An introduction to dynamic optimization optimal control. It can be seen that the necessary conditions are identical to the ones stated above for the hamiltonian. Majors need to take calculus before they take certain intermediatelevel courses but do not need to have taken any economics in high school. Explain the firstorder condition of sufficiency of.
Using dynamic constrain t, simplify those rst order conditions. Inspired by, but distinct from, the hamiltonian of classical mechanics, the hamiltonian of optimal control theory was developed by lev pontryagin as part of his maximum principle. Nonlinear h infinitycontrol, hamiltonian systems and. The book can be used for a specialized or seminar course in robust and it is written for practicing professionals, educators, researchers and graduate students in electrical, computer, mechanical, aeronautical, chemical, instrumentation, industrial and systems engineering, as well as applied mathematics, economics and management. Solows growth model most modern dynamic models of macroeconomics build on the framework described in solows 1956 paper. Business situations are further complicated by constraints, which can be accounted for in managerial economics using the lagrangian function. Sufficiency of consumers firstorder conditions this firstorder condition is merely essential conditions for a local optimum. Moreover, it is often useful to assume that the time horizon is in. Introduction to hamiltonian fluid dynamics and stability.
Practical and accessible, elements of numerical mathematical economics with excel. Macroeconomic studies emphasize decisions with a time dimension, such as various forms of investments. Wyrwoll a dictionary of climate change and the environment bridges the gap between the many disciplines encompassing climate change, environmental economics, environmental sciences, and environmental studies. The hamiltonian method ilarities between the hamiltonian and the energy, and then in section 15. Dynamic optimization joshua wilde, revised by isabel ecu,t akteshi suzuki and maria jose boccardi august, 20 up to this point, we have only considered constrained optimization problems at a single point in time. But, at this point, you do not know if they yield a maximum or a minimum. Ill try to cover some motivation for the hamiltonian a rough description of what it is what the rules are for how to use it some of what it tells us about mechanics. The first order condition to the hamiltonian, equation 2 is very.
When the problem is formulated in discrete time, the hamiltonian is defined as. Optimal control theory 4 the maximum principle is unlike the euler equation which is a single rst order condition. This is a summary of some basic mathematics for handling constrained optimiza. The hamiltonian approach to dynamic economics focuses on the. Introduction to hamiltonian dynamical systems and the nbody. Economic growth current and present value hamiltonian september 27, 2005 to derive the formulae for the current hamiltonian knowing the present value hamiltonian goes as follows.
Can obtain necessary and sufficient conditions for an optimum. Under what conditions is the profit function strictly concave. Exponential lawson integration for nearly hamiltonian systems. Autonomous or nonautonomous optimal control system. This is an openbook exam, meaning that it is fine for you to consult your.
They are first order, 2k in number for k degrees of freedom, and show a remarkable the function hq,p,t is the hamiltonian and eqs. An equivalent way of solving this problem which at first may seem more difficult. Nov 09, 2016 in outline, the method of solution is this. Growth model, dynamic optimization in continuous time. We define the first two terms of the integrand as the so called hamiltonian. Alagrangian function for which the legendre transformation is applicable is said to be. Economic theory and mathematical economics kindle edition by cass, david, shell, karl. How would you define a hamiltonian in laymans terms. The firstorder necessary conditions for a maximum are given by. As before, the conditions are called firstorder conditions and a point that satisfies them is a stationary point.
Inparticular, we want to know the con ditions under which the legendre transformation can be used. This is the book i started learning mechanics with and especially for people unfamiliar with the subject it gives a good, but slowlypaced introduction. In economics, the objective function in dynamic optimization problems often depends directly on time. It is a wellorganized and accessible introduction to the subject. The discussion of topics such as invariance, hamiltonianjacobi theory, and actionangle variables is especially complete. The hamiltonian is a useful recipe to solve dynamic, deterministic. The book s tutorial approach and plentiful exercises combine with its thorough presentations of both subjects to make introduction to hamiltonian fluid dynamics and stability theory an ideal reference, selfstudy text, and upper level course book. Fortunately, you dont have to derive them from first principles for every problem.
