A function is concave if the convex combination of the images of flies below the function at the convex combination. It is partially based on a twosemester course, held by the author for thirdyear students in physics and mathematics at the university of salerno, on analytical mechanics, differential geometry, symplectic manifolds and integrable systems. Demand and supply of individual goods and services, the price elasticity sensitivity of demand for goods and services, production, cost functions, business behavior and profit maximization in various. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Construction of mappings for hamiltonian systems and their by. Emphasis is placed on the stability of longrun steadystate equilibrium in models of heterogeneous capital accumulation.
What makes hamiltonian systems so important is the fact that the hamiltonian function is a first integral or constant of the motion. C 1 a number of factors of work conditions as well as whole work activities affect to workers, from which a many of them enter to the mutual interactions. Pdf a simple proof of the constancy of the pontryagin hamiltonian. Pdf controllability of a nonholonomic macroeconomic system. Growth model, dynamic optimization in discrete time eco 503. The hamiltonian function is also used to write the schroedinger equation of quantum mechanics.
By advanced i mean a book considering directly lagrangian and hamiltonian formulation, and also providing a firm basis in the geometrical consideration related to these to formalism like tangent bundle, cotangent bundle, 1form, 2form, etc. Book for hamiltonian and lagrangian mechanics physics forums. Clarity comes in part from the intuitive presentation style, but we have also integrated a number of pedagogical features that we believe make learning economic concepts and principles easier and more fun. The term microeconomics and macroeconomics were first given by ragner frisch in 1933. We can express that fact with mathematical formula. There is a large literature on hamiltonian systems. Chapter 2 lagranges and hamiltons equations in this chapter, we consider two reformulations of newtonian mechanics, the lagrangian and the hamiltonian formalism. Evans department of mathematics university of california, berkeley. Keynes, the general theory of employment, interest, and money 1936.
Both are good, but both have a really annoying flaw. You may find the following text books useful to read krusell note will be used substantially in the first section of the course per krusell, lecture notes for macroeconomics i, 2004 k jerome adda and russell cooper, mit press, 2003 ac nancy stokey and robert e. This workbook shows how to use economic theories, statistical and econometric methods for conducting research to find answers to puzzling issues in modern economies. The state of the system at a given time t is determined by six numbers, the coordinates of the position q 1,q 2,q 3 and the momentum p 1,p 2,p 3. Hamiltonian potential functions for differential games. The condition of strict concavity of the maximized hamiltonian function h 0 x. Firstly, to solve a optimal control problem, we have to change the constrained dynamic optimization problem into a unconstrained problem, and the consequent function is known as the hamiltonian function denoted. Macroeconomicsoptimal growth wikibooks, open books for an. Such functions are, of course, nondecreasing and strictly concave. Definition 1 a control is a bounded measurable function u. We find this path by setting up the currentvalue hamiltonian dropping time subscripts. Pdf we provide a new, simpler, and more direct proof of the well known fact that for autonomous optimal.
For a problem to be a concave problem, fmust be concave and the constraint set must be convex. General formulation consider the general optimal control problem two slides back. This chapter presents an introduction to hamiltonian dynamics in economics. Macroeconomic theory i benjamin moll princeton university fall 2014 6. Since articles are the primary means through which economists communicate, you should read at. Object of interest is a single or small number of household or. Recognizing that a course in economics may seem daunting to some students, we have tried to make the writing clear and engaging. It should have some historical context explaining the need to change the approaches for solving equation of motions. Example 2 conservation of the total linear and angular momentum we consider a system of nparticles interacting pairwise with potential forces depending on the distances of the particles. Bayesian inference3 requires that we express our prior beliefs about. It is thus not necessary to read all or even most of the articles on the list. 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. The hamiltonian approach to dynamic economics 1st edition.
Hamiltonian graph a connected graph g is called hamiltonian graph if there is a cycle which includes every vertex of g and the cycle is called hamiltonian cycle. References in the notes refer to articles given on the reading list. The differences between the lagrange and hamilton equations result. Since our boundary conditions rt1 r1 and rt2 r2 should still hold, it directly follows that. Please keep in mind that the readings, exercises, and course packet materials for this years econ 502 may differ in various ways from the materials assigned for econ 502 in previous years, and these differences may be reflected in differences in the form and content of the exams. Intuitively, the rm always has the option not to reoptimize as the price changes and secure a linear pro t function. Of all the possible paths along which a dynamical system may more from one point to another in configuration space within a specified time interval consistent with any constraints, the actual path followed is that which minimizes the time integral of the lagrangian function for the system. I am looking for a book about advanced classical mechanics.
