In the last post, I gave vent to all my left-wing righteousness against the growing income disparity. तो यह मेरे लिए हुआ — a totally uniform wealth distribution is stochastically unlikely. वास्तव में, it is over seven billion times less likely than one person in the whole world holding all the wealth in the world. That brings me to the topic of this post – what is the most likely wealth distribution?
How do we attempt to answer this question? ध्यान रहे, it is a mathematical question, not a moral one. The most moral distribution, one might think, would be to take all the wealth in the world and divide it equally among the 7.4 billion people. Borrowing the concept of phase space from statistical mechanics, this particular distribution will correspond to just one point in a 7.4 billion dimensional space. The other extreme situation – any one person holding all the wealth in the world, और कुछ नहीं के साथ बाकी सब पर – corresponds to 7.4 billion points, and is so much more likely, which is the statement I made in the first paragraph. लेकिन सबसे ज्यादा संभावना है कि इन दो चरम सीमाओं के बीच में एक विशेष वितरण नहीं होना चाहिए. So I sat down in front of my trusted iMac, trying to figure it out.
इस सवाल का जवाब दे विभाजित करने के तरीकों की संख्या ढूँढ़ बाहर करने के लिए नीचे फोड़े n में वस्तुओं को समूहों (तथाकथित सितारों और सलाखों समस्या), and then sorting and counting the number of points in the phase space, और अधिक से अधिक. Not a very hard problem, सिवाय इसके कि n और को काफी बड़े हैं, and the combinatorics involves factorials of large numbers. So you have to use Stirling’s approximation. खैर, I will spare you the mathematical details and get straight to what I found.
What I found was quite surprising. The most likely situation seems to be the one in which the wealth distribution is highly skewed. The richest person would own more than 60% of the wealth, and the top 50 people would own practically everything!
So any wealth redistribution effort will have to fight the tremendous stochastic pressure in addition to our innate reluctance to part with the green stuff. Does this constitute a mathematical proof that socialism cannot work? I guess the right-wingers might take it that way. मुझे, it shows a couple of other things: As the total wealth grows, the inequality will grow faster. दूसरे, those who find themselves on the right side of the income divide should thank their stars more than their so called talent, hard work etc. ज़रूर, यह सब सही है कि पक्ष के लिए अपने रास्ते बिल के लिए ले जाता है (unless you were born there, which is statistically quite unlikely), but talent, proclivity for hard work, नरम कौशल, आदि कठिन कौशल. परिस्थितियों के सभी दुर्घटनाओं हैं. By the same token, if you do find yourself on the right side, एक साथ चारों ओर चलने के लिए बहुत ज्यादा कारण नहीं है परेशान विवेक — hey, somebody had to be here. Better you than somebody else, सही?