Introduction to hamiltonian dynamical systems and the n. Malkevitch, 8 this theory is named after leonhard euler, an outstanding mathematician during the 18th century. The nbody problem is used as the primary example of a hamiltonian system, a touchstone for the theory as the authors develop it. For the first time in the literature, this book presents a detailed account of the theory of reduction by stages of hamiltonian systems with symmetries.
How to use the langrangian function in managerial economics. The hamiltonian is a function used to solve a problem of optimal control for a dynamical system. It is therefore a useful tool in computing reduced spaces and the authors illustrate it with many physical examples. Introduction cass and shell 3 have shown how a wide range of problems in the theory of economic dynamics may be formulated in terms of hamiltonian dynamical systems. This is the first of three necessary conditions we need to solve any optimal control. Nonlinear hinfinity control, hamiltonian systems and. The hamiltonian approach to dynamic economics focuses on the application of the hamiltonian approach to dynamic economics and attempts to provide some unification of the theory of heterogeneous capital. Search for library items search for lists search for contacts search for a library. Lecture notes for macroeconomics i, 2004 yale economic. Fertile area of research, excellent dissertation topics. An introductory textbook exploring the subject of lagrangian and hamiltonian dynamics, with a relaxed and selfcontained setting. Iournal of economic theory 12, 110 1976 introduction to hamiltonian dynamics in economics david cass and karl shell department of economics, university of pennsylvania, 3718 locust walk cr, philadelphia, pennsylvania 19174 received december 20, 1975.
Explain the firstorder condition of sufficiency of consumer. Using the hamiltonian, you derive various necessary or firstorder conditions that any solution must satisfy. The hamiltonian approach to dynamic economics 1st edition. Perhaps the business has signed a contract to produce 1,000 units of the good daily, or the business has certain inputs, such as the factory size, that cant be changed. It can be understood as an instantaneous increment of the lagrangian expression of the problem that is to be optimized over a certain time horizon. Exponential lawson integration for nearly hamiltonian.
The hamiltonian economic program was the set of measures that were proposed by american founding father and first secretary of the treasury alexander hamilton in four notable reports and implemented by congress during george washingtons first administration. R s denotes the average discount rate between time zero and. Static and dynamic optimization increases the computing power of economists worldwide. The hamiltonian can be written in terms of present value. The hamiltonian and the maximum principle conditions c. To solve these problems we usually set up the hamiltonian function. Although the hamiltonian is most often thought of as a device to remember the optimality conditions, it does have an interpretation related to the rate of accumulation of. Thus the hamiltonian can be understood as a device to generate the first order necessary conditions. These reports outlined a coherent program of national mercantilism governmentassisted economic development.
Daron acemoglu mit economic growth lectures 6 and 7 november 15 and 17, 2011. Definition let f be a function of n variables defined on a set s. At a minimum, dynamic optimization problems must include the objective function, the state equations and initial conditions for the state variables. First ignore 11 and set up the currentvalue hamiltonian. Euler had been the first person to study this category of circuits.
The hamiltonian approach to dynamic economics sciencedirect. This book is the result of many lectures given at various institutions, including the. Using the secondorder condition, establish whether the critical value corresponds to a maximum or minimum. Approach to modeling in macroeconomics well summarized by following. Introduction to lagrangian and hamiltonian mechanics. This makes dynamic optimization a necessary part of the tools we need to. Introduction to hamiltonian dynamics in economics sciencedirect. An euler circuit is a circuit that reaches each edge of a graph exactly once. Using the firstorder condition, obtain the critical value of q. Partial differential equations in economics and finance.
