Wednesday, August 27, 2014

On the global warming "pause"

The latest popular right-wing talking point on climate change is that we've entered a global warming "pause" which somehow proves that global warming is a hoax, or that humans aren't the cause. What is more disappointing is that I'm seeing more mainstream media outlets reporting about various theories of what might have "caused" the global warming "pause." The theories these mainstream reports cite are about oceanic warming. Skeptical Science explains perfectly why we should worry about oceanic warming:
It certainly is possible for the oceans to continue warming while measured surface temperature remains relatively constant, but that's not what the theory is about! Oceanic warming is not being proposed as an explanation for the global warming pause, because in reality there is no global warming pause that needs explaining.

The alleged "global warming pause" hype is an example of what is actually an excruciatingly common logical fallacy whereby people argue on the basis of perceived structural breaks in time trends which are really just white noise. It's a fallacy that's extremely familiar to macroeconomists, who spend their days staring at time-series data not unlike the global temperature data. The fact is that when you have a time-series that follows a statistical process known as a "random walk," deviations from the long-run trend are likely, large, and long-lasting. As I explained in a post about the stock market, a random walk with drift has a clear directionality--to move in the direction of the drift term--but no inherent tendency to regress to a One True Trend line.

The fallacy is this: people think that if global warming is real, then being below the historical trend should make a temperature increase over the next year more likely. Hence, the longer we remain below trend, the less likely it is that global warming is real. This is true of many data generating processes that are not random walks, but the existence of random walks proves that this type of reasoning is fallacious--in a random walk, being below the perceived time-trend this year has no impact on the probability of a temperature rise next year whatsoever (see unit root hypothesis).

That temperature rise follows a random walk with drift is almost surely the case. Because, what's the mechanism for the presumed regression to the mean? The earth's atmosphere either traps more of the sun's heat this year or it doesn't. There's a lot of reasons why it might not--perhaps cloud cover has incidentally reflected more light than usual, or maybe sunspots or other solar activity have reduced the amount of light radiating towards earth. Whatever the reason, one thing is clear: there is no particular reason to expect the environment to compensate next year--lower than average solar energy this year does not imply that next year we'll have higher than average solar activity! That is what it means to be a random walk. It does not contradict global warming at all--on average, temperatures are rising 0.03C per year and those increases are permanent, but also stochastic.

I've run some simulations to illustrate the structural break fallacy. Here's a random walk [$$]y_t=y_{t-1}+0.03+\epsilon_t[$$] where [$]y_t[$] is the global mean surface temperature in year [$]t[$]. The term [$]0.03+\epsilon_t[$] is the increase in temperature in year [$]t[$], where [$]\epsilon_t[$] represents the difference between the actual temperature increase this year and the average 0.03 per year increase over all years (hence, [$]\epsilon_t[$] is a normally distributed random variable with mean 0 and standard deviation of 0.2). I simulated global warming over 100-year spans 1000 times. One such simulation is plotted below
A simulated century of climate change.
Notice that if you were an observer in, say, year 50 of the graph above, it would appear that global warming had "paused" over the previous 20 years. But it didn't! We know it didn't because we controlled the time series process. With a random walk, such deviations from perceived trends are likely, and imply nothing about the veracity of global warming.

In the real world, commentators are claiming that we've experienced a pause based on only about 10 or so years worth of data. So I've run the above simulation 1000 times (script file here), and checked to see how often 10 years of constant or declining temperatures occur in a random walk like the one above, in which we know that global warming actually continues with no structural breaks. Here's the results:
Summary statistics showing how often perceived 10-year "pauses" might occur in a global warming process with no structural breaks.
As you can see, on average we could expect to see false global warming "pauses," like the one commentators are currently debating, about 15 times every 90 years. Almost 17 percent of the time it will look like global warming has "paused" over the previous decade, even though there were no actual structural breaks in any of these simulations!

So, I'm labeling this the structural break fallacy. When someone argues that there's been a structural break in a time-series like global mean temperature data, challenge them--their logic is incomplete unless they can first establish that the time-series should exhibit regression to the mean. If not, then large and persistent deviations from "trend" are not only possible, but likely.

Monday, August 25, 2014

Monday Morning Music

I'm finishing off my posts from Mozart's opera Abduction from the Seraglio with my favorite: a quartet between the Konstanze, Belmonte, Blondechen, and Pedrillo Olga Peretyatko really stole the show, in my opinion, which is impressive considering this is a production that includes Diana Damrau. I think that poor artistic choices on the director's part really hindered Damrau's performance.

