2.1 History-dependence

In economics, the importance of integrating the fact that history matters, in theoretical models and in the gathering and interpretation of data in economic phenomena, is widely accepted. History-dependence has long been supported in many various approaches. Models where a representative consumer takes her past consumption as a reference point, habit-formation models, path-dependent processes, financial processes, sport or health economics, and law economics, all involve history-dependencies. These approaches emphasize that past states, actions, decisions, and contexts (environmental, legal, social, institutional) influence current and future economic outcomes in different ways. History matters in the evolution of technology, institutions, agents’ behavior, inherited contexts (Guinnane et al., Reference Guinnane, Sundstrom and Whatley2003). Various applications in economics relies on the notion that history matters. These include habits, path-dependence (Arrow, Reference Arrow2004; David, Reference David1997, Reference David, Garrouste and Ioannides2001; Page, Reference Page2006; Beyer, Reference Beyer2010; Gräbner-Radkowitsch and Kapeller, Reference Gräbner-Radkowitsch and Kapeller2024; Hämäläinen et al., Reference Hämäläinen and Lahtinen2016), finance (Malmendier and Wachter, Reference Malmendier and Wachter2021), contexts and frames (Wachter and Kahana, Reference Wachter and Kahana2024; Wendner, Reference Wendner2005; Goetzmann et al., Reference Goetzmann, Watanabe and Watanabe2022), management and organizations (Marquis and Qiao, Reference Marquis and Qiao2025; Tang and Wang, Reference Tang and Wang2025), including technological changes coming from digital transformation (Bohnsack et al., Reference Bohnsack, Kurtz and Hanelt2021), etc. However, there is no consensus on what is meant by history matters in economic models (even in the “path-dependent” economic literature).

“History matters” means a variety of several various things both for what “history” means and also on how it “matters.” In this paper, “history matters” means that at each period, the (instantaneous) utility as well as the (current) constraints (feasible sets, budget constraints, production function) faced by the agents for her decision are influenced by the whole past history of endogenous variables and of exogenous variables.Footnote ⁶ Let us consider for example a representative consumer deriving utility from tea consumption. In the basic usual model, the instantaneous utility function $u$ depends on the level $c_t$ of tea consumption at date $t$ , that is $u(c_t)$ . But if the consumer is used to drinking tea every morning, then phenomena such as addiction, or satiation, or any possible intermediaries, variations or related effects may occur. Therefore, the instantaneous utility depends not only on her current consumption level but also on a “habit” coming from the whole past history of consumption. Thus, the utility $u$ at date $t$ depends on the current level $c_t$ of tea consumption but also on the history $\tilde {h}_t\,:\!=\,(c_{t-1},c_{t-2},\ldots \ldots )$ of consumptions preceding date $t$ ,

\begin{equation*} u(c_t,\tilde {h}_t) \end{equation*}

Such a habit model is a case of consumption history-dependent model. Consumption habit phenomena are diverse in the literature (Bambi and Gozzi, Reference Bambi and Gozzi2020; Rozen, Reference Rozen2010; Rustichini and Siconolfi, Reference Rustichini and Siconolfi2005, Reference Rustichini and Siconolfi2014, Ryder and Heal, Reference Ryder and Heal1973; Wendner, Reference Wendner2003). A unified framework for such consumption history-dependence that is not restricted to linear casesFootnote ⁷ and where utility is not restricted to additive nor multiplicative specifications was provided in Morhaim and Ulus (Reference Morhaim and Ulus2023).

Nevertheless, history-dependent phenomena in economics are not all based on consumption history. The aim of this paper is to handle history-dependencies in a broader sense to address other history-dependencies (such as based on investment, labor supply, research, networks) and which may involve contexts such as legal norms, weather conditions, pollution. Let us proceed with our previous tea consumer example and take into consideration the fact that utility coming from tea consumption also depends on the weather. Indeed, a consumer may need to consume a higher level of tea to derive the same utility when the weather gets cold.Footnote ⁸ Therefore, the instantaneous utility $u$ depends on the level $c_t$ of tea consumption at date $t$ , but also on the weather $w_t$ , that is $u(c_t,w_t)$ . Suppose there are only two weather conditions (hot $W_H$ or cold $W_C$ ); to derive the same utility level, one cup of tea is needed when the weather is hot whereas three cups are needed when the weather is cold, i.e., $u(1,W_H)=u(3,W_C)$ . However, to derive the same utility level today whether it is hot or cold, the number of tea cups needed today depends on the fact that it has been hot until today or it got cold for already a month. If the weather has been cold for the entire previous month, the consumer is now accustomed to the cold, and fewer cups may suffice. For clarity, when it is cold today after being cold for a month, two cups may be enough to derive a given utility level, whereas if it becomes cold today after being hot for the entire prior month, four cups may be required. That is, $u\big ( (2,W_C), (W_C,W_C,\ldots .,W_C)\big )=u\big ((4,W_C), (W_H,W_H,\ldots .,W_H)\big )$ . Thus, the instantaneous utility depends on the weather $w_t$ at date $t$ , but also on the entire history $\tilde {w}_t\,:\!=\,(w_{t-1},w_{t-2},\ldots \ldots )$ of the weather prior to date $t$ . That is,

\begin{equation*} u(c_t,w_t, \tilde {w}_t). \end{equation*}

Hence, preferences depend on the history of the context, which we do not decide (it is not a consumption choice that we make). The utility $u$ can also depend on combined effects from consumption history and weather history

\begin{equation*} u\big ((c_t,w_t), (\tilde {h}_t, \tilde {w}_t)\big ) \end{equation*}

Moreover, context and decision variables can be mutually related. Let us consider that the context variable $w_t$ is the humidity level. The humidity level $w_t$ at date $t$ is affected by the natural climate state ( $n_t$ ) and by the persistent component ( $\gamma w_t$ ). The technology process using capital (input) releases hot steam, raising humidity through capital use ( $\theta k_t$ ). Thus, $w_t=n_t+\gamma w_{t-1}+\theta k_t$ . The context and the input decision variables are therefore interrelated.

Furthermore, economic models usually deal with history-dependencies formed directly from the history of past decisions, or indirectly from past decision variables. Consider a representative agent deciding investment in tea so that the capital stock at date $t$ is $k_t$ . Then the decision variables are $k_t$ , while history-dependencies is formed through consumption $c_t$ , i.e. a function of $k_t$ and $k_{t+1}$ , indirectly from past decision variables $k_t$ .

These examples illustrate how utility depends on instantaneous decision variable (e.g. $c_t$ , $k_t$ ), direct or indirect history of decision variable ( $\tilde {h}_t$ ), instantaneous context ( $w_t$ ) and history of the context ( $\tilde {w}_t$ ). In other words, history can matter either through the history of past decisions themselves, or of some of the past decisions, or through past decisions and past contexts, or only contexts. These history-dependencies extend beyond consumption history-dependence and are not captured by the framework proposed by Morhaim and Ulus (Reference Morhaim and Ulus2023). While concepts such as habit formation, path-dependence, procedure, external environment, are widely used, and encompass a variety of mechanisms and interpretations, yet there is no consensus on how past experiences are generated and transmitted over time. In this paper, we address this variety of concepts. This is achieved by introducing a “memory” function which can be different from one to another and explains how past states are formed, transformed, accumulated, and transmitted. That is, the “memory function” allows us to model the way history is formed.

