Unobtainium and Walras' Law

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.

Comments

[Jim Sentance]

My first thought was, how can you expect anyone to think clearly enough to address this, sedated on turkey as we are.

My second thought was how and why did you come up with it, sedated on turkey? Or have you not been following socially mandated consumption practices?

[Nick Rowe]

Hi Jim! I was reading this post, http://unqualified-reservations.blogspot.com/2010/12/monetary-reconstruction-presented.html and saw the reference to "unobtainium" in the comments. Then I realised: unobtainium was just the metaphor I was searching for, but couldn't find.

I think I deserve a drink now.

[Jon]

Doesn't Walras's law assume that the supply and demand functions are continuous over the positive real-space, and hence by implication that all quantities are infinitely divisible.

So this is hardly a disproof of Walras's "law". Its a cheap shot.

[Nick Rowe]

Jon: No, Walras' Law doesn't assume that, to my knowledge. And anyway, who's to say the demand and supply functions for unobtainium aren't continuous over the non-negative real space?

[Mike Moffatt]

I need to think about this for a second, but does this "Suppose people want to spend 30% of their income on unobtainium." violate the preference assumptions of Walras' Law. (Need to drag out my copy of Mas-Colell et. al.)

[Adam P]

I think this argument is incomplete. Nick hasn't what the price of unobtainium is or how the demand for it depends on it's price.

Do people want to spend 30% of their income on it no matter what its price is?

Are you now an MMTer who thinks relative prices have no effect at all on relative demands?

[Mike Moffatt]

Yeah.. I can't see how you can get convex preferences out of: "Suppose people want to spend 30% of their income on unobtainium."

[Adam P]

yep.

[Stephen Gordon]

If a consumer has Cobb-Douglas preferences, then expenditure shares would be constant. If U = x^a y^b, then the expenditure share of x is a/(a+b).

[Mike Moffatt]

Wow.. look what happens when you go a term without teaching micro. Thanks!

[Adam P]

Stephen, the expenditure on unobtainium is zero.

[Stephen Gordon]

Sure. But if the price were finite, the expenditure share would still be 30%. I think the point of the post is what happens out in the limit as the price goes to infinity. Usually, we get around the problem by assuming that preferences are defined only over goods that exist and for which prices are - or will be - finite.

[Aaron K.]

The only application of Walras' law here would be, if you assume that all other markets are clearing, that the market for unobtanium is also clearing. That seems like a limited, reasonable conclusion.

On the other hand, we are given an assumption that an economy wants to spend 30% of its income on one good. Gordon points out that a Cobb-Douglas function would generate this prediction, but that suggests that we look at the underlying preferences that this utility function represents. Here, agents are indifferent between any allocation that does not include some positive unit of unobtantium. This is, of course, a very silly assumption to make, so it is similarly silly to spend much time on the conclusions.

[Adam P]

"But if the price were finite, the expenditure share would still be 30%"

No because it's unobtainable, so actual expenditure remains zero. *Desired* expenditure may still be 30% but the differenc matters, and Nick is wrong here.

Even with Cobb-Douglas preference we get downward sloping demand curves don't we?

And is it not the case that the *way* we get downward sloping demand curves is that we impose the budget constraints of consumers?

Basically, the expenditure share is constant and the actual amount demanded is then determined by the price and the budget constraint. Higher price leads to lower demand by the fact that the same percentage of the same total income now buys less, so downward sloping demand.

So, if you take the interpretation that the price of unobtainium is infinity then the actual quantity demanded, under your Cobb-Douglas preferences, is zero by this mechansim. So, in that interpretation Nick is wrong.

On the other hand, if you assume it has a finite price but then consumers spend 100% of their income on other things, as Nick says they would, then again their expenditure on unobtainium falls to zero so actual demand for unobtainium is also zero in this case and again Nick is wrong.

What say you professor Gordon?

[Michael Smart]

Dude, just set the prices of your imaginary goods to zero. Trying to disprove identities over the holidays ... I think you worry too much!

(If people had C-D preferences over unobtainable goods, wouldn't their heads explode?)

[Mike Moffatt]

"Usually, we get around the problem by assuming that preferences are defined only over goods that exist and for which prices are - or will be - finite."

