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Atlantic Review of Economics 

            Revista Atlántica de Economía

Colegio de Economistas da Coruña
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Volumen 4 Número 08: A Technical Remark on the Problem of the Optimum income taxation

Xabier Ruiz del Portal*  
Universitat de Lleida

Reference: Received 21st February 2005; Published 13rd May 2005.
ISSN 1579-1475

Este Working Paper se encuentra recogido en DOAJ - Directory of Open Access Journals http://www.doaj.org/



Este artículo presenta un método para tratar problemas del tipo Pontriyagin, cuando éstos pueden contener restricciones para que ciertas funciones resulten de variación acotada. Entre la lista de modelos económicos incluyendo este tipo de restricción, la formulación por Mirrlees del problema de la imposición óptima sobre la renta ha sido analizada bajo el enfoque propuesto. Las condiciones necesarias resultantes apenas difieren de las que aparecen en la literatura, exceptuando los intervalos donde la función monótona z(n) resulta, a un tiempo, singular y estrictamente creciente.


This paper presents a method for treating problems of the Pontriyagin type, when we allow them to contain constraints restricting some functions to be of bounded variation. Among the list of economic models including this kind of constraints, Mirrlees´ formulation of the optimum income tax problem has been analyzed under the proposed procedure. The necessary conditions found hardly differ from those in the literature, excepting intervals where the monotonous z(n) happens to be, simultaneously, singular and strictly increasing.

1.- Introduction

   It has become customary, when dealing with monotonous functions in optimization processes1, to adopt continuity and differentiability assumptions which reduce the scope of application in the results obtained. These are then said to have only a heuristic validity.

   This working paper is aimed at finding a simple procedure, when continuity and differentiability restrictions are relaxed, for dealing with functional monotonicity constraints defined over compact intervals, within the context of problems that can be treated through Pontriyagin Maximum Principle techniques.

   The Maximum Principle has a number of advantages compared with other approaches such as the explicit use of the calculus of variations. First, the whole number of investigations and their degree of generality constitute a growing field, where the set of topics analysed expands everyday.2 Second, the existence of theorems, providing both necessary and sufficient conditions for the question studied, gives more reliability to the results than if they had been obtained through other "ad hoc" methods. Also, the way one arrives at the particular conclusions becomes much more accessible to the public, as well as manageable in introductory surveys of the different sources of literature where dynamic optimization is required3 . Lastly, Maximum Principle allows an easier interpretation of the findings reached in contrast to the often ambiguous results of the variational analysis.

   As a second issue, the use of monotonous functions on maximization processes has its own interest for several reasons, especially when they are defined over a compact interval. In this way, monotonous correspondences, or monotonous set-valued functions, are remarkably "functions-like" for they coincide, possibly excepting an enumerable (finite or not) or countable set of points, with monotonous functions 4. Therefore, when monotonicity is guaranteed we do not have to worry for the presence of multiform functions, so frequently found on economic grounds [cf. Green and Heller (1981, pg.46)] when facing any particular investigation.

   Moreover, monotonous functions over compact intervals have specific properties which are very well known 5 . Thus, they have at most a countable set of discontinuity points of the first kind, and a finite derivative almost everywhere, i.e. with the only possible exception of a set of measure zero. Further, being measurable and bounded, monotonous functions fulfil the standard requirements to be treated as control variables on dynamic optimization problems of the Pontriyagin tradition.

   In the following Section we explain the foundations of our method. Though the analysis here lacks a large economic content, it seems to me necessary in order to understand the remaining parts of the paper. Section 3 is devoted to applying our approach to the optimum income taxation model with the goal, in so doing, to compare the results with those of Mirrlees (1971) that gave rise to the now classical theory on redistribution, and whose derivation was made by using "ad hoc" variational techniques; such comparison will serve, in Section 4, to elucidate a number of relative advantages arising from the use of the proposed procedure as applicable on other economic grounds.

   Conclusions can be found in Section 5, while an appendix with some analytical details is included in Section 6. The Appendix represents the only part of the paper which is not self contained, as it relates with the application of a theorem from the optimal control theory to a problem of maximization; notwithstanding this, in order to help those who do not want to lose much time reading the paper with the theorem, the same Section also contains a non-rigorous derivation of the main conditions of optimality.

