Reporters often call me for insight into various issues or just a quotable comment. Yesterday, however, I had a hard time believing the policy the reporter wanted to talk about was real.

Tacoma, Washington wants to install boxes around the city so people can drop off their guns, no questions asked. But, but, but…seriously?

After the reporter assured me this was real I offered the following:

George Mason University economics professor Alex Tabarrok said [gun buyback] programs have been proven ineffective, and he predicted the same result for a drop box scheme. Anyone in the port city who wants to get rid of a gun discreetly, Tabarrok said, has an easy option: toss it in Puget Sound.

Not really something you need a PhD for but when you are tossed a softball you may as well hit it.

There is something to be said for simply reducing the hassle of disposing a gun for those who don’t want to sell but as a crime control measure this is a bust. Regardless of your views on guns not much will change if you don’t do something nationally about the big boxes where people can buy guns (otherwise known as stores).

I recently ran across a problem using the Frechet probability bounds. The bounds weren’t immediately obvious to me and Google didn’t enlighten so I wrote an intuitive explanation. Super wonkiness to follow. You have been warned.

Suppose that we have two events A and B which occur with probability P(A) and P(B). We are interested in saying something about the joint event P(A∩B), the probability that both events occur, but we don’t know anything about whether the events are correlated or independent. Can we nevertheless say something about the joint event P(A∩B)? We can. The Frechet bounds or inequalities tell us:

max[0,P(A)+P(B)-1] ≤ P(A∩B) ≤ min[P(A),P(B)]

In words, the probability of the joint event can’t be smaller than max[0,P(A)+P(B)-1] or bigger than min[P(A),P(B)]. Let’s give an intuitive explanation.

The events themselves are not important only the probabilities so let’s use an intuitive model for the events. Let x be a random number distributed between 0 and 1 with each number equally likely (i.e. a uniform distribution). Suppose that the event A occurs whenever x is in some region between a and b. For example, we might say that event A occurs if x is between .4 and .6 and event B occurs if x is between .1 and .7. Notice also that since any number between 0 and 1 is equally likely the probability that event A occurs is just the width of the A region, b-a. In this case, for example, P(A)=.6-.4=.2 and similarly P(B)=.7-.1=.6.

Now suppose that a demon arranges the events A and B to minimize the probability that both events occur. What is the smallest the demon could make P(A∩B)? To minimize the probability that both A and B occur we can think of the demon as placing the two events “as far apart” as possible. For example, the demon will begin one event at 0 and define it as the region moving right until the width is equal to P(A) and the other event the demon will begin at 1 and move left until the width is equal to P(B). If P(A)=.2 and P(B)=.4, for example, then the demon will define the events so that A∈[0,.2] and B∈[.6,1]. Here’s a picture. Notice that by beginning one event at 0 and the other at 1 the demon minimizes the overlap which is the probabiity that both events occur.

In this case, the events don’t overlap at all and so the demon has ensured that the probability of P(A∩B)=0.

But what if the two events must overlap? Suppose for example that P(A)=.5 and P(B)=.7 then the demon’s logic of minimization leads to the following picture:

We can see from the picture that the overlap is the region between .3 and .5 which we know has probability .5-.3=0.2. Thus we have discovered that if P(A)=.5 and P(B)=.7 then the joint probability has to be at least .2, i.e. .2≤P(A∩B).

Now let’s generalize. The probability of any event occuring is the length of its region b-a but since the demon always begins the event A at 0 then the probability of P(A) is simply b the rightmost point of the A region. Thus, on the diagram the rightmost point of the A region is labelled P(A) (.5 in this case). Also since the demon always begins the B region at 1, P(B)=1-a′ where a′ is the leftmost point of the B region so rearranging we have that the leftmost point a′=1-P(B) as shown in the picture at .3. Thus, we can read immediately from the figure that the overlap region has length P(A)-(1-P(B)). Notice that if P(A)<1-P(B) then there is no overlap and the difference is a negative number. Rearranging slightly, the length of the overlap region is P(A)+P(B)-1. So now we have two cases, either there is no overlap at all in which case P(A∩B)=0 or there is overlap and P(A∩B)=P(A)+P(B)-1. So putting it all together we have proven that:

max[0,P(A)+P(B)-1] ≤ P(A∩B)

The Frechet bounds also say:

P(A∩B) ≤ min[P(A),P(B)]

This is much easier to show. If minimization is accomplished by minimizing the overlap then maximization is accomplished by maximizing the overlap. To maximize the joint probability the demon starts both regions at the same point, say at the 0 point, so the picture looks like this:
The joint probability is then the region of overlap. But the regions can’t overlap by more than the smallest of the two regions! In this case the A region is smaller than the B region and since the regions start at 0 we have P(A∩B)≤P(A). Generalizing we have that:

P(A∩B) ≤ min[P(A),P(B)]

Thus we have proven the Frechet bounds:

max[0,P(A)+P(B)-1] ≤ P(A∩B) ≤ min[P(A),P(B)]

Addendum: Here is Heckman on the theory of Frechet Bounds and Heckman and Smith and Heckman, Smith and Clements with applications.

