Universal Basic Income: Some Theoretical Aspects

In this paper, we review possible theoretical justifications of a universal basic income (UBI) scheme and also examine the determinants of its feasibility and scope. We begin by contrasting the unconditionality of UBI with the many conditions that typically accompany other welfare policies. Possible justifications for an unconditional UBI range from pure normative reasons to practical reasons due to the problem of screening beneficiaries and imperfections in institutions in charge of implementing tax and welfare policies. We also explore theoretically the conditions that determine the feasibility and size of a UBI. The broad picture that emerges from our review is that both normative and practical considerations make UBI easier to defend as a tool of poverty alleviation in poor economies than a tool to achieve social justice in rich ones.


Introduction
A Universal Basic Income (UBI) is a universal and unconditional stream of cash income paid by the government to every full member of society-it is paid irrespective of whether an individual is working, of his or her existing income, and whoever he or she lives with (Van Parijs, 1995). 1,2 The concept of UBI has a distinguished intellectual tradition starting from radical thinkers, liberals, and utopian socialists in the eighteenth and nineteenth century, such as Thomas Paine, Thomas Spence, Charles Fourier, Joseph Charlier, and John Stuart Mill. 3 It is an idea that appears under various labels such as "social dividend", "citizen's income", "demogrant", and "basic income". It has drawn support from both the left and the right end of the political spectrum -for example, it has been discussed by left-of-centre economists like James Tobin and James Meade, and it also seems to have inspired the negative income tax proposal of Milton Friedman. In recent years the idea of UBI has gained traction again in the debate on reforming the welfare state in major market economies on both sides of the Atlantic and has received support from mainstream politicians in the British Labour Party, the US Democratic Party, the French Socialist Party, and the Green Party in several European countries, the UK, Canada, and Australia. It was put up for a referendum (and was defeated) in Switzerland in 2016, has been tried out on an experimental basis with mixed initial reports in various European countries like Finland and the Netherlands, and the Canadian province of Ontario. The idea of UBI also resonates with a recent policy shift in many developing countries as well as international development assistance organizations like the World Bank and the DFID towards direct cash transfers which involves rolling all subsidies into a single lump-sum cash transfer to households. 4 This paper has two objectives -to discuss the desirability and feasibility of UBI from the point of view of economic theory.
First, we explore the desirability of UBI by examining normative justi…cations of it in di¤erent economic environments. As we are also interested in understanding the case for UBI as a policy tool in both developed and developing countries, it is useful to clarify how we di¤erentiate between their economic environments.
The key di¤erences we focus on are the size of the population living close to the margin of subsistence, the degree of formality of certain markets (such as labor markets) that determines the extent to which income levels and hours worked are observable to the policymaker, and state capacity in terms of implementing tax and welfare policies.
Second, a UBI proposal typically (but often implicitly) assumes a change in the tax system or in overall government spending for it to be a budget-balanced or revenue-neutral proposal. We analyze the feasibility of a UBI scheme assuming it is funded by a linear income tax, taking into account the behavioural response of such a scheme on labor supply.
We restrict our attention to studying UBI purely as a mechanism for redistribution (or transfer), as opposed to a substitute to policies to address speci…c market failures, such as micro…nance in the context of credit markets or unemployment insurance in the context of labor markets, or of policies to fund the provision of public goods and services. 5 The natural comparison is with other transfer policies that are not in the form of cash or involve targeting to speci…c groups or conditional on some form of compliance criteria being met.
Among the several conditions that characterize a UBI compared to other redistribution policies, the absence of targeting or means-testing is the one that we concentrate on the most. Any universal scheme is expensive, and a natural question to ask is why should those who do not earn any labor income receive the same net transfer independently of their ability to earn income, of their own non-labor income, and of the income of those who live with them? How should the tax burden of UBI be spread among the net contributors to the redistribution system? 6 To answer these questions, we start with the observation that an individual's total income is the sum of labor and non-labor income, and labor income in turn is equal to an individual's labor time times her wage. Income inequality can then come from di¤erences in labor time or di¤erences in wages or di¤erences in non-labor income.
We distinguish between three types of arguments, which we label as …rst-best, second-best, and third-best arguments.
In a …rst-best world, the benevolent policymaker is assumed to have full information about the characteristics of individuals and can taylor tax and transfer schemes according to them. We …nd that while it is possible to justify UBI on normative grounds even in a …rst-best world, it is much easier to defend it based on considerations of poverty alleviation than on the notions of fairness that are at the heart of optimal taxation. For example, transferring the same amount to all idle individuals independently of why they do not work -whether it is due to low wages, or low willingness to work -requires the optimal allocation to be one where there is no redistribution from high-wage to low-wage earners, combined with a large redistribution from high to low-willingness-to-work individuals, which is hard to justify on normative grounds. 7 In a second-best world, the characteristics of agents are no longer assumed to be observable to the policymaker. Taxation is assumed to be based on pretax incomes only. In this case it is much easier to justify UBI based on some egalitarian objective than in a …rst-best world because information asymmetries bene…t individuals who have high wages but low willingness to work. However, if there are formal labor markets (as is the case in developed countries) then a screening device exists that can be used by the benevolent policymaker to identify, among these who do not work, these who have the ability to earn income on their own. This consists of monitoring of individuals bene…ting from unemployment or social assistance, with …rms asked to reveal whether they would be ready to hire them or not. It is interesting to note that all existing unemployment insurance or social assistance programs in developed countries are in fact based on such devices, something that has not been su¢ ciently studied in the theoretical literature. In the presence of such a device, a UBI can still be justi…ed, including on the grounds of poverty alleviation. However, some quali…cations to this should be added when the maximum feasible level of UBI from the …scal point of view is not su¢ cient to guarantee a basic income equal to the poverty line for all. In this case, the goal of poverty alleviation may imply not treating all those who do not work equally generously but to provide some incentives to work so that their overall after-tax income is above the poverty line. In this case, the arguments in favor of UBI also justify complementing it with means-tested social assistance programs targeted towards low-wage individuals. 8 In a third-best world, we introduce imperfections in the tax and welfare system into the picture. By imperfections, we mean that labor income may be imperfectly observed, and conditional social assistance may require the involvement of ine¢ cient or corrupt local agencies. In such a world, we argue that UBI may be a way to circumvent these imperfections under the same egalitarian objective as these who do not necessarily lead to UBI absent these imperfections.
In our classi…cation of the …rst, second, and third-best environments, we do not directly deal with frictions in speci…c markets, such as labor, credit, and insurance, other than the reference to the formality of labor markets in developed countries mentioned above. In the presence of market failures, cash transfers can have e¢ ciency-enhancing e¤ects by relaxing borrowing constraints or allowing individuals to smooth consumption or income. A satisfactory treatment of this angle is beyond the scope of this paper and besides, Banerjee et al (2019) deal with it fairly comprehensively in the context of developing countries in this symposium. 9 Next we turn to examining the feasibility of a UBI scheme funded by a linear tax (as in the formulation of Atkinson, 1995) across environments where the fraction of the population that is very poor, average income levels, the degree of inequality as well as the e¤ectiveness of the tax and bene…ts system varies. Taking into account the labor supply responses to taxation of working individuals to fund a UBI, we show that the case for a UBI even from the point of view of feasibility may be stronger for poorer countries.
The plan of the paper is as follows. In Section 2, we discuss in detail various aspects of UBI proposals and identify the ones we will focus on. In Section 3, we begin our review of the theoretical arguments for and against UBI by comparing cash and in-kind bene…t systems. In Sections 4, 5 and 6, we review the possible justi…cations of non-means tested transfers in …rst-best (full information), secondbest (asymmetric information about bene…ciaries), and third-best environments (imperfect enforcement of tax and bene…ts systems). In Section 7 we carry out a positive analysis of a UBI scheme that is funded by a linear tax in terms of what it implies in terms of its e¤ects on labor supply and the determinants of budgetary 8 By low-wage individuals we refer to those that would be unable to earn an after-tax income above the poverty line even if they work full time. In our formal analysis we take them to be zero-wage individuals for simplicity. 9 See also Baird, McKenzie, and Özler (2018) and Ghatak (2015) for discussions of some of these mechanisms related to cash transfers in the context of developing countries. feasibility. We conclude in Section 8.

Characterization of a UBI
UBI is a redistribution scheme that has three components. First, it is a cash transfer as opposed to in-kind transfer like food, housing, or fuel; second, it is universal, i.e., it is not targeted to any speci…c group based on socioeconomic or demographic criteria and third, it is unconditional and not contingent on the recipient satisfying any compliance criteria.
Unconditionality of UBI may be de…ned in reference to the various conditions that exist in current social bene…t systems: First, it is not conditioned on means. We can distinguish three types of means unconditionality of basic income: a) it does not depend on the bene…ciary's ability to earn labor income, whereas a large spectrum of current social bene…ts are restricted to involuntary unemployed (those whose e¤ective temporary wage rate is zero), b) it does not depend on the bene…ciary's non-labor income, in particular, capital income, c) it does not depend on the income of the people living with the bene…ciary, whereas current bene…ts are often conditional on the income of the spouse or the parents; Second, it is not conditioned on any demographic criteria like age, gender, marriage or family status, and family composition; Third, it is not conditioned on special needs, using criteria like health, handicap, etc; Fourth, it is not conditioned on whether the bene…ciary is deserving or worthy which goes against conditional transfer schemes that are contingent on parents sending their kids to school, on the bene…ciary's criminal records, etc.
It is true that many UBI proposals in practice allow some demographic criteria like age -for example, they are typically aimed at people in the working age group. 10 Also, they are typically not expected to replace bene…ts based on special needs, such as health or handicap, which makes the overall bene…ts contingent on some aspect of needs. Given this, the absence of conditionality e¤ectively relates to absence of means-testing, whether the recipient deserves it, and considerations of family composition (e.g., married or single, having children).
From the economic point of view, the absence of means-testing is the most controversial aspect of UBI. Accordingly, we will spend a large part of this review discussing the possible justi…cations of non-means tested transfers. Before that, we brie ‡y review the debate between transfers in cash versus in-kind in the next section.