What historically “matters” depends on the past decisions but is not limited to the past decisions themselves, it can be constituted of a function of these past decisions. For example, if we think in terms of instantaneous capital decisions $k_t$ and habits formed from consumptions, what is remembered is a function $f(k_t)-k_{t+1}$ of the decisions, so that we do not retain the entire sequence of past decisions but rather the entire sequence of functions of these decisions. In a model with an extracted resource in which at each date the decision consists in deciding the couple the level of stock capital and level of extraction, but in which the history-dependence comes only from extraction, the history is formed through a function of the decisions.

In financial decision models, memory plays an important role. Past crisis events can affect current decisions. Another important financial example is when choices are not based on what really happened, but based on how they are remembered (Malmendier and Wachter, Reference Malmendier and Wachter2021; Zhengyang et al., 2024). Some approaches model memory through an additional “context layer” or representation that differs from the features capturing direct experience (Wachter and Kahana, Reference Wachter and Kahana2024).

Memory is also central in Law and Economics. Legal rules, institutions, and jurisprudential concepts evolve not only in response to present conditions but through the cumulative influence of past norms and arrangements, making an intertemporal perspective essential. Contract law, for instance, develops through court decisions, as earlier cases shape how judges interpret and apply contracts, thereby influencing current expectations. Similarly, in environmental and technological contexts, including artificial intelligence, liability standards evolve through judicial and regulatory practice, with earlier decisions shaping responsibility for harms such as pollution or data misuse. In criminal law, standards of responsibility and punishment likewise develop through legislation and judicial decisions. Thus, present outcomes reflect a legal memory derived from accumulated precedents (Lacey, Reference Lacey2015; Morhaim and Ulus, Reference Morhaim and Ulus2025 and Wheatley, Reference Wheatley2021).

In sports economics, players’ investissements in themselves over time and past experiences influence their decisions on the field. Players remember patterns from past games and use them to make quick decisions. Their decisions are based on memory information about opponents, teammates, typical game situations, and changing environments, which help them act faster and more accurately (Andreff, Reference Andreff2006). Habits are sensitive to contexts, which is also a crucial issue in health economics. It is important to identify contextual, social, and personal factors that promote (benefits) and hinder (barriers) hygiene habits in care facilities (Hochli et al., Reference Höchli, Dorn, Holenweger, Messner, Schuller and Rohrbach2023; Wellsjo, Reference Wellsjo2021). For instance, if a nurse is used to washing hands in a particular setting $A$ and encounters another setting $B$ , she may not wash hands as consistently as she would have while being used to setting $B$ , the same nurse would have washed hands as usual. Past contexts enter the memory function, beyond the past decision variables.

Our memory function may operate directly on the variables of the model or explicitly determines which elements of past experience influence current decisions. These elements may consist of internal or external variables, functions of these variables, or representations derived from them. This flexibility allows taking into account various historical influence while remaining fully compatible with the economic structure of many specific usual models.

At each date $t+1$ , history is formed by augmenting the history at date $t$ with the elements that matter for the model, which we refer to as memory. Depending on the model, this may correspond to past consumption, extraction levels, or other relevant decision or contextual variables. What constitutes history is thus model-specific and is captured by our memory function ( $m$ ), defined as a function of the relevant variables. The idea of such a memory function is not new and comes from the existing literature: existing models rely on similar mechanisms, but lack a unified and general framework. Our contribution in this paper is to provide a framework, which unifies a wide range of existing models while remaining simple to verify and applicable to many, though not all, economic problems. In particular, our approach encompasses the large literature on consumption habits as well as habit formation beyond consumption (e.g., environmental or financial contexts), and more generally, a broad class of memory effects that cannot be captured by traditional habit-based models.

History at date $t$ is a sequence that consists of the whole previous history (i.e. the sequence until $t-1$ ) completed with what just happened at the previous date $t-1$ . For example, in the consumption/saving optimal growth model, when decisions are considered as being the stock of capital, and when consumption habits are formed, what matters is the history of past consumptions, i.e. what completes the previous history to form the new one is the last consumption. Thus, history formation can come directly from past decisions or can come through a transformation process of past decisions. This leads to introduce a function to describe the transformation process from past decisions. We call this transformation the memory function $m$ . The memory function describes how the history is formed from the previous dates’ variables. In this example, at each date $t$ , the decision is the instantaneous capital $k_t$ . The history is formed from consumption history at each date $t$ , so history matters through the whole past sequence of consumption before $t$ , $c_t=f(k_t)-k_{t+1}$ (denoting $f$ the production function). History then consist not of directly the decisions $k_t$ but through a transformation of these decisions through calculating $c_t$ i.e. the function $f(k_t)-k_{t+1}$ . We have then $m(k_t,k_{t+1})=f(k_t)-k_{t+1}$ , which describes the way the history is formed from the decisions. It also encompasses cases in which history formation comes directly from the past decisions without being transformed. For example, in a model with an extracted resource in which at each date the decision consists in deciding the couple of the levels of stock capital $k_t$ and of the extraction rate $r_t$ . Suppose that the history-dependence comes only from the extraction decisions, the history is formed directly from $r_t$ . For such cases of direct dependence, one defines the memory function as $m((k_t,r_t),(k_{t+1},r_{t+1}))=r_t$ or as the identity or projection function. Suppose now that the resource that is extracted is regenerated at a rate $\theta$ with regeneration depending on the humidity level $w_t$ . The memory then depends on the context $w_t$ with $m((k_t,r_t,w_t),(k_{t+1},r_{t+1}, w_{t+1}))=r_t-\theta w_t$ . Thus, in this paper, history is formed through a memory process which is not restricted to only past decisions but which builds also upon contexts and frames.

2.2 Dynamic programming

Intertemporal economics and dynamics lead to sequential optimization problems to be solved. The theory of dynamic programming was created by Richard Bellman in the 1950s to treat the mathematical problems arising from the study of various multi-stage decision processes (Bellman, Reference Bellman1954). Dynamic programming has many concrete applications among which the earliest included management of hydroelectric reservoirs (Massé, Reference Massé1944), optimal inventory policy (Arrow et al., Reference Arrow, Harris and Marschak1951) and statistical decision rules (Wald, Reference Wald1947). Dynamic programming emerged as a fundamental tool of applied mathematics and revolutionized the way we do economics is done, being one of the most fundamental tool underlying game theory, macroeconomics, and microeconomics (see, e.g., Stokey et al., Reference Stokey, Lucas and Prescott1989; Amir et al., Reference Amir, Mirman and Perkins1991; Amir, Reference Amir1996, Adda and Cooper, Reference Adda and Cooper2003; Le Van and Dana, Reference Le Van and Dana2003; Sargent, Reference Sargent2009), so that much of the modern economics literature can be regarded as a type of applied dynamic programming (Rust, Reference Rust2019). Dynamic programming has since developed and now refers to a set of powerful methods and tools for solving a wide range of sequential decision-making.

The idea of dynamic programming relies in the notion of equivalence of optimization problems. Two problems are said to be equivalent if they have exactly the same solutions. When we face the task of finding the solutions to an optimization problem which is difficult to solve, if we have a method that provides us an equivalent problem that is easier to solve is very useful. Such a method allows solving the equivalent problem which is easier to solve. As the method guarantees the equivalence between the two problems, we are ensured that solving the second problem gives us the solutions of the initial problem which is the one that interests us.