Friday, August 22, 2014

Profit-sharing is not the solution to inequality

Profit-sharing forces poor workers to put all their eggs in a very risky basket.
In Fortune, sociologist Joseph Blasi thinks he has a better solution to inequality than Piketty's proposed global wealth tax. His solution is this: instead of taxing wealth as a way to give the non-rich an advantage in capital markets, we should simply promote "profit-sharing" between firms and their employees, by having firms compensate employees in stocks and dividends. But here's the problem: don't bring a sociologist to an economics fight profit-sharing is bad for workers. Especially the poorest ones.

You don't need a mathematical model to see why. Individual corporations are extremely risky. They may make a mistake--a flaw in the computer chip design, or a contaminate in their food products--that will put them out of business tomorrow, or at least cause their stocks to plunge shortly before it's time for you to retire. And even when they do everything right, corporations are still extremely risky--a mad cow scare in Virginia can easily put your beef packing plant in California out of business, or a new technology can come along and make your paper company obsolete. Merely by working at a company, workers are subjected to huge amounts of risk because they could loose their future wages at any moment if they are laid off. But, at least in that eventuality their savings will not also be lost when they loose their future wages.

Except when they have a profit-sharing agreement with their company. These agreements require that part of worker's wages will be paid not in cash but in stocks, with strings attached about when and how these stocks can be traded. The result is that profit-sharing arrangements require workers to invest not just their future wage incomes in their extremely risky employer, but also a large share of their savings too. Now when their employer goes bankrupt, they loose both their source of income and their life savings. Profit-sharing magnifies risks to workers, and no one is harmed more by this than the poor.

I'm not saying that we shouldn't encourage the poor to invest in equities. I'm saying that they are much better off being able to diversify that investment to insure themselves against the extremely high amount of risk that any single firm represents.

We can examine this with math. Households value consumption according to [$]U=E\left[u\left(C\right)\right][$] where [$]u[$] is a continuous strictly concave increasing function--that is, households want to consume more, but experience diminishing returns. The household's wage income is [$]y[$], and there are two identical firms that earn [$]\pi_i[$] profits. Each firm faces a probability [$]p[$] that it will have a disaster (say, a recall) that will reduce profits by [$]D[$] so that [$]\pi_i=\pi[$] with probability [$]1-p[$] and [$]\pi_i=\pi-D[$] with probability [$]p[$]. The two firms are identical, but their disaster risks are independent. A household works for just one of these firms.

Under profit sharing, household utility is [$$] U=pu\left(\pi-D+y\right)+\left(1-p\right)u\left(\pi+y\right)[$$] because the individual is forced, by the profit sharing agreement, to own the firm's equity and therefore bears the firm's disaster risk. Without profit sharing, total worker's compensation is the same, but the worker is no longer required to invest in his own firm's equity. He therefore buys a diversified portfolio of both firm's equity, such that his portfolio earns [$]\pi[$] with probability [$]\left(1-p\right)^2[$], [$]\pi-\frac{1}{2}D[$] with probability [$]2\left(1-p\right)p[$], and [$]\pi-D[$] with probability [$]p^2.[$] The expected returns on this portfolio are identical to the expected returns in the profit-sharing case, but with a much smaller variance--he is less likely to earn [$]\pi[$], but a lot more likely to earn more than [$]\pi-D[$]. The utility without profit sharing is [$$]U=\left(1-p\right)^2u\left(\pi+y\right)+2\left(1-p\right)pu\left(\pi-\frac{1}{2}D+y\right)+p^2u\left(\pi-D+y\right).[$$] The only assumption needed to show that [$]U[$] is higher (and therefore the household is better off) without profit-sharing than with is that [$]u[$] is strictly concave, which is equivalent to saying that people are risk averse which is equivalent to saying people experience diminishing returns.

