Question

Given the matrix

D = [.07, .20, .31, .19, .27; .35, .24, .16, .13, .05; .09, .25, .21, .29, .23; .26, .25, .06, .24, .15; .22, .14, .18, .08, .21]

Using the Leontief Input - Output Model

with idustries

1. Auto
2. Steel
3. Electricity
4. Coal
5. Chemical

How would you decipher the following question.

Determine the industry on which the steel industry is most dependent.

That is to say, will steel be the input or output?

The way I am reading it it could be either or.
I read it as

Steel is most dependent on Electricity for its production, and/or is most dependent on Auto for its consumption.

To answer the question as it is stated should I then compare it its 2 largest dependencies against each other...

That is to say Steel is most dependent on Auto, because Auto > Electricity ( .35>.25) ?

P.s This is also posted in the science, but I think it will get a better response in here

After reading up on the Leontief, and on input-output models, and on matrixes, I subtracted the matrix from I, and then took the inverse of it. If you multiply the the inverse with a (1,5) matrix of five one's, if I understand correctly it tells you what is needed to keep the wheels spinning and get a net output of one of each. That is, if A is the matrix you provided, d the net output, x the stuff put into the whole thing, and ^(-1) the inverse of a matrix, we get

x = AX + d
x - Ax = d
(I - A)x = d
x = (I-A)^(-1) d

I'm really rusty when it comes to matrices, but if I got it right it would seem that steel is most dependent on chemistry?

Originally posted by talzamir
After reading up on the Leontief, and on input-output models, and on matrixes, I subtracted the matrix from I, and then took the inverse of it. If you multiply the the inverse with a (1,5) matrix of five one's, if I understand correctly it tells you what is needed to keep the wheels spinning and get a net output of one of each. That is, if A is the matrix y es to matrices, but if I got it right it would seem that steel is most dependent on chemistry?

Its actually a little simpler than that... The inputs and outputs of the model are respectively interpreted as the rows and columns of the matrix I provided.

So, a question like

How many units of electricity are required to produce 1 unit of steel...steel is the output and electricity the input, so the element d_3,2 is what we are looking for.

So my problem lies with the interpretation of the original question I posed in that context.

Originally posted by joe shmo
Its actually a little simpler than that... The inputs and outputs of the model are respectively interpreted as the rows and columns of the matrix I provided.

So, a question like

How many units of electricity are required to produce 1 unit of steel...steel is the output and electricity the input, so the element d_3,2 is what we are looking for.

So my problem lies with the interpretation of the original question I posed in that context.

I have a question. Because electricity is produced partly using coal, is the dependence of steel on coal jacked up by the degree it is used in electricity production?

Hence the above zero numbers all over. The production of each of the five consumes some amount of all five.

Originally posted by JS357
I have a question. Because electricity is produced partly using coal, is the dependence of steel on coal jacked up by the degree it is used in electricity production?

I'm not sure how to quauntify that exact amount, but I think you'll notice that the summation of any output column is less than 1, which probably accounts for the sum total of interdependecies in relative porportion..., but thats just a hunch, as this was only supposed to be an excercise in Linear Algebra for me.

It seems to me, for any one output industry the summation approaching 1 would mean that its input industies are less interconnected, and increased interconnectedness the summation would approach zero...but as I said, just a hunch.

Incidentally, when I added up steels I got 1.08, so there must be a transcription error.

The bigger the value, the more of something is required to produce one unit of something else. But does it speak of a level of interconnectedness or dependence? In the above model, all five products are fully dependent on each other. For example, what if there would be no electricity? That would reduce the production of all five to zero, including that of electricity itself => the whole thing shuts down.

I wonder if looking at a simpler model would help.

Let's say that I produce a nifty new toy that consists of one piece of plank and three nails. My job is to buy planks for $3 each, nails for $2 apiece, assemble the whole thing, and sell it at profit. The material costs are 1 x $3 + 3 x $2 = $9. In addition I pay my cousin $2 to paint the whole thing with glossy colors, so the costs are $11, and I make a profit when I sell the whole thing for $12.

There are three kinds of resources here. Planks, nails, and cash.

One plank consumes zero planks, zero nails, and $3.
One nail consumes zero planks, zero nails, and $2.
One dollar costs 1/12 planks, 3/12 nails, and 2/12 dollars.

That gives the matrix

(0, 0, 3; 0, 0, 2; 1/12, 1/4, 1/6).

How can I say that one resource would be more dependent on the others than any other? If I have no nails, I can't make stuff, and thus can't sell it, and thus can't get planks, so aren't planks fully dependent on nails?

Looking at the numbers as such, getting dollars is dependent on on nails more than planks of dollars as three nails are needed per $12 dollars of income, while two dollars and one plank, that is, fewer, are needed. But I could replace the product "nail" by the product "sixpack of nails", half of which is consumed per toy, and suddenly it becomes least crucial..?