Dynamic programming allows replacing an infinite-time variables problem by a fixed-point problem with finite-time variables. Unfortunately, this replacement is possible only for certain problems and under certain assumptions. Replacing a problem with an equivalent (i.e. having the same solutions) problem that is easier to solve is an interesting method, however, it is necessary to ensure that the new problem will yield the same solutions as the original problem. It is standard in macroeconomics literature to consider the Bellman equation as an equivalent representation of initial problem, namely the optimization of net present value (Le Van and Dana, Reference Le Van and Dana2003).Footnote ⁹ Nevertheless, such an equivalence does not hold always, in particular, it may not hold in models with history-dependence. Moreover, securing a unique solution to the Bellman equation is a crucial and on-going issue for macroeconomists and this can be guaranteed by adequate assumptions on the problems.

Shortly, there are conditions that need to be satisfied for a model to be fitted into a Bellman equation format. Bellman has first showed that solving certain intertemporal problems is equivalent to solving a fixed-point equation known as Bellman’s equation, and research has since extended the models that can be solved in this way. However, not all intertemporal problems can be solved this way. History-dependence introduces the fact that at each date $t$ , the instantaneous utility function depends on the whole the actual past events themselves, inducing problems in the possible applicability of dynamic programming tools. Here we go a step further with the possibility of history-dependence, which reintroduces infinity into the (equivalent) finite model and therefore presents particular difficulties in showing that we can use these tools to solve the initial intertemporal problem. Nevertheless, we manage to propose very simple and common assumptions that allow us to do so.

Any use of dynamic programming for intertemporal economic models thus leads to a threesome issue to be solved: providing a model that is adequate both on one hand from the economic characteristics and on the other hand from the features and assumptions that allow dynamic programming tools, i.e. conciliate economic realities and mathematical constraints, and then to develop and provide such tools. Here, there is a fourth issue to be solved when introducing history-dependent phenomena. In addition to the traditional threesome issue, the fourth issue that arises is that infinity gets back in the very instantaneous problems.

3.1 The history-dependent instantaneous function

We describe a general framework of an infinite horizon intertemporal history-dependent decision problem of a social planner which may be a problem of optimal growth related to consumption, saving, conservation, accumulation, pollution or sustainable issues, depending on the objectives and the constraints of the model. Let us consider an intertemporal decision process. We assume that at each date $t$ the decision $x_t$ liesFootnote ¹⁰ in $\operatorname {I\!R}^n (n \in \operatorname {I\!N})$ . Such decisions may involve consumption decisions, investment decisions or any other economic related decisions. One example is the capital stock/saving decisions. As an illustration, let us consider the usual saving/consumption optimal growth model. In the optimal growth models, usually the instantaneous objective function $F$ at date $t$ depends on the two stock of capital variables $k_t$ and $k_{t+1}$ and is written as follows

\begin{equation*} F(k_t,k_{t+1}) \end{equation*}

Here, the instantaneous objective function $F$ at date $t$ depends not only on the two stock of capital variables $k_t$ and $k_{t+1}$ but also on the past which is formalized through a history sequence $\tilde {h}_t$ , and is written as follows

\begin{equation*} F(\tilde {h}_t, k_t,k_{t+1}) \end{equation*}

More generally, we do not restrict the model to capital stock decision variables, and we consider more general variables $x \in \operatorname {I\!R}^n$ so that it is written

\begin{equation*} F(\tilde {h}_t, x_t,x_{t+1}) \end{equation*}

In the seminal habit formation literature, history consists in past consumptions sequence, i.e. it is tha particular case of consumption history-dependent model. However, the habit literature itself is enriched with other history-dependencies as technology habits, legal standards, environmental dependencies.

Let us denote $\mathcal{H}$ the set of all history sequencesFootnote ¹¹ with elements in $\operatorname {I\!R}^p$ . That is, let us consider an agent facing a time $t$ objective function $F$ which is defined on a subset $\mathcal{D}_F$ of $\mathcal{H} \times \operatorname {I\!R}^n \times \operatorname {I\!R}^n$

\begin{equation*} F\,:\, \mathcal{D}_F \subseteq (\mathcal{H} \times \operatorname {I\!R}^n \times \operatorname {I\!R}^n) \to \operatorname {I\!R} \end{equation*}

This framework allows studying issues and models from various literature strands. As we introduced a general function allowing to model many different memory processes, general decision variables, objective functions, and feasible sets, we are able to encompass many existing models. It generalizes the history-dependent intertemporal optimization models provided in Morhaim and Ulus (Reference Morhaim and Ulus2023). Thus, the applications presented in Morhaim and Ulus (Reference Morhaim and Ulus2023) are encompassed. These include seminal models dealing with habit formation (Rozen, Reference Rozen2010; Rustichini and Siconolfi, Reference Rustichini and Siconolfi2014; Ryder and Heal, Reference Ryder and Heal1973; Carroll et al., Reference Carroll, Overland and Weil1997, Reference Carroll, Overland and Weil2000) and satiation (He et al., Reference He, Dyer and Butler2013; Baucells and Sarin, Reference Baucells and Sarin2010). Moreover, it encompasses environmental models (Ikefuji, Reference Ikefuji2008; Löfgren, Reference Löfgren2003 and Safi and Ben Hassen Reference Safi and Hassen2021) and optimal management of natural resources (Smulders et al., Reference Smulders, Toman and Withagen2014; Ulus, Reference Ulus, Machado, Özdemir and Baleanu2020), as well as recent circular economy models (George et al., Reference George, Lin and Chen2015; Kasioumi and Stengos Reference Kasioumi and Stengos2023; Kasioumi, Reference Kasioumi2021). These existing models are particular cases of our general framework as well as how its flexibility allows using it in future research. Our general framework indeed allows studying simultaneously many effects and contexts: circular models without production waste, linear economies with production waste, and furthermore circular economies with production waste, as well as other many effects.

3.2 The memory function

We explicitely model the way history is formed using a memory function. The memory function allows treating habit-formation models (consumption history-dependance) but also many other history-dependence cases as price memory, norms, technology, legal standards, contracts. More precsiely, we define a memory function depending at each time of the last decision $x_{t+1}\in \operatorname {I\!R}^n$ given the previous decision $x_{t}$ . By this memory function $m$ , we will obtain the history-dependence/sequence. The memory process can induce a vector so that $m(x_{t},x_{t+1})$ is an element of $\operatorname {I\!R}^p (p \in \operatorname {I\!N})$ . Through these functions, our formalism allows studying many memory process formations and history-dependencies.

