some are ill and use theinsurance to cover their medical expenses; others are healthy and do not usethe insurance. Let S denote the random set of ill policyholders, v ( S ) be thetotal medical expenses with copays deducted, and ˜ δv ( S ) be the surcharge paidto the insurance company. Let δ = − ˜ δ . Then the total expenses except for thecopays, (1 − δ ) v ( S ), are billed to all insurance policyholders.If τ = 1 − ω + δω and v is symmetric among the two types of policyholders,respectively, then by the equality of outcome, the cost to buy the insurance pol-icy would be (1 − δ ) E [ v ( S )] n per policyholder. We take expectation on (1 − δ ) v ( S ) n because the policyholders pay it upfront. On the contrary, the unemploymentwelfare and employment benefits payments come after the production.In this example, patients pay the predetermined copays. In the labor mar-ket studied before, the labor costs are exempt from the distribution of the netproduction; they, however, have an indirect effect on τ through the corporateearning ratio and δ . In the next example, we use the equality of outcome toderive the same type of payments as copays and labor cost. DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 29 ξ ( n, ω ). Here n isthe number of cars in the segment of the highway, and ω is the percentage ofsolo drivers in the traffic.Let g ( n ) be the average carpool driver’s cost in the traffic when the trafficvolume is n cars. It is likely a nonlinear increasing function of n . An excellentchoice of g ( n ) is the expected driving time in hours multiplied by the averagehourly pay rate, plus expenses on gas and vehicle depreciation. Also, let S de-note the random set of solo drivers. Then, v ( S ) = ng ( n ) − n (1 − ω ) g ( n (1 − ω )) − nωξ ( n, ω ) is the total cost of over-traffic generated by the solo drivers, withtoll fees deducted.The production function v is symmetric among all solo drivers and alsosymmetric among all carpool drivers. By the equality of outcome, each drivershares the same cost v ( S ) n . As a carpool driver pays no toll, his or her sharedcost should exactly offset the extra cost caused by the solo drivers, which is g ( n ) − g ( n (1 − ω )). Finally, equation v ( S ) n = g ( n ) − g ( n (1 − ω )) implies ξ ( n, ω ) = g ( n (1 − ω )) . In this example, an administration surcharge δ may apply; the carpooledpassengers are free-riders, but their costs are exempt from v ( S ). In this paper, a fair-division solution is proposed to allocate the unem-ployment welfare in an economy, where the heterogeneous-agent productionfunction is almost unknown. We interpret “fairness” as equal employmentopportunity and model it by a Beta-Binomial probability distribution. Our“sustainability” is meant to be free of debt and free of surplus in the taxa-tion budget. To justify the value of the unemployed labor, we capitalize onthe D-value concept in Hu (2018). The D-value specifies how much of the netproduction to be retained with the employed labor, and what portion to bedistributed to the unemployed. Finally, we postulate that the labor marketis static and identify a sustainable tax policy by minimizing the asymptoticvariance of the posterior employment rate. The policy can also be uniquely de-termined by minimizing the asymptotic posterior mean of the unemploymentrate, or minimizing the downside risk of the posterior employment rate, orminimizing the posterior mean absolute deviation. Surprisingly, the tax ruleis not only simple enough for practical use but also motivates the unemployedto seek employment and the employed to improve productivity.One could extend this framework in several ways. One way is to re-specifythe probability distribution of equal employment opportunity, for example, byany of the following re-specifications. First, we can replace the two-parameterBeta distribution with a four-parameter one or a Beta rectangular distribution.Secondly, we can let θ and ρ be some functions of other unknown parameters.Thirdly, we could substitute the Beta-Binomial distribution with a Dirichlet- DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 31
