Lecture 3 - Math Tools | Lecture Note - Edubirdie

University:
Massachusetts Institute of Technology
Course:
14.03 | Microeconomic Theory and Public Policy
Academic year:

2020
Views:

296

Pages:

17
Author:

mason mc44

Lecture 3 Math Tools - 14.03/14.003 Fall 2016 Agenda • Optimization single variable multi-variable Implicit Function Theorem and comparative statics • Envelope Theorem: constrained and unconstrained Constrained optimization (Lagrangian method) • Duality 1 Single Variable Optimization Say (q) is the profit function and we choose q to maximize (4) Citaph 15 Profits Slope = 0 q Quantity First Order Condition (FOC): =0 θη Q: Is q necessarily the profit max? A: No, FOC is necessary, but not sufficient for profit maximization. The point that satisfies FOC could well be a minimum. Find x* through single variable optimization and then substitute. [\frac{dy}{dx} = -2x + a = 0] [x^{} = \frac{a}{2}] [y^{} = -(\frac{a}{2})^2 + a(\frac{a}{2}) = \frac{a^2}{4}] [\frac{dy^{}}{da} = \frac{a}{2} = x^{}] Find the derivative using the envelope theorem: [y^{} = -(x^{})^2 + ax^{}] [\frac{dy^{}}{da} = \frac{\partial y}{\partial a}|_{a=x^{}} = x^{}] 5.1 Visual Explanation of the Envelope Theorem Remember that y = -x² + ax and y* = f(a, x*(a)) = a²/4 y* [y^{*}(a_{3})] [y^{*}(a_{2})] [y^{*}(a_{1})] a a₁ a₂ a₃ The function drawn in this graph is such that its derivative at each point is [\frac{dy^{}}{da} = x^{}(a)] Remember: Envelope Theorem is multi-variate [y^{} = f[x_{1}^{}(a), x_{2}^{}(a), ..., x_{n}^{}(a); a]] [\frac{dy^{*}}{da} = \frac{\partial f}{\partial x_{1}}\frac{\partial x_{1}}{\partial a} + ... + \frac{\partial f}{\partial x_{n}}\frac{\partial x_{n}}{\partial a} + \frac{\partial f}{\partial a}] [\frac{dy^{*}}{da} = \frac{\partial f}{\partial a}] 6 Constrained Maximization Most maximization problems in economics are subject to constraints: maximize utility subject to budget constraint maximize social welfare subject to a resource constraint maximize profits subject to a technological constraint The tool for maximizing constrained functions is the Lagrangian Method. This a "trick" that turns out to have very useful economic content. 12

Related Documents

Lecture 3 - Math Tools

Lecture 3 - Math Tools - Page 1

Lecture 3 - Math Tools - Page 2

Lecture 3 - Math Tools - Page 3

Lecture 3 - Math Tools - Page 4

Lecture 3 - Math Tools - Page 5

Lecture 3 - Math Tools - Page 6

Lecture 3 - Math Tools - Page 7

Lecture 3 - Math Tools - Page 8

Lecture 3 - Math Tools - Page 9

Lecture 3 - Math Tools - Page 10

Lecture 3 - Math Tools - Page 11

Lecture 3 - Math Tools - Page 12

Lecture 3 - Math Tools - Page 13

Lecture 3 - Math Tools - Page 14

Lecture 3 - Math Tools - Page 15

Lecture 3 - Math Tools - Page 16

Lecture 3 - Math Tools - Page 17

of 17

Recommended Documents

Private Information, Adverse Selection and Market Failure

Private Information, Adverse Selection and Market Failure

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Lecture Note

General Equilibrium Practice Problem

General Equilibrium Practice Problem

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Answer Key

14.03/14.003 Fall 2016 Lecture 8 Part 1 Notes

14.03/14.003 Fall 2016 Lecture 8 Part 1 Notes

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Lecture Note

Lecture 4 - Utility Maximization

Lecture 4 - Utility Maximization

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Lecture Note

The Market for Risk

The Market for Risk

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Lecture Note

Demand Functions: Income Effects, Substitution Effects, and Labor Supply

Demand Functions: Income Effects, Substitution Effects, and Labor Supply

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Lecture Note

Axioms of Consumer Preference and the Theory of Choice

Axioms of Consumer Preference and the Theory of Choice

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Lecture Note

The Gains from International Trade: Empirical Evidence Using the Method of Instrumental Variables

The Gains from International Trade: Empirical Evidence Using the Method of Instrumental Variables

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Lecture Note

14.03 Fall 2012 Problem Set 4 Solutions

14.03 Fall 2012 Problem Set 4 Solutions

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Answer Key

General Equilibrium in a Pure Exchange Economy

General Equilibrium in a Pure Exchange Economy

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Lecture Note

Linking Compensated and Uncompensated Demand: Theory and Evidence

Linking Compensated and Uncompensated Demand: Theory and Evidence

Massachusetts Institute of Technology 14.03 | Microeconomic Theory and Public Policy

Lecture Note