# Dynamic stochastic optimization for sustainability and management of overconsumption

Rosella Castellano, Roy Cerqueti

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Rosella Castellano, Roy Cerqueti. Dynamic stochastic optimization for sustainability and management of overconsumption. Communications in Nonlinear Science and Numerical Simulation, 2025, 149, pp.108960. 10.1016/j.cnsns.2025.108960. hal-05108867

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Submitted on 12 Jun 2025

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Research paper

Dynamic stochastic optimization for sustainability and
management of overconsumption

Rosella Castellano<sup>a</sup>, Roy Cerqueti<sup>b,c,*</sup>

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ARTICLE INFO

Keywords:

Stochastic optimal control

Simulations

Sustainability

Natural resources

A B S T R A C T

This paper deals with an intergenerational utility maximization problem for consuming a naturally exhaustible resource. In this context, we are at odds with the unfair standard procedure of applying a time-dependent factor for discounting the utility and introducing a suitable function for penalizing overconsumption. A finite-time horizon dynamic stochastic optimization problem is presented to achieve the desired target. We provide a closed-form solution to the problem which does not lead forcefully to the classical golden rule outcome of not consuming now to save the future. The theoretical findings are validated through extensive numerical experiments, with a parameter set selected based on empirical data. Such experiments highlight that optimal consumption depends strongly on the natural resource regeneration rate and the initial value of the stock.

**1. Introduction**

Human life depends on the availability of natural resources. Some of them are renewable – such as wind and sun – and their availability is granted now and for the future. Unfortunately, another class of natural resources is exhaustible. For instance, fossil fuels are characterized by a finite stock available on the Earth. Exhaustible resources become everyday scarcer than the day before, and the entire world is experiencing a period of growing poverty and conflicts. In this framework, it is a must to reflect on the fact that the so-called *overconsumption* of an exhaustible resource today (or shortly) – i.e., an excessive consumption with detrimental effects on the environment and society (on this, see the analysis provided by [1] and the discussion of the sustainable use of ecosystems in [2]) – can cause serious harm to future generations. We have in mind an institutional supranational policymaker that should act to push Governments to fairly balance today’s needs in terms of consumption with the well-being of future generations. To this aim, the policymaker can solve the classical *intergenerational utility maximization* problem, which is composed of three main steps: first, we introduce a time variable for labeling and ordering the generations, being $t = 0$ the current generation and a large value of $t$ associated to a far future from now; second, we define a utility function associated with the consumption at time $t$ of the considered exhaustible natural resource; third, we maximize the generation-based aggregation of the utility functions by taking consumption as control variable. Intuitive economic reasonings suggest that the policymaker is more interested in the wellness of the present rather than that of the future. Following this viewpoint, a time-decreasing discount factor is often applied to the utility functions in the classical intergenerational utility maximization problem (see e.g., [3]). The reduction of the relevance of the well-being of future generations in the decision processes has led to the well-known critical essay on “time preference” by the Cambridge economist Frank Ramsey [4]. He argued that the application of a time-decreasing discount to the utility of future generations is “a practice

<footnote><sup>a</sup>UnitelmaSapienza University of Rome, P.zza Sassari 4, 00161, Roma, Italy</footnote>

<footnote><sup>b</sup>Sapienza University of Rome – Department of Social and Economic Sciences, Piazzale Aldo Moro, 5 I-00185, Rome, Italy</footnote>

<footnote><sup>c</sup> GRANEM – Université d'Angers, 49036 Angers, CEDEX 01, France</footnote>

<footnote>* Corresponding author at: Sapienza University of Rome – Department of Social and Economic Sciences, Piazzale Aldo Moro, 5 I-00185, Rome, Italy.
E-mail addresses: rosella.castellano@unitelmasapienza.it (R. Castellano), roy.cerqueti@uniroma1.it (R. Cerqueti).</footnote>
<caption>**Table 1**</caption>

Analysis of some relevant recent literature, with discrepancies and analogies with the present research.

<table><thead><tr><td>Paper</td><td>Similarities</td><td>Deviations</td></tr></thead><tbody><tr><td>Asheim et al. [11]</td><td>(1) Analysis of intergenerational utility and sustainability;<br/>(2) Discussion of the time-dependent utility discounting</td><td>(1) Axiomatic approach;<br/>(2) Not an explicit definition of the intergenerational utility</td></tr><tr><td>Chichilnisky [12]</td><td>(1) Time-discounted aggregated intergenerational utility is not sustainable;<br/>(2) Inclusion of present and long-run future utilities</td><td>(1) Axiomatic approach;<br/>(2) Dictatorship of present and future is not allowed</td></tr><tr><td>Goodland [13]*</td><td>Analysis of sustainability</td><td>(1) Survey paper based on literature review;<br/>(2) Not a specific focus on mathematical models</td></tr><tr><td>Huang and Hsu [14]</td><td>Analysis of the discounting the utility of the future generation</td><td>Proposal of a modified version of the discount rate by including also the adjustment of individuals' preferences</td></tr><tr><td>Khan and Stinchcombe [15]</td><td>(1) Theoretical framework for building intergenerational equity;<br/>(2) Criticism of discounting the utility of the future generation</td><td>(1) Discussion of the case of finite generations;<br/>(2) Theoretical preferences- and decision theory-based approach</td></tr><tr><td>Padilla [16]*</td><td>Criticism of discounting the utility of the future generation</td><td>Not a specific focus on mathematical models</td></tr><tr><td>Pearce et al. [17]</td><td>Criticism of discounting the utility of the future generation</td><td>Not a focus on an optimization problem for sustainable decisions</td></tr><tr><td>Phillips [18]</td><td>Definition of sustainability as dependent on time</td><td>(1) Synthesis of four existing theories of sustainability;<br/>(2) Not aggregation of the time-dependent utility</td></tr><tr><td>Stavins et al. [19]</td><td>Criticism of discounting the utility of the future generation</td><td>Implicit conditions on intergenerational equity based on a nondecreasing utility over time</td></tr><tr><td>Sumaila and Walters [20]</td><td>Intergenerational utility model</td><td>Sustainability is achieved by generalizing time-discount by including the wellness of the future generations</td></tr><tr><td>Tol [21]*</td><td>Discussion of the discounting the utility of the future generation</td><td>Review paper for discussing the optimal policies for climate change</td></tr><tr><td>Weitzman [22]</td><td>Analysis of the discounting the utility of the future generation</td><td>Proposal of a stochastic discount rate</td></tr></tbody></table>

