regressions on selected populations

Suppose there are two different sets of factors that affect some kind of outcome, and each moves toward optimizing the outcome, but in a noisy way.  I'm imagining two gradual processes, each affecting a subset of factors that affect the outcome, drifting generally toward values of those factors that increase some real-number-valued function of the factors; the function itself may move, somewhat gradually and smoothly, requiring the processes to chase it.  We might imagine them as multi-dimensional factors, but the language might be easier and more concrete if we imagine that each of them is simply a real number as well; x and y follow stochastic processes that drift in a direction of higher f(x,y).

Suppose, though, that x tends to move more quickly than y; it will, at any given time, tend to be closer to its optimal value.  Imagine, now, an ensemble of these, each trying to optimize the same function but otherwise moving independently of each other.  Cross-sectional differences in y will tend to capture differences in the extent to which the y values have adapted to the most recent function; if the optimal value of y has been increasing, there will be a pronounced tendency for higher values of y in the ensemble to be closer to the optimal value of y, with perhaps some overshooting, but, if y adjusts slowly compared to the rate at which its optimal value changes, probably relatively few points that have too high a value of y.  Values of x, on the other hand, will be scattered more or less randomly around the optimal value of x; if the noise now exceeds the residual mismatch between the average value of x and its current optimum, then the highest values of f will be associated with values of x near the middle of the distribution, rather than near one of its ends.

This is all assuming more or less the right ratios — that adjustment of x isn't too slow compared to changes in its optimum, that adjustment in y is fairly slow compared to changes in its optimum, and that in some practical sense they are each intrinsically of similar importance to f(x,y) relative to the noise in each variable.  The upshot, though, is that if we do a standard linear regression on how f varies with x and y within the observed cross-section, we find that x has little to no linear effect, while y does have one; even if we add nonlinear terms to the regression, x contributes only second-order terms, while y has both first- and second-order terms, and in general if you can pick to know only either x or y from a datapoint sampled from the population, knowing only y will enable you to predict f(x,y) with much more accuracy than knowing only x would.

There's literature suggesting that parenting is less important to a child's outcomes than genes are.  This is, however, conditional on most parents' trying to be decent parents; it is certainly the case that environment can matter, as we see with the victims of Romanian orphanages and other examples of violence and neglect.  It seems likely to me that parenting isn't, in some sense, "less important" than genes; it's just that most people have mostly done a decent job of adjusting their parenting to modern times, such that observed variations tend to have second-order effects, while our genes haven't caught up in the same way, and that differences there tend to be first-order.

privacy

Three and a half years ago, I was looking into the idea of doing research on people's desire for privacy.  I kind of gave up, though others have not, and perhaps I'll go back in that direction some day.  Today's post is triggered by recent news from my school that a student has a strain of meningitis, which information the school may be required to make public, though of course other regulations require that they not announce which student.  This seems like a useful example in which to lay out some spare thoughts I've had over the years.

The essential point is that there would be some public health justification for releasing the name of the building in which the student lives, the classes the student attends, possibly even student organizations in which the student has been active.  If you release all of those, it seems likely that the student would be uniquely identified.  Privacy is, at first order, less of a concern if you release any single one of those pieces of information; what I mostly want to note here is that, however you might try to place value on the student's privacy, one of the primary costs of releasing one of those pieces of information is that you thereby make it more expensive to release one of the others (and, similarly, more costly if one of them comes out by accident).  While there is perhaps some extent to which different information can be released to different people, insofar as the information is likely to spread, it seems likely that their decision not to release any of those extra bits of identifying information is probably the right cost-benefit decision, regardless of what legal liability rules HIPPA might impose as well.

unravelling

I'm making less of an effort than usual with this post to be interesting to people who aren't me.  I'm just formalizing the idea of "unravelling" of certain employment markets, in particular markets where there's some sense in which it's best for all firms and workers to be making offers that are in some sense simultaneous, but where individual agents might have an incentive to cheat, making deals before the thick market; in particular, under what conditions might a firm make a take-it-or-leave-it offer to a prospective employee before the market rather than waiting for the market?  In particular, it should be clear that, if the offer were really enticing, the employee would have accepted it in the regular market, and if it's not enticing enough, the employee won't accept it anyway.

