Leontiff Economic Models

APPM 2460 Spring 2011
Lab/Project 2

Wassily Wassilyovitch Leontief won the Nobel Prize in 1973 for his work on how changes in one economic sector affect other economic sec- tors. His simple input-output model uses linear algebra extensively, and is therefore an excellent candidate for an APPM 2360 Project 2. In this project, you will examine:

A simple two-industry Leontief system.

A more complex five-industry Leontief system

The existence and uniqueness of Leontief solutions

The effect of technology on Leontief solutions

It is recommended that you read pages 152-154 of your textbook and that you familiarize yourself with Matlab or Mathematica to expedite your matrix computations!

1. Instructions

This lab is due Friday March 18, 2011 at the beginning of lecture. TA's will hold their regular office hours in ECCR 143(computer lab) during that week before it is due. The TAs in the lab are there to help you with Mathematica, MVT and Matlab syntax; they are not there to debug your files. Please keep this in mind.

Three (3) students is the maximum group size. You need to hand in only one lab per group.
Use a cover page
No loose leaf paper, please use printer paper for all calculations included in an appendix
Refer to the lab writing guidelines. They are important and you will be held to them; up to 20% of your final grade can be based on things like writing, introduction, conclusion, etc. Use "Equation Editor" in Microsoft Word for any equations that you include.
Late projects are not accepted.

On your title page, clearly mark

Name(s),
Student ID number(s)
Semester (Spring, 2011)
Professor
TA and section number
If you do not include this information, as much as 5% will be deducted from your final score

Format is worth 10% of your grade. Please refer to the following writing guidelines for the expository sections of this report. You are required to know and follow the Writing Guidelines for all labs. Guidelines and good and bad sample labs can be found on the web page for APPM 2460.

DO NOT LEAVE ANY OF YOUR FILES SAVED ON THE COMPUTERS IN THE LAB!

You can either E-mail them to yourself, use webfiles, or save on a USB 'memory stick'

For this lab, you should be using Matlab or Mathematics. Both are installed on the computers in computer labs in ECCR.

2. Background and Tasks

Section I: A simple two-industry Leontief system

Suppose that we have two industries: electricity and transportation (E and T). The first step of the Leontief input-output analysis looks at the inter- dependency of these two industries on each other. For example, it is reasonable to assume that the production of electricity requires a small amount of electricity (to power the power plant) and some transportation (to bring coal or natural gas to the power plant). Similarly, it is reasonable to assume that the production of transportation requires both transportation (to supply parts or fuel) and electricity (to power manufacturing of transportation elements).

We call this cost for resources to produce more or the same resource internal consumption and we can place this information in a technological matrix, A, where A_X→Yis the amount of X required to produce 1 of Y:

Now, suppose there is an external demand for electricity, d_E, and for transportation, d_T , which we can express in vector d. The question that we attempt to answer is: How much E and T should the industries produce to satisfy both demand and internal consumptions? We will call this vector x with components x_E and x_T. In other words:

The model equation is as follows:

Total output = External Demand + Internal Consumption

Then x = d + Ax

Task 1: Solve for x in terms of d and A. (Hint: your solution will also involve the identity matrix, I.)

Now you will examine a specific example, and solve numerically. Suppose that for every $1.00 of electricity produced, $0.50 of electricity and $0.30 of transportation are used. Similarly, for every $1.00 of transportation produced, $0.09 of transportation and $0.30 of electricity are required.

Task 2: Write the matrix A for this system. Be sure to label your matrix

The world that surrounds this example industry demands $500 of Electricity and $600 of Transportation.

Task 3: How much Electricity and Transportation should be produced?

Section II: A more complex five-industry Leontief system

We now expand on the simple concept and examine a system that involves 5 industries. Suppose that there are 5 dimensions to our state space: Electricity, Trucks, Widgets, Computers, and Human Power.

The electric company sees that it takes 0.15 units of electricity to make 1 unit of trucks; 0.23 units of electricity to make 1 widget; 0.23 units of electricity to make 1 unit of computers; 0.05 units of electricity to provide 1 unit of human power. It also takes 0.10 units of electricity to produce 1 unit of electricity, due to internal consumption.
The truck company sees that it takes 0.10 units of trucks to produce 1 unit of electricity; 0.20 units of trucks to make 1 widget; 0.01 units of trucks to produce 1 unit of computers; 0.10 units of trucks to provide 1 unit of human power. It also takes 0.20 units of trucks to produce another unit of trucks.
The widget company sees that it takes 0.05 widgets to produce 1 unit of electricity; 0.25 widgets to produce 1 unit of trucks; 0.01 widgets to produce 1 unit of computers; 0.10 widgets to provide 1 unit of human power. It also takes 0.05 widgets to produce another widget.
The computer company sees that it takes 0.02 units of computers to provide 1 unit of electricity; 0.17 units of computers to produce 1 unit of trucks; 0.10 units of computers to produce 1 widget. It also takes 0.13 units of computers to provide another unit of computers, and furthermore, computers are not involved in the provision of human power at all.
The HR staffing company sees that it takes 0.15 hundreds of employees to provide 1 unit of electricity; 0.20 hundreds of employees to provide 1 unit of trucks; 0.15 hundred employees to provide 1 widget; 0.10 hundred employees to produce 1 unit of computers. Also, the HR staffing company employs 0.15 people per person that it staffs out to other industries.