The * on some papers points to contributions leading to a clear view of the problem we deal with, along with the papers therein contained.

which is ethically indefensible and arises merely from the weakness of the imagination". In the same line, Solow [5] discussed the exploitation of exhaustible resources by considering the theme of the well-being of future generations. Several years later, Schelling [6] provided a twofold view of the intergenerational discount which is driven by time and consumption preference. In general, mainly in the environmental context, the reduction of the relevance of future well-being in the decision processes is a serious issue. In this context, Ray [7] proposed an altruistic perspective by considering future generations' wellness as a current-generation utility driver.

Other classical contributions to this debate are Sidgewick [8], Pigou [9] and Harrod [10]. Referring to a more recent large strand of literature, it is worth mentioning Asheim et al. [11], Chichilnisky [12], Goodland [13], Huang and Hsu [14], Khan and Stinchcombe [15], Padilla [16], Pearce et al. [17], Phillips [18], Stavins et al. [19], Sumaila and Walters [20], Tol [21] and Weitzman [22]. Such contributions are properly described in Table 1, where we discuss their similarities and deviations from the present paper.

This paper is part of this debate. It moves from the setting presented in Castellano et al. [23] present a finite time horizon intergenerational utility maximization problem for the consumption of an exhaustible – and, likely, scarce – natural resource in the frame of sustainability – where sustainability is here intended to be a criterion for avoiding the time preference mentioned by Ramsey and, rather than this, introduce a consumption-based preference rule.

Under the adopted sustainability perspective, we apply a discount factor to the utility functions which is independent of time but such that overconsumption is a habit of being penalized.

We proceed in two directions. First, we develop a stochastic optimal control problem, which formalizes the utility maximization and introduces a discount factor based on the level of consumption; this model is representative of a more general class of nonlinear optimization problems to be conveniently formulated and solved by dynamic programming. Second, we implement extensive simulations in a wide set of scenarios to get an intuitive view of the connections between the model's main theoretical features and the optimal consumption path. The simulation experiments are carried out based on the empirical data related to oil.

The adopted approach of stochastic optimal control in a dynamic programming setting (see e.g. [24,25] for an overview) is particularly appropriate for the analysis of environmental issues and sustainable consumption. Such an approach insists on the line traced by authoritative scholars – like the Nobel Laureate Vernon Smith's (1977) contribution – on resource management developed by using stochastic optimization models (on this, see the Handbook by [26]).

In this respect, we can mention some important recent contributions to this field. Tang and Zhang [27] present a stochastic optimal control model for carbon sequestration by afforestation. Tirkolaee et al. [28] acknowledge that stochastic optimal control
theory can be used for sustainable waste management. Hou et al. [29] use control theory to investigate the effects of the heterogeneity of human activities on the patterns of the vegetation. Srinivasu and Kumar [30] present an optimal harvesting problem in the context of exploiting a renewable resource. Maier [31] presents a stochastic optimization problem for real options-based evaluation of natural resources investments. Pinheiro [32] is close to us in that the author discusses the optimal management of resources of renewable type in a dynamic programming framework. However, the quoted paper does not present an intergenerational model with infinite generations. Hence, there is no need to introduce a discount factor in Pinheiro's model — while here we need to provide a discounting for the convergence of the aggregate utility function. We are philosophically close to Yoshioka and Yoshioka [33], who provide a stochastic optimization model through dynamic programming for managing the reservoir of dams. The authors claim that they focus on simple mathematical models because deeply analyzing simplified problems can often provide useful and non-trivial insights into more complicated and practical cases. We adopt the same strategy, to provide a clear illustration of the nature of the results and their practical usefulness. In this line, we offer extensive numerical experiments that complete the theoretical study of the mathematical problem (see e.g. 34,35).

The novelties of the present paper are two. First, in presenting a finite-time horizon stochastic optimal control problem we overcome the limitations of the infinite horizon context proposed by Castellano et al. [23]. Indeed, in the quoted paper we have a total agreement with the golden rule, for which sustainability means that one must avoid consumption now for storing resources for the future. In the present framework, we obtain less trivial solutions. Second, we obtain a closed-form solution to the stochastic optimization problem even in the cases of no-dictatorship of the present and the future.

The results of our study show that the optimal consumption of an exhaustive resource is remarkably dependent on its regeneration rate and the initial value of the available stock. Moreover – differently from what the time-preference utility context suggests – the proposed sustainability approach leads to the possibility of high consumption levels even for generations furthest away from the current one.

The remaining part of the paper is organized as follows. Section 2 contains the theoretical formalization and the solution of the optimal control problem, along with the introduction of the overconsumption-based discount term. Section 3 is devoted to numerical experiments for the validation of theoretical results and to provide more insights on the optimal consumption path. Section 4 offers some conclusive remarks.

## 2. The model

We introduce a filtered probability space $(\Omega, \mathcal{F}, \{F_t\}_{t \ge 0}, \mathcal{P})$, which is assumed to contain all the random terms of the model.

