The diamond and water paradox and the law of diminishing marginal utility

Water is more useful than diamonds so why are diamonds more expensive than water? Surely people would pay more for diamonds than water. This is the diamond and water paradox and I will attempt to answer this question at the end of my post.

Let us imagine that there is a barn with 5 animals; 3 horses and 2 cows, there is now a fire and you only have time to save 4, what do you save? Well in order to do that you have to list out how useful each animal is.
So in this hypothetical scenario, your usefulness is in this order.

1. horse for ploughing the field
2. horse to act as backup for heavy equipment
3. cow for milk
4. cow for meat
5. a horse for pleasure riding.
   (We assume that all horses are identical and so are the cows)
   The answer should be obvious from this list, you let one of the horses die. But something interesting to note here is that even though horses may be more useful on average than cows you still sacrifice the horse as what is important is not average but marginal utility.

The marginal utility is the value that someone assigns to the last unit of a good so in this example, the marginal utility of a horse is the value of the last horse.

In economics, there is something called the law of diminishing marginal utility which states that as the quantity of a good you have the value of the last unit of that good falls, provided that the goods are used to serve independent uses.

Why is this? Well, when you are given the first unit of a good you allocate it to serve your most valued want that can be satisfied with that good, the second unit to serve the second most valuable want etc...
And so the value that you attach to each new unit is lower than the value of the unit that came before.
This is an interesting idea because you won't pay for something based on the average utility of the good but on the utility of the last unit of the good. Let us say that good A has an average usefulness to you equal to $10 but the value of the last one that you buy is only $5, would you pay $10 for the last unit? of course not since it's only worth $5 to you. The inverse is also true, if the price was $4 then you wouldn't stop where the utility of the last unit is $5 since buying another would be worth more than the $4 paid and so you would buy more until the value of the last (or marginal) unit was equal to $4. If the marginal utility less than $4 to you, you would buy less until it was worth $4 and if it was worth more you would buy more until the marginal utility was equal to $4.
Here we get another law of economics:
the price of a good is equal to its marginal utility (measured in dollars).

Now we are equipped to answer the diamond and water paradox: the average utility of water may be higher than a diamond but the marginal utility of water (the utility of the last unit sold) is less than the marginal utility of a diamond. just think how much you value the least valued litre of water that you use.

Economist Eugen Böhm von Bawerk was one of the first people to come up with this idea in part of what is known as the marginal revolution.