Cash versus in-kind transfers
Suppose individuals care about c (essential consumption) and x (inessential consumption) and their preferences are represented by the utility function u (c; x). 11 Suppose the budget constraint is c + px y where y is income, and p is the price of the inessential consumption good (the price of essential consumption is normalized to 1). Let us denote the individual demand functions for c and x as c (y; p) and x (y; p). In this world, if a cash transfer of amount b is to be given to an individual, then the new budget constraint is c + px y + b, where y now stands for pre-transfer income. Assuming both essential and inessential consumption to be normal goods, we would expect c (y + b; p) > c (y; p) and x (y + b; p) > x (y; p). In Fig. (1), we depict the two budget lines with and without the transfer, and the corresponding choices of the consumer by the points E 0 and E 0 0 . Denoting the indirect utility function by v(y; p), it follows directly that v (y + b; p) exceeds the utility that the individual can obtain from any other form of transfers since a lump-sum transfer of b makes the choice problem unconstrained.
Consider now an alternative transfer of value b that is either in-kind, in the form of a coupon or voucher, or electronic bene…t transfer (EBT) cards. For example, in India, fair price shops (FPS), also known as 'ration shops", distribute food and essential items at a subsidized price to the poor. In contrast in the US the Food Stamp programme, or its current version called the Supplemental Nutrition Assistance Program (SNAP), bene…ts are directly deposited into the household's EBT card account, which can be used to pay for food at various retail outlets. 12 The new budget line can be written as max fc b; 0g + px y. For c b , the budget line shifts parallelly as in the case of cash transfers. However, under this scheme if the bene…ciary wants c < b, then the amount b c cannot be converted to x, something that is possible under cash transfers, and the upper bound for x is set by the pre-transfer income y. 13 In Fig. (1), the budget line under this scheme is the same as the one for the unconditional cash transfer of b for c b 11 We interpret essential goods as those that are indispensable, like water or food, in the sense that zero consumption of them makes the marginal utility of all other goods zero (see Matsuyama and Ushchev, 2017 for a formal treatment). This is di¤erent from the distinction between necessities and luxuries that has to do with the income elasticities being smaller or greater than one. It is related but di¤erent from the distinction between merit vs demerit goods that re ‡ects the notion of the policymaker as to what kind of goods are potentially deserving of societal support. These could re ‡ect need as is the case with essential goods, but it can also re ‡ect positive externalities or behavioral biases associated with them that could lead to underconsumption. 12 In fact, India has experimentally introduced a Direct Bene…t Transfer (DBT) scheme in a limited number of areas starting in 2013, that transfers subsidies directly to the bank account of bene…ciaries, with the hope that this will reduce leakages and delays associated with the existing public distribution system. Some of the initial evaluations present a mixed picture. For example, Muralidharan et al (2017) conclude that DBT-based reform holds long-term promise, and that over time bene…ciaries prefer DBT to in-kind transfer via the ration shops. However, in the short-run there is not enough evidence to support a universal expansion of DBT, and responses from bene…ciaries do not unambiguously establish a revealed preferences for cash transfers. 13 See Hoynes and Schanzenbach (2009) for a detailed discussion. and for c < b it is a vertical line with the intercept y p on the horizontal axis. Letĉ(y + b; p) andx (y + b; p) denote the choices of c and x under the inkind scheme. Given the nature of the change in the budget constraint, as long as c(y +b; p) b;ĉ(y +b; p) = c(y +b; p) andx(y +b; p) = x(y +b; p): the condition on which good to buy has no bite and the choice of the consumer is depicted by the point E 0 0 . Ifĉ(y + b; p) = b (we assume free disposal post receiving the transfer) thenx(y + b; p) = y p . In this case, the outcome would be di¤erent from the previous case. We have highlighted this with the points E 1 and E 0 1 -clearly, the consumer would have chosen a di¤erent point with an unconditional cash transfer, one where she would consume c < b and x > y p but is constrained to choose the point E 0 1 which involveŝ c(y + b; p) = b andx(y + b; p) = y p . But even in this case the consumption of x is higher than in the pre-transfer situation, as x(y; p) < y p . What this means is, either way, there is greater consumption of x compared to the pre-transfer period. So contrary to what one might think, even with a restricted form of transfer intended to increase essential consumption, there will be an increase in inessential consumption due to an income e¤ect. The extent of the increase in inessential consumption, of course, would be less than the case of an unconditional cash transfer. This simple analysis therefore suggests that only for bene…ciaries who are marginal the impact of cash and in-kind transfers are di¤erent -for inframarginal bene…ciaries, there is no di¤erence. Evidence in fact seems to suggest most recipients happen to be inframarginal (Hoynes and Schanzenbach, 2009).
From the welfare point of view, clearly the individual is better o¤ with a ‡at cash transfer of b and any transactions costs involved in making an in-kind transfer of the amount b will make the logic stronger from the social e¢ ciency point of view. Only when the in-kind transfer is completely fungible the two forms of transfers will be equivalent.
Therefore, we conclude from this discussion that the case for in-kind transfers has to rest on departures from this framework.
First, in-kind transfers can be used for self-targeting, if these bene…ts are targeted to the poor as opposed to being universal, if the goods involved are necessities of very basic quality. 14 Second, if the benevolent policymaker has paternalistic preferences that are di¤erent from that of the bene…ciary, then in-kind transfers may be preferred to cash transfers. This would depend on to what extent there are positive externalities or the bene…ciaries are subject to behavioral biases that would lead them to underconsume merit goods. It would also depend on to what extent intrahousehold resource allocation considerations are relevant (e.g., gender discrimination, insu¢ cient altruism towards children) so that we cannot automatically assume that the head of the household will necessarily spend the cash income in a way that is most bene…cial for the whole family.
Third, if there are transactions costs in accessing markets, in-kind transfers are e¤ectively larger in real terms. Clearly in remote rural areas of developing countries having cash is less useful than in more urban areas. 15 Fourth, there is also a political economy argument for in-kind transfers: if voters dislike the fact that their taxes are used by the bene…ciaries to buy inessential (or more broadly, demerit) goods, then a restriction to essential or merit goods makes sense because that makes voters more willing to pay and it does not e¤ectively decrease the utility of the bene…ciaries so long as they are inframarginal.
In the end, it is an empirical matter as to how cash transfers tend to get spent by bene…ciaries. Evidence from developing countries actually suggest that, on average, cash transfers to the poor do not cause them to spend their money on inessential consumption (see, Evans and Popova, 2017).
Before we end this section, we would like to mention a possible reinterpretation of the model of this section as a model of labor supply, which is the one we will focus on for the rest of the paper. We can think of y, as an individual's full income, i.e., the money value of her available time evaluated at her wage, together with her non-labor income. Then the allocation of y into c and x can be thought of as the allocation of full income into consumption, c, and leisure, x, the price of which 14 In section 5, when we discuss workfare, the issue of self-targeting in a second-best environment comes up again. 15 In a study of a bicycle scheme for school children in Bihar (Ghatak, Kumar, and Mitra, 2016), it was found that those living in remote areas do in fact prefer in-kind to cash transfers, with the opposite holding for those who live in urban areas. Muralidharan et al (2017) in their study of cash transfers replacing food-ration entitlements for the poor, the Direct Bene…t Transfer (DBT) programme, found that costs varied across bene…ciaries depending on access to banking. is the wage. Under this reinterpretation, a paternalistic policymaker would be one who thinks that people are likely to consume more leisure than what they should. An in-kind policy would then be to impose a minimal amount of work requirement (or, job-seeking e¤ort) on an individual. We do in fact observe policies that try to incentivize people to work or look for jobs in developed countries, and also workfare programs in developing countries, such as the employment guarantee scheme in India (see Ravallion, 2018). 16

In a …rst-best world
Under what circumstances would a benevolent policymaker wish to transfer the same amount of money to di¤erent types of individuals? We begin exploring this question in a …rst-best world. By …rst-best, we mean that the policymaker has full information about all the relevant characteristics of individuals and can tailor taxes and transfers to these, and there are no direct costs or frictions associated with the implementation of these policies. 17 Given that most arguments in favor of UBI refer to informational asymmetries or incentive issues or administrative costs, the common intuition is certainly that UBI is an implausible …rst-best policy. We demonstrate this point explicitly in this section, which acts as a benchmark and a point of departure for our analysis of the desirability of UBI in more realistic second and third-best worlds.
We assume that the benevolent policymaker is interested in maximizing a social welfare function that is consistent with Pareto e¢ ciency, which is a natural starting point. In a …rst-best world, all Pareto e¢ cient allocations can be obtained by …rst designing a lump-sum tax and transfer system, and then letting individuals choose their labor supply given that labor is rewarded at their actual productivity. 18,19 Choosing the optimal allocation then boils down to choosing the optimal list of lump-sum taxes and transfers.
We begin by distinguishing two de…nitions of basic income that come up in debates on UBI. Sometimes basic income is de…ned as the transfer that an individual would get on top of all her other incomes, with the amount being independent of these other incomes. It is universal when all individuals in the economy receive 16 Arguments to justify such policies are sometimes grounded on the assumption of behavioral biases, such as present bias that may induce people to overestimate the cost of looking for jobs. The consequence of this kind of bias is studied, for instance, by Lockwood (2016). 17 As mentioned in the introduction, we do not deal with market imperfections except in a very limited way that has to do with formality of labor markets. This in turn is largely related to the ability of the policymaker to use certain screening mechanisms in the design of the tax-transfer policies. 18 Taxes and transfers are lump-sum when they do not depend on the behaviour of individuals. 19 Technically speaking, we are in a setting in which the second fundamental welfare theorem holds: each Pareto e¢ cient allocation can be decentralized by an appropriate design of lumpsum taxes and transfers. Note that the ability of the policymaker to observe the types of the individuals and make the lump-sum transfers depend on these types implies that observing labor times becomes redundant: optimizing individuals freely choose the labor time that the policymaker would assign to them. the same amount of it. This is the notion of UBI that lies behind some policy experiments, such as the ones conducted in Namibia and India, funded by the German United Evangelical Mission and the UNICEF respectively. 20 This will be the topic of Section 4.1.
In general, though, basic income typically refers to the income of individuals who do not work, and it is an income guarantee in the sense that individuals would automatically get this income in case they stop working. 21 It is funded by taxes paid by working individuals. It is universal when all individuals who do not work receive the same amount of it, independently of whether they do not work because of their low wage or their low willingness to work. This will be the topic of Section 4.2.
We will study two possible justi…cations of such a UBI. First, we will explore whether there exist good normative reasons to treat all individuals who do not work identically, whether it is due to low productivity re ‡ected in low wages or low willingness to work (Section 4.2.1). Second, we will explore the conditions under which the more direct objective of poverty alleviation leads to granting the same income to all idle individuals (Section 4.2.2).
We adopt a simple framework where individuals have Cobb-Douglas preferences over net or after-tax income and leisure time. All through we stick to a static framework of income-leisure choice, and so after-tax income equals consumption. The preferences of an individual are represented by a utility function: where`stands for her leisure time and y for her disposable income. We assume there is an upper bound on`, which we denote by T , the amount of available time, assumed to be equal among individuals. The di¤erence between available time and leisure is the labor supply. Preferences are heterogeneous among individuals, and this is captured by , the weight that income receives relative to leisure in the utility function. A higher means a greater willingness to work. Given this simple preference shifter, we denote utility as u(y;`; ). We assume that takes values in an interval [ ; ].
Each individual has an ability to earn income, or wage, w > 0, which may di¤er across individuals. We assume all through that w is exogenous for simplicity. Finally, we assume that individuals have some exogenous amount of non-labor income, m 0. As a result, the set of individuals in the economy is fully described by the distribution of the parameters , w; and m re ‡ecting the preferences, the ability to earn income, and non-labor income. This distribution is represented by a density function f ( ; w; m).
When we consider the di¤erence between developed and developing countries in this section that will be in terms of only one factor -a subsistence consumption 20 See Van Paris and Vandenborght (2017) for detailed descriptions of such experiments and related references. 21 This, for example, is the formulation of Tobin et al (1967), Friedman (1968), and Atkinson (1995). level. For a developing country we will modify the above set up by introducing an additional constraint that y y where y > 0 is the subsistence consumption level. This can be incorporated by a slight modi…cation of the above utility function to: u (y;`) = log y y + (1 ) log`for y > y = V , otherwise where V is some large positive number. In a …rst-best world, in which the policymaker is assumed to observe all parameters of the economy, whether individuals earn income in the formal or the informal market does not matter, a further distinction between developed and developing countries that we will return to in the two subsequent sections.
As mentioned above, all Pareto e¢ cient allocations can be obtained through lump-sum transfers based on characteristics of individuals. Let t( ; w; m) 2 R denote the tax paid by an individual of type ( ; w; m), which can be positive or negative. The t's have to satisfy the government budget constraint: where B stands for the net expenditure of the government. We can restrict our attention to situations in which an individual of type ( ; w; m) chooses the bundle of goods (consumption and leisure time) that she prefers among all bundles she can reach given her tax t and the reward to her labor, w, to solve: This leads to the optimal consumption and leisure time y ( ; w; m) = (wT + m t( ; w; m)) (4) This yields the following indirect utility function: Otherwise we get a corner solution: with the corresponding indirect utility function v(w; t; m; ) = log( t + m) + (1 ) log T: This gives us the following expression for labor supply: As this expression makes it clear, the set of individuals who do not work (L ( ; w; m) = 0) is composed of individuals of several types. Some may have a very low wage w, others a very low willingness to work , and yet others a very high non-labor income m. As we will see below, what is an ethically justi…ed way to treat of these di¤erent types of individuals is a central question for UBI.
If we consider a developing economy, an additional case can arise, the case in which the subsistence constraint is binding. In this case, the optimality conditions are and the indirect utility function is given by v(w; t; m; ) = log y + (1 ) log T y m + t( ; w; m) w : In this …rst-best world, the policymaker has the freedom to treat di¤erently income inequalities that come from di¤erences in labor time from these that arise from di¤erences in wages and these that arise from di¤erences in m. As a result, it is not necessarily optimal for the policymaker to allocate transfers t( ; w; m) that monotonically decrease with labor income wL ( ; w; m): the policymaker may well want to distinguish between those who work a lot for a low wage from those who work less with a higher wage, even if they end up with the same labor income.
Inequalities may also come from di¤erences in m. If the benevolent policymaker considers inequalities in m normatively undesirable, then she should tax m at 100% and redistribute the proceeds. If the policymaker does not consider these inequalities normatively problematic, then she can simply disregard them. In both cases, there is no loss of generality in developing our analysis under the assumption that m = 0 for all individuals and focus on heterogeneity in terms of and w. This is what we do for the remaining of this section. We will bring inequalities in m back in the analysis in the next section.
The two de…nitions of basic income that we introduced at the beginning of Section 4 can be translated into our model as, …rst, a transfer amount of t, and, second, a guarantee on y. We study them in turn in the following subsections.

