Suppose people want to spend 30% of their income on unobtainium. But the supply is zero, because unobtainium is unobtainable. So there's an excess demand for unobtanium equal to 30% of GDP.
What does that imply, all you students of economics?
By Walras Law, the sum of the values of all excess demands must equal zero. If there's an excess demand for one good, there must be an equal and offsetting excess supply for at least one other good. So if there's an excess demand for unobtainium, it is impossible, even given perfect price flexibility, for there to be equilibrium in the markets for all the other, obtainable, goods. There must be an excess supply of obtainable goods, equal to 30% of GDP.
An excess supply equal to 30% of GDP would cause a really big recession. Prices for all the obtainable goods would fall without limit, but can never reach equilibrium. They can't reach equilibrium until the excess demand for unobtainium disappears. But that can never happen.
God, but I hope nobody ever wants to buy unobtainium. I hope nobody even thinks about unobtainium. Who knows how desirable they might imagine unobtainium to be, and how much of their income they might want to spend on unobtainium? They might want to spend 100% of their income on unobtainium, so the excess supply of all the obtainable goods would be 100% of GDP. Nobody would be able to sell anything, if we let people's imaginations get going. GDP would fall to zero.
Is it morally responsible for me to publish this post, given what I know about Walras' Law? Try very hard not to think about unobtainium! If you think about unobtainium, you might want to buy some, and that would cause a Great…..CRAAAAAAAASH! Too late!
Of course it is. I know that Walras' Law is total rubbish. If we want to buy more of something, but can't, because the sellers can't or won't sell more of it, we will have to spend our income on something else instead. We know we can't buy any unobtainium, so we spend the income we can't spend on unobtainium on obtainable things. There's an excess demand for unobtainium, but no excess supply for all the obtainable things. Walras' Law fails.
Money is the only dangerous form of unobtainium. If we want more money, but can't buy more money by selling other things, like our labour, we get more money by selling less money — by buying less of other things. That's what causes an excess supply of all the obtainable things.
That's about as clear as I can get it.
Worthwhile Canadian Initiative 
Joe: I haven’t read Alan Rabin.
My guess is there’s more of a disagreement on what precisely “Walras’ Law” means.
If you define WL as “the sum of the values of the n excess notional demands is zero”, then I say that WL is true.
But if we are talking about effective (constrained) excess demands then: there aren’t n excess demands. The number of excess demands depends on the market structure, on whether it’s a barter or monetary exchange economy.
In a monetary exchange economy, in each of the n-1 markets the value of the excess effective demand for apples (or whatever) in that market equals the excess effective supply of money in that market. In other words, a version of WL holds in each market individually. But there are n-1 different excess effective demands for money, one for each market.
Aaron: OK, your last line in 2.23 “P_uZ_u + Z_o = .3W – .3W = 0″ is what I mean by Walras’ Law. So we are on the same page.
What you have done is used standard micro to prove that WL is true in an “unobtainium economy”. Yep. But doesn’t that result strike you as empirically implausible? We do in fact live in an unobtainium economy, because everyone can imagine a good that they would like to spend 30% (or more) of their income on, if it existed. But we do not in fact observe an excess supply of 30% of GDP.
So, maybe there’s something wrong with standard micro theory?
Yes, there is.
We start out by maximising U() subject to the budget constraint. The result is a set of “notional” demand functions (which is what you solved for). But if the price vector is not in equilibrium, we hit quantity constraints, where we can’t buy or sell as much of some goods as we want to. We then revise our optimal plans, taking those quantity constraints into account. And when we do this, we have to be careful to specify the market structure – what goods can be exchanged for what other goods. Is it monetary exchange? Or barter?
In each market, we max U() subject to the standard budget constraint, plus subject to all the quantity constraints in all the other markets.
So, in your example, Z_u = .3W/P_u – 0 = .3W/P_u would still be correct. But Z_o = .7W – W = -.3W would no longer be true. It is the notional demand function, and ignores the constraint on the amount of unobtainium we are actually able to buy.
Again, I think your utility functions are implausible. If we take them to be literally true, then in “unobtainium economy” everyone has a utility of zero and for every person is indifferent to every possible bundle of the non-unobtainium goods.
