Irving fisher 1930 first analyzed the optimization problem of a consumer who faces no uncertainty and lives for two periods. 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. The book is completed by an extensive index which helps finding topics of interest very quickly. The role of risk aversion and intertemporal substitution in dynamic consumptionportfolio choice with recursive utility 1introduction recursive utility functions kreps and porteus,1978. The fundamentals of intertemporal optimization in the. Anticipated shocks in continuoustime optimization models.
Continuous optimization nonlinear and linear programming. Optimal consumption choice with intertemporal substitution jstor. Time preferences, intertemporal optimization, and the permanent incomelife. Optimal growth in continuous time ucsb department of. Mertons widelyused text provides an overview and synthesis of finance theory from the perspective of continuoustime analysis. For direct numerical optimization, a continuoustime infinite horizon model. Let denote the probability of surviving until age t, which is a strictly positive and decreasing function. 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.
Nonlinear intertemporal general equilibrium models are hard to solve because of the dimensionality of the optimization problem involved. 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. But this kind of linearity is only local, so it does not allow one easily to. In continuoustime optimization problems, the analogous equation is a partial differential equation that is usually called the hamiltonjacobibellman equation. In this case, the program is more easily solved in. Formulating and solving problems under continuous time uncertainty has never been explained in such a nontechnical and highly accessible way. Applied researchers have been slow to adopt the intertemporal paradigm because it can impose formidable computational requirements. Whether cast in optimization or equilibrium form, most discrete time continuous state dynamic economic models pose in. Pdf the fundamentals of intertemporal optimization in. Discretetime finite horizon approximation of infinite. Dynamic optimization and optimal control columbia university. The index is normally the time, but can be a spatial parameter as well. Applied intertemporal optimization by klaus walde is a very very nice book, even for those who are not really familiar with mathematics.
Intertemporal pricing with strategic customer behavior. The seller adjusts prices dynamically to maximize revenue. N, since we can always transform the problem to this. It covers individual finance choice, corporate finance, financial intermediation, capital markets, and selected topics on. Intertemporal asset pricing without consumption data. As a preparation for this, the present chapter gives an account. It covers optimization methods and applications in discrete time and in continuous time, both in worlds with certainty and worlds with uncertainty. All wage income not consumed flows into the individual financial. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. 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. To calculate the corresponding real interest rate let the nominal price of a consumption good be. In the model, there is a monopolist who sells a finite inventory over a finite time horizon.
Epstein and zin,1989, in contrast to expected utility functions, enable one to separate cleanly an investors risk aversion and elasticity of intertem. Davide dragone dynamic optimization and lab on mathematica. Following 1 and 2, time is continuous and denoted by t. Siam journal on control and optimization siam society for. The second part of the course teaches how to use the mathematica software to solve dynamic optimization problems. 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. 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. 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.
An intertemporal optimal allocation must obey the following conditions, which, given. 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. 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. Mertons widelyused text provides an overview and synthesis of finance theory from the perspective of continuous time analysis. I would really say for this book dynamic optimization for dummies. The computation of intertemporal general equilibria therefore calls for time aggregation assumptions. Notes on intemporal optimization economics 205a fall 2014 k. A criterion for time aggregation in intertemporal dynamic models. Optimal time aggregation of infinite horizon control. On intertemporal optimization and dynamic efficiency uio. A criterion for time aggregation in intertemporal dynamic. Intertemporal optimization and olg models in continuous time iv. Full text of on intertemporal preferences in continuous time. By the theory of the optimum, if a timepath of the control is optimal, a marginal increase in.
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. The fundamentals of intertemporal optimization in the continuous time modelling of consumer behaviour. Jeanphilippe garnier, kazuo nishimura, alain venditti. Acontinuousfunctioncanbenowheredifferentiabletake,forexample,abrownian motionsamplepath. Chapter 9 the intertemporal consumptionsaving problem in. The computation of intertemporal general equilibria therefore calls for timeaggregation assumptions. The value of this optimal behaviour or optimal program is denoted by. When it comes to stochastic methods in continuous time, the applied focus of this book is the most useful.
This chapter provides an introduction to the theory of discrete time continuous state dynamic economic models. 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. View notes notes on intemporal optimization from econ 205a at university of california, santa cruz. Doing this in effect linearizes by taking the decision interval as infinitely small, so that the model becomes linear over this interval. From discrete to continuous time the maximum principle1. 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. 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.