A lump-sum transfer to all
If basic income is interpreted as a lump-sum transfer t received by individuals, then it is universal if it does not depend on or w, i.e., if t( ; w) = t for all and all w.
Clearly, under this formulation the basic income can be positive only if the government has a revenue to match the spending on basic income from other sources than taxes. An example is the well-known case of the Alaska Permanent Fund, in which a share of the pro…t of the state-owned oil industry is evenly redistributed among the citizens of the state. Such a basic income is non distortionary, i.e., it does not impose any ine¢ ciency on the economy.
If the public sector of the economy does not have any pro…t to redistribute, or, as it is typically the case, if running the government requires some resources, then a basic income in this sense is simply not feasible given the budget constraint (2). As a result, this way of de…ning basic income is not very promising from the point of view of justifying UBI, and hence the Namibian and Indian experiments we mentioned earlier are of limited interest.

A lower bound on consumption
Let us now de…ne basic income as an income guarantee. Here we need to distinguish between developed and developing countries. We begin with the former, where those who have the lowest income (or consumption, as income and consumption are equivalent in our static model) are to be found among those who do not work. Making a basic income universal, that is guaranteeing a minimal y independently of individual type ( ; w) raises several normative questions. Observe that individuals who do not work consume the lump-sum transfer that they get, and nothing more. Let b denote this universal minimal consumption level, i.e., b = t( ; w) for all the individuals ( ; w) who do not work, i.e., choose`= T . From our earlier analysis, it means that all individuals such that do not work at an e¢ cient allocation in which they receive a basic income of b. Remember that the group of these idle individuals includes low-wage-highwillingness-to-work individuals and high-wage-low-willingness-to-work individuals. The key normative question is, is it possible to de…ne social welfare in a way that suggests treating all individuals who do not work, i.e., whose types satisfy (12), identically? There are two ways of answering yes to this question, which we review in the next two subsections. The …rst way is based on normative principles of fairness consistent with interpersonal welfare comparison among individuals with heterogenous preferences. The second way is consistent with the goal of poverty alleviation.

Interpersonal welfare comparison and heterogeneity of preferences
The feature of UBI that is the most di¢ cult to justify is that it treats those who do not work identically, whether they don't work because they are not able to earn income or because they would be able to work but they prefer not to. As Van Parijs and Vanderborght (2017) put it: "Of all objections to a basic income, one sticks out above all othersand is more emotional, more principled, and more decisive in the eyes of many. It relates to its being unconditional in the sense of being obligation-free, of not requiring its recipients to work or be willing to work. To deal with this question, we need to develop a notion of social welfare in a world characterized by heterogeneity in productivity and willingness to work. For this, we will use the double heterogeneity (in and in w) introduced in our basic model at the beginning of this section. De…ning social welfare in such a context happens to have been at the center of optimal taxation theory in the recent years. 22 The brief discussion that follows builds on these developments.
Assume, to simplify, that we have two types of preferences, parameterized by and , with < , and several types of wages: w 2 f0; : : : ; wg. The tax function can be decomposed as follows: such that: t e (w; ) is the amount of redistribution within sets of individuals having the same preferences; in the optimal taxation literature following Mirrlees's (1971) seminal contribution, individuals are assumed to have the same preferences; income redistribution then only aims at decreasing income inequality arising from di¤erences in wages; the social welfare function used by Mirrlees was de…ned in terms of individual utility and not incomes, so that we can treat the choice of t e (w; ) to pursue the objective of equalizing utilities among individuals with the same preferences; 23 22 Starting with the important contribution of Boadway et al (2002), a general answer to this question has been recently provided by Fleurbaey and Maniquet (2011) in the framework of the theory of fair allocation. See also Fleurbaey and Maniquet (2018) for a survey of the contribution of the fairness approach to optimal taxation. 23 Mirrlees did actually de…ne social welfare as the sum of utilities, leaving the equalization objective to be contained in the concavity of the utility function. We do not discuss the consequence of this de…nition here and consider the simpler social objective of decreasing utility inequality instead. t( ) is the correction factor to the above component that takes heterogeneity in preferences into account, i.e., t( ) is the amount paid by all individuals of preferences , and it determines the amount of redistribution from high to low willingness-to-work individuals; t(w) is the correction factor based on wages, that is t(w) is the amount paid by all individuals of wage w, and it determines the amount of redistribution from high to low wage individuals, after utility equalization within classes of individuals with the same preferences.
We start with the following properties of t e (w; ). Given that we are in a …rst-best world, we can assume that full equality is achieved among individuals having the same preferences. 24 To simplify the exposition, let us assume that, for each preference parameter , we have only two individuals, one with a low and one with a high wage. Let us further assume that, within each preference groups, the low wage is so low that the individual remains idle, by e¢ ciency, which means that (12) holds: the marginal rate of substitution of leisure to income exceeds wage at zero labor supply.
Let us look at both 's in turn and let us concentrate on redistribution among the corresponding two individuals. Let t be the tax paid by the working individual, and let b be the transfer received by the idle individual. To begin with, we ignore budget balance issues, and assume that b is calibrated in such a way that both individuals reach the same utility level. This means that b depends on t according to the following equation, in which the left-hand side gives us the (indirect) utility of the working individual, maximizing her preferences given her wage w and tax t, and the right-hand side gives us the utility of the idle individual, consuming bundle (b; T ): This can be rewritten as: It is clear that b is a strictly decreasing function of t: if the working individual is asked to pay a larger tax, then a lower transfer is needed to keep the idle individual at the same level of utility.
Let us now bring budget balance back in the picture. If t = 0, then a positive b is needed, but that leads to a de…cit. If t = wT , then b = 0 is su¢ cient to equalize utilities, but the budget runs a surplus. That means that a unique …xed point exists to (13) that balances the budget. At this point, b = t. 25 The key point in this reasoning is that b is increasing in , i.e., the transfer obtained by the idle individual at the utility equalizing allocation is increasing in the willingness to work of the corresponding group of individuals. This is easily seen by computing This expression is positive as both 1 < 1 and wT t wT < 1, so that the logarithms of the terms in the right hand side are negative.
This result is intuitive. Let us assume that preferences are such that even very productive individuals do not work much. Then not much can be redistributed and utility is equalized among these individuals when idle individuals only receive a small transfer. If, on the other extreme, we look at a group of individuals with a high willingness to work, so that only very unproductive individuals remain idle, then a lot is available for redistribution and utility equalization results in idle individuals receiving large transfers.
Let us now merge the two sets of individuals, and let us consider two idle individuals (both of whom by the e¢ ciency conditions relating to labor supply do not work) ( ; w) and ( ; w 0 ) such that w > w 0 . The level of consumption of these two individuals is equal to the transfer they get, t( ; w) and t( ; w 0 ). A UBI is the outcome only if t( ; w) = t( ; w 0 ). By the property of t e (w; ) highlighted above, we know that t e ( ; w) < t e ( ; w 0 ), i.e., individual ( ; w) receives less from individuals with the same preferences as hers than does individual ( ; w 0 ). Therefore, UBI requires that at least one of the following two conditions holds. Either t(w) t(w 0 ); which means that high-wage individuals should pay no more taxes than low-wage individuals. Or t( ) t( ) > t e ( ; w) t e ( ; w 0 ): We know from (14) that individual ( ; w 0 ) receives more from individuals with the same preferences as hers than does individual ( ; w). Therefore, the right-hand side of the inequality is positive. Consequently, the left-hand side of the inequality has to be positive and larger which means that redistribution should take place from high to low willingness-to-work individuals, and the di¤erence between the tax paid by the former and that paid by the latter needs to more than o¤set the di¤erence in the transfers that take place within groups of individuals with the same preferences.
To sum up, UBI requires the combination of very low or even no redistribution from high to low-wage individuals and a large redistribution from high to low willingness-to-work individuals, two rather counter-intuitive norms of justice. 26 It is hard to imagine policymakers redistributing from low to high wage individuals or from high to low willingness-to-work individuals and so UBI cannot be viewed as a …rst-best policy.
It is useful to make one observation here. What the argument above has shown is that the normative intuitions that justify redistributing income all point toward transferring a larger amount to the low-wage-high-willingness-to-work individual than to the high-wage-low-willingness-to-work ones. This shapes the question we need to address in Sections 5 and 6 as follows: in spite of the normative desire to transfer larger amounts to low-wage-high-willingness-to-work than to high-wage-low-willingness-to-work individuals, can issues such as incentives, or administrative imperfections force us to treat them equally?
To conclude this subsection, let us turn to the case of developing countries. Remember that under the …rst-best, the only di¤erence between developed and developing countries is that in the latter the behavior of some individuals is determined by subsistence considerations: the constraint that consumption should be above a subsistence level is binding. This di¤erence turns out to have two drastic consequences for the discussion we just carried out, which can immediately be derived from Eqs. (9) and (10).
First, in an economy in which subsistence constraints are binding, individuals with the lowest consumption are not those who do not work and wait until they are employed at a positive wage in a formal sector. It is indeed what studies from developing countries suggest -namely, that very poor people in developing countries are not idle but actually work long hours often in several di¤erent forms of work to make ends meet (see, for example, Banerjee and Du ‡o, 2011).
Second, the labor time of these individuals is independent of their preferences but only depends on their wage, their non-labor income and the tax and transfer system.
As a consequence, the question that we raised in this section, based on what Van Parijs and Vanderborght (2017) posed as a key objection to UBI, namely, that it may subsidize those who could work but simply do not because of their preferences for leisure, does not appear very relevant when we consider developing countries.

