Envelope theorems in dynamic programming envelope theorems in dynamic programming zhao, fuan. Dennis cook and xin zhangy march 18, 2014 abstract envelopes were recently proposed as methods for reducing estimative variation in multivariate linear regression. The envelope theorem in dynamic optimization monash. The envelope theorem in dynamic optimization, journal of economic dynamics and control, elsevier, vol. The style of presentation, with its continual emphasis on the economic interpretation of mathematics and models, distinguishes. It is a fundamental result in the calculus of variations and is therefore often used in large deviations research. Envelope theory for constrained optimization lecture notes, econ 210a, ucsb, fall 20 envelope theory shows us how to deal with the interplay of direct and indirect e ects of parameters in a constrained maximization or minimization problem. It is aimed at firstyear and secondyear phd students in economics, agricultural and resource economics, operations research, management science, and applied mathematics. This can be found for example in the convex optimization book of boyd and vandenberghe. Envelope theorem is a general parameterized constrained maximization problem of the form. We consider recursive preferences and dispense with interiority assumptions. Effect of a parameter change on the maximized value. Completing the one line proof of the dynamic envelope theorem. Tucker theorem and the maximum principle in both discrete and continuous time.
Accordingly, motivated and economically revealing proofs of the transversality conditions come about by use of the dynamic envelope theorem. Request pdf an envelope theorem and some applications to discounted markov decision processes in this paper, an envelope theorem et will be established for optimization problems on euclidean. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. Leonardo felli 23 october, 2002 microeconomics ii lecture 3 constrained envelope theorem consider the problem. We show in this paper that the class of lipschitz functions provides a suitable framework for the generalization of classical envelope theorems for a broad class of constrained programs relevant to economic models, in which nonconvexities play a key role, and where the primitives may not be continuously differentiable. Clausen and strub 2010 show that the property is sufficient for the. This approach is aided dramatically by introducing the dynamic envelope theorem and the method of comparative dynamics early in the exposition. From what i understand, the intent of the envelope theorem is to make a shortcut from indirect utility to the expenditure function. In this paper we apply the envelope theorem to dynamic programming, in particular to resource allocation and inverse problems.
Journal of economic dynamics and control 15 1991 355385. In this case, we can apply a version of the envelope theorem. We postulate some sufficient conditions stemming from the static optimization theory. This can be found for example in the convex optimization book of boyd and vandenberghe see chapter 3. Lecture notes on dynamic programming economics 200e, professor bergin, spring 1998 adapted from lecture notes of kevin salyer and from stokey, lucas and prescott 1989 outline 1 a typical problem 2 a deterministic finite horizon problem 2. Envelope theorem in dynamic economic models with recursive. An introduction to dynamic programming jin cao macroeconomics research, ws1011 november, 2010. Consumer theory and the envelope theorem 1 utility maximization problem the consumer problem looked at here involves two goods.
Convex envelopes for quadratic and polynomial functions over. Envelope theorem in static optimization problem consider the optimization problem max fx. Course emphasizes methodological techniques and illustrates them through applications. Foundations of dynamic economic analysis presents a modern and thorough exposition of the fundamental mathematical formalism used to study optimal control theory, i. N2 the dynamic envelope theorem is presented for optimal control problems with nondifferential constraints. That being said, what you can do once you have obtained your indirect utility, is derive it in terms of i income and this will give you the expenditure function hicksian demand. Application of envelope theorem in dynamic programming saed alizamir. Is optimization a ridiculous model of human behavior. On the other hand, dynamic programing, unlike the kuhntucker theorem and the maximum principle, can be used quite easily to solve problems in which. Chapter 8 discrete time continuous state dynamic models.
Northholland the envelope theorem in dynamic optimization jeffrey t. The relevant point of the lemma is the differentiability of the felicity functional with respect to present and future assets. Foundations of dynamic economic analysis by michael r. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has. Microeconomics ii lecture 3 constrained envelope theorem. Please contact us if you think this content is not open access according to the boai definition. Lecture notes on dynamic programming economics 200e, professor bergin, spring 1998.
The most basic form of the envelope theorem concerns maximizing a su ciently smooth function fx. Optimal control theory and static optimization in economics kindle edition by leonard, daniel, long, ngo van. The existing literature is full of them and the reason is that most families of optimal value functions can produce them. Completing the one line proof of the dynamic envelope. The envelope theorem, euler and bellman equations, without.
Envelope theorems for nonsmooth and nonconcave optimization. Lafrance montana state university, bozeman, mt 59717, usa l. Dynamic programming university of texas at san antonio. Consumers maximize utility ux,y which is increasing in both arguments and quasiconcave in x,y. Foundations of dynamic economic analysis presents an introductory but thorough exposition of optimal control theory. An introduction to dynamic optimization optimal control and dynamic programming agec 642 2020 i.
The tree below provides a nice general representation of the range of optimization problems that. This is the reason why neoclassical economics, which assumes optimizing behaviour, has been the most successful of social sciences. Lecture 7 envelope theorems, bordered hessians and kuhntucker conditions eivind eriksen bi norwegian school of management department of economics october 15, 2010 eivind eriksen bi dept of economics lecture 7 october 15, 2010 1 20. Envelope theorems in dynamic programming springerlink. The envelope theorem allows us to reduce the secondorder difference equation system of euler equations to a. This chapter may be used for a course in static optimization. While the maximum principle lends itself equally well to dynamic optimization problems set in both discrete time and continuous time, dynamic programming is easiest to apply in discrete time settings. While we are not going to have time to go through all the necessary proofs along the way, i will attempt to point you in the direction of more detailed source material for the parts that we do not cover. We show that for general quadratic functions the computation can be carried on through a copositive problem, but in some cases including all the twodimensional ones we can solve a semide. Some of these constraints may switch from binding to nonbinding, or vice versa, along the optimal path. Solution methods for microeconomic dynamic stochastic.
