A lot of my conclusions here aren't much different from those of a previous discussion, but I'm going to frame/derive them slightly differently.
Suppose two agents are exogenously matched and given an exogenous date in the future for which they can construct a bilateral derivative; maybe we even require zero NPV, or maybe we allow for some cash transfer now, but either way if they're infinitely clever (and assuming e.g. that they don't anticipate future opportunities to insure before that date, etc.) then I believe the negotiations should leave the ratio of marginal costs of utility for the two agents at that date pre-visible. If we add constraints we modify that, but probably in relatively intuitive ways; if we can only condition on certain algebras of events (coarser than what would in principle be measurable at the final date), for example, then there's an expected value on each of those events that should give the same ratio, and if in some states an agent is unlikely to be able to make a payment, that agent is allowed to be better off than the other agent in that state relative to the usual ratio. Further, if there are a bunch of pairs of agents doing this, and the agents can be put into large classes, but need then to have very similar contracts, I'm probably doing more averaging over agents in each class.
I don't know whether this gets me any closer to an answer, but perhaps this is a useful framework for thinking about monetary policy as the medium of exchange consists increasingly of electronic (even interest-bearing) accounts and centrally-managed money is mostly about the unit of account. Buyers and sellers and debtors and lenders still referencing a given unit of account will tend to have certain risk similarities intraclass and differences interclass that one can try to optimize; if a surprise causes borrowers more pain than lenders, I try to weaken the unit of account, and if a surprise causes sellers more pain than buyers[1], then I try to strengthen the unit of account, and everyone ex ante looks at this and says "doing my contract (legal and explicit or customary and implicit) in this unit of account affords me a certain amount of insurance".
[1] A further note on the inclusion of "buyers" and "sellers" here: on some level this only matters for forward contracts, i.e. if we're entering an agreement to an immediate transaction there's none of this sort of uncertainty that resolves itself between the creation of the contract and its conclusion. Parties to a forward contract take on a lot of the properties of borrowers and lenders, insofar as there is a (say) dollar-denominated transfer in the future to which they've committed. Further, in principle borrowers, lenders, and parties to forward contracts can, as above, create their own risk-sharing contract. As a practical matter, of course, this is likely to be impossible to do perfectly, and it's likely that the extent to which it can be done practically leaves a lot of room for a central bank to come in and improve things. This is a usual theory-meets-practice kind of dynamic, especially in monetary theory; somewhat famously, perfect Walrasian economies don't need money, so a useful theory of money will have to figure out what parts of reality outside of Walrasian economics matters, and incomplete contracts would seem to be a biggie.
I believe, though, that more important than difficulties in contracting formally are informal contract-like substances that result from various incompletenesses in information. Buyers and sellers form long-term relationships that may be "at will" for each party, but are formed typically because one or both parties would incur some expense in looking anew for a counterparty each time a similar transaction was to take place. It seems likely to me that this would result in similar long-term dynamics to a contract, and is likely to involve prices that are sticky in some agreed-upon unit of account, whereupon a benevolent manager of that unit of account would again be trying to optimize as discussed above.
Tuesday, November 10, 2015
Wednesday, November 4, 2015
LQRE and quasi-strict equilibrium
A not-terribly-standard but conceivably interesting refinement of Nash equilibrium is "the rational limit of logistic quantal response equilibria". Last night I had myself convinced that it was related to quasi-strict equilibrium; if there is a relationship between the two, though, it's complicated. The simplest precise conjectures that I don't know to be false are (1) any quasi-strict NE has the same payoff as some RLLQRE and (2) any connected set of quasi-strict NE includes a RLLQRE. It may also be that there is a tighter relationship in 2 player games than more generally; in particular, every game has an RLLQRE, and every 2-player game has a quasi-strict NE, but not every larger game has a quasi-strict NE; it wouldn't surprise me if every RLLQRE in two-player games is quasi-strict, but that obviously can't be true in games that lack a quasi-strict equilibrium.
Monday, November 2, 2015
supply,demand, and fundamental value of assets
There is a sort of investor (who typically regards Warren Buffett as his idol) who insists that the true value of a security is the amount of cash can be expected to spin off, with some appropriate discounting of cash flows that are far off or uncertain. When deciding whether to buy a stock, the thing to do is figure out this value, and then to buy if the price is much lower than that value, and to sell if it is above that value, and simply stay away if it is in between. Short-term movements in the price are noise, and should simply be ignored.
There are other investors, more typical of hedge funds or (especially before Dodd-Frank) proprietary trading desks on Wall Street, who do worry about the short-term movements; "The long term is just a series of short terms," they might say, and the more prudent of them will repeat Keynes's dictum, "The market can remain irrational longer than you can remain solvent." "The price of a stock is determined by supply and demand" is not a view against which it is easy to argue.
It seems as though something "fundamental" ought to enter the demand for a stock at at least some level, but it also seems as though new shelf offerings by companies or even the expirations of lock-up periods should reduce the equilibrium price of the stock. How do these reconcile?
