Munich Personal RePEc Archive
Lecture Notes on The Optimal Growth Problem
Keister, Todd
ITAM
January 2005
Online at https://mpra.ub.uni-muenchen.de/1461/ MPRA Paper No. 1461, posted 14 Jan 2007 UTC LECTURE NOTES ON ECONOMIC GROWTH
The Optimal Growth Problem
Todd Keister Centro de Investigación Económica, ITAM [email protected]
January 2005
These notes provide an introduction to the study of optimal growth in the one-sector neoclassical growth model in continuous time. The model is de- veloped using the analogy of Robinson Crusoe living on a deserted island. Both the Hamiltonian method and the phase diagram are presented and ex- plained on an intuitive level. Some familiarity with optimization theory and differential equations, as well as a thorough understanding of intermediate- level microeconomics, is assumed.
Rough draft. Comments and corrections would be would be greatly appreciated. These notes may be distributed freely, provided that theyaredistributedintheirentirety. T. Keister: Notes on Economic Growth
Lecture Outline
1 A “Robinson Crusoe” Model 3
1.1 Technology 4
1.2 Intensive Form 6
1.3 Preferences 9
1.4 Feasible consumption plans 12
1.5 The optimal growth problem 13
2 The Hamiltonian Method: 14
2.1 A 3-step recipe 14
2.2 Intuition for the Hamiltonian approach 16
2.3 Working with the FOC 24
3 Finding the Solution 26
3.1 The phase diagram 26
3.2 Transversality and feasibility 30
3.3 Drawing time paths 36
4 Comparative Dynamics Exercises 37
4.1 Different levels of impatience 38
4.2 Different depreciation rates 39
2 The Optimal Growth Problem
1 A “Robinson Crusoe” Model
We are going to start with what is in many ways a rather simple model. It is going to look nothing like a model of a modern economy, for instance. Nevertheless, we are going to work through this model completely, going over all the details of how to solve it. Some of this may already be familiar to you, but I want you to go over it carefully because it is the foundation of everything we will do in this course. In addition, we will see later that, by reinterpreting the variables in the model and making a few minor changes, we can interpret the model as a representation of a modern, market-based economy. I am going to set up the model by telling a story about a family of people who live alone in a very primitive environment. The goal of this analogy is to help you see very clearly what the variables and the equations in the model mean, and to gain solid intuition for the solution. Once we understand the model very well, we will see that different interpretations can be given to these same equations. For now, we will talk about a person named Robinson Crusoe,1 who has been shipwrecked with his family on a small, deserted island. The family will live in isolation on this island forever.