Don't you have to here if you also assume C-D preferences. With U(0,x)=0 for all x, preferences are no longer strictly convex.

[Matt Young]

The transaction cost for the next purchase of unobtanium went much higher than its value, hence the must remain a finite quant of unobtanium somewhere.
The error is simple. If we like unobtanium, then we must have purchased some in the past, hence, there must have been a production system which retains a finite 'forecasting error', a quant of the stuff.
Sorry Nick, you solved this one with the thermosatat.

[Nick Rowe]

As Stephen says, Cobb Douglas preferences would give you that demand function. Or, simply choose whatever demand function you like, as long as you always demand some unobtainium.

Walras Law is supposed to hold at *any* price vector. Not just at equilibrium. Any finite price for unobtainium is a disequilibrium price. So what? Walras' Law is supposed to hold at equilibrium and at disequilibrium price vectors. It drops straight out of the budget constraint. Remember micro?

[RSJ]

"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."

Why? If an individual wants more money, they can

1) buy fewer goods, increasing their money holdings
2) sell some of their existing bonds, increasing their money holdings
3) borrow money, increasing their money holdings

Of these options, if *all* households attempt 1), then they will fail, in aggregate. But if *all* households attempt 2) or 3), then they will succeed.

Now if an increased demand for money was even remotely responsible for recessions, then you would assume that households would attempt all 3 methods, right? That means, in particular, during those recessions caused by an excess demand for money, the non-financial sector would shift their claims on banks to be more in the form of money (deposits) and less in the form of bonds. They would do this by selling their bonds and purchasing deposits, meaning that banks, in aggregate, need to sell fewer bonds in order to attract funding and are able to rely more on deposits, so that the non-financial sector's deposit claims on banks go up and their bond/commercial paper claims on banks go down.

Yet you do not see the above happening in recessions -- ever. This suggests that, at least in our financial architecture, an excess demand for money is never a factor in causing recessions.

If you re-define money to include any bonds that could be sold in the period in question, then 1), 2), 3) collapses to option 1). But here the definition of money will depend on the accounting period of your model, and will in general include all financial assets. I'm assuming you don't want to do that, but that is the only way you can argue than an excess demand for money can remain unmet, particularly as the non-financial sector is sitting on large quantities of financial sector debt that can be converted into deposits whenever the non-financial sector wants. The financial sector would be more than happy to accommodate such a trade.

Why are you ignoring this possibility in your analysis?

[Aaron K.]

You assumed a ridiculous utility function and Walras' law lead you to an equally ridiculous conclusion. This is garbage in, garbage out.

It is not enough to wave your hands and announce that agents have preferences that can be represented by a Cobb-Douglas utility function, you must justify this! This means you must explain why agents should be indifferent between any bundle that includes 0 unobtanium and $0 of other goods and 0 unobtanium and $1,000,000,000 worth of other goods.

[RSJ]

And I would add that the only "unobtanium" here is an increased net-worth, or savings. Households will fail, in aggregate, to increase their savings by either making portfolio shifts or by purchasing fewer goods. They can increase, in aggregate, the proportion of their savings held as deposits whenever they want. In our current financial system, deposits are not unobtanium.

[Greg Ransom]

Doesn't revealed preference imply that there can never be a genuine "excess demand" for money?

Just asking.

[Sergei]

What is the problem for banks to generate more "money"? That is their business, isn't it?

Banks can give loans. Price is not infinite.
Banks can buy financial assets (at discount). Discount can not always be infinite (100%).

[Sergei]

RSJ has the comment in the same vein with a nice conclusion that it is all about financial net worth.

[Adam P]

"Remember micro?"

Do you?

I already explained (@7:24pm above) that you've given no violation of Walras' Law in this example. Demands have to respect budget constraints.

Once people have gone and spent 100% of their income on other goods then they can no longer be counted as having a demand for unobtainium.

[Mike Moffatt]

Let's throw some equations and numbers on this.

Two agents. Each has a Leontief demand curve of U=max{.7x,.3y}, which satisfies Nick's 30% on x condition.

Each agent has an initial endowment of (0,B).