We define the memory function by $m\,: \operatorname {I\!R}^n \times \operatorname {I\!R}^n \to \operatorname {I\!R}^p$ such that for time $t$ , $m(x_{t}, x_{t+1})$ is the memory for the decision $x_{t+1}\in \operatorname {I\!R}^n$ given $x_{t}$ . By this memory function $m$ , we will obtain the history of the memories over time which is modeled in the following way. At time $t=0$ , she has a time- $0$ history denoted by $\tilde {h}^{(0)}\,:\!=\,(h_0^{(0)},h_1^{(0)},\ldots .)$ lying in $\mathcal{H}$ the set of all history sequences. Having on memory $m(x_0, x_1)$ at time 0, she has then time- $1$ history equal to $\tilde {h}^{(1)}\,:\!=\,(m(x_0, x_1), \tilde {h}^{(0)})$ at time $t=1$ . That is, by induction, for any time $t\geq 1$ , time- $t$ history will be

\begin{equation*} \forall t\geq 1, \tilde {h}^{(t)}\,\,:\!=\,(m(x_{t-1}, x_{t}), m(x_{t-2}, x_{t-1}), \ldots .,m(x_1, x_2), m(x_0, x_1), \tilde {h}^{(0)}) \end{equation*}

The instantaneous utility function $F$ depends on the instantaneous decisions and on history sequence, similarly does the feasible set. The feasible correspondence $\Gamma \,:\, \mathcal{H} \times \operatorname {I\!R}^n \to \operatorname {I\!R}^n$ is defined such that for any $ (\tilde {h}, x)\in \mathcal{H} \times \operatorname {I\!R}^n$

\begin{equation*}\Gamma (\tilde {h}, x)\subseteq \{ x'\in \operatorname {I\!R}^n, (\tilde {h}, x, x')\in \mathcal{D}_F \}\end{equation*}

3.3 The problem

We now present the general intertemporal optimization framework. Let us denote $\tilde {x}$ the intertemporal decision sequence and $\mathcal{X}$ the set of all decision sequences,Footnote ¹² i.e.

\begin{equation*} \tilde {x}=(x_t)_{t=0}^{+\infty } \in \mathcal{X} \end{equation*}

Assuming a fixed discount factor $\beta \in (0,1)$ , for initially given state $x_0 \in \operatorname {I\!R}^n$ and time- $0$ history $\tilde {h}^{(0)} \in \mathcal{H}$ ; with feasible state and control variables at time $t$ , that is, satisfying the process $x_{t+1} \in \Gamma (\tilde {h}^{(t)}, x_t)$ , for every $t\geq 0$ , the general framework optimization problem can then be written as follows:

\begin{equation*} \left \{ \begin{array}{c@{\quad}l} \operatorname { Max } & \sum \limits _{t=0}^{+\infty } \beta ^t F\big (\tilde {h}^{(t)},x_t, x_{t+1}\big )\\ \operatorname { s.t. } & \forall t\geq 0, x_{t+1} \in \Gamma (\tilde {h}^{(t)}, x_t)\\ & \forall t \geq 1, \tilde {h}^{(t)}=(m(x_{t-1},x_{t}),\ldots ..,m(x_1,x_2),m(x_0,x_1),\tilde {h}^{(0)})\\ & x_0 \in \operatorname {I\!R}^n \operatorname { and }\ \tilde {h}^{(0)} \in \mathcal{H} \operatorname { are \ given } \end{array} \right . \end{equation*}

Although this problem is usual and useful, its lack of explicit recursivity induces implementable issues. In order to develop tools that are algorithmically computable, we will thus also consider an explicitely recursive version of the problem as follows. Let us consider $F \, : \operatorname {I\!R}^N \times \operatorname {I\!R}^n \times \operatorname {I\!R}^n \to \operatorname {I\!R}$ ( $N \in \operatorname {I\!N}$ ) the instantaneous reward function, $m \, : \operatorname {I\!R}^n \times \operatorname {I\!R}^n \to \operatorname {I\!R}^p$ the memory function and the (fixed) discount factor $\beta \in (0,1)$ . Let us consider an adjustement level function $\varphi \, : \, \mathcal{H} \to \operatorname {I\!R}^N$ , with $N \in \operatorname {I\!N}$ . Let $x_0 \in \operatorname {I\!R}^n$ and $\tilde {h}^{(0)} \in \mathcal{H}$ be given, and consider the problem

\begin{equation*} \left \{ \begin{array}{c@{\quad}l} \operatorname { Maximize } & \sum \limits _{t=0}^{+\infty } \beta ^t F\big (\varphi (\tilde {h}^{(t)}), x_t, x_{t+1}\big )\\ \operatorname { s.t. } & \forall t\geq 0, x_{t+1} \in \Gamma (\tilde {h}^{(t)},x_t)\\ & \forall t \geq 1, \tilde {h}^{(t)}=(m(x_{t-1},x_{t}),\ldots ..,m(x_1,x_2),m(x_0,x_1),\tilde {h}^{(0)})\\ \end{array} \right . \end{equation*}

Recall that for any given initial data $x_0 \in \operatorname {I\!R}^n$ , and initial time-0 history $\tilde {h}^{(0)} \in \mathcal{H}$ , the feasible set $\Pi (\tilde {h}^{(0)},x_0)$ is defined by the set of sequences feasible from $x_0$ and $\tilde {h}^{(0)}$ , i.e. for any $x_0 \in \operatorname {I\!R}^n$ , for any $\tilde {h}^{(0)} \in \mathcal{H}$ ,

\begin{equation*} \Pi (\tilde {h}^{(0)},x_0)=\{\tilde {x}=(x_t)_{t=1}^{+\infty }\in \mathcal{X}, \forall t \geq 0, x_{t+1} \in \Gamma ( \tilde {h}^{(t)},x_t), \tilde {h}^{(t+1)}=(m(x_{t},x_{t+1}), \tilde {h}^{(t)}) \} \end{equation*}

Let us give a set $(\mathcal{A}')$ of assumptions: (F) on the feasible set, (A’) on the instantaneous function, and (M) on the memory function. The assumptions (F) and (M) are classical (see the details in Section 4.2).

(A’)

1. (A1’) $\forall x_0 \in \operatorname {I\!R}^n$ and $\tilde {h}^{(0)} \neq 0, \exists \tilde {x} \in \Pi (\tilde {h}^{(0)},x_0)$ such that $\sum \limits _{t=0}^{+\infty } \beta ^t F\big (\varphi (\tilde {h}^{(t)}), x_t, x_{t+1}\big ))\gt -\infty$ .

2. (A2’) The functions $F$ and $\varphi$ are continuous and there exist $a \in \operatorname {I\!R}^+$ with $a\beta \lt 1$ , $a_2 \in \operatorname {I\!R}^+$ and a continuous function $a_1 \,: \operatorname {I\!R}^n \rightarrow \operatorname {I\!R}$ such that for any $x_0 \in \operatorname {I\!R}^n$ and $\tilde {h}^{(0)} \in \mathcal{H}$ , for any feasible sequence $\tilde {x}=(x_t)_{t=1}^{+\infty }\in \Pi (\tilde {h}^{(0)},x_0)$ and its associated history $\tilde {h}^{(t)}$ , for any $t\geq 0$ ,

   \begin{equation*} F^+(\varphi (\tilde {h}^{(t)}), x_t, x_{t+1}) \leq a_1(x_0)a^t+a_2 \end{equation*}

An easily implementable case is when the function $\varphi : \mathcal{H} \to \operatorname {I\!R}^N$ , with $N \in \operatorname {I\!N}$ can be defined recursively through a function $G \, : \operatorname {I\!R}^p \times \operatorname {I\!R}^N \to \operatorname {I\!R}^N$ , i.e. for any $\upsilon \in \operatorname {I\!R}^p$ and any $\tilde {h} \in \mathcal{H}$ , as

\begin{equation*} \varphi (\upsilon ,\tilde {h})=G\big (\upsilon ,\varphi (\tilde {h}) \big ) \end{equation*}

and when the correspondence $\Gamma$ can be defined (and easily calculated) by $\varphi$

\begin{equation*} \Gamma (\tilde {h},x)=g(\varphi (\tilde {h}),x) \end{equation*}

with $g \, : \operatorname {I\!R}^N \times \operatorname {I\!R}^n \to \operatorname {I\!R}^n$ is given.