Consider a natural resource, whose stock varies according to the following stochastic differential equation:

$$M_t = m + \int_r^t (a - c_s) M_s ds + \int_r^t b M_s dW_s, \quad (1)$$

where $r \ge 0$ is the time when the accumulation of resource starts, $M_r = m > 0$ is the initial level of the stock of resource, $t \ge r$ is time; $a \in \mathbb{R}$, $b \in (0, +\infty)$ are the rate of regeneration of the available resource and its instantaneous volatility, respectively, $W_t$ is a standard 1-dimensional Brownian motion at time $t$, $\{c_t\}_{t \ge r} \subseteq [0, 1]$ is a stochastic process representing the consumption of the natural resource evolving as in (1), with $c_r = c \in (0, 1)$. For the sake of notation, the dependence on $r$ of $M_t$ in (1) is omitted. We assume that the stochastic process $\{c_t\}_{t \ge r}$ lives in a set $\mathcal{A}$ defined as follows:

$$\mathcal{A} = \{c : [r, +\infty) \times \Omega \to [0, 1], F_t - \text{progressively measurable processes}\}. \quad (2)$$

Also, the dependence on $r$ of $\mathcal{A}$ is omitted. In a continuous-time setting and aligned with a large strand of mainstream economic literature (see e.g. 5,36, just to cite a few), we introduce a utility function independent from time $u : [0, 1] \to \mathbb{R}$ assigning a utility to the level of consumption. We have in mind an altruistic policymaker, which acts under a sustainability criterion. She/he is in the position of planning the consumption of an exhaustible resource for the current and future generations. Her/his incentives can be found in the will to achieve the maximum level of satisfaction for all generations by treating them fairly – i.e., by not using a time preference approach and by avoiding the excessive consumption of the considered natural resource by some of them.

Under this perspective, the objective function of the optimal control problem is the expected aggregate intergenerational utility defined as follows:

$$J(r, m, \{c_t\}_{t \ge r}) = \mathbb{E} \left[ \int_r^T h(c_t)u(c_t)dt + \Gamma(T, M_T) \mid F_T \right], \quad (3)$$

where $\mathbb{E}$ is the expected value operator; $T > 0$ is the time horizon of the problem and represents here the limit generation for which a sustainable policy is implemented; $\Gamma(T, M_T)$ is the terminal utility associated with the stock of resource, $M_T$, at the achievement of the horizon $T$; $h : [0, 1] \to [0, +\infty)$ is a decreasing function of consumption which formalizes the discounting of the instantaneous utilities of consumption, $u$. In other words, the discount factor $h$ penalizes overconsumption rather than time. We notice that in general $u$ and $h$ do not mutually interact. Indeed, $h$ is the discount factor applied by the policymaker, while the utility function $u$ measures the well-being of humans deriving from the exploitation of the resource. Therefore, $h$ and $u$ have different natures. The policymaker is free to set a function $h$ – and, of course, results will be affected by such a choice. Differently, the utility function $u$ comes from the analysis of the satisfaction levels of the people when consuming the considered natural resource.
The intergenerational expected utility maximization problem comes out from (1) and (3). Its solution is the value function of the problem, and can be written through a function of the initial value of the dynamics, denoted by $V: [0, T] \times [0, +\infty) \to \mathbb{R}$, as follows:

$$V(r, m) = \sup_{\{c_t\}_{t \ge r} \in \mathcal{A}} J(r, m, \{c_t\}_{t \ge r}). \quad (4)$$

The horizon $T$ here represents the last generation considered by the policymaker in her/his analysis so that we deal with a finite time without paying attention to the case of an infinite number of generations. This is a quite reasonable condition: it leads to easily interpretable results and it is not too restrictive because a large value of $T$ might be always interpreted as the attention paid by the policymaker to a very far future.

We find the theoretical solution of the stochastic optimal control problem (3) by adopting a dynamic programming approach. To this aim, we impose a technical assumption on $h, u$, and $\Gamma$. We select these functions to guarantee that they are twice differentiable in their arguments everywhere and satisfy the following growth conditions:

$$\begin{cases} |h(c)u(c)| \le L_1(1 + |c|^{\ell_1}), \\ |\Gamma(T, M)| \le L_2(1 + |M|^{\ell_2}), \end{cases} \quad (5)$$

for some constants $L_1, L_2, \ell_1, \ell_2$. Moreover, we observe that the drift and the diffusion coefficients in (1) are linear with respect to $M_s$ and $c_s$. Thus, they easily satisfy the regularity conditions and the growth requirements of Eq. (2.2) in Fleming and Soner [24], Section IV.2. Therefore, we observe that the proposed optimal control problem can be attributed to a large class of very general problems of this type under the mild requirements in (5). In this respect, refer to Fleming and Soner [24], Section IV.2, where the stochastic control problem in the case of finite-time horizon is presented (an overview of such a class of models can be found also in Section III.6). Thus, we limit the discussion here to the statement of the main results related to the solution of the problem, and we refer to the quoted reference for the adaptations of the proofs to our specific context.

By applying the Dynamic Programming Principle, we can write the Hamilton-Jacobi-Bellman equation (HJB, hereafter), which is an ordinary differential equation satisfied by the value function in (3), so that the following result holds true:

**Theorem 1 (HJB).** Assume that $V: [0, T] \times [0, +\infty) \to \mathbb{R}$ is continuous in $[0, T] \times [0, +\infty)$ and twice differentiable in $(0, T) \times (0, +\infty)$. Then $V$ satisfies the following equation:

$$\frac{b^2 m^2}{2} \frac{\partial^2}{\partial m^2} V(r, m) + \sup_{c \in (0,1)} \left\{ (a-c)m \cdot \frac{\partial}{\partial m} V(r, m) + h(c)u(c) \right\} = -\frac{\partial}{\partial r} V(r, m), \quad (6)$$

for each $m \in (0, +\infty)$, with $V(r, m) = \Gamma(r, m)$ for $(r, m) \in \partial([0, T] \times [0, +\infty))$, where $\partial(\cdot)$ is the boundary of the set $\cdot$.

Eq. (6) is a particular case of equation (3.3) in Section IV.3 in Fleming and Soner [24].