Poverty reduction
There is a second way to justify imposing a universal lower bound on consumption. It is related to poverty reduction. Poverty, in this case, is de…ned as living with means that are considered, for whatever reasons, insu¢ cient. If poverty is de…ned in an unconditional way, i.e., if being poor is de…ned only as a function of the means of existence that one individual has independently of why this individual lies below the means threshold, then the objective of de…ning a minimal consumption level can be immediately justi…ed.
In a …rst-best world, this objective translates into a UBI proposal. Nobody can gain by trying to incentivize individuals whose types satisfy (12). It is e¢ cient to let them not work and allocate them the minimal consumption level b.
If the objective is to prevent all individuals from consuming less than b, then some individuals will receive a positive (lump-sum) transfer even if they work. From (4), the consumption level at an e¢ cient allocation is equal to (wT + m t( ; w; m)). It is important now to bring m back in the picture, as the policymaker does not have any reason to tax it all away. Rather, it should be taken into account as it in ‡uences the minimal consumption. All individuals of type ( ; w; m) should then receive a lump sum transfer satisfying so that their consumption is bounded below from b. This argument holds for both developed and developing countries. The main di¤erence between them, of course, has to do with the targeted level of b: The most natural candidate is the income poverty line. In developing countries, this line is typically absolute and is similar to what we called the subsistence level above. In developed countries, this line is typically relative and it is …xed in most OECD countries at 60% of median income (considerably above the subsistence level).
To sum up, poverty reduction may justify UBI in a …rst-best world, provided poverty is de…ned as a minimal consumption level that does not depend on individual characteristics. This, however, calls for an important quali…cation regarding conditionality. We had characterized means unconditionality earlier as having three components, namely, no conditions on the ability to earn income, non-labor income, and incomes of other household members. While poverty reduction justi…es the absence of condition on one's ability to earn income, it does not justify making the transfer independent of non-labor income or of incomes of other household members, two variables that can be captured by m.
To conclude, the main lesson from this section on the …rst-best is that UBI can be normatively justi…ed even in a …rst-best world, but it is much easier to defend based on poverty alleviation than on the values that are at the heart of optimal taxation. If the benevolent policymaker is able to distinguish income di¤erences that arise from di¤erences in ability as captured by wages from those that arise from di¤erences in willingness to work, only extreme and somewhat implausible normative stances can justify treating individuals who do not earn income independently of whether it is because their wages or because their willingness to work is too low.

In a second-best world
In the previous section we discussed conditions that a social welfare function has to satisfy to justify UBI in a …rst-best world, i.e., when the benevolent policymaker is perfectly informed of the characteristics of the individuals. In most debates, however, UBI is presented as a second-best policy. It aims at reforming an existing tax and welfare system in which the policymaker does not observe types and, in particular, where "means" cannot be costlessly tested. In this section, we drop the assumption that the characteristics of individuals, ( ; w; m), are perfectly observable, and assume that only pre-tax incomes are. Of course, the assumption of observable incomes is mostly valid in economies where the labor market is formal and the …scal administration works well, i.e., developed countries. In the next section, we consider additional departures from the …rst-best assumption, and look at problems due to the informality of the labor market and frictions on the administrative side.
In the optimal labor income tax literature, it is customary to summarize such a tax-transfer system by the equation where y g denotes gross labor income, y n denotes net (that is after-tax) labor income and aggregates all the policies that transform pre-tax into post-tax incomes (labor income tax, social security contribution, in-work bene…ts, family bene…ts, housing bene…ts, social assistance, unemployment insurance and unemployment assistance). Notice that it is possible for (y g ) to be negative, in which case it represents a net transfer, and also, that y n is an increasing function of y g . 27 Is it easier to justify UBI in such a setting? This is the question we address in this section.
Before we discuss this question, we need to justify our focus on (15). There are two reasons for this.
First, the celebrated Atkinson-Stiglitz theorem (see Atkinson and Stiglitz, 1976) states that provided preferences are separable in labor time and consumption and identical across individuals, any utility pro…le that is implementable by using indirect taxation is also implementable if we restrict ourselves to direct (that is labor income) taxation. It implies that indirect taxation can be dispensed with if the goal is to maximize any social welfare function. Of course, in reality preferences are often not separable between labor time and consumption. Think, for example, of day care, the demand for which increases with labor supply. These goods are easily identi…ed, though, and optimally taxing or subsidizing them is not too di¢ cult. What is key for our purpose is the assumption that funding a generous UBI through indirect taxation is not justi…ed. The e¤ect of the UBI reform on social welfare is much more transparent if we restrict ourselves to studying who should bear the …nancial cost of the reform. Of course, if pre-tax incomes are not well observed, with the consequence that it is fairer to tax consumption, then social welfare maximization may require to use indirect taxation. We do not address this question here.
Second, by restricting our attention to (15), we disregard capital-income taxation, which can be a part or all of non-labor income, m. Contrary to the point we made about indirect taxation, this simpli…cation does not imply that it is not justi…ed to fund a generous UBI with capital income taxation. We do not address this question in this section and come back to the issue of taxing m in Section 7 below. In this section, we focus on how the tax and transfer system should treat individuals with the lowest incomes. As it is reasonable to assume that these individuals do not have much capital income, we disregard capital income taxation.
The literature on tagging (e.g., Akerlof, 1978, Cremer et al, 2010, and Weinzierl, 2011 is relevant in our second-best world, where the characteristics of an individual, namely productivity or wages, preference for leisure, and non-labor income are not known to the tax or welfare authorities. The key lesson from this literature is, if there are characteristics of the individual that are not directly relevant from the economic point of view (e.g., gender, race, age) but are correlated with an individual's earning potential, then making the tax-cum-transfer scheme partly dependent on them may help in the screening problem. Therefore tagging may be e¢ cient both directly by economizing on scarce tax revenue available for redistribution but also, indirectly by not having to use screening mechanisms like workfare. The problem with tagging, of course, is that it violates horizontal equity.
The universal aspect of UBI may look the very opposite of tagging, but it is not. First, defenders of UBI sometimes think of it as depending on observable and non-manipulable variables that are correlated with need, such as age. Tagging on age, in this case, would certainly be acceptable. Second and more fundamentally, in the way we de…ned it, UBI requires that the transfer to zero-income earners be unconditional, and this does not preclude the use of tagging, provided it only a¤ects taxes and transfers beyond UBI. Of course, if horizontal equity is part of the set of values that UBI is supposed to be based on, then clearly tagging is not consistent with it.

Does no means-test mean that there is no phasingout?
A popular interpretation of UBI is that it will change (15) into where b is the UBI, that is everybody, including Serena Williams and Floyd Mayweather, would receive an additional income, b, and taxation needs to be adjusted from to so as to collect the necessary funds. To put it di¤erently, universality is understood as meaning that there is no phasing out of UBI, everybody bene…ts from it. We begin this section by underlining that Eqs. (15) and (16) are perfectly equivalent, as can be seen by …xing b = (0) (17) (y) = (y) (0): When (0) < 0, any labor income tax scheme that can be described as (15) is a UBI system.
What this implies is that the UBI proposal is not about everybody receiving an additional income, it is about making (0) unconditional. Remember that unconditionality of UBI requires that it is not means tested, and, in particular, that it does not depend on the ability to earn income. That requires the following two characteristics of the new tax system, which, as we will see in the next section, are absent of all existing redistribution systems.
1. If employed individuals earning y g > 0 and consuming y n = y g (y g ) decide to stop working, they are entitled to receive (0) independently on their type ( ; w).
2. Unemployed individuals, bene…tting from unemployment insurance, unemployment assistance or social assistance, are entitled to (0) independently of whether they are ready to take a job, should one become available.
How this unconditionality can be practically implemented is an important but di¤erent issue. Will b be transferred on a monthly basis from the social security budget to all individuals before they earn any income, with employers paying b back to the social security and not to the worker at the end of the month? Will it rather be paid ex post yearly by the …scal authorities after they observe all the y g 's? Or, will individuals earning less than b have to apply to receive it monthly? These are some di¢ cult and important practical policy questions to think about. We do not deal with these questions here. Rather, we focus on the consequences of UBI on the ex post relationship between earning and disposable income.