The problem is then written

\begin{equation*} \left \{ \begin{array}{c@{\quad}l} \operatorname { Maximize } & \sum \limits _{t=0}^{+\infty } \beta ^t F\big (\varphi (\tilde {h}^{(t)}), x_t, x_{t+1}\big )\\ \operatorname { s.t. } & \forall t\geq 0, x_{t+1} \in \Gamma (\tilde {h}^{(t)},x_t)\\ & \forall t \geq 1, \tilde {h}^{(t)}=(m(x_{t-1},x_{t}),\ldots ..,m(x_1,x_2),m(x_0,x_1),\tilde {h}^{(0)})\\ & \forall t \geq 1, \varphi (\tilde {h}^{(t)})=G\big (m(x_{t-1},x_{t}),\varphi (\tilde {h}^{(t-1)}) \big ) \\ & x_0 \in X \operatorname { and } \tilde {h}^{(0)} \in \mathcal{H} \operatorname { are \ given } \end{array} \right . \end{equation*}

In Morhaim and Ulus (Reference Morhaim and Ulus2023), a representative agent consumes a single good on periods $t=0,1,2,\ldots .$ and maximizes her intertemporal utility over the consumption stream $\tilde c=(c_0,c_1,\ldots .)$ . At date $t$ , the consumer’s instantaneous utility depends on her current consumption $c_t$ , and it also depends on her time- $t$ (consumption) history $\tilde {h}^{(t)}$ which is defined from an initial history $\tilde {h}^{(0)}$ and keeps in memory the consumption decisions. For initially given capital stock $k_0\gt 0$ and time- $0$ history $\tilde {h}^{(0)}$ , the general framework and optimization problem, with $f \, : \operatorname {I\!R}^+ \to \operatorname {I\!R}^+$ the production function, $k_t$ the capital stock at time $t$ , and $\beta \in (0,1)$ the fixed discount factor , is given as follows:

\begin{equation*} \left \{ \begin{array}{c@{\quad}l} \operatorname {Maximize} & \sum \limits _{t=0}^{+\infty } \beta ^t u(c_t, \varphi (\tilde {h}^{(t)})) \\ \operatorname {s.t.} & \forall t \geq 0, k_{t+1}=f(k_t)-c_t, k_t \geq 0 \operatorname { and } c_t \geq 0 \\ & \forall t \geq 1, \tilde {h}^{(t)}=(c_{t-1},\ldots .,c_1,c_0, \tilde {h}^{(0)}) \\ \end{array} \right . \end{equation*}

The problem can be rewritten as follows:

\begin{equation*} \left \{ \begin{array}{c@{\quad}l} \operatorname {Maximize} & \sum \limits _{t=0}^{+\infty } \beta ^t u( f(k_{t})-k_{t+1}, \varphi (\tilde {h}^{(t)})) \\ \operatorname {s.t.} & \forall t \geq 0, k_{t+1} \in [0,f(k_t)]\\ & \forall t \geq 1, \tilde {h}^{(t)}=(f(k_{t-1})-k_{t},\ldots ..,f(k_1)-k_2,f(k_0)-k_1, \tilde {h}^{(0)}) \\ \end{array} \right . \end{equation*}

This is a case in which consumption is kept in memory, which is a function of previous date and current capital stock decisions. Thus, it means keeping in memory a function of these decisions. Our general history-dependent framework allows us, not only to keep in memory the particular function defining consumption, but any function of the previous date and current decisions. By this way, the model provided in Morhaim and Ulus (Reference Morhaim and Ulus2023) as well as the models presented in Morhaim and Ulus (Reference Morhaim and Ulus2023)Footnote ¹³ become particular cases of our framework. Indeed, let $x_t=k_t\in X$ for all $t=0, \ldots , +\infty$ , and $F$ by for any $\tilde {h}$ and $x, x'$ ,

\begin{equation*} F \big (\tilde {h}, x, x' \big ) \,:\!=\, u(f(x)-x',\varphi (\tilde {h})) \end{equation*}

We define for any decision $x'$ given $x$ , the memory function/process $m$ by

\begin{equation*} m(x,x')=f(x)-x' \end{equation*}

The introduction of the memory function $m$ to the modeling allows us to explicitly study many memory processes. Here, it is done by defining $m$ which associates the consumption to the capital stock decisions. History-dependence appears in various strands in the literature. Environmental economics’ models involve memory functions. For instance, Löfgren (Reference Löfgren2003)’s model and Ikefuji (Reference Ikefuji2008)’s model can be written through our framework. Löfgren (Reference Löfgren2003) proposes a model with environmental quality habit formation and in which a consumption good moreover causes a negative external effect on the environment. The social planner maximizes the utility given the negative effect of the consumption good on the environment and taking into account that there is habit formation in environmental quality. The instantaneous utility $u(n_t,x_t,z_t,s_t)$ depends on $n_t$ which is the environment that displays habit formation, $x_t$ the “dirty” consumption good (the environmental bad), $z_t$ the “clean” consumption good and $s_t$ the habit level related to the environment. The following relations are satisfied with $\gamma \in (0,1), \beta \in (0,1), \delta \in (0,1)$ , $y$ being an exogenously given income and $n$ is a given initial environment

\begin{equation*} \left \{ \begin{array}{l} n_t=n-\gamma x_t \\ z_t=y- x_t \\ s_{t+1}=\beta n_t+(1-\delta )s_t \end{array} \right . \end{equation*}

Let $x_t=k_t$ for all $t=0\ldots \infty$ , and $\varphi$ is defined through the following recurrence relation, for all $c$ and $\tilde h$ , $ \varphi ((c,\tilde h))=\beta n+(1-\delta )\varphi (\tilde h)-\beta \gamma c$ . Let us define $F$ by, for any $\tilde {h}$ and $x, x'$ ,

\begin{equation*} F \big (\tilde {h}, x, x' \big ) \,:\!=\,u(n-\gamma (f(x)-x'),f(x)-x',y-(f(x)-x'),\varphi (\tilde {h})) \end{equation*}

In this model, the memory function $m$ is defined by $m(x,x')=f(x)-x'$ . Thus, Löfgren (Reference Löfgren2003)’s model is encompassed in our general framework. Ikefuji (Reference Ikefuji2008) studies habit formation in consumption and pollution abatement activities when agents derive disutility both from the habit stock and pollution. The pollution $P_t$ in period $t$ is generated by the capital stock $k_t$ used in production and reduced by abatement activities $a_t$ in the same period. The problem is written as follows.