2.- Adaptation to the optimal control framework

   In addition to the properties referred, and according to the so-called Lebesgue´s Decomposition Theorem [Cf. for example Stromberg (1981, pp.381)], every non-decreasing function p(t), defined on an interval , can be uniquely represented in the form:

   We shall come back to this point at the end of the present Section. Of course, if is absolutely continuous on any subinterval of I, it is a constant function all over such range.

   As we shall have to work with essentially bounded controls, let us construct for a given positive number k the function:

   Here is absolutely continuous and, since a.e., also non-decreasing.6

   The function , being equal to , turns out therefore to be non-decreasing. At the same time, x(t) consists of an absolutely continuous non-decreasing function whose derivative is bounded where it exists, i.e. almost everywhere. Thus, we have obtained (8) which is the kind of expression we were seeking for, as it may represent a monotonous function with an absolutely continuous part satisfying, notably, the requirements needed in the control theorem used below.

   We shall make the assumption that the function is known in advance. With it we mean the corresponding to the optimal function q*(t) that solves any particular control problem, one of whose constraints is of the form:

   If instead of this constraint we write (5)-(6) and (8) with replacing , the new problem then happens to be quite easy to deal with, provided both q(t) and u(t) are treated as control variables, and x(t) as a state variable. Once we see as representing a given parametric function 7, whose unique role reduces to help finding optimal trajectories for q(t) and x(t), it can easily be checked that the solution for the new problem will exactly coincide with that of the original one, i.e. q*(t).

   Such assertion rests heavily on the fact that with the method here suggested, all admissible x(t) will always be non-decreasing trajectories, thus implying by (6) that the resulting q(t) will be non-decreasing too. Therefore, with the second problem still we keep on seeking for the optimal non-decreasing q(t), but this time through x*(t), since is given.

   However, and this is important to be stressed, the admissible trajectories x(t) will continue being non-decreasing if, instead of (6), we write its right-hand side inequality alone, i.e. we want to put u(t)0 replacing (6). In this way, the rationale just followed to justify our method with an almost bounded u(t) will remarkably serve, again, whenever we let such function to be unbounded. Moreover, as this implies that the optimal solution among those candidates u(t) not necessarily bounded will precisely coincide with u*(t), something that we know in advance, there will exist no mistake if we choose, for the characterization of the optimal solution, a theorem whose admissible control variables are necessarily (measurable) almost bounded functions.

   Summarizing, once carried out the conversion of a problem with a general monotonicity constraint, as (9), into another where the choice is made only among all possible non-decreasing absolutely continuous functions with an almost bounded derivative, as in (5), (6) and (8), we may delete the previous upper bound on the function defining a.e. such derivative and solve for the second problem now with (6) thus transformed 8. The solution will necessarily remain being the same, and such circumstance will allow us to apply an optimization theorem where the only controls admitted are those represented by (measurable) essentially bounded functions.

   Now, let I´ be any subinterval of I. Below we shall make use of the following:

   PROPOSITION 1: q(t) is a singular function over I´ if and only if x(t) turns out to be constant along the same subinterval.

   Proof: x(t) and being both non-decreasing their derivatives are non-negative a.e. Thus, a singular q(t) requires a constant x(t). As for the first part of the Proposition, constancy of x(t) implies through (5), (4) and (2) that is also constant, and then by (7) that becomes constant, as well; this fact, together with (3), proves that defines a singular function in this case. Of course, q(t) is constant if and only if both x(t) and are constant, too.

   Let us suppose that we face a problem which, except for its constraints of the type in (9), may be included within the problem setting studied by Makowski and Neustadt (1974), satisfying therefore their so-called conditions .Then, there will always exist the possibility of using (5)-(6), instead of (9), and apply Theorem 12.1 in Makowski and Neustadt (1974, Sections 13).

