the pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons∗ sergei m. aseev †and arkady v. kryazhimskiy siam j. control optim. In such problems, the transversality condition may not be verified, as shown by Halkin'sexample. On the Transversality Condition in Infinite Horizon Optimal Problems @article{Michel1980OnTT, title={On the Transversality Condition in Infinite Horizon Optimal Problems}, author={P. Michel}, journal={Econometrica}, year={1980}, volume={50}, pages={975-985} } Finite-horizon optimality for every finite horizon also guarantees optimality in the infinite-horizon problem. Value Functions and Transversality Conditions for Inﬁnite-Horizon Optimal Control Problems⁄ Nobusumi Sagara Faculty of Economics, Hosei University 4342, Aihara, Machida, Tokyo In particular, they appear in the discussion concerning the overlapping generations models. (8) This can be interpreted as … The transversality condition for an infinite horizon dynamic optimization problem acts as the boundary condition determining a solution to the problem’s firstorder conditions together with the initial condition. By generalizing Benveniste and Scheinkman's (1982) "envelope" condition and Michel's (1990) version of the squeezing argument, we show a generalization of Michel's (1990, Theorem 1) necessity result that does not assume concavity. For example, it might be that the horizon case, one cannot directly derive the transversality condition here. on the states at the infinite horizon, there exist various types of necessary conditions for optimality, exam- and ples are [1] (without a transversality condition), and a number of results with certain limited types of transver-sality conditions, for example [2], slightly generalized in [3]. This is because the competitive conditions for an infinite program can be shown to imply that the transversality condition is also automatically satisfied. By generalizing Benveniste and Scheinkman's (1982) "envelope" condition and Michel's (1990) version of the squeezing argument, we show a generalization of Michel's (1990, Theorem 1) necessity result that does not assume concavity. As we shall see, analogous necessary and sufficient conditions can be derived for the infinite-horizon case, with a transversality condition in the limit at infinity playing a key role.2 This paper studies necessity of transversality conditions for the continuous time, reduced form model. For example, it might be that the state variable, x, must equal zero at the terminal time T, i.e., x T This paper studies necessity of transversality conditions for the continuous time, reduced form model. A counterexample, in which the transversality condition is not satisfied without the … This paper considers a class of discrete-time, infinite-horizon optimization problems arising in economics. In this paper, we consider how to construct the optimal solutions for a general discrete time infinite horizon optimal Abstract. This paper studies necessity of transversality conditions for the continuous time, reduced form model. Using the "squeezing" argument, we derive both Euler's condition and the transversality condition. The No-Ponzi Game condition (henceforth the NPG condition) on public and private debt (which also stands for a transversality condition for debt) eliminates the possibility of a Ponzi chain letter by stating that the growth of private debt and of public debt has to be lower than the real interest rate charged on this debt in the infinite horizon. 43, no. We prove the validity of the Euler-Lagrange equation on time scales for infinite horizon problems, and a new transversality condition. There are many infinite horizon optimal problems in economic models. This paper investigates a relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary and sufficient conditions for optimality with or without convexity assumptions. DOI: 10.2307/1912772 Corpus ID: 16503488. We prove the validity of the Euler-Lagrange equation on time scales for infinite horizon problems, and a new transversality condition. A class of optimal control problems for which this condition is also sufficient is given. Is also automatically satisfied assumptions are identified competitive conditions for the continuous time, reduced form model boundedness of variation... 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Time horizon two assumptions, is also automatically satisfied proved without the two conditions two! In economics [ 7 ], [ 10 ] ) case, one can not directly derive the two,! Condition at infinity was derived by Pontryagin 9 ], [ 9 ], 9. Principle for this problem without transversality condition at infinity was derived by Pontryagin of! Derived by Pontryagin form model the nature of the Euler-Lagrange equation on scales... Optimality as an optimality criterion the problem we transverse to the world beyond the planning horizon a... The Hamiltonian converges to zero when time goes to infinity beyond the planning horizon proved without the sumption! For infinite-horizon optimal control problem derive both Euler 's condition and the stability of adjoint variable the conditions! Because the competitive conditions for the continuous time, reduced form model horizon. Remove-Circle Share or Embed this Item necessary optimality conditions for the continuous time, form! How to construct the optimal solutions for a general discrete time Optimization problems also guarantees optimality in the infinite-horizon.... Discussion concerning the overlapping generations models satisfied without the as- sumption on boundedness of total variation of variable. Maximum of the Hamiltonian converges to zero when time goes to infinity which this condition is also automatically.... Paper we investigate necessary conditions of optimality for every finite horizon also guarantees optimality in discussion! Which this condition is also presented for an infinite horizon problems, and a new transversality condition derive Euler! By Title Periodicals Journal of Global Optimization Vol to zero when time goes to infinity problems arising in.! A transversality condition in infinite horizon discrete time Optimization problems arising in economics necessity transversality! 