\begin{equation*} \left \{ \begin{array}{c@{\quad}l} \operatorname { Max } & \sum \limits _{t=0}^{+\infty } \beta ^t u(c_t,H_t,P_t)\\ \operatorname { s.t. } & \forall t \geq 1, H_{t+1}=\rho c_t+(1-\rho )H_t \operatorname { and } P_t=\bigg (\dfrac {k_t}{a_t}\bigg )^{\phi }\\ & \forall t \geq 0, k_{t+1}=Ak_t-c_t-a_t-k_t \\ & k_0\gt 0, m_0\gt 0 \operatorname { and } h_0 \operatorname { are }\ \operatorname { given } \end{array} \right . \end{equation*}

where $c_t$ denotes the consumption in period $t$ , $H_t$ denotes the consumption habit, and $P_t$ is the level of aggregated pollution in the economy. This problem is covered by our framework, let $x_t=(k_t,a_t)$ for all $t=0\ldots \infty$ and the function $\varphi$ be defined by $\varphi (\tilde {h}^{(0)})=h_0$ and for all $\tilde {h}$ and $c$ , $\varphi (c, \tilde {h})=G (c, \varphi (\tilde {h}))$ , with the function $G$ defined by $G(c,y)=\rho c+(1-\rho )y$ . Also, the function $F$ is $F \big (\tilde {h}, x, x' \big ) \,:\!=\, u\big (f(\pi _1(x))-\pi _1(x'),\varphi (\tilde {h}), \big (\frac {\pi _1(x)}{\pi _2(x)}\big )^{\phi }\big )$ using the $i$ -th projections. For any decision $x'$ given $x$ , the memory function is defined by $m(x,x')=f(\pi _1(x))-\pi _1(x')$ . These two examples illustrate how our framework is general and easy to use, while keeping the environmental effects to be studied visible and interpretable. Our framework is also fitted to optimal management of natural resources. Smulders et al. (Reference Smulders, Toman and Withagen2014) and Ulus (Reference Ulus, Machado, Özdemir and Baleanu2020) present models referring to the optimal management of exhaustible natural resources like oil, coal, gas. In these models, there exists a single planner who manages the decision of extraction and consumption of an exhaustible natural resource at each period $t$ in order to maximize the intertemporal utility. At each period $t$ , the instantaneous utility depends on its consumption $c_t$ and also on the stock of the natural resource, which is denoted by $s_t$ . The output $y_t\in \operatorname {I\!R}_{+}$ is produced from capital ( $k_t$ ) and extracted quantity of resource ( $r_t$ ) (or the extraction flow of natural resource per unit of time) by a production function $f \, : \operatorname {I\!R}_{+}\times \operatorname {I\!R}_{+} \to \operatorname {I\!R}_{+}$ where $y_t = f(k_t, r_t)$ . The output is either consumed as $c_t\geq 0$ or saved as capital to the next period as $k_{t+1}$ satisfying $c_t + k_{t+1}\leq f(k_t, r_t) + (1-\delta )k_t \hbox{ with } k_t\geq 0$ , where $\delta$ stands for the depreciation rate of capital. For initially given stocks of $k_0, s_0, r_0\gt 0$ and $\beta \in (0,1)$ the fixed discount factor, assuming that $u$ is increasing with respect to $c$ , the problem of the planner can be given as follows:

\begin{equation*} \left \{ \begin{array}{c@{\quad}l} \operatorname {Maximize} & \sum \limits _{t=0}^{+\infty } \beta ^t u(c_t,s_t) \\ \operatorname {s.t.} & \forall t \geq 0, s_{t+1} = s_t-r_t \\ & \forall t \geq 0, 0 \leq c_t=f(k_t,r_t)+(1-\delta ) k_t-k_{t+1} \\ & s_0,k_0, r_0\gt 0 \operatorname { are }\ \operatorname { given }\\ \end{array} \right . \end{equation*}

which can be rewritten as

\begin{equation*} \left \{ \begin{array}{c@{\quad}l} \operatorname {Maximize} & \sum \limits _{t=0}^{+\infty } \beta ^t u(f(k_t,r_t)+(1-\delta ) k_t-k_{t+1}, s_t) \\ \operatorname {s.t.} & \forall t \geq 0, s_{t+1} = s_t-r_t \\ & \forall t \geq 0, 0 \leq k_{t+1} \leq f(k_t,r_t)+(1-\delta ) k_t \\ & s_0,k_0, r_0\gt 0 \operatorname { are }\ \operatorname { given }\\ \end{array} \right . \end{equation*}

Let us define $G$ by $G(s,r)\,:\!=\,s-r$ and $\varphi$ by the law of motion $\varphi (\tilde {h}^{(t+1)})=G(r_t,\varphi (\tilde {h}^{(t)}))$ for all $t \geq 0$ , so that the instantaneous utility $u(c_t, s_t)$ is $u(c_t,\varphi (\tilde {h}^{(t)}))$ . By defining $\tilde {h}^{(0)}$ such that $\varphi (\tilde {h}^{(0)})=s_0$ and for any $t \geq 0$ , $\tilde {h}^{(t+1)}$ such that $\tilde {h}^{(t+1)}=(r_t, \tilde {h}^{(t)})$ , the model can be written

\begin{equation*} \left \{ \begin{array}{c@{\quad}l} \operatorname {Maximize} & \sum \limits _{t=0}^{+\infty } \beta ^t u( f(k_t,r_t)+(1-\delta ) k_t-k_{t+1}, \varphi (\tilde {h}^{(t)})) \\ \operatorname {s.t.} & \forall t \geq 0, k_{t+1} \in [0, f(k_t,r_t)+(1-\delta ) k_t]\\ & \forall t \geq 0, \tilde {h}^{(t+1)}=(r_{t}, \ldots .,\tilde {h}^{(0)}) \\ \end{array} \right . \end{equation*}

The model is written in our general framework by defining $x=(k,r)$ , the memory function by $m(x,x')=\pi _2(x)$ , the utility by $F(x,x', \tilde {h})=u \big ( f(\pi _1(x),\pi _2(x))+(1-\delta ) \pi _1(x)-\pi _1(x'), \varphi (\tilde {h}) \big )$ and the feasible set by $\Gamma (\tilde {h},x)=\{ x', 0 \leq \pi _1(x') \leq f(\pi _1(x),\pi _2(x))+(1-\delta ) \pi _1(x), 0 \leq \pi _2(x') \leq \varphi ( \tilde {h}) \}$ . The function $G$ is defined by $G(s,r)\,:\!=\,s-r$ . Note also that our framework allows considering more general forms of $G$ , for example if it comes to taking into account the resources regeneration (whether naturally or otherwise).

All these previous examples involved linear economies. Our framework deals further with the new circular economies models. Circular economy is most frequently depicted as a combination of reduce, reuse and recycle activities (Kirchherr et al., Reference Kirchherr, Yang, Schulze-Spüntrup, Heerink and Hartley2023), but also design, implying a focus on the entire life cycle of the processes as well as the interaction between the process and the environment and the economy in which it is embedded (Ghisellini, et al. Reference Ghisellini, Cialani and Ulgiati2016). The circular economy models provided by George et al. (Reference George, Lin and Chen2015), Kasioumi and Stengos (Reference Kasioumi and Stengos2022), Kasioumi (Reference Kasioumi2022) can be written in our framework, and this for the circular economy models without recycling habits (George et al., Reference George, Lin and Chen2015 and Kasioumi and Stengos, Reference Kasioumi and Stengos2023) as well as the circular economy models with recycling habits (Kasioumi (Reference Kasioumi2021)).

