Technical rambly post.
So last night I started reading MWG on microeconomics. One of the things which struck me was their use of a rather artificial-feeling mathematical framework, with consumption being a function of prices (a vector in an L-dimensional space) and of wealth (a single real number). Various bits of math follow from the statement that consumption is homogenous of degree zero as a function of these two sets of variables, which is just the statement that prices are only meaningful relative to overall wealth.
What’s a bit unnatural is the division of price and wealth into two separate variables, and the equations all reflect this. It seems a great deal more natural to merge these into an L+1-dimensional vector, with “commodity zero” being money. This is nice both mathematically (the equations are suddenly a lot more compact) and conceptually (it makes it a lot easier to think about, say, multiple kinds of money flowing around in a system) The Walras axiom then takes the form that the aggregate consumption of money over time is equal to total wealth, i.e. ultimately people spend all of their money.
But this led me to two questions which I think still need some pondering.
- In this context, the Walras axiom no longer seems so obvious, especially when you consider that there could be multiple “money-like” commodities in the system. What is special about money that causes people to ultimately spend all of it? In a utility model, I could see that money would be a utility-zero commodity, so if there’s anything with positive value to spend it on you would probably do so. (At least, so long as all interactions are linear — but I think that you can prove that they always are) But this non-obviousness suggests that there may be a more interesting way to phrase the axiom which ties more directly to the way that people relate to money.
- Once you start to treat money as Yet Another Commodity, the arbitrariness of using it as the scale for all the other variables seems significantly more obvious. Not in the moral sense, where it was pretty obvious to begin with, but simply mathematically; the choice of a preferred axis in commodity space seems almost perverse. One interesting alternative way to model things (which fits more naturally with choice models) would be to think about pairwise exchange costs rather than overall numerical costs — i.e., to think of everything as barter, with money simply a highly fungible good. What’s interesting is that this is significantly more general than numerical costs, in the same way that choice models are more general than preference models; it lets you model things such as nonfungible goods. (Money can buy time, but can’t necessarily buy loyalty; on the other hand, loyalty can buy loyalty) I suspect that there are some interesting techniques possible here — has this area been explored?
Yonatan Zunger’s Blog 
Yeah, the mathematical formalism is a bit much. I think usually what happens is that one of the prices in the price vector gets normalized to 1 and is considered the numeraire/money. How would you write out a bartering system? You’d have to define relative prices for every pair…which would be quite cumbersome. Maybe you can choose a few base commodities as your scale to deal with non-corporeal goods like loyalty.
(I say money can buy loyalty…for some people 😛 just set p = Infinity for everyone else).
Yeah, the mathematical formalism is a bit much. I think usually what happens is that one of the prices in the price vector gets normalized to 1 and is considered the numeraire/money. How would you write out a bartering system? You’d have to define relative prices for every pair…which would be quite cumbersome. Maybe you can choose a few base commodities as your scale to deal with non-corporeal goods like loyalty.
(I say money can buy loyalty…for some people 😛 just set p = Infinity for everyone else).
Um, wow… your post is Google result #4 for “walras axiom”. I’m not sure what that says.
Now that I looked up Walras’ Law again (been a while…), I recall it’s part of general equilibrium theory. This is one of those places where you have to be very very careful trying to figure out what microeconomics “means” by fiddling with the math.
A lot of this stuff is known to be totally wrong, and sort of intentionally – they’re hugely simplified models. If you rearrange it, I’m sure you’ll find that looking from another angle makes it seem a little odd.
In the case of general equilibrium theory, the real economy is actually never in equilibrium anyway. This is a huge point of contention between neoclassical microeconomists and Keynsian macroeconomists. Walrasian equilibrium “proves” that booms, busts, and depressions do not exist. Actually, there are rationalizations for things that look like booms and busts, but Keynsian macro is basically saying “Walras’ Law is rubbish, aggregate demand can in fact be too high or too low”.