   It should be emphasized that the choice of Makowski and Neustadt (1974), among other theorems in the optimal control literature allowing for equality-type phase constraints 9, could be explained for several reasons. Above all, it includes a highly general set of necessary conditions within a wide frame of terminal conditions, equality together with inequality constraints defined, not exclusively throughout the whole interval of existence I, but possibly over certain subintervals, or even measurable subsets of I as well. Moreover, unlike other theorems imposing piecewise or even global differentiability on the controls, Makowski and Neustadt (1974) do not require any special assumption other than measurability, which is a typical property of monotonous functions 10. In fact, Problem III below consists only of an extremely particular case of the general system of equations and constraints examined in Makowsky and Neudstadt (1974).

   In the following paragraphs we shall use indistinctly expressions like "intervals of singularity" and "singular intervals" to denote, of course, ranges where the non-decreasing function in question is singular, i.e. where its derivative equals zero almost everywhere 11. In similar terms, "intervals of constancy" and "constant intervals" will express the same meaning in the text, being therefore alternatively interchanged as well.

   Furthermore, with a "non-constant singular function" we shall refer to the non-absolutely continuous case, a particular example of which is the strictly increasing singular function, here denoted as "increasing singular function". Such apparently odd mappings can even be continuous 12.

   Finally, words like "increasing" or "increase" will be always applied in a strict sense. Just with emphasizing purposes we shall write them, from time to time, after the adjective "strict".

3.- The optimal nonlinear income tax

   The same notation as that of Mirrlees (1971) will be used throughout, so that any confusion that might arise would be avoided. It is assumed that individuals share the same utility function U(x, y), representing their preferences between consumption x and time worked y.

   U(x, y) is strictly concave, three times continuously differentiable, strictly increasing in x and strictly decreasing in y, defined for x > 0 and 0 y < 1. As x tends to 0 from above, or y tends to 1 from below, U(x, y) tends to - .

   Every person possesses an ability level n, the distribution of which follows a specific density function f(n), defined on the interval [0, N].

   The function f(n) is supposed continuous and strictly positive 13for every n [0, N]. Likewise, denotes labour in efficiency units.

   The choice of an n-person is represented by {x(n), y(n)}when maximizing utility under a budget constraint xc(z) imposed by the government. The public planner chooses arbitrarily the corresponding consumption function , which is taken to be upper semi-continuous.

   There exist a production function H(Z) continuously differentiable with respect to total work Z available in the economy, i.e.:

   Inequality (12) means that total labor supplied (in efficiency units) must produce, at least, total quantity of goods demanded.

   Let represent the social utility function, whose integral over all individuals symbolizes the so-called social welfare function 14:

   is assumed twice continuously differentiable, increasing and concave.

   The government chooses the function so as to maximize the social welfare function in the light of two constraints: the production possibilities and the fact that each individual, in turn, tries to optimize his utility function subject to the budget restriction, which itself -and here underlies the special complexity of the income tax problem- depends on . This public policy can be formally expressed as, what from now on we shall call:



   Dealing directly with Problem I seems to be extremely hard 15, so we need to transform it into another more manageable characterization, in which the function c(.) does not appear in a explicit form. Let us first introduce the following notation:

Note that, as > 0, the second relation is legitimate.

Although no complete proof has yet been published for the general case
16, Mirrlees (1969, 1971) has shown (nearly) that if preferences satisfy the single-crossing property > 0, Problem I is equivalent to:



   Just to comment in passing, one way of dealing with Problem II is to maximize after deleting c), i.e. with constraints a)-b) only; call this new setting Quasi-problem II. If the y(n) found is such that n.y(n) comes out non-decreasing, it is then also a solution for the whole Problem II. This is the approach suggested in Mirrlees (1976, pg. 336) and adopted, for a different kind of models, by Laffont (1987, pg. 141) and Spence (1977, pp. 10-4) among others. However, the resulting necessary conditions would be only quasi-necessary for the optimum income tax problem, since we do not have any guarantee that the y*(n) of Problem II will also solve Quasi-problem II. We shall come back to this point later.

   Of course, the best alternative when seeking for necessary conditions of Problem II is to take explicit account of constraint c). For doing this task successfully, the best method among other possible approaches such as Mirrlees´ variational argument turns out to be the one suggested in Section 2, and that is what we are going to do all at once.

