Yuri Yatsenko

Houston Baptist University · Dunham College of Business

Hundreds of meticulously crafted mathematical problems and puzzles in this book are incorporated into fascinating stories about our world. These wor(l)d problems are grouped by their mathematical concepts such that the titles of sections and chapters combine both mathematical and applied terms, hinting at the topics covered. Detailed solutions conc...

A nonlinear optimal control problem in the Lotka–McKendrick population model is studied. It describes rational management of age-structured farmed populations in aquaculture and indoor farms. Employing generalized functions, we prove the impulse nature of optimal harvesting. Exact analytic solutions for sustainable harvesting strategies are obtaine...

We analyze the optimal lockdown in an economic–epidemic model with realistic infectiveness distribution. The model is described by Volterra integral equations and accurately depicts the COVID-19 infectivity pattern from clinical data. A maximum principle is derived, and a qualitative dynamic analysis of the optimal lockdown problem is provided over...

The integral model with finite memory is employed to analyze the timeline of COVID-19 epidemic in the United Kingdom and government actions to mitigate it. The model uses a realistic infection distribution. The time-varying transmission rate is determined from Volterra integral equation of the first kind. The authors construct and justify an effici...

Pollution mitigation and environmental adaptation are two major strategies to fight for environmental sustainability. Both adaptation and mitigation require enormous resources, investments, and careful planning. Most related studies focus on coordinating adaptation and mitigation policies at global international scales. However, a variety of adapta...

This paper briefly surveys age- and size- structured linear and nonlinear population models and reveals their practical meaning. It focuses on optimal control problems with two types of harvesting and their impact on the sustainable harvesting.KeywordsAge- and size-structured population modelsHarvesting rate and effortTwo-dimensional optimal contro...

The paper focuses on modeling of public health measures to control the COVID-19 pandemic. The authors suggest a flexible integral model with distributed lags, which realistically describes COVID-19 infectiousness period from clinical data. It contains susceptible–infectious–recovered (SIR), susceptible–exposed–infectious–recovered (SEIR), and other...

We demonstrate that use of the Dirac delta-function simplifies qualitative optimization analysis of two contemporary population models. The first problem maximizes the discounted profit from harvesting in the linear age-structured Lotka–McKendrick model. The second problem describes optimal investments in new and old equipment under evolving techno...

We consider a nonlinear profit maximization problem in the Lotka–McKendrick model of age-structured harvested population describing farmed populations in agriculture and aquaculture. The control functions are time- and age-dependent harvesting rate and time dependent supply of newborns. We establish the existence of optimal controls with measure-va...

The article is devoted to modeling the rational renewal of capital assets of an enterprise. The developed software system makes it possible to solve the problem of rational replacement of an asset in conditions when the enterprise does not have complete data on future technological changes. Replacing an asset on time can help reduce capital and ope...

We formulate and study a nonlinear game of n symmetric countries that produce, pollute, and spend part of their revenue on pollution mitigation and environmental adaptation. The optimal emission, adaptation, and mitigation investments are analyzed in both Nash equilibrium and cooperative cases. Modeling assumptions and outcomes are compared to othe...

This article surveys recent developments in the optimal renovation of capital assets and introduces new asset replacement algorithms under limited forecast about changing uncertain costs. The costs include operating costs of the current asset in use and the replacement cost (price of new assets). The evolution of those costs depends on external tec...

Purpose Despite the existence of multiple asset replacement theories, the economic life replacement method remains a major practical technique for making rational machine replacement decisions. The purpose of this paper is to bridge this method with comprehensive data analytic tools and make it applicable it to modern business reality with abundant...

We offer a simple analytic framework for modelling an asymmetric multi-country world that faces negative impacts of climate change. A two-player asymmetric game combines economic efficiency and environmental vulnerability in the North-South competitive system. Comparison of competitive (Nash equilibrium) and cooperative strategies sheds a new light...

The paper analyzes a profit-maximizing serial replacement problem under variable asset productivity, operating cost, and replacement cost that depend on asset age and installation time. The majority of modern asset replacement models minimize costs. The authors examine and highlight essential differences between profit-maximizing and cost-minimizin...

This paper studies the cost-minimizing serial asset replacement problem under changing operating and replacement costs. Technological improvements affect these costs in different ways, which leads to different duration of sequential replacement cycles of the same asset. Therefore, a firm should optimize a chain of consecutive asset replacements usi...