For any price vector p, each person has an excess supply of y of b = .3B/(.7p + .3) and an excess demand for x of a = .3Bp / (.7p + .3). The sum of the values of all excess demands must equal zero, which they in fact do. So what's the issue?

[Mike Moffatt]

Err.. U=min{.7x,3y}. Sorry.

[anon]

"the non-financial sector would shift their claims on banks to be more in the form of money (deposits) and less in the form of bonds. They would do this by selling their bonds and purchasing deposits, meaning that banks, in aggregate, need to sell fewer bonds in order to attract funding and are able to rely more on deposits"

Forget it. This is naive.

It won't work unless households sell bank bonds back to banks, and banks don't make markets in their own bonds in this way. They issue them in the first place for liquidity protection. They're not going to redeem this protection in the middle of a balance sheet recession.

[Maxine Udall (girl economist)]

If the unobtainium you want to buy is less risk in a market with an oversupply of it and no reliable insurance against it, then you must self-insure, even if you can still sell your labor for money. As you say, people will buy less in order to hold more money. They almost certainly won't borrow if they can avoid it. It's not clear what one would do with less illiquid assets (most likely hold if they can manage it.) My point being that it won't be enough simply to increase employment in order to reduce the excess demand for money (although it will help). Excess risk (financial shocks, credit freezes, foreclosures (including wrongful foreclosures) and uninsured health losses (in the US)) must also be credibly reduced for those not in the top 5% of the income distribution in order to lower the excess demand for money (or less risk).

[Nick Rowe]

1. Remember: Walras' Law is about excess *demands*. Demand is the quantity people *desire/want* to buy. It is not the quantity they *actually* buy. If Walras' Law were about the quantities that people *actually* bought and sold, it would be true trivially. Since quantity of apples bought must equal quantity of apples sold, and the value of apples bought must equal the value of apples sold, we can sum over all goods and Walras' Law would simply say that the sum of n zeros must equal zero. Walras' Law is about *desired* expenditures, not *actual* expenditures.

2. I only assumed a constant (desired) expenditure share demand function for simplicity. Sure, Cobb-Douglas preferences have that weird property that utility is zero if actual consumption of one of the goods is zero. But economists often use Cobb Douglas preferences for simplicity, and I don't hear people saying it's illegitimate to use them. What preferences do economists normally use when they are analysing new goods? How do you handle, say, CES preferences, when each firm produces one good, and the number of firms can vary?

3. Don't assume that the price of unobtainium is infinite. Sure, the equilibrium price would be infinite, but Walras' Law is supposed to be true at *all* price vectors, not just at the equilibrium price vector. Again, Walras' Law would be true trivially at the equilibrium price vector. If each excess demand is zero, then the sum of n zeros is zero.

[Nick Rowe]

Ah! I think Mike has got it. So x is unobtainium, and y is regular GDP, and p is the price of unobtainium, right?

And Mike has derived the excess demand and supply functions, and they satisfy Walras' Law, as they should.

So, very basic micro theory says that Walras' Law should work with unobtainium too. There is nothing in Walras' Law that says that an endowment of zero is ruled out.

[Jon]

Jon: No, Walras' Law doesn't assume that, to my knowledge. And anyway, who's to say the demand and supply functions for unobtainium aren't continuous over the non-negative real space?

No it does. It part of the proof the Walrasian auctioneer can find a solution. Its absolutely required, and walras's law breaks down when its violated. For example, if the quantity of something is capped, the functions are discontinuous and a solution may not pertain.

Money is one particular instance of this problem since when discussing these problems we discuss the ceiling of money available as being well within the bounds of the auction. For all other goods we assume that the quantities available for trade are well in excess of the quantities traded.

Now I understand the root of your confusion!

[RSJ]

"Money is one particular instance of this problem since when discussing these problems we discuss the ceiling of money available as being well within the bounds of the auction."