   Nevertheless, just before doing it we would like to clarify some possible misunderstanding that might arise after reading Mirrlees (1971), in part due to the presentation in that paper of results whose derivation had been carried out on an earlier unpublished work, and also because of the somewhat obscure argument used to justify the allowable variations in the presence of a restriction on the utility function. The confusion in question would consist in that one might think, prima facie, that Mirrlees´ assumption (B), i.e. property > 0 above, directly intervenes from Problem II onwards in the derivation of the necessary conditions for an optimum, precisely those that characterize the structure of the optimum income tax 17.

   But the answer is clearly negative; the role of the single-crossing property ends up in the transformation of constraint 2), in Problem I, into constraint c), in Problem II. Of course, it also serves in the rationale to justify the non-negative sign of the optimal marginal rates, though this is something that is done once the demonstration of the necessary first order conditions has been fulfilled, not before. Neither Mirrlees´ deduction of such necessary conditions using variational methods, nor our own derivation made in the Appendix through Problem III below, have had at all the need to resort to the referred key property, as we shall have the opportunity to see on the next pages.

   Therefore, introducing ex novo measurable, essentially bounded functions R(n) and r(n), the following step will consist in expressing constraint c) in the terms explained in Section II as:

, where the value of entails as before some specific positive number k previously chosen which, being in reality an upper bound of r*(n), we do not need to make it stand out explicitly in the analysis of the optimal solution.

   After the substitution of c) by c´) and, in order to treat the resulting new formulation of Problem II as an optimal control problem of Mayer, we again introduce new variables W(n), Z(n), X(n), which are assumed to be absolutely continuous functions. After some minor changes to convert what is indeed a maximization problem into an equivalent minimization problem, we shall have fully completed the adaptation of our problem so that it can be dealt with within the frame of the optimal control theorem below. The final setting that arises is given by:



   Notice that (14)-(16) is the form that (13) adopts here; in a similar way, (17)-(20) together with (27) exhibit identical contents 18 as (10)-(12). Hence, Problem III is readily seen to be equivalent to Problem II, and the same could be said then in connection to Problem I as well.

   In addition, W(n), Z(n), X(n) and R(n) have been taken to be absolutely continuous functions and, because of constraint b) in Problem II, u(n) exhibits the same property as well. On the other hand, we have seen that there is no loss of generality if, apart from y(n), which is measurable and bounded from the definition of U(x, y), we suppose r(n) also measurable and (essentially) bounded.

   Therefore, the various functions satisfy all the technical requirements demanded by the control theorem to be applied in Appendix 1, where a proof can be found for the following necessary conditions 19:

   iii) At points of increase 20contained in intervals where z(n), in addition to being non-singular, either is continuous, or/ and, shows no subinterval of singularity:

   If is a non-decreasing function of y for constant u, and n is a point as in iii), z(n) results continuous.

   Comparing these necessary conditions with those in Mirrlees (1971, Theorem 2), one can easily check that the only remarkable differences are:

   I) Our statement v) only works for discontinuity points satisfying our statement iii), failing also to display the additional condition:

   Contrarily, condition (v) in Mirrlees is stated for whatever discontinuity point that may exist.

   II) In Mirrlees (1971), the point-wise condition (30) in his condition (iii) seems to remain in force for all kinds of points of increase, even for those belonging to intervals of singularity, unlike here where for them only the interval-wise condition iv) has been shown to hold. This becomes the major difference with our results since:

   - Subintervals of non-constancy of z(n), where z´(n) = 0 a.e., are included in iv) while for Mirrlees (1971) they were comprised in iii).

   - The first part of v) only faces discontinuities produced outside subintervals of singularity for z(n).

   At first sight, it seems as though these cases of a non-constant singular z(n) could have been disregarded in Mirrlees´ work, as representing unimportant, or unusual, pathological situations. If this were the whole story, the present paper would at least have served, in the context of the optimal taxation literature, to add a minor correction towards greater generality.

   The consequence might be, therefore, that widening the field of application of the income tax model cannot be done without a slight loss of strength in the necessary conditions. In this sense, notice that iv) is a much weaker condition than iii), since the first condition is always implied by the second. Moreover:

   The well known "bunching case" [cf. Seade (1977), Mussa & Rosen (1978), Picard (1987), etc] would no longer constitute the only conflicting case, having to share that label with the increasing singular case 21, among other kinds of singularities.