It is worth remarking here that **Theorem 1** states that the value function can be viewed as a solution of the differential Eq. (6) with its boundary conditions in a formal way, in the sense that the regularity of the value function is required. If such a regularity is granted, then one can formalize that (6) with its boundary conditions leads to the solution of the optimal control problem. Specifically, a Verification Theorem can be proved. In our context, we are in the position of stating the Verification Theorem. For the details of the proof, see Fleming and Soner [24], Theorem 3.1 in Chapter IV.

**Theorem 2 (Verification Theorem).** Assume that $v: [0, T] \times [0, +\infty) \to \mathbb{R}$ is continuous in $[0, T] \times [0, +\infty)$, twice differentiable in $(0, T) \times (0, +\infty)$ and it is a solution of the differential equation with boundary condition in (6). Then we have:

(a) $v(r, m) \ge J(r, m, \{c_t\}_{t \ge r})$, for each $(r, m) \in [0, T] \times [0, +\infty)$ and any $\{c_t\}_{t \ge r} \in \mathcal{A}$.

(b) If there exists $(\{c_t^*\}_{t \ge r}, \{M_t^*\}_{t \ge r})$ such that

$$c^*(r, M^*(t)) \in \operatorname*{argsup}_{c \in (0,1)} \left\{ (a-c)M_t^* \cdot \frac{\partial}{\partial m} v(r, M^*(t)) + h(c)u(c) \right\}$$

and $c_t^* = c^*(r, M^*(t))$, for each $t$, then $v(r, m) = J(r, m, \{c_t^*\}_{t \ge r})$, for each $(r, m) \in [0, T] \times [0, +\infty)$.

The optimal control $c^*$ can be sometimes expressed in a closed formula, based on the behavior of functions $h$ and $u$. We now present a particular case in which the argument of the argsup in point (b) of **Theorem 2** has a simple shape, hence allowing the direct computation of the optimal control by using standard calculus. The following result holds true.

**Theorem 3.** Define $\ell: [0, 1] \to \mathbb{R}$ such that $\ell(c) = h(c)u(c)$ and assume that $\ell$ is strictly concave
Define

$$\bar{c}(r, m) = I \left( m \cdot \frac{\partial}{\partial m} V(r, m) \right).$$

where $I$ is the inverse function of $\ell'$. Denote as $\bar{M}$ the solution of the closed-loop equation, given by

$$\bar{M}_t = m + \int_r^t (a - c^*(r, \bar{M}_s)) \bar{M}_s ds + \int_r^t b \bar{M}_s dW_s,$$

where $c^*(r, m)$ is the optimal consumption, and belongs to the set $\{\bar{c}(r, m), 0, 1\}$. If we set $\bar{c}_t := c^*(r, \bar{M}_t)$, we have $V(r, m) = J(r, m, \{\bar{c}_t\}_{t \ge r})$, for each $(r, m) \in [0, T] \times [0, +\infty)$.
Function $\ell$ has a clear financial meaning. Indeed, it represents the utility of consumption filtered by the action of the altruistic policymaker, who penalizes overconsumption. The concavity of $\ell$ in Theorem 3 is a condition stating that the well-being of the people – as interpreted by the policymaker, who applies the form of consumption-based discount $h$ – has decreasing increments as consumption grows. Such a regularity allows us to derive the optimal consumption and the optimal trajectory in closed form.

To gain more significant insights into the proposed theoretical optimization model and its solutions, we propose some numerical experiments (see e.g. 23,37).

### 3. Numerical experiments

In this section, we conduct a numerical analysis to investigate the management of overconsumption within the sustainability framework discussed above. The goal is to determine the optimal intertemporal consumption strategy, $\{c_t^*\}_{t \ge r} \subseteq [0, 1]$ where $c_r = c \in (0, 1)$, for the stock of natural resources whose dynamics are governed by the stochastic differential Eq. (1), as defined by the theoretical optimization problem in (4). We take hereafter $r = 0$ and present a discrete-time setting. To this aim, we fix a time step $\Delta t$ and we set $\Delta t = 1$. Such a choice is consistent with the intergenerational nature of the model, where each unit time $t = 1, \dots, T$ represents a full generation. Similar approaches are found in intergenerational and macroeconomic models [38–41], reinforcing the validity of this choice. Additionally, adopting $\Delta t = 1$ enables extensive simulations while maintaining computational feasibility, balancing accuracy and efficiency in the optimization process.

We determine the optimal path for the control variables by implementing a multi-step procedure to optimize the objective function in (3). The analysis is conducted over a finite time horizon, $T = 16$, where one generation corresponds to approximately 25 years. This choice allows for an intuitive interpretation of the results while reflecting the policymaker’s long-term perspective on resource management.

To provide a concrete example, we focus on oil as an exhaustible resource and calibrate the key parameters of the numerical experiments accordingly. Initially, calibration is performed on real data. Based on the calibrated data, the lattice is constructed and then normalized to ensure that $c_i \in [0, 1]$.

The initial value, $c_0$, is derived from the forecasts provided by the International Energy Agency (IEA), which projected global oil demand at 0.099 billion barrels per day in 2020 (see [42]). Converting this into an annual figure, we set $c_0 = 36.46$ billion barrels in the simulations. To account for uncertainty, the IEA considers two alternative scenarios: one where global oil demand decreases by 730,000 barrels per day, and another increases by 825,000 barrels per day.

