With an imperfect **income** **tax** base assessment, a new term appears in the numerator: ( ) w dF 0
y
ò > , where ( ) y w º ò Y ¢ U dH C h . This term depicts the marginal social costs from the **tax** system’s inaccuracy, i.e., the increase in the variance of after-**tax** **income** generated by a marginal increase in the **tax** rate. The new term takes account of two effects. First, with a higher **tax** rate, individuals bear more risk from a given degree of inaccuracy. 7 Second, due to inaccurate **tax** base assessment there is a larger dispersion of after-**tax** incomes within each w -class. Both effects run against the (strictly positive) first term in the numerator, the welfare gains from redistribution reflected in the covariance term. 8 Put simply, inaccuracy reduces the effectiveness of progressive taxation in generating a more equal after-**tax** **income** distribution. Hence, the **optimal** **tax** rate (and the **optimal** demogrant) depend not only on the manipulability of the **tax** base, but also on its **observability**. While a higher manipulability increases the deadweight loss of a given **tax** rate per dollar raised, more **observability** decreases the social costs from inaccurate taxation.

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Since the penalty rule (43) refers to a sufficient condition for the no-evasion condition (11) to be satisfied, one would expect the penalty needed to satisfy (11) to be smaller for risk averse agents. Indeed, when agents are risk-adverse, and thus endowed with a concave utility function, a somewhat smaller penalty may be sufficient to comply the no-evasion condition. Eventually, the penalty could be equal to that imposed by the US system if risk aversion is large. In this case, however, independently of the class of strictly concave utility function chosen to model agents’ preferences, it is not generally possible to explicitly solve equation (11) for the penalty. Numerical simulations suggest that the concavity of the utility function produce a significantly different penalty only when the sheltered **income** is a large proportion of earned **income**. In our example, the difference in the penalty to be applied is negligible even for extremely risk averse agents.

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Our results are closely related to those in the existing literature. Aiyagari and McGrattan (1998) developed a heterogeneous agents model with incomplete markets and analyzed the **optimal** debt policy under the utilitarian social welfare functions. However, while the **income** **tax** schedule is **linear** in their model, we allow for progressive **income** taxes. HSV (2014) provided a tractable model of **optimal** **tax** progressivity. The endogenous labor supply and skill accumulation limits high progressivity—the **optimal** rate in their model is lower than the current progressivity. Our model embodies capital **tax** rates and consumption **tax** rates as well as **income** **tax** and focuses on the **optimal** **tax** rate for the Korean economy. Our results imply that the **optimal** **tax** rate in Korea is much higher and more progressive than the current **tax** schedule. Chang et al. (2015) computed the **optimal** **income** **tax** rate for each of the 31 OECD countries, including Korea. 1 Under the

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There are important limitations that we share with many **optimal**-taxation models. First, there is no account of capital, although it should be even more mobile than high-skilled labor. We focus on **income** taxation because we want to clearly identify the e¤ect of combining competition with the principal– agent framework that underlies **optimal** taxation models. Second, due to the simple **linear** production technology in one good economy, there are no general-equilibrium or trade e¤ects of the wage changes that could lead to repercussions on the e¤ects discussed.

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What the analysis shows is the following result: any qualitative structure for the **optimal** **tax** function can be supported by some skill distribution. Expressed alternatively, except for the fact that the marginal rate cannot rise between the second to highest and highest skill consumers, there are no a priori restrictions on the qualitative properties of the **optimal** **tax** function. So the structure of the “classical” **optimal** **tax** function is just a consequence of the restricted set of simulation specifications and does not capture some deeper feature of **optimal** taxation. The model used here assumes utility is **linear** in labor supply. Diamond (1998) has already exploited a **linear**-in-consumption model to show **tax** rates may increase above the modal **income** for some skill distributions.

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and it increases as G increases. In contrast the quadratic part of the **income** **tax** structure τ 2 moves from a positive number to increasingly negative numbers. This indicates that the **tax** structure moves from one which is progressive to ones which are more and more regressive. The rate of money creation λ increases as G increases, but somewhat surprisingly it changes very little and remains negative for all values of G. Also surprising is that the changes in policy decision variables, as well as the changes in other variables in the table, indicate approximate linearity with respect to G. This is surprising because while the model is **linear**-quadratic with respect to consumption and leisure, it appears to be highly nonlinear with respect to the policy decision variables.