Di¤erent consequences of removing conditionality
Most existing transfers to low-income individuals are means tested. Following our discussion of UBI in Section 2, we can distinguish between three levels at which transfers are conditional on means: they can be conditional on a low ability to earn income, on the lack of personal non-labor incomes, and/or on the lack of income of other households members. The …rst level of conditionality is typical of both unemployment bene…ts and social assistance. The second and third levels are more typical in the case of social assistance, which is consistent with the fact that unemployment insurance is an individual insurance against the risk of losing one's job and one's labor income, independently of other means of subsistence. Let us illustrate these conditionality schemes.
First, transfers to low-income individuals can be conditional on a low (or even zero) ability to earn income. Implementing this level of conditionality requires to screen, among individuals with very low incomes, those who have a low ability to earn income from those who have a low willingness to work. This screening is typically done by delegating to …rms the evaluation of the employability of the candidates. Candidates are requested to look for jobs and they are considered as able to earn income as soon as one or several …rms are ready to hire them. Countries di¤er in their ways of implementing this level of conditionality, but the general spirit is clear: if it appears that an individual could earn income by working but refuses to take available jobs, he or she is typically excluded from unemployment insurance or social assistance.
The wider heterogeneity has to do with the eligibility for this bene…t of those who have voluntarily quit their jobs. A large number of countries do not pay unemployment bene…ts in case of voluntary unemployment, with the number of legitimate reasons that justify a voluntary quit varying (see Langenbucher, 2015). Then there are countries on the other end of the spectrum, like Slovakia, who do not even examine the reason why the previous employment ended (see http://ec.europa.eu/social/BlobServlet?docId=13773&langId=en).
There is less heterogeneity in the condition that bene…ciaries be looking for jobs and ready to take one when it is available. In fact, with the exception of Switzerland and Spain, all countries in Europe require a "willingness to work" for unemployment assistance schemes. Exceptions, such as those depending on the family situation, and what constitutes a "willingness to work," vary across countries (see MISSOC Analysis, 2011).
Second, transfers to low-income individuals can be conditional on lack of personal non-labor incomes.
Third, transfers to low-income individuals can be conditional on the lack of incomes of other household members. In the Netherlands, for instance, The Supplementary Bene…ts Act (TW) which is part of their unemployment insurance scheme) provides assistance to people who get a bene…t from one of the employee insurance schemes if their income plus that of their partner falls below the minimum guaranteed income. In Belgium, the level of bene…t of unemployment insurance from year two on is adjusted according to how many earners there are in the household (see http://www.oecd.org/els/soc/benefits-and-wagescountry-specific-information.htm).
To conclude, all current unemployment insurance or social security systems in developed countries impose conditions on the bene…ciaries, and these conditions, in spite of di¤erences, all boil down to limiting eligibility to those who are ready to take jobs when they are available. 28 Introducing UBI would be a clear departure from all existing systems, with the consequence that those who decide not to participate in the labor force, typically as the result of a rational decision at the household level, would suddenly be entitled to basic income. This would hardly incentivize them to go back to the labor force, and granting them UBI is likely to be a major additional cost for social security budgets.

UBI in a Mirrleesian framework
Let us begin our analysis of optimal second-best taxation with the classic framework of Mirrlees (1971). Mirrlees pioneered the study of income taxation by being the …rst to derive theoretical results on how labor incomes have to be taxed if the objective is to maximize a social welfare function. The novel features of his approach were: taxation is not necessarily linear, which means that di¤erent income levels could be taxed at di¤erent rates; the policymaker is assumed to have an objective in terms of distribution of utilities rather than the utility of a representative agent; information constraints are explicitly taken into account by the assumption that only gross income (as opposed to wages or labor times) are observed; and individuals are assumed to freely choose their labor time (and, therefore, their pre-tax income) knowing the tax function and being able to compute how much net income they will be able to derive.
The most restrictive assumption, on the other hand, was that all individuals have the same preferences. This assumption has the nice consequence that it is normatively appealing to use the same utility function for all individuals, so that interpersonal utility comparisons are not problematic. As we already studied in the previous section, the natural model in which justi…cations to UBI should be discussed is one in which preferences are heterogenous, but it is worth identifying the relationship between the optimal taxation system when all individuals have the same preferences and UBI.
If we stick to our assumption of Cobb-Douglas preferences, the Mirrlees model amounts to assuming that all individuals have the same and all non-labor incomes, m, are equal to 0. The result that is of interest for our purpose is that at the second-best optimal taxation scheme, all individuals who are idle in the optimal …rst-best allocation continue to be idle in the second-best optimal taxation and many more are induced by the tax system not to work. 29 This is the distortive e¤ect of taxation in the second-best: the larger is the redistribution from the rich to the poor, the larger is the incentive of the rich to mimic the poor by working less and earning less. 30 The only way to decrease this incentive to mimic the poor is to incentivize the poor to work less and earn less, which makes the resulting situation of the poor less appealing for the rich.
As a result, at an optimal allocation, there is a threshold w > 0, such that all individuals with a wage below this threshold do not work and receive (0). As only gross incomes are observed, no discrimination can be made on the basis of di¤erences in w. Given that all individuals have the same utility function, all idle individuals also have the same utility.
The optimal Mirrlees taxation scheme is therefore a UBI scheme. By de…nition of the optimum, all individuals prefer the bundle they get (because they choose it freely and the set of available gross income/net income bundles is identical for all individuals) compared to the bundle of any other individual, including those who do not work. Consequently, (0) can be thought of as a minimal basic income (all other individuals earn and consume more than that amount) and a universal one, because no individual strictly prefers not working and getting (0) over their actual situation.
In conclusion, the optimal taxation scheme of Mirrlees and the ones of many 29 See, for instance, Piketty and Saez (2013) and Boadway (2012) for excellent surveys of this literature. 30 Mimicking an individual here means to earn the same gross income as that individual.
of his followers do include UBI, but the universality of the basic income is derived from the assumptions that: …rst, all individuals have the same preferences; and second, no screening device exists that would allow the policymaker to make the basic income conditional on a very low (or even zero) ability to earn income. If such a device was introduced in the literature, then conditionality would be optimal. We view the failure to take conditionality of unemployment bene…t or social assistance on ability to earn income into account to be a serious shortcoming of the optimal taxation literature. We try to take it into account in the next section in which we reintroduce preference heterogeneity.

Interpersonal welfare comparison and heterogeneity of preferences
We studied in Section 4 the desirability to UBI in a …rst-best world. In Section 4.2.1, we raised the question of whether a benevolent policymaker could wish to allocate the same lump-sum transfer to low-wage-high-willingness-to-work and high-wage-low-willingness-to-work individuals. We saw that such a policy could only be seen desirable by a policymaker who refrains from redistributing from high to low-wage individuals but wishes to redistribute large amounts from high to lowwillingness-to-work individuals. The conclusion, though, was that most intuitive normative stances would lead to the opposite objective, that is to the objective of redistributing more to low-wage-high-willingness-to-work individuals than to high-wage-low-willingness-to-work individuals. Assume the policymaker endorses the latter, more intuitive, normative stance. The question that we address in this section is: could the information asymmetry faced by this policymaker lead her to …nd it optimal to transfer the same amount to all idle individuals? As explained in the previous paragraph, social assistance and unemployment bene…ts in all OECD countries are awarded under the condition that bene…ciaries are ready to take jobs whenever they become available. We have interpreted this condition as a screening device between the low-wage-high-willingness-towork individuals, who will not …nd jobs and the high-wage-low-willingness-to-work individuals, who will …nd jobs (if properly monitored in their job search) and will then be excluded from social assistance or unemployment bene…t if they refuse them. Here again, the answer to our question will crucially depend on whether such a device is available or not.
Let us begin by assuming that such a screening device is not available. Then, given the information asymmetry between individuals and the policymaker, individuals choose their labor time, and, therefore, their pre-tax income by taking (15) into account. As a result, all individuals who do not work consume the bundle (0; (0)). Consequently, as soon as the optimal allocation has (0) > 0, UBI is optimal. That means that adding heterogeneity of preferences into the Mirrlees framework does not change anything as soon as the interpretation of (0) as a UBI is concerned. Note that under these assumptions there is no difference between UBI and the negative income tax proposal of Friedman. However, such screening devices do exist and the important question is then: should the policymaker use them or not?
Let us assume, following Atkinson (1995), that wages are either 0 or above some minimal legal wage w. 31 The tax system can then be described by a function describing the tax paid, as a function of their income, by those who have a positive wage w w, and b, the bene…t allocated to the other individuals, having w = 0. 32 Adapting (12), we can derive that all non-constrained individuals having a type ( ; w) 33 such that prefer not to work, where 0 (0) stands for the derivative of at income level 0, that is the marginal tax rate at the zero earning level. 34 In this condition, one essential feature of the …rst-best optimal tax and transfer scheme immediately applies to the second-best optimal tax scheme -if the policymaker adopts the normative view (which we found counter-intuitive) that all idle individuals should be treated identically, then the screening device should not be used, b = (0), and UBI is optimal. If, on the contrary, the policymaker considers that individuals with the lowest wage (in this case w = 0) should be assigned a strictly larger amount, then it cannot be optimal to impose b = (0), because it would amount to imposing an additional constraint to the social welfare maximization problem, with the necessary consequence that social welfare would be lower. In all the cases in which b > (0), the optimal tax system can take the form of a basic income of (0) transferred to all those who do not work, complemented with a conditional additional bene…t of b ( (0)) for those who agree to go through the screening device (i.e., to look for jobs) and whom no employer is ready to hire, thereby revealing a zero wage.
This proves that the optimal second-best taxation scheme can take one out of three possible forms: The …rst form is a pure UBI system, replacing all other social assistance program: b = (0) > 0. It is optimal only when the social welfare function requires not to make any di¤erence among the idle individuals.
The second form is an hybrid system in which a smaller UBI coexists with a conditional supplement, available only to those who have a zero wage, and this 31 In Atkinson's work, however, individuals with a zero wage are "the sick and retired" and not the workers (temporarily) unable to …nd jobs. 32 We could generalize this discussion to the case in which the constrained individuals have pre-tax incomes in an interval [0; y], with y < wT . We would then have to design a function b : [0; y] ! R. 33 Given that we do not study the optimal design of the tax on m (for the reasons explained at the beginning of this section), we assume, for the sake of simplicity, that m = 0 for all individuals. 34 This …rst-order-condition is necessary but it is actually not su¢ cient, as su¢ ciency requires that the budget set is convex. The additional condition is that marginal tax rates on low incomes are non-decreasing. supplement implies the use of the screening device that is used in existing conditional social assistance systems: b > (0) > 0. 35 All those who refuse to go through the screening device because they prefer not to work will be treated identically, independently of their true ability to earn income, as it is not observable (this is the main di¤erence with a …rst-best system, in which the policymaker could discriminate among them, by giving lower and lower transfers to higher and higher-wage individuals).
The third form is a pure conditional system, in which only those who have a zero wage are eligible for social assistance: b > (0) = 0. This is optimal when the recommended transfer to those who could work but prefer not to work is not positive. This corresponds to most current systems.
In conclusion, UBI is easier to justify in a second-best world. The easier system to justify, however, is an hybrid system in which a plausibly modest UBI is combined with a supplemental bene…t conditional on a zero ability to earn income. In this case, UBI means that a certain amount of money is allocated to all non-working individuals, independently of the reason why they do not work, but UBI does not mean that all individuals who do not work are treated identically: a part of the transfers dedicated to non-working individuals remains conditional on a zero-ability-to-work, and the same screening device as the one existing in current systems is used to discriminate among non-working individuals.
We should emphasize that our conclusion heavily depends on the assumption that the redistribution tools available to the policymaker include the possibility of distinguishing between those whom no employer wishes to hire, revealing a (perhaps, temporary) zero wage, and those whom some employers are ready to hire, and who can choose not to work only by giving up on conditional transfers. This assumption re ‡ects the monitoring of unemployment status that almost all developed countries implement. It is interesting that optimal income taxation theory does not take it into account. In our view, only by taking it into account one can undertake a thorough analysis of UBI.

