The structure and existence of solutions of the problem of consumption with satiation in continuous time

With the help of the method of Lagrange multipliers and KKT theory, we investigate the structure and existence of optimal solutions of the continuous-time model of consumption with satiation. We show that the differential equations have no solutions in the C1 class but that solutions exist in a wider space of functions, namely, the space of functions of bounded variation with non-negative Borel measures as controls. We prove our theorems with no additional assumptions about the structure of the control Borel measures. We prove the conjecture made in the earlier literature, that there are only three types of solutions: I-shaped solutions, with a gulp of consumption at the end of the interval and no consumption at the beginning or in the interior; U-shaped solutions, with consumption in the entire interior of the interval and gulps at the beginning and the end; and intermediate (J-shaped) solutions, with an initial interval of abstinence followed by a terminal interval of distributed consumption at rates and a gulp at the end. We also establish the criteria that permit determination of the solution type using the problem’s parameters. When the solution structure is known, we reduce the problem of the existence of a solution to algebraic equations and discuss the solvability of these equations. We construct explicit solutions for logarithmic utility and CRRA utility.


Introduction
often called systems with impulsive controls, more general maximum principles are discussed in [7] and [8], and extensively in [9] and in papers quoted therein. All of these works use a solution-dependent change of the time variable which was introduced in [10]. In [11] the Bellman principle is extended to impulsive systems. In all of these works it is explicitly assumed that the measure controls have atoms. All of these general theories are very complex, hence we convert Problem 1 into a variational problem, and we use the simpler and more straightforward method of Lagrange multipliers. The approach presented in this paper uses very little beyond the finite-dimensional case described in [12]. Our method is simpler than any proof of a maximum principle, especially when unbounded controls are involved. The additional novelties of our approach are as follows: Instead of assuming a priori that the control measures have atoms, we prove that they do. In addition, we use no transformation of the time variable, and we do not require any additional differential equations (such as Eq (4.6) and later equations in [8]) to be satisfied by the atoms of the control measures.
In this work, Problem 1 is transformed into a problem that is easier to handle: Problem 3, with a different utility (see Eq (9)). We first prove that this problem has no solutions with continuously differentiable satiation s. Then we extend the solution space to the BV space (the space of functions of bounded variation). This implies that we must allow the consumption c, the controlling variable, to be a general Borel measure; it could even be as exotic as the derivative of the Cantor function. An important lemma (Lemma 23) is proved using only the assumption that c is a non-negative Borel measure.
We conduct a detailed investigation of the structure of optimal solutions for general utilities. Our general results are proved under the following assumptions: (i) the original utility V and the transformed utility are both concave down, or −V and −V S are both convex in the classic sense; and (ii) additional barrier conditions (Eqs (36) and (37)) hold. Further, we assume two pairs of inequalities, where one pair (Eqs (32) and (40)) corresponds to a high future discount n, and the other pair (Eqs (33) and (41)) corresponds to a low future discount n. Each pair is sufficient for the existence of solutions; however, the corresponding solutions differ remarkably. In the case of a high future discount, all the consumption must take place in a single gulp, or burst, at time t = 0, while in the case of a low future discount a terminal gulp of consumption at t = T is necessary. In [4] the proofs of many theorems use an additional assumption: that there are no gulps of consumption in (0, T). We prove (in Theorems 21 and 25) that this assumption is correct, namely, that the measure c is continuous relative to Lebesgue measure in (0, T), which means there are no gulps of consumption there, and that gulps of consumption can occur only at the endpoints of the interval [0, T]. We prove that at most one interval with non-zero consumption is possible and that at most one interval with no consumption is possible. This is done in Theorem 21 for the case of a large future discount and in Theorem 25 for a small future discount. We extend an earlier observation in [4], which was stated only for special cases of utilities and selected parameters, to general utilities and prove that under one of the aforementioned pairs of inequalities, which essentially corresponds to a large future discount, the only type of solution is that in which all the consumption takes place in a single gulp at t = 0 (as discussed in Section 6.1), while under the other pair of inequalities, which essentially corresponds to a small future discount, there are three types of solutions: Jshaped ones that correspond to poor or over-satiated agents; U-shaped ones that correspond to rich or under-satiated agents, with two gulps of consumption, one at the beginning and the other at the end; and intermediate solutions with an initial period of abstinence, a terminal gulp of consumption, and a terminal interval of non-gulpy consumption.
The general theory presented in Section 5 is used in Section 8 to reduce the problem of the existence of solutions to simple non-differential equations, and the solvability of these equations is proved. In the case of CRRA utility (Section 8.2), only one of these equations requires numerical methods, while in the case of logarithmic utility all of them are solved explicitly (Section 8.1).