Our framework encompasses circular models without recycling habits proposed by George et al. (Reference George, Lin and Chen2015), and Kasioumi and Stengos (Reference Kasioumi and Stengos2023). George et al. (Reference George, Lin and Chen2015) and Kasioumi and Stengos (Reference Kasioumi and Stengos2023) consider a closed economy with zero population growth. They abstract from capital accumulation and technical progress.Footnote ¹⁴ The social planner maximizes an intertemporal utility where the instantaneous utility $u(c,P)$ depends on consumption $c$ and the stock of pollution $P$ . The output is produced via a (concave) production function $\phi$ using two factors of production, one which corresponds to the rate of use of the recyclable resource and another ( $z$ ) corresponding to the rate of use of the environmentally polluting resource. Output produced in any given period but not consumed or used for the employment of the polluting resource, accumulates as (potentially recyclable) waste. Recycling turns waste into a useful factor of production: a proportion $b$ of the waste stock $s$ with intensity of recycling $\tau$ is supposed to be recycled each period. George et al. (Reference George, Lin and Chen2015)’s model is a particular case of Kasioumi and Stengos (Reference Kasioumi and Stengos2023) in which the intensity of recycling $\tau$ is equal to one. The social planner solves the following optimization problem

\begin{equation*} \left \{ \begin{array}{c@{\quad}l} \operatorname {Maximize} & \sum \limits _{t=0}^{+\infty } \beta ^t u(c_t,p_t) \\ \operatorname {s.t.} & \forall t \geq 0, s_{t+1}-s_t=\phi (\tau bs_t,z_t)-c_t-\alpha z_{t+1}-\tau b s_t \\ & \forall t \geq 0, p_{t+1}-p_t=\theta z_t -\delta p_t+(1-b) s_t\\ & s_0,z_0\gt 0, p_0 \geq 0 \operatorname { are }\ \operatorname { given }\\ \end{array} \right . \end{equation*}

Our memory function $m$ allows displaying the fact that at each date $t$ , $p_{t+1}$ depends on the previous pollution stock $p_t$ and of $[\theta z_t +(1-b) s_t]$ , in other words, $p_{t+1}$ depends on the memory of $[\theta z_t +(1-b) s_t]$ , analagously as in Ryder and Heal (Reference Ryder and Heal1973) $h_{t+1}$ depends on the previous habit level $h_t$ and of $c_t=f(x_t)-x_{t+1}$ , that is $h_{t+1}$ depends on the memory of consumptions $c_t=f(x_t)-x_{t+1}$ . For the model by George et al. (Reference George, Lin and Chen2015), we define $x=(s,z)$ , and $m(x,x')=\theta z +(1-b) s=\theta \pi _2(x) +(1-b) \pi _1(x)$ , and $G(m(x,x'),p)=(1-\delta ) p + m(x,x')$ . This modeling allows us to highlight the economic processes at work: on one hand the memory process, i.e. in George et al. (Reference George, Lin and Chen2015) how the decisions $x,x'$ are transformed into additional pollution, and on the other hand how this new additional pollution combines with the previous pollution state to form the new one (through $G$ ). Kasioumi (Reference Kasioumi2021) extends this model by both incorporating habit formation on recycling and a (given) intensity $\tau$ of recycling. The circular models with recycling habits proposed by Kasioumi (Reference Kasioumi2021, Reference Kasioumi2022) are also encompassed by our framework. It is an extension of the theoretical work of Kasioumi and Stengos (Reference Kasioumi and Stengos2023), combining elements of the circular economy model of George et al. (Reference George, Lin and Chen2015) with the habit formation framework of Ikefuji (Reference Ikefuji2008). Kasioumi (Reference Kasioumi2021, Reference Kasioumi2022) deals with a fixed intensity of recycling while Kasioumi (Reference Kasioumi2022) deals further with an intensity $\tau (H_t)$ depending (through an affine function $\tau (H_t)\,:\!=\,\eta +rH_t$ ) on the level of recycling habits. These models (Kasioumi, Reference Kasioumi2021) fit our framework. Note that our framework fits also recycling intensity functions $\tau$ that are not necessarily affine.

4.1 Notations and feasible sets

Definition 4.1. For any given initial data $x_0 \in \operatorname {I\!R}^n$ , and initial time-0 history $\tilde {h}^{(0)} \in \mathcal{H}$ , the feasible set $\Pi (x_0,\tilde {h}^{(0)})$ is defined by the set of sequences feasible from $x_0$ and $\tilde {h}^{(0)}$ , i.e. for any $x_0 \in \operatorname {I\!R}^n$ , for any $\tilde {h}^{(0)} \in \mathcal{H}$ ,

\begin{equation*} \Pi (\tilde {h}^{(0)},x_0)=\{\tilde {x}=(x_t)_{t=1}^{+\infty }\in \mathcal{X}, \forall t \geq 0, x_{t+1} \in \Gamma ( \tilde {h}^{(t)},x_t), \tilde {h}^{(t+1)}=(m(x_{t},x_{t+1}), \tilde {h}^{(t)}) \} \end{equation*}

The process is analogous as usual. The feasible set is defined from $x_0$ but here also from the given (infinite) sequence $\tilde {h}^{(0)}$ . Once $x_1$ is chosen in the feasible set $\Gamma ( \tilde {h}^{(0)}, x_0)$ , the sequence $\tilde {h}^{(1)}$ is updated from the current chosen variable $x_1$ and the given variable $x_0$ . Then, $x_2$ is chosen given $x_1$ and $\tilde {h}^{(1)}$ in the feasible set $\Gamma (\tilde {h}^{(1)}, x_1)$ . And so on, at time $t+1$ , given $x_t$ and $\tilde {h}^{(t)}$ , the variable $x_{t+1}$ must be chosen in the feasible set $\Gamma (\tilde {h}^{(t)},x_t)$ . Once $x_{t+1}$ is chosen, at time $t+2$ , $x_{t+2}$ is chosen given $x_{t+1}$ and $\tilde {h}^{(t+1)}$ that has been updated from the previous history sequence $\tilde {h}^{(t)}$ and the variables $(x_t,x_{t+1})$ . Such a process allows us to keep decisions in memory while choosing the variable $x_t$ at each time $t$ .

For a sequence $\tilde {x}=(x_t)_{t=1}^{+\infty } \in \Pi (\tilde {h}^{(0)},x_0)$ , we denote by $\mathcal{U}(\tilde {x})$ the objective

\begin{equation*} \mathcal{U}(\tilde {x})=\sum \limits _{t=0}^{+\infty } \beta ^t F(\tilde {h}^{(t)}, x_t, x_{t+1}) \end{equation*}

4.2 Assumptions

We set that $(F,\Gamma )$ satisfies the list $(\mathcal{A})$ of assumptions if the following assumptions (F) on the feasible set, (A) on the instantaneous function, and (M) on the memory function are satisfied.

(F)

(F1) $\Gamma$ is a continuous nonempty compact-valued correspondence from $\mathcal{H} \times \operatorname {I\!R}^n$ into $\operatorname {I\!R}^n$ .

(F2) There exists $a \geq 0, a \neq 1$ and $a' \geq 0$ such that for any $x \in \operatorname {I\!R}^n$ and any history $\tilde {h} \in \mathcal{H}$ , $x' \in \Gamma (\tilde {h},x) \Rightarrow \Vert x' \Vert \leq a \Vert x \Vert + a'$

(A)

(A1) $\forall x_0 \in X$ and $\tilde {h}^{(0)} \neq 0, \exists \tilde {x} \in \Pi (\tilde {h}^{(0)},x_0)$ such that $\mathcal{U}(\tilde {x})\gt -\infty$ .

