This post will have less content than usual, so I'll try to keep it short. This post is mostly here to jog my memory if I look at it in the future.
One of my interests is in how heterogeneity has important effects on aggregate economic variables that get lost with the "representative agent" framework. One somewhat-well known example is borrowing constraints; if they bind different agents differently those agents may behave, in aggregate, in a way that is very different from how any single agent might be expected to behave. There's a lot of literature on the idea that many of the homebuyers driving the recent housing price bubble were, in fact, acting at the time they did in part because they had recently had borrowing constraints eased. Other agents may, in this sort of model, still play a role in magnifying what might be a small bubble if it were left only to the agents whose borrowing constraint was loosened; agents bid up prices in anticipation of each other. Depending on the response functions, you may only need a small initial impetus to cause a dramatic change. (The multiplier may even be locally infinite in some sense where you have a no-longer stable equilibrium.)
Thursday, April 23, 2015
Thursday, April 16, 2015
Interpretations of Probability
I've been doing some reading (at Stanford's philosophy portal, among other places) and thinking about the meaning of probability — well, to some large degree on-and-off for at least 15 years, but a bit more "on" in the last month again. The page I linked to groups concepts into three groups, which they describe as "a quasi-logical concept", "an agent's ... graded belief", and "an objective concept" that I will conflate with one of their examples, the "frequentist" idea. My own interpretation of these ideas is that they form a nexus around "subjective" and "frequentist" ideas, with the formal mathematical calculus of probability connecting ideas to each other in important ways. What follows are mostly my own thoughts, though clearly building on the ideas of others; that said, I'm sure there is a lot of relevant philosophical literature that I have never seen, and even much that I have seen that I have not understood the way it was meant.
I'll start by referencing a theorem related to rational agent behavior. The upshot is that under reasonable assumptions, rational agents behave in such a way as to maximize "expected utility", where by "reasonable" I mean not that anybody behaves that way, but that if you could demonstrate to a reasonably intelligent person that they had not behaved that way, they would tend to agree that they had made a mistake somewhere. "Utility" is some numerical assignment of values to outcomes, and "expected utility" is its "expected value" under some mathematically consistent system of probabilities. The theorem, then, is that if a person's decisions are all in some normatively-appealing sense consistent with each other, there is some way of assigning probabilities and some way of assigning "utility" values such that those decisions maximized expected utility as calculated with those probabilities and utilities.
A related result that gets a lot of use in finance is that if "the market" isn't making any gross mistakes — again, in a normatively-appealing way, but also in a way that seems likely to at least approximately hold in practice — then there is some system of probabilities and payouts such that the price of an asset is the expected value of the discounted future cash flows associated with that asset. In finance texts it is often emphasized that this system of probabilities — often called the "risk-neutral measure" — need not be the "physical measure", and indeed most practitioners expect that it will put a higher probability on "bad outcomes" than the "physical measure" would. The "physical measure" here is often spoken of as an objective probability system in a way that perhaps sits closer to the "frequentist" idea, but if the market functions well and is made up mostly of rational agents whose behaviors are governed by similar probability measures, the "physical measure" used in models will tend to be similar to those. The point I like to make is that the "physical measure", in a lot of applications, turns out not to matter for finance; the risk-neutral measure is all you need. Further, the risk-neutral measure seems philosophically clearer; it's a way of describing the prices of assets in the market, and, implicitly, even prices of assets that aren't in the market.[1] It should be noted, though, that the "physical measure" is what people prefer for econometrics, so when one is doing financial econometrics one often needs both.
These contexts, in which a set of numbers on possible events has all of the mathematical properties of a probability system but need not correspond tightly to what we think of as "probability", play a role in my thinking.[2]
I think the most common definitions you would get for "probability" from the educated layman would fit into the frequentist school; the "probability" of an event is how often it would occur if you ran the same experiment many times. Now, the law of large numbers is an inevitable mathematical consequence of just the mathematical axioms of probability; if a "draw" from a distribution has a particular value with an assigned probability, then enough independent draws will, with a probability as close to 1 as you like, give that particular value with a frequency as close to the assigned probability as you like. If you and I assign different probabilities to the event but use the laws of probability correctly, then if we do the experiment enough times, I will think it "almost impossible" that the observed frequency will be close to your prediction, and you will think it "almost impossible" that it will be close to my prediction. Unless one of us assigns a probability of 0 or 1, though, any result based on a finite number of repetitions cannot be completely ruled out; inferring that one of us was wrong requires at some point deciding that (say) 1×10-25 is "practically 0" or 1-1×10-25 is "practically 1". For any level of precision you want (but not perfect precision), and for as small a probability (but not actually 0) as you insist before declaring a probability "practically zero", there is some finite sample size that will allow you to "practically" determine the probability with that precision. So this is how I view the "frequentist" interpretation of probability: the laws of probability are augmented by a willingness to act as though events with sufficiently low probability are actually impossible.[3]
More often, my own way of thinking about probabilities is closer to the "subjective" probability; "a probability" is a measure of my uncertain belief, and the tools of probability are a set of tools for managing my ignorance. It is necessarily a function of the information I do have; if you and I have different information, the "correct" probability to assign to an event will be different for me than for you.[4] If one of us regularly has more (or more useful) information than the other, then one of us will almost certainly, over the course of many probability assessments, put a higher probability on the series of outcomes that actually occurs than the other will; that is to be expected, insofar as my ignorance as to whether it will rain is in part an ignorance of information that would allow me to make a better forecast. There is a tie-in here to the frequentist interpretation as I cast it in the previous paragraph, related to Mark Twain's assertion that "history doesn't repeat itself, but it rhymes": not only is it impossible to take an infinite number of independent draws from a distribution, it is impossible to take more than one with any reliability. At least sometimes, however, we may do multiple experiments that are the same as far as we know — that is, we can't tell the difference between them, aside from the result. If we count as a "repetition" those events that looked that same in terms of the information we have[5], then we might have enough "repetitions" to declare that it is "practically impossible" that the probability of an observation, conditional on the known information, lies outside of a particular range.
One last interpretation of probability, though, is on some level not to interpret probability. (One might call this the "nihilist interpretation".) A fair amount of the "interpretations of probability" program seems oriented around the idea that whether an event "happens" or not, or whether something is "true" or not, is readily and cleanly understood, and there is some push to get probabilities close to 0 or 1, since we feel like we understand those special cases. We know, though, that our senses and minds are unreliable; everything we know about the world outside ourselves is with a probability that is, in honesty, strictly between 0 and 1. As we get close to 0, or close to 1, as a practical matter, the remaining distance will make no practical difference — it can't. But those parts of the world that are practically described by probabilities are in reality on a continuum with those we can practically treat differently, and consistently follow the laws of mathematics and nature, with 0 and 1 as, at the very best, special cases.
[1] If there are a lot of relevant possible assets that "aren't in the market", the risk-neutral measure may not be unique, i.e. there may be several different systems of probability that are consistent with the mathematical rules of probability and market prices; the conditions for existence are more practically plausible than the conditions required for uniqueness. Sometimes you might wish to a price a hypothetical asset whose price depends on which of the available risk-neutral measures you use, in which case existing prices will not fully guide you.
[2] As is noted at the Stanford link, there is some sense in which mass and volume can be made to behave according to the laws of probability; it is probably important to my philosophical development that the systems of "probability" I give in the text are closer in ineffable spirit to the common idea of "probability" than that.
[3] To some extent I'm restricting my discussion to "discrete" probability distributions to avoid having to talk much about "measurability", and to some extent I have failed here; if you flip a fair coin 100 times, any series of outcomes has a probability of less than 1×10-25. There are 161700 different series that contain 3 heads and 97 tails; if I don't distinguish between any of those 161700 different outcomes, then the probability of that single aggregate "3 heads" event is bigger than 10-25, even though any single way of doing it is not. If I insist on rounding the probability of each possible outcome to 0, then it is certain that an "impossible" outcome will result, but if I say "there are 101 measurable events, one for each possible number of heads," then the probability of an "impossible" outcome is extremely low (in this case, there are 6 such "impossible" outcomes, and they are, taken together, "impossible"). Ultimately you would probably want to take account of how many different events you want to distinguish when you're deciding what threshold you're rounding to 0; if you want to distinguish 1025 different events, then a probability threshold substantially smaller than 10-25 should be used.
