1 Introduction

Recently, Baucells and Sarin [1, 2] described a new and interesting discrete-time model of consumer behavior: the satiation model. The psychological justifications of the model are explained in plain language in [3]. In [4, 5] this model was extended to continuous time, and the solutions were constructed for CRRA utilities.

The satiation model presented in [4, 5] is mathematically described as an optimal control problem (Problem 1) in this paper. In this problem the control (consumption c(t)) enters the evolution equation (Eq (2)) in this paper linearly and may be unbounded, hence the classical Pontryagin Maximum Principle (see [6]) is not directly applicable. For problems of this type, 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: J-shaped 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).

2 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 ) (1) under the following constraints: (2) (3) (4) (5) (6)

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: (7) (8) where (9) The following will be used later, in the proofs of Propositions 8 and 14 and Theorem 21: (10) where (11) 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: (12) Similarly to the method used in obtaining (7), we can transform the wealth constraint (6) with the help of integration by parts: (13)

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 ∈ C¹ ([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 (14) 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):

Proposition 5 If (4) holds and (5) holds for all t ∈ [0, T], then s(t) > 0 for all t ∈ [0, T].

Proof. The solution of Eq (2) is (15) 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 s₀ = 0; see Remark 36.

Following the finite-dimensional KKT theory (see [12, 13]), we introduce the Lagrangian functional (16) Here λ_W ∈ R is the Lagrange multiplier that corresponds to constraint (13), and λ_c ∈ C¹ ([0, T] → R) corresponds to (5). The KKT method also involves additional constraints on λ_c(t): (17) 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: (18) The first-order conditions for an extremum are obtained as follows: If s(t) is an optimal solution and Δs(t) ∈ C¹([0, T] → R) with Δs(0) = 0 (hence for every real number ε, s + ε ⋅ Δs satisfies the initial condition (3): s(0) + ε ⋅ Δs(0) = s₀), then which must be 0, hence also 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), (19)

In the interior of the interval (i.e., for t ∈ (0, T)), (20)

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:

Problem 7 (The C¹ version) Find [s (t), λ_c (t), λ_W] ∈ C¹ ([0, T] → R) × C¹ ([0, T] → R) × R that satisfies (2), (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.

3 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.

Proposition 8 If (r − n + γ ⋅ α(s)) ≠ 0 and for all s ≥ 0, then there is no continuously differentiable solution [s, λ_c, λ_W] of the boundary value problem (Problem 7).

Proof. Assume that there is a continuously differentiable solution. First, we assume that consumption c(T) > 0. Since s ∈ C¹ ([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: (21) (22) One can eliminate λ_W by subtraction, obtaining, with the help of (10),

However, 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) ≠ 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 explicitly. This suggests that the result on the absence of a Lavrentiev gap in [14] can be proved without the coercivity condition.

4 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 well-known 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:

1. The one-sided limits φ(τ⁻) = lim_t→τ,t<τ φ(t) and φ(τ⁺) = lim_t→τ,t>τ φ(t) exist whenever τ is 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.
3. Integration by parts is permitted. If φ is a BV function on [a, b] and g is C¹ on R, then is a Borel measure and, if φ(a) and φ(b) are defined, we can integrate by parts as follows: (23)

If φ is C¹ on R except for some point τ ∈ (a, b) and the limits φ(τ⁺) and φ(τ⁻) exist, then with the help of simple calculus we can proceed as follows: (24) In Eq (24), denotes the classical derivative, which is defined everywhere except at τ. By comparing (23) to (24), we conclude that the Borel measure is given by (25) where is the classical derivative and (φ(τ⁺) − φ(τ⁻)) ⋅ δ_τ(t) 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 into the sum of the part that is continuous with respect to Lebesgue measure, , and the part that is singular with respect to Lebesgue measure, . 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 C¹ 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 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.