Under the assumption that oil demand coincides with consumption, we construct a set of $n = 2^T$ paths using a recombining dynamic lattice for the control variable $\{c_t\}_{t \ge 0}$. During the transition from time $t$ to $t+1$, consumption evolves as $c_{t+1} = \zeta_t c_t$, where $\{\zeta_t\}_{t \ge 0}$ is an i.i.d. stochastic process having generic realization taking value $\delta_u$ with probability $p$ and $\delta_d$ with probability $1-p$. We conveniently set $p = 0.5$. Moreover, based on the IEA’s scenarios, we set $\delta_u = 1.21$ and $\delta_d = 0.82$, corresponding to the projected growth and decline rates of oil demand. As already said above, the resulting consumption trajectory will be suitably normalized, to maintain $c_i \in [0, 1]$. We have

$$c_t = c_0 \cdot \prod_{i=0}^{t-1} \zeta_i \qquad (7)$$

To ensure that $c_i \in [0, 1]$, an affine transformation is applied:

$$\tilde{c}_t = \frac{c_t - c_{\min}}{c_{\max} - c_{\min}}$$

where $c_{\min}$ and $c_{\max}$ represent the minimum and maximum values of $c_t$ over the values obtained in the evolution of $c$ described above, respectively. The theoretical maximum and minimum for $c_t$ are $c_0 \cdot \delta_u^{-1}$ and $c_0 \cdot \delta_d^{-1}$, respectively. This normalization preserves the empirical structure of the data while ensuring consistency with model constraints. Fig. 1 presents the normalized consumption.

The trajectory construction of the consumption follows a two-step process: a recombining tree based on real-world data is first generated to capture empirical resource consumption dynamics, and then values are normalized within $[0, 1]$ to maintain consistency with theoretical constraints. These steps guarantee numerical stability, as supported by previous studies [43–46].

Regarding the resource stock, the parameters $a \in \mathbb{R}$, $b \in (0, +\infty)$ in (1) are calibrated using annual data on oil reserves from major oil-producing countries [47]. We notice that the choice of $\Delta t = 1$ in the discretization of the SDE does not affect its stability or reliability, as the Euler–Maruyama scheme remains consistent for larger time steps, provided that the coefficients of the stochastic differential equation (e.g., $a$ and $b$) satisfy the standard Lipschitz conditions, which hold in our model. Under these conditions, the SDE (1) simplifies, allowing parameter estimation through Maximum Likelihood techniques.

In the experiments, we adopt a CRRA utility function:

$$u(c_i) = \frac{c_i^{1-\gamma} - 1}{1-\gamma}, \quad \gamma \neq 1,$$

which simplifies to $u(c_i) = \ln(c_i)$ when $\gamma = 1$. Overconsumption is penalized using $h(c_i) = \frac{p}{\ln(c_{n,i})}$, with $P > 0$, reflecting the negative externalities associated with excessive resource use. The following pseudo-code outlines the algorithm implemented to determine the optimal consumption path: Fig. 2 presents the optimal consumption trajectory obtained using the pseudo-code described above.

The optimal consumption trajectory shown in Fig. 2 exhibits a three-phase structure: (i) an initial sharp decline, (ii) a prolonged near-zero consumption phase, and (iii) a final increase. This reflects a resource conservation strategy that prioritizes long-term
<figure><img src="image_5.png" /><figcaption>Fig. 1. Lattice of the normalized consumption.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. Optimal consumption path, $(\tilde{c}_{i,\pi})_{i=1}^T$.</figcaption></figure>

sustainability. Initially, high consumption is rapidly adjusted downward to prevent premature depletion. A subsequent conservative phase allows the resource stock to accumulate, followed by a recovery phase where consumption increases as sustainability is met.

Fig. 3 shows the dynamics of the stock of resources with the optimal consumption. Interestingly, the selected scenarios share a rather evident increasing behavior. This confirms that a sustainable approach leads to a positive growth rate of the stock of resource.

### 3.1. Sensitivity analysis

To better understand the robustness of the results, we analyze the sensitivity of the optimal consumption path to key parameters, including the initial stock of the resource $M_0$, its regeneration rate $a$, and volatility $b$.

**Effect of $M_0$ and $a$.** We here consider some values of $a$ (see Fig. 4) and $M_0 = 16 : 10 : 76$.

The analysis reveals that changes in the initial stock of the resource $M_0$ do not influence the relative consumption trajectory. This stems from the structure of the stochastic differential equation governing $M$, where the drift term $(a-c_i)$ is proportional to $M_{i-1}$. Consequently, variations in $M_i$ simply scale the entire trajectory of $M_i$, without affecting the fundamental trade-off between resource growth and consumption.

By contrast, the regeneration rate $a$ plays a crucial role in shaping the consumption trajectory. When $a$ is low, the system adopts a conservative approach, prioritizing resource accumulation over immediate consumption. Conversely, as $a$ increases,
1. Set the initial parameters: $\Delta t = 1$, $T = 16$, $m = c_0 = 36.46$, $a = 1.295$, $b = 0.1025$, $\delta_d = 0, 82$, $\delta_u = 1, 21$, $\gamma = 0.5$, $P = 1$; $K = 10, 000$, $n = 2^T$.

2. Construct the admissible region of controls C as a matrix with n rows and T columns. The generic element $c_{n,t}$ of the set C for $n = 1, \dots, 2^T$ and $t = 1, \dots, T$ satisfies:

$$c_{n,t} = \text{Normalize} (\{\delta_u c_{n,t-1}, \delta_d c_{n,t-1}\}),$$

where the normalization is defined as:

$$\tilde{c}_{n,t} = \frac{c_{n,t} - c_{\min}}{c_{\max} - c_{\min}},$$

with $c_{\min} = \min_{n,t} c_{n,t}$ and $c_{\max} = \max_{n,t} c_{n,t}$, ensuring that $c_{n,t} \in [0, 1]$ for all $t$ and $n$. Initialize $c_{n,0} = c_0$ for each $n$.

