There's a natural connection between set theory and logic that can be more or less drawn by considering the set of possible universes, and making a correspondence between a binary statement ("watermelon is a fruit") and the set of universes in which it's true. The statement "A and B" is true in exactly those universes in the intersection of the two sets; logical "and" is equivalent to set intersection. "or" is the union. "not" is the complement relative to the set of possible universes. "A implies B" means "either A is false or B is true".[1]
We can extend this common notion of logic and sets by introducing probability theory. For any probability distribution on the set of universes, there's a probability that A is true, and a probability that B is true. If we know (for all elements of the set) that A implies B, then we know that for any probability distribution, the probability that A is true is less than or equal to the probability that B is true; perhaps less obviously, the converse is also true, at least for finite sets of universes: if it is the case that the probability that A is true is less than or equal to the probability that B is true no matter what valid probability distribution is used, then A implies B. If we restrict to one probability distribution, or a proper subset of all possible probability distributions, then there might be more to say; in particular, with one distribution, we can do Bayesian inference, and since P(A|A)=1, we have that if A implies B, P(B|A)=1.
Suppose we ask, "what would have happened if the Axis had won World War II?" To some extent the answer necessarily depends on how we fill out the counterfactual. In settings where we feel as though we have a reasonable answer to a counterfactual question like this, I think it's because we think there is some distribution (or distributions) of universes that is somehow "reasonable", and that, conditional on the information provided in the counterfactual, that answer is more likely than its complement. For questions that are particularly ill-formed[2] may suffer from being conditional on far-fetched possibilities, but also may suffer from conditioning on information that is relatively independent of other interesting information; A is interesting about B if P(B|A) is close to 0 or 1 and substantially different from P(B).
[1] You might ultimately know which universe you're in, or that you're in one of a restricted set of universes, but that's separate from the concerns of formal logic.
[2] I have in mind, in particular, the sort of question my son asks, e.g. "What if a baby beat a grandmaster in chess?", which tend to have the additional flaw that it's not clear what about the proposed reality is being asked.
Monday, December 30, 2019
Wednesday, November 27, 2019
trading on the blockchain
Bitcoin is regarded as a digital currency, to some extent, but I often find it useful to think of blockchain as a system for indelibly publishing messages. In the case of bitcoin, these messages are largely of the form "I am taking the bitcoin I got from [provenance] and giving X of it to [bitcoin address] and Y of it to [other bitcoin address]", and as part of the system of maintaining the blockchain it is verified that the sender has bitcoin from that provenance in a quantity that is no less than X+Y.
Actually buying bitcoin involves, as do all transactions, two legs: you give someone dollars or euros or pizza, and they publish a message on the blockchain publicly relinquishing some bitcoin to you. There are exchanges that get together people who want to trade dollars for bitcoin and people who want to trade bitcoin for dollars. When a match is found, the dollars are conveyed in some usual dollar-conveyance manner, and a bitcoin conveyance is published on bitcoin's blockchain. I'm starting, though, to sort of envision a system in which the blockchain itself serves as the exchange.
Consider a blockchain on which the messages took the form of "I trade W units of asset A and X units of asset B, from [provenances], for Y units of asset C and Z units of asset D." The process of incorporating a new block of such messages into the blockchain would require verifying that the person submitting the message has at least W units of A and X of B from the stated provenances, and also verifying that the entire block gives up at least as much of every asset as it conveys. If there is a very small set of prices that clear the market, then calculating how to put such trades together into a valid block gets computationally hard if a lot of these bids are very close to worth zero, but if there aren't too many assets, and there are a fair number of orders that give up a nonnegligible amount of value for some set of market prices, it becomes practically tractable, and certainly sufficiently tractable to reasonably incorporate into the "proof of work" that is part of bitcoin mining.