An optimal solution to the program cp above is a triplet. Kletzer a primer on intertemporal optimization in continuous time a. Full text of on intertemporal preferences in continuous. Notes on intemporal optimization economics 205a fall. Their critique focuses on the very basis of continuoustime preference theory and applies not. Most of the material for part 1 is based on the textbook applied intertemporal optimization by klaus walde. 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. An euler equation is an intertemporal version of a firstorder condition characterizing an optimal. Bavarian graduate program in economics, the universities of dortmund. Since well be looking at an intertemporal utility function, well need to dis tinguish between overall.
Chapter 9 solving the intertemporal consumptionsaving. 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. Intertemporal preferences with a continuous time dimension. The role of risk aversion and intertemporal substitution in. 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. Dynamic optimization in continuoustime economic models a. We analyze the intertemporal utility maximization problem under uncertainty for the. Continuous optimization nonlinear and linear programming stephen j. The intertemporal consumptionsaving problem in discrete and continuous time a bundle is reduced to one consumption good. The models simply assume there is only one consumption good in the economy.
In this model, the intertemporal tradeoff involves a choice between higher. Intertemporal pricing with strategic customer behavior abstract this paper develops a model of dynamic pricing with endogenous intertemporal demand. Sloanschoolofmanagement unintertemporalrrererencesinuontinuoustime. Dynamic economic optimization of a renewable resource the canonical example the faustman model of optimal forest rotation an example of optimizing over multidimensional states. Here we see how taxes and a forced saving program affect utility and decisions. We are interested in recursive methods for solving dynamic optimization. At the same time, there are many problems in macro with uncertainty which are easy to formulate in continuous time. 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. Generalpurpose software for intertemporal economic models. The book treats deterministic and stochastic models, both in discrete and continuous time.
This chapter provides an introduction to the theory of discrete. Theoretically, by not consuming today, consumption. Ive always been fascinated by the work of blackscholes and merton from the good old days. The focus of this textbook is on learning through examples and gives a very quick access to all. Pdf time preferences, intertemporal optimization, and the. Introduction to intertemporal optimization yulei luo sef of hku september 6, 20 luo, y. Chapter 8 discrete time continuous state dynamic models. The role of risk aversion and intertemporal substitution. Hjbs from continuoustime in finance are a sham economics. Continuous linear programs have wide applicability as models of many real world situations that are intertemporal in nature. 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. 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. 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. This stepbystep approach is especially useful for the transition from.
It covers individual finance choice, corporate finance, financial intermediation, capital markets, and selected topics on the interface between private and public finance. Hamiltonian method and pontryagins maximum principle as a tool to analyze. Under this procedure, the policy maker gives a commitment to maintain the same. The role of risk aversion and intertemporal substitution in dynamic consumptionportfolio choice with recursive utility. This textbook provides all tools required to easily solve intertemporal optimization problems in economics, finance, business administration and related disciplines. We derive the wellknown continuity principle for adjoint variables for preannounced or anticipated changes in parameters for continuoustime, infinitehorizon, perfect foresight optimization models. 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.
The basic structure of this book is simple to understand. 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. 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. The intertemporal approach to the balance of payments, and. For simplicity we will assume that the index is, i. However, what passes as continuous time theory work in finance is simply a sham. Sloanschoolofmanagement onintertemporalpreferencesincontinuoustime thecaseofcertainty by chifuhuanganddavidkreps wp203788 july1988. Now suppose that the consumers utility is timeseparable. Read, highlight, and take notes, across web, tablet, and phone. Economic theories of intertemporal consumption seek to explain peoples preferences in relation to consumption and saving over the course of their lives. This paper proposes a novel method that enhances numerical approximation of infinite horizon optimal control problems. 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.
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. Adobe acrobat reader for free links are current as of january 25, 2007. 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. Intertemporaldynamic optimization in static optimization, the task is to nd a single value for each. A marketdetermined constant wage per labor efficiency unit, is w. An important special case is the all linear optimal control problem with mixed state and control variable constraints. An economic term describing how an individuals current decisions affect what options become available in the future.
1283 880 674 924 1221 308 1408 374 658 772 569 485 1282 1286 47 342 386 1426 980 1125 950 675 850 335 1076 10 1562 1074 147 303 1354 1140 1045 221 518 1433 937 1360 684 105 1201 847 1272 781 103