oil extraction: (capital) + (skilled labour) + (machinery) + (oil fields) -> (crude oil)
one legitimate possibility (with a nice match between units and circumstances) would be
1 + 1 + 1 + 1 -> 4,
and that another legitimate possibility would be
1 + 1 + 1 + 0.1 -> 3.1,
or even
1 + 1 + 1 + 0 -> 3.0?
Pretty much everything is locally linear, but the locale is sometimes too small. If we can exploit computers and forget about academic "general equilibrium" (!) ideas, why use a linear model? "System dynamics" is more expressive, but I'd like to know what economists are using these days. Words and ideas I offer here may be used freely and without attribution.
Anyway, the equation
(oxygen) + (hydrogen) -> (water)
O<sub>2</sub> + 2 H<sub>2</sub> -> 2 H<sub>2</sub>O
One of the interesting insights that allowed Leontief to make a useful model is that he reversed the flow of the "production" equations/processes. Instead of asking "how much of the output can be produced given the inputs?" which, if the quantities are not right will result in some of the inputs being left over, he asked "how much of (input) is needed to produce one unit of (output)?". This is the inverse of the productivity of the input, by the way. The cool thing is that, given the inputs, how much and which outputs arre produced is an indeterminate problem, and not all the inputs may end up being used; but, given the outputs, you know exactly how much of the inputs has been used.
This is why actually none of the equations you has written really is a correct interpretation of how to turn the "process equations" into linear equations. But, then again, all this is true of chemical reaction equations. The symbols '+', '->' and the coefficients have the same meaning. The 1/2 in
(1/2) O<sub>2</sub> + H<sub>2</sub> -> H<sub>2</sub>O
So, there is a linear model at the end, but it is not obvious from the presentation of the "process" equations, not even if the "productivity" (stoichiometric) coefficients are given.
As an aside, I think this is clear why this was an insane choice of ancillary topic in the course I was assisting. Maybe there is a diary in there where I pick apart college level mathematics education in the US.
Anyway, the assumption of linearity is implicit in the question "how much of (input) is needed to produce one unit of (output)?", but you can of course replace that with a nonlinear function. I am more interested, however, in the "process equations", which might be represented as a directed graph (with nodes representing processes and arrows representing inputs and outputs). Those whom the Gods wish to destroy They first make mad. -- Euripides
I am more interested, however, in the "process equations", which might be represented as a directed graph (with nodes representing processes and arrows representing inputs and outputs).
H'mmmm.
What you need is a Neural Net.
If the model isn't linear then presumably it's pretty much unstable, almost by definition. I suppose a subset of small problems might convege towards something resembling equilibrium. But won't a reality-sized model that assumes time-based relationships between inputs and outputs wobble chaotically all over the place?
I agree, though, that as soon as you introduce time and, more importantly, delays, you get an entirely different order of complexity. Delayed feedback is really, really nasty in that respect. Those whom the Gods wish to destroy They first make mad. -- Euripides
Perhaps I should actually break down and look at Leontief's methods. But since I heard that he treated "technology" as a scalar input, I've had a disgust toward the topic.
...Hmmm... Google doesn't think that "process equations" have much to do with "Leontief" (10 hits)....Further digging suggests that Leontief was indeed saying what you and I have trouble believing he'd say:
The neoclassical production function is an explicit function Q = f(K,L), where Q = Quantity, K = Capital, L = Labor... Leontief, the innovator of input-output analysis, uses a special production function which depends linearly on the total output variables.... Widipedia: Input-output model
Q = f(K,L),
where Q = Quantity, K = Capital, L = Labor... Leontief, the innovator of input-output analysis, uses a special production function which depends linearly on the total output variables....
Widipedia: Input-output model
I'm puzzled. Words and ideas I offer here may be used freely and without attribution.
The inimitable book by Leontief himself remains the best exposition of input-output analysis. See bibliography.
The wikipedia article also says
Despite the clear ability of the input-output model to depict and analyze the dependence of one industry or sector on another, Leontief and others never managed to introduce the full spectrum of dependency relations in a market economy.
Irving Hock at the Chicago Area Transportation Study did detailed forecasting by industry sectors using input-output techniques. At the time, Hock's work was quite an undertaking, the only other work that has been done at the urban level was for Stockholm and it was not widely known. Input-output was one of the few techniques developed at the CATS not adopted in later studies. Later studies used economic base analysis techniques.
In 2003, Mohammad Gani, a pupil of Leontief, introduced Consistency Analysis in his book 'Foundations of Economic Science', which formally looks exactly like the input-output table, but explores the dependency relations in terms of payments and intermediation relations. ... In a technical sense, input-output analysis can be seen as a special case of consistency analysis without money and without entrepreneurship and transaction cost.
...
In a technical sense, input-output analysis can be seen as a special case of consistency analysis without money and without entrepreneurship and transaction cost.
Understanding the layout of the fluxes aids in predictive qualities - similar to the Leontief matrix as you describe. One difference I see seems that Leontief uses economic components as building blocks, and not reservoirs. Somewhat logical, as geology constantly mucks with the influence of time and kinetics.