Over at Econlog Bryan Caplan responds to a Scott Alexander post on the bargaining power of workers. Bryan makes excellent points. I want to focus on two larger issues. Many people look at big firms and little workers and they see an imbalance and can’t imagine how the firms are not in control. Even framing the issue as the bargaining power of workers makes it seem like a David and Goliath battle.

The firm v. worker framing focuses attention on the threat to the worker of unemployment. From this perspective it seems as if the firm can “bargain” the worker down to the least the worker is willing to accept and, given the threat of unemployment, that isn’t much.

The firm versus worker framing, however, obscures a point that Tyler and I make in Modern Principles: Buyers don’t compete against sellers, buyers compete against other buyers (and sellers compete against other sellers). Firms buy labor and they are competing primarily not against workers but against other firms. Firms versus Firms! Now that is a real battle!

When firms are thinking about wages what they are thinking about is the threat from other firms. When a firm is hiring it knows it must pay the worker at least as much as other firms are willing to pay.

The other-firm threat is very real. In fact, more often that not when a worker-firm match breaks up, it’s the worker who leaves, usually for another firm (the hot, young startup?), rather than the firm who leaves the worker with a layoff. The figure below shows data on quits divided by layoffs. Most of the time quits exceed layoffs (even during some recessions) with only brief windows during severe recessions when quits are fewer than layoffs.

Using the firms versus firms frame it becomes clear that rather than a battle between firms and workers, firms are a worker’s best friend. To be sure, it’s the firms that the workers don’t work for who are their best friend not necessarily the firm they do work for! Indeed, another problem with the firm versus worker frame is that in their eagerness to win the “battle” with their firm workers sometimes support policies which harm all firms, including their friends. A case of cutting off the nose to spite the face. Remember, firms are buyers and sellers benefit when there are lots of rich, successful buyers.

Hat tip: Justin Merrill.

Addendum: Bargaining can be important when the hiring firm is willing to pay the worker considerably more than are other firms–this can happen for highly-skilled workers with specific talents and unique firm-complementarities. Note that in these cases it’s more a case of the worker “bargaining up” to grab surplus than the firm bargaining down. For workers as a group, this kind of surplus is gravy. But there’s nothing wrong with gravy so if this applies to you do read Getting to Yes and learn to bargain well.

Here is a great little story by Danny Vinik from Politico’s The Agenda on how so-called marketing boards are surreptitiously turned into lobbying boards.

Industries with a large number of producers find it difficult to organize collectively because of the free rider problem. Mostly, that’s a good thing because it prevents cartels. Collective action, however, could also be used to perform research or marketing that’s good for the industry as a whole but too expensive for any small subset of producers. In theory, therefore, some type of collective action could be beneficial and in agriculture governments have created checkoff programs which force producers to pay a tax to fund collective goods.

Checkoffs exist for dairy farmers, mushroom producers, and even popcorn processors. Critics say they violate economic freedom and distort the market; big corporate farmers, they allege, easily find ways to influence the boards and siphon the money off to push their own causes.

“In one sense, it’s a classic case of the larger producers are the more powerful political forces within these organizations,” said Dan Glickman, the Agriculture Secretary at the end of the Clinton administration who largely supports checkoff programs.

For the unhappy hog farmers, the current problem started with the 1985 Pork Law, when Congress set up the National Pork Board and required all farmers to contribute. Today, hog farmers must hand over 40 cents out of every $100 in revenue from pork sales. The board uses the money, totaling nearly $100 million a year, to conduct research and promote the pork industry, but is not allowed to lobby.

But as Adam Smith said “People of the same trade seldom meet together, even for merriment and diversion, but the conversation ends in a conspiracy against the public, or in some contrivance to raise prices.” Quite so. And in this case by creating a National Pork Board the government is providing the meeting hall and paying for the conversation. According to the law, the money from the checkoff program isn’t supposed to go for lobbying but here is where the story gets interesting.

You may recall the slogan, “Pork: The Other White Meat.” The slogan hasn’t been used for years but the National Pork Board still pays $3 million a year every year for the rights. Why would the Pork Board pay millions for an unused slogan? The key is who they are paying. The slogan is owned by National Pork Producers Council. The NPPC is a lobby group and you won’t be surprised to know that it is closely connected with the NPB (having once even shared offices).

…critics say the two groups have never been as separate as the law calls for, and now are essentially colluding through a deal that lets the Pork Board funnel money to the NPCC by assigning an absurdly inflated value to the “other white meat” slogan; the money then goes to promote the NPPC’s lobbying agenda.

A neat trick. The story is also a good object lesson in Mancur Olson’s thesis about how special interest groups grow in power over time, slowly choking off innovation as they cartelize the economy.