3. Generate a random matrix $W$ whose generic element $\omega_{k,t}$ is drawn from $N(0, 1)$.

4. Initialize the indices: set $n = 1, k = 1$, set $t = 1$.

5. Compute the resource stock dynamics:

$$M_t^{n,k} = M_{t-1}^{n,k} + (a - \tilde{c}_{n,t-1}) M_{t-1}^{n,k} \Delta t + M_{t-1}^{n,k} b \sqrt{\Delta t} \cdot \omega_{k,t}.$$

6. Compute the penalized utility:

$$hu_t^n = \frac{P}{\ln(c_{n,t})} \cdot \frac{c_{n,t}^{1-\gamma} - 1}{1 - \gamma}$$

7. Update the time step: set $t = t + 1$. If $t = T + 1$, go to step 8; otherwise, return to step 5.

8. Update the Monte Carlo iteration: Set $k = k + 1$. If $k = K + 1$, go to step 9; otherwise, return to step 4.

9. Compute the intergenerational aggregated utility for each admissible path:

$$J^n = \left[ \frac{1}{T} \sum_{t=1}^{T} hu_t^n \right] + \ln \left( 1 + \frac{T}{K} \sum_{k=1}^{K} M_T^{n,k} \right).$$

10. Update the path index: set $n = n + 1$. If $n = N + 1$, go to step 11; otherwise, return to step 3.

11. Identify the optimal path: select $\tilde{n}$ such that

$$V^{\tilde{n}} = \max_{n=1,\dots,N} J^n.$$

12. The optimal control is given by:

$$\{c_{t,\tilde{n}}\}_{t=1}^T.$$

<figure><img src="image_7.png" /><figcaption>Fig. 3. Randomly selected scenarios for $\{M_t\}_{t=1}^T$ with $\{\tilde{c}_{n,t}\}_{t=1}^T$. The parameters set specified in the algorithm is used.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 4. Optimal consumption trajectories {$\tilde{e}_{i,\pi}^T$}$_{t=1}^T$ for different values of a.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Optimal consumption trajectories {$\tilde{e}_{i,\pi}^T$}$_{t=1}^T$ for different values of b and $\gamma = 0.5$.</figcaption></figure>

consumption begins earlier and grows more steeply. This result highlights the fundamental trade-off between sustainability and resource exploitation: a higher regeneration rate allows for a faster transition from conservation to active consumption. Fig. 4 illustrates the analysis.

*Effect of volatility b*. We consider some values of *b*, as shown in the legend of Fig. 5.

While the regeneration rate significantly impacts consumption dynamics, the effect of volatility *b* is more limited, except at extreme levels. For most values of *b*, the optimal consumption path remains unchanged, suggesting that moderate fluctuations in resource availability do not require major adjustments in consumption strategy.

However, when volatility reaches exceptionally high levels ($b = 1.5$), the system shifts from a precautionary approach, where uncertainty reduces consumption, to a pro-cyclical strategy, where consumption begins earlier and increases more aggressively. Rather than treating uncertainty as a risk, the decision-maker exploits it as an opportunity for resource utilization. This suggests that only extreme uncertainty disrupts the stability of the optimal consumption plan. Results are shown in Fig. 5.

The role of *b* is closely linked to the risk aversion coefficient $\gamma$. With moderate risk aversion ($\gamma = 0.5$), the system does not exhibit strong precautionary behavior, allowing the consumption trajectory to remain relatively stable for moderate values of *b*. However, at extreme volatility levels, a higher $\gamma$ would likely lead to a more conservative strategy, reducing consumption rather than increasing it.

*Effect of $P > 0$*. We consider several values of $P$ as reported in Fig. 6.
<figure><img src="image_10.png" /><figcaption>Fig. 6. Optimal consumption trajectories {$\tilde{e}_{\epsilon,\bar{\epsilon}}$}$_{t=1}^T$ for different values of P and $\gamma = 0.5$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 7. Optimal consumption trajectories {$\tilde{e}_{\epsilon,\bar{\epsilon}}$}$_{t=1}^T$ for different values of $\gamma - G$, in the legend - and $P = 0.5$.</figcaption></figure>

The sensitivity analysis on *P* in the penalization function highlights its strong influence on the timing and intensity of consumption. Even small increases in *P* significantly alter the optimal trajectory: (i) for low *P* (*P* = 0.2), the system maintains minimal consumption for an extended period, prioritizing long-term resource accumulation; (ii) as *P* increases, consumption begins earlier, and the system moves toward a more balanced utilization strategy. This suggests that the penalization function plays a non-linear role as even minor changes in *P* can shift the trade-off between conservation and immediate consumption. Fig. 6 shows the results of the analysis.

**Effect of risk aversion** $\gamma$. A further sensitivity test examines the role of risk aversion $\gamma$ in shaping optimal consumption. We present two values of $\gamma$, as in Fig. 7

For low-risk aversion ($\gamma = 0.5$), consumption remains relatively stable, with an initial decline followed by a moderate and near-constant phase. This indicates a preference for smoothing consumption over time rather than adopting a precautionary approach. In contrast, for high-risk aversion ($\gamma = 2.0$), the system prioritizes extreme prudence, keeping consumption minimal for most of the period before a final surge near the terminal horizon. This behavior reflects the decision maker's tendency to defer consumption to secure future resources, even at the cost of lower utility in earlier periods as it is shown in Fig. 7.

**Joint effect of P and γ.** Finally, we examine how penalization *P* and risk aversion *γ* interact in shaping optimal consumption strategies. The values of *P* and *γ* are those in Fig. 8. Fig. 8 highlights distinct patterns in consumption behavior depending on the combination of *P* and *γ*. A high penalization and high-risk aversion lead to an extremely cautious strategy, with consumption
<figure><img src="image_12.png" /><figcaption>Fig. 8. Optimal consumption trajectories for different combinations of *P* and *γ*. The red diamond marks the initial consumption level.</figcaption></figure>

remaining minimal for most of the period before increasing sharply at the end of the considered time range. Conversely, when both parameters are low, the system follows a more uniform trajectory, with no prolonged conservation phases. The effects become more nuanced when one parameter is high and the other is low. With high *P* and low *γ*, consumption increases progressively over time, as the penalization discourages excessive accumulation. On the other hand, with low *P* and high *γ*, the system maintains a precautionary stance, delaying consumption but without the extreme deferral observed in the high-*P*, high-*γ* case. These findings confirm that the interaction between penalization and risk aversion plays a crucial role in determining the trade-off between early and late consumption. The degree of resource accumulation and utilization is not dictated by a single factor but rather by their combined effect, shaping the optimal trajectory over time.