In fact, most economists would just say that my result follows directly from the definition of concavity, though this may be less obvious to non-math folks. Indeed, what constitutes a mathematical "proof" depends entirely on the knowledge level of the reader. If you are aware, for example, that a continuous function [$]f[$] is strictly concave on a set [$]C[$] if and only if for any [$]x,y\in C[$] [$$]f\left(\frac{x+y}{2}\right)\gt\frac{f\left(x\right)+f\left(y\right)}{2}[$$] I could simply start there. But if you don't know that, then a proof that assumes that doesn't really prove anything, does it? I had planned on simply linking you to the wikipedia page to assert that this is true--you can bug them if you remain unconvinced--but it turns out wikipedia only asserts the weak form of concavity ([$]\geq[$]) and not the strict form I'm using here ([$]\gt[$]). So I'm just going to tell you that this result is easily shown by taking wikipedia's definition of strict concavity and setting [$]t=\frac{1}{2}[$]. That let's us jot down the following proof:

Proof:
Recall that utility in the non-profit-sharing case was given by [$$]U=\left(1-p\right)^2u\left(\pi+y\right)+2\left(1-p\right)pu\left(\pi-\frac{1}{2}D+y\right)+p^2u\left(\pi-D+Y\right).[$$] Therefore,
\begin{align}
U&=\left(1-p\right)^2u\left(\pi+y\right)+2\left(1-p\right)pu\left(\pi-\frac{1}{2}D+y\right)+p^2u\left(\pi-D+Y\right)\\
&\gt\left(1-p\right)^2u\left(\pi+y\right)+2\left(1-p\right)p\frac{u\left(\pi+y\right)+u\left(\pi-D+y\right)}{2})+p^2u\left(\pi-D+Y\right)\\
&=\left(1-p\right)u\left(\pi+y\right)+pu\left(\pi-D+Y\right)
\end{align}
which was the utility in the profit sharing case. So, profit sharing is worse.

Thursday, August 21, 2014

Racial profiling of sickle cell patients

Scientists have identified four separate evolutionary origins of the sickle cell characteristic. Three of them were in Africa.
A news report from Seattle suggests that black sickle cell patients that go to emergency departments (EDs) for pain are often discriminated against because of their race:
"Karim Assalaam and Joehayward Wilson are both living with sickle-cell anemia. This genetic blood disorder is most common among people of African and Mediterranean descent. It can create intense internal pain. But when these two 24 year-old African-American go to emergency rooms for help, they are often met with suspicion.

"'I think it is because of being a young adult and African-American,' Wilson told us.

"The pair are among the 12,000 people in Washington State who carry the genetic trait for Sickle Cell Anemia. Wilson and Karim Assalaam are among the 450 living with the disease.

"'It's hard, people assume you're only out for the drugs or that's the only thing you're coming to there for,' Assalaam said.

"Sickle-cell sufferers frequently use Oxycontin for the pain. The drug is a narcotic that’s often abused by addicts who crush the slow-acting pills and then use the powder for a quick high."
There's now a fair amount of research establishing that medical staff tend to underestimate pain levels of black patients relative to white ones and that while this bias is larger for white doctors, black doctors also underestimate pain of black patients relative to white patients. Hence, it is not terribly surprising that ED staff--who do not specialize in sickle cell--would suspect black patients of abusing drugs, because they probably underestimate the amount of pain in black patients.

Sickle cell disease is not particularly common, affecting less than 3 in every 10,000 Americans. This means dedicated 24 hour sickle cell clinics staffed with doctors with experience with sickle cell are not financially viable, even though sickle cell patients could have severe episodes of pain or even life-threatening complications at any hour of the day. As a result many sickle cell patients rely at least in part on non-specialists in EDs to help manage their chronic condition, which is why this discrimination in pain prescriptions can be particularly problematic in this population.

Tuesday, August 19, 2014

On law enforcement


Yup. You should be an economist.
The crisis in Ferguson has prompted a national dialogue about law enforcement tactics and the unfair targeting of innocents through "tough-on-crime" policies like racial profiling, mandatory minimums, and criminal procedures that make it easier to convict. However, economics tells us that "tough-on-crime" tactics do not always maximize law and order. The unfair targeting of innocent people for criminal investigation through racial profiling and stop-and-frisk style tactics actually increases the incidence of criminality at the margins. Here's how.

A Model in two parts

1   Modeling the incidence of criminality

We consider a model with a large number of households with preferences over consumption [$] C [$] and jail [$] J [$] according to [$] U=E_B\left[u\left(C\right)-J\right] [$] where [$] u [$] is strictly concave increasing in [$] C. [$] The household is endowed with lawful income of [$] y [$] and has the option of committing a burglary to steal [$] B [$] units of consumption. If the household is convicted, he serves jail time that yields [$] J [$] units of disutility (we can think of disutility from jail as being a function [$] j\left(s\left(B\right)\right) [$] where [$] s\left(B\right) [$] is a policy function prescribing sentences based on the magnitude [$] B [$] of the crime, and [$] j\left(\cdot\right) [$] as the utility function of jail time. However, we consider the extensive margin where [$] B [$] is fixed.)

