The importance of transversality conditions more on the ramsey model there has been some fun and interesting discussion of the ramseycasskoopmans rck model on this post of mine. Dynamic optimization and optimal control columbia university. Without relying on dynamic programming, we directly prove the necessity of the transversality condition using only an elementary perturbation argument. This condition, called transversality condition tvc, is very reminiscent of the. Recall the general setup of an optimal control model we take the casskoopmans growth model as an example. Put di erently, if the rm has twice as much capital, it will want twice as much labor. It is shown that, when the discount rate is sufficiently large, the problem admits normal multipliers and a strong transversality condition holds. Sastry revised march 29th there exist two main approaches to optimal control and dynamic games. Necessity of the transversality condition for stochastic models with bounded or crra utility. There exist two main approaches to optimal control and dynamic games.
The rst order necessary condition in optimal control theory is known as the maximum principle, which was named by l. It writes the value of a decision problem at a certain point in time in terms of the payoff from some initial choices and the value of the remaining decision problem that results from those initial choices. Necessity of transversality conditions for stochastic problems. Lectures notes on deterministic dynamic programming. The optimal control is one of the possible controllers for a dynamic system, having a linear quadratic regulator and using the pontryagins principle or the dynamic programming method. Transversality even though the programming problem is concave, the. This paper shows stochastic versions of i michels 1990, econometrica 58, 705723, theorem 1 necessity result, ii a generalization of the tvc results of weitzman 1973, manage. We now change the problem described above in the following way. Transversality condition ensures individual would never want to have positive wealth asymptotically, so noponzigame condition can be strengthened to though not necessary in general. There is an appendix on measuretheoretic questions that arise in dynamic programming. We assume throughout that time is discrete, since it leads to simpler and more intuitive mathematics.
Dynamic programming computer science and engineering. Optimal control requires the weakest assumptions and can, therefore, be used to deal with the most general problems. I lets put the income process back into the problem. Sorry to those in the discussion that i havent gotten back to the comments yet ive been taking my time to think about whats been brought up. We are going to begin by illustrating recursive methods in the case of a. Then i will show how it is used for innite horizon problems. We shall stress applications and examples of all these techniques throughout the course. Growth model, dynamic optimization in discrete time. Section 3 introduces the euler equation and the transversality condition, and then explains their relationship. A simple proof of the necessity of the transversality condition. Jun 29, 2015 the importance of transversality conditions more on the ramsey model there has been some fun and interesting discussion of the ramseycasskoopmans rck model on this post of mine.
Transversality condition plays the role of the second condition. Dynamic programming involves taking an entirely di. Research supported in part by the national science foundation, under grant nsfdms0601774. This paper shows that the standard transversality condition stvc is necessary for optimality in stochastic models with bounded or constantrelativerisk aversion crra utility under fairly general conditions.
Lecture notes 7 2 assumecontinuoustradingandconsumption,thenbybellmansprinciple, 0 maxdriftofj. Both approaches involve converting an optimization over a function space to a pointwise. We then study the properties of the resulting dynamic systems. Rs ch 15 dynamic optimization summer 2019 6 11 the setup we used is one of the most common dynamic optimization problems in economics and finance.
It was developed by inter alia a bunch of russian mathematicians among whom the central character was pontryagin. The importance of transversality conditions more on the. To understand the form and the reason for the transversality condition, consider the. Here we explore the connections between these two characterizations. Rather than getting the full set of kuhntucker conditions and trying to solve t equations in t unknowns, we break the optimization problem up into a recursive sequence of optimization problems. The more general problem of bolza in which the final time is defined implicitly and in which the expression to be extremized is. There is an appendix on measure theory in dynamic programming. An euler equation is a local condition that no gain be achieved by slightly. Transversality conditions and dynamic economic behavior transversality conditions are optimality conditions often used along with euler equations to characterize the optimal paths plans, programs, trajectories, etc of dynamic economic models. On our way to deriving the transversality condition, we also provide a somewhat more formal derivation of conditions. The reason why we may need the transversality condition is that the firstorder conditions. In these notes, both approaches are discussed for optimal control. The necessity of the transversality condition at in nity.
Because of the in nite dimension of the optimization problem, we also have to consider the transversality condition. This paper shows that the standard transversality condition stvc is necessary for optimality in stochastic models with bounded or constantrelativerisk aversion crra utility under fairly. These notes represent an introduction to the theory of optimal control and dynamic games. Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. In this paper, it will be shown that the functional equation. At the end of time, you would never want to leave any utility on the table, so to speak, since there is no more tomorrow. It means that in net present value terms, the agent should not have capital left. While similar arguments are used in kamihigashi 2000a, 2000b, 2000c, these papers do not provide a direct proof of the necessity of the transversality condition. The tree of problemsubproblems which is of exponential size now condensed to a smaller, polynomialsize graph. Introduction to dynamic programming xin yi january 5, 2019 1. The transversality condition for an infinite horizon dynamic optimization problem is the boundary condition determining a solution to the problems firstorder conditions together with the. Dynamic programming and the calculus of variations. Kenichi tamegawa meiji university, school of commerce, 11 kandasurugadai, chiyodaku, tokyo 1018301, japan in an optimal consumption choice problem, in which households have assets yielding interest rates, it is dif.
The additional requirement that the second derivative of 3. Hence, the solution to the original dynamic optimization problem can be characterized by nding solutions to a system. The proof makes it clear that, contrary to common belief, the necessity of the transversality condition can be shown in a. Transversality condition in general, dynamic programming problems require two boundary conditions. Lecture notes for macroeconomics i, 2004 yale economic. For dynamic programming, the optimal curve remains optimal at intermediate points in time. This note provides a simple proof of the necessity of the transversality condition for the differentiable reducedform model. This model was set up to study a closed economy, and we will assume that there is a constant population. I will illustrate the approach using the nite horizon problem. A maximum principle in terms of differential inclusions with a michel type transversality condition is given.
Now i should introduce dynamic programming in more formal settings. The weierstrass condition even the legendre condition is not strong enough since. Dynamic programming is an approach to optimization that deals with these issues. Most modern dynamic models of macroeconomics build on the framework described in solows 1956 paper. A relationship between dynamic programming and the maximum principle is also given. It is quite strong because it requires the limit to converge to zero for any feasible policy. Consider the following maximum path sum i problem listed as problem 18 on website project euler. Douglas production technology, the rst order condition for the choice of labor, given the capital stock, wage rate, and level of tfp, is. A key aspect of such known examples is that utility is not bounded. Transversality conditions and dynamic economic behavior. Dynamic programming, optimality, thriftiness, equalization, euler equations, envelope equation, transversality condition. Fortunately, the transversality condition helps us here.
Nevertheless, there are a couple of examples where we know how. To ensure the equivalence of the sequential and recursive problem, we also need then a transversality condition. Uniqueness of the steady state, on the other hand, is more elusive, but is a necessary condition for global stability. It was developed by inter alia a bunch of russian mathematicians among whom the. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. 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.
We should emphasize that the transversality condition is a su cient condition, but not a necessary one. Irigoyen, claudio, esteban rossi hansberg, and mark l. This makes dynamic optimization a necessary part of the tools we need to cover, and the. In this lecture we focus primarily on condition 4, the transversality condition. Chapter 2 optimal control optimal control is the standard method for solving dynamic optimization problems, when those problems are expressed in continuous time.
848 1376 434 682 1207 1379 859 403 990 1292 670 1145 1264 308 1390 188 1240 191 1245 16 674 1440 1527 273 1342 754 1506 282 78 255 572 95 969 878 973 1154 244 1039 654