[4] In some sense, this is what the Stanford site calls "objective probability", insofar as I'm asserting a "right" and "wrong" notion. What might be a conditional probability from the standpoint of the "objective" probability idea — that is, the probability conditional on the information we know — is what I'm thinking of here as my "prior" probability, along with an assertion that what from the "objective probability" standpoint would be a "prior" probability isn't actually meaningful.
[5] This, too, is basically "measurability", which is perhaps unavoidable in any non-trivial treatment of probability, even with finite "sample spaces".
I'll start by referencing a theorem related to rational agent behavior. The upshot is that under reasonable assumptions, rational agents behave in such a way as to maximize "expected utility", where by "reasonable" I mean not that anybody behaves that way, but that if you could demonstrate to a reasonably intelligent person that they had not behaved that way, they would tend to agree that they had made a mistake somewhere. "Utility" is some numerical assignment of values to outcomes, and "expected utility" is its "expected value" under some mathematically consistent system of probabilities. The theorem, then, is that if a person's decisions are all in some normatively-appealing sense consistent with each other, there is some way of assigning probabilities and some way of assigning "utility" values such that those decisions maximized expected utility as calculated with those probabilities and utilities.
A related result that gets a lot of use in finance is that if "the market" isn't making any gross mistakes — again, in a normatively-appealing way, but also in a way that seems likely to at least approximately hold in practice — then there is some system of probabilities and payouts such that the price of an asset is the expected value of the discounted future cash flows associated with that asset. In finance texts it is often emphasized that this system of probabilities — often called the "risk-neutral measure" — need not be the "physical measure", and indeed most practitioners expect that it will put a higher probability on "bad outcomes" than the "physical measure" would. The "physical measure" here is often spoken of as an objective probability system in a way that perhaps sits closer to the "frequentist" idea, but if the market functions well and is made up mostly of rational agents whose behaviors are governed by similar probability measures, the "physical measure" used in models will tend to be similar to those. The point I like to make is that the "physical measure", in a lot of applications, turns out not to matter for finance; the risk-neutral measure is all you need. Further, the risk-neutral measure seems philosophically clearer; it's a way of describing the prices of assets in the market, and, implicitly, even prices of assets that aren't in the market.[1] It should be noted, though, that the "physical measure" is what people prefer for econometrics, so when one is doing financial econometrics one often needs both.
These contexts, in which a set of numbers on possible events has all of the mathematical properties of a probability system but need not correspond tightly to what we think of as "probability", play a role in my thinking.[2]
I think the most common definitions you would get for "probability" from the educated layman would fit into the frequentist school; the "probability" of an event is how often it would occur if you ran the same experiment many times. Now, the law of large numbers is an inevitable mathematical consequence of just the mathematical axioms of probability; if a "draw" from a distribution has a particular value with an assigned probability, then enough independent draws will, with a probability as close to 1 as you like, give that particular value with a frequency as close to the assigned probability as you like. If you and I assign different probabilities to the event but use the laws of probability correctly, then if we do the experiment enough times, I will think it "almost impossible" that the observed frequency will be close to your prediction, and you will think it "almost impossible" that it will be close to my prediction. Unless one of us assigns a probability of 0 or 1, though, any result based on a finite number of repetitions cannot be completely ruled out; inferring that one of us was wrong requires at some point deciding that (say) 1×10-25 is "practically 0" or 1-1×10-25 is "practically 1". For any level of precision you want (but not perfect precision), and for as small a probability (but not actually 0) as you insist before declaring a probability "practically zero", there is some finite sample size that will allow you to "practically" determine the probability with that precision. So this is how I view the "frequentist" interpretation of probability: the laws of probability are augmented by a willingness to act as though events with sufficiently low probability are actually impossible.[3]