Poverty reduction
In a …rst-best world, we saw in the previous section that the …ght against poverty in the sense of guaranteeing all individuals, including those who do not work, a minimal consumption, was a clear and almost direct justi…cation of UBI. It turns out that when applied to a second-best world, there is a literature that has challenged this simple justi…cation. We review this literature in this section in order to check under which conditions UBI can still be justi…ed by the objective of poverty reduction.
The literature on the ability of second-best labor income taxation to reduce poverty can be divided in two sets. In the …rst set, the policymaker does not try to maximize a social welfare function satisfying Pareto e¢ ciency. In Coate (1992, 1995), an income poverty line is …xed and the policymaker wishes to minimize the total amount of money transferred to the poor under the constraint that they all reach the poverty line. In Kanbur et al (1994aKanbur et al ( , 1994b, the policymaker maximizes a combination of a Paretian social welfare function and a statistic of income poverty, again de…ned as having an income below a …xed poverty line. In all these papers, the optimal policy consists in incentivizing poor individuals to work hard and earn as much as they can so that the income increment that is needed to help them reach the poverty line is not too high. In both cases, this policy does not help increase the utility of the poor, and in Kanbur et al (1994aKanbur et al ( , 1994b it even decreases it compared to the policy that would be optimal without this poverty reduction objective.
One key assumption in these papers is that the poor individuals are productive, which we could capture in our model by assuming that their wage is low but strictly positive. As we argued above, this is a reasonable assumption for developing countries in which poor individuals work a lot in the informal sector to reach the subsistence level. If we want to apply the argument to developed countries, we need to introduce in their model individuals with a zero wage. In this case, again, an essential assumption would be whether a screening device exists that allows the policymaker to distinguish between those who are unable to earn income and the others. Absent such a device, the optimal policies would be to transfer money to the most needy and maybe to complement this transfer with a lower transfer to those who earn some income in such a way as to neutralize their incentive to mimic the needy. 36 If such a screening device is available, then, like in most of the cases above, it is optimal to use it, with the consequence that the transfer to those who do not earn income on their own becomes conditional, and individuals with a strictly positive ability to earn income should be treated like in these papers, with the consequence that these individuals should be incentivized to work more and earn more, without an impact on their utility.
In conclusion, this literature proposes redistribution systems aimed at reducing poverty that sharply di¤er from UBI. However, it does not follow that UBI may not be the optimal policy given the objective of poverty alleviation. The desirability of the policies studied in Coate (1992, 1995) comes from the implicit assumption that individuals care about the income of the poor and have a utility function such as: where y p is the income of the poor and h( ) is an increasing function, which means that these individuals are willing to give up their own income provided it increases the income of the poor. Notice that the utility of these individuals does not depend on the utility of the poor (in which case the income support policy would not be desirable) but on the income of the poor, that is the consumption of the poor has a positive externality on the utility of the rich.
In his in ‡uential 1970 contribution, Sen has shown that when preferences exhibit this kind of consumption externalities, imposing Pareto e¢ ciency may yield outcomes in which freedom of choice is impeded. As a consequence, it has become customary to develop welfare analysis after preferences are cleansed of these consumption externalities as we have implicitly done in the previous sections. If the policymaker has good reasons to respect these consumption externalities, though, then Pareto e¢ ciency should be applied to preferences represented by (20).
Suppose we adopt the latter point of view and think of how to increase the utility of rich individuals with the utility function (20). These individuals care about the income of the poor independently of how much poor individuals have had to work to reach such an income level, Therefore, these rich individuals are ready to treat, among the poor, high-wage-low-willingness-to-work and low-wagehigh-willingness-to-work individuals in the same way, which seems like a promising way to justify UBI. What contributions like Coate (1992, 1995) teach us, then, is that in spite of the rich having this kind of preferences, UBI is not the optimal policy.
Most of the defenders of UBI, however, do not seem to argue in favor of UBI based on its ability to satisfy the preferences of rich individuals who care about the income levels of the poor with utility functions like (20). Rather, they adopt more standard normative positions, often related to poverty alleviation (see, for instance, Tobin et al, 1967, or Galbraith, 1969. 37 Consequently, we assume away consumption externalities and stick to our benchmark model and ask the question that under this assumption, does the goal of poverty alleviation justify UBI in a second-best world? A second strand of literature proposes a more direct challenge to UBI. Saez and Stantcheva (2016) and Maniquet and Neumann (2017) study the taxation scheme that can be derived from maximizing a social welfare function that favors transfers from rich to poor individuals as long as they do not decrease the utility of the poor. In both cases, if an allocation exists that allow all individuals to receive incomes above the income poverty line, this allocation is optimal and we can think of it as o¤ering a UBI to all individuals who do not earn income. The relevant case, however, seems to be the one in which such an allocation does not exist. All OECD countries turn out to have (conditional) income guarantees that are below the income poverty line. 38 If this income guarantee becomes unconditional, it is quite likely that the amount will decrease. As a result, it is plausible that the maximization of a universal income guarantee will not reach the income poverty line.
If an allocation in which all individuals get incomes above the poverty line does not exist, then both Saez and Stantcheva (2016) and Maniquet and Neumann (2017) show that a UBI lower than the poverty line is not optimal. In both cases, the optimal taxation scheme includes negative marginal tax rates on low incomes, thereby incentivizing individuals to earn income and reach the poverty line. In Maniquet and Neumann (2017), the optimal taxation scheme is the one that minimizes the labor time it takes to minimal wage individuals to reach the poverty line, and this may require very low transfers to those who do not work.
These two papers, however, share the assumption that the minimal wage is positive: by working, individuals earn some income even if not su¢ cient to reach the poverty line. They do not cover the case in which some individuals are involuntarily unemployed and (temporarily) have a wage equal to 0. Again, we can look at what the optimal taxation scheme would be if we add a group of zero wage individuals and a screening device aimed at identifying them. Not surprisingly, the optimal scheme would be to transfer money directly to the zero wage individuals to complement what Saez and Stantcheva (2016) or Maniquet and Neumann (2017) identify as the optimal scheme. That is, the transfer received by those who do not earn any income should be conditional on not being able to earn income and, therefore, is not a UBI. This illustrates that the ability of a tax and transfer system to alleviate poverty is not necessarily measured by the distance between the poverty line and the lowest income in the population. In Maniquet and Neumann (2017), for instance, the criterion that is proposed to evaluate tax schemes is the labor time it takes individuals working at the legal minimal wage to reach the poverty line.
To sum up, poverty alleviation becomes more di¢ cult to use as a justi…cation of UBI in a second-best compared to a …rst-best world. This is true under the (plausible) assumption that guaranteeing a basic income equal to the poverty line is infeasible. As a result, minimizing poverty does not necessarily require to be generous to those who do not work and to treat them all identically. It may be optimal, on the contrary, to incentivize these individuals to work, by applying negative marginal tax rates on very low incomes, so that they choose the labor time that allow them to obtain an after-tax income above or equal to the poverty line.
Before we move on to studying the third-best context, let us stress again that the analysis in this section builds on the assumption, common in the optimal labor income taxation literature, that those who earn the lowest income are to be found among those who do not work. This captures the formality of the labor market in developed economics but not the widespread informality of the labor market in developing countries .

The stigmatization cost of conditionality
An argument often put forward to justify the unconditionality of social assistance is that it removes the stigma associated to requesting help and revealing one's lack of resources. We do not fully address this question here, but make some observations that raise some reservations about the absence of stigma argument.
The conditionality of social assistance at least in theory must lead to some stigma, with the resulting decrease in utility of those who request assistance and low take-up of the others. However, the literature on take-up has not been able to prove that stigma is a major explanation of low take-up (see, for instance, the review of Currie, 2006). Also, even if UBI removes stigma, it is likely to decrease utility in another, symmetric, way -all those who live in rich families and who have decided to quit the labor force may be ashamed to be entitled to UBI and they may su¤er a disutility because of this form of stigma. Some of them may even decide to go back to the labor force to avoid this cost, thereby decreasing their utility and a¤ecting the labor time or wages of others.
Because of the absence of enough evidence on UBI, it is impossible to estimate the extent of utility loss among those who prefer not to work and not to receive any transfer. What we already know, however, is that people who currently do not work and do not request any transfer because their partner is su¢ ciently rich are much more numerous than those who bene…t from social assistance. 39 To sum up, the stigma argument seems much more of an argument to make social assistance an indisputable right rather than an argument to justify transferring large amounts of money to those who have decided to quit the labor force because they live in rich families.