Multinomial distribution or a Beta-Geometric distribution. Fourthly, we couldrandomize p without involving ( θ, ρ ), by generating two independent three-parameter Gamma random variables X and Y . Then, the ratio X/ ( X + Y ) isa Beta random variable. In any of the four cases, however, we need additionalidentification restrictions to figure out a τ ( ω, δ ) precisely.From other angles, we could apply other identification schemes or otherobjective functions to find a unique fair tax rule. From a statistical viewpoint,one could try the maximum likelihood estimation using the twelve months’data prior to determining the policy tax rate, or minimize the ex-ante risk of ω , or apply the statistical methods mentioned in Section 3.2. From an economicviewpoint, one could minimize the Gini coefficient of the Beta distribution of˜ p n,ω . From a strategic game-theoretical viewpoint, one could seek a bargainingsolution from the feasible solution set Ω n,ω,δ , which may be particularly usefulwhen n is small. Finally, a policymaker could treat the reserve ratio δ asendogenous, for example, letting it be an increasing function of ω . He or shecould also place a heavier weight on the marginal gain than on the marginalloss, in order to stimulate employment.The simple static model, however, ignores several important aspects ofa real labor market. First, it does not capture the dynamic features of theincome inequality, nor its rational response to the tax rule. Secondly, whilepreventing the fungibility of borrowing funds from the future reduces the riskthat a government administration piles up its national debt, it impairs thatadministration’s ability (especially, monetary policy) to intervene in the econ- omy. The government, however, can still moderately stimulate the economyduring a recession by adjusting the reserve ratio δ . Thirdly, the postulation oflabor market efficiency does neglect the recent development of the incomplete-market theory (e.g., Magill and Quinzii 1996). Fourthly, a multi-criteria ob-jective function may be a viable alternative to the minimum-variance one,especially when there is a high unemployment rate 1 − ω or a large δ . Lastly, asingle tax rule τ ( ω, δ ) could have overly simplified the complexity of the tax-ation system, which is also affected by other determinants. These are just afew challenges our framework introduces, which require further development.In summary, the fair and sustainable tax policy studied here has a solidtheoretical underpinning, together with simplicity in practical use, consistencywith productivity and employment incentives, and robustness to similar ob-jectives. When applying this framework to a real fair-division problem, oneshould also consider the benefits of alternative probability distributions forequal opportunity, alternative objective functions, alternative restrictions, anddynamic thinking. DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 33
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APPENDIX
Our focus in this paper is to study the relationship between the fair taxrate τ and the employment rate ω when the labor market size n is large.To analyze the limit behavior of τ and ω for a large n , we only need therelevant asymptotic approximations. We say two functions f ( n ) = O ( g ( n )) iflim sup n →∞ (cid:12)(cid:12)(cid:12) f ( n ) g ( n ) (cid:12)(cid:12)(cid:12) < ∞ ; and we say f ( n ) ≈ g ( n ) if lim n →∞ f ( n ) g ( n ) = 1. For simplicity, letus denote the following shorthands: ∆ ≡ ω + τ − δω − ,∆ ≡ δω − ω − τ + 2 ≡ − ∆,∆ ≡ δωτ − δω − ωτ + 2 ωτ + τ − ,∆ ≡ (1 − τ )( δ − τ ) = δ − τ − δτ + τ ,∆ ≡ − δωτ + δω + δτ − δ + ωτ − ωτ + ω − τ + 3 τ − ,∆ ≡ − δω + δτ − δ + ω − τ + 4 τ − ≡ ∆ + ∆ ,∆ ≡ − ωδ + ωτ + τ − ≡ ∆ + ω∆ ,∆ ≡ − δ − ωτ + 2 τ + ω − ≡ ∆ + (1 − ω ) ∆ ,∆ ≡ − δω − δ + ω + 3 τ − ≡ ∆ + ∆ ,∆ ≡ − δω − δ + 2 ω + 4 τ − ≡ ∆ + ∆ . The following lemma, to be used in other proofs, re-writes Eq.(8) andEq.(9).