The judicial authority investigates individuals with probability [$] q [$] and the investigation leads to a conviction rate of [$] r_1\lt 1 [$] among those who committed the crime (that's a sensitivity of [$]r_1[$]), and [$] r_2\lt r_1 [$] among individuals who are innocent (that's a specificity of [$]1-r_2[$]). Therefore, the probability of being convicted given that the household commits the crime is [$] p_1=qr_1 [$] and the probability of conviction given hat the household does not commit the crime is [$] p_2=qr_2. [$]

The household's budget constraint is [$] C\leq y+B [$] if he commits the crime, and [$] C\leq y [$] otherwise. The household choice has two regimes, one where he commits crimes with probability 1, and one where he commits crime with probability 0, where utility from the former is [$] u\left(y+B\right)-p_1 J [$] and the utility from the latter is [$] u\left(y\right)-p_2 J. [$]

Borrowing from the indivisible labor literature1 and assuming a functional form for the utility function, we can rewrite the above in terms of a representative agent that chooses a probability of committing crime [$] \alpha [$] according to
\begin{align*}
\max_{\alpha}~ ln\left(C\right)-\alpha p_1J-\left(1-\alpha\right)p_2J&\\
subject~to~C\leq y+\alpha B&
\end{align*}
Solving yields [$$] \alpha^*=\frac{1}{J\left(p_1-p_2\right)}-\frac{y}{B} [$$] This model captures our intuitions about criminal justice. For example, it is plainly apparent from the solution that increasing penalties [$] J, [$] all else equal, reduces crime rates. It is often assumed that stepping up investigations--that is, targeting a larger share of the population for investigations--will result in a reduction in crime rates. To examine this we take the derivative of [$] \alpha [$] with respect to the investigation rate [$] q: [$] [$$] \frac{\partial \alpha}{\partial q}=\frac{J}{\left(J\left(p_1-p_2\right)\right)^2}\left(\frac{\partial p_2}{\partial q}-\frac{\partial p_1}{\partial q}\right) [$$] which is less than zero if and only if [$$] \frac{\partial p_1}{\partial q}>\frac{\partial p_2}{\partial q}. [$$]

2   Modeling the profiling decision

To understand that last derivative, we need a model of police profiling. Mixing models of heterogeneity with a representative agent framework can be problematic, but lets assume that utilities are such that this is valid. Assume that individuals are heterogeneous in such a way that their probability [$] \tilde{\alpha} [$] of committing a crime--as measured by a hypothetical social planner--is distributed according to a continuously differentiable distribution function [$] F\left(\tilde{\alpha}\right) [$] with support [$] \left[0,1\right]. [$] The judicial authority prioritizes investigations of individuals so that individuals with the highest [$] \tilde{\alpha} [$] probabilities are investigated first, followed by progressively lower probability types until they've exhausted their investigative resources--that is, until the share of the population being investigated equals the policy parameter [$] q. [$] Thus we can write [$$] q\equiv 1-F\left(\bar{\alpha}\right) [$$] where [$] \bar{\alpha} [$] is the lowest probability type to be investigated. Therefore, we have that
\begin{align*}
p_1&=\underbrace{\left(1-F\left(\bar{\alpha}\right)\right)}_{q}\underbrace{\int^{1}_{\bar{\alpha}}\tilde{\alpha}f\left(\tilde{\alpha}\right)d\tilde{\alpha}}_{\alpha}r_1\\
p_1&=\underbrace{\left(1-F\left(\bar{\alpha}\right)\right)}_{q}\underbrace{\left(1-\int^{1}_{\bar{\alpha}}\tilde{\alpha}f\left(\tilde{\alpha}\right)d\tilde{\alpha}\right)}_{1-\alpha}r_2
\end{align*}
where [$] f\left(\tilde{\alpha}\right) [$] denotes the density function of [$] \tilde{\alpha} [$] and therefore the first derivative of [$] F\left(\tilde{\alpha}\right). [$] Hereafter we will write [$] E_\bar{\alpha} [$] to denote [$] \int^{1}_{\bar{\alpha}}\tilde{\alpha}f\left(\tilde{\alpha}\right)d\tilde{\alpha}, [$] which is the expected criminality of the population being investigated.