I agree with Mike. This example only works because the supply and demand curves are so unrealistically defined. In this case, I don’t think it even makes sense to ask whether the market for unobtainium is in equilibrium. The net excess demand is simply undefined. More specifically, for any real good, the portion of total wealth devoted to that good should go to zero as real price goes to infinity. Secondly, for any real good, quantity supplied will rise above zero for some price. Without those conditions being met, I don’t think we can put much stock in our model.
Given your assumptions, I wonder what the model would predict to happen if some finite amount of unobtainium were brought to market. What would happen to the market in that case? Might the original equilibrium be considered to have been depressed relative to the new one?
“But doesn’t that result strike you as empirically implausible?”
Yes, but this just tells me that the price of unobtanium should be endogenous. The problem disappears in a general equilibrium set up. As for the idea to “revise our optimal plans,” there is no point given the preference relations underlying the utility function you specified. Any level of consumption for non-unobtanium goods maximizes utility. That was my original criticism, that the utility function made no sense.
Full disclosure: I now desperately want unobtanium after spending so much time trolling the comments in here.
” now desperately want unobtanium ”
Oh no! Nick, you’ve killed us all!
Um… not having any supply of unobtainium just raises the equilibrium price of unobtainium.
You made a freshman mistake. The supply of unobtanium not an amount, thats quantity supplied. Supply is a curve.
Supply and demand equilibrate even if supply of a good is 0.
The way you get excess demand or supply is market restrictions. That suggests by your analogy that regulatory policy causes recessions.
It’s a ridiculous question. No rational actor would demand something they knew was unobtainable. Sheesh.
RN: Have you ever been to Cuba? people demand things they can’t buy the whole time. Excess demand for lots of goods. If they were able to buy as much as they wanted, those goods wouldn’t be in excess demand, would they? Are you saying excess demand is logically impossible?
Doc: Come on. I know the difference between supply functions and quantity supplied. (But for unobtainium there is no difference: both are zero.)
I’m happy to go with market restrictions. So, in your example, if the government put price controls on goods that don’t exist, like unobtainium, by Walras’ Law you get a recession. We absolutely have to ensure there’s a free market in goods that don’t exist?
No, regulations only have economic effects when they change economic outcomes. The quantity of unonbtainium supplied will be zero no matter what the government does. The key issue that you seem to doubt is that the quantity demanded is also zero. The amount supplied is zero. The quantity demanded (and supplied) is value of q where the supply and demand meet. Therefore, the quantity demanded is zero. Your model failed to account for that rule.
Blik: “The quantity demanded (and supplied) is value of q where the supply and demand meet.”
I just can’t accept that. It’s a self-contradictory definition of quantity demanded. If quantity demanded is defined as the quantity where supply and demand meet, then what does “demand” mean at prices where supply and demand don’t meet?
“Demand” means the whole curve (or function) that relates quantity demanded to price. We can talk about what quantity demanded would be if the price were not where demand and supply curves cross. If we can’t talk about what quantity demanded would be at various prices, then we can’t even talk about the demand curve, so it would be meaningless to talk about where the demand curve meets the supply curve.
We can talk about excess demand, and excess supply. We do it all the time.
There can be cases in which supply and demand can’t balance for some reason, but this isn’t one of them. In this case, the supply function and the demand function do meet up, so there’s no excess demand. If the quantity supplied can come into balance with the quantity demanded (as it can in this case at q=0), then that value will be the quantity supplied and the quantity demanded. That’s all I was trying to say.
Blikk, you seem to be assuming that if any equilibrium exists (the functions meet up, as you put it) then the system must always be in equilibrium.
“If the quantity supplied can come into balance with the quantity demanded (as it can in this case at q=0), then that value will be the quantity supplied and the quantity demanded.”
I think that sentence most clearly summarizes what I’m trying to say. Do you disagree with anything in that sentence?
I think I see where Blikk is coming from. Let me re-formulate my point this way.
Suppose there is some imaginary good for which Qd would be positive at any price below (say) $1 Billion per kg. If the government passed a law setting a price ceiling on all imaginary goods of $1 million per kg, that would create an excess demand for imaginary goods and, by Walras’ Law, create a recession.
Maybe the problem is that the definition of excess demand you (and many other economists) use is not the one implicit in Walras’ Law, which I’m trying to follow. Maybe the problem is semantic.
I would claim that the market in your example IS in equilibrium, with Qd=Qs=0 and price indeterminate, but higher than $1 Billion per kg. I don’t think the price ceiling will create excess demand because it can’t and won’t actually move the price or quantity away from their prior levels.