May 04, 2015 hamiltonian government focuses on productive investment rather than on consumptive spending. First order necessary optimality conditions for a class of infinite horizon optimal control problems. Emphasis is placed on the stability of longrun steadystate equilibrium in models of heterogeneous capital accumulation. Bavarian graduate program in economics, the universities of dortmund, dresden, frank. The present value hamiltonian evaluates the optimal behavior in terms of period 0 value. In economics it is much more common to start with inequality constraints of the. Characterization of equilibrium household maximization household maximization v. A key technique of environmental economics economic droplets. The rst order condition to the hamiltonian, equation 2 is very much like the rst order conditions of the lagrangian, where they are obtained di erentiating the. Where im going stochastic models of distribution and growth or heterogeneous agent models or micro to macro want to get you started on building these kind of models why should you be interested in this. Pdf first order necessary optimality conditions for a class. Use features like bookmarks, note taking and highlighting while reading the hamiltonian approach to dynamic economics. Lecture notes for macroeconomics i, 2004 yale university.
Moreover, conditions are sought on the geometry of the hamiltonian. The initial and terminal conditions on k t pin then do wn the optimal paths. Existence of solutions to hamiltonian dynamical systems of. Control, hamiltonian systems and hamiltonjacobi equations was written for practicing professionals, educators, researchers and graduate students in electrical, computer, mechanical, aeronautical, chemical, instrumentation, industrial and systems engineering, as well as applied mathematics, economics and management. This book takes the student from the newtonian mechanics typically taught in the first and the second year to the areas of recent research. Notes for macroeconomics ii, ec 607 university of michigan. Inspired by, but distinct from, the hamiltonian of classical mechanics, the hamiltonian of optimal control.
It starts with wellknown applications to consumer and producer theory, and to the theory of option pricing and then introduces new applications that emerge from current research some of which is the authors own in bounded rationality. In order for the hamiltonian to be equal to the total energy, certain conditions must apply such as there must be no time dependence of the system. The transversality condition says that the discounted value of the limiting capital stock cannot be positive. Using the hamiltonian, you derive various necessary or first order conditions that any solution must satisfy. Microeconomics assignment help, explain the firstorder condition of sufficiency of consumer, explain the firstorder condition of sufficiency of consumer.
Elements of numerical mathematical economics with excel. The reason why we may need the transversality condition is that the rstorder conditions only determine what is optimal from period to period, but might ignore the overall picture. Using the first order condition, obtain the critical value of q. Paul schweinzer school of economics, statistics and mathematics birkbeck college, university of london. Growth model, dynamic optimization in discrete time. Two examples of math we use on a regular basis are euler and hamiltonian circuits. Economics majors are required to complete five mandatory courses and four electives within the department. Using the second order condition, establish whether the critical value corresponds to a maximum or minimum. American school of economics the hamiltonian is a certain breed of capitalist, one who respects capitalism not for its own sake but for the sake of the dynamism and prosperity of the nation itself. Obtain the secondorder derivative of the profit function. Theproblem of optimal growth with zero discounting is. Comprised of eight essays, this book begins with an introduction to hamiltonian dynamics in. Constrained optimization, shadow prices, inefficient markets, and. An economic interpretation of optimal control theory.
Obtain the second order derivative of the profit function. Lagrangian and hamiltonian dynamics is the continuation of newtons classical physics into new formalisms, each highlighting novel aspects of mechanics that gradually build in complexity to form the basis for almost all of theoretical physics. This book is intended to support a first course at the graduate level for mathematics and engineering students. Essay v existence of solutions to hamiltonian dynamical systems of optimal growth r. Random productivity in the solow model one could model a more carefully and be speci. Instead, we construct a way of writing down the optimal control. Download it once and read it on your kindle device, pc, phones or tablets. A hamiltonian statement of principles abiasedperspective. Nonetheless, for the particular problem at hand, such. In the literature, this problem is analyzed numerically by using rungekutta scheme based on exponential lawson integration for nearly hamiltonian systems 11. Pontryagin proved that a necessary condition for solving the optimal contro. Drawing on the objective function and state equations, you set up an expression known as a hamiltonian, which will contain one or more additional variables known as costate variables. The book also covers special complementary topics such as inventory modelling, data analysis for business and economics, and the essential elements of monte carlo analysis.
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