The basic flaw in all this is that illustrative examples are only useful if your audience already knows about the subject matter. For instance: I can use exponential growth or an elastic spring to illustrate/introduce ordinary differential equations provided that my audience knows about population growth or mechanics so I can breeze through the modelling and see how the equations arise. But I always found that the ancillary topic that was used to ease in the main mathematical topic was harder for the students than the actual math. It would have been preferable to just do the math in the abstract, because the examples were just confusing.
And the worst were the pre-business students.
But please, don't get me started on pedagogy or on busines majors. I want to have a nice modelling (if incredibly unreadable) thread. Those whom the Gods wish to destroy They first make mad. -- Euripides
We just did it with 2x2 matrices so their head wouldn't explode, and they still couldn't understand what it was all about, or more importantly why they should care.
In all honesty, in my econometric theory class, I, too, often think "Why the hell do I need to know this given that the damned computers do it for me?" when the lecturer frustrates me. But that usually only happens when I've begun tuning out after classmates begin asking stupid questions.
I've certainly had all of my prejudices about education across the world confirmed by this program. The business majors can't do economics to save their lives. Students from East and South Asia are all incredibly strong with mathematics, but they have no intuition whatsoever, whereas Westerners -- all six of us in my program (five Brits and me) -- are in the exact opposite position. We stumble through the math, but we understand the essential ideas and have actually developed some of our own, which is, in my opinion, far more important in econ. Conservatives want live babies so they can raise them to be dead soldiers. - George Carlin
We just did it with 2x2 matrices so their head wouldn't explode, and they still couldn't understand what it was all about, or more importantly why they should care. In all honesty, in my econometric theory class, I, too, often think "Why the hell do I need to know this given that the damned computers do it for me?" when the lecturer frustrates me.
In all honesty, in my econometric theory class, I, too, often think "Why the hell do I need to know this given that the damned computers do it for me?" when the lecturer frustrates me.
Anyway, I means that the students couldn't understand a 2x2 leontieff model, or why they should care. The sad part is that the "practical example" was there for the sole purpose of increasing the "relevance" of the material. In actual fact, the material was absolutely irrelevant. And the reason is not that linear algebra is useless necessarily, but that whatever economics courses that these business majors had in their program that used algebra or calculus did not actually have their dependence enforced as a prerequisite. So, people just left the math courses for "later" and took the econ courses or whatever it was, and the professors in those courses just introduced the calculus or algebra they needed and got on with it. So I basically had a freshman class with a roomfull of seniors who had taken the courses which used algebra at least two years prior and forgotten about them because ot their irrelevance to Fraternity life and likely irrelevance to business administration. So the course was useless. And the textbook writer though Leontief's model was a good excuse for 2x2 matrices, and my professor thought that was cute, and I ended up learning something interesting, but my students? No way!
Oh, and what you say about math and intuition reminds me of how I felt taking an advanced Partial Differential Equations in my math degree. I had a good physics background and I could see my pure-math classmates had no idea what that was all about, but had the "professional" traning in functional analysis to follow the class at the ttechnical level. But meaning? No, sir. Same thing with differential geometry.
Bah. Those whom the Gods wish to destroy They first make mad. -- Euripides
Oh, you've got to see the level of achievement in stupidity for yourself to fully appreciate it. If it were an Olympic sport, these people would take home the gold everytime. Some of my classmates can run through mathematics like nothing I've ever seen. But trying to get them to make a statement even somewhat bold-sounding about (say) household behavior in multi-period models or the implications of imperfect competition on outcomes in the aggregate is like pulling teeth. They don't even know where to begin.
My "favorite" classes are the (unfortunately mandatory) tutorials, which largely involve the lecturer or professor working through only the first half of one question -- one out of five or six on a problem set -- because of the aforementioned classmates having presumably not looked at the sheet or put in the extra studying time with the lecture notes. (This is what happens when professors spoon-feed students by printing off handouts with all of the notes on them. Arrogance sets in, followed by laziness -- and, in the end, the idiots learn nothing and drag the rest of us down.) Conservatives want live babies so they can raise them to be dead soldiers. - George Carlin
Garbage In, Garbage Out (abbreviated to GIGO) is an aphorism in the field of computer science. It refers to the fact that computers, unlike humans, will unquestioningly process the most nonsensical of input data and produce nonsensical output. It was most popular in the early days of computing, but has fallen out of use as programs have become more sophisticated and now usually have checks built in to reject improper input.
... computers, unlike humans, will unquestioningly process the most nonsensical of input data and produce nonsensical output.
and this line is completely false.
Homo sapiens is a branch of the ape family, a branch so confident of its superiority that it names itself "Man the wise," flatly denying the evidence of most of its own daily experience. Cohen and Stewart, The Collapse of Chaos
Cohen and Stewart, The Collapse of Chaos
The computers will certainly give you an answer. But will the answer they give be 1) correct 2) relevant? You won't know.
GIGO is not used much nowadays, because, well, it was a sort of joke--like the old wireless operators' "after careful investigation we discovered the radio transmition was being prevented by a short between the ears."
But with the popularization and multiplication of computers, GIGO is not a joke anymore--it is just everyday life. The Fates are kind.