## 3.2. Discussion on the role of the time horizon

An important consideration in the analysis of optimal consumption trajectories is whether extending the time horizon *T* significantly alters the observed dynamics. While increasing *T* could, in principle, reveal new patterns in consumption strategies, the results obtained suggest that the fundamental structure of the trajectories remains largely unchanged for sufficiently large *T*. Across different values of the penalization parameter *P* and the risk aversion coefficient *γ*, the key dynamics – characterized by an initial drop in consumption, a prolonged phase of near-zero consumption, and a final surge – are consistently observed. We acknowledge the relevance of assessing how a large *T* reflects a policymaker's attention to the distant future. In our study, *T* is set to 16 generations, which represents a significantly long horizon in the context of intergenerational decision-making [11,12]. This choice ensures that the optimization framework inherently captures the trade-offs between immediate resource use and long-term sustainability. From a theoretical perspective, the optimization model embeds *T* into the objective function, ensuring that a larger *T* amplifies the weight assigned to long-term outcomes. This property aligns with established results in intergenerational dynamic programming [4,36], where larger planning horizons naturally prioritize sustainability. Moreover, conducting a full sensitivity analysis on *T* would impose a substantial computational burden, given the exponential growth in the complexity of the optimization problem as *T* increases. The lattice-based structure used in the numerical procedure generates $2^T$ consumption paths, meaning that increasing *T* beyond the chosen value would rapidly escalate computational costs without necessarily providing additional insights. Given that *T* spans 16 generations, its impact is already substantial within the model, making further numerical validation unnecessary. The robustness of this interpretation is further supported by the mathematical structure of the optimization, which ensures convergence and stability even for large *T* [16,22]. The results confirm that the model exhibits stable consumption behavior, where optimal decisions are primarily driven by the trade-off between immediate utility and precautionary savings, rather than by the absolute length of the time horizon.

## 4. Conclusions

This paper contributes to the literature on optimization models for sustainability by proposing a framework that addresses the intertemporal management of an exhaustible natural resource across generations. A key issue in intergenerational resource alloca-
tion is the inherent unfairness of traditional models based on intertemporal equivalence, which often favors present generations at the expense of future ones. To overcome this limitation, our model replaces time-based discounting with a consumption penalization mechanism, ensuring a more balanced treatment of present and future generations. This approach provides policymakers with an effective tool to encourage responsible resource management, mitigating the risks associated with overconsumption while preserving long-term sustainability. The main findings align with fundamental principles of sustainable development: optimal consumption trajectories should account for the regeneration rate of the resource. Our results confirm that higher regeneration rates enable more resource-intensive consumption in close generations while still ensuring long-term availability. Additionally, the model reveals that violating the time-preference axiom leads to peak consumption at intermediate generations, emphasizing the need for policies that smooth consumption patterns over time. A critical insight from our analysis is the strong interaction between penalization and risk aversion in shaping optimal consumption decisions. The sensitivity analysis demonstrates that higher risk aversion encourages extreme precautionary savings, whereas a strong penalization of overconsumption promotes a more gradual resource allocation. Furthermore, the robustness of the consumption strategy for sufficiently large planning horizons suggests that beyond a certain threshold, increasing the time horizon does not significantly alter the results. This reinforces the model's ability to capture long-term sustainability trade-offs without requiring arbitrarily large simulation periods.

From a policy perspective, the results underscore the importance of technological innovation and alternative resource development. Policymakers should prioritize investments that enhance resource regeneration or facilitate the transition toward renewable substitutes, ensuring long-term economic and environmental sustainability. While our model implicitly captures the effects of regeneration, the broader challenge of replacing exhaustible resources with renewable alternatives remains an open question that warrants further investigation.

Methodologically, the independence of the penalization and utility functions from time offers two key advantages. First, it enhances the mathematical tractability of the model. Second, it allows for a broader discussion of how different functional forms of preferences influence optimal strategies. However, the assumption of constant preferences and penalization may represent a limitation, as real-world decision-making often involves generational shifts in consumption behavior and risk attitudes. Future research could extend the model by incorporating preference heterogeneity across generations to better capture the dynamic nature of intergenerational resource use.

### CRediT authorship contribution statement

**Rosella Castellano:** Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Investigation, Formal analysis, Conceptualization.
**Roy Cerqueti:** Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Investigation, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

No data was used for the research described in the article.

### References

[1] Hakansson A. What is overconsumption?–A step towards a common understanding. Int J Consum Stud 2014;38(6):692–700.

[2] Morse-Jones S, Luissetti T, Turner RK, Fisher B. Ecosystem valuation: some principles and a partial application. Environmetrics 2011;22(5):675–85.

[3] Becker RA. On the long-run steady state in a simple dynamic equilibrium model with heterogeneous households. Q J Econ 1980;95(2):375–82.

[4] Ramsey FP. A mathematical theory of saving. Econ J 1928;38:543–59.

[5] Solow R. The economics of resources or the resources of economics. Am Econ Rev 1974;15:1–14.

[6] Schelling TC. Intergenerational discounting. Energy Policy 1995;23(4–5):395–401.

[7] Ray D. Nonpaternalistic intergenerational altruism. J Econom Theory 1987;41(1):112–32.

[8] Sidgewick H. The methods of ethics. London: Macmillan; 1907.

[9] Pigou AC. Wealth and welfare. London: Macmillan; 1912.

[10] Harrod RF. Towards a dynamic economics. London: Macmillan; 1948.

[11] Asheim GB, Buchholz W, Tungodden B. Justifying sustainability. J Environ Econ Manag 2001;41(3):252–68.

[12] Chichilnisky G. An axiomatic approach to sustainable development. Soc Choice Welfare 1996;13:231–57.