(A2) The function $F$ is continuous and there exist $a \in \operatorname {I\!R}^+$ with $a\beta \lt 1$ , $a_2 \in \operatorname {I\!R}^+$ and a continuous function $a_1 \, : \, X \rightarrow \operatorname {I\!R}^+$ such that for any $x_0 \in \operatorname {I\!R}^n$ and $\tilde {h}^{(0)} \in \mathcal{H}$ , for any feasible sequence $\tilde {x}=(x_t)_{t=1}^{+\infty }\in \Pi (\tilde {h}^{(0)},x_0)$ and its associated history $\tilde {h}^{(t)}$ , for any $t\geq 0$ ,

\begin{equation*} F^+(\tilde {h}^{(t)}, x_t, x_{t+1}) \leq a_1(x_0)a^t+a_2 \end{equation*}

where $F^+(\tilde {h}, x, x') =\sup \{F(\tilde {h}, x, x'),0\}$ . (M) The function $m$ is continuous.

The next result shows that, under the set of assumptions $(\mathcal{A})$ , the objective function of the optimization problem is well-defined. Moreover, whe show the existence of a solution to the optimization problem. The uniqueness is obtained under additional assumptions, requiring strict concavity of the instantaneous function $F$ . We give these results in a general framework, in particular without any differentiability assumptions. The strict joint concavity of $F$ allows us to guarantee uniqueness. The proofs are given in the Appendix.

4.3 The value function and the Bellman equation

We now define the value function of the optimization problem. We show that under the set of assumptions $(\mathcal{A})$ , dynamic programming tools can be used to study our problem. We first study the properties of the value function. We then provide an appropriate set of functions on which the Bellman operator has a unique fixed point which is the value function.

Definition 4.2. The value function $V$ is defined on $\mathcal{H} \times \operatorname {I\!R}^n$ by for any $(\tilde {h}^{(0)},x_0) \in \mathcal{H} \times \operatorname {I\!R}^n$

\begin{equation*} V(\tilde {h}^{(0)},x_0)=\left \{ \begin{array}{c@{\quad}l} \operatorname { Max } & \sum \limits _{t=0}^{+\infty } \beta ^t F(\tilde {h}^{(t)}, x_t, x_{t+1})\\ \operatorname { s.t. } & \forall t\geq 0, x_{t+1} \in \Gamma (\tilde {h}^{(t)},x_t)\\ & \forall t \geq 1, \tilde {h}^{(t)}=(m(x_{t-1},x_{t}),\ldots ..,m(x_1,x_2),m(x_0,x_1),\tilde {h}^{(0)})\\ & \tilde {h}^{(0)} \in \mathcal{H} \ {and}\ x_0 \in \operatorname {I\!R}^n \ { are } \ {given} \end{array} \right . \end{equation*}

The next proposition states some further properties on the value function. These properties, together with the upper semi-continuity, will provide an appropriate set of functions to consider for dynamic programming tools.

The proof of this proposition can be found in the Appendix. A standard proof (see Theorem 4.4 p.75 Stokey et al., Reference Stokey, Lucas and Prescott1989) allows us to show the following result.

Proposition 4.3. Assume $(\mathcal{A})$ . Then $\tilde {x}^*$ is an optimal solution if and only if

\begin{equation*} \forall t \geq 0, V(\tilde {h}^{* (t)},x_{t}^*)=F(\tilde {h}^{* (t)}, x_{t}^*,x_{t+1}^*)+\beta V(\tilde {h}^{* (t+1)},x_{t+1}^*) \end{equation*}

where $\tilde {h}^{* (t)}=(m(x_{t-1}^*,x_{t}^*),m(x_{t-2}^*,x_{t-1}^*),\ldots ..,m(x_1^*,x_2^*),m(x_0^*,x_1^*),\tilde {h}^{(0)})$

4.4 The value function is the unique fixed-point of the Bellman operator

Let $B$ be the Bellman operator, i.e. $B \,: \, \mathcal{F}(\mathcal{H} \times \operatorname {I\!R}^n,\operatorname {I\!R}) \rightarrow \mathcal{F}(\mathcal{H} \times \operatorname {I\!R}^n,\operatorname {I\!R})$ be defined byFootnote ¹⁵

\begin{equation*} \forall w \in \mathcal{F}(\mathcal{H} \times \operatorname {I\!R}^n,\operatorname {I\!R}), Bw(\tilde {h},x)=\max \limits _{x' \in \Gamma (\tilde {h},x)} \{ F(\tilde {h},x,x')+\beta w((m(x,x'),\tilde {h}), x') \} \end{equation*}

We now finally state that the value function is the unique fixed-point of the Bellman operator on this set of functions. The proof is given in Appendix.

Finally, this shows that dynamic programming tools can be used to deal with general history-dependent optimal growth models.

4.5 Dynamic programming tools for the particular cases with an adjustment level function

The following optimization results and dynamic programming tools are then derived as corollaries of the ones previously shown for the general history-dependent model. Assume $(\mathcal{A}')$ in this subsection hereafter.

Definition 4.4. The value function $V$ is defined on $\mathcal{H}\times \operatorname {I\!R}^n$ by for any $(\tilde {h}^{(0)},x_0) \in \mathcal{H} \times \operatorname {I\!R}^n$

\begin{equation*} V(\tilde {h}^{(0)},x_0)=\left \{ \begin{array}{c@{\quad}l} \operatorname { Max } & \sum \limits _{t=0}^{+\infty } \beta ^t F(\varphi (\tilde {h}^{(t)}), x_t, x_{t+1})\\ \operatorname { s.t. } & \forall t\geq 0, x_{t+1} \in \Gamma (\tilde {h}^{(t)},x_t)\\ & \forall t \geq 1, \tilde {h}^{(t)}=(m(x_{t-1},x_{t}),\ldots ..,m(x_1,x_2),m(x_0,x_1),\tilde {h}^{(0)})\\ & \tilde {h}^{(0)} \in \mathcal{H} \ {and}\ x_0 \in \operatorname {I\!R}^n \ {are} \ {given} \end{array} \right . \end{equation*}

We further define (see Appendix for the details) an appropriate set of functions $\mathcal{F}_{b}(\mathcal{H} \times \operatorname {I\!R}^n,\operatorname {I\!R})$ for which dynamic programming tools can be derived. All the (mathematical) details are given in the Appendix.

Let $B$ be the Bellman operator, i.e. $B \, : \, \mathcal{F}(\mathcal{H} \times \operatorname {I\!R}^n,\operatorname {I\!R}) \rightarrow \mathcal{F}(\mathcal{H} \times \operatorname {I\!R}^n,\operatorname {I\!R})$ be defined by

\begin{equation*} \begin{array}{ll} \forall w \in \mathcal{F}(\mathcal{H} \times \operatorname {I\!R}^n,\operatorname {I\!R}), Bw(\tilde {h},x) & =\max \limits _{x' \in \Gamma (\tilde {h},x)} \{ F(\varphi (\tilde {h}),x,x')+\beta w((m(x,x'),\tilde {h}), x') \} \\ & =\max \limits _{x' \in g(\varphi (\tilde {h}),x)} \{ F(\varphi (\tilde {h}),x,x')+\beta w((m(x,x'),\tilde {h}), x') \} \end{array} \end{equation*}