There are two big technological barriers that occur to me: the simpler one is that there has to be a way to cancel an order that doesn't get executed. It seems to me that bitcoin must have a way to deal with this — that, if I publish "I give Sam 2 bitcoins" and it doesn't go through within a reasonable amount of time that there must be a way to withdraw it or for it to expire — but I don't know what it is. Probably the message should include some sort of timestamp and/or expiration time, along with a hash of the message that includes the expiration. An actual cancel may be impossible.
The other, perhaps bigger, issue, is how the assets get on the relevant blockchain in the first place. If the only messages convey bitcoin, and all bitcoin originate at some level of indirection from bitcoin mining, then you have a fully closed system, and it's all fine. I can really only trade things that are on the blockchain, and for this to be useful they have to be able to somehow get there.
One possibility goes back to an older idea I had, and one that I later came to be was largely Ripple's initial idea, which is essentially to let each person have an asset that they can create out of thin air, simply by being them. I can trade "Dean's dollars" in any quantity for anything I can persuade other people to sell me; the problem is just in getting them accepted.. The provenance is just me. Other people can then trade them as they will, once I have put them out there. Maybe some of my friends would be willing to accept a certain amount of Dean's dollars among themselves; widespread acceptance would probably only come to a few currencies issued by a few people who are in some sense trusted (perhaps trusted to back their currency at some ratio with some basket of off-blockchain asset). You could imagine State Street publishing a list of blockchain addresses it maintains in which it promises to keep the "currency" of each address linked to a corresponding ETF.
Actually buying bitcoin involves, as do all transactions, two legs: you give someone dollars or euros or pizza, and they publish a message on the blockchain publicly relinquishing some bitcoin to you. There are exchanges that get together people who want to trade dollars for bitcoin and people who want to trade bitcoin for dollars. When a match is found, the dollars are conveyed in some usual dollar-conveyance manner, and a bitcoin conveyance is published on bitcoin's blockchain. I'm starting, though, to sort of envision a system in which the blockchain itself serves as the exchange.
Consider a blockchain on which the messages took the form of "I trade W units of asset A and X units of asset B, from [provenances], for Y units of asset C and Z units of asset D." The process of incorporating a new block of such messages into the blockchain would require verifying that the person submitting the message has at least W units of A and X of B from the stated provenances, and also verifying that the entire block gives up at least as much of every asset as it conveys. If there is a very small set of prices that clear the market, then calculating how to put such trades together into a valid block gets computationally hard if a lot of these bids are very close to worth zero, but if there aren't too many assets, and there are a fair number of orders that give up a nonnegligible amount of value for some set of market prices, it becomes practically tractable, and certainly sufficiently tractable to reasonably incorporate into the "proof of work" that is part of bitcoin mining.
There are two big technological barriers that occur to me: the simpler one is that there has to be a way to cancel an order that doesn't get executed. It seems to me that bitcoin must have a way to deal with this — that, if I publish "I give Sam 2 bitcoins" and it doesn't go through within a reasonable amount of time that there must be a way to withdraw it or for it to expire — but I don't know what it is. Probably the message should include some sort of timestamp and/or expiration time, along with a hash of the message that includes the expiration. An actual cancel may be impossible.
The other, perhaps bigger, issue, is how the assets get on the relevant blockchain in the first place. If the only messages convey bitcoin, and all bitcoin originate at some level of indirection from bitcoin mining, then you have a fully closed system, and it's all fine. I can really only trade things that are on the blockchain, and for this to be useful they have to be able to somehow get there.
One possibility goes back to an older idea I had, and one that I later came to be was largely Ripple's initial idea, which is essentially to let each person have an asset that they can create out of thin air, simply by being them. I can trade "Dean's dollars" in any quantity for anything I can persuade other people to sell me; the problem is just in getting them accepted.. The provenance is just me. Other people can then trade them as they will, once I have put them out there. Maybe some of my friends would be willing to accept a certain amount of Dean's dollars among themselves; widespread acceptance would probably only come to a few currencies issued by a few people who are in some sense trusted (perhaps trusted to back their currency at some ratio with some basket of off-blockchain asset). You could imagine State Street publishing a list of blockchain addresses it maintains in which it promises to keep the "currency" of each address linked to a corresponding ETF.
