I mentioned it on my blog the other day as well as here, but I thought I'd translate the snippet Duncan Black flagged as being an example of overly mathematical economics into a more readable form. I'll even emphasize the point by not using LaTeX.
Here's my translation of the mathematical symbols ...
We assume that there are a finite number of commodities ℓ. The consumption of agent i is defined by the set of ℓ positive numbers labeled xᵢ. Agent i's utility function in state tᶰ is uᵢ(xᵢ, tᶰ). We sometimes assume a quasi-linear economy: the ℓth quantity is a numeraire and the consumption of agent i is defined by ℓ-1 positive numbers and one positive or negative number xᵢ. The utility function uᵢ becomes uᵢ = vᵢ(xᵢ-ℓ, tᶰ) + xᵢℓ. The endowment of agent i of type ti is set of ℓ numbers labeled ωᵢ consistent with the definition of xᵢ (assumed to be independent of the state -- with this assumption, all private information concerns agents' preferences and beliefs.)
These quantities now define an admissible exchange economy E.
Or maybe we can go simpler ...
There is stuff. Each person has certain amounts of stuff and they like it to varying degrees. Sometimes one type of stuff is money and they like that differently. Everyone starts with some stuff. This defines an economy.
If we say the paragraph that Black identified is 100 on the mathiness scale, and your ending paragraph about stuff is 0, then your 1st translation (the LaTeX free one) is about 80 in my view. Can you create one at about the 40 or 50 mark? The idea that some of that stuff is money completely escaped me in both the 100 and 80 mathiness versions. More wordiness is fine.
ReplyDeleteMy point about missing the money is just one example of what got lost in translation for me.
DeleteI just wanted to reduce the level of the mathematical symbols to a minimum. Eliminated "element of", "real numbers" (&c), "Cartesian product", "maps to", "cardinality". My main point wasn't to explain the paragraph, but remove the unnecessary symbols (some are still necessary because you'd end up changing the meaning) allowing others to figure out the meaning if they'd like. I left the economics jargon because every field has jargon.
DeleteI added the end piece as a joke. Overall, the paragraph is just a bunch of definitions of symbols, so doesn't really have much content if you eliminate the symbols completely (as demonstrated in the end paragraph).
I think the only piece you are missing is that numeraire = money (or specifically, commodity money).
numeraire = money. Yes, I should have understood that as I've seen it before.
Delete