An optional tax system

An optional tax system

Journal of Public Economics 24 (1984) 389-395. AN OPTIONAL North-Holland TAX SYSTEM A.A. SAMPSON* University ofShej,Eeld, Sheffield SIO ZTU, U...

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of Public


24 (1984) 389-395.




A.A. SAMPSON* University ofShej,Eeld, Sheffield SIO ZTU, U.K. Received

May 1983, revised version received



The model is of choice of hours worked and net of tax income by workers who vary in their skill levels. All workers face the same initial linear income tax schedule. We construct a set of linear income tax schedules, differeing in their degree of progressivity, and then allow each worker to choose under which schedule he shall be taxed. No worker is worse off under the option, and no worker pays less tax. Most workers are better off, and pay more tax. Hence, the optional tax system Pareto dominates the initial tax system.

1. Introduction In a world where all workers have identical preferences and identical skills or productivities, and the social welfare function is utilitarian, a lump-sum poll tax is optimal, Where skills differ but are directly observed by the government, a lump-sum skill-dependent poll tax is optimal. Where skills are observable only from hourly wage rates, such a tax will provide an incentive to disguise one’s skills, and Dasgupta and Hammond (1980) prove that a first-best tax system is not attainable. Any feasible tax system must therefore involve distortions. In their studies of the properties of the optimal nonlinear income tax, Atkinson (1973) and Mirrless (1971) find with a specific example that the optimal tax schedule may be approximately linear. Seade (1977) finds, however, that the optimal marginal tax rate at the highest level of earnings is zero, but it is not clear what implications this result has for the shape of the optimal tax schedule at lower incomes. In this paper we set up a simple model of choice between comsuption and effort and discuss a piecemeal improvement to a tax system that initially is linear. The new system is equivalent to a non-linear system, but its method of implementation is novel. Specifically we construct a sequence of linear income tax schedules, including the initial schedule. Each element except the first has a higher lump-sum element and lower marginal tax rate than the succeeding element in the sequence, which has a finite number of elements. Each worker is faced with the set of tax schedules, and has to choose one *The author

is particularly




to two referees

1984, Elsevier Science Publishers

B.V. (North-Holland)




An optional tax system

element of the sequence as the schedule under which he will be taxed. As the real gross of tax wage rate increases, a worker will opt for a tax schedule with a higher lump-sum element and a lower marginal tax rate. All workers will opt for a scheme under which they pay more tax than under the initial scheme. Hence, the optional system Pareto dominates the initial system.

2. The model Let there be a finite number m of ability levels in the working population. We measure ability by gross of tax earnings w per unit of effort, and index w in decreasing order of ability: wr>w2>...>wm-1>wm.


We assume that all individuals of a given ability level Wi have identical strictly convex preferences over consumption Ci (a good) and level of effort Li (a bad). We describe Li as ‘Labour supply’, and M$ as the gross hourly ‘wage rate’. Li could however be interpreted as intensity of effort under selfemployment, or level of output under piece-work. In these interpretations the government would not be able to observe & directly, and we assume below that w is unobservable for any individual. We asume that the same initial tax structure faces all workers, and consists of a lump-sum tax a (which might be negative) and a constant marginal tax rate t. A worker of ability level w has to choose consumption Ci and labour suply Li subject to ci = --a+(1




Hence, we can write total earned income x(a, t, K) and 7Ja, t, w) as functions of the parameters of the tax structure:



(3) We allow preferences over consumption and between ability levels but impose the restriction & > wj ---fY(u, t, fq > E;(u, t Wj),






so that an individual of higher ability will choose to earn more than an individual of lower ability. As stated, we assume that the government cannot identify the ability of any individual, The government is assumed to know how an individual of ability level q would respond to a change in the tax structure.

A.A. Sampson, An optional tax system


3. Optional tax system Consider an initial tax system (a’, to) where all workers pay the same rate of tax to, so that their lump-sum tax a’, and the same marginal consumption-labour supply locus is

ci= --aO+(l-P)H$L,. We construct and marginal

a set of tax structures tax rate, such that

(a’,t’) which differ in the lump-sum


All workers are offered the choice of the original structure (a’, to) or one element in the set (a’, t’). We prove that some workers will opt for a tax structure different from (a’, to), and that any such worker will pay more tax than he did under (a’, to). The construction rests on the following elementary proposition: Let the lump-sum tax a be increased and the marginal tax rate t be reduced, such that an individual’s original choice of lubour supply and consumption lies on the boundary of his new opportunity set: If his preferences are strictly convex, he will become better off, earn more income and pay more tax. The construction is diagrammatic. Fig. 1 is a diagram in consumption-gross income space, and for an individual of type @ can be interpreted as a


'i s3




= W.L.