Solutions from non-smooth functional spaces, including generalized functions and measures, often appear in optimal control theory but are avoided in applications. They are however useful in finding the optimal distribution of investments into new and old capital equipment under improving technology. The corresponding economic problem involves optim...

The owner of an asset often transfers the right to use or exploit that asset to an agent in exchange for a rent. A limited time of the license and the failure of the owner's commitment to compensate the agent for any asset improvement are likely to lead to underinvestment (holdup). In this study, we analyze the optimal length a contract would need...

The authors examine contemporary integral dynamic models of biological, environmental, and technological systems with heterogeneous components. The models contain two-dimensional control functions, nonlinearities, and delays in system inputs. The paper demonstrates the versatility of integral equations and their importance to various applications....

We study the impact of technological progress on the level of employment in a vintage capital model where: (i) capital and labor are gross complementary; (ii) labor supply is endogenous and indivisible; (iii) there is full employment, and (iv) the rate of labor-saving technological progress is endogenous. We characterize the stationary distribution...

The infinite-horizon cost minimization and the real-option stopping problem are used to analyze serial replacement of a single asset under improving technology. It is shown that both techniques produce the same (equal life) replacement policy when exponential technological improvements affect the operating cost and new asset cost in the same way. N...

An optimal control problem for nonlinear integral equations of special kind is analyzed. It considers a firm’s investment into age-dependent capital under improving technology and limited substitutability among capital of different ages. We prove the existence of solutions and analyze their structure. It is shown that the initially bang-bang optima...

We suggest analytic economic-environmental models that aim to investigate the multi-faced interplay between economic production, industrial pollution, and environmental protection. The considered static and dynamic optimization problems assess a country’s sustainable environmental policy that involves the optimal combination of adaptation and mitig...

The paper analyzes the strategic behavior of several countries engaged in capital accumulation, pollution mitigation, and environmental adaptation in the context of an environmental common good. Both cooperative and non-cooperative strategies are discussed. The non-cooperative strategy is a dynamic game in which each country makes its own environme...

We analyse the serial asset replacement problem under incomplete data about technological change that that affects the capital cost, operating cost and salvage value of newer assets. We construct an efficient discrete-time algorithm for this problem in the case where only partial information is available about future costs of new assets. The algori...

Optimal asset replacement is analysed under non-exponential technological improvement, whereby the asset capital cost and the operational cost reduce for newer assets, and the relative rate of this decrease varies with time. We establish that the optimal lifetime of sequentially replaced assets increases if the relative rate of technological change...

A hierarchical economic–environmental model is formulated to analyze the sustainable management of farming production in a landlord–tenant system. The problem under study is interdisciplinary and combines various agricultural, economic, social, and environmental factors. To maximize profit, the landlord chooses a rental payment and the duration of...

The paper combines analytic and numeric tools to investigate a nonlinear optimal control problem relevant to the economics of climate change. The problem describes optimal investments into pollution mitigation and environmental adaptation at a macroeconomic level. The steady-state analysis of this problem focuses on the optimal ratio between adapta...

Work significantly affects human life and health. Overworking may decrease the quality of life and cause direct economic losses. Technological innovations promote the modernization of firms' capital and lead to the improvement of labor productivity in the workplace. This paper investigates the optimal individual choice of work intensity under impro...

This paper performs a systematic study of some nonlinear integral equations with unknown lower limits of integration, which arise in economics, operations research, population biology and environmental sciences. A general investigation technique is developed for systems of such equations and several theorems about the existence and uniqueness of a...

The study of development and interaction of biological species is an important direction of modern research. As one of the central parts of this study, mathematical modeling assists in understanding the behavior of populations and provides reliable forecasts and recommendations for sustainable policies and management. A variety of mathematical mode...

This chapter analyzes optimization problems in the economic models with heterogeneous capital and labor of Chap. 4. Such models are important in explaining economic modernization under improving technology. Section 5.1 provides a qualitative analysis of the continuous-time optimization problem of one-machine replacement from Sect. 4. 4 using standa...

This chapter is devoted to an important economic–environmental problem: the modeling of the optimal extraction and utilization of nonrenewable (exhaustible) resources. Nonrenewable resources are natural resources that cannot be replaced as quickly as they are being consumed. Examples of such resources include fossil fuels (petroleum, coal, and natu...

Age-structured population models, considered in Chap. 6, have become a traditional tool in biological modeling and are widely used in other disciplines. They possess well-developed investigation techniques. However, the size of individuals is a more important parameter than their age for some species. Section 7.1 considers two size-structured model...

Human activities can cause negative global changes in the environment, such as global warming, excessive pollution of the environment, extinction of some species, degradation of entire geographic areas, and others. Section 12.1 discusses the modeling of global changes as a scientific problem of great complexity and importance. Section 12.2 describe...

This chapter provides a brief overview of the goals, general principles, and specific tools of mathematical modeling, specifically for economic and environmental systems. Section 1.1 analyzes the role and structure of the modeling process in scientific research and decision-making practice. Section 1.2 explores and compares different types of mathe...

This chapter is devoted to analytic modeling of the relations between the economy, society, and the environment, with a focus on combating climate change. Section 11.1 describes simple models of these relationships and discusses mitigation and adaptation as two major human responses to environmental damage. Sections 11.2 and 11.3 investigate econom...

This chapter explores economic growth models with heterogeneous capital and labor described by the integral or partial differential equations. Such models are imperative in explaining economic development under embodied technological change. Section 4.1 describes the well-known macroeconomic growth models with vintage capital of R. Solow and L. Joh...

We consider the optimal control of nonlinear integral equations with endogenous delay and state constraints, which describe a developing economy subjected to resource constraints. The economy invests in new resource-efficient technologies, invests in new capital, and scraps obsolete capital. We derive the optimality condition and determine long-ter...

We develop an aggregated model to study rational environmental adaptation policies that compensate negative consequences of climate change. The model distinguishes three categories of adaptation measures that (a) compensate the decrease of environmental amenity value, (b) compensate the decrease of total productivity, (c) develop and introduce new...

The Kyoto protocol and subsequent agreements allow Annex I countries to deduct carbon sequestered by land use, land-use change and forestry from their national carbon emissions. From this point of view the determination of the optimal logging regime in the presence of timber and carbon sequestration benefits is of utmost importance. Thornley and Ca...

A summary of the results of a study that compared the effects of climate change on the forest ecosystem, mitigation costs, and timber production under several climate change scenarios. Forest carbon sequestration is an important technique for the capture of carbon dioxide (CO2) from the atmosphere. A large number of studies 1 have determined the co...

Climatic changes will affect the dynamics of a forest ecosystem. Consequently, carbon sequestration costs can only be estimated correctly if changes in climatic conditions are considered. This article determines the changes in mitigation costs of an optimal forest management regime in the presence of climatic changes and varying prices, and takes a...

We prove that the introduction of endogenous indivisible labor supply into the vintage capital growth model does not rule out the turnpike and optimal permanent regime properties, notably the non-monotonicity properties of optimal paths, inherent in this model.

This textbook offers an attractive collection of a wide range of models ranging from the classic Cobb-Douglass production function, Solow models of economic growth, Lotka-Volterra and McKendrick-McCamy popula-tion models, Hotelling and Dasgupta-Heal models of exhaustible resource, Forrester and Meadows models of world dynamics, to modern models of...

We perform the steady-state analysis of a nonlinear partial differential equation model that describes the dynamics of a managed size-structured forest. The harvesting policy is to maximize the net benefits from timber production over an infinite planning horizon. The existence and uniqueness of the steady-state trajectories are analysed. Closed-fo...

This work investigates the impact of global climate change on the sustainable growth of forest, namely, on its aggregated characteristics such as the number of trees, the basal area, and the amount of carbon sequestrated in the stand. The forest dynamics is described by a nonlinear size-structured population model. The existence of a steady state r...

The paper examines long-term strategies of capital modernization under different assumptions about embodied technological change, energy regulation, and substitutability between energy and capital. To describe modernization of physical capital, the authors use a vintage capital model with the constant elasticity of substitution between capital and...

The paper analyzes a fleet replacement problem under general assumptions about technological improvement. The corresponding optimization model considers the variable lifetime of assets in deterministic infinite-horizon settings. The optimal dynamics of the asset lifetime and investment is investigated analytically and numerically under continuous t...

An optimal harvesting problem is analyzed in the Lotka–McKendrick model of age-structured populations. Depending on the imposed constraints, this problem possesses bang-bang or impulse regimes, which have meaningful interpretations and relevant policy implications. A steady state analysis of the model produces closed-form solutions for the optimal...

SUMMARYA nonlinear optimal control problem describes the capital modernization strategy of a firm under improving technology and restrictions on available resources. The restricted resource may be space, land, labor, finance, or environmental pollution. The authors reveal a complete qualitative picture of optimal regimes and show that the only prof...

We study an economy in which firms use labor and various vintages of capital in a CES production function for the final good. We explicitly solve for the investment in capital of a given vintage as a function of its age, and for the resulting stocks of capital. We show that for reasonable parameter values, inverted-U-shaped dynamics of investment a...

We consider an optimal growth model of an economy facing an exogenous pollution quota. In the absence of an international market of pollution permits, the economy has three instruments to reach sustainable growth: R&D to develop cleaner technologies, investment in new clean capital goods, and scrapping of the old dirty capital. The R&D technology d...

The economic life (EL) method of asset replacement is analyzed under improving technology that impacts the maintenance cost, new asset cost, and salvage value. In particular, we prove that the asset EL is constant when all these costs decrease with the same rate. If these costs decrease geometrically, then the EL method with a corrected capital rec...

Vintage capital growth models have been at the heart of growth theory in the 60s. This research line collapsed in the late 60s with the so-called embodiment controversy and the technical sophisitication of the vintage models. This paper analyzes the astonishing revival of this literature in the 90s. In particular, it out-lines three methodological...

We develop an aggregated model to study rational environmental adaptation policies that compensate negative consequences of certain environmental hazards and changes. The model distinguishes three categories of adaptation measures that (a) compensate the decrease of the environmental amenity value, (b) compensate the decrease of total productivity,...

This paper studies to which extent a firm using a scarce resource input and facing environmental regulation can still manage to have a sustainable growth of output and profits. The firm has a vintage capital technology with two complementary factors, capital and a resource input subject to quota, the latter being increasingly scarce through an exog...

Optimal replacement of a firm's capital is described in the framework of Solow-type vintage capital models. The firm controls the investment into new capital and scrapping of obsolete capital. The embodied technological change involves a continuous component and technological innovations (breakthroughs, technology shocks). The provided analytic and...

Optimal control of partial differential equations arises in population ecology, economics, and demography. The consistency of mathematical treatment is demonstrated for the Lotka-McKendrick model and its nonlinear modifications of increasing complexity. The obtained qualitative optimal dynamics show that the models have either the bang-bang structu...

The paper analytically explores the optimal policy mix between mitigation and environmental adaptation against climate change at a macroeconomic level. The constructed economic-environmental model is formulated as a social planner problem with the adaptation and abatement investments as separate decision variables. The authors prove the existence o...

We consider a firm with the following characteristics: (i) it has a vintage capital technology with two complementary factors, capital and an input subject to quota; (ii) scrapping of obsolete capital is endogenous; (iii) technological progress allows to save the regulated input, and is endogenous through purposive R&D investment; (iv) the innovati...

In operations research, the equipment replacement process is usually modeled in discrete time. An alternative approach is represented by continuous-time vintage capital models that explicitly involve the equipment lifetime and are described by nonlinear integral equations. The paper introduces and analyzes a model that unites both these approaches....

The Kyoto protocol allows Annex I countries to deduct carbon sequestered by land use, land-use change, and forestry from their national carbon emissions. J. Thornley and M. Cannell (2000. Managing forests for wood yield and carbon storage: A theoretical study. Tree Physiol. 20:477-484) have demonstrated that the objectives of maximizing timber and...

Several R&D-based models of endogenous economic growth are investigated under the Solow-like assumption of fixed allocation of resources across activities. We identify model parameters that lead to explosive dynamics and analyze various economic techniques to avoid it. The techniques include adding stricter constraints on model trajectories and lim...

Age-structured and size-structured population models with a distributed harvesting control are considered and their qualitative behaviors are compared. The maximum principle and bang–bang structure of the optimal control are established for a size-structured forestry model with carbon sequestration benefits.

The paper analyzes the optimal maintenance of a "fully manageable" biological population with no natural reproduction. Examples of such populations include dairy farms, horse breeding, gardening, and forestry. Mathematically, the problem leads to the nonlinear optimal control of a first-order partial differential equation with pointwise state const...

The paper is devoted to theoretic and numeric investigation of a nonlinear integral equation with an unknown function in the upper limit of integration. This equation appears in optimal replacement problems of engineering and production economics. Its solution is essential for finding the optimal policy of equipment replacement under technological...

Optimization problems with nonlinear utility and endogenous capital lifetime are investigated in one- and two-sector modifications of the Solow vintage capital model. Both models have the same balanced growth path in the case of exponential technological change and labour. Turnpike theorems in normal form are proved for the optimal capital lifetime...

The paper analyzes nonlinear optimal control of integral-differential equations that describe the optimal management of a forest taking into account intra-species competition and carbon sequestration. The objective function includes the revenues from timber production, operational expenses, and the net benefits from carbon sequestration. A dual sys...

Several R&D-based models of endogenous economic growth are investigated under the Solow-like assumption of fixed allocation of resources across activities. We identify model parameters that lead to explosive dynamics and analyze various economic techniques to avoid it. The techniques include adding stricter constraints on model trajectories and lim...

a b s t r a c t The variable lifetime of assets is analyzed in a serial replacement problem. Technological change impacts the maintenance cost and new asset cost. The optimal asset lifetime appears to be constant only when both costs decrease with the same rate. We identify cases when the technological change decreases or increases the optimal life...

This paper studies the optimal vintage specific investment and disinvestment decision of a firm. The decision is formulated as a distributed optimal control problem with a constraint on the state variable. The economic model takes into accounts maintenance and operating costs, technological progress and learning. It allows firms to purchase not onl...

The paper analyzes the structure of optimal trajectories in the one-sector vintage capital model that optimizes the endogenous lifetime of age-structured capital under technological change. A new repetition pattern (anticipation echoes) is demonstrated in both optimal investment and optimal capital lifetime in the case of finite-horizon optimizatio...

An optimization problem with nonlinear utility is investigated in the well-known Ramsey vintage capital model described by nonlinear Volterra integral equations with an unknown function in the lower integration limits. The necessary and sufficient condition for an extremum is proven and a qualitative analysis is provided. A balanced growth case is...

The authors develop a new approach to model the optimal policies of machine replacement under technological change. They consider a single-machine replacement problem in continuous time and reduce it to a nonlinear integral equation for the variable optimal service life of machine. A qualitative analysis of this equation leads to new dynamic proper...

The paper analyzes a nonlinear integer-valued optimization problem that describes the optimal machine replacement under improving technology. The problem is extended into the classes of real-valued discrete-time optimization problems and continuous-time optimal control. The results include some qualitative properties of optimal management policies...

A model of the governmental control of age at retirement uses population age structures, productivity, employment, and money income. The sustainability criterion is a sufficient level of individual money income for all ages. The decision variables are the variable age at retirement and the intensity of technological change. The model is calibrated...

The Kyoto protocol allows Annex I countries to deduct carbon sequestered by land use, land-use change and forestry from their national carbon emissions. Thornley and Cannell (2000) demonstrated that the objectives of maximizing timber and carbon sequestration are not complementary. Based on this finding, this paper determines the optimal selective...

The authors analyze the optimal replacement of assets under continuous and discontinuous technological change. They investigate the variable lifetime of assets in an infinite-horizon replacement problem. Due to deterioration, the maintenance cost increases when the asset becomes older. Because of technological change, both maintenance and new capit...

In Operations Research, the equipment replacement process is usually modeled in discrete time. The optimal replacement strategies are found from discrete (or integer) programming problems, well known for their analytic and computational complexity. An alternative approach is represented by continuous-time vintage capital models that explicitly invo...

The paper examines the Porter and induced-innovation hypotheses in a firm model where: (i) the firm has a vintage capital technology with two complementary factors, energy and capital ; (ii) scrapping is endogenous; (iii) technological progress is energy-saving and endogenous through purposive R&D investment; (iv) the innovation rate increases with...

The paper analyzes optimal harvesting of age-structured populations described by the Lotka-McKendrik model. It is shown that the optimal time- and age-dependent harvesting control involves only one age at natural conditions. This result leads to a new optimization problem with the time-dependent harvesting age as an unknown control. The integral Lo...