In a third-best world
In this section, we allow for imperfections in the tax and bene…ts system and examine the case for a UBI from a "third-best" point of view -namely, when not only there is possibility of various informational and incentive issues that arise on the side of bene…ciaries but also on the side of those in charge of administering the tax and bene…ts system.
An argument often made in favour of a UBI is that it cuts administrative costs -namely, as it does not involve targeting, it avoids inclusion and exclusion errors associated with screening as well as administrative costs (direct costs, ine¢ ciencies of various kinds including delays and waste, and corruption). 40 In an economy in which public agencies cannot be relied to deliver bene…ts to the targeted groups, due to corruption (or lack of accountability more broadly) or due to limited state capacity, there is a risk that those who need transfers most are at risk of being excluded from the bene…ts. In such a setting, the less is the scope for discretion, as is the case with any uniform and universal policy, the lower will be such risks. Accordingly, an egalitarian policymaker may prefer to divide the budget equally rather than ask public agencies to target. 39 In the U.S., for instance, the Bureau of Labor Statistics estimates that the non-participation rate in the labor force is currently around 36.8% (see https://www.bls.gov/news.release/pdf/empsit.pdf), whereas the Census Bureau estimates that 13.5% of all households received SNAP bene…ts at some point in 2013. 40 It is not that there are no administrative costs associated with cash transfers or that there is no potential for corruption. Also, for cash transfers to be feasible, a well-functioning …nancial infrastructure is necessary. This is often not the case in developing countries, though mobile banking is making a dent in the problem.
As we discussed earlier, if y g is perfectly observable, then any tax system , such that y n = y g (y) can be reinterpreted as a UBI system if (0) < 0 in which everybody receives a UBI equal to (0) and pays (y) (0). In such contexts, unless there is a way to discriminate among those who do not earn any labor income, any proposal in which there is a positive transfer at zero income is by de…nition a UBI. In this case the debate about UBI is clearly not very interesting.
If there is no way to discriminate among individuals other than income (or any other measure of means) and that in turn is noisily measured, is there a stronger case for UBI? The literature on targeting (see Besley and Kanbur, 1990 for an overview) is relevant here, as it deals with situations where actual income is not costlessly observable. 41 Let us consider an economy where income is not perfectly observed. Suppose the tax authorities observe a noisy signal about gross income y g , which we denote by s and any tax and transfer scheme has to be based on s. With probability p(y g ), s = y g where p (y g ) 2 [0; 1] and p(0) < 1. We consider two forms of noisiness in the measurement of income when s 6 = y g .
First, suppose that with probability 1 p (y g ) a completely uninformative signal s = ' about y g is obtained, which has the same support as y g but is uniformly distributed. 42 In this case, for a given y g , the signal s takes two values y g and ' where s = ' denotes a null or uninformative signal. If an uninformative signal is observed, the level of income could be anything, from the lowest possible to the highest possible and so the question arises as to what should be the transfer in this case.
Suppose our goal is to transfer a net amount (0) to those whose income does not exceed a certain threshold y g 0. Do we make the same net transfer (0) to all individuals whose observable income is y g y g as well as to individuals for whom we do not observe y g but observe the uninformative signal or do we make a net transfer of a di¤erent amount (which can be zero) to the latter group? 43 Or, should we adopt a UBI scheme with the same net transfer to those with y g y g as well as y g = ' while for others, namely whose income is observed and exceeds y g , (0) is a gross transfer accompanied by a tax (y g )? Clearly the desirability of UBI will increase with the degree of inequality aversion of the policymaker. Alternatively, for a given level of inequality aversion, as the noisiness of income measurement goes up, UBI would appear more attractive. After all in the limit case where income cannot be measured at all (i.e., p (y g ) = 0), UBI is the only possible …scal instrument for making a transfer that is available to the policymaker, although in this extreme case, it cannot be funded by direct income 41 A "proxy-means test" (PMT) is usually used to estimate the income or consumption when precise measurements are not available or di¢ cult to obtain. This typically involves collecting information on assets owned by the household (such as, type of house, ownership of livestock, and various durable consumer goods) as proxies for income or consumption. 42 This is the same formulation as in the model of supervision in Tirole (1986). 43 We do not provide a full characterization of the tax schedule (y g ) here. Presumably taxes can only be levied on those whose observable income signal y g is positive and above a certain threshold. taxes and must be funded from other sources, such as indirect taxes.
Second, consider a variant of frictions in the tax and bene…ts system above and suppose that with probability p(y g ), s = y g where p (y g ) 2 (0; 1) and p(0) < 1 as before, but with probability f1 p (y g )g, a value of income y g (1 + ) that is higher than the true value y g is measured, while with probability (1 ) f1 p (y g )g, a value of income y g (1 ) that is lower than the true value is measured. If the goal is to make sure that those with an income level below a threshold receives a net transfer of (0), a similar dilemma arises. Unless the income measurement process is relatively accurate (i.e., p (y g ) is high), or the likelihood of incorrectly overestimating someone's income is low (i.e., and are small), there is a risk of denying bene…ts to a deserving bene…ciary. A ‡at transfer does not have this problem, but of course, is more expensive.
The solution to this problem can be less stark than posed above. The central government could tag regions based on characteristics that cannot be manipulated or monitored by these agencies (see Ravallion, 2018). In other words, UBI could be adjusted to regional characteristics such as average income levels or household characteristics such as families with children. Moreover, as a large literature on using various screening devices (such as workfare or in-kind transfers) points out (see Kanbur, 1990, Currie andGahvari, 2008) one can make claiming bene…ts costly for the non-poor. To the extent these cannot be manipulated, these would ameliorate the stark trade o¤ between a UBI and targeted schemes, namely, the former has no screening costs but a larger bill because it is universal.

Positive Aspects of a UBI Scheme
After discussing the normative justi…cations for a UBI in Sections 4-6, in this section, we take the basic income-leisure choice framework introduced in Section 4 and examine some positive implications of introducing a UBI scheme that is funded by an additional linear income tax. In particular, taking into account the e¤ect of a UBI funded by additional taxes on the labor supply decision of both net recipients and net payees, we ask what factors will determine the size of the basic income.

Labor supply responses to UBI
To address the question what is level of UBI that is feasible, an important consideration is the labour supply e¤ects of a UBI that is funded by additional taxes since that will determine overall income levels and the tax revenue available to redistribute as UBI after such a scheme is introduced.
Compared to the existing tax and bene…t systems in developed countries, the introduction of UBI is expected to make it attractive to stop working for some, because of the lack of conditionality, and also the additional taxes needed to …nance the additional spending will create incentives for working less. However, for certain groups, this conclusion has to be modi…ed.
First, consider individuals who are rich. Let us take the model of section 4, and set t ( ; w; m) = 0 in the budget constraint to capture a situation prior to the introduction of a balanced-budget UBI scheme, namely, y w(T `)+m. We can see from (5) that leisure being a normal good, as income grows, the consumption of leisure will go up. However, for those with high levels of non-wage income there is a possibility of a corner solution, namely, it is possible that`= T . Setting t ( ; w; m) = 0 in (6) and simplifying, the condition for this to happen would be: wT: This will occur when m is relatively high with respect to w, namely, the marginal cost of not working is low, while overall income is high. Slightly stretching the model, this case can be thought of as representing partners of spouses with higher incomes. Suppose that m in the model can also stand for the labor income of a partner. Then the model illustrates that spouses may prefer to stay home rather than participate in the labor force as a consequence of having a large wage spouse. Of course staying home may mean producing household goods and engaging in social and charitable activities as well as what is typically interpreted as leisure in the sense of a private good.
The second type of individuals/households whose labor supply is likely to remain una¤ected by UBI are the very poor. Now let us modify our simple set-up in Section 4 and introduce subsistence considerations not only for consumption but also for leisure in the following way: u (y;`) = log y y + (1 ) log (`+`) for y > y and` 0 = V , otherwise where` 0 represents the minimal level of non-paid-labor time required for subsistence and y and V have the same signs and interpretations as in Section 4. This formulation allows`to take the value of 0 at an optimum, something that the Cobb-Douglas functional form does not permit.
Let us now turn to individuals who are inframarginal. In the case of interior solutions we have:`= For` 0 we need wT + m y + 1 w`: Similarly, for y > y we need wT + m > y w`: The …rst condition is more strict than the second. Therefore if y w`< wT + m < y + 1 wt hen we have a corner solution with`= 0 and y = wT + m: This needs to be greater than y for the subsistence constraint to be met for income (which means it also satis…es the condition wT + m y w`that applies for interior solutions).
If wT + m < y then there is no solution to the optimization problem that satis…es both the budget constraint and the subsistence constraint and whatever is the choice, the individual receives a payo¤ of V . For simplicity we assume the individual continues to choose`= 0 and y = wT + m.
Using the notation z to denote full income and de…ning z y 1 w`+ 1 1 wT , we can sum up the above analysis as: y = z for 0 z y + 1 w= z + (1 )y `for z 2 y + 1 w`; z = m for z z and`= 0 for 0 z y + 1 w= We assume`< 1 2 T and so z > y + 1 w`. The take away from this exercise is that in situations where income levels are so low that subsistence considerations are important (i.e., w and m are low relative to y and`) a good proportion of the population will be working very hard with = 0 (equivalently, L = T ). For them, a UBI that is not large in size may not have any e¤ect on the labor supply. It should also be noted that for those who are below the level of subsistence (namely, wT + m < y) the utility gains from a UBI that pushes them above the subsistence level is high.
To sum up, even with the classical model of the labor supply there are some theoretical reasons to think that the potential disincentive e¤ect of a UBI on labor supply is more likely to be an issue in developed countries in contrast to developing countries. There are additional channels to this simple framework such as missing markets, price e¤ects from conditions attached to transfers, and dynamic and general equilibrium e¤ects that would tend to reinforce this general conclusion in the context of low and middle income countries. 44 While we do not have much direct evidence regarding the e¤ect of a UBI scheme on labor supply yet, Banerjee et al (2019) and Hoynes and Rothstein (2019) review the evidence from related studies on the likely labor supply e¤ects of a UBI. The general picture that emerges is consistent with our analysis, namely, for developed countries, a UBI would be expected to lead to lower labor supply, at least in the short run, while in developing countries there is no systematic evidence of various cash transfer programs having a negative e¤ect on labour supply. 45