Lemma 1
In terms of ( n, δ, τ, ω ) , we can solve ( θ, ρ ) from Eq.(8) and Eq.(9)as θ = n ω∆ + n∆ + ∆ n∆ + ∆ ,ρ = n (1 − ω ) ∆ + n∆ + ∆ n∆ + ∆ . (A.1) A1. Proof of Theorem 1In this proof, we use the following relation about Beta functions: β ( x − , y + 1) = yx − β ( x, y ) , x > , y > β ( x + 1 , y −
1) = xy − β ( x, y ) , x > , y > . First, the expected aggregate marginal gain and loss are: E (cid:20) P i ∈ S (cid:0) v ( S ) − v ( S \ i ) (cid:1)(cid:21) = P T ⊆ N P ( S = T ) P i ∈ T (cid:2) v ( T ) − v ( T \ i ) (cid:3) = P i ∈ N P T ⊆ N : i ∈ T P ( S = T ) (cid:2) v ( T ) − v ( T \ i ) (cid:3) = P i ∈ N γ i [ v ] , E " P i ∈ N \ S (cid:0) v ( S ∪ i ) − v ( S ) (cid:1) = P T ⊆ N P ( S = T ) P i ∈ N \ T (cid:2) v ( T ∪ i ) − v ( T ) (cid:3) = P i ∈ N P T ⊆ N : i T P ( S = T ) (cid:2) v ( T ∪ i ) − v ( T ) (cid:3) = P i ∈ N λ i [ v ] . Next, by Eq.(3) − (5), we re-write the expected marginal gain and loss as: γ i [ v ] = P T ⊆ N : T i β ( θ + t,ρ + n − t ) β ( θ,ρ ) [ v ( T ) − v ( T \ i )] Z = T \ i = P T ⊆ N : T i β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T ) − P Z ⊆ N \ i β ( θ + | Z | +1 ,ρ + n −| Z |− β ( θ,ρ ) v ( Z ) ,λ i [ v ] = P Z ⊆ N \ i β ( θ + | Z | ,ρ + n −| Z | ) β ( θ,ρ ) [ v ( Z ∪ i ) − v ( Z )] T = Z ∪ i = P T ⊆ N : T i β ( θ + t − ,ρ + n − t +1) β ( θ,ρ ) v ( T ) − P Z ⊆ N \ i β ( θ + | Z | ,ρ + n −| Z | ) β ( θ,ρ ) v ( Z ) . (A.2) DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 37
By Eq.(A.2), the aggregate value of the employed labor P i ∈ N γ i [ v ] is P i ∈ N P T ⊆ N : T i β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T ) − P i ∈ N P Z ⊆ N \ i β ( θ + | Z | +1 ,ρ + n −| Z |− β ( θ,ρ ) v ( Z ) T = Z = P T ⊆ N : T = ∅ P i ∈ T β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T ) − P T ⊆ N : T = N P i ∈ N \ T β ( θ + t +1 ,ρ + n − t − β ( θ,ρ ) v ( T )= P T ⊆ N : T = ∅ tβ ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T ) − P T ⊆ N : T = N ( n − t ) β ( θ + t +1 ,ρ + n − t − β ( θ,ρ ) v ( T )= nβ ( θ + n,ρ ) β ( θ,ρ ) v ( N ) − nβ ( θ +1 ,ρ + n − β ( θ,ρ ) v ( ∅ )+ P T ⊆ N : T = ∅ ,T = N tβ ( θ + t,ρ + n − t ) − ( n − t ) β ( θ + t +1 ,ρ + n − t − β ( θ,ρ ) v ( T )= nβ ( θ + n,ρ ) β ( θ,ρ ) v ( N ) − nθρ + n − β ( θ,ρ + n ) β ( θ,ρ ) v ( ∅ )+ P T ⊆ N : T = N ,T = ∅ h t − ( n − t )( θ + t ) ρ + n − t − i β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T )= nβ ( θ + n,ρ ) β ( θ,ρ ) v ( N ) + P T ⊆ N : T = N t ( θ + ρ − − nθρ + n − t − β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T ) . Also by Eq.