Mathematical Economics for Theory of Social Common Capital(Chapter4: Ramsey Optimal Consumption Model and Phase Diagram)
1 Introduction
1.1 Summary
In constructing his theory of social common capital, Hirofumi Uzawa modeled it through mathematical economics as well as he discussed in a institutional-school analysis through natural language. The basic concept of his model is “how to make evaluate something that cannot be priced, such as the natural environment, in a price manner.” Uzawa approached this problem using the idea of “shadow pricing” in dynamic optimization.
Shadow pricing is an application of the Lagrange multiplier method which we often use real analysis. In this lecture, starting from the Lagrange multiplier method with high school mathematics as a prerequisite, we will discuss the theory of dynamic optimization and then read Uzawa’s mathematical paper on the theory of social common capital.
1.2 Schedule
Lagrange multiplier method and imputed price
• Apr 30, 2023Kuhn-Tucker conditions
• May 20, 2023Dynamic programming
• June 17, 2023Optimal investment and Penrose function
• Sep 17, 2023Global Warming
• March 17, 2024Saddle Point Solution
• May 19, 2024Fishing commons
Forest commons
Agriculture commons
2 Chapter4: Ramsey Optimal Consumption Model and Phase Diagram
Introduction
In the third lecture, we introduced the Hamiltonian and considered dynamic optimization problems in discrete time. In this lecture, we will learn about Pontryagin’s maximum principle and discuss dynamic optimization problems in continuous time. Finally, as a specific example of an optimization problem, we will examine the optimal growth model.
In this course, we first solved constrained static (single-period) optimization problems using Lagrange’s method of undetermined multipliers and its application, the Kuhn-Tucker method. For example, we sought to maximize utility within a budget constraint, such as ”What should be purchased while holding a 1000 yen bill in hand at a supermarket?” in situations where the problem concludes within one period. Then, in the previous lecture, we discussed dynamic optimization problems extended over multiple periods and learned about the Hamiltonian as a solution method. These dynamic problems involved questions like ”Based on the 1000 yen in hand, how much should be spent today and how much should be invested for tomorrow and beyond?” The dynamic optimization problem could be expressed as follows:
*Please see PDF file for full text and details.（Full text file revised on September 27, 2024）