Unconstrained and constrained optimizations the envelope theorem comparative statics static and dynamic optimization most problems in econ 200201 are static optimization that is we are interested in minimizing or maximizing an objective i. This paper studies how envelope theorems have been used in economics, their history and also who first introduced them. In addition, these same two results provide foundations for the work on the maximum principle and dynamic programming that we. These optimal values of the choice variables are, in turn, functions of the exogenous variables and parameters of the problem. Envelope theorem, euler, and bellman equations without. The envelope theorem is explained in terms of shepherds lemma. The envelope theorem in dynamic optimization sciencedirect. We establish an envelope theorem in concave dynamic problems. As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in the objective function. Mar 05, 2020 note the parallel between this trick and the fundamental insight of dynamic programming. One of its important aspects is the envelope property discussed in this appendix. Envelope theorems in dynamic programming, annals of.
Appendix iii the envelope property optimization imposes a very strong structure on the problem considered. Some of these constraints may switch from binding to nonbinding, or vice versa, along the. Review static optimization unconstrained and constrained. Carroll department of economics, the johns hopkins university, baltimore md, 212182685, usa. The dynamic envelope theorem is presented for optimal control problems with nondifferential constraints. Use features like bookmarks, note taking and highlighting while reading optimal control theory and static optimization in economics. This paper investigates the firstorder differentiability properties of the value function in dynamic economic models with recursive preferences where the optimal policy may lie at the boundary of the feasible set under several regular assumptions originating from the static optimization theory plus an additional asymptotic condition. Envelope theorem, euler, and bellman equations without differentiability ramon marimon y jan werner z february 15, 2015 abstract we extend the envelope theorem, the euler equation, and the bellman equation to dynamic constrained optimization problems where binding constraints can give rise to nondifferentiable value functions. We illustrate this here for the linearquadratic control problem, the resource allocation problem.
Mathematical optimization alternatively spelt optimisation or mathematical programming is the selection of a best element with regard to some criterion from some set of available alternatives. Dynamic programming techniques transform a multiperiod or in. Business e number research eulers numbers mathematical optimization models optimization theory. Now the problem turns out to be a oneshot optimization problem, given the transition equation. Now the problem turns out to be a oneshot optimization problem. I provide both simple purely mathematical and economic examples of each to illustrate how they are used. The maximum principle peter ireland econ 772001 math for economists boston college, department of economics fall 2019 here, we will explore the connections between two ways of solving dynamic optimization problems, that is, problems that involve optimization over time.
Envelope theorem kevin wainwright mar 22, 2004 1 maximum value functions a maximum or minimum value function is an objective function where the choice variables have been assigned their optimal values. An envelope theorem for dynamic programming consider a dynamic programming dp problem. The results are then demonstrated on the onesector growth model. This chapter provides an introduction to the theory of discrete time continuous state dynamic. Ramon marimon jan werner april, 2017 abstract we extend the envelope theorem, the euler equation, and the bellman equation to dynamic constrained optimization problems where binding constraints can give rise to non. Macro theory b dynamic programming ofer setty the eitan berglas school of economics tel aviv university march 12, 2015 1 dynamic optimization with nite horizon the economy has a social planner whose horizon is nite, t. Download it once and read it on your kindle device, pc, phones or tablets. A dynamic consumption problem like the static envelope theorem, the dynamic envelope theorem can be applied to a variety of problems. Estimation of an envelope usually involves optimization over grassmann manifolds. Lecture 7 envelope theorems, bordered hessians and kuhn. This chapter provides an introduction to the theory of discrete time continuous state dynamic economic models. The bellman equation and an associated lagrangian e.
We extend the envelope theorem, the euler equation, and the bellman equation to dynamic constrained optimization problems where binding constraints can give rise to nondifferentiable value functions. The method of endogenous gridpoints for solving dynamic stochastic optimization problems christopher d. Chapter 4 introduction to dynamic programming an approach to solving dynamic optimization problems alternative to optimal control was pioneered by richard bellman beginning in the late 1950s. The envelope theorem, euler and bellman equations, without differentiability. The envelope theorem is a statement about derivatives along an optimal trajectory. The envelope theorem is a result about the differentiability properties of the objective function of a parameterized optimization problem. In this section the dynamic envelope theorem is applied to a problem of a consumer maximizing discounted utility from consumption subject to a lifetime budget constraint. Find materials for this course in the pages linked along the left. Whether cast in optimization or equilibrium form, most discrete time continuous state dynamic economic models pose in.
An introduction to dynamic optimization optimal control. Dynamic envelope theorems in optimal control can, for example, be found in lafrance and barney 1991 and the most general results known to the authors appeared in milgrom and segal 2002. Maximum value functions and the envelope theorem a maximum or minimum value function is an objective function where the choice variables have been assigned their optimal values. Application of envelope theorem in dynamic programming. Constrained optimization discusses the kuhntucker algorithm, the implicit function theorem, and the envelope theorem. We propose a fast and widely applicable onedimensional 1d al. The method of endogenous gridpoints for solving dynamic. We illustrate this here for the linearquadratic control problem, the resource allocation problem, and the inverse problem of dynamic programming. In dynamic programming the envelope theorem can be used to characterize and compute the optimal value function from its derivatives. Overview of optimization optimization is a unifying paradigm in most economic analysis. An envelope theorem and some applications to discounted. We give sufficient conditions for the value function of a lipschitz program. Dwayne barney boise state university, boise, id 83725, usa received november 1988, final version received march 1990 the dynamic envelope theorem is presented for optimal control problems with. Dynamic optimization, precautionary saving, stochastic growth model, en.
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