Remark 9 Eq (23) for open sets takes the form (26) Eqs (26) and (23) imply that

5 Sufficiency and necessity

In Section 3 it was shown that the space C¹([0, T] → R) × C¹ ([0, T] → R) × R is inadequate for investigation of solutions of Problem 7, hence we modify that problem as follows:

Problem 10 (BV version) Find [s (t), λ_c (t), λ_W] ∈ BV ([0, T] → R) × BV ([0, T] → R) × R that satisfies (2), (20), initial condition (3), terminal condition (19), constraints (4) and (5), constraint (13) with W > 0, and KKT conditions (17).

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(t₁) ⋅ dt₁, and the second term inside the parentheses in Eq (15) is still positive for all t ∈ (0, T].

In Section 2 it was proved that if [s, λ_c, λ_W] ∈ C¹ ([0, T] → R) × C¹ ([0, T] → R) × R is a stationary point of the Lagrangian functional 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 [s, λ_c, λ_W] ∈ BV ([0, T] → R) × BV ([0, T] → R) × R be a solution of Problem 10. Then the functional (8) attains a maximum value on the set (27) at s.

Proof. Let Δs ∈ BV ([0, T] → R) be any function such that s + Δs satisfies (2), initial condition (3), constraint (4), constraint (13) with W > 0, and (5). Then for all x ∈ [0, 1] the function s_x = (1 − x) ⋅ s + x ⋅ (s + Δs) = s + x ⋅ 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 Since [s, λ_c, λ_W] is a solution of Problem 10, we may use (20) and (19), and we obtain Now constraint (13) implies that so Integration by parts produces since Δs(0) = 0, hence Now note the following: λ_c(t) ≥ 0 and c(t) = 0 when t ∈ supp(λ_c), thanks to KKT conditions (17). Therefore, when t ∈ supp(λ_c), since s + Δs satisfies (5). Hence on supp(λ_c), and we conclude that . This implies that Θ(x) attains a maximum at x = 0, that is, at s, since the function Θ is concave down on [0, 1]. But the functional BV ([0, T] → R) ∋ s → U_S(s) is continuous on BV ([0, T] → R), 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) and (4). Then s ∈ BV ([0, T] → R). Suppose [c, s] maximizes the functional (8) under constraints (5), (6) with W > 0, (2), (3), and (4). Then there exist a non-negative Lipschitz continuous function λ_c ∈ Lip ([0, T] → R) and a non-negative number λ_W ∈ R such that [s, λ_c, λ_W] ∈ BV([0, T] → R) × Lip ([0, T] → R) × R is a solution of Problem (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 is a Borel measure thanks to (2), and so s ∈ 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 ∈ Lip ([0, T] → R) and a λ_W ∈ 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 c₁ be any non-negative Borel measure on [0, T] that satisfies (6), and let s₁ be the corresponding solution of the initial-value problem that consists of Eq (2) together with (3) and (4). In addition, for all x ∈ [0, 1], let c_x = (1 − x) ⋅ c + x ⋅ c₁ and s_x = (1 − x) ⋅ s + x ⋅ s₁. Then c_x is non-negative and also satisfies (6), and s_x ∈ BV ([0, T] → R). Consider the function of x ∈ [0, 1] given by (28) where the integrals above are Lebesgue-type integrals with respect to Borel measures. Since s and s₁ 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 Since U_S(s_x, c_x) attains its maximum at x = 0, we have The second integral may be transformed as follows: hence thanks to (10), Since we have Thanks to Fubini’s theorem, ∫_[0,T] ∫_[0,t]… ⋅ dt₁ ⋅ dt = ∫_[0,T] ∫_{[t, T]}… ⋅ dt ⋅ dt₁, hence so where (29) since and . Hence we obtain which means that the Borel measure c maximizes ∫_[0,T] M(t) ⋅ c₁(t) ⋅ dt on the set of all non-negative Borel measures c₁ that satisfy constraint (6). Since M > 0, the intuition behind the solution of this problem is as follows: Let Then the optimal c is any non-negative Borel measure that satisfies (6) and in which case If c₁ is any non-negative Borel measure on [0, T] that satisfies (6), then Thus c maximizes ∫_[0,T] M(t) ⋅ c₁(t) ⋅ dt on the set of all non-negative Borel measures c₁ on [0, T] that satisfy (6). Now let (30) Obviously, λ_c ≥ 0 and λ_c(t) = 0 for t ∈ supp(c), hence λ_c satisfies KKT condition (17). Next, (2), (3), (4), and (5) imply that s > 0 on the compact interval [0, T], hence inf_t∈[0,T] (s(t)) > 0. Thanks to (which is equivalent to α_S > 0), this implies that is bounded on [0, T], which in turn implies that M ∈ Lip ([0, T] → R), where M is given by (29). Thus λ_c ∈ Lip ([0, T] → R), where λ_c is given by (30). We need to show that λ_c satisfies the boundary condition (19) and the differential Eq (20). Indeed, hence so λ_c satisfies the boundary condition (19). Next, so which is the differential Eq (20).

The two proofs above use ideas from [18]. A few minor gaps in that material are covered here.

6 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 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 (31) 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⁻) ≠ 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:

Proposition 14 Let [s, λ_c, λ_W] ∈ BV ([0, T] → R) × BV ([0, T] → R) × R be a solution of Problem 10. If and for all s, then the following hold:

1. There must be a gulp of consumption at t = T if for all s (32)
2. A gulp of consumption at t = T is impossible if for all s (33)

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 (34) hence . Thus there is a discontinuity, s(T) ≠ s(T⁻), if n − r − α(s) ⋅ γ ≠ 0 for all s.

If (32) holds, then , hence s(T) − s(T⁻) > 0 because .

This discontinuity of s(t) produces the singular part of ; see Section 4. Now Eq (2) implies that the consumption c has singular part , 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.

Lemma 15 If [s, λ_c, λ_W] ∈ BV ([0, T] → R) × BV ([0, T] → R) × R is a solution of Problem 10, then λ_c(t) ∈ Lip ([0, T] → R).

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) and the help of the Gronwall lemma, implies boundedness of λ_c(t) on [0, T]. This property, together with (20), implies boundedness of , hence λ_c(t) is Lipschitz continuous.

Continuity of λ_c permits us to deduce the following result:

Proposition 16 Let [s, λ_c, λ_W] ∈ BV ([0, T] → R) × Lip ([0, T] → R) × R be a solution of Problem 10. Then there is a family of at most countably many (relatively) open subintervals of [0, T] such that λ_c > 0 and c = 0 in these subintervals, and λ_c = 0 and c ≥ 0 at the isolated points as well as on the closed subintervals that separate these open subintervals.

Proof. Thanks to Lemma 15, λ_c(t) is continuous, hence the inverse image of (0, +∞) is relatively open in [0, T]. Since the set of rational numbers, which is countable, is dense in the set of all real numbers, there is at most a countable set of (relatively) open subintervals of [0, T] in which λ_c > 0 and c = 0, while λ_c = 0 and c > 0 on each closed subinterval and at each single point that separates any two consecutive (relatively) open subintervals of [0, T] in which λ_c > 0.

In the remaining part of this section, we prove that there can be at most one (relatively) open subinterval in which λ_c > 0 and c = 0, and at most one closed subinterval on which λ_c = 0 and c > 0; see Theorems 21 and 25. In the next lemma, we construct a basic block of the explicit solution of Problem 10 in open subintervals where λ_c = 0 and c > 0. In this case, Eq (20) simplifies to (35) with λ = λ_W ∈ R. Eq (35) does not contain , and this simplifies forthcoming proofs.

Lemma 17 If and α_S(s) > 0 for all s ≥ 0, and if both of the following hold, (36) (37) then Eq (35) has a unique bounded, continuous positive solution for all t ≥ 0 and 0 < λ ∈ R that satisfy Eq (39). The corresponding optimal consumption is given by (38)

Proof. The conditions and λ > 0 ensure existence of a solution σ(t, λ) of (35), and (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 Since , we have (39) and now we transform (2) as follows: This completes the derivation of (38).

Remark 18 In [4] a less restrictive assumption is used, namely with 0 < s_M < ∞ 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 17, is called the general solution.

Now with the help of Lemma 17 we can derive the properties of subintervals of [0, T] in addition to those we derived in Proposition 16.

