Non-Renewable Resource Extraction Model with Uncertainties

A differential game with random duration is considered. The terminal time of the game is a random variable settled using a composite distribution function. Such a scenario occurs when the operating mode of the system changes over time at the appropriate switching points. On each interval between switchings, the distribution of the terminal time is characterized by its own distribution function. A method for solving such games using dynamic programming is proposed. An example of a non-renewable resource extraction model is given, where a solution of the problem of maximizing the total payoff in closed-loop strategies is found. An analytical view of the optimal control of each player and the optimal trajectory depending on the parameters of the described model is obtained.

Trends in the extraction of non-renewable resources: The case of fossil energy

We consider the class of differential games with random duration. We show that a problem with random game duration can be reduced to a standard problem with an infinite time horizon. A Hamilton-Jacobi-Bellman-type equation is derived for finding optimal solutions in differential games with random duration. Results are illustrated by an example of a game-theoretic model of nonrenewable resource extraction. The problem is analyzed under the assumption of Weibull-distributed random terminal time of the game.

Discontinuous extraction of a nonrenewable resource

This paper examines the sequence of optimal extraction of nonrenewable resources in the presence of multiple demands. We provide conditions under which extraction of a nonrenewable resource may be discontinuous over the course of its depletion.

Advances in economic theory and optimal control methods have considerably improved the prescriptions for the extraction of renewable and nonrenewable resources. While the classic analyses, going back to Hotelling (1931), proceeded under assumptions of price certainty, recent work addresses the considerable uncertainty in resource prices (which is of the same order as that of stock prices). For tractability, this newer literature assumes a constant marginal cost of extraction which ignores the considerable emphasis in the older literature on increasing marginal costs. We frame and solve the nonrenewable resource extraction problem (in the context of a typical mine) that jointly accounts for the significant price uncertainties as well as the dependence of extraction costs on the extraction rate and on the cumulative amount extracted. We find that ignoring cumulating cost when determining extraction strategy can lead to significant loss of value. Our analysis also establishes the general tools to pursue, for instance, the optimal taxation of natural resource sector that is of significant policy interest to many developing countries.

Background When implementing new statistical procedures, there is often a need for simple—and yet computationally efficient—ways of numerically evaluating composite distribution functions. If the statistical procedure must support calculations for censored and noncensored cases, those calculations should be carried out using efficient computational implementations of both definite and indefinite integrals (e.g., calculation of tail areas of distribution functions). Method We developed a generic function evaluator such that users may specify a function using reverse Polish notation. As its argument the function evaluator takes a matrix of pointers and then applies the rows of this matrix to its internally defined stack of pointers. Accordingly, each row of the argument matrix defines a single operation such as evaluating a function on the current element of the stack, applying an algebraic operation to the two top elements of the stack, or manipulating the stack itself. Defining new composite distribution functions from other (atomic) distribution functions then corresponds to joining two or more function-defining matrices vertically. This approach can further be used to obtain integrals of any defined function. As an example we show how the density and distribution function for the minimum of two Weibull distributed random variables can be numerically evaluated and integrated. Results The procedure provides a flexible and extensible framework for implementing numerical evaluation of general, composite distributions. The procedure is numerically relatively efficient, although not optimal.

When implementing new statistical procedures, there is often a need for simple--and yet computationally efficient--ways of numerically evaluating composite distribution functions. If the statistical procedure must support calculations for censored and noncensored cases, those calculations should be carried out using efficient computational implementations of both definite and indefinite integrals (e.g., calculation of tail areas of distribution functions). We developed a generic function evaluator such that users may specify a function using reverse Polish notation. As its argument the function evaluator takes a matrix of pointers and then applies the rows of this matrix to its internally defined stack of pointers. Accordingly, each row of the argument matrix defines a single operation such as evaluating a function on the current element of the stack, applying an algebraic operation to the two top elements of the stack, or manipulating the stack itself. Defining new composite distribution functions from other (atomic) distribution functions then corresponds to joining two or more function-defining matrices vertically. This approach can further be used to obtain integrals of any defined function. As an example we show how the density and distribution function for the minimum of two Weibull distributed random variables can be numerically evaluated and integrated. The procedure provides a flexible and extensible framework for imple- menting numerical evaluation of general, composite distributions. The procedure is numerically relatively efficient, although not optimal. Copyright 2007 by StataCorp LP.

This paper studies the optimal extraction and taxation of nonrenewable natural resources. It is well known that the market values of the main strategic resources such as oil, natural gas, uranium, copper,..., etc, fluctuate randomly following global and seasonal macroeconomic parameters, these values are modeled using Markov switching L\'evy processes. We formulate this problem as a differential game. The two players of this differential game are the mining company whose aim is to maximize the revenues generated from its extracting activities and the government agency in charge of regulating and taxing natural resources. We prove the existence of a Nash equilibrium. The corresponding Hamilton Jacobi Isaacs equations are completely solved and the value functions as well as the optimal extraction and taxation rates are derived in closed-form. A Numerical example is presented to illustrate our findings.

We derive a measure of national income, defined in terms of maximum sustainable per capita consumption. If population and interest rates are constant, the income generated by natural resource extraction is the return on resource wealth, defined as the present value of future resource rents. With a growing population or declining interest rate, sustainability requires net increases in wealth over time. The income from extraction of nonrenewable resources can exceed resource rents, contrary to the conclusions of El Serafy (1989) and Repetto et al. (1989). Finally, we present an example illustrating the hazard of consuming the full increment of Hicksian income.

This paper examines a class of cooperative differential games with continuous updating. Here it is assumed that at each time instant players have or use information about the game structure defined for a closed time interval with fixed duration. The current time continuously evolves with the updating interval. The main problems considered in a cooperative setting with continuous updating is how to define players' cooperative behaviour, how to construct a cooperative trajectory, how to define the characteristic function and how to arrive at a cooperative solution. This paper also addresses the properties of the solution and presents some techniques to fix the process by which a cooperative solution is constructed. Theoretical results are demonstrated on a differential game model of non-renewable resource extraction, initial and continuous updating versions are also considered. Comparison of cooperative strategies, trajectories, characteristic functions and corresponding Shapley values is presented.

The paper considers and describes the class of cooperative differential games with continuous updating. Such a class of differential games is new, at the moment only the classnoncooperative game models with continuous updating have been studied. This paper describes the process of constructing cooperative strategies, cooperative trajectory, characteristicfunction and cooperative solution with continuous updating. Cooperative case of limited resource extraction game model with continuous updating is considered. Optimal strategies,characteristic function and cooperative solution are constructed. The Shapley vector is used as a cooperative solution. The numerical simulation results are demonstrated in the Matlabenvironment.

Molecular weight and composition heterogeneity of multicomponent copolymers

Sustainable solution for hybrid differential game with regime shifts and random duration

A note on nonrenewable resource extraction under discontinuous price policy