Keynes is known as father of modern macroeconomics. Working papers series from swedish entrepreneurship forum in2009swedishentrepreneurshipforumstartedpublishinganewseriesofworkingpapers. Hamiltonian dynamics world scientific publishing company. Introduction to hamiltonian mechanics hamilton theory or more particularly its extension the hamiltonjacobi equations does have applications in celestial mechanics, and of course hamiltonian operators play a major part in quantum mechanics, although it is doubtful whether sir william would have recognized his authorship in that. Lucas, recursive methods in economic dynamics, cambridge. Macroeconomics qualifying examination january 2012 department of economics unc chapel hill instructions. Necessary and sufficient conditions which lead to pantryagins principle are stated and elaborated.
Hamiltonian dynamics operates on a ddimensional position vector, q, and a ddimensional momentum vector, p, so that the full state space has 2d dimensions. Macroeconomicsoptimal growth wikibooks, open books for. Example the hamiltonian often has a physical meaning for the sysem of odes that is modelling a. 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. Driscoll brown university and nber1 december 21, 2003. Inverse and implicit function theorems 827 chapter 31. Eulerian and hamiltoniangraphs there are many games and puzzles which can be analysed by graph theoretic concepts. Musthave math tools for graduate study in economics william neilson department of economics university of tennessee knoxville september 2009. The direction of the arrows shows that in the goods and services market, households receive goods and services and pay firms for them. This symmetry leads to very flexible transformation properties between sets of. Overview hamiltonians and phase diagrams princeton. Inparticular, we want to know the con ditions under which the legendre transformation can be used. Of all possible paths between two points along which a dynamical system may move from one point to another within a given time interval from t0 to t1, the actual path followed by the system is the one which minimizes the line integral of. As in the 1d case, time dependence in the relation between the cartesian coordinates and the new coordinates will cause e to not be the total energy, as we saw in eq.
Inspired by, but distinct from, the hamiltonian of classical mechanics, the hamiltonian of optimal control theory was developed. Applications to economics optimal control theory has been extensively applied to the solution of economics problems since the early papers that appeared in shell 1967 and the works of arrow 1968. In practice, the time scale of the electron dynamics is usually much shorter than that of the nuclear spin dynamics, which might therefore be described by a state function for the nuclear spin system and a hamiltonian. The classical harmonic oscillator is described by the hamiltonian function 8. Because historical episodes allow diverse interpretations, many conclusions of macroeconomics are not coercive. Having established that, i am bound to say that i have not been able to think of a problem in classical mechanics that i can solve more easily by hamiltonian methods than by newtonian or lagrangian methods. Hamiltonian mechanics and mathematics this is a wikipedia book, a collection of wikipedia articles that can be easily saved, imported by an external electronic. Ramsey, with significant extensions by david cass and tjalling koopmans. They are strongly connected to the possibility of describing a nonholonomic macroeconomic system via a gibbspfaff equation or by four associated vector fields, based on a contact structure of.
We want to get all of the binding constraints attached to multipliers. Saddle points of hamiltonian trajectories in mathematical economics. Most modern dynamic models of macroeconomics build on the framework described in solows 1956 paper. In other macroeconomic topics, such as monetary economics, labor. The hamilton equations have an elegant symmetry that the lagrange equations lack.
Hamiltonian walk in graph g is a walk that passes througheachvertexexactlyonce. An introduction to mathematical optimal control theory version 0. Finally it is shown how the pontryagins principle fits very well to the theory of hamiltonian systems. In economics, the objective function in dynamic optimization problems often depends directly on time only through. An aggregate is a multitude of economic subjects that share some common features.
Example 1 conservation of the total energy for hamiltonian systems 1 the hamiltonian function hp,q is a. Hamiltonian function an overview sciencedirect topics. From wikibooks, open books for an open world function. The heisenbergdiracvan vleck hamiltonian of molecular magnetism. Other readers will always be interested in your opinion of the books youve read. Diracs theorem if g is a simple graph with n vertices, where n. The hamiltonian approach to dynamic economics economic theory and mathematical economics and a great selection of related books, art and collectibles available now at.
Note that we say a hamiltonian function because we can always add a constant to any hamiltonian function without changing the equations. Introduction to hamiltonian dynamics in economics sciencedirect. Recent advances in applied mathematics proceedings of the american conference on applied mathematics americanmath 10 harvard university, cambridge, usa january 2729, 2010 mathematics and computers in science and engineering a series of reference books and textbooks published by wseas press. 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 rst order necessary condition in optimal control theory is known as the maximum principle, which was named by l. Macroeconomics became popular after great depression of 1929 33. Answering only two questions or providing a cursory answer to the third question will greatly reduce your chances of passing the exam. 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. Hamiltonian as a function of generalized co ordinates and momen ta. Request pdf the pontryagin maximum principle and optimal economic growth. The point of departure circular relation between aggregate income and aggregate expenditure and two alternative approaches to macroeconomics supply side approach and demand based approach. Hamiltonian walk in graph g is a walk that passes through each vertex exactly once. 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.