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When we extend the basic model of **optimal** **income** taxation to include a more realistic treatment of the labor market, a number of new interesting mechanisms arise. In the case with exogenous wages we Þnd that a positive external eﬀect from longer work hours gives the government incentive to lower the marginal **tax** for both skill types. In the endogenous wage model the new mechanisms are more intricate, since the **tax** system now also aﬀect the bargained wages and thereby also the unemployment rates. Perhaps the most surprising insight is that the government now will, to some extent, use the **tax** system to redistribute through the unemployment rates. Lower risk of unemployment gives — ceteris paribus — higher expected utility. To transfer utility in the form of low risk of unemployment is a very reÞned way to redistribute, since it does not cause any adverse behavioral eﬀects; the potential mimicker does not enter the low skilleds’ labor market and she therefore cannot beneÞt from lower low skilled unemployment.

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Thus, given the **optimal** choice of **tax** brackets and of the lump sum a, the **tax** rates are set optimally over the sub-populations within each bracket. The advantage over a strictly **linear** **tax** is therefore that the **tax** rates can more closely take account of di¤erences in the relationships between **income** and the marginal social valuation of **income**, and in the average deadweight losses, across the subsets of the population. This suggests the intuition that there would be little to gain from deviating from a **linear** ("‡at") **tax** when the ratio of the equity e¤ect to the e¢ ciency e¤ect remains constant as we move through the wage type distribution. As we show in the numerical analysis in Section 4 below, however, given realistic wage distributions the two bracket progressive **tax** does deliver higher social welfare than the **linear** **tax**, even when the elasticity on gross **income** with respect to the **tax** rate is constant throughout the population.

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A case for over-insurance of low-skilled workers arises when the so called self selection constraint (SSC) binds. The SSC ensures that the high-skilled workers prefer the consumption bundle that the government intends for them rather than the consumption bundle intended for the low-skilled workers. This constraint is due to the government’s lack of information about each worker’s skill type. If the high-skilled workers earn the same **income** as the low-skilled workers, the government cannot levy diﬀerent **tax** rates, nor dif- ferent unemployment beneÞts since the beneÞt level is based on the **income**. In the **optimal** taxation literature this is called ”mimicking”; the high-skilled workers may mimic the low-skilled workers in order to avoid high taxes. The attractiveness of mimicking generally increases with the level of resources redistributed from high-skilled workers to low-skilled workers. But if unem- ployment is mostly a low-skilled phenomenon, increased low-skilled beneÞts will not increase the attractiveness of mimicking to a large extent. The reason for this is that the mimickers do not beneÞt from the low-skill unemployment beneÞts to the same extent as the low-skilled workers do.

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In this paper we investigate how accounting for the ability of individuals to avoid **tax**, as well as to evade **tax**, alters the conclusions for **optimal** enforcement of models in which only **tax** evasion is possible. In our model individuals can engage in **tax** evasion by under-reporting their **income**, but can also, at a cost, participate in a **tax** avoidance scheme that permits them to further lower reported **income**. Additional to the financial cost of avoidance, both forms of non-compliance are assumed, when detected, to impose psychic harm in the form of a social stigma cost. The nature of the avoidance scheme is not unambiguously prohibited by law, but is unacceptable to the **tax** authority. Accordingly, if the **tax** authority learns of the scheme, it will move to outlaw it ex-post. If a taxpayer is audited the **tax** authority observes whether they are using a **tax** avoidance scheme and also the extent of any **tax** evasion. The taxpayer is fined on the evaded **tax**, but the **tax** authority has no grounds to impose a fine on the avoided **tax** (it can only take measures to outlaw the scheme and then recover the **tax** owed on the avoided **income**). In this context we characterise the audit function first for a **linear** penalty function, and later for a general penalty function. The **tax** authority can condition its audit function only on the amount of **income** declared; it does not observe the amount of non-compliance or how it is split between evasion and avoidance. We therefore look for a taxpayer such that, if this taxpayer (weakly) prefers to report truthfully rather than hide an amount of **income**, then all other taxpayers will also wish to report truthfully.

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As a first step, we needed to consider the specific structure of the labor supply curve. According to relevant literature, the elasticity of supply is mostly positive. [6] estimated elasticity of labor supply to be about 0.12. [7] accounted for the wage differential between part time and full time work, and applied four differ- ent estimation procedures to estimate uncompensated wage elasticities of wom- en’s labor supply. They found elasticities that were always positive. [8] provided a review of estimates of labor supply elasticity. They found several examples of negative wage elasticity estimates. [9] reported backward bending labor supply schedule for self-employed sea-scallop fishermen, who could freely adjust their working hours, unlike employees who are often demand constrained. However, [10] reported that many estimates of elasticities might be biased downwards as a result of the failure to take into account discontinuities in the labor supply func- tion, taxes and non-**linear** budget constraints.