The problem and preliminaries
In this paper, we address the following problem, which is a continuous-time version of the satiation problem in [1]: Problem 1 Let V be a concave-down, twice differentiable function of s. Maximize the functional (the sum of instantaneous utilities dV ds � c � dt) under the following constraints: sð0Þ ¼ s 0 ð3Þ fsubscript 0 addedd heregs 0 � 0 ð4Þ In this model, s(t) is the satiation level caused by consumption c(t), and W is the initial wealth; n, r, γ, and φ are positive parameters representing respectively the agent's discount rate, the risk-less interest rate, the satiation decay rate, and the satiation generation rate caused by consumption c.
Remark 2 In Problem 1 we deliberately do not specify the space in which we search for solutions. It is more convenient to do this in the transformed Problem 3. However, in Section 3 we argue that the classical space used in Problem 3 does not contain any solutions of the problem. The proper space of BV functions is described in Section 4 and is used in the final version of the problem (Problem 10).
It is helpful to eliminate c(t) from (1). By constraint (2), Substituting this into (1) and then using integration by parts, we obtain an equation for the functional f � U S in which the consumption function c does not explicitly appear: where The following will be used later, in the proofs of Propositions 8 and 14 and Theorem 21: The expression for α(s) looks very similar to the expression for relative risk aversion. The term relative risk aversion is appropriate in the context of investments, where every investment carries some risk. However, in this paper the risk of consumption is ignored, hence the term relative satiation aversion seems more appropriate. Thus we will call α(s) the relative satiation aversion. The relative satiation aversion for V S is denoted by α S : Similarly to the method used in obtaining (7), we can transform the wealth constraint (6) with the help of integration by parts: We assume that V S given by (9) is an increasing, concave-down function of the satiation s. In Section 8.2 it is demonstrated that this is always the case for CRRA utility. Therefore, Problem 1 can be replaced by the following problem: Problem 3 (The smooth version) Maximize the functional f � U S in (8) on the space of all continuously differentiable functions s 2 C 1 ([0, T] ! R) that satisfy the constraints (4), (5), and (3), and constraint (13) with W > 0, under the assumption that V and V S are increasing, concave-down, twice continuously differentiable functions of s. Remark 4 If V is an increasing function, then it follows from (10) that V S is increasing if Downward concavity of V S requires that α S > 0, and downward concavity of V requires that α > 0. Therefore, these two conditions are assumed to be satisfied in everything that follows.
One of the objectives of this paper is to discuss the solvability conditions for Problem 3. Necessary and sufficient conditions are obtained by generalization of the Lagrange multipliers (Karush-Kuhn-Tucker (KKT) theory for a finite number of dimensions; see [12]). We first note an important consequence of (5): hence s(t) > 0 for t � 0, thanks to (4) and (5).

Remark 6
Later it is shown that s(t) > 0 in (0, T] when W > 0 even if s 0 = 0; see Remark 36. Following the finite-dimensional KKT theory (see [12,13]), we introduce the Lagrangian functional Here λ W 2 R is the Lagrange multiplier that corresponds to constraint (13), and λ c 2 C 1 ([0, T] ! R) corresponds to (5). The KKT method also involves additional constraints on λ c (t): Since s(t) and λ c (t) are assumed to be differentiable on [0, T], they are continuous on that interval. Thus after integration by parts, the functional (16) takes the following form: The first-order conditions for an extremum are obtained as follows: If s(t) is an optimal solution and Δs(t) 2 C 1 ([0, T] ! R) with Δs(0) = 0 (hence for every real number ε, s + ε � Δs satisfies the initial condition (3): which must be 0, hence also @L @sðÞ ¼ 0 by the DuBois-Reymond lemma, as Δs(t) is arbitrary (except for Δs(0) = 0, which follows from (3)). This leads to the following set of first-order conditions: At the boundary (i.e., at t = T), In the interior of the interval (i.e., for t 2 (0, T)), KKT theory guarantees that if Problem 3 has a solution s, then that solution is the first of the three entities in [s(t), λ c (t), λ W ] that make up the solution of the following problem: (20), initial condition (3), terminal condition (19), constraints (4) and (5), constraint (13) with W > 0, and KKT conditions (17).
We show in Section 3 that Problem 7 has no (everywhere differentiable) solution. The remedy, discontinuous functions with bounded variation, is described in Section 4. Problem 7 is reformulated as Problem 10 in the BV space. The sufficient conditions are derived in Theorem 11, and in Theorem 13 it is shown that the sufficent conditions are also necessary. The structure of these discontinuous solutions is investigated in greater detail in Section 6. Explicit solutions for logarithmic and CRRA utilities are constructed in Sections 8.1 and 8.2, respectively.

