# A simple model of gun ownership and crime rates

Does gun ownership increase or reduce violent crime? I’m not sure whether a model for this already exists, but in this post, I attempt to model the causal effect of gun ownership on violent crimes in this post (I’ll share any relevant literature I find, if I find any, in subsequent posts). I’ll jump straight into the model setup, with explanations injected as I go along.

Model setup

• We assume that a person makes the decision about whether or not to commit a violent crime based on two variables: their innate desire to commit the crime ( $\beta_{i} \sim U(0,1)$) and their probability of success ( $\chi_{i}$). We represent this with $\kappa_{i} = 0.5(\beta_{i} + \chi_{i})$.
• Their probability of success depends on two factors: some innate ability to use force ( $a_{i} \sim U(0,1)$) and the ability of the other party to use force ( $a_{-i} \sim U(0,1)$). Their innate ability is known to them. The ability of the other party is not. We represent this with $\chi_{i} = a_{i} - a_{-i} + 0.5$, i.e. if you are better able to use force than the other party ( $a_{i} > a_{-i}$), it is likely that you successfully commit the crime and subdue them with force, with your probability of success increasing linearly as the gap between your ability and the other party’s ability grows. If you are less able to use force than the other party ( $a_{i} < a_{-i}$), the converse applies. If your ability is equal to that of the other party ( $a_{i} = a_{-i}$), your probability of success is $\chi_{i} = 0.5$, i.e. you have a 50% chance of success and 50% chance of failure.
• If $\kappa_{i}$ crosses some threshold, $\bar{\kappa}$, $1 > \bar{\kappa} > 0.5$, then the person will commit a crime. If it doesn’t, then they will not. We set the upper bound on $\bar{\kappa}$ such that there will always be people who commit violent crimes ( $\bar{\kappa} < 1 =$ maximum $\kappa$). We set the lower bound on $\bar{\kappa}$ such that there will always be people who deeply want to commit violent crimes ( $\beta_{i} = 1$) but who are completely unable to ( $\chi_{i} = 0$) and so $\kappa_{i} = 0.5 < \bar{\kappa}$, and these people do not commit violent crimes. Similarly, people who are very able to commit violent crimes ( $\chi_{i} = 0$) but really don’t want to ( $\beta_{i} = 0$) have $\kappa_{i} = 0.5 < \bar{\kappa}$ and don’t commit violent crimes.

Baseline case: No guns

The individual decides on whether or not to commit a violent crime, i.e., he considers his desire and his probability of success. Not knowing the ability of the other party, he takes the expected ability of the other party ( $E[a_{-i}] = 0.5$, since we are considering a uniform distribution over the interval $[0,1]$), i.e. $\kappa_{i} = 0.5E[\beta_{i} + a_{i} - a_{-i} + 0.5] = 0.5(\beta_{i} + a_{i})$

If $0.5(\beta_{i} + a_{i}) \geq \bar{\kappa}$, the individual attempts a crime (this doesn’t say anything about whether or not he succeeds, only that he attempts it).

Everyone has a gun

Now we consider the case in which guns are provided to everyone. What happens when a gun adds $\alpha$ to each person’s ability to be violent? We note that this does not change the number of violent crimes committed, since $\chi_{i} = 0.5(a_{i} + \alpha - (a_{-i} + \alpha) + 1) = 0.5(a_{i} - a_{-i} + 0.5)$

Let’s say that the value a gun adds to someone’s ability to be violent is increasing in $\kappa_{i}$, i.e. people who have decided to or have very high likelihood of deciding to commit a violent crime are more able to use a gun to their advantage than people who don’t want to commit a violent crime. Intuitively, this seems reasonable, because violent criminals have the advantage of surprise, and their victims, even when carrying guns, are unable to reach for their guns in time or don’t want to because they don’t want to risk angering their attackers and being shot. For simplicity, let’s say $\alpha_{i}$ is a function that takes the form $\gamma\kappa_{i} + c$, where $\gamma < 1$, $c > 0$.

What we then have is $\kappa_{i} = 0.5E[\beta_{i} + (a_{i} + \gamma\kappa_{i} + c - (a_{-i} + \gamma\kappa_{-i} + c) + 0.5] = 0.5(\beta_{i} + a_{i} + \gamma\kappa_{i} - 0.5\gamma)$ $\kappa_{i} - \gamma \kappa_{i} = 0.5(\beta_{i} + a_{i} - 0.5\gