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For nearly a quarter of a century, the US has persistently run significant current account deficits that transformed it from the world’s largest net creditor to its largest debtor. The stochastic optimal control/dynamic programming approach provides a framework to derive the optimal debt ratio, based upon both objective variables and preferences. The following set of questions are answered: What is a sustainable ratio of external debt/GDP? What are the consequences of an “excessive” debt? By how much does the dollar need to decline in order to achieve a sustainable current account position for the US and rest of world? The deviation of the actual debt ratio from the derived optimum increases the vulnerability of the economy to external shocks.

Early Warning Signals of a default or debt crisis are derived by drawing upon the stochastic optimal control model of an optimal and excessive short-term debt developed in chapter 2. Operational benchmarks for optimal foreign debt and a quantitative measure of the maximum debt which will not be defaulted in the event of adverse shocks are established. Insofar as the actual debt exceeds the benchmark, there is an excess debt, the risk of default is increased. Two sets of emerging market countries are considered: one set renegotiated/defaulted and the other set did not. The countries that defaulted/renegotiated had significant excess debt, whereas the countries that did not default/renegotiate did not have significant excess debt. An Early Warning Signal of a debt crisis is the excess debt, and not the level of the debt/GDP ratio per se.

This chapter analyses the stochastic optimal control problem. The problem considers an economic agent over a fixed time interval [0, T]. At time t = 0, the agent is endowed with initial wealth x₀, and the agent’s problem is how to allocate investments and consumption over the given time horizon. The agent must choose a portfolio-consumption strategy that will maximize the total utility over [0, T]. Practice exercises are included.

The Asian financial crises were unexpected by the market and many countries in the region experienced it at about the same time. Drawing upon the theoretical analyses in chapters 2-4, an operational theory is provided to answer the following questions for the Asian countries: Was a currency crisis produced by an overvalued real exchange rate? Was a debt crisis produced by an “excessive/unsustainable” external debt? What was the interaction between the two? The models imply a set of objective, theoretically-based warning signals and empirical analysis allows the assessment of which countries were or were not highly vulnerable to shocks.

This chapter introduces the notion of a controllable system, which is a system that can be moved using available controls from one state to another. The chapter is organized as follows. Section 3.2 provides a characterization of controllability for linear systems, both with bounded and unbounded sets of feasible controls. Section 3.3 discusses the simplest version of an optimal control problem and presents necessary as well as sufficient conditions for its solution. Section 3.4 considers the general finite-horizon optimal control problem with endpoint constraints and state-control constraints, which can be effectively dealt with using a more general version of the Pontryagin maximum principle (PMP), the proof of which is discussed in Section 3.6. Section 3.5 deals with optimal control problems where the decision horizon T goes to infinity. Section 3.7 discusses the existence of solutions to optimal control problems, notably the Filippov existence theorem. Exercises are included at the end of the chapter.

This chapter aims to determine precisely how macroeconomic and sectoral policies influence soil conservation with the use of an optimal control model. It shows that in general, it is not possible to predict how policy reforms affect soil conservation. Knowledge of the technical details of agricultural production is needed to predict how policy reforms affect soil conservation. Though this conclusion may seem unsatisfying, it compels us to rethink the premise that policy reforms are good if they conserve soil and bad if they don't. The flaw in this view is that our concern should not lie with soil per se. Unlike many other environmental resources, soil does not contribute to well-being directly; its value is as an input in agricultural production. Even if policy reforms cause further depletion, the ability of the land to yield an income may still be greatly enhanced. This conclusion relates only to the on-site impacts of soil erosion. There could be other externalities due to eroded soil to downstream and other users. Correcting such externalities may be difficult because shadow prices are not directly observable and are location-specific.

Optimal control theory is a mathematical discipline for studying the neural control of movement. This chapter presents a mathematical introduction to optimal control theory and discusses the following topics: Bellman equations, Hamilton-Jacobi-Bellman equations, Ricatti equations, and Kalman filter. It also examines the duality of optimal control and optimal estimation, and, finally, describes optimal control models and suggests future research directions.

This chapter describes a planning problem that generates competitive equilibrium allocations and compares two methods for solving it. The first method uses state- and date-contingent Lagrange multipliers; the second uses dynamic programming. The first method reveals a direct connection between the Lagrange multipliers and the equilibrium prices in a competitive equilibrium to be analyzed in Chapter 7. The second method provides good algorithms for calculating both the law of motion for the optimal quantities and the Lagrange multipliers. The chapter also describes a set of MATLAB programs that solve the planning problem and represent its solution in various ways. These programs are used to solve the planning problem for six sample economies formed by choosing particular examples of the ingredients from Chapter 4.

This chapter considers how economists model biodiversity in coupled social ecological systems, taking two polar cases along with a more general problem. Economists assume that all human decisions are purposive: people are assumed to optimize some objective function subject to some set of initial conditions, to some set of resource constraints, and to the dynamics of the system being used. The chapter considers two polar cases and one intermediate case. One polar case involves the preservation of wilderness areas or protected parks in 'close to natural' states. A second involves the exploitation of ecosystems to produce foods, fuels and fibers. The intermediate case involves the management of ecosystems to achieve a balance between non-consumptive cultural services with consumptive provisioning services. While the constrained optimization technique applied in all three cases may be unfamiliar, the chapter tries to give the intuition behind it. It also provides a verbal description of each of the three model structures developed. In all cases the social and biogeophysical components of the coupled system are interdependent — connected through a series of feedback loops. Economists refer to such systems as 'general equilibrium systems'. That is, the dynamics of the system in some state are driven by a tendency towards the equilibrium corresponding to that state, and any perturbation has the potential to stimulate responses across the system.

This chapter's appraisal places models of time and uncertainty in household allocation of resources at the center of the Samuelson Contribution. It assesses Samuelson's contributions to the areas of efficient market theory and risk analysis, portfolio selection, and option and warrant pricing. Samuelson found that efficient markets do not allocate resources the way casinos do. Rather, asset prices vary randomly around an optimal path that can be discerned mathematically. The theory links spot price with future prices in order to forge a solution where spot prices are determined by optimal control theory. Samuelson's work in warrants and option pricing provides a bridge between early and later option pricing models, thanks to his insights on hedging and mathematical analysis that were incorporated into subsequent theories.

This chapter focuses on approaches for efficient monitoring and management of established invasive species. Essential variables for monitoring and reporting and the requirements for practical monitoring strategies are discussed. Based on information from monitoring, experts estimate the impact of invasive species using different metrics, and the invasion debt for established species that are continuously progressing along the introduction–naturalization–invasion continuum. Bioeconomic models are used to devise optimal management strategies and address questions to ascertain different types of intervention. Commonly formulated as a mathematical model for optimal control, dynamic programming, and spatial optimization, bioeconomic models rely on the elucidation of the dynamics of impact, economic cost or valuation of control, and the effect of control on impact. When there are multiple management targets or options, or when conflicts of interest exist, prioritization protocols need to be formulated to identify highly sensitive sites, species, and pathways for optimal monitoring and management.

This chapter gives a self‐contained introduction to optimal control of stochastic differential equations. We derive the Hamilton‐Jacobi‐Bellman equation as well as a verification theorem. The general theory is then applied to optimal consumption and investment problems.

This chapter uses a three-equation model based on a New Keynesian Phillips curve, real-time data on the unemployment gap, and forecasted survey data on expected inflation to test the efficiency of the Federal Reserve’s pursuit of an optimal control approach to monetary policy that approximates the fine-tuning views of the New Economics prevalent in the 1960s and 1970s. It shows that the fine-tuning approach to monetary policy, with its emphasis on stabilizing the level of real activity, might have succeeded in stabilizing the economy if policymakers had possessed accurate real-time assessments of the natural rate of unemployment. In the event they did not, and failed to account for their imperfect information regarding the economy’s potential and the effects of these misperceptions on the evolution of inflation expectations and inflation.

This chapter introduces a theoretical framework, Optimal Control Theory, which will enable a phonology-extrinsic-timing-based, three-component model to determine values of controlled variables, and to model the influence of multiple factors on these parameter values. Key features of Optimal Control Theory models are discussed, as well as evidence for types of movement costs (including the cost of time) used in the models, and predictions of the models for the coordination of multiple effectors and hierarchical control. Finally, the chapter reviews Optimal Control Theory models currently used to account for timing phenomena in speech.

This chapter first sets out the book’s purpose, which is to introduce continuous-time systems and methods for solving dynamic optimization problems at three different levels: single-person decision making, games, and mechanism design. It presents a brief history of optimal control. An overview of the subsequent chapters is also presented.

This chapter focuses on electronic phenomena that occur when atoms and molecules are exposed to strong and short laser pulses. Under these conditions, the external potential cannot be treated as a small perturbation. Several highly nonlinear effects are introduced and reviewed: multiphoton ionization, above-threshold ionization, and high-harmonic generation. The performance of time-dependent density-functional theory for these phenomena is discussed. A particular challenge is the enhancement of double ionization in rare-gas atoms, such as the so-called helium knee, due to strong correlation effects. This puts stringent demands on the exchange-correlation functional, and on the way in which one calculates ionization yields. Time-dependent density-functional theory has been successfully used to calculate high-harmonic spectra in atoms and molecules. Finally, the basics of optimal control theory are reviewed, and illustrated with an example. It is shown how control theory and time-dependent density-functional theory can be combined.

This chapter reviews the basics of static mechanism design in settings where a principal faces a single agent of uncertain type. The aim of the resulting screening contract is for the principal to obtain the agent’s type information in order to avert adverse selection, maximizing her payoffs. Nonlinear pricing is discussed as an application of optimal control theory.