Sep 23, 20 a brief introduction to the hamiltonian and its relationship to the lagrangian. The value function of the generic optimal control problem satis es the hamiltonjacobibellman equation. Variational arguments variational arguments iv di culty lies in two features. The latter implies the diminishing marginal productivity of capital. Daron acemoglu mit economic growth lectures 6 and 7 november 15 and 17, 2011.
The hamiltonian is a function used to solve a problem of optimal control for a dynamical system. If you cannot derive the mathematical expressions, you are still encouraged to give the economic intuition and comment. Notice the objective function is now discounted in the hamiltonian, whereas before it was not. To see that the hamiltonian must be weakly convex, note that for xed x. The book begins by applying lagranges equations to a number of mechanical systems. To solve these problems we usually set up the hamiltonian function. Inspired by, but distinct from, the hamiltonian of classical mechanics, the hamiltonian of optimal. The initial and terminal conditions on k t pin then do wn the optimal paths. Chapter 2 optimal control optimal control is the standard method for solving dynamic optimization problems, when those problems are expressed in continuous time. Macroeconomicsintroduction wikibooks, open books for an. Yrather than a vector or a nite dimensional object.
Macroeconomics, on the other hand, is the field of economics that studies the behavior of the economy as a whole and not just on specific companies, but entire industries and economies. The hamiltonian approach to dynamic economics 1st edition. General formulation consider the general optimal control problem two slides back. First that we should try to express the state of the mechanical system using the minimum representation possible and which re ects the fact that the physics of the problem is coordinateinvariant. Local theory 02102011 6 hamiltonian and langrangian ormfulations 6. The text is long but answers should be quite short. It was developed by inter alia a bunch of russian mathematicians among whom the central character was pontryagin. As in physical modelling, for these systems, the dynamics is the result of the interaction. Macroeconomicsoptimal growth wikibooks, open books for an. The present value hamiltonian evaluates the optimal behavior in terms of period 0 value units, that is, at the time the optimization is actually done. 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. Notes for macroeconomics ii, ec 607 university of michigan. Oct 06, 2014 introducing lagrangian and hamiltonian mechanics is quite confusing because the functions that are involved i.
This paper aims at extending the porthamiltonian formalism to a simple class of macroeconomic systems. Consider the dynamic optimization problem max z 1 0 e rtfxt. This looks at economywide phenomena such as gross domestic product gdp and how it is affected by changes in unemployment, national income, rate of growth. The notes here heavily borrow from stokey, lucas and prescott 1989, but simplify the exposition. These reports outlined a coherent program of national mercantilism governmentassisted economic development. A students guide to lagrangians and hamiltonians a concise but rigorous treatment of variational techniques, focusing primarily on lagrangian and hamiltonian systems, this book is ideal for physics, engineering and mathematics students. Note that any old function wont solve the bellman equation. To evaluate this condition for the integral functional jabove, we rst com. This is an article about hamiltonian mechanics, which is a branch of physics and not related to economics in any way. Jeffersonian thought is the main competitor of the hamiltonian school, especially since both men were actually alive during the same time. I think that hamiltonian function shouldnt redirect tho hamiltonian mechanics, so other areas would only look for the use of hamiltonian function.
Econ 4925 resource economics lecture note 1b, michael hoel. 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. Inspired by, but distinct from, the hamiltonian of classical mechanics, the hamiltonian of optimal control theory was developed by lev pontryagin as. In macroeconomics we look at both the long run and short run changes in the economy. In a classical growth model, it represents the utility value of having one extra unit of capital. Sastry revised march 29th there exist two main approaches to optimal control and dynamic games. An introduction to lagrangian and hamiltonian mechanics. 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 production function in the economy is nondecreasing. Thus, in his paper in this volume, relating properties of the set t to properties of the function hp, q, k, 1, lau quite properly refers to the hamiltonian function as the restricted profit function. Using dynamic constrain t, simplify those rst order conditions. Introducing lagrangian and hamiltonian mechanics is quite confusing because the functions that are involved i. That is, the rule specifies the choice as a function of the current state of the economy.
About the hamiltonian function, it is the same thing as the hamiltonian, and hence it has to redirect here or if hamiltonian function can mean more things, a. On the other hand, macroeconomics is the study of an entire economy. Emphasis is placed on the stability of longrun steadystate equilibrium in models of heterogeneous capital accumulation. This is a summary of some basic mathematics for handling constrained optimization problems. The optimal control problem is solved using a hamiltonian that reads. Mod01 lec35 hamiltonian formulation for solution of. The function u ux is the candidate extremizing function and the functions x represent perturbations from u ux which are parameterized by the real variable. The function h is called the hamiltonian function for the system. Its original prescription rested on two principles.
Econ 4925 resource economics lecture note 1b, michael hoel optimal control theory this note gives a brief, nonrigorous sketch of basic optimal control theory, which is a useful tool in several simple economic problems, such as those in resource economics. You will see the corresponding planners problem later with david romer. 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. Of all the schools, american foreign policy stays the most consistent w hamiltonian thought. This paper aims at extending the port hamiltonian formalism to a simple class of macroeconomic systems. Let fk be the output rate of the economy when k is the capital stock. This payoff function depends, at each instant of time, on the value of the control variable c t the variable that the planner can directly control, for example. H a mathematical function that can be used to generate the equations of motion of a dynamic system, equal for many such systems to the sum of the.
Hamiltonian definition of hamiltonian by the free dictionary. 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. But are not straightforward to check since neither concavity nor convexity of the g function would guarantee the concavity of the. If im not mistaken necessary conditions are sufficient if maximized hamiltonian is concave in state arrow sufficiency if i recall correctly see e. 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.
Example the hamiltonian often has a physical meaning for the sysem of odes that is modelling a. Pdf porthamiltonian formulation of simple macroeconomic. How would you define a hamiltonian in laymans terms. It looks at aggregate inputs and outputs with the goal to understand the flows of total consumption, employment rates, wages, prices and many other aggregate parts of an economy. This chapter presents an introduction to hamiltonian dynamics in economics. Extending the model in this direction achieves three purposes. Master in economics advanced international macroeconomics homework 1 due at lecture 6 march 16th. We can do this at least in theory by inverting the definition of p, namely p.
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. Macroeconomicsoptimal growth wikibooks, open books for. Now we have an expression determining the value of the choice variable, xt, and an. 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. 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. Lectures 6 and 7, neoclassical growth daron acemoglu mit november 15 and 17, 2011. Thus, all trajectories are needed to evaluate the quantum propagator. Optimal control theory derives the optimality conditions of the problem. Our goal in general will be to solve for such a function, called a policy function. The scheme is lagrangian and hamiltonian mechanics. The present aluev hamiltonian would be formulated this way. Keynes, the general theory of employment, interest, and money 1936. Newtonian, lagrangian and hamiltonian mechanics reading. A brief overview of the history of macroeconomics i classics smith, ricardo, marx did not have a sharp distinction between micro and macro.
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