Modern asset pricing has in some ways moved closer to traditional economics, and in particular the idea that trade should (to some extent) be driven by mutual gains from trade; while many people view financial markets as zero-sum, even in the short-run that is only true ex post; different market participants may have different risk preferences, whether that means more or less risk-averse or exposed to one set of risks instead of another, and at least one socially valuable service of markets is to allow people who own a stock and find that current information implies that it is likely to be correlated with other risks they have to be paired up with people who don't own it who find that its expected return compensates them for any marginal risk it would give to their portfolios. The value of the asset in this model is in fact the sum of the cash flows it will yield, with appropriate discounts for cash flows that are far out or uncertain — but the appropriate "discounting" depends on each agent's own risk preferences and exposures. Even if everyone takes a Buffett-style "fundamental" approach, not everyone will agree on which stocks are "overpriced" and which are "cheap"; the various stocks, in equilibrium, will migrate to the people who are best equipped to handle the associated risks, and away from those who are least able.
If a particular stock has a risk-profile that is very different from that of any other stock — if it is not strongly correlated with any other stock, and hedges a risk for which no other stock provides a good hedge — then the agents whose risks have the lowest (signed) correlation with the stock are likely to find it uniquely valuable. If a stock is fairly strongly correlated with many others available, those will effectively be substitutes (in the demand-theory sense of the word) for it; the demand curve for any particular stock, with the others at fixed prices, will be less elastic; the price of the stock should be relatively insensitive to its own supply. It seems reasonable to me to think that, with realistic microstructure and transaction cost assumptions, stocks (or just about anything, really) will tend to be better substitutes for each other in the longer-run than in the shorter-run; in the short run, various frictions will more likely gum up the ability of agents to substitute between stocks, so that an exogenous, uninformative increase in supply of a stock will cause the price to drop in the short run (relative to the other stocks) and then tend back toward its previous parity, just compensating the marginal new buyer for the related transaction costs (/ liquidity provision).
Addendum: Related to this is the topic of stock buybacks. Recently some politicians have implied that stock buybacks are a sign of focus on the short-term at the expense of the long-run; if the price of the stock is its fundamental value, though, buying back stock is equivalent to issuing a dividend, and will reduce stock prices relative to what they would have been if the money had instead gone into useful investment. I think the model people have in mind is on some level not dissimilar to the one I've presented here, if perhaps less fleshed out: buying pressure lifts stock prices as people are slow to diversify out of it into similar investments, but ultimately the long-term shareholders are worse off than if real investments had been made instead.
There are other investors, more typical of hedge funds or (especially before Dodd-Frank) proprietary trading desks on Wall Street, who do worry about the short-term movements; "The long term is just a series of short terms," they might say, and the more prudent of them will repeat Keynes's dictum, "The market can remain irrational longer than you can remain solvent." "The price of a stock is determined by supply and demand" is not a view against which it is easy to argue.
It seems as though something "fundamental" ought to enter the demand for a stock at at least some level, but it also seems as though new shelf offerings by companies or even the expirations of lock-up periods should reduce the equilibrium price of the stock. How do these reconcile?
Modern asset pricing has in some ways moved closer to traditional economics, and in particular the idea that trade should (to some extent) be driven by mutual gains from trade; while many people view financial markets as zero-sum, even in the short-run that is only true ex post; different market participants may have different risk preferences, whether that means more or less risk-averse or exposed to one set of risks instead of another, and at least one socially valuable service of markets is to allow people who own a stock and find that current information implies that it is likely to be correlated with other risks they have to be paired up with people who don't own it who find that its expected return compensates them for any marginal risk it would give to their portfolios. The value of the asset in this model is in fact the sum of the cash flows it will yield, with appropriate discounts for cash flows that are far out or uncertain — but the appropriate "discounting" depends on each agent's own risk preferences and exposures. Even if everyone takes a Buffett-style "fundamental" approach, not everyone will agree on which stocks are "overpriced" and which are "cheap"; the various stocks, in equilibrium, will migrate to the people who are best equipped to handle the associated risks, and away from those who are least able.
If a particular stock has a risk-profile that is very different from that of any other stock — if it is not strongly correlated with any other stock, and hedges a risk for which no other stock provides a good hedge — then the agents whose risks have the lowest (signed) correlation with the stock are likely to find it uniquely valuable. If a stock is fairly strongly correlated with many others available, those will effectively be substitutes (in the demand-theory sense of the word) for it; the demand curve for any particular stock, with the others at fixed prices, will be less elastic; the price of the stock should be relatively insensitive to its own supply. It seems reasonable to me to think that, with realistic microstructure and transaction cost assumptions, stocks (or just about anything, really) will tend to be better substitutes for each other in the longer-run than in the shorter-run; in the short run, various frictions will more likely gum up the ability of agents to substitute between stocks, so that an exogenous, uninformative increase in supply of a stock will cause the price to drop in the short run (relative to the other stocks) and then tend back toward its previous parity, just compensating the marginal new buyer for the related transaction costs (/ liquidity provision).
Addendum: Related to this is the topic of stock buybacks. Recently some politicians have implied that stock buybacks are a sign of focus on the short-term at the expense of the long-run; if the price of the stock is its fundamental value, though, buying back stock is equivalent to issuing a dividend, and will reduce stock prices relative to what they would have been if the money had instead gone into useful investment. I think the model people have in mind is on some level not dissimilar to the one I've presented here, if perhaps less fleshed out: buying pressure lifts stock prices as people are slow to diversify out of it into similar investments, but ultimately the long-term shareholders are worse off than if real investments had been made instead.
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