Thursday, August 8, 2019
bond ratings
Matt Levine mentions a front-page Wall Street Journal article in today's newsletter, and notes, interestingly,
The rational equilibrium story you'd like to tell, provided you have a degree from the University of Chicago, is that an issuer would pay for a credible bond rating because bond buyers are risk-averse and will, on average, underpay for a bond that would be rated (say) BBB+ if it is instead unrated because they don't know that it should be rated BBB+. Because I'm in recent possession of an "ambiguity" hammer, I see an "ambiguity" nail here; "BBB+" is itself essentially a probability of default, and while one can construct purely Bayesian models with risk-averse agents in which credibly revealing additional information is of positive value to sellers, I look at this and wonder whether ambiguity aversion, in which buyers don't know the right probability and are more willing to take on quantified than unquantified risks, can play an additional role.
[1] There's some level on which "one more level of rationality than the conventional wisdom" makes a lot of sense; this suggests as a rule of thumb that you should try to be two levels more rational than the conventional wisdom, provided you have a good idea what that is.
Moody’s Corp., for instance, gave lower grades to some tranches of commercial mortgage-backed securities than its competitors did, with the result that “by 2015, issuers ‘essentially stopped soliciting our ratings’ on those slices.” And investors priced those tranches accordingly:So this looks like one more level of rationality than in what I perceive to be the conventional wisdom: in particular, the issuers do have an incentive to solicit more credible, less lenient ratings, but they don't realize (or act on) them.[1]
Investors demanded higher yields on triple-B portions of deals without Moody’s ratings than on triple-B slices that included Moody’s during 2011 to 2014, according to a Journal analysis of Commercial Mortgage Alert data. The difference was about three-tenths of a percentage point more, on average, than benchmark triple-B rated CMBS—which means it was costlier to borrow than comparably rated debt.
The rational equilibrium story you'd like to tell, provided you have a degree from the University of Chicago, is that an issuer would pay for a credible bond rating because bond buyers are risk-averse and will, on average, underpay for a bond that would be rated (say) BBB+ if it is instead unrated because they don't know that it should be rated BBB+. Because I'm in recent possession of an "ambiguity" hammer, I see an "ambiguity" nail here; "BBB+" is itself essentially a probability of default, and while one can construct purely Bayesian models with risk-averse agents in which credibly revealing additional information is of positive value to sellers, I look at this and wonder whether ambiguity aversion, in which buyers don't know the right probability and are more willing to take on quantified than unquantified risks, can play an additional role.
[1] There's some level on which "one more level of rationality than the conventional wisdom" makes a lot of sense; this suggests as a rule of thumb that you should try to be two levels more rational than the conventional wisdom, provided you have a good idea what that is.
Monday, August 5, 2019
money supply and the business cycle
I ended my previous post with the suggestion that the ability to borrow be viewed as part of the money supply; it's worth noting that this is even more pro-cyclical[1] than usual measures of the money supply.
[1] As far as I know. Fact-checking isn't strictly opposed to the spirit of this blog, but while this is an assertion I'm making based on intuition that is less informed than most of the assertions I make here, I don't feel like trying to figure out whether it's true.
[1] As far as I know. Fact-checking isn't strictly opposed to the spirit of this blog, but while this is an assertion I'm making based on intuition that is less informed than most of the assertions I make here, I don't feel like trying to figure out whether it's true.
Sunday, July 14, 2019
free banking
Bitcoin and, to a lesser extent, Ethereum are pretty well-known cryptocurrencies at this point; only slightly behind them in scale is something called Ripple. My understanding, though, is that Ripple started as something else, though not a lot different. A cryptocurrency system is essentially a system for indelibly publishing statements of the form "I Alice have 42.3 units of currency from Source X of which I hereby give 24.9 units to Bob and retain 17.3 units for myself," and I won't here go into why the numbers don't quite add up. My understanding is that Ripple collected statements of the form "I Alice owe Bob 24.9 units of currency," with some capacity for detecting cycles, so that if Bob owed Carol 24.9 or more units and Carol owed Alice 24.9 or more units those might be cancelled against each other, and if Bob wanted to transfer the debt that Alice owed him to Carol (so that now she owes Carol the money instead of owing it to Bob), there might have been a capacity for that, too.
