Abstract pdf 1178 kb 2012 a unified numerical scheme for linearquadratic optimal control problems with joint control and state constraints. We can compare the candidate path with any other feasible path as in the proof of theorem 2, however at the moment of integrating by parts 7 we are left. The book is completed by an extensive index which helps finding topics of interest very quickly. The index is normally the time, but can be a spatial parameter as well. In the model, there is a monopolist who sells a finite inventory over a finite time horizon. The role of risk aversion and intertemporal substitution in. Ive always been fascinated by the work of blackscholes and merton from the good old days. The intertemporal consumptionsaving problem in discrete and continuous time a bundle is reduced to one consumption good. The earliest work on the subject was by irving fisher and roy harrod, who described hump saving, hypothesizing that savings would be highest in the middle years of a persons life as they saved for retirement. Notes on intemporal optimization economics 205a fall. The role of risk aversion and intertemporal substitution in dynamic consumptionportfolio choice with recursive utility. Mertons widelyused text provides an overview and synthesis of finance theory from the perspective of continuous time analysis. The book treats deterministic and stochastic models, both in discrete and continuous time.
Solving the intertemporal consumptionsaving problem in discrete and continuous time in the next two chapters we shall discuss the continuoustime version of the basic representative agent model, the ramsey model, and some of its applications. Jeanphilippe garnier, kazuo nishimura, alain venditti. Time preferences, intertemporal optimization, and the permanent incomelife. The computation of intertemporal general equilibria therefore calls for timeaggregation assumptions. Most of the material for part 1 is based on the textbook applied intertemporal optimization by klaus walde. Chapter 8 discrete time continuous state dynamic models. C r, which we will assume is continuous and bounded in the consumption problem, this would just be the utility of consumption in a given period, but. Optimal time aggregation of infinite horizon control. We will study an optimization problems with the following features. In this model, the intertemporal tradeoff involves a choice between higher. Dynamic optimization and optimal control columbia university. Siam journal on control and optimization siam society for.
Whether cast in optimization or equilibrium form, most discrete time continuous state dynamic economic models pose in. Theoretically, by not consuming today, consumption. Intertemporal pricing with strategic customer behavior abstract this paper develops a model of dynamic pricing with endogenous intertemporal demand. Intertemporal choice is the process by which people make decisions about what and how much to do at various points in time, when choices at one time. The computation of intertemporal general equilibria therefore calls for time aggregation assumptions. This chapter provides an introduction to the theory of discrete.
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. The overarching objective of this chapter is to provide a better understanding of macro marketing optimization methods to interested marketing analysts and highlevel marketing decisionmakers. For the given values of y 1, y 2, r, and b, the families choose c 1 and c 2 to maximize the value of. Formulating and solving problems under continuous time uncertainty has never been explained in such a nontechnical and highly accessible way.
A criterion for time aggregation in intertemporal dynamic models. Following 1 and 2, time is continuous and denoted by t. An economic term describing how an individuals current decisions affect what options become available in the future. As a preparation for this, the present chapter gives an account. Given that discrete and continuous time problems are given equal attention, insights gained in one area can be used to learn solution methods more quickly in other contexts. Doing this in effect linearizes by taking the decision interval as infinitely small, so that the model becomes linear over this interval. Wilcoxen department of economics the university of texas at austin this is a reprint of impact preliminary working paper ip45 1989, which was written while the author was at the impact research centre at the university of melbourne. Pdf the fundamentals of intertemporal optimization in. Adobe acrobat reader for free links are current as of january 25, 2007. Acontinuousfunctioncanbenowheredifferentiabletake,forexample,abrownian motionsamplepath. The focus of this textbook is on learning through examples and gives a very quick access to all. Bavarian graduate program in economics, the universities of dortmund. The optimization software will deliver input values in a, the software module realizing f will deliver the computed value f x and, in some cases, additional. An euler equation is an intertemporal version of a firstorder condition characterizing an optimal.
The role of risk aversion and intertemporal substitution. Hamiltonian method and pontryagins maximum principle as a tool to analyze. Kletzer a primer on intertemporal optimization in continuous time a. On intertemporal optimization and dynamic efficiency uio. Anticipated shocks in continuoustime optimization models. At the same time, there are many problems in macro with uncertainty which are easy to formulate in continuous time. When it comes to stochastic methods in continuous time, the applied focus of this book is the most useful. The focus of this textbook is on learning through examples and gives a very quick access to all methods required by an undergraduate student, a phd student and an experienced researcher who wants to explore new. Now suppose that the consumers utility is timeseparable. Chapter 9 solving the intertemporal consumptionsaving. The basic structure of this book is simple to understand. In continuoustime optimization problems, the analogous equation is a partial differential equation that is usually called the hamiltonjacobibellman equation. The fundamentals of intertemporal optimization in the continuous time modelling of consumer behaviour. The euler equation is a central result in intertemporal optimization theory, and will be used again and again as the course progresses.
Full text of on intertemporal preferences in continuous time. Full text of on intertemporal preferences in continuous. Solving the intertemporal consumption saving problem in discrete and continuous time this is the actual utility rate of return, a kind of nominal interest rate. In this case, the program is more easily solved in. These policies are derived from intertemporal reoptimization each period for that period and future ones, with an unchanged intertemporal objective function, which is maximized subject to the longrun or multiperiod constraints specified by the structure of the economy. We analyze the intertemporal utility maximization problem under uncertainty for the. Continuous optimization nonlinear and linear programming stephen j. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. All wage income not consumed flows into the individual financial. An optimal solution to the program cp above is a triplet. Sloanschoolofmanagement onintertemporalpreferencesincontinuoustime thecaseofcertainty by chifuhuanganddavidkreps wp203788 july1988.