(A.2), the aggregate value of the unemployed labor P i ∈ N λ i [ v ] is P i ∈ N P T ⊆ N : T i β ( θ + t − ,ρ + n − t +1) β ( θ,ρ ) v ( T ) − P i ∈ N P Z ⊆ N \ i β ( θ + | Z | ,ρ + n −| Z | ) β ( θ,ρ ) v ( Z ) T = Z = P T ⊆ N : T = ∅ P i ∈ T β ( θ + t − ,ρ + n − t +1) β ( θ,ρ ) v ( T ) − P T ⊆ N : T = N P i ∈ N \ T β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T )= P T ⊆ N : T = ∅ tβ ( θ + t − ,ρ + n − t +1) β ( θ,ρ ) v ( T ) − P T ⊆ N : T = N ( n − t ) β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T )= nβ ( θ + n − ,ρ +1) β ( θ,ρ ) v ( N ) − nβ ( θ,ρ + n ) β ( θ,ρ ) v ( ∅ )+ P T ⊆ N : T = ∅ ,T = N tβ ( θ + t − ,ρ + n − t +1) − ( n − t ) β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T )= nρθ + n − β ( θ + n,ρ ) β ( θ,ρ ) v ( N ) − nβ ( θ,ρ + n ) β ( θ,ρ ) v ( ∅ )+ P T ⊆ N : T = ∅ ,T = N h t ( ρ + n − t ) θ + t − − ( n − t ) i β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T )= P T ⊆ N : T = ∅ t ( θ + ρ − − n ( θ − θ + t − β ( θ + t,ρ + n − t ) β ( θ,ρ ) v ( T ) − nβ ( θ,ρ + n ) β ( θ,ρ ) v ( ∅ ) . A2. Proof of Lemma 1We can re-write Eq.(8) and Eq.(9) as a linear system of unknowns ( θ, ρ ): (1 − τ )( ρ + n − s −
1) = s ( θ + ρ − − nθ, ( τ − δ )( θ + s −
1) = s ( θ + ρ − − n ( θ − . As a consequence, the symbolic solution of ( θ, ρ ) is unique.Let us assume Eq.(A.1) and verify that it satisfies both Eq.(8) and Eq.(9)by the following identities, some of which are used in other proofs: θ + s = n ω∆ + n∆ + ∆ n∆ + ∆ + nω ( n∆ + ∆ ) n∆ + ∆ = n ω + n∆ + ∆ n∆ + ∆ ,θ + s − n ω + n∆ + ∆ n∆ + ∆ − n∆ + ∆ n∆ + ∆ = n ω + n ( ∆ − ∆ ) n∆ + ∆ = n ω + nω ( τ − n∆ + ∆ ,θ + ρ = n ω∆ + n∆ + ∆ n∆ + ∆ + n (1 − ω ) ∆ + n∆ + ∆ n∆ + ∆ = n ∆ + n∆ +2 ∆ n∆ + ∆ ,θ + ρ − n ∆ + n∆ +2 ∆ n∆ + ∆ − n∆ + ∆ n∆ + ∆ = n ∆ + n ( δτ − δ − τ +3 τ − ∆ n∆ + ∆ ,θ + ρ + n = n ∆ + n∆ +2 ∆ n∆ + ∆ + n ( n∆ + ∆ ) n∆ + ∆ = n + n∆ +2 ∆ n∆ + ∆ ,θ + ρ + n + 1 = n + n∆ +2 ∆ n∆ + ∆ + n∆ + ∆ n∆ + ∆ = n + n∆ +3 ∆ n∆ + ∆ ,θ + ρ + n − n + n∆ +2 ∆ n∆ + ∆ − n∆ + ∆ n∆ + ∆ = n + n ( ∆ − ∆ )+ ∆ n∆ + ∆ ,θ + ρ + n − n + n∆ +2 ∆ n∆ + ∆ − n∆ + ∆ ) n∆ + ∆ = n + n ( ∆ − ∆ ) n∆ + ∆ ,ρ + n − s = n (1 − ω ) ∆ + n∆ + ∆ n∆ + ∆ + n (1 − ω )( n∆ + ∆ ) n∆ + ∆ = n (1 − ω )+ n∆ + ∆ n∆ + ∆ ,ρ + n − s − n (1 − ω )+ n∆ + ∆ n∆ + ∆ − n∆ + ∆ n∆ + ∆ = n (1 − ω )+ n (1 − ω )( τ − δ ) n∆ + ∆ . DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 39
Thus, s ( θ + ρ − − nθ = nω [ n ∆ + n ( δτ − δ − τ +3 τ − ∆ ] n∆ + ∆ − n ( n ω∆ + n∆ + ∆ ) n∆ + ∆ = n [ ω ( δτ − δ − τ +3 τ − − ∆ ]+ n ( ω − ∆ n∆ + ∆ = n (1 − ω )(1 − τ )+ n ( ω − ∆ n∆ + ∆ ,s ( θ + ρ − − n ( θ −
1) = [ s ( θ + ρ − − nθ ] + n = n (1 − ω )(1 − τ )+ n ( ω − ∆ n∆ + ∆ + n ( n∆ + ∆ ) n∆ + ∆ = n ω ( τ − δ )+ nω∆ n∆ + ∆ . Therefore, s ( θ + ρ − − nθρ + n − s − = n (1 − ω )(1 − τ )+ n ( ω − ∆ n (1 − ω )+ n (1 − ω )( τ − δ ) = 1 − τ, s ( θ + ρ − − n ( θ − θ + s − = n ω ( τ − δ )+ nω∆ n ω + nω ( τ − = τ − δ, which are equivalent to Eq.(8) and Eq.(9), respectively.A3. Proof of Theorem 2For any integer z ≥