More often, my own way of thinking about probabilities is closer to the "subjective" probability; "a probability" is a measure of my uncertain belief, and the tools of probability are a set of tools for managing my ignorance. It is necessarily a function of the information I do have; if you and I have different information, the "correct" probability to assign to an event will be different for me than for you.[4] If one of us regularly has more (or more useful) information than the other, then one of us will almost certainly, over the course of many probability assessments, put a higher probability on the series of outcomes that actually occurs than the other will; that is to be expected, insofar as my ignorance as to whether it will rain is in part an ignorance of information that would allow me to make a better forecast. There is a tie-in here to the frequentist interpretation as I cast it in the previous paragraph, related to Mark Twain's assertion that "history doesn't repeat itself, but it rhymes": not only is it impossible to take an infinite number of independent draws from a distribution, it is impossible to take more than one with any reliability. At least sometimes, however, we may do multiple experiments that are the same as far as we know — that is, we can't tell the difference between them, aside from the result. If we count as a "repetition" those events that looked that same in terms of the information we have[5], then we might have enough "repetitions" to declare that it is "practically impossible" that the probability of an observation, conditional on the known information, lies outside of a particular range.
One last interpretation of probability, though, is on some level not to interpret probability. (One might call this the "nihilist interpretation".) A fair amount of the "interpretations of probability" program seems oriented around the idea that whether an event "happens" or not, or whether something is "true" or not, is readily and cleanly understood, and there is some push to get probabilities close to 0 or 1, since we feel like we understand those special cases. We know, though, that our senses and minds are unreliable; everything we know about the world outside ourselves is with a probability that is, in honesty, strictly between 0 and 1. As we get close to 0, or close to 1, as a practical matter, the remaining distance will make no practical difference — it can't. But those parts of the world that are practically described by probabilities are in reality on a continuum with those we can practically treat differently, and consistently follow the laws of mathematics and nature, with 0 and 1 as, at the very best, special cases.
[1] If there are a lot of relevant possible assets that "aren't in the market", the risk-neutral measure may not be unique, i.e. there may be several different systems of probability that are consistent with the mathematical rules of probability and market prices; the conditions for existence are more practically plausible than the conditions required for uniqueness. Sometimes you might wish to a price a hypothetical asset whose price depends on which of the available risk-neutral measures you use, in which case existing prices will not fully guide you.
[2] As is noted at the Stanford link, there is some sense in which mass and volume can be made to behave according to the laws of probability; it is probably important to my philosophical development that the systems of "probability" I give in the text are closer in ineffable spirit to the common idea of "probability" than that.
[3] To some extent I'm restricting my discussion to "discrete" probability distributions to avoid having to talk much about "measurability", and to some extent I have failed here; if you flip a fair coin 100 times, any series of outcomes has a probability of less than 1×10-25. There are 161700 different series that contain 3 heads and 97 tails; if I don't distinguish between any of those 161700 different outcomes, then the probability of that single aggregate "3 heads" event is bigger than 10-25, even though any single way of doing it is not. If I insist on rounding the probability of each possible outcome to 0, then it is certain that an "impossible" outcome will result, but if I say "there are 101 measurable events, one for each possible number of heads," then the probability of an "impossible" outcome is extremely low (in this case, there are 6 such "impossible" outcomes, and they are, taken together, "impossible"). Ultimately you would probably want to take account of how many different events you want to distinguish when you're deciding what threshold you're rounding to 0; if you want to distinguish 1025 different events, then a probability threshold substantially smaller than 10-25 should be used.
[4] In some sense, this is what the Stanford site calls "objective probability", insofar as I'm asserting a "right" and "wrong" notion. What might be a conditional probability from the standpoint of the "objective" probability idea — that is, the probability conditional on the information we know — is what I'm thinking of here as my "prior" probability, along with an assertion that what from the "objective probability" standpoint would be a "prior" probability isn't actually meaningful.
