Mathematical Economics for Theory of Social Common Capital(Chapter3: Dynamic Programming)
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, 2023Fishing commons
Forest commons
Agriculture commons
Global warming
2 Chapter3:Dynamic Programming
Introduction
So far, we have solved constrained optimization problems using Lagrangian multiplier method and its application, the Kuhn-Tucker method. For example, we have been solving consumption plans that maximize utility within budget constraints. These problems are static optimization problems at one point in time. They focus on a consumer in a supermarket at a certain point in time and considered how to make a good purchase using 1,000 yen. On the other hand, this chapter discusses a dynamic optimization problem that spans multiple points in time. The problem is to determine how much to spend today and how much to invest for future, using the 1,000 yen that the consumer has today. The problem is written like this:
*Please see PDF file for full text and details.
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