[13] Goodland R. The concept of environmental sustainability. Annu Rev Ecol Syst 1995;26:1–24.

[14] Huang YS, Hsu CZ. An anticipative hyperbolic discount utility on intertemporal decision making. European J Oper Res 2008;184(1):281–90.

[15] Khan U, Stinchcombe MB. Intergenerational equity and sustainability: a large population approach. Social Choice and Welfare; 2023, http://dx.doi.org/10.1007/s00355-023-01450-w.

[16] Padilla E. Intergenerational equity and sustainability. Ecol Econom 2002;41:69–83.

[17] Pearce D, Groom B, Hepburn C, Koundouri P. Valuing the future. World Econ 2003;4(2):121–41.

[18] Phillips J. The sustainability dynamics framework–A holistic approach to define and evaluate sustainability and unsustainability in the anthropocene. Environ Impact Assess Rev 2020;84:106436.

[19] Stavins RN, Wagner AF, Wagner G. Interpreting sustainability in economic terms: dynamic efficiency plus intergenerational equity. Econom Lett 2003;79(3):339–43.
[20] Sumaila UR, Walters C. Intergenerational discounting: a new intuitive approach. Ecol Econom 2005;52(2):135-42.

[21] Tol RS. Targets for global climate policy: An overview. J Econom Dynam Control 2013;37(5):911-28.

[22] Weitzman ML. Why the far-distant future should be discounted at its lowest possible rate. J Environ Econ Manag 1998;36(3):201-8.

[23] Castellano R, Cerqueti R, Spinesi L. Sustainable management of fossil fuels: A dynamic stochastic optimization approach with jump-diffusion. European J Oper Res 2016;255:288-97.

[24] Fleming WH, Soner HM. Controlled Markov processes and viscosity solutions, vol. 25, Springer Science & Business Media; 2006.

[25] Yong J, Zhou XY. Stochastic controls: Hamiltonian systems and HJB equations. vol. 43, Springer Science Business Media; 1999.

[26] Bjørndal T, Epstein R, Romero C, Weintraub A. In: Handbook of operations research in natural resources, vol. 99, Berlin Heidelberg: Springer-Verlag; 2007.

[27] Tang W. Zhang S. Optimal decision model and solution for carbon sequestration by afforestation. Comput Math Appl 2018;76(10):2484-95.

[28] Tirkolaee EB, Goli A, Gütmen S, Weber GW, Szwedzka K. A novel model for sustainable waste collection arc routing problem: Pareto-based algorithms. Ann Oper Res 2022. http://dx.doi.org/10.1007/s10479-021-04486-2.

[29] Hou LF, Sun GQ, Perc M. The impact of heterogeneous human activity on vegetation patterns in arid environments. Commun Nonlinear Sci Numer Simul 2023;126:107461.

[30] Srinivasu PDN, Kumar GK. Bio-economics of a renewable resource subjected to strong Allee effect. Commun Nonlinear Sci Numer Simul 2014;19(6):1686-96.

[31] Maier S. Re-evaluating natural resource investments under uncertainty: An alternative to limited traditional approaches. Ann Oper Res 2021;299(1):907-37.

[32] Pinheiro S. Optimal harvesting for a logistic growth model with predation and a constant elasticity of variance. Ann Oper Res 2018;260(1):461-80.

[33] Yoshioka H, Yoshioka Y. Regime-switching constrained viscosity solutions approach for controlling dam-reservoir systems. Comput Math Appl 2020;80(9):2057-72.

[34] Castellano R, Cerqueti R. Optimal consumption/investment problem with light stocks: A mixed continuous-discrete time approach. Appl Math Comput 2012;218(12):6887-98.

[35] Castellano R, Cerqueti R. Mean-variance portfolio selection in presence of unfrequently traded stocks. European J Oper Res 2014;234(2):442-9.

[36] Abel AB, Blanchard OJ. An intertemporal model of saving and investment. Econometrica 1983;51(3):675-92;
Aishwarya A, Kumar P. A perspective view of sustainability of natural resources: the governance challenge. In: Rani M, Chaudhary BS, Jamal S, Kumar P, editors. Towards Sustainable Natural Resources. Cham: Springer; 2022, p. 3-9.

[37] Cerqueti R, Marazzina D, Ventura M. Optimal investment in research and development under uncertainty. J Optim Theory Appl 2016;168(1):296-309.

[38] Vitousek PM, Howarth RW. Nitrogen limitation on land and in the sea: How can it occur? Biogeochemistry 1991;13:87-115.

[39] Howarth RB. Intergenerational competitive equilibria under technological uncertainty and an exhaustible resource constraint. J Environ Econ Manag 1991;21(3):225-43.

[40] Möst D, Keles D. A survey of stochastic modelling approaches for liberalised electricity markets. European J Oper Res 2010;207(2):543-56.

[41] Ekeland I, Karp L, Sumaila R. Equilibrium resource management with altruistic overlapping generations. J Environ Econ Manag 2015;70:1-16.

[42] IEA. (2020) https://www.iea.org/news/global-oil-demand-to-decline-in-2020-as-coronavirus-weighs-heavily-on-markets.

[43] Du Baigang, Tan Tian, Guo Jun, Li Yibing, Guo Shunsheng. Energy-cost-aware resource-constrained project scheduling for complex product system with activity splitting and recombinating. Expert Syst. Appl. 2021;173.

[44] de Lamare Bastian-Pinto C. Modeling generic mean reversion processes with a symmetrical binomial lattice-applications to real options. Procedia Comput Sci 2015;55:764-73.

[45] Felix Bastian Joachim, Weber Christoph. Gas storage valuation applying numerically constructed recombinating trees. Eur J Oper Res 2012;216(1):178-87.

[46] Dai TS. Efficient option pricing on stocks paying discrete or path-dependent dividends with the stair tree. Quant Finance 2009;9(7):827-38.

[47] OPEC. (2020) https://www.opec.org/opec_web/en/publications/202.htm.