A hypothetical economy has the following industries: farming, fishing, quarrying, and mining.
Question 1:
Does maximizing the output of all industries will yield optimum economic value for that particular economy?
I have been pondering upon this when I was stuck in front of an ultra-long waiting time traffic light on my way to badminton.
What do you think?
Tentative Solution:
For now we assume
- The activities of each industries will have no side-effects i.e. no environmental problems due to over-farming, no landslide due to over-mining, etc.
Even if we extend the time horizon, it seems optimized subsystems will yield a globally optimum system. However this is just my gut feeling.
Question 2 (Renewable Resource):
Zoom into the fishing industry and consider it as a system. Subsystems are the individual fishermen.
Will optimize outputs of all fishermen will yield maximum output for the fishing sector?
Tentative Solution:
In the short run, yes, maximizing individual fisherman will maximize the output. However in the long run, indiscriminated fishing will deplete the sea resources, therefore ultimately the fishermen will have diminishing returns, which is not desirable.
In this case, maximizing local subsystems will not yield a global optimum.
And we are bordering on game theory. Let's look at Prisoner's Dilemma:
2 felons have committed a crime and are caught. The police can convict them for 10 year jail time only if they testify each other. Or else the police can only charge them with a minor crime which will jail them for 5 years.
The police put the two persons in different locations and offers each of them a deal: if he testifies against his accomplice, he will be freed on the spot while the other will get the full sentence (10 years of jail time). Each of them is told the other person is offered the same deal as well.
For each of these felons, he is only interested in his own welfare, and he is aware of the deal is offered to the other person. In order to optimize his own welfare, he will choose to testify against his accomplice. However, optimizing of individual will make the situation worse, which is similar to the fishing example we see above. Therefore local optimization is not equal to global optimization in this case.
Question 3 (Non renewable resources):
Homework question. :P
4 comments:
In general, no.
Let's do da math, doo-daa day.
When you say optimize, you mean maximize the result of a metric. So make it easy on us, it means
max(f(x)) where f(.) is the activity, and x is the 'resources'.
Global max, would be the opposite of minimax -- the maximax, which means
max_f_i max(f_i(x)) where x of f_i is contained in region i.
So the question is ...
max_f_i max(f_i(x)) ?= max(f(x)
where x of f is contained in the sum of all regions i.
Tired liao.
I've got this wonderful answer, but the comment box does not have enough space to write it in...
I try to understand this problem with intuitive reasonings.
If we were to frame this using maths, I will say it may look like the following:
Let F(X) be the system
and X is 1xN matrix of the form (x_1 x_2 ... x_N) where x_i stands for resource i
By itself F(X) consists some number of functions f_i(.) representing the subsystems and there is no restriction on what these functions maybe. My gut feeling will be they are non-linear and may even coupled with each other to model the interdependences exist in some industries.
My question is:
Does
max(F(X)) ?= Sum(max f_i(x_i))?
Hope your above theorem isn't your last one ;)
Let x in Rieman Space and f(.) bounded, continuous.
Then YES, local optimization => global optimization.
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