From discretetime to continuoustime we may treat the continuoustime case as a limiting case of the discretetime case as the time interval dt goes to 0. View notes notes on intemporal optimization from econ 205a at university of california, santa cruz. But this kind of linearity is only local, so it does not allow one easily to. Mertons widelyused text provides an overview and synthesis of finance theory from the perspective of continuoustime analysis. The fundamentals of intertemporal optimization in the. It covers individual finance choice, corporate finance, financial intermediation, capital markets, and selected topics on. Applied intertemporal optimization by klaus walde is a very very nice book, even for those who are not really familiar with mathematics. Let denote the probability of surviving until age t, which is a strictly positive and decreasing function. This note explains the fundamentals of intertemporal optimization for consumer behaviour modelled in continuous time when preferences are recursive as in uzawa and epstein. State space modeling permits intertemporal planning and optimization as well as myopic. To calculate the corresponding real interest rate let the nominal price of a consumption good be. The role of risk aversion and intertemporal substitution in dynamic consumptionportfolio choice with recursive utility 1introduction recursive utility functions kreps and porteus,1978. In this paper we address this problem by presenting flexible, efficient new software which can solve complex intertemporal models in a fraction of the time required by conventional approaches. However, what passes as continuous time theory work in finance is simply a sham.
This stepbystep approach is especially useful for the transition from. This paper proposes a novel method that enhances numerical approximation of infinite horizon optimal control problems. Davide dragone dynamic optimization and lab on mathematica. The second part of the course teaches how to use the mathematica software to solve dynamic optimization problems. Intertemporal asset pricing without consumption data. This chapter provides an introduction to the theory of discrete time continuous state dynamic economic models. Many of the recently added listings were suggested by alexandr stepanov of the state university higher school of economics in moscow. Applied researchers have been slow to adopt the intertemporal paradigm because it can impose formidable computational requirements. An important special case is the all linear optimal control problem with mixed state and control variable constraints. Sloan school of management on intertemporal preferences in continuous time the case of certainty by chifu huang and david kreps wp 203788 july 1988 massachusetts institute of technology 50 memorial drive cambridge. Read, highlight, and take notes, across web, tablet, and phone.
Introduction to intertemporal optimization yulei luo sef of hku september 6, 20 luo, y. Intertemporal preferences with a continuous time dimension. Intertemporaldynamic optimization in static optimization, the task is to nd a single value for each. Chapter 9 the intertemporal consumptionsaving problem in. I would really say for this book dynamic optimization for dummies. Optimal growth in continuous time ucsb department of. Students will become familiar with the methods used in discrete and continuous time optimization under certainty and stochastic optimization in discrete time including additional examples and.
We are interested in recursive methods for solving dynamic optimization. N, since we can always transform the problem to this. Multiobjective optimization also known as multiobjective programming, vector optimization, multicriteria optimization, multiattribute optimization or pareto optimization is an area of multiple criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. From discrete to continuous time the maximum principle1. Their critique focuses on the very basis of continuoustime preference theory and applies not. An intertemporal optimal allocation must obey the following conditions, which, given. After the differential in utility and the differential in the value of assets have been defined and computed, the optimization principle is summarized and fully set out in one equation equating the former with the utility value of the latter. It covers optimization methods and applications in discrete time and in continuous time, both in worlds with certainty and worlds with uncertainty. We derive the wellknown continuity principle for adjoint variables for preannounced or anticipated changes in parameters for continuoustime, infinitehorizon, perfect foresight optimization models. The use of optimization software requires that the function f is defined in a suitable programming language and connected at compile or run time to the optimization software. Dynamic economic optimization of a renewable resource the canonical example the faustman model of optimal forest rotation an example of optimizing over multidimensional states. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. The seller adjusts prices dynamically to maximize revenue. Intertemporal optimization and olg models in continuous time iv.
By the theory of the optimum, if a timepath of the control is optimal, a marginal increase in. Sloanschoolofmanagement unintertemporalrrererencesinuontinuoustime. A criterion for time aggregation in intertemporal dynamic. Here we see how taxes and a forced saving program affect utility and decisions. Wright computer sciences department, university of wisconsin, madison, wisconsin, usa 1 overview at the core of any optimization problem is a mathematical model of a system, which could be constructed from physical, economic, behavioral, or statistical principles. Epstein and zin,1989, in contrast to expected utility functions, enable one to separate cleanly an investors risk aversion and elasticity of intertem. Continuous linear programs have wide applicability as models of many real world situations that are intertemporal in nature.
The models simply assume there is only one consumption good in the economy. The intertemporal approach to the balance of payments, and. The first part of the course provides an introduction to dynamic optimization methods in continuous and discrete time optimal control problems, hamiltonjacobibellman equations and bellman equations. Pdf time preferences, intertemporal optimization, and the. Generalpurpose software for intertemporal economic models. For easy and intuitive numerical computation of the resulting multi point boundary value problem we suggested to simulate the resulting differential algebraic system representing the. Dynamic optimization in continuoustime economic models a. Irving fisher 1930 first analyzed the optimization problem of a consumer who faces no uncertainty and lives for two periods. Under this procedure, the policy maker gives a commitment to maintain the same. Intertemporal pricing with strategic customer behavior.
Discretetime finite horizon approximation of infinite. Optimal consumption choice with intertemporal substitution jstor. It covers individual finance choice, corporate finance, financial intermediation, capital markets, and selected topics on the interface between private and public finance. This textbook provides all tools required to easily solve intertemporal optimization problems in economics, finance, business administration and related disciplines.
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