0, by the proof of Lemma 1, θ + s + zθ + ρ + n + z = n ω + n∆ ∆ n∆ + ∆ + z ( n∆ + ∆ n∆ + ∆ n n∆ ∆ n∆ + ∆ + z ( n∆ + ∆ n∆ + ∆ = n ω + n ( ∆ + z∆ )+(1+ z ) ∆ n + n ( ∆ + z∆ )+(2+ z ) ∆ → ω, as n → ∞ . As a function of η , the characteristic function of ˜ p n,ω (e.g., Johnson et al.1995, Chapter 21) is E (cid:2) e iη ˜ p n,ω (cid:3) = 1 + ∞ X k =1 ( iη ) k k ! k − Y z =0 θ + s + zθ + ρ + n + z where i is the unit imaginary number, i.e., i = −
1. We let n → ∞ ,lim n →∞ E [ e iη ˜ p n,ω ] = 1 + ∞ P k =1 ( iη ) k k ! lim n →∞ k − Q z =0 θ + s + zθ + ρ + n + z = 1 + ∞ P k =1 ( iηω ) k k ! = exp( iηω ) . Therefore, as n → ∞ , ˜ p n,ω converges in distribution to the degenerate distri-bution with mass at ω , which has the characteristic function exp( iηω ).A4. Proof of Theorem 3As ˜ p n,ω has a Beta distribution with parameters ( θ + s, ρ + n − s ) (seeSection 2.1), its variance is ( θ + s )( ρ + n − s )( θ + ρ + n ) ( θ + ρ + n +1) (e.g., Gupta and Nadarajah2004, page 35). By the proof of Lemma 1, the variance of √ n ˜ p n,ω is n n ω + n∆ ∆ n∆ + ∆ n − ω )+ n∆ ∆ n∆ + ∆ (cid:16) n n∆ ∆ n∆ + ∆ (cid:17) n n∆ ∆ n∆ + ∆ = n ( n∆ + ∆ )( n ω + n∆ + ∆ ) [ n (1 − ω )+ n∆ + ∆ ]( n + n∆ +2 ∆ ) ( n + n∆ +3 ∆ )= (cid:0) ∆ + ∆ n (cid:1)(cid:2) ω (cid:0) ∆ nω + ∆ n ω (cid:1)(cid:3)h (1 − ω ) (cid:16) ∆ n (1 − ω ) + ∆ n − ω ) (cid:17)i(cid:0) ∆ n + ∆ n (cid:1) (cid:0) ∆ n + ∆ n (cid:1) = ω (1 − ω ) (cid:16) ∆ + ∆ n (cid:17) h ∆ nω + ∆ n (1 − ω ) − ∆ n − ∆ n + O (cid:0) n (cid:1)i = ω (1 − ω ) ∆ + ω (1 − ω ) n h ∆ + ∆ ( ∆ ω + ∆ − ω − ∆ − ∆ ) i + O (cid:0) n (cid:1) → ω (1 − ω )( ω + τ − δω − , as n → ∞ . (A.3)To minimize lim n →∞ VAR( √ n ˜ p n,ω ) = ω (1 − ω ) ∆ while lim n →∞ VAR( √ n ˜ p n,ω ) ≥ ∆ = 0. Thus, argmin τ lim n →∞ VAR ( √ n ˜ p n,ω ) = 1 − ω + δω. DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 41
However, when applying τ = 1 − ω + δω to a labor market with finite n , wehave ∆ = 0, ∆ = 1 and ∆ = − ω (1 − ω )(1 − δ ) . And Lemma 1 reduces to θ = n ω + n∆ − ω (1 − ω )(1 − δ ) − ω (1 − ω )(1 − δ ) ,ρ = n (1 − ω )+ n∆ − ω (1 − ω )(1 − δ ) − ω (1 − ω )(1 − δ ) , which converge to −∞ as n → ∞ . In theory, therefore, for a large but finite n ,we need to choose τ to be 1 − ω + δω plus a small positive number to ensurethat θ > ρ >
0. To estimate the small positive number, let us first trythe higher-order approximation of τ = 1 − ω + δω + cn , for some constant c >