The matrix thing that wiki suggests looks at what is needed to keep the wheels turning. So if A = stock, x = the above consumption matrix, and d is profit, I get

xA = A + d

that is, to keep the business running, I need to have enough capital and stock to work, and the rest is for paying another monthly bill of the Lamborghini in my yard. Solving that gives

(x - I)A = d

from which I can solve the needed amount of goods for any level of profit I like, assuming that I can sell all that I produce and produce as much as I wish. All I need is the inverse matrix of (x - I), with which I would multiply the desired profit, and get the capital I require.

Puzzling, all in all. It feels like asking whether populating the world is more dependent on men, women, boys, or girls. Or whether making a Big Mac depends more on meat or buns.

Originally posted by talzamir
The bigger the value, the more of something is required to produce one unit of something else. But does it speak of a level of interconnectedness or dependence? In the above model, all five products are fully dependent on each other. For example, what if there would be no electricity? That would reduce the production of all five to zero, including that of e n men, women, boys, or girls. Or whether making a Big Mac depends more on meat or buns.

I don't think simplifying the model is that easy

just some observations, but the determinant of your input output matrix is 0, which may or may not mean something.

also any element in the model has to be between 0 and 1, which voids validity in your case.

I dont think cash can be an industry, cash should be the common value between industries for comparison.

You can well be right. Planks and nails are not industries either, but I don't see how why it would matter whether we speak of industries or commodities, and for the latter a dollar bill does just as well as a nail or a plank. Nor do I see of what use the determinant, whether zero or not, is here - but it can simply be that there is something about it that I don't get.

The way I understand is that you have A1 ... An kinds of stuff. You can label them as you like, the matrix doesn't care. They can be industries or whatever. All that the matrix tells you is how much of each kind of stuff you need to get more of that kind of stuff, such as, you need 0.07 units of "auto" stuff to get one unit of "auto" stuff. Or zero units of "planks" stuff to get one unit of "planks". How is that different from saying you have

* engineers who use 7 cars, 20 steel bars, 31 batteries, 19 lumps of coal and 27 bottles of moonshine to make 100 more cars;
* metallurgists that use 35 cars, 24 steel bars, 16 batteries, 13 lumps of coat, and 5 bottles of moonshine to make 100 steel bars;
* pikachus who use 9 cars, 25 steel bars, 21 batteries, 29 lumps of coal and 23 bottles of moonshine to make 100 batteries;
* miners who use 26 cars, 25 steel bars, 6 batteries, 24 lumps of coal and 15 bottles of moonshine to make 100 lumps of coal; and
* distillers who use 22 cars, 14 steel bars, 18 batteries, 8 lumps of coal and 21 bottles of moonshine to make 100 bottles of moonshine?

Perhaps I'm simply not getting something. Wiki uses as example a 2x2 matrix, that could well represent two kinds of cavemen. Let's say they like to eat a piece of meat and two fish each day. Fishermen in the tribe can use the energy to get five more fish; hunters to get two pieces of meat.

1: fishing industry; 2f + 1m -> 5f
2: hunting industry; 2f + 1m -> 2m

that gives the matrix (0.40, 0.20; 1.00; 0.50).

So who is most dependent on whom?

To make the question sensible, one could ask questions like how much food the tribe can store, or how large a % of the tribe is needed to provide everyone with fish and meat. In this case, two fishermen provide five for ten, so twenty provide fish for 100. One hunter feeds two people with meat, so 50 provide meat for 100. Therefore, in a tribe of 100 with 20 fishermen and 50 hunters, the other 30 can do other things.

The same could be done for the industry model, though with a trick at the end. One can fairly easily find how much investment is needed in each of the five industries to make it self-sustaining, but since the consumption differs for the industries some questions would need to be made about that too. Such as, the rest of the "tribe" is called profit, and consumes all five kinds of resources evenly, or it doesn't care what kind it gets and just wants as much as possible of it, or how to optimize the system so that it sustains itself and provides extra electricity to the outside.

Originally posted by talzamir
You can well be right. Planks and nails are not industries either, but I don't see how why it would matter whether we speak of industries or commodities, and for the latter a dollar bill does just as well as a nail or a plank. Nor do I see of what use the determinant, whether zero or not, is here - but it can simply be that there is something about it that ...[text shortened]... system so that it sustains itself and provides extra electricity to the outside.

Well, maybe some more people should be working on the model...I tried to get down to the nuts and bolts, but I cant tell if its the mathematics or the economics of the model that is keeping me from getting there, (probably a good bit of both), but I've like exploring it!

Aye, that's fun stuff. Though it reminds me of one of the most horrible experiences ever in my life, getting random questions from about 3,500 pages of books about international economics in an oral examination by the professor and his main assistant, with me the only student there, trying to explain Markoff chains on the blackboard in a 39C fever. ^_^