   But it may be the case that the unpublished work promised in Mirrlees (1971), using the cumbersome variational argument, applies to all kinds of non-decreasing z(n) functions, as it happens here. Then i)-vii) in Mirrlees (1971) would have the same ambit of attention as the one studied in the present work, displaying however slightly stronger necessary conditions as a counterpart of arising from a much harder technical analysis.

   I suspect that the reason for our weaker conditions iii)-iv), versus those in Mirrlees (1971) if really they applied to all kind of non-decreasing z(n) functions, rests in the fact that the Maximum Principle approach only benefits from one of the two functions appearing in the right hand side of (9): specifically the one that is absolutely continuous with a derivative which is bounded almost everywhere. Functions never intervene, not even when they are themselves absolutely continuous.

   Finally, it should be stressed that, according to Propositions 1-3, our necessary conditions i)-vii) become identical to those in Mirrlees (1971) if and only if y(n), or equivalently z(n), is absolutely continuous within the maximal intervals of n where z(n) is singular.

4.- Conclusions

   The main goal of this paper has been to derive a method for dealing with functional monotonicity constraints, when searching for necessary conditions in optimization problems of the Pontriyagin Maximum Principle type. As is well known, this sort of constraint is very often found in models investigating both public and private economic behaviour.

   Once the corresponding procedure has been stated, we have applied it to the widely known optimal non-linear income tax formulation outlined by Mirrlees in 1971, though a bounded distribution of skills was taken in consideration. The results are very close to those of Mirrlees, differing only in the characterization of the optimal solution in the ranges of n where the monotonous z(n) turns out to be non-constant, but singular. The difference is therefore very technical.

   As a counterpart of the faint loss in the strength of necessary conditions for those increasing singular ranges, there appear many important advantages stemming from the straightforward character of our methods, more based on the application of well-known existing theorems, than in the use of "ad hoc" optimization techniques. Among them, there is the possibility of extending the typical characterization arising from the analysis here, represented through both the point-wise and the integral-wise first order conditions, to a larger number of economic problems, far beyond the exploration of the optimum income tax.

   After considering other models with non-decreasing function constraints, the general finding that one might probably deduce would be that the key distinction (when we do not impose special assumptions of piecewise continuity, almost differentiability, etc.) ought to be between singular and non-singular intervals, rather than between increasing versus constant intervals.

   Such conclusion certainly diminishes the relative importance of the so-called "bunching case", expanding the spectrum of those ranges where necessary conditions are weaker. To obtain stronger conditions in these ranges of the monotonous functions one may perhaps appeal to "ad hoc" more detailed analyses, not always available in the particular optimization process under study, or else, adopt more stringent assumptions and seek heuristic solutions. In the end, it seems that the key distinction, in order to achieve point-wise necessary conditions rather than those much weaker ones with an integral shape, should no longer be between increasing versus constant , but between singular and non-singular intervals. This implication certainly increases the range of difficulties somehow above the well-known "bunching case", when the only awkward intervals are those of constancy, but it represents the price for relaxing the customary continuity and differentiability assumptions.


   Since the proof here closely follows Theorem 12.1 in Makowski and Neustadt (1974), it would be convenient for the reader to have in hand a copy of the paper, though only pages 186-90 and 213-6 of it are really needed to get a good understanding of the various steps that complete the analysis below.

   From now on, instead of etc. we will write etc., respectively.

   Of course, it can easily be checked that Problem III fulfills by far conditions A-1/A-6, as well as C1-C2 in Makowski and Neustadt (1974) 22.

   Then, according to the mentioned Theorem 12.1 in connection with Problem III, now transformed into an equivalent minimization problem, for each of our boundary conditions (14), (16), (18), (20), (23) y (27), there exist respectively one parameter: and p, not all zero; for each of the inequalities and equality constraints (24), (25) and (26) above, there exist respectively one essentially bounded function: and for each of our differential equations (13), (15), (17), (21) and (22), there are respectively absolutely continuous functions: such that, by virtue of the differential equations (12.21) above:

   Both, the first order conditions in terms of each control variable (12.28) above, and the two requirements for each control variable established in statement (viii) of Theorem 12.1 above, yield:

   Readers familiar with the classical theory on optimal control 23will recognize (38)-(40) as the multiplier equations; also, (41)-(44) represent the optimality conditions; and similarly, (45)-(50) coincide with the transversality conditions. Of them, (38)-(44) may be generated, heuristically, in a similar way as that of solving a nonlinear programming problem consisting in the maximization of the Hamiltonian:

   , subject to the so-called nonnegativity control constraints (24)-(25) and the equality-phase constraint (26). As usual, represent the costate variables.