0. Then ∆ = cn ,∆ = (cid:2) ω (1 − δ ) − cn (cid:3) (cid:2) − (1 − ω )(1 − δ ) − cn (cid:3) = − ω (1 − ω )(1 − δ ) + O (cid:0) n (cid:1) . (A.4)By the next to the last step in (A.3),VAR (cid:16) √ n ˜ p n,ω (cid:12)(cid:12)(cid:12) τ = 1 − ω + δω + cn (cid:17) = ω (1 − ω ) c − ω (1 − ω ) (1 − δ ) n + O (cid:18) n (cid:19) . To minimize the above variance, we let c = ω (1 − ω )(1 − δ ) . Thus, τ = 1 − ω + δω + ω (1 − ω )(1 − δ ) n is a higher-order approximation for 1 − ω + δω . Furtherhigh-order approximations, if necessary, could be found similarly.When τ = 1 − ω + δω + cn , let us use Eq.(A.4) to calculate n∆ + ∆ = c + (cid:2) ω (1 − δ ) − cn (cid:3) (cid:2) − (1 − ω )(1 − δ ) − cn (cid:3) = (1 − ω )(1 − δ ) cn + c n (A.5) which is negative when ω > . n ≥ (cid:12)(cid:12)(cid:12) ω (1 − ω )(1 − δ )1 − ω (cid:12)(cid:12)(cid:12) . As the numerators of θ and ρ in Lemma 1 are both positive for a large n , θ and ρ are negative when n is large and ω > . τ = 1 − ω + δω + cn . Similar to Eq.(A.5), n∆ + ∆ = 2 c + (cid:2) ω (1 − δ ) − cn (cid:3) (cid:2) − (1 − ω )(1 − δ ) − cn (cid:3) = c + − ω )(1 − δ ) cn + c n . In this case, both θ and ρ are larger than n when n is large. Note from thelast step of Eq.(A.3) that VAR ( √ n ˜ p n,ω ) is an increasing function of τ when n is large; thus, the small positive number to be added to 1 − ω + δω is between cn and cn . In practice, however, as the number is too small for a large n , thereis no necessity to exactly calculate it and add it to 1 − ω + δω .Let ˜ τ n = argmin τ VAR (cid:0) √ n ˜ p n,ω (cid:12)(cid:12) θ, ρ ≥ n (cid:1) . We add the restriction θ, ρ ≥ n to ensure the existence of ˜ τ n . As VAR ( √ n ˜ p n,ω ) is an increasing function of τ when n is large, ˜ τ n is unique and less than 1 − ω + δω + cn when n is large. ByEq.(A.3),VAR( √ n ˜ p n,ω (cid:12)(cid:12)(cid:12) τ = ˜ τ n ) = ω (1 − ω )( ω + ˜ τ n − δω −
1) + O (cid:18) n (cid:19) . Finally, we use the relation0 ≤ VAR (cid:16) √ n ˜ p n,ω (cid:12)(cid:12)(cid:12) τ = ˜ τ n (cid:17) ≤ VAR (cid:18) √ n ˜ p n,ω (cid:12)(cid:12)(cid:12) τ = 1 − ω + δω + 2 cn (cid:19) to get 0 ≤ ω (1 − ω )( ω + ˜ τ n − δω −
1) + O (cid:18) n (cid:19) ≤ O (cid:18) n (cid:19) . Letting n → ∞ in the above inequality, we get ω (1 − ω )( ω + ˜ τ n − δω −
1) = O ( n ),i.e., lim n →∞ argmin τ VAR (cid:0) √ n ˜ p n,ω (cid:12)(cid:12) θ, ρ ≥ n (cid:1) = 1 − ω + δω . DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 43
A5. Proof of Theorem 4In this proof, we constantly apply the identities in the proof of Lemma 1.Let µ n = θ + sθ + ρ + n be the mean of ˜ p n,ω , and let σ n be the variance of ˜ p n,ω . Thelower semivariance of ˜ p n,ω is calculated as σ n − = Z µ n ( x − µ n ) x θ + s − (1 − x ) ρ + n − s − β ( θ + s, ρ + n − s ) d x. We apply Chebychev’s inequality in terms of the lower semivariance (e.g.,Berck and Hihn 1982) to get P (˜ p n,ω ≤ µ n − a n σ n − ) ≤ a n , ∀ a n > . By the proof of Theorem 3, ˜ p n,ω has the variance σ n = ω (1 − ω ) ∆n + O ( n ). Welet a n = p ω (1 − ω ) ∆n σ n − , then P ˜ p n,ω ≤ µ n − r ω (1 − ω ) ∆n ! ≤ nσ n − ω (1 − ω ) ∆ . (A.6)Let κ n = θ + s − θ + ρ + n − and ε n be the mode and median of ˜ p n,ω , respectively.The mode κ n maximizes the density function x θ + s − (1 − x ) ρ + n − s − β ( θ + s,ρ + n − s ) , < x < | µ n − ε n | ≤ | µ n − κ n | = (cid:12)(cid:12)(cid:12) θ + sθ + ρ + n − θ + s − θ + ρ + n − (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) n + ρ − θ − s ( θ + ρ + n )( θ + ρ + n − (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) θ + s ( θ + ρ + n )( θ + ρ + n − (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) n + ρ − s ( θ + ρ + n )( θ + ρ + n − (cid:12)(cid:12)(cid:12) = O ( n ) . In the following lower-bound estimation of Eq.(A.6), we use Gamma function,denoted by Γ ( · ), and its Stirling’s approximation Γ ( z + 1) ≈ √ πz (cid:16) z exp(1) (cid:17) z . P (cid:18) ˜ p n,ω ≤ µ n − q ω (1 − ω ) ∆n (cid:19) = P (˜ p n,ω ≤ ε n ) − ε n R µ n − p ω (1 − ω ) ∆n x θ + s − (1 − x ) ρ + n − s − β ( θ + s,ρ + n − s ) d x ≥ − (cid:12)(cid:12)(cid:12)(cid:12) ε n − (cid:18) µ n − q ω (1 − ω ) ∆n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) κ nθ + s − (1 − κ n ) ρ + n − s − β ( θ + s,ρ + n − s ) = − (cid:12)(cid:12)(cid:12)(cid:12)q ω (1 − ω ) ∆n + O ( n ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) θ + s − θ + ρ + n − (cid:1) θ + s − (cid:0) − θ + s − θ + ρ + n − (cid:1) ρ + n − s − Γ ( θ + s ) Γ ( ρ + n − s ) Γ ( θ + ρ + n ) = − (cid:12)(cid:12)(cid:12)(cid:12)q ω (1 − ω ) ∆n + O ( n ) (cid:12)(cid:12)(cid:12)(cid:12) ( θ + s − θ + s − Γ ( θ + s ) ( ρ + n − s − ρ + n − s − Γ ( ρ + n − s )( θ + ρ + n − θ + ρ + n − θ + ρ + n − Γ ( θ + ρ + n − ≈ − q ω (1 − ω ) ∆n exp( θ + s − √ π ( θ + s −
1) exp( ρ + n − s − √ π ( ρ + n − s − θ + ρ + n − θ + ρ + n − √ π ( θ + ρ + n − = − q ω (1 − ω ) ∆n ( θ + ρ + n − √ θ + ρ + n − √ π ( θ + s − ρ + n − s − ≈ − q ω (1 − ω ) ∆n n∆ √ n∆ q π nω∆ n (1 − ω ) ∆ = − √ π . Finally, we re-write Eq.(A.6) as (cid:18) − √ π (cid:19) ω (1 − ω ) ∆ + O (cid:18) √ n (cid:19) ≤ nσ n − ≤ nσ n = ω (1 − ω ) ∆ + O (cid:18) n (cid:19) . Letting n → ∞ , we get (cid:18) − √ π (cid:19) ω (1 − ω ) ∆ ≤ lim inf n →∞ nσ n − ≤ lim sup n →∞ nσ n − ≤ ω (1 − ω ) ∆. Therefore, ∆ = 0 minimizes the limit of lower semivariance of √ n ˜ p n,ω .We can apply similar arguments to the upper semivariance of √ n ˜ p n,ω . DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 45
A6. Proof of Theorem 5Note that ∆ − ω∆ = (1 − τ )(2 ω −
1) + ω ( δ − . By the proof of Lemma1, E [˜ p n,ω ] = θ + sθ + ρ + n = n ω + n∆ + ∆ n + n∆ +2 ∆ = ω + n ( ∆ − ω∆ )+(1 − ω ) ∆ n + n∆ +2 ∆ = ω + (1 − τ )(2 ω − ω ( δ − n + O ( n ) . In the above approximation, the mean reacts negatively with an increasing τ , when n is finitely large and ω > .