Thanks to the Leibniz rule, we can differentiate this to get
\begin{align*}
\frac{\partial p_1}{\partial q}&=E_\bar{\alpha}r_1+\bar{\alpha}\left(1-F\left(\bar{\alpha}\right)\right)r_1\\
\frac{\partial p_2}{\partial q}&=\left(1-E_\bar{\alpha}\right)r_2-\bar{\alpha}\left(1-F\left(\bar{\alpha}\right)\right)r_2
\end{align*}
Therefore, using the result derived in the first section, increasing enforcement decreases crime only if
\begin{equation}
E_\bar{\alpha}+\bar{\alpha}\left(1-F\left(\bar{\alpha}\right)\right)\gt \frac{r_2}{r_1+r_2} \label{conditions}
\end{equation}
We can now state two propositions.

Proposition 1.

It is optimal to investigate everyone if and only if [$] E_0\geq \frac{r_2}{r_1+r_2}. [$]

Proof: Sufficiency follows immediately from \eqref{conditions} with [$] \bar{\alpha}=0. [$] Necessity follows from Proposition 2.


Proposition 2.

If [$] E_0\lt\frac{r_2}{r_1+r_2}, [$] then there exists [$] q^*\gt 0 [$] such that [$] \frac{\partial \alpha}{\partial q}\geq 0 [$] for all [$] q\gt q^*, [$] with strict inequality whenever [$] f\left(\tilde{\alpha}\right)\gt 0. [$] That is, there exists a point beyond which further increasing enforcement actually increases crime.

Proof: Denote [$] G\left(\bar{\alpha}\right)\equiv E_\bar{\alpha}+\bar{\alpha}\left(1-F\left(\bar{\alpha}\right)\right). [$] Then [$] G\left(1\right)=1\gt\frac{r_2}{r_1+r_2} [$] because [$] E_1=1 [$] and [$] F\left(1\right)=1. [$] Moreover, we postulated that [$] G\left(0\right)=E_0\lt \frac{r_2}{r_1+r_2}. [$] Since [$] F\left(\bar{\alpha}\right) [$] is continuously differentiable, [$] G\left(\bar{\alpha}\right) [$] is continuous and by the intermediate value theorem there exists a nonempty set [$] A [$] such that for [$] \hat{\alpha}\in A [$] we have [$] G\left(\hat{\alpha}\right)=\frac{r_2}{r_1+r_2}. [$] Let [$] \bar{\alpha}=\min \hat{\alpha}\in A, [$] and [$] q^*=1-F\left(\bar{\alpha}\right), [$] then for all [$] \tilde{\alpha}\in \left[0,\bar{\alpha}\right) [$] we have [$] G\left(\tilde{\alpha}\right)\leq \frac{r_2}{r_1+r_2} [$] with strict inequality if [$] f\left(\bar{\alpha}\right)\gt 0. [$] Furthermore, note that [$] q\equiv 1-F\left(\tilde{\alpha}\right) [$] is monotonically decreasing in [$] \tilde{\alpha}, [$] and is one-to-one when [$] f\left(\tilde{\alpha}\right)\gt 0, [$] which implies that for [$] q\in \left[0,q^*\right) [$] we must have [$] G\left(\tilde{\alpha}\right)\leq \frac{r_2}{r_1+r_2} [$] with strict inequality when [$] f\left(\bar{\alpha}\right). [$] This concludes the proof.


So what do these propositions say about stop-and-frisk? We are compelled to draw conclusions contrary to the beliefs of the New York City police commissioner: economics tells us that we can actually reduce crime by not investigating those individuals who are least likely to commit crimes, because this will reduce the wrongful conviction rate and increase the incentive to avoid committing crime. Stop-and-frisk policies do precisely the opposite: they target investigations indiscriminately at the public, innocent and guilty alike, which will increase the wrongful convictions and obviate the disincentive our justice system aims to place on criminal acts.

So there you have it. Economics tells us that stop-and-frisk causes crime.

1. I'm probably not the first one to have applied the indivisible labor literature to criminality in this way, though I did not do a search. If you know of any papers that I have incidentally duplicated, let me know so I can give credit here.