[5] This, too, is basically "measurability", which is perhaps unavoidable in any non-trivial treatment of probability, even with finite "sample spaces".
monetary policy and the theory of money
I have several dollars on top of my dresser, but most of my money (in pretty much any sense in which economists regularly use the word) exists as electronically-recorded liabilities of financial institutions. For most of my bills, it is more convenient to pay them out of such intangible money than the tangible money. Supposing we can still count the zero-interest-rate environment that has persisted for more than six years "abnormal", we have mostly shifted to a medium of exchange that pays interest, and the trajectory of technology (both information technology and financial technology) is toward more of that.
Traditional explanations of how monetary policy work often run more or less like this: the fed controls short-term interest rates, which affect the trade-off people make between holding their money in more liquid versus less liquid forms, and if they increase the amount they have in more liquid forms they spend more.[1] As the most liquid form of money starts to pay interest at a rate that moves more or less one-to-one with other interest rates, we face something of a paradox; the interest rate is effectively zero in terms of the actual medium of exchange, and the "interest rate" that the fed targets simply measures the rate at which the value of the dollar declines relative to that.
If people at that point are largely using interest-bearing deposits and funds as the actual store of value and medium of exchange in the economy, to what extent does this "dollar" whose value declines relative to it even matter? At least at first, it can continue to serve as a unit of account. Indeed, at this point it seems to have retained that function in the United States, even as it has largely lost the others; even where you see contracts with "indexing" of some sort, it's far more often to a price index than to something connected to interest rates per se. Perhaps over time contracts could start to have future cash flows stipulated in terms of the amount of money that would be in a bank account at that point in time if a specified amount had been deposited at the beginning of the contract, but there's no logical reason why the new medium of exchange would need to take over this last function of money.
Thus the dollar, increasingly, serves only as a unit of account, and will maintain its relevance only if it continues to serve for many purposes as a better unit of account than some alternative.[2] What makes a good unit account is not necessarily entirely the same thing that makes a good store of value or medium of exchange. To the extent that it does not, this new separation is in fact liberating for the Fed; it can focus on making the dollar a good unit of account, possibly allowing more volatility in its value than would be optimal if it were also a widespread store of value.
A business, for example, will typically have inputs that it purchases as it goes along, but will also require long-term inputs into the production process — a lease on a retail store, for example. (Employees who may, in principle, be freely dischargeable at-will employees, are probably in practice at least somewhat long-term inputs due to firm-specific knowledge and training and the costs of hiring and firing.) It also will produce products that may include short-term sales, longer-term contracts to supply clients, or both. It is likely that there will be some "duration mismatch" between inputs and outputs. In each case where the company is locked in to a decision years ahead of time, it risks a change in circumstances; if it is mostly selling as it goes along, it might wish to respond to an unexpected drop in demand by finding a way to cut production costs, but if it is selling mostly by long-term contract but has to buy its inputs day-to-day, it is subject to an increase in costs that it can't pass along. To the extent that it can specify prices in long-term contracts in terms of a unit of account that will drop in value if demand for its product goes down, or increase in value as competition for its supplies goes up, it will be easier for the company to responsibly engage in this business. A central bank that is trying to optimize its currency for use as a unit of account, therefore, will tend to devalue its currency when the economy in general is slowing down and increase its value (at least relative to expectations) when the economy is especially robust. These kinds of fluctuations in the value of actual holdings of the currency — long-term, as a store of value, or even short-term, as a medium of exchange, between the sale of one good or service and the purchase of another — will tend to make it less useful for those purposes. In a world where the central bank doesn't have to trade off these costs against the benefits of a countercyclical unit of account, it can focus on a better unit of account, while the other functions of money are provided elsewhere.
[1] There are (perhaps more compelling) arguments related to intertemporal substitution as well, but note that those explanations implicate the real interest rate rather than the nominal interest rate. You therefore need a story about how inflation and interest rates are simultaneously determined, and in particular why a decision by the fed to raise interest rates would reduce inflation expectations. These stories and explanations typically themselves come back to a "liquidity effect", so we're left with the same conundrum as the role of non-interest-bearing money atrophies.