5. To maximize the mean, we thus minimize τ ∈ (1 − ω + δω ) such that θ > ρ >
0. Particularly, using the proof ofTheorem 3, we can make it smaller than 1 − ω + δω + cn when n is largeenough, i.e.1 − ω + δω < argmax τ MEAN (cid:18) ˜ p n,ω (cid:12)(cid:12) θ, ρ ≥ n (cid:19) ≤ − ω + δω + 2 cn . Finally, let n → ∞ in the above inequalities to getlim n →∞ argmax τ MEAN (cid:18) ˜ p n,ω (cid:12)(cid:12) θ, ρ ≥ n (cid:19) = 1 − ω + δω. for any ω ∈ ( . , n is large, θ, ρ ≥ n implies τ ( ω, δ ) ∈ (1 − ω + δω, θ + s → ∞ and ρ + n − s → ∞ as n → ∞ . Applying Stirling’sformula, Johnson et al. (1995, page 219) derive the following approximationfor the ratio of the variance and the squared MAD around the mean:lim θ + s →∞ ,ρ + n − s →∞ ( E [ | ˜ p n,ω − E (˜ p n,ω ) | ]) VAR(˜ p n,ω ) = 2 π . Thus, minimizing the MAD around the mean is equivalent to minimizing thevariance of ˜ p n,ω when n is large. By Theorem 3, we have proved Theorem 6.A8. Proof of Theorem 7For any i, j ∈ N and i = j , if i uniformly outperforms j and they have equalemployment opportunity, then γ j [ v ] = P T ⊆ N : j ∈ T P ( S = T )[ v ( T ) − v ( T \ j )]= P T ⊆ N : j ∈ T,i ∈ T P ( S = T )[ v ( T ) − v ( T \ j )]+ P T ⊆ N : j ∈ T,i T P ( S = T )[ v ( T ) − v ( T \ j )] ≤ P T ⊆ N : j ∈ T,i ∈ T P ( S = T )[ v ( T ) − v ( T \ i )]+ P T ⊆ N : j ∈ T,i T P ( S = T )[ v ( T ) − v ( T \ j )] Z = T \ j = P T ⊆ N : j ∈ T,i ∈ T P ( S = T )[ v ( T ) − v ( T \ i )]+ P Z ⊆ N : j Z,i Z P ( S = Z ∪ j )[ v ( Z ∪ j ) − v ( Z )] ≤ P T ⊆ N : j ∈ T,i ∈ T P ( S = T )[ v ( T ) − v ( T \ i )]+ P Z ⊆ N : j Z,i Z P ( S = Z ∪ i )[ v ( Z ∪ i ) − v ( Z )] T = Z ∪ i = P T ⊆ N : j ∈ T,i ∈ T P ( S = T )[ v ( T ) − v ( T \ i )]+ P T ⊆ N : j T,i ∈ T P ( S = T )[ v ( T ) − v ( T \ i )]= P T ⊆ N : i ∈ T P ( S = T )[ v ( T ) − v ( T \ i )]= γ i [ v ] . Neither the Beta-Binomial distribution nor Eq.(3) is required in the proof.Also, the equality of employment opportunity is not required for other players
DICHOTOMOUS ANALYSIS OF UNEMPLOYMENT WELFARE 47 in N , except that P ( S = Z ∪ i ) = P ( S = Z ∪ j ) for any Z ⊆ N \ i \ j . This identityimplies that i and j have equal chance to be hired by Z , when both areunemployed.Similar arguments can be used to prove λ j [ v ] ≤ λ i [ v ]. In this case, we usethe identity P ( S = Z \ i ) = P ( S = Z \ j ) for any Z ⊆ N such that i, j ∈ Z . Thisidentity implies both i and j have equal opportunity to be laid off from Z ,when both are employed in Z .A9. Proof of Theorem 8We distribute v ( S ) to N : (1 − τ ) v ( S ) to S and ( τ − δ ) v ( S ) to N \ S . When theemployed individuals are symmetric in v , Theorem 7 claims that each employedperson receives (1 − τ ) v ( S ) | S | as his or her employment benefits. Similarly, anyunemployed person receives ( τ − δ ) v ( S ) n −| S | as his or her unemployment welfare.Finally, (1 − τ ) v ( S ) | S | = ( τ − δ ) v ( S ) n − | S | is equivalent to (1 − τ ) ω = ( τ − δ )1 − ω , which itself is equivalent to τ = 1 − ω ++