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Thursday, 14 February 2013

Purchasing As Knapsack Problem

In response to:

I was wandering about the idea of measuring happiness of buyers with demand surplus. If I want to pay 100$ for a disc and I can buy it at 20$ market has created a surplus of 80$. It's a simple and straightforward model, easy to apply but it has some limits.
  • But if I earn 160$/week this surplus is equal to half of my week, it's a lot for me and I'm vary happy.
  • On the other side if I earn 800$/week it's equal to half of a day, not really a big deal.
Maybe with a CD the example is a little bit silly, but if we use food in war time instead it becomes more interesting. In this case it seems more reasonable to share equally the resources (food) and avoid that rich people take more food than what they really need and poor die. In this case what we want to maximize is not a "fictitious" demand surplus but a real "wealthy surplus": how useful the resource is. And we maximize this when everybody can survive.
Do someone knows if there is some different way to consider happiness of consumers, maybe that takes into account also their possibility?

This makes me think that allocating your income is an unbounded knapsack problem ( http://en.wikipedia.org/wiki/Knapsack_problem) where the amount a person has to spend is the size of the knapsack, the value is the marginal utility of an item and obviously cost is cost. It seems to be straightforward to calculate if you can only purchase one item but marginal utility makes multiple purchases very complex.
Anyway, when we consider the question like this we can see that how much you are willing to pay depends on the following: - the cost can only be as large as the room left in your knapsack (this is obvious and explains why the greater income - for someone to buy something it must give them the highest ratio of utility to price
From the last point I think that the person who has more money must get more utility from the purchase unless :
1) they have already made many purchases such that this option is way down the list (the $100 vs $20 valuation is only with say a multi-millionaire)
2) the person with lesser income is faced with options that have sharply decreasing marginal utility such as food

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