Nonexistence of continuously differentiable solutions
By assumption, V and V S are concave down and all the constraints are linear, hence there can be no more than one solution. We will now show that the existence of a solution of the most tractable form (continuous on [0, T] and continuously differentiable on (0, T)) will lead to the contradictions described in the proof of the proposition below.
Proof. Assume that there is a continuously differentiable solution. First, we assume that consumption c(T) > 0. Since s 2 C 1 ([0, T] ! R), (2) implies that c is also continuous and therefore c(t) > 0 for all t in an open neighborhood of T. In this case, as a consequence of the first KKT condition in (17), we have λ c (t) = 0 for all t close to T, hence the terminal condition (19) and condition (20) with t = T take the following form: One can eliminate λ W by subtraction, obtaining, with the help of (10), However, n À r À a � g ð Þ � dV ds s T ð Þ ð Þ cannot be 0, by the premises of this proposition. Therefore, the assumption that s(t) is continuously differentiable, and that c(T) > 0, is false. The problem seems to be caused by the fact that Eq (20) has to be satisfied with t ! T. We could try to get around this problem by assuming that there is no consumption at t = T, hence that λ c (T) 6 ¼ 0, in which case we have one additional variable, λ c (T), that can help us to satisfy Eq (20) at t = T. If c(T) = 0 and c(t) > 0 for all t close to T, then λ c (t) = 0 for all these t, hence λ c (T) = 0 since λ c is continuous in [0, T], so we do not have the additional variable λ c (T). In order to obtain it, we need to assume that there is some T e < T such that c(t) = 0 for all t > T e and c(t) > 0 for t � T e . But in such a case the solution must be optimal in the interval [0, T e ] and at t = T e the same boundary condition must be satisfied, hence we obtain the same contradiction. Therefore, the assumption that s(t) is continuously differentiable is false.
Proposition 8 implies that Problem 1 does not have a continuously differentiable solution. Note, however, that all of the results proved in this section for continuous, almost everywhere differentiable functions (except for the nonexistence of a continuously differentiable solution) remain valid for discontinuous solutions with bounded variation, which are considered in the rest of the paper. This can be deemed as an extreme case of the Lavrentiev phenomenon (see [14]): In one Banach space there is a solution, and in another there are none. In [15] there appears the following remark: "Lavrentiev's phenomenon is related to the existence of singular minimizers, i.e., absolutely continuous minimizers that are not Lipschitz." In the case discussed in this work, the maximizers are of bounded variation but discontinuous, hence we can expect a Lavrentiev phenomenon. However, the optimal value can be approximated by a suitably chosen Lipschitz continuous function when α > 0, and probably not when α < 0. The functional (7) does not satisfy the coercivity condition (A4) from [14], especially not on any space that imposes restrictions on derivatives (of satiation), for the functional (8) does not even contain the derivative ds dt explicitly. This suggests that the result on the absence of a Lavrentiev gap in [14] can be proved without the coercivity condition.