[2]To some extent, as long as the government is using it as a unit of account — specifying tax liabilities, contract payments, and social security benefits in dollars, and even taxing the deviation between our new electronic currency and the dollar as "interest income" — it can be kept relevant by fiat.
Traditional explanations of how monetary policy work often run more or less like this: the fed controls short-term interest rates, which affect the trade-off people make between holding their money in more liquid versus less liquid forms, and if they increase the amount they have in more liquid forms they spend more.[1] As the most liquid form of money starts to pay interest at a rate that moves more or less one-to-one with other interest rates, we face something of a paradox; the interest rate is effectively zero in terms of the actual medium of exchange, and the "interest rate" that the fed targets simply measures the rate at which the value of the dollar declines relative to that.
If people at that point are largely using interest-bearing deposits and funds as the actual store of value and medium of exchange in the economy, to what extent does this "dollar" whose value declines relative to it even matter? At least at first, it can continue to serve as a unit of account. Indeed, at this point it seems to have retained that function in the United States, even as it has largely lost the others; even where you see contracts with "indexing" of some sort, it's far more often to a price index than to something connected to interest rates per se. Perhaps over time contracts could start to have future cash flows stipulated in terms of the amount of money that would be in a bank account at that point in time if a specified amount had been deposited at the beginning of the contract, but there's no logical reason why the new medium of exchange would need to take over this last function of money.
Thus the dollar, increasingly, serves only as a unit of account, and will maintain its relevance only if it continues to serve for many purposes as a better unit of account than some alternative.[2] What makes a good unit account is not necessarily entirely the same thing that makes a good store of value or medium of exchange. To the extent that it does not, this new separation is in fact liberating for the Fed; it can focus on making the dollar a good unit of account, possibly allowing more volatility in its value than would be optimal if it were also a widespread store of value.
A business, for example, will typically have inputs that it purchases as it goes along, but will also require long-term inputs into the production process — a lease on a retail store, for example. (Employees who may, in principle, be freely dischargeable at-will employees, are probably in practice at least somewhat long-term inputs due to firm-specific knowledge and training and the costs of hiring and firing.) It also will produce products that may include short-term sales, longer-term contracts to supply clients, or both. It is likely that there will be some "duration mismatch" between inputs and outputs. In each case where the company is locked in to a decision years ahead of time, it risks a change in circumstances; if it is mostly selling as it goes along, it might wish to respond to an unexpected drop in demand by finding a way to cut production costs, but if it is selling mostly by long-term contract but has to buy its inputs day-to-day, it is subject to an increase in costs that it can't pass along. To the extent that it can specify prices in long-term contracts in terms of a unit of account that will drop in value if demand for its product goes down, or increase in value as competition for its supplies goes up, it will be easier for the company to responsibly engage in this business. A central bank that is trying to optimize its currency for use as a unit of account, therefore, will tend to devalue its currency when the economy in general is slowing down and increase its value (at least relative to expectations) when the economy is especially robust. These kinds of fluctuations in the value of actual holdings of the currency — long-term, as a store of value, or even short-term, as a medium of exchange, between the sale of one good or service and the purchase of another — will tend to make it less useful for those purposes. In a world where the central bank doesn't have to trade off these costs against the benefits of a countercyclical unit of account, it can focus on a better unit of account, while the other functions of money are provided elsewhere.
[1] There are (perhaps more compelling) arguments related to intertemporal substitution as well, but note that those explanations implicate the real interest rate rather than the nominal interest rate. You therefore need a story about how inflation and interest rates are simultaneously determined, and in particular why a decision by the fed to raise interest rates would reduce inflation expectations. These stories and explanations typically themselves come back to a "liquidity effect", so we're left with the same conundrum as the role of non-interest-bearing money atrophies.
[2]To some extent, as long as the government is using it as a unit of account — specifying tax liabilities, contract payments, and social security benefits in dollars, and even taxing the deviation between our new electronic currency and the dollar as "interest income" — it can be kept relevant by fiat.
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