Suppose there are two prospective firms hiring an employee, and let's examine the consequences of their making offers sequentially or simultaneously (in the sense of game theory).  Ex ante, the employee attaches probabilities p₁ and p₂ of getting offers from them, and gets utility u₁ or u₂ from accepting an offer. (The value of unemployment is set to 0, and may be assumed lower than either u_i.)  If firm 1 makes an offer that must be accepted or declined before receiving a response from firm 2, then declining the offer is worth p₂u₂; conditional on firm 1's offer, the expected payoff is
p₂u₂.

max{u1,p2u2}	or	p2u2.
If the offers are simultaneous, then those conditional values are
max{u1, u1(1-p2)+p2u2}	or	p2u2.

If the first firm fails to make an offer, the structure of the game is immaterial. Similarly, if the first firm is the worker's first choice, the structure is immaterial. If the worker gets an offer from the first firm but u₂>u₁, then the simultaneous game payoff, which can be written as u₁+max{0, p₂(u₂-u₁)}, is u₁+p₂(u₂-u₁)=p₂u₂+u₁(1-p₂); the simultaneous game payoff in these circumstances is higher by min{p₂(u₂-u₁),u₁(1-p₂)}.

The first firm gets the hire in the sequential game whenever it does in the simultaneous game, but also when

• It makes an offer
• firm 2 makes or would have made an offer
• u₂ > u₁ > p₂u₂

where the last condition is the condition in which the worker would choose firm 2 over firm 1 if given both choices but will choose firm 1 rather than take the chance that the other offer isn't coming.  This condition essentially captures the intuition of the last sentence of the first paragraph; by exploiting the worker's uncertainty as to whether a better offer would be forthcoming, the employer can induce the worker to take the "bird in the hand", but as the probability of another offer gets close to 1, there's an increasingly narrow range of offers the worker would accept now but not later.

the value of money

Let's continue on the idea of eggs as a medium of exchange.  In particular, while they make a decent store of value for a short period of time, they're terrible at storing value for very long; even if you boil them, they won't be a way to stockpile wealth on the scale of a generation or something.  They hold their value well enough, though, that one could reasonably use them to facilitate a couple of trades before they end up with their final consumer; eventually, though, their value is that of a straight consumable.

Now consider an agricultural economy with limited currency of other sorts such that eggs pick up at least some of the slack.  The consumer of the eggs presumably sees some gain from trade in the final transaction; that person values the eggs at least as highly as what is being exchanged for them.  The penultimate owner presumably values what is being received from the final owner more highly.  Now consider the trade between the antepenultimate owner and the penultimate owner of the eggs.  We suppose

• the transaction couldn't have happened had the eggs not been available as a medium of exchange
• there was a substantial gain from trade in the eyes of both parties
• if it is common knowledge that the eggs are deteriorating, the terms of trade should reflect this.

Young eggs, in particular, with three or four transactions left in them, might be worth more than eggs with only one or two transactions left in them — "eggs" may not be a unit of account (by which I really mean "standard of value"), exactly, even though they're a medium of exchange. At all both points, though, the eggs can acquire a liquidity premium in excess of the value that any consumer puts on them, essentially incorporating some of the gains from the trades they will facilitate into their initial value.

Now let's go back to money that is expected to last, in some practical sense, forever.  The social value of a dollar is the value of the transactions it can facilitate; a dollar that is return-dominated is, in some neutral unit of account, depreciating, but as long as the present discounted value of the gains from trade of the trades it will mediate is at least (say) a few dollars, the dollar can kind of steal that.  The more slowly the dollar deteriorates, the more marginal trades it can intermediate; a dollar that depreciates quickly will only be accepted as payment if the gains from trade are large.

barter and money

One of the classic virtues of money, as related by Jevons, is the reduction of the "double coincidence of wants" problem; I have noted, in this regard, that good markets attempt to mitigate the remaining "single coincidence of wants" problem.

Today is my grandmother's 90th birthday, and I was talking to her on the phone this evening when conversation turned to the low level functioning of the economy of Iowa in the 1930's.  One of the things she noted was that barter was more common in the rural areas than in town, in part because — very much not her words — a lot of farm products are sufficiently widely of at least some value that the double coincidence problem doesn't have an insuperable amount of bite; chicken eggs might not be your top choice of good to receive in exchange for what you're selling (likely labor), but it's likely to be of some positive use value to you, and, even if it isn't, it is similarly likely to be of even higher use value to someone from whom you would like to buy something, so that you might well be willing to accept it in trade anyway.