Functions of bounded variation
If a model has no solutions, as was shown for the satiation model in Section 3, it has to be modified-we can "regularize" it by introducing an artificial viscosity or limiting the analysis to bounded consumption, as in [5],-or it could be abandoned in favor of another model. Alternatively, the definition of a solution could be changed. Intuition would suggest that the satiation model does have a solution and, as a result, an artificial regularization is not necessary. One alternative would be to relax the requirement that the satiation level s(t) be continuous, and instead allow s to have some points of discontinuity. If such discontinuities are permitted, the satiation s(t) is not differentiable in the classical sense at the points of discontinuity. However, s(t) is present in differential Eq (2), hence we need to use non-classical derivatives. In order to accommodate this relaxation, we permit s to be a function of bounded variation (BV); see [16] or [17]. In general, functions belonging to the BV space could be discontinuous, and their derivatives could be Borel measures. The Heaviside function, which is defined by h(x) = 0 for x � 0 and h(x) = 1 for x > 0, is an example of a BV function. Its derivative is the wellknown Borel measure popularly known as the Dirac delta function. For this reason, when satiation s(t) is a BV function, Eq (2) forces consumption c to be a Borel measure. Whenever satiation s(t) has a discontinuity, the singular part of the Borel measure c is the Dirac delta measure. In order to avoid this non-descriptive term, we use a more intuitive term for it: consumption gulp, following [4]. The mathematical description of a "gulp" or a gulp at a point of discontinuity of a function is given in Eq (25).
Functions of a single variable with bounded variation (BV functions) are continuous almost everywhere, and their derivatives are finite Borel measures. The three most important properties of BV functions are as follows: in the closure of the interior of the domain of φ.
2. The derivatives exist even if the functions are discontinuous. The downside is that the derivatives are permitted to be Borel measures.

Integration by parts is permitted. If φ is a BV function on [a, b]
and g is C 1 on R, then dφ dt is a Borel measure and, if φ(a) and φ(b) are defined, we can integrate by parts as follows: Z If φ is C 1 on R except for some point τ 2 (a, b) and the limits φ(τ + ) and φ(τ − ) exist, then with the help of simple calculus we can proceed as follows: In Eq (24), dφ dt denotes the classical derivative, which is defined everywhere except at τ. By comparing (23) to (24), we conclude that the Borel measure dφ dt is given by dφ dt is the Dirac delta Borel measure with strength equal to the jump (φ(τ + ) − φ(τ − )) in the value of φ at the discontinuity located at t = τ. This can be interpreted as a decomposition of the Borel measure dφ dt into the sum of the part that is continuous with respect to Lebesgue measure, dφ dt t ð Þ � � classical , and the part that is singular with respect to Lebesgue measure, dφ As mentioned earlier, in this paper the latter is referred to as a "gulp." The theory of BV functions provides us with derivatives of discontinuous functions and formulas for integration by parts which are applicable to discontinuous functions, even if the singular parts of their derivatives are much more complicated than in the example above (e.g., the singular part of the derivative of the Cantor function). The use of integration by parts to derive formulas (18) and (13), as well as formula (69) in Section 7.3, is justified by the formula in (23) when s(t) is a BV function. Hence all of the identities involving integrals derived in the preceding sections are also valid when C 1 differentiability is relaxed to that in the BV sense. In this paper the discontinuous function is satiation s(t). If s(t) is discontinuous at t = T, then Eq (2) implies that this contributes sðTÞÀ sðT À Þ f � d T t ð Þ to consumption. If we know that the discontinuities in satiation can occur only at the endpoints of the interval [0, T], we can avoid BV theory altogether, as was done in [4]. However, in order to prove this, one needs to initially permit the derivative of the satiation to be a general Borel measure on [0, T] and then to show that it is continuous with respect to Lebesgue measure in (0, T). Also, a discontinuity of s(t) at t = T allows us to remove the contradiction obtained in the proof of Proposition 8. This is the approach used in the proof of Proposition 14 in the next section.
In spite of discontinuities and exotic derivatives, the use of integration by parts is valid for the functions in BV ([0, T] ! R), hence all the equations and conclusions obtained in Sections 2 and 3, including Eq (18) and the conditions (19) and (20), remain valid under the premises of Problem 10. Proposition 5 is also valid, and the proof is almost the same: When the consumption c is a non-negative measure, it replaces c(t 1 ) � dt 1 , and the second term inside the parentheses in Eq (15) is still positive for all t 2 (0, T]. In (16), then [s, λ c , λ W ] must be a solution of Problem 7. With the help of the BV theory, one can easily prove sufficiency of conditions (19) and (20) in Problem 10.