If I'm wrong about Ripple's historical origins, I don't really care. Let's discuss such a system anyway.
The bitcoin blockchain has a mechanism by which a positive number of bitcoins actually exist — in fact, it's an important part of the way the blockchain works — but we don't really need that to be the case. As long as there are a lot of people who are each willing to lend some amount of money to each of several other people, you can run the system entirely on credit, where the total amount of currency in circulation is zero. If I'm willing to lend Bob up to 25 units, I can "sell" him a good for up to that amount of money, with his payment going into the system; I've sold it for the IOU. If Carol is willing to lend Bob up to 15 units, and has an item I wish to buy (and she wishes to sell) in exchange for 15 units worth of debt from Bob, we can use Bob's debt to mediate our exchange.
Of course, if Bob and Carol and I don't really know each other, there's an issue. If there's someone we all trust for some reason — let's call him Uncle Sam — and I (by some means or another) am owed money by him, then we're in a fine (and very familiar) place; I can buy things from Bob and Carol by novating to them the debt from Sam.[1]
At this point, I've perhaps worked backward a bit; my point is that the supply of money doesn't depend on any monetary base, and certainly not any commodity (or other) backing. "Medium of exchange" is essentially created wherever credit is extended.[2] We can, in principal, have "net cash" of zero in an economy, with every unit of currency representing the entirely unsecured liability of a private individual.
In closing, I'll note that there are many measures of "money supply", the broadest of which include commercial paper; I think it probable that there are purposes for which lines of credit, credit card credit limits, and pledgeable assets should be counted as contributing to the supply of money.
[1] Hyman Minsky is reported to have observed, "Anyone can create money; the problem lies in getting it accepted."
[2] This is more or less one of the best points that the adherents of "Modern Monetary Theory" have made.
If I'm wrong about Ripple's historical origins, I don't really care. Let's discuss such a system anyway.
The bitcoin blockchain has a mechanism by which a positive number of bitcoins actually exist — in fact, it's an important part of the way the blockchain works — but we don't really need that to be the case. As long as there are a lot of people who are each willing to lend some amount of money to each of several other people, you can run the system entirely on credit, where the total amount of currency in circulation is zero. If I'm willing to lend Bob up to 25 units, I can "sell" him a good for up to that amount of money, with his payment going into the system; I've sold it for the IOU. If Carol is willing to lend Bob up to 15 units, and has an item I wish to buy (and she wishes to sell) in exchange for 15 units worth of debt from Bob, we can use Bob's debt to mediate our exchange.
Of course, if Bob and Carol and I don't really know each other, there's an issue. If there's someone we all trust for some reason — let's call him Uncle Sam — and I (by some means or another) am owed money by him, then we're in a fine (and very familiar) place; I can buy things from Bob and Carol by novating to them the debt from Sam.[1]
At this point, I've perhaps worked backward a bit; my point is that the supply of money doesn't depend on any monetary base, and certainly not any commodity (or other) backing. "Medium of exchange" is essentially created wherever credit is extended.[2] We can, in principal, have "net cash" of zero in an economy, with every unit of currency representing the entirely unsecured liability of a private individual.
In closing, I'll note that there are many measures of "money supply", the broadest of which include commercial paper; I think it probable that there are purposes for which lines of credit, credit card credit limits, and pledgeable assets should be counted as contributing to the supply of money.
[1] Hyman Minsky is reported to have observed, "Anyone can create money; the problem lies in getting it accepted."
[2] This is more or less one of the best points that the adherents of "Modern Monetary Theory" have made.