Proposition 20 Let [s, λ_c, λ_W] ∈ BV ([0, T] → R) × Lip ([0, T] → R) × R be a solution of Problem 10. In every subinterval of [0, T] in which λ_c > 0, the satiation is of the form s(t) = const ⋅ e^−γ⋅t and the following hold:

1. In every open subinterval of [0, T] in which λ_c = 0, the optimal satiation is s(t) = σ(t, λ_W) if for all s (40)
2. Consumption c(t) is 0 everywhere in (0, T) if for all s (41)

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).

7 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) and (40), the original problem (Problem 1) does have a unique solution and there are only three possible types of solutions of Problem 10, described as follows:

Definition 26 A solution is I-shaped (poor or over-satiated agent) when c = 0 and λ_c > 0 in [0, T), and all consumption is done in a single gulp at t = T; J-shaped (intermediate agent) when c = 0 and λ_c > 0 in [0, T_s), and c > 0 and λ_c = 0 in [T_s, T], and there is a gulp at t = T; U-shaped (rich or under-satiated agent) when there is a gulp at t = 0, c > 0 and λ_c = 0 in all of [0, T], and there is a gulp at t = T.

Section 6 contains almost a complete proof of existence of a solution of Problem 10. The only missing part is the proof of the second KKT condition in (17): λ_c ≥ 0. We now reduce this question to a simple non-differential inequality.

Lemma 27 Suppose (40) holds for all s and that s₀ > 0. Let [s, λ_c, λ_W] ∈ BV ([0, T] → R) × Lip([0, T] → R) × R be an I-shaped solution (T_s = T) or a J-shaped solution (0 < T_s < T) that satisfies all the requirements of Problem 10, except possibly the KKT condition λ_c ≥ 0 (see (17)) in [0, T_s) with 0 < T_s ≤ T. Then λ_c(t) > 0 in [0, T_s) iff (55)

Proof. Eq (20) for λ_c implies that in [0, T_s) (56) We need to prove that λ_c > 0 in [0, T_s). Multiplying (56) by e^−γ⋅t, we obtain hence it suffices to show that (λ_c(t) ⋅ e^−γ⋅t) > 0 in [0, T_s). The derivative of the expression in square brackets can be written as which is positive, thanks to and (40), hence the expression in square brackets, takes its largest value in [0, T_s) at t = T_s, and that value is If this is positive, then for all t close to but less than T_s, hence for all such t. Therefore, condition (55) is necessary for λ_c(t) ≥ 0. It remains to show that this is also a sufficient condition. Suppose condition (55) holds. Since , we obtain hence for t ∈ [0, T_s). Therefore, (λ_c(t) ⋅ e^−γ⋅t) is a decreasing function, so for all t ∈ [0, T_s). Thus condition (55) is necessary and sufficient for λ_c(t) > 0 in [0, T_s).

Remark 28 If s₀ → 0, then condition (55) cannot be satisfied, since that would imply that , thanks to (36). Hence λ_W would not be real, so the interval [0, T_s) must be empty, that is, T_s = 0.

8 Construction of explicit optimal solutions

In the previous section we described how to reduce the problem of finding optimal solutions of Problem 1 to solving non-differential equations. With the help of this theory, in this section we construct explicit solutions for logarithmic and CRRA utilities.

9 Conclusions and closing remarks

We investigated the structure and existence of solutions of the problem of consumption with satiation in continuous time for general utilities. We proved that the satiation function s(t) cannot be continuously differentiable, hence we assumed s to be a function of bounded variation. We proved that and c must be continuous with respect to Lebesgue measure in (0, T) but that there may be a gulp of consumption at either or both endpoints. Since such gulps take place in continuous time, they are not artifacts of discrete time and could be observed experimentally.

We described how to construct optimal solutions.

In Section 6.1, we considered solutions under the assumptions of a high discount rate (assumptions (33) and (41)). This happens in societies characterized by high conflict. The solution we obtained for that case can be described as follows: It is optimal to consume all of one’s wealth at once and not wait, no matter what one’s utility function is. Indeed, in societies with 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 death-row 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.