Theorem 11 (on sufficiency) Suppose V and V S are concave-down, continuously differentiable functions, and let
Δs also satisfies these conditions. Consider the function where U S is given by (8). Since s and Δs are bounded, the bounded convergence theorem ensures that Θ is differentiable with respect to x and dY dx is a solution of Problem 10, we may use (20) and (19), and we obtain dY dx Now constraint (13) implies that Now note the following: λ c (t) � 0 and c(t) = 0 when t 2 supp(λ c ), thanks to KKT conditions (17). Therefore, dDs dt þ Ds � g , thanks to continuity of V S , and this ensures that s is also a maximum of the convex functional U S (s) on the set in (27). Thus s is unique, thanks to the downward concavity of U S .
Remark 12 A proof of necessity of the conditions in Problem 10 is not needed for explicit construction of solutions in the sections of the paper that follow. For the sake of completeness, however, we present a proof of the necessity.
Theorem 13 (on necessity) Let V be a twice continuously differentiable function, let c be a non-negative Borel measure on [0, T] that satisfies (6), and let s be the solution of the differential Eq (2) that satisfies (3) (10).
Proof. If c is a Borel measure, then the solution of the initial-value problem (Problem 2) with the initial condition (3) is given by (15), hence the solution is bounded, which means that ds dt is a Borel measure thanks to (2), and so s 2 BV ([0, T] ! R). Suppose that under the constraints listed in the Theorem, the functional (8) attains its maximum value on BV ([0, T] ! R) at s. We need to construct λ c 2 Lip ([0, T] ! R) and a λ W 2 R, and show that λ c � 0 and λ W > 0, and that [s, λ c , λ W ] satisfy the differential Eq (20) and the terminal condition (19). Let c 1 be any non-negative Borel measure on [0, T] that satisfies (6), and let s 1 be the corresponding solution of the initial-value problem that consists of Eq (2) together with (3) and (4). In addition, for all Then c x is non-negative and also satisfies (6), and s x 2 BV ([0, T] ! R). Consider the function of x 2 [0, 1] given by where the integrals above are Lebesgue-type integrals with respect to Borel measures. Since s and s 1 are both bounded, with the help of the bounded convergence theorem one can prove that U S (s x , c x ) is differentiable with respect to x and that The second integral may be transformed as follows: ðsince s and s 1 satisfy ð2Þ with c and c 1 ; respectivelyÞ hence thanks to (10),   (6). Now let Obviously, λ c � 0 and λ c (t) = 0 for t 2 supp(c), hence λ c satisfies KKT condition (17). Next, (2), . We need to show that λ c satisfies the boundary condition (19) and the differential Eq (20). Indeed, so λ c satisfies the boundary condition (19). Next, which is the differential Eq (20).
The two proofs above use ideas from [18]. A few minor gaps in that material are covered here.

Structure of the optimal solutions
In this section the structure of solutions of Problem (10) is investigated, the main results being Theorems (21) and (25).
Discontinuities complicate the proofs, but they do not destroy the conclusions of Section 2. To a large extent, functions of bounded variation that are not continuously differentiable can be treated formally as if they were (e.g., in terms of differentiation and integration by parts).
Note also that condition (5) and Eq (25) require the gulp of consumption cðtÞ ¼ sðt þ ÞÀ sðt À Þ φ at each jump in the satiation to be positive, that is, they require the singular part of c to be positive as well as the part that is continuous with respect to Lebesgue measure.
We first show that a discontinuity in the satiation at t = T is essential, as it allows one to resolve the contradiction that arose in the proof of Proposition (8). When functions with bounded variation are considered, Eq (20) reduces to where s(T − ) = lim t!T,t<T s(t). Eq (31) replaces Eq (22). If there is a discontinuity at t = T, then s(T − ) 6 ¼ s(T); this provides an extra degree of freedom, which removes the contradiction between Eqs (21) and (31) at t = T that eliminated continuous solutions. We can prove even more: 1. There must be a gulp of consumption at t = T if for all s n < r þ a � g: ð32Þ