Except that money is not capped at all. Each time a loan is made a deposit is created for the amount of the loan. The problem is that those funds do not remain in the form of deposits. By withdrawing the funds, the creditors of the banks force banks to compete on the liability side of their balance sheet, first by offering certificates of deposit, and then by selling commercial paper, and finally by selling bills and bonds. In this way, the loan amount is converted into higher maturity assets other than deposits. But this happens because households don't want to hold all of their wealth in the form of deposits. Money being "capped" is a non-problem. The problem is that households want more income and more savings in general, not that they are unable to sell their non-deposit financial assets and leave the proceeds as deposits within the banking system. Even Nick, if he really wanted more money, could liquidate his portfolio anytime he wanted to, except that he doesn't want to, and other households don't want to, either. An "excess demand" for a stock of money is a non-issue. But an excess demand for a stock of net-worth may be an issue, or an excess demand for a flow of savings may be an issue.

When will economists join the 19th Century and include banks and deposits in their models?

[Nick Rowe]

Jon: there's Walras Law, and then there's the existence of competitive equilibrium. Continuity is required for the latter, but not the former.

[anon]

"Even Nick, if he really wanted more money, could liquidate his portfolio anytime he wanted to, except that he doesn't want to, and other households don't want to, either. An "excess demand" for a stock of money is a non-issue. But an excess demand for a stock of net-worth may be an issue, or an excess demand for a flow of savings may be an issue."

True, almost.

An excess demand for money is an issue if the excess demand for saving includes a target mix of money and non-money within that saving.

So savers may not be willing to convert non-money to money, given a self-imposed constraint on asset mix for saving.

So Nick's demand for money remains true in that context. Your two views are compatible in that sense.

[Mike Moffatt]

"Ah! I think Mike has got it. So x is unobtainium, and y is regular GDP, and p is the price of unobtainium, right?"

Exactly. I made y an aggregate 'everything else' good.

It's funny - when we were discussing this in words, I found the discussion hopelessly confusing - I kept thinking about what the equilibrium would look like (even though it has nothing to do with WL), the whole '30% of income' thing, etc. etc. But once I translated the problem into algebra, it became straight forward. Sometimes modeling really is the way to go.

[Aaron K.]

I'll try a new approach:

Statement 1) "Walras' Law is total rubbish."
Supporting assumption: "Don't assume that the price of unobtainium is infinite. Sure, the equilibrium price would be infinite..."
Conclusion: "By Walras Law, the sum of the values of all excess demands must equal zero.... There must be an excess supply of obtainable goods, equal to 30% of GDP."

Response 1) Ok, got it: at non-equilibrium prices, markets will not be in equilibrium. So, clearly there is nothing wrong with Walras' law in this instance.

Statement 2) "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."
Interjecting a comment: This is a standard argument about searching for an equilibrium. At the first set of prices we considered in statement 1, markets did not clear, so the price for unobtanium rises and we try again...
Statement 2 continued: "There's an excess demand for unobtainium, but no excess supply for all the obtainable things. Walras' Law fails."

Response 2) Rhetoric aside, here is where the blunder occurs. We are clearly talking about a set of prices and allocations that are a candidate for an equilibrium. If we are not in equilibrium, then repeat the analysis in part 1: Walras' law continues to work well and predicts that neither market clears. If we are in equilibrium, then clearly the price of unobtanium is infinite, so the excess demand of unobtanium = 0, and this market clears. The demand for all other goods is actually a correspondence to R+ (any value maximizes utility under these preferences), including the equilibrium allocation that you describe (characterized by the market for all other goods clearing). In this case, the excess demand of all other goods = 0. This, combined with the observation about the excess demand in the market for unobtanium, is the prediction of Walras' law.

Long live micro!

[Adam P]

I'm confused, here's Nick in the comments:

"Ah! I think Mike has got it. So x is unobtainium, and y is regular GDP, and p is the price of unobtainium, right?

And Mike has derived the excess demand and supply functions, and they satisfy Walras' Law, as they should.

So, very basic micro theory says that Walras' Law should work with unobtainium too. There is nothing in Walras' Law that says that an endowment of zero is ruled out."

and here's Nick in the original post:

"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."

And here's Mike:

"The sum of the values of all excess demands must equal zero, which they in fact do. So what's the issue?"

Nick in the post is saying Walras' law fails, Mike gives an example where Walras' law holds and Nick says "Mike's got it".

Nick are you saying Walras' law fails or not?