   If we were able to find a suitable constraint qualification for (24)-(26), then by introducing the functional shadow prices for each of such constraints, respectively, Lagrange´s method of undetermined multipliers would be applicable, for every value of n. This should allow us to achieve conditions for optimality, by setting equal to zero the derivatives with respect to y(n) and r(n) of the Lagrangean:

   The operation just described leads to the main condition (41), together with the second stationary condition (43); in addition, expressions (42) and (44) are the Kuhn-Tucker complementary slackness conditions of non-linear programming. But (41)-(44) may also be obtained by treating L as though it indeed were a generalized Hamiltonian 24, so that they could come out as the first order conditions for a maximum of the Lagrangean with respect to the control variables y(n) and r(n); the remaining standard conditions usually met with in control theory, i.e. linking the rate of change of each costate variable to the partial derivatives of the Hamiltonian with respect to the state variables, would be the differential equations:

, yielding the supplementary conditions that complete our non-rigorous derivation of (38)-(44).

   Now, (38) allows us to write w and q instead of Using the integrating factor:

   The first conclusion is If not q = 0 from (52) and then, by virtue of (38), (45), (46) and (47): Therefore, we would contradict Theorem 12.1 in Makowski and Neustadt (1974).

Due to (47) and (50): q 0 and hence, from (52): q < 0. Thus, we can make q = -1 and rewrite (46) and (50) as:

   Effectively, as (n) is continuous, either (n´) 0 or 0 are impossible options, for if not r(n), because of (57), and hence z(n), because of Proposition 1, would be both singular around or .

   In addition, if , or belong to an open interval none of whose points are contained in an open subinterval of singularity 25for z(n), no point of the interval will violate the integral equation in (58).

   This, combined with the fact that A(n) is a regulated function because z(n) and x(n) also are, together imply A(n) = 0 for every point of the referred interval. The remaining part of statement iii) above, providing conditions for the application of equation (30), is given by:

   PROPOSITION 2: If and are contained in open intervals where z(n) is both continuous and non-singular, then (30) holds at such points.

   Proof: Suppose for instance that A(n´) > 0. Hence, the same inequality should hold along an open interval containing . But this would mean, due to the integral equation in (58), that is the only point not contained in an open interval of singularity for z(n), which results clearly impossible.

   On the other hand, if represents a maximal interval of singularity (constancy or not) for z(n):

   Again, the justification derives from the continuity of (n), together with (66) and our Proposition 1. It explains statement iv) in the text, when (56) and (57) are taken into account.

   Focusing now on v), condition (33) comes from (56), together with the continuity of expression and the fact that? ´(n) = 0 a.e. outside intervals of singularity. Referring to the second part of v), notice that if is a non-decreasing function of y for constant u, then (33) cannot be satisfied, unless y(n) = (n), due to the strict concavity of the function U(x, y).

   Obviously, (42) represents (34) after making q = -1. Moreover, (53)-(54) are (36)-(37) respectively.


1 See, among others, Spence (1977), Roberts (1979), Picard (1987), and also the literature quoted in section 3 of this paper.

 2 Neustadt (1974, pp. 398-412) offers an interesting survey on the subject.

 3 For an example, see Atkinson and Stiglitz (1980, pp. 415-6) or Arrow and Kurtz (1970).

4 Rockafellar and Wets (1998, pp. 536-7).

5 See Natanson (1964, chapter VIII) for an excellent overview.

6 The absolute continuity of comes from the fact, obviously, that it is the difference of two absolutely continuous functions. Concerning its monotonicity, see for instance Bruckner (1978, pg. 174).