How generous can UBI be?
Now we turn to the analysis of a UBI scheme taking into account its funding through taxation. Such a scheme will lead to some redistribution in net terms within the population, which implies that the sum of the income e¤ects is likely to be low. Also an increase in the tax rates on labor income facing all individuals will result in a standard substitution e¤ect leading to a decrease in the labor supply. This has important consequences on the level of the basic income that an economy can a¤ord. We illustrate the major trade-o¤s related to this issue here, …rst in the case of a linear income tax system (as in Atkinson, 1995), and then in the case of a non-linear tax. 46 Let us go back to the model without subsistence considerations, namely where u (y n ;`) = log y n + (1 ) log`. Then the revised optimization problem is: where b is the basic income and t is the linear tax rate that applies to total income. With b = 0 and t = 0 we have the benchmark model and so we have the same …rst-order conditions adjusting for the new budget constraint under a UBI scheme:`= y n = fb + (1 t) (wT + m)g : Given y n , we can solve for y g = y n b 1 t = (wT + m) (1 )b 1 t : Using the notation of z we can write: Observe that z is the value of gross income in the absence of a balancedbudget UBI scheme. For simplicity, we assume away heterogeneity in and T: Each individual is then characterized by a pair (m; w). Let the joint distribution of m and w in the population be denoted by the probability density function f (m; w). Without loss of generality, we assume w 2 (0; 1) and m 2 (0; 1). disincentive for work, especially for the poor. However, careful analysis shows that there was no evidence of reduced labor supply and if anything, the labor supply of women and self-employed men actuall went up (Salehi-Isfahani and Mostafavi-Dehzooei, 2018). 46 As mentioned earlier, we do not deal with the issue of imperfections in labor or credit markets (other than the formality of labor markets) for simplicity. In their presence, as is well known in the development economics literature, the usual equity-e¢ ciency trade o¤ is muted, if not overturned, because cash transfers can relax liquidity constraints faced by small enterprises.
Let the associated cumulative distribution function be F (m; w). Given that full income z = wT + m is a linear function of m and w, we can derive the distribution of z across individuals in the population from f (m; w) (even when m and w are not independently distributed). Henceforth we work with the pdf and CDF of z; de…ned over z 2 (0; 1), g (z) and G (z) respectively, which are given by: Let us de…ne average full income asz R 1 0 zg (z) dz: Since both gross and net incomes are functions of z, this allows us to derive the personal (as opposed to functional) distribution of net and gross incomes from the personal distribution of full income.
For the budget to be balanced on aggregate, we must have Let us de…ne average gross and net income as: y j (z)g (z) dz; j = g; n: We can deriveỹ g andỹ n as functions ofz, using (24) and (23): The fact that gross and net incomes are linear functions of full income is a consequence of our assumption that preferences are Cobb-Douglas, which will give us closed-form solutions to some key variables. Substituting the expression forỹ g in the budget balance (25) condition above and solving we get a key budget-balance equation for a UBI scheme: An obvious implication of this expression is, the higher the average income of a country, the easier it is to fund a basic income scheme so long as b does not rise proportionally with average income. A recent report by the IMF (2017) provides a calculation of the …scal cost of a UBI program as percentage of GDP when the basic income is set at 25% of the per capita median income. The ratio of median to mean income is larger in richer countries re ‡ecting less inequality, and so the …scal cost of UBI is expected to be larger. This is con…rmed by the calculations that are provided in this report. For example, the cost of UBI as a percentage of GDP is 6.4% and 6.7% for the USA and the UK, while it is 3.7% and 2.3% for Mexico and South Africa. In this context, however, we have to keep in mind that the …scal capacity of poorer countries is more limited and so despite these calculations, raising the relevant tax revenue could be much harder.
A lot of debate about UBI concerns what is the appropriate level of b. Clearly it cannot be the same absolute level (even controlling for purchasing power) across countries that have di¤erent levels of average income since standard of living changes with the level of prosperity. So long b does not increase proportionally with average income levels, (26) suggests that it is easier to fund a UBI scheme, the richer is the country.
The formula for basic income in (26) gives an aggregate trade-o¤ between b and t given the need for budget balance, the formula of the so-called La¤er curve. Again, the fact that it only depends on the average full income and not on its distribution is particular and follows from our choice of Cobb-Douglas type of preferences.
The formula in (26) allows us to characterize the largest possible level of b. Di¤erentiating (26) with respect to b and rearranging, we get the following …rstorder condition: This gives us If, for instance, = 0:5 (which means that individuals like to spend half of their full income in consumption and devote the other half to their leisure), then the income tax rate that maximizes the basic income is equal to 58.58%. Taxing income at a higher rate would be detrimental for everybody in the economy. The equation (28) also shows that the largest t compatible with e¢ ciency is an increasing function of . This comes from the fact that a larger is associated with a lower elasticity of the labor supply. 47 If we take (26) and we …x the average gross income in the absence of taxation, z, we see that b is an increasing function of : if individuals are more sensitive to taxation (a lower for a …xed z), then the same tax rate t leads to a lower UBI. All tax rates between zero and 1 p 1 lead to more or less generous basic incomes, and some individuals, typically the poorer, gain from an increase in t, whereas the others lose. Another question that (26) allows us to consider is the preferred tax rate of the average individuals, namely, those with full incomez. Remember that several types of individuals (m; w) have full incomez. We know from (24) that the gross 47 The (uncompensated) elasticity of the labor supply, @L @w w L can be computed from Eq. income of each individual decreases as t increases, and so does the gross income of the average individuals. As the average individuals are precisely those whose net income is equal to their gross income, clearly their net income also decreases with higher t. What is unclear, though, is the e¤ect of t on their utility as the decrease in net income goes together with a decrease in labor time.
The indirect utility of an individual with full incomez and wage w is given by v(w;z; ) = ( ) + log(b + (1 t)z) (1 ) log(w(1 t)): where ( ) = log + (1 ) log(1 ). Using the value of b from (26) we get: v(w;z; ) = ( ) + logz (1 ) log w + log so that the sign of the derivative of the indirect utility v(w;z; ) with respect to the tax rate boils down to the sign of the derivative of (1 t) 1 t with respect to t, i.e., the sign of: This is negative because < 1. This proves that the average full income individual always prefers a lower tax rate, independently of whether her full income is large because of her wage or non-labor income. Income distributions are always skewed, so that median income is typically smaller than average income. The result above is independent of the distribution of incomes, which means that the median income may be arbitrarily close to the average one, with the consequence that the median individual would also prefer a lower tax rate. That illustrates the fact that there is no guarantee that a majority of people would bene…t from UBI, should it be …nanced by a linear tax, as proven by Romer (1975), who pioneered the study of voting on the labor income tax when behavioral responses are taken into account. 48 If labor income tax is allowed to be non-linear, and if the policymaker wishes to implement a generous UBI, what should be the shape of the tax system? The literature suggests that the optimal non-linear tax should be convex, at least on low incomes, which means that individuals earning very low incomes should face higher marginal income tax rates. 49 The intuition of this result is as follows: as the basic income becomes larger, the amount of tax that needs to be collected increases. That requires an increase in the average tax rates. In order not to deter high wage individuals from working hard, this increase in the average tax rates should be accompanied as much as possible by low marginal tax rates on large incomes. This is accomplished by having large marginal tax rates on low incomes, thereby increasing the average tax rates on the whole income distribution. This has the drawback that it discourages low wage individuals from working but given 48 Romer also studies the case in which part of the tax return goes to …nancing …xed government spendings, in which case the preferred tax rate of the median voter is even lower than in the pure redistributive case that we study here. 49 See Boadway and Jacquet (2008) for a comprehensive treatment of this question.
the lower marginal tax rates on larger incomes, the very productive individuals continue to work and a su¢ ciently large amount of tax is collected. The main lesson of this subsection is that the amount of UBI and the labor income tax system that needs to be designed to …nance it depend strongly on the behavioral responses of the tax payers. The simple linear tax example above has shown that there is a maximal feasible amount of UBI and a maximal rate of taxation beyond which everybody loses. The example has also illustrated the result that even if low income earners necessarily bene…t from an increase in UBI when it is …nanced by a linear tax, a majority of individuals may strictly prefer a lower tax and a lower UBI. Finally, a quick review of the optimal non-linear taxation literature has shown that the optimal way of …nancing UBI was typically not through a linear tax but a convex one, in which low income earners face higher marginal tax rates. This is likely to even decrease the set of individuals bene…tting from UBI.
The arguments in this subsection are mainly relevant for economies in which all individuals either earn income and pay tax or do not earn income and then receive some form of social bene…ts. Developing countries, on the contrary, are characterized by low fractions of the population who are income tax payers as well as welfare bene…ciaries. We have also shown in the previous subsection that the e¤ect of UBI and its …nancing on the labor supply is likely to be modest in such economies. However, given that the institutions in charge of tax and social welfare in these economies do not work as well as they do for developed countries, we now introduce imperfections in these institutions along the lines of section 6.

Allowing Frictions in the Tax System
We now analyze the feasibility of UBI allowing for noisiness in measuring income. We address the following question: if economies di¤er in terms of both the extent to which the administrative capacity is subject to frictions, as well as the distribution of income, under what circumstances a UBI scheme can provide the same amount of bene…ts with lower taxes, or higher bene…ts for the same amount of taxes?
To fund the UBI scheme, taxes would need to be collected and the question arises, what happens in this regard when income cannot be measured? For simplicity, we assume that taxes can only be collected when true gross income is accurately measured, the probability of which is given by p (y g ), and not when an uninformative signal about income is received, with probability 1 p (y g ). The modi…ed budget-balance condition for a UBI is: t Z 1 0 y g (z)p (y g (z)) g (z) dz = b: If p (y g ) = p for all y g then tpỹ g = b whereỹ g = + z + b 1 t . Therefore, the tax rate has to be higher to fund the same b compared to an economy where income is measured accurately.
Let y and y be the lowest and highest value of gross income in the economy, corresponding to z and z, the lowest and highest possible values of full income. Also, let us de…ne (z) y (z) p (y (z)).
To consider more interesting possibilities, suppose y g p (y g ) is increasing in y g -i.e., as true income goes up, the expected value of the signal of income also goes up. As a lower t is needed to fund a given b, it follows directly that it is easier to fund the same level of UBI in richer countries, and those countries where tax enforcement is better, i.e., the function p (y g ) shifts out.
More interestingly, suppose we compare two economies where the mean value of y g p (y g ) is constant but one of them has more inequality in the sense of secondorder stochastic dominance. Suppose y g p (y g ) is concave (resp. convex) which implies y g p 00 (y g ) + 2p 0 (y g ) is negative (resp. positive) assuming p (y g ) is di¤erentiable. In the case of concavity (resp. convexity) the more unequal economy will yield lower (resp. higher) tax revenue on average. Now consider the e¤ect of increasing income inequality. Suppose there is a shift of g (z) in the sense of second-order stochastic dominance. In the case of concavity (resp. convexity), the less (resp. more) unequal economy will have a higher value of R z z (z) g (z) dz and so a lower (resp. higher) t will be needed to fund a given b: We can think of concavity (resp. convexity) of y g p (y g ) as the case where the richer people …nd it easier (resp. harder) to evade taxes and so the curvature of y g p (y g ) can be interpreted as a measure of progressivity of tax enforcement. The case of concavity may be relevant in developing countries where the …scal capacity of the state is limited and the rich may be better able to hide income. With good tax enforcement, greater inequality raises greater revenue for a UBI while the opposite holds for bad tax enforcement. 50 We summarize our main conclusion as : when income is noisily measured, this tightens the …scal budget constraint for funding a UBI-scheme. However, if tax enforcement is progressive then greater inequality in the distribution of income relaxes the …scal budget constraint, while the opposite holds if tax enforcement is not progressive.

Conclusion
Despite our attempt to be broad, our overview falls well short of being comprehensive. For example, we have not paid su¢ cient attention to dynamics and uncertainty by focusing all through on a static deterministic model. Clearly, a welfare system has important impacts on savings, skill formation, and intergenerational e¤ects such as through human capital investments. Also, by providing a steady ‡ow of income, a UBI is likely to a¤ect risk-taking and entrepreneurship. Also, we did not address the speci…cs of a UBI scheme, such as in what frequency should a basic income be paid out (monthly, yearly), should it be paid before they earn any income or paid ex post yearly by the …scal authorities after they observe gross incomes, should employers be involved in paying out the basic income as they are for withholding taxes and then deal with the social security authorities, will individuals earning less than the basic income have to apply to receive it monthly.
We conclude by making a few points to take away from the debate between UBI and other forms of welfare programmes.
First, UBI does not look like the proposal that all egalitarian policymakers should wish to implement. There are many egalitarian social welfare functions that do not call for closing one's eyes on the reason why people have low incomes.
Second, among the normative values that may be called for to justify redistribution policies, poverty alleviation seems to be the ultimate value to justify UBI. That suggests …rst to compare UBI with other programs dedicated to the poor. That also suggests that UBI might be more appropriate in developing countries, especially those in which UBI could help circumvent the imperfections of government institutions in charge of helping the poor.
Third, we do not see any reason why guaranteeing a UBI and, through it, a universal minimal consumption, should necessarily be coupled with the crowdingout of other transfer policies. Complementing UBI with other, conditional income support policies are likely to be better than UBI alone.
Finally, in our theoretical framework, we did not allow for the role of public goods and services, or the role of policies that would lead to greater income growth (e.g., better infrastructure). As we argued, a UBI or a cash transfer will provide some relief to the poor. However, we do not suggest that UBI will provide a long-term solution to the problem of poverty. Therefore, even if UBI is accepted to be better than in-kind or conditional or targeted transfers, that does not mean that the entire budget of poverty alleviation or social welfare should be devoted to transfer programs.