Task 4: Write out the A matrix for this industry. Be sure to label your matrix so it is well understood which rows and columns correspond to which industries.

Suppose that the World demands 1000 units of electricity, 2000 units of trucks, 3000 widgets, 4000 units of computers, and 500,000 human employees.

Task 5: How much of each should be produced to satisfy this demand?

Task 6: Could these industries satisfy any amount of demand that the world required? Why or why not?

Section III: The existence and uniqueness of Leontief solutions

For this section, suppose that our industrial system consists again (as in Part I) of only two industries: electricity and transportation. Suppose that for every $1.00 of electricity produced, $0.50 of electricity and $0.30 of transportation are used. Similarly, for every $1.00 of transportation produced, $0.82 of transportation and $0.30 of electricity are required.

Task 7: Write out the A matrix for this system. Again, do not forget labels.

Suppose there exist two countries, each with the industrial setup that you found above, in Task 7. In one country, the King of Bortzania demands $10,000 of electricity and $20,000 of transporation. In the other country, el Jefe of Curtisburg demands $100,000 of electricity but donates $60,000 in transportation.

Task 8: Can the demands of the King of Bortzania be met? Can the demands of el Jefe of Curtisburg be met? Explain your conclusions here using the power of linear algebra!! Hint: attempt to make your explanation understandable to both the Count of Jovakistan, who understands economics but doesn't know math, and the Mayor of Ruthopolis, who is clueless about economics but loves a good math text.

Section IV: The effect of technology on Leontief solutions

In this section, we start to examine the effects of technology. We accept as our premise that technological development happens and that technology improves the efficiency with which one element of an economy can be converted into another.

Here's the story. Say there are two industries, X and Y. The first time that the economy was observed, it took 0.3 units of X to make one unit of X, and 0.3 units of Y to make one unit of X. Similarly, it took 0.2 Units of X to make one unit of Y, and 0.1 units of Y to make one unit of Y. However, economists continued to observe this economy, and found that due to technological increases, the amount of X it takes to produce one unit of Y is actually a decreasing function of time. The relationship they found was:

Price(t) = Price(0)
Technology(t)

where Technology(t) could be called α(t) for the production of Y using X.

What's more, the economists noticed an interesting trend: the growth rate of technology α, with respect to time, was proportional to the current technology level α and furthermore, doubled every 10 years. This is a realistic model, in fact. For more information on an actual trend in technology development, read up on Moore's Law.

Furthermore, the economists noticed that the amount of Y used to produce X was also a function of time, due to a seasonal dependence on the number of daylight hours in a day (also realistic in agricultural markets). The relationship they found was:

Price(t) = 0.3 cos((2π/365) * t)

where t is in days.

*** Hint: since in this section some of your matrix entries will be functions of time, Mathematica may be useful in solving!! ***

Task 9: What is the A matrix for this economy when the economists first observed it? Don't forget labels.

Task 10: What is the A(t) matrix for this economy, that holds for any time, t? Don't forget labels.

Task 11: Suppose that external demand is static (not changing with time), asking for 150 units of X and 175 units of Y. Plot the productions of X and Y with respect to time, and be sure to include labels. Comment on your results.

Task 12: What happens when the system runs for a long time? (At least 50 years!) Be sure to justify this both graphically and using your matrix from Task 10.

Section V: Other things to write about:

Task 13: What is realistic about the Leontief models? What isn't?

Task 14: Give another application outside of economics for which a similar setup might be involved. Explain...

4. Comments

Do include a printed version of your numerical programs (if you have any; Matlab or Mathematica) in the printed copy of your write up that you hand in. DO NOT include pages of number-output from these programs. You can (and should) include PLOTS of output from these programs.
Be sure to discuss your findings and results in your write-up.

Created: October 2008 by Dan Larremore and Dan Kaslovski
Last updated: January 10, 2011 by Sebastian Skardal