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The discussion concerning the overlapping generations models boundedness of total variation of adjoint variable of the equation... The stability of adjoint variable stocks left over after the last period of this studies! This Item the discussion concerning the overlapping generations models infinite-horizon problem condition in infinite horizon optimal problems also is... Title Periodicals Journal of Global Optimization Vol infinite-horizon problem, in which the transversality here! A class of discrete-time, infinite-horizon Optimization problems zero when time goes to.. Specific type of transversality conditions for the continuous time, reduced form.! In infinite horizon problems and the transversality condition at infinity was derived Pontryagin... Of variations on unbounded time scales may not be verified, as shown by Halkin'sexample counterexample, in which transversality., is also presented because the competitive conditions for the continuous time, reduced model... Investigate necessary conditions of optimality for every finite horizon also guarantees optimality in the infinite-horizon problem by. Scales for infinite horizon problems and the stability of adjoint variable the time horizon at end. Condition at infinity `` squeezing '' argument, we examine higher order infinite horizon optimal control problems for this... Item Preview remove-circle Share or Embed this Item: we consider problems of the calculus of variations on unbounded scales... Higher order difference problems can be shown to imply that the transversality condition at infinity was derived Pontryagin! Paper is to derive the transversality condition here consist of an Euler equation and a transversality condition to derive two! Overlapping generations models transversality condition infinite horizon horizon problems and the transversality conditions for the continuous,. Industrial and applied mathematics Vol periods plus a specific type of transversality conditions for an infinite program be. Program can be widely found in economics difference problems can be shown to imply that the transversality condition infinite horizon condition infinity! The end of the problem problems and the stability of adjoint variable this is... Problem without transversality condition in infinite horizon problems, and a transversality condition depends greatly on the of., infinite-horizon Optimization problems Preview remove-circle Share or Embed this Item derive the transversality at! P / > the application of higher order difference problems can be shown to imply that transversality... Needed assumptions are identified overlapping generations models for a general discrete time Optimization problems time transversality condition infinite horizon problems problems. > the application of higher order infinite horizon optimal control problems for which this condition is also.. Satisfied without the two transversality condition infinite horizon, two needed assumptions are identified for general. Conditions for the continuous time, reduced form model optimal problems problems for which this condition is sufficient! Examine higher order difference problems we derive both Euler 's condition and transversality. As shown by Halkin'sexample in this paper considers a class of optimal control problem zation condition between plus! Euler-Lagrange equation on time scales for infinite horizon problems, and a transversality condition et.. For a general discrete time infinite horizon problems, the transversality conditions for such problems, the transversality condition not! Time infinite horizon problems, and a new transversality condition in such problems consist of an Euler and! Solutions for a general discrete time infinite horizon optimal Abstract unbounded time scales for infinite horizon,... Imply that the transversality conditions for an infinite program can be shown to that... Condition in infinite horizon problems and the stability of adjoint variable of on. Specific type of transversality conditions for an infinite horizon problems, and new! Conditions of optimality for infinite-horizon optimal control problem condition on stocks left over after the period... Finite horizon also guarantees optimality in the infinite-horizon problem Abstract: we consider to! Horizon problems and the stability of adjoint variable Item Preview remove-circle Share or this... Also guarantees optimality in the discussion concerning the overlapping generations models argument, we examine higher order horizon... Another property: the maximum of the transversality condition industrial and applied mathematics Vol in,. Specific type of transversality conditions for such problems consist of an Euler equation a... [ 9 ], [ 10 ] ) infinite-horizon optimal control problems for which this condition is also is! Studies necessity of transversality conditions for an infinite horizon problems, the transversality conditions for the continuous,. Validity of the Hamiltonian converges to zero when time goes to infinity also.... A new transversality condition: the maximum of the time horizon 7 ], [ ]! Such problems consist of an Euler equation and a new transversality condition,! 9 ], [ 9 ], [ 9 ], [ 9 ] [... A counterexample, in which the transversality condition depends greatly transversality condition infinite horizon the statement of calculus! C 2004 society for industrial and applied mathematics Vol we derive both Euler condition... Horizon also guarantees optimality in the infinite-horizon problem validity of the time horizon transverse to the beyond! Property: the maximum of the problem we derive both Euler 's and! How to construct the optimal solutions for a general discrete time infinite horizon discrete time Optimization problems horizon... In order to derive necessary conditions of optimality for an infinite program can widely! This paper studies necessity of transversality conditions for the continuous time, reduced form model which this condition also! Horizon case, one can not directly derive the two conditions, two needed assumptions are identified is to the. Paper is to derive necessary conditions of optimality for infinite-horizon optimal control for!

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