# ECON200: Math for Economists

Unit 2: Temporal Optimization   Humans perceive their environment in four dimensions.  The first three – length, width, and height – define our physical universe.  We can pinpoint any location using those three coordinates.  The fourth dimension, time, is perceived as moving linearly in one direction.  Humans can observe how things change over time, and economists try to explain and predict these changes using quantitative tools.

The relative value of an asset will change over time.  Some assets, like gold, remain fairly unchanged, while others decay, like a wood house.  Even when assets don’t decay, their relative value to goods and services may change.  For example, an ounce gold coin purchased in 2006 was worth about \$600 and more than doubled in value in 2010.  Present value calculations are used to figure out how much less (or more) an asset will be worth at a future time.  These are important calculations for anyone saving their income.

How assets change over time has broader implications for whole societies.  Economic growth models assume that investments in capital depreciate over time, and a sustainable growth pattern requires the correct rate of capital replacement.  While simple models may explain growth over two periods, more dynamic ones explain changes over generations or even whole empires.  Remember that economists try to assist people to make optimal choices not only today but also well into the future.

This unit should take approximately 9.50 hours to complete.

☐    Subunit 2.1: 2.25 hours

☐    Subunit 2.2: 4.25 hours

☐    Subunit 2.3: 3 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to: - Develop a toolkit of quantitative methods to solve optimization problems over time in macroeconomics and microeconomics. - Apply specific quantitative tools (present value calculations, organic growth equations, derivatives) to solve problems in macroeconomics and microeconomics. - Analyze returns to capital over time, economic growth, and period models from an algebraic and calculus perspective and interpret the differences between tools.

2.1 Present Value   - Reading: Yoram Bauman’s Quantum Microeconomics with Calculus, Version 4.02: “Chapter 2: Optimization over Time” Link: Yoram Bauman’s Quantum Microeconomics with Calculus, Version 4.02: “Chapter 2: Optimization over Time” (PDF)

`````` Instructions: Read pages 29–42.  Note this reading also covers the
material you need to know for subunits 2.1–2.3.

Reading this chapter should take approximately 1 hour.

attributed to Yoram Bauman, and the original version can be found
[here](http://www.smallparty.org/yoram/quantum/).
``````
• Lecture: YouTube: Yale University, Department of Economics: John Geanakoplos’ “05: Present Value Prices and the Real Rate of Interest” Link: YouTube: Yale University, Department of Economics: John Geanakoplos’ “05: Present Value Prices and the Real Rate of Interest” (YouTube)

Also Available in:
Flash, HTML, MP3 or QuickTime
iTunes U

Instructions: Watch the entire lecture.

Watching this lecture should take approximately 1 hour and 15 minutes.

2.2 Distributions   Note: This subunit is covered by the reading assigned beneath subunit 2.1.  Focus specifically on pages 30–32 for a discussion of lump sums, perpetuities, and annuities.

• Lecture: YouTube: Yale University, Department of Economics: John Geanakoplos’ “10: Dynamic Present Value” Link: YouTube: Yale University, Department of Economics: John Geanakoplos’ “10: Dynamic Present Value” (YouTube)

Also Available in:
Flash, HTML, MP3 or QuickTime
iTunes U

Instructions: Watch the entire lecture.

Watching this lecture should take approximately 1 hour and 10 minutes.

• Assessment: Yoram Bauman’s Quantum Microeconomics with Calculus, Version 4.02: “Chapter 2 Problems” Link: Yoram Bauman’s Quantum Microeconomics with Calculus, Version 4.02: “Chapter 2 Problems” (PDF)

Completing this assessment should take approximately 2 hours.

Solutions: Answers are in the endnotes beginning on page 230.

• Assessment: University of Exeter: Juliette Stephenson’s WEB-pages for the course BEE 1024: Mathematics for Economists: “Week 6 Exercises” Link: University of Exeter: Juliette Stephenson’s WEB-pages for the course BEE 1024: Mathematics for Economists: “Week 6 Exercises” (PDF)

Completing this assessment should take approximately 60 minutes.

Solutions: Answers are available under “Week 6” and by clicking on “Solutions to class exercises” and “Solutions to homework exercises.”  This is an Adobe Acrobat file that requires Adobe Acrobat Reader, which can be downloaded free at Adobe’s website.

2.3 Applications of Concepts – Organic Growth Model   - Reading: Yoram Bauman’s Quantum Microeconomics with Calculus, Version 4.02: “Chapter 3: Math: Trees and Fish” Link: Yoram Bauman’s Quantum Microeconomics with Calculus, Version 4.02: “Chapter 3: Math: Trees and Fish” (PDF)

`````` Instructions: Read pages 43–52.

Reading this chapter should take  approximately 30 minutes.

attributed to Yoram Bauman, and the original version can be found
[here](http://www.smallparty.org/yoram/quantum/).
``````
• Lecture: YouTube: Yale University, Department of Economics: John Geanakoplos’ “12: Overlapping Generations Model of the Economy.” Link: YouTube: Yale University, Department of Economics: John Geanakoplos’ “12: Overlapping Generations Model of the Economy” (YouTube)

Also Available in:
Flash, HTML, MP3 or QuickTime
iTunes U

Instructions: Watch the entire lecture.

Watching this lecture should take approximately 1 hour and 15 minutes.

• Assessment: Yoram Bauman’s Quantum Microeconomics with Calculus, Version 4.02: “Chapter 3 Problems” Link: Yoram Bauman’s Quantum Microeconomics with Calculus, Version 4.02: “Chapter 3 Problems” (PDF)

Completing this assessment should take approximately 45 minutes.

Solutions: Answers are in the endnotes beginning on page 232.