[james wilson]

When we buy happiness with money we are buying unobtainium, but this does not prevent us from spending and borrowing to reach for this state. Therefore those peculiar creatures that buy happiness by saving money are the ones selling unobtainium, and should be reviled and taxed accordingly. Obama's Law.

[Nick Rowe]

Adam: my argument is a reductio ad absurdam of Walras' Law.

An "unobtanium economy" is an economy where we can imagine a good on which we would all like to spend 30% (or whatever) of our income, but that good does not exist.

I submit that the actual world we live in is an unobtainium economy.

Mike has verified my claim, using standard micro theory, that Walras' Law ought to hold as well in an unobtainium economy as it does in any other economy. So we ought to observe an excess supply of 30% of GDP in the actual economy, since the actual economy is an unobtainium economy.

But we do not observe an excess supply of 30% of GDP in the actual economy. Therefore Walras Law is false.

[Nick Rowe]

aaron: you lost me. Walras' Law says that at *any* price vector (equilibrium or disequilibrium) the sum of the values of excess demands must equal zero.

As a matter of fact, there is an excess demand for unobtainium equal to (at least) 30% of GDP. But there is not an excess supply of other goods equal to 30% of GDP. So Walras' Law fails.

[RSJ]

"An excess demand for money is an issue if the excess demand for saving includes a target mix of money and non-money within that saving."

If, in a given period, your financial assets are $1000 and your earning power is $50, then at most you can demand $1050 worth of deposits in that period. Demand is not a wish, it's a wish backed with the ability to pay at the current set of prices. Now if there really was a sudden demand for more deposits, then you would not only see people refrain from purchasing -- e.g. not spending the $50 of their earnings power on goods, but they would also go ahead and convert some of the $1000 of their assets into deposits as well. That is easy to do, particularly once you include a financial sector in your model.

But we do not see the non-financial sector converting their assets into deposits during recessions, and therefore you cannot argue that there is an unmet demand for deposits. The household sector in the U.S. holds about 40-50 Trillion in assets, of which about 7 Trillion is deposits. It would be very easy for the household sector to increase that to 8 Trillion if they wanted.

But they don't want to. There is no unmet demand for deposits. There is a demand to convert the 40 Trillion in net worth to, say, 60 Trillion, which is what it was before the assets were re-priced.

But that is a "demand" for high asset prices, or equivalently, for financial assets in general. I don't think that there is a word for a "demand" that other people demand something else. But that's how I look at excess savings demands. Everyone wants to defer consumption, which means that they demand that others pull-forward consumption. But no one is willing to go ahead and pull forward consumption. This has nothing to do with money at all, but with a desire to defer consumption, which the non-financial sector as a whole cannot do. But the non-financial sector as a whole can easily shift its claims on banks from bonds to deposits. They can increase their money holdings in aggregate, but they cannot in aggregate defer consumption, without some of the consumption being thrown away.

[Mike Moffatt]

"As a matter of fact, there is an excess demand for unobtainium equal to (at least) 30% of GDP. But there is not an excess supply of other goods equal to 30% of GDP. So Walras' Law fails.|

Our the indifference curves are not reflective of real-world wants.

[Mike Moffatt]

Errr.. *or*.

[Adam P]

"But we do not observe an excess supply of 30% of GDP in the actual economy. Therefore Walras Law is false."

But the unobtainium you have in mind is money right?

So, back when the money supply was constrained by Golden Fetters we did in fact experience a roughly 30% excess supply of labour (I guess it was actually like 25%).

Now that we can supply as much liquidity as we want we don't see that anymore.

Perhaps Walras' Law holds in reality and we've simply gotten better at making unobtainium?

[Jon]

Nick writes:

Walras' Law says that at *any* price vector (equilibrium or disequilibrium) the sum of the values of excess demands must equal zero.

That's a mis-generalization that happens to be true sometimes, but it isn't a statement of Walras's law. The formal statement of Walras's law is not the sum of the excess demands; it is that if n-1 markets clear, the nth market must clear (given that supply and demand are continuous functions over the positive real space and given another condition we'll discuss below).

When Walras's law holds, then we can take by implication that the sum of the excess demands equals zero. It's a corollary, but as such its conditioned under the primary theorem, and amounts to some nonsense about 0 + 0 + 0 + a = 0 so a = 0. Not very interesting and a distraction.