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In the preceding sections source- and origin-based taxes are presented as an indirect means of taxing labour at diﬀerent rates when the labour **tax** is **linear**. However, the informational constraints do not bind the taxation on labour **income** to be **linear**: the government does not need to know individual wages and the labour supply in order to implement a nonlinear **tax** on labour **income**. Are source- and origin-based taxes still desirable when an **optimal** nonlinear **income** **tax** is levied?

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This appears to be an attractive route because, unlike the usual approach to op- timal **tax** modelling, the conditions can be expressed in terms of what appear to be empirically observable counterparts such as elasticities. However, given the consider- able complexities introduced by nonlinear budget constraints — unlike the consump- tion **tax** case where **linear** pricing is a reasonable assumption — any clear results need strong assumptions and could only be regarded as illustrative rather than of practical relevance. Nonlinear budget sets make it diﬃcult to generalise regarding labour sup- ply responses and welfare changes even for workers with similar preferences and with relatively simple **tax** structures. In practice, populations display considerable hetero- geneity in preferences and household circumstances, and **tax** and transfer structures are extremely complex.

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papers leave the literature in a rather un…nished state, despite the fact that the paper by Slemrod et al gives a very thorough and insightful discussion of the results of its simulation analysis of the nonconvex case, as well as of the problem of piecewise **linear** taxation in a model consisting of only two types. The contribution by Sheshinski was the …rst to formulate and solve the problem of the **optimal** two-bracket piecewise **linear** **tax** system, including the choice of the bracket threshold, for a continuum of worker/consumer- types. However, he claims to have shown that, under standard assumptions, marginal rate progressivity, the convex case, must always hold: in the social

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Productivities are Pareto distributed where f (n) = µn µ /n µ+1 ∀ n ≥ n. The Pareto distribution captures well the upper tail of observed **income** dis- tributions and its adoption in the more recent **optimal** **tax** literature has supported increasing marginal **tax** rates on higher earners. 5 It would there- fore seem appropriate for simulating progressive piecewise **tax** schedules. To capture how the spread of abilities affects the results, we consider two alter- native distributions: (i) n = 1, µ = 4 ; and (ii) n = 1.067, µ = 5, where n is adjusted so that the average productivity is 1.333 in both cases. The second distribution has a smaller spread of abilities than the first.

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In fact, many developed countries of today rely on environmentally motivated taxes, in order to improve the quality of the environment and there has been an ongoing discussion of how these environmental taxes affect the rest of the **tax** system. According to economic theory the Pigouvian solution results in a first-best outcome, if there are no other distortions that affect the allocation of resources in the economy. If the government is not able to set the rest of the **tax** system in an **optimal** way, that is, no lump-sum taxes or transfers are available, then the government faces a second-best world.

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Thus, given the **optimal** choice of **tax** brackets and of the lump sum a, the **tax** rates are set optimally over the sub-populations within each bracket. The advantage over a strictly **linear** **tax** is therefore that the **tax** rates can more closely take account of di¤erences in the relationships between **income** and the marginal social valuation of **income**, and in the average deadweight losses, across the subsets of the population. This suggests the intuition that there would be little to gain from deviating from a **linear** ("‡at") **tax** when the ratio of the equity e¤ect to the e¢ ciency e¤ect remains constant as we move through the wage type distribution. As we show in the numerical analysis in Section 4 below, however, given realistic wage distributions the two bracket progressive **tax** does deliver higher social welfare than the **linear** **tax**, even when the elasticity on gross **income** with respect to the **tax** rate is constant throughout the population.

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On the other hand, several authors favor the use of housing subsidies. Cremer and Gahvari (1998) prove that the differential **tax** treatment of housing may be justified on grounds of **optimal** **tax** policy, creating the con- ditions under which consumption of housing by the poor must be subsi- dized. Also Nakagami and Pereira (1995) show that first-time home buy- ers would suffer great utility loss from the elimination of mortgage-interest deductibility.

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We would argue that the economic circumstances of households in this type of sample are, at least to some extent, consistent with two assumptions of **optimal** **tax** theory - that productivities are innate and cannot be observed and, therefore, that wage rates representing productivities can be treated as exogenous and unobservable. These assumptions cannot plausibly be con- sidered to hold in the case of recipients of disability pensions or long term unemployment bene…ts. Many types of disabilities are observable and disabil- ity pensions are individual-speci…c and not part of the general **tax** system. In the case of the long term unemployed, the available empirical evidence sug- gests that their earnings possibilities re‡ect the need for further education, training and work experience, implying that a broader set of policy instru- ments than **income** taxation are relevant, and indeed are in use. We therefore regard the results for the above set of distributions as being the most relevant for the general analysis of **tax** systems in present-day economies.

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