A gulp of consumption at t = T is impossible if for all s
Proof. When λ W is eliminated between (21) and (31) for t = T, one obtains where s(T − ) = lim t!T,t<T s(t). With the help of (10), this can be transformed into This discontinuity of s(t) produces the singular part of ds dt j t¼T ¼ s T ð Þ À s T À ð Þ; see Section 4. Now Eq (2) implies that the consumption c has singular part sðTÞÀ sðT À Þ f , which we call a consumption gulp. This gulp must be positive, since consumption c must be positive; see (5). However, (33) implies that the gulp of consumption must be negative. Therefore, a gulp of consumption at T is impossible if (33) holds. Proposition 14 resolves the non-existence contradiction from Section 3: The optimal solutions must be discontinuous at t = T if (32) holds for all s. The next lemma provides the first step in deciphering the structure of the optimal solutions.
Proof. The satiation s is bounded (although it may be discontinuous if c is not absolutely continuous with respect to Lebesgue measure), because every function of bounded variation is bounded; see [16]. The boundedness of s and condition (19) imply boundedness of λ c (T), which, together with (20) with λ = λ W 2 R. Eq (35) does not contain ds dt , and this simplifies forthcoming proofs.

Lemma 17
If dV S ds s ð Þ > 0 and α S (s) > 0 for all s � 0, and if both of the following hold, then Eq (35) has a unique bounded, continuous positive solution for all t � 0 and 0 < λ 2 R that satisfy Eq (39). The corresponding optimal consumption is given by Proof. The conditions dV S ds s ð Þ > 0 and λ > 0 ensure existence of a solution σ(t, λ) of (35), and d 2 V S ds 2 < 0 (which is equivalent to α S > 0) ensures uniqueness and continuity. The two "barrier" requirements, (36) and (37), ensure positivity and boundedness of σ(t, λ). In order to derive (38), we differentiate the expression on the left-hand side of (35) with respect to t and obtain and now we transform (2) as follows: This completes the derivation of (38).

Remark 18
In [4] a less restrictive assumption is used, namely dV S ds s M ð Þ ¼ 0 with 0 < s M < 1 instead of our (37). Our results could be extended, with some modifications, to that case too.

Definition 19
The solution of Eq (35), σ (t, λ), whose existence and basic properties are as described in Lemma Proof. If λ c > 0 in a (relatively) open subinterval, the first KKT condition in (17) implies that c = 0 in that subinterval, hence the solution of (2) is of the form s(t) = const � e −γ�t . If λ c = 0 on a closed subinterval, Eq (20) simplifies to (35) with λ = λ W , hence s(t) = σ(t, λ W ). If (40) holds for all s, the corresponding consumption is positive. However, if (41) holds for all s, then (38) implies that c < 0, which violates requirement (5), hence consumption must be everywhere 0 in (0, T).
Proposition 20 also shows that s(t) must be continuously differentiable everywhere, except possibly at the endpoints of the open subintervals in which λ c = 0, where there may be discontinuities in satiation and gulps of consumption; however, as will be proved in Lemma 24, this possibility is excluded. Proposition 20 also shows that the solutions can be divided into two groups. Each of these two groups corresponds to a particular pair of sufficient conditions, one pair being the conditions given in inequalities (32) and (40), and the other pair being the conditions given in inequalities (33) and (41). Hence these two groups of solutions will be investigated separately. The structure of solutions of Problem 10 under (32) and (40) is as described in Theorem 25 (Section 6.1), and the structure of solutions of Problem 10 under (33) and (41) is as described in Theorem 21 (Section 6.2).

The structure and existence of solutions for large future discount
When the future discount rate n is large, intuition suggests that all the consumption should take place at the beginning of the interval [0, T]. This is confirmed by the following theorem: where s + = f � W + s 0 (see (44)); λ W is given by (49), λ W > 0; and λ c is given by (53), λ c > 0. All the consumption takes place in a single gulp at t = 0. Proof. By (33), the second part of Proposition 14 implies that there is no gulp of consumption at t = T, and Lemma 20 states that there is no consumption in (0, T), hence all the consumption must happen in a gulp at t = 0. Therefore, satiation is of the form (42), and the corresponding consumption is given by The constant s + can be calculated using the wealth constraint (13): The condition s + > s 0 ensures that the gulp of consumption at t = 0 is positive. We need to show that λ W > 0 and λ c > 0 in (0, T). For λ W and λ c , we have the following equations from Problem 10: For t = 0, by the first KKT condition in (17) and the fact that there is a gulp of consumption (c(0) > 0): For t 2 (0, T), from (20): For t = T, from (19): We need to prove that λ c (t) > 0 in (0,T] and λ W > 0. The solution of (46), together with the initial condition (45), is After substitution of this into the boundary condition (47), we obtain an equation for λ W : This implies that λ W > 0, since dV ds > 0 and dV S ds > 0. This expression can be simplified. First, note that hence the first term on the right-hand side of (49) can be transformed as follows: Therefore, by (10), Now (48) takes the form Integration of (50) on [0, t] produces and now (52) takes the form Using transformations similar to those used in the proof of (51), we can prove the following: which is positive, thanks to (33) and the fact that dV S ds > 0.