Refer to f as the instantaneous objective function or the payoff function and to g as. Introduction to dynamic programming applied to economics. By contrast, microeconomics treats economic processes that concern individuals. If a graph has a hamiltonian walk, it is called a semihamiltoniangraph. An example of a function satisfying these assumptions, and that will be used repeat. The system is described by a function of q and p known as the hamiltonian, hq,p. Extra resources for construction of mappings for hamiltonian systems and their applications pd sample text this property will allow us to turn the difference equation of a digital filter into an algebraic equation and then determine the mathematical description of a digital filter, called its transfer function. Introduction to lagrangian and hamiltonian mechanics. A brief overview of the history of macroeconomics i classics smith, ricardo, marx did not have a sharp distinction between micro and macro. For help with downloading a wikipedia page as a pdf, see help. The hamiltonian formalism well now move onto the next level in the formalism of classical mechanics, due initially to hamilton around 1830. Partial differential equation models in macroeconomics.
The pontryagin maximum principle and optimal economic growth. This model was set up to study a closed economy, and we will assume that there is a constant population. The ramseycasskoopmans model, or ramsey growth model, is a neoclassical model of economic growth based primarily on the work of frank p. In addition we will derive a cookbookstyle recipe of how to solve the optimisation problems you will face in the macropart of your economic theory. Microeconomics includes those concepts that deal with smaller components of the economy. Roussel october 25, 2005 1 introduction todays notes will deviate somewhat from the main line of lectures to introduce an important class of dynamical systems which were. Just about all articles prior to 1989 are available via the internet at the. Alagrangian function for which the legendre transformation is applicable is said to be. Examples are domestic and international in their subject matter and are of the modern era. These problems can generally be posed as hamiltonian systems, whether dynamical systems on finite dimensional phase space as in classical mechanics, or partial. Growth model, dynamic optimization in discrete time. Economic growth current and present value hamiltonian september 27, 2005 to derive the formulae for the current. There is an alternative representation of static technological opportunities that is more congenial to dynamic analysis, that is, the representation of the technology by its hamiltonian function. We will focus on the bellman approach and develop the hamiltonian in both a deterministic and stochastic setting.
The notes here heavily borrow from stokey, lucas and prescott 1989, but simplify the exposition. Keynesian economics, both of which will eventually be in the bookstore. In the labor market, households provide labor and receive payment from firms through wages, salaries, and benefits. Macroeconomics qualifying examination august 2010 department of economics unc chapel hill instructions. All these topics are complementary to what is dealt with in existing books on fractional calculus and its applications. Classically, if one starts from a point q, p in the phase space at an initial instant of time, then subsequently q and p vary. 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. Macroeconomics ii course description this course contains a basic set of tools and models in dynamic macroeconomics, required to either take up more advanced courses in macroeconomics or to follow the standard contemporary literature in otherareasofeconomics.
Hamiltonian cycle in graph g is a cycle that passes througheachvertexexactlyonce. Fertile area of research, excellent dissertation topics. Ecn 275375 natural resource and environmental economics. A viscosity solution of 5 is a continuous function vsuch that the following hold. This is a summary of some basic mathematics for handling constrained optimiza. Aspects of each topic are treated in more detail by making use of advanced mathematics and setting them in a broader context than is the case in standard undergraduate text books. Lecture notes in macroeconomics university of houston. It only describes a few lowlying multiplets of the complete spectrum. Can rewrite the necessary condition in terms of hamiltonians as follows. There are two general formalisms used in classical mechanics to derive the classical equations of motion.
Notes for macroeconomics ii, ec 607 university of michigan. The history, principles, practice, and results of the hamiltonian system. Ct allows for simplifying the current value hamiltonian and eliminate emissions, mt. This chapter argues that hamiltonian mechanics is a much better base from which to build more advanced methods. These methods can be applied in finance, to model tickbytick logprice fluctuations, in insurance theory, to study ruin, as well as in macroeconomics as prototypical growth models. The function h is called the hamiltonian function for the system. In fact, the two early discoveries which led to the existence of graphs arose from puzzles, namely, the konigsberg bridge. With few exceptions, the articles are also summarized in romer or blanchard and fischer. Solving this using the first order conditions for a hamiltonian which, after some math, yields us a set of differential equations for dkt and. Principles of macroeconomics open textbook library. Introduction to hamiltonian mechanics oxford scholarship. Ifagraphhasahamiltoniancycle,itiscalleda hamiltoniangraph. Suppose that the position of a mechanical system with d.
Lecture notes for macroeconomics i, 2004 yale economic. Over 2 million books and magazines booksee download books for free. The hamiltonian is a function used to solve a problem of optimal control for a dynamical system. Keynes wrote the book general theory of employment, interest and money in 1936. We introduce the concept of hamiltonian potential function for noncooperative openloop differential games with n players, n controls and n states, and characterise a sufficient condition for its existence.
394 784 941 20 1542 881 1469 1393 647 866 702 964 1522 62 164 28 245 580 755 559 690 1534 1092 156 1257 1334 1154 709 741 438 94 1037 539 441 977 335 606 1217 982 1468 969 135 627