Tuesday, June 18, 2019
Facebook's stablecoin and interest
Per Matt Levine, Facebook is setting up a cryptocurrency backed by a variety of low-risk assets, but they're not stabilizing it the way I would.
Note 1: I would like to see a fixed interest rate; I use 2% in the example, but another number might be better. It should be high enough that the costs are covered, which might be a tricky number to come up with if those costs don't scale linearly with the size of the pool; probably you should pick a conservative size and expect that the early backers may have to subsidize it while it's small. Conditional on its being "large enough", though, I'd rather it be as small as possible, though of course my money's not on the line here.
Note 2: If the interest rate is close to 0, and you're in a society in which "interest" is repugnant, then denominating transactions in this currency allows you to avoid the problems this creates for positive interest currencies. This note highlights that "interest" isn't some absolute economic phenomenon; it's a property of the unit of account, and in particular of its failure over time to capture the true market rate at which value at different points of time are being traded.
Users of Libra do not receive a return from the reserve. The reserve will be invested in low-risk assets that will yield interest over time. The revenue from this interest will first go to support the operating expenses of the association — to fund investments in the growth and development of the ecosystem, grants to nonprofit and multilateral organizations, engineering research, etc. Once that is covered, part of the remaining returns will go to pay dividends to early investors in the Libra Investment Token for their initial contributions.If one were to use Libra as a unit of account, and ask what the "risk-free" interest rate in Libra would be, the answer should be more or less equal to the average return on the assets being used to back it. I would propose that instead they retain all returns in the reserve, take out a fixed fee (say 2% per year) to manage the ecosystem and pay out returns to investors, and allow the value of the coin to follow the pro-rata share of the reserve. This fixes the "risk-free" interest rate for Libra at 2% — if it takes off and becomes a significant unit of account for long-term transactions, this will increase its suitability for that purpose by eliminating interest rate risk. In particular, even if some of the currencies in the basket start to hyperinflate badly, and their interest rates go way up, the coin remains stable in a more absolute sense, and the rate at which it depreciates compared to risk-free investments is relatively unaffected.
Note 1: I would like to see a fixed interest rate; I use 2% in the example, but another number might be better. It should be high enough that the costs are covered, which might be a tricky number to come up with if those costs don't scale linearly with the size of the pool; probably you should pick a conservative size and expect that the early backers may have to subsidize it while it's small. Conditional on its being "large enough", though, I'd rather it be as small as possible, though of course my money's not on the line here.
Note 2: If the interest rate is close to 0, and you're in a society in which "interest" is repugnant, then denominating transactions in this currency allows you to avoid the problems this creates for positive interest currencies. This note highlights that "interest" isn't some absolute economic phenomenon; it's a property of the unit of account, and in particular of its failure over time to capture the true market rate at which value at different points of time are being traded.
Wednesday, June 12, 2019
urbanization and land value
Suppose land has two uses, urban and farming, and that some land is more suitable than other land for cultivation, but the only thing that determines suitability for urban use is proximity to other urban land. Suppose there's a single city in the middle of
our universe that is growing at some exogenous rate; farming land right next to urban land is presumably worth more than similar land farther away, but I would strongly guess that fertile land next to the city is worth at least as much as infertile land next
to the city. If your land is 10 miles away, though, your land is going to be worth more if the land in between your land and the city is infertile than if it's fertile. With some kind of spatial correlation in fertility, there's probably some distance away
from the city where land value correlates negatively with fertility.
Friday, February 22, 2019
national debt
In the past couple of years, politics seems largely to have given up on the idea that the federal deficit is of any importance; some people go so far as to actively say that it doesn't matter. There is some truth to the statement that the budget constraint is a bit different for a government than for an individual, but I do believe that it represents some kind of (fuzzy) constraint, with some kind of attendant (hard-to-price) cost associated with incurring more debt. Right now, though, I want to think about it from a different angle than the idea that, at some point, we would be unable to borrow more money or buy goods with the money we can print.