Fig. 1



A.A. Sampson, An optional tax system

diagram in consumption-labour supply space. Let A’S0 represent consumption as a function of gross income under the initial tax structure (a’, to). Since K is indexed in decreasing order of ability, and income rises with ability, the most able individuals of type WI will choose A’ under (a”, to). The next most able, of type W,, will choose a point A2 to the left of A’, and so on. The line A’S’ depicts the opportunity locus of any individual under a tax structure (u’, t’), with u’ >a’, t’ t’s tl, which would be rejected by individuals of ability W, in favour of (a’, t’), but preferred to both (a’, to) and (a’, t’) by individuals of ability W,. If allowed, such individuals would choose (u’, t”) and a consumption-income point B,, at which they paid more tax than they did under (a’, to). Note that the structure (a’, t2), which is preferred to (a’, to) by individuals of ability W, and under which they pay more tax than under (a’, to), must be less attractive than (a’, t’) to individuals of ability W,. This is to avoid an individual of ability W, opting for (a2, t2) and then choosing a point like B2 at which he would pay less tax than he did under (a’, to). It is achieved by having A2S2 below his indifference curve through B’. We can similarly construct a schedule A3S3, representing a tax structure (u3, t3) with a3 t3. This would be preferred to (a’, to) by individuals of ability W,, and would be rejected in favour of (u2, b2) by individuals of ability W,. This procedure can be described in terms of an algorithm: for each ability level Wj, construct a tax schedule (uj, tj) such that individuals of ability level Wj prefer (uj, tj) to (a’, to) and pay more tax under (a’, t’) than under (a’, to). (a’, tj) in such a way that For each ability level Wj < W’ construct individuals of ability greater than W-” do not choose (a’, tj). This can be done for all ability levels, since for any point Aj in fig. 1 we can construct a line AjSj through A’ steeper than A’S’, which represents a tax structure (a’, ti) preferred to (a’, to) by individuals of ability Wj. We can then construct a line through Aj+l steeper than A’S0 but flatter than A’S’, which represents a tax structure rejected by individuals of ability level Wj. Hence we can construct a series of tax structures (a’, t’) commencing with (a’, t’) which has a high lump-sum tax and low marginal rate, and with the lump-sum tax falling and the marginal tax rate rising, as the series progresses. As constructed, each individual will opt for one element in the

A.A. Sampson, An optional tax system


set. Under his chosen tax structure, he will be better off, and pay more tax, than under the initial linear tax system (a”,to) faced by all workers. The optional tax system therefore Pareto dominates a common linear tax system. 4. Remarks 4.1. A continuum of ability levels The only result one can prove with a continuous distribution of skill levels is that a system with two options will Pareto dominate a linear system. The options are the initial tax structure (a”,to) and a structure (a*,~*) with a lump-sum tax equal to the amount paid by the most able worker under (a’, to) and a zero marginal tax rate. Individuals close in ability to the most able would choose the second option and pay more tax than they would under (a’, to). When one tries to construct a structure preferred to (a’, to) by individuals who prefer (a’,t’)*to (a*,~*), some individuals who prefer (a*,t*) to (a’, to) would then opt for this structure, and might pay less in tax than they did under (a’, to). 4.2. Rationale of the optional tax system This paper is about a piecemeal improvement to an existing linear income tax system, and only indirectly about optimal taxation. Since marginal tax rates are almost everywhere positive, distortions occur; hence, no first-best optimum is achieved. Mirrlees (1971) showed that where skill levels are directly and perfectly observable by the government, a skill-dependent lumpsum tax is optional in a utilitarian framework. Since the lump-sum tax will increase as skill increases, workers might have an incentive to disguise their skills. Viewing the skill of an individual as the highest wage he could achieve, a worker could disguise his skill by working for a lower wage than the maximum he could achieve. If skill levels are not directly observable by the government, but have to be inferred from wage rates, Dasgupta and Hammond (1981) point out that the first-best optimum of Mirrlees is not incentive compatible, since some workers will have an incentive to disguise their skills. They put forward an incentive-compatible second-best scheme that is Pareto efficient, which requires lump-sum taxes on revealed skills. In their paper individuals reveal their skill by choice of an observable wage rate. In the model of this paper, they reveal their skills by choice of a tax structure. One would expect this outcome to be Pareto dominated by an outcome attainable by the Dasgupta-Hammond system, if the government could observe wage rates directly. There are many cases, however, where a government cannot observe wage rates, or more precisely reward per unit of effort, since effort is not

A.A. Sampson, An optional tax system


observable. About the only observations on effort a government could make is time spent at work, and in many cases this poor measure of effort. One could instance artists and entertainers, professional and scientific workers, self-employed craftsmen and entrepreneurs as cases where effort is not observable. Similar remarks could be made for workers on piece-rates, and the principal-agent problem itself arises because effort is not readily observable. Also, the reward structure, which determines how effort is measured (e.g. time or output), may itself vary as the parameters of the tax structure vary. The informational requirements of the optional tax system are less stringent. In principle all a government would need to know is the likely response of an individual, who earns a given income under an existing linear tax structure, to a different tax structure. This response will allow a move towards an ideal system of skill-dependent lump-sum taxes.

4.3. Income distribution In its operation the optimal tax system would be formally equivalent to a uniform tax system with non-increasing marginal tax rates. In fig. 1 the sequence A’S’ of consumptionincome loci (and hence tax structures) is depicted. Individual choice of tax structures would yield the upper envelope of these segments as the effective consumption-income locus. Hence tax per worker as a function of his income would be as depicted in fig. 2. If a0 < 0 so that the initial tax system were the social dividend version of the negative



Fig. 2



A.A. Sampson, An optional tax system


income tax, the average rate of tax would fall with income at first and then rise. We note that the gain in income for the skilled individuals will be larger than the gain for the unskilled. Post-tax income inequality would increase, but social welfare on any utilitarian criterion (including maximin) would rise.

Atkinson, Anthony, B., 1973, How progressive should income tax be?, in: Michael Parkin, ed., Essays in modern economics (Longmans, London) 9&108. Dasgupta, Partha and Peter Hammond, 1980, Fully progressive taxation, Journal of Public Economics 13, 141-154. Mirrlees, James A., 1971, An essay in the theory of optimum taxation, Review of Economic Studies 38, 175-208. Mirrlees, James A., 1976, Optimal tax theory: A synthesis, Journal of Public Economics 6, 327358. Seade, Jesus M., 1977, On the shape of the optimal income tax schedule, Journal of Public Economics 7.203-235.