None of which matters to your more basic claim that Walras's law holds as absolute only when given k goods, there are k*(k-1)/2 markets. Your critique then that a monetary exchange economy has too many goods and too few markets undermines the preconditions of Walras's law, and thus yes it does not necessary hold (and does not hold).

That critique is sound. The critique about unobtainium is not since it neglects something which has always been a part of Walras's law.

[anon]

“The household sector in the U.S. holds about 40-50 Trillion in assets, of which about 7 Trillion is deposits. It would be very easy for the household sector to increase that to 8 Trillion if they wanted.”

The first point is that their desired cash ratio may well be 7/50. So they have an excess demand for both cash and assets in total, but they won’t convert the cash portion because their desired ratio constraint is binding.

The second point is that they CAN’T increase their deposits unless they sell those assets to the banks. But the banks may not want those assets, for many reasons, including capital constraints. You seem to be overlooking this constraint as well.

[Aaron K.]

"aaron: you lost me. Walras' Law says that at *any* price vector (equilibrium or disequilibrium) the sum of the values of excess demands must equal zero.

As a matter of fact, there is an excess demand for unobtainium equal to (at least) 30% of GDP. But there is not an excess supply of other goods equal to 30% of GDP. So Walras' Law fails."

Ok, numbers.

Notation: P are prices (exogenous), E are endowments, Z is an excess demand function, u and o refer to unobtanium and other goods.

P = (P_u, P_o) and normalize P_o = 1

E = (E_u, E_o) = (0,W)

Utility(u,o) = u^.3*o^.7
s.t. P_u * u + P_o * o <= Pu*E_u + Po*E_o = Po*E_o = W
simplified: P_u * u + o <= W

This gives the desired demand functions:

u*(P) = .3*W
o*(P) = .7*W

Z_e = .3*W - 0 = .3*W
Z_o = .7*W - W = -.3*W

Z_e + Z_o = 0

Excess demand = 0 given an arbitrary price ratio between unobtanium and all other goods.

[Aaron K.]

Minor edits that fix the math, but don't change the conclusions:
u*(P) = .3*W/P_u
Z_u = .3*W/P_u - 0 = .3*W/P_u (replace the line that begins "Z_e = ...")
P_u*Z_u + Z_o = .3*W - .3W = 0 (replace the line that begins "Z_e + ...")

(Initial version treated W = 1 and P_u = 1, but those were unnecessary simplifications).

[joe]

Peofessor Rowe,

Quick question. I've been reading a book called "monetary Theory" by Alan Rabin, who is a student of Leland Yeager. The entire book is just a summation of yeager's theory. In the chapter on Walras's Law he discusses all these things of notional and effective demand, and the idea of recessions as an excess demand for money. But he states that the Law holds. He does not agree with you that it breaks down.

Is there some sort of conflict between disequilibirum economists? Is it that Clower, Leijonhufvud, and Barro and you of course, say that it breaks down, where as Yeager says it holds?

Thanks,

Joe

[Nick Rowe]

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.

[Nick Rowe]

Aaron: OK, your last line in 2.23 "P_u*Z_u + Z_o = .3*W - .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 = .3*W/P_u - 0 = .3*W/P_u would still be correct. But Z_o = .7*W - W = -.3*W 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.

[Mike Moffatt]

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.

[Blikktheterrible]

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?

[Aaron K.]

"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.

[Patrick]

" now desperately want unobtanium "

Oh no! Nick, you've killed us all!

[Doc Merlin]

1. Um... not having any supply of unobtainium just raises the equilibrium price of unobtainium.
2. You made a freshman mistake. The supply of unobtanium not an amount, thats quantity supplied. Supply is a curve.
3. 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.

[RN]

It's a ridiculous question. No rational actor would demand something they knew was unobtainable. Sheesh.

[Nick Rowe]

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?

[Blikktheterrible]

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.

[Nick Rowe]

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.

[Blikktheterrible]

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.

[Kevin Donoghue]

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.

[Blikktheterrible]

"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?

[Nick Rowe]

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.

[Blikktheterrible]

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.