For a closed economy, total savings equals investment; investment in new capital is one of the important factors in the long-run increase in real wages. With an open economy, people from other countries can invest in US capital as well, so we get an accounting identity
Presumably it depends on the details -- at the coarsest level, a tax hike would probably have a different effect than a spending cut, and it seems likely to matter which tax you hike or which spending you cut. It seems like we ought to be able to say something useful about a general tendency, though, perhaps with caveats. Twentieth century macroeconomics would suppose that this is intermediated through interest rates, and would probably expect all three to absorb some of the change, with the portion depending on how sensitive investment, private consumption/savings decisions, and foreign trade partners (including foreign exchange markets) were to interest rates. The identity holds regardless of mechanism, though, and interest rates don't seem to have been obviously responsive to government deficits; perhaps more importantly, I don't think the people who deny any meaning to the deficit could be persuaded by this mechanism. The identity itself, however, is pure arithmetic. Something must give.
So what happens when the government gives a bunch of money to a group of people, whether in exchange for goods or services or not? In the latter case, those goods or services and/or the scarce resources that went into their production are no longer available; a Keynesian might believe that some nonscarce resource is being newly employed, in which case that's not a factor. If you're removing a consumer good from the market and prices aren't changing, it seems that you're forcing private savings or the trade deficit to change; if you're removing more of a capital good, perhaps you expect investment or the trade deficit to change. If you hand out the cash and it doesn't go anywhere, private savings is what automatically absorbs it at the first instance, but it seems likely that a lot of it would quickly go to investment or trade deficit.
While correlation is, of course, not causation, if I really wanted to pursue this a first thing to do would probably be to look at data on these four variables and see both which linear combinations change the most and what the time behavior looks like over the course of a year or two. I would think that the federal deficit is the most exogenous of the four, and that interpreting the correlations as exhibiting the causal effect of federal deficits would be a good first pass, but I would hope more generally that some regularity would suggest a next step in the investigation.
For a closed economy, total savings equals investment; investment in new capital is one of the important factors in the long-run increase in real wages. With an open economy, people from other countries can invest in US capital as well, so we get an accounting identity
federal deficit + investment = private savings + trade deficit
If the federal deficit were to come down, investment would go up, or private savings would go down, or the trade deficit would go down. What I'm wondering is how much of each would be likely to happen?Presumably it depends on the details -- at the coarsest level, a tax hike would probably have a different effect than a spending cut, and it seems likely to matter which tax you hike or which spending you cut. It seems like we ought to be able to say something useful about a general tendency, though, perhaps with caveats. Twentieth century macroeconomics would suppose that this is intermediated through interest rates, and would probably expect all three to absorb some of the change, with the portion depending on how sensitive investment, private consumption/savings decisions, and foreign trade partners (including foreign exchange markets) were to interest rates. The identity holds regardless of mechanism, though, and interest rates don't seem to have been obviously responsive to government deficits; perhaps more importantly, I don't think the people who deny any meaning to the deficit could be persuaded by this mechanism. The identity itself, however, is pure arithmetic. Something must give.
So what happens when the government gives a bunch of money to a group of people, whether in exchange for goods or services or not? In the latter case, those goods or services and/or the scarce resources that went into their production are no longer available; a Keynesian might believe that some nonscarce resource is being newly employed, in which case that's not a factor. If you're removing a consumer good from the market and prices aren't changing, it seems that you're forcing private savings or the trade deficit to change; if you're removing more of a capital good, perhaps you expect investment or the trade deficit to change. If you hand out the cash and it doesn't go anywhere, private savings is what automatically absorbs it at the first instance, but it seems likely that a lot of it would quickly go to investment or trade deficit.
While correlation is, of course, not causation, if I really wanted to pursue this a first thing to do would probably be to look at data on these four variables and see both which linear combinations change the most and what the time behavior looks like over the course of a year or two. I would think that the federal deficit is the most exogenous of the four, and that interpreting the correlations as exhibiting the causal effect of federal deficits would be a good first pass, but I would hope more generally that some regularity would suggest a next step in the investigation.
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