Remark 22
Since the objective function in Problem 10 is concave down, the sufficient conditions are also necessary, and there is only one solution. This result establishes the existence and uniqueness of solutions of the equivalent of Problem 1 in the class of BV functions, independent of the utility.

Small future discount rate and solutions with positive distributed consumption
In this section we discuss the structure of solutions under assumptions (32) and (40). The results of this section permit us to prove the existence of solutions in Section 7. We begin with a lemma, which we will use in the proof of Theorem 25, that permits us to reduce the number of intervals as described in Propositions 16 and 20 (with or without consumption) to no more than two.
Proof. In (t 1 , t 2 ), s(t) is a solution of Eq (20), while the general solution σ(t, λ W ) is a solution of (35). We eliminate λ W from these two equations by subtraction to obtain Multiplying this equation by (s(t) − σ(t, λ W )), and with the help of the Fundamental Theorem Structure and existence of solutions of the problem of consumption with satiation in continuous time of Calculus, we obtain Next, we integrate this over (t 1 , t 2 ) to obtain since the boundary terms ½ðsðtÞ À sðt; l W ÞÞ � l c ðtÞ� t¼t 2 t¼t 1 vanish, thanks to the fact that λ c (t 1 ) = λ c (t 2 ) = 0. Next, thanks to (2) and (39), hence the second integral in (54) can be rewritten as follows: Z since c(t) � λ c (t) = 0, thanks to the first KKT condition in (17). Hence we obtain Z The left-hand side is less than or equal to 0, thanks to the fact that d 2 V S ds 2 < 0, while the righthand side is greater than or equal to 0, thanks to (40). Moreover, σ(t, λ W ) > 0 by Lemma 17, and λ c (t) � 0. Hence both sides must be equal to 0. This is possible only if (s(t) − σ(t, λ W )) = 0, hence c > 0, thanks to (38) and (40), so λ c (t) = 0 in (t 1 , t 2 ).
Lemma 23 implies that between any two open intervals with c(t) > 0 and λ c = 0 there cannot be any interval with λ c > 0, which in turn implies that there can be at most one interval with λ c > 0 and at most one interval with λ c = 0. This fact is used in the proof of Theorem 25. The next lemma, which is also used in that proof, permits us to reduce the number of gulps of consumption to no more than two. Proof. If there is a gulp of consumption sðT þ s ÞÀ sðT À s Þ f 6 ¼ 0 at T s , then inequality constraint (5) implies that it must be non-negative. Since f > 0, it must be the case that sðT þ s Þ � sðT À s Þ. Therefore, in order to prove that sðT þ s Þ ¼ sðT À s Þ, we have to prove that sðT þ s Þ � sðT À s Þ. In (T s − ε, T s ) satiation s(t) satisfies Eq (2) with c = 0, hence In [T s , T s + ε) the satiation s(t) satisfies Eq (20) with λ c (t) = 0, that is, it satisfies Eq (35), hence In (T s − ε, T s ) the Lagrange multiplier λ c (t) satisfies Eq (20) with λ c (t) � 0, thanks to the second KKT condition in (17), hence for λ c (T s ) = 0, because λ c (t) is continuous and λ c (t) = 0 for t > T s ; therefore, dV S ds s T þ s À � À � � dV S ds s T À s À � À � . Since d 2 V S ds 2 < 0, the above is possible only if sðT þ s Þ � sðT À s Þ, which, together with the opposite inequality (sðT þ s Þ À sðT À s Þ � 0), implied by c > 0 and f > 0, implies that sðT þ s Þ ¼ sðT À s Þ, and hence that s(t) is continuous at t = T s . The consumption c(t) is 0 in every relatively open interval where λ c (t) > 0, hence it is trivially continuous with respect to Lebesgue measure there, and similarly it is continuous with respect to Lebesgue measure in intervals where c(t) > 0 by Lemma 23. Now continuity of satiation s at the boundaries between these two types of intervals, which is as described in Proposition 16, implies continuity of consumption c with respect to Lebesgue measure throughout the interval (0, T). Now we can prove the main result of this section.