7 The treatment of is further illustrated in footnote 13, as the parametric part of an equality constraint

8 The reason for deleting u(t) k lies in that, at intervals where u*(t)= k , instead of a point-wise necessary condition we would achieve a, much weaker, interval necessary condition. Thus, the change aims only to derive a better characterization of the optimal solution.

9 Hestenes (1965), Guinn (1965), Virsan (1970, 1971) are some of the most well known contributions on the topic. They are analyzed on comparison basis in Makowski and Neustadt (1974, pp.323-4).

10 Seierstad and Sydsaeter (1987, pg. 277) single out the theorem by Makowski and Neustadt (1974), among those to be considered for economic applications. Well-known examples of continuous singular functions are the Lévesque´s singular function and, of course, the absolutely continuous case of the constant function.

11 The typical discontinuous singular function is the jump function, or equivalently the saltus function, a particular case of which is the so-called step function. For these concepts see, for intance, Natanson (1964).

12 See Natanson (1964, chapter VIII) for an excellent overview. Lots of references are found in Takács (1978). Mirrlees (1969) contains the proofs for most of the results published in Mirrlees (1971).

13 The same can be said about the existence of continuous third partial derivatives that, for the utility function, we have explicitly supposed above.

14 Here: u(n) = U[x(n), y(n)].

15 For Mirlees (1986, pg. 1228), the difficulty with using the function c(.) as a control variable lies in the fact that: "… variations in it can have complicated effects on the variables of the problem".

16 Mirrlees (1986) only contains some partial demonstration, while the complete proof in Mirrlees (1971, 1976) is given under strong differentiability assumptions. As I want to show elsewhere, Theorem 1 in Mirrlees (1971) does not guarantee the complete equivalency between Problems I and II, unless some additional condition be assumed.

17 In Mirrlees (1971) the separation between two different formulations, Problem I and Problem II, is not made clear; it appears just as being implicitly understood.

18 This type of transformations has been taken from Brito and Oakland (1977), of whose approach the present paper can be considered a generalization since they suppose a differentiable z(n).

19 The suppression of the asterisk in the optimal trajectories u*(n), z*(n), together with the parallelism observable in the exposition of the corresponding conditions i)-vii) respect to those in Mirrlees (1971, Theorem 2), only intends to facilitate the comparison.

20 According to Mirrlees (1971, pg. 183), z(n) is said to be a point of increase if, for all n´, n´´ such that n´ > n > n´´, always: z(n´ ) > z(n) > z( n´´).

21 Examples of non constant singular functions, continuous or not, and strictly monotonous or not, may be found in Burrill and Knudsen (1969, example 13-1), Natanson (1964, pp.213-4), and Stromberg (1981, pp.210).

22 Using the formal setting in Section II, the most stringent among the regularity conditions is the requirement that the matrix of partial derivatives of in the l equality constraints (10), respect to the m control variables u, has rank l.

23 Of course, Problem III shows the typical structure of an optimal control problem, with (16), (18), (20) and (23) as the initial conditions; (14) and (27) as the terminal conditions; (15), (17), (19), (21) and (22) as the transition equations.

24 See Kamien and Schwartz as an introduction to control theory with economic applications. For this line of analysis, see Arrow and Kurtz (1970, pp. 39-41) and, again, Kamien and Schwartz (1981, pp. 183-4).

25 Note that the condition is equivalent to asserting that the points of the interval where z´(n) > 0 form a dense set of positive measure.



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About the Author

Autor: Xabier Ruiz del Portal. Universitat de Lleida
Dirección: c/Mallorca, 56 4º5ª. 08029 Barcelona (Spain)
Correo electrónico: Rportal@econap.udl.es

*"I am grateful to James Poterba and Javier Prado Domínguez for suggesting some comment, included in the conclusions, on the difficulties  to achieve economic insights when applying Pongriyagin maximum principe".

"Agradezco el comentario de James Poterba y Javier Prado Domínguez, incluido al final de las conclusiones, sobre algunas dificultades que surgen con el Principio del Máximo de Pontriyagin a la hora de llegar a ciertas interpretaciones económicas".




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Venancio Salcines. (Universidade da Coruña)
Andrés Blancas. Instituto de Investigaciones Económicas (UNAM)
Editor Asociado para America Latina:
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