You’ve restated things into “people evetually spend all their money”, and of course they don’t, and that’s a huge part of the disagreement. So far, it’s something that I don’t think has bee reconciled between micro and macro models.
Keynsian macro and also monetarist macro (Milton Friedman et. al.) are about how money isn’t like any commodity, and how that determines how macroeconomics works.
Money is basically imaginary. Even when a physical good is used as money (such as gold), its usefulness as a unit of exchange substantially raises its price above the normal commodity value. I remember my intro macro professor calling the Fed chairman (Greenspan at the time) the “high priest”, because his real job was keeping the faith that those numbers mean something.
This is the most fundamental job of a bank – money from (nearly) nothing. Fiat currencies let banks do something you cannot do with a commodity, which is loan out a lot more of it than you actually have.
In general, microeconomics is full of mathematical models that are very complex for the amount of reality that they actually describe. Macro models are a bit better to deal with because they’re more like economic rules of thumb. But economics isn’t a mathematical science, or shouldn’t be. The interesting stuff is highly experimental/evidentiary/historical, and a lot of trite stuff is concealed by wrapping it in much cleaner equations.
To put it more up the scientific alley, economics is far more complicated than a 17-body orbital dynamics problem where all measurements have an error of +/- 5%. Beware of spending too much time on the calculus.
Rechecking, I need to clarify that I was blatantly misleading about the history of economics there. “Walrasian” and “Keynesian” are very broad terms used for many years. Classification of economic models is really hard to follow. Here’s the less misleading version.
This is long but I hate to risk having misled anyone.
In the early 20th century, economists tried to make models accounting for many things – such as individual behavior, the supply of goods, and the demand for goods. Some of them were fairly psychological, not treating individual behavior as simple cardinal utility maximization. None of them worked very well mathematically.
“Walrasian” economics, which we now know as formulated by later theorists such as Kenneth Arrow, created a nice-looking theory of general equilibrium for microeconomics. It did it by basing things mainly on demand. “Individuals” were just quite simple utility functions. Later on it became tied up with somewhat politicized ideas such as rational choice theory.
This is how microeconomics textbooks start up to the present day. Economists don’t use it – it fell out of popularity by 1980 or so because it wasn’t useful in explaining things about the real world. Nobody came up with a single superior general theory that I know of, but economics has loads of models these days and the behavioral economists come up with the coolest results.
Keynesian economics has three phases. Keynes himself came up with a bunch of models and a bunch of very important psychology to go with them. Other economists, in the “neoclassical synthesis”, used Keynesian economics as their macro and Walrasian economics as their micro. They used the math and discarded the psychology. Around the 70s, Keynesian economics lost favor as a macro model and successors started with monetarism. Then there’s a fourth very recent phase in which the Depression suddenly looks very relevant and the original Keynes (including and especially on investor and consumer psychology) looks highly relevant.
So basically for the middle of this century, what I said about Walrasian and Keynesian economics being opposed was wrong. It’s only from the 80s on that Walrasian economics became strongly associated with monetarism and rational choice and generally the view that the market produces optimal outcomes (optionally with proper Fed intervention in the money supply). So it’s associated with the idea that the response to a depression is to stand back and let the markets handle themselves.
The people in that camp now hate Keynesian, because they say that the market left to its own devices can be catastrophically stupid and the best solution is for the government to go into debt giving truckloads of free money to people. If you are of a certain ideological persuasion this is not appealing. For decades they haven’t studied Keynesian economics and their students rarely heard Keynes’ name.
Modern Keynesian economics is associated with thinking micro models anything like Walrasian models are silly, and being more likely to look at the influence of paths (basically, how different initial positions in a market can lead to the world’s tech industry
Economists themselves don’t have good terms for these divisions. The closest they have is Freshwater vs. Saltwater (because the very mathematical market optimality types epitomized by the Chicago School are closer to the center of the country, whereas the Keynesian are on the coast).
BTW if you want a macro textbook I recommend Macroeconomics (Krugman & Wells) based solely on hearsay. Saltwater.