Existence of solutions for small future discount
With the help of the results of Section 6, we can reduce the problem of the existence of solutions to solving non-differential equations, and we prove their solvability here. We prove theorems that, with the help of the diagnostic profiles described in Section 7.1, allow us to use the problem data to determine the type of the solution.
Theorem 25 implies that under assumptions (32) (17)) Proof. Eq (20) for λ c implies that in [0, T s ) We need to prove that λ c > 0 in [0, T s  where σ(0, λ) is as described in Definition 19, s(T) = s T (λ W ), s T (λ W ) is a solution of Eq (57), and λ W is to be determined from the wealth constraint (13) reduced to the non-differential equation where w r (λ W ) is defined as follows: f � w r ðl W Þ ¼ e À r�T � s T ðl W Þ À s 0 þ ðg þ rÞ � Z ½0;T� e À r�t � sðt; l W Þ � dt: Therefore, the existence of solutions in this case reduces to solving Eq (62), and solvability of this equation is established in the following proposition: Proposition 35 If W � W u , there is exactly one solution λ W of Eq (62). In addition, the solution given by (61) is a solution of Problem 10 and also an optimal solution of Problem 1. If W = W u , this solution coincides with the upper diagnostic profile; see Definition 31.
Proof. We need to prove the existence of a unique solution of Eq (62), and we need to prove that the consumption gulps, sð0;l W ÞÀ s 0 f at t = 0 and s T À sðT;l W Þ f at t = T, are positive. In order to prove uniqueness, we find the derivative d dl W w r l W ð Þ ð Þ: This implies existence and uniqueness of a solution of w(λ W ) = W with l W � l u W whenever W � W u , since lim l w !0 ðsðt; l w ÞÞ ¼ þ1 and lim l w !0 ðs T ðl w ÞÞ ¼ þ1. Positivity of the gulp of consumption at t = T and the result that λ W > 0 follow from Proposition 14. If W = W u , the solution coincides with the upper diagnostic profile from Definition 31, thanks to the uniqueness proved earlier, and there is no gulp of consumption at t = 0. Since l W < l u W , it follows low safety levels, when life expectancy is short (e.g., during famines in ancient times), or when an agent expects to lose his wealth to robbers, it seems reasonable to consume all of the wealth at once. This conclusion is independent of the time horizon. The analysis of consumption with satiation when the discount rate is low (conditions (32) and (40)) is more complicated. It turns out that a gulp of consumption at the end of the consumption period is always optimal (see Proposition 14), because no further penalty can be imposed by the growth of satiation. In fact, there exists anecdotal evidence of addicts having final binges before going into rehab (as in the movie "The Wonderful Whites of Virginia"), smokers taking the last puff before quitting (as in the movie "A Good Woman"), and deathrow inmates accepting the offer of the last wish. According to [19], empirical data indicate that such gulps of consumption take place when people consume cultural goods, such as music. Detailed analysis in Section 7.4 shows that when the wealth to be consumed is small and/or satiation is high, the agent consumes all of his wealth at the very end of the consumption period. When the wealth is large and/or satiation is low, the agent consumes some portion in a gulp of consumption at the beginning of his consumption period, as shown in Section 6.2. This can be explained as follows: Wealthy agents consume a lot at the beginning in order to curtail their desires by increasing their satiation level, and then they consume at a moderate rate throughout the rest of their consumption period. They also leave some wealth to be consumed at the very end. There is also an intermediate pattern, where an agent begins consumption after some delay and then enjoys a gulp of consumption at the very end of his consumption interval. Although the terminal gulp of consumption is non-negotiable in our model, we rarely observe it in the real world. This is especially true when non-cultural goods are consumed. A possible explanation is that agents are not aware of the exact time at which their consumption life will end, and as a result they optimize beyond the actual time of termination of their consumption. The finding regarding the optimality of the terminal gulp of consumption may explain the large sums some people spend on funerals and funerary monuments. One extreme example of such behavior is that of Caterina Campodonico, who lived in Italy in the 19th century and is said to have saved for most of her life in order to afford an elaborate funerary monument in the famous Staglieno Cemetery in Genoa; see [20]. Leaving an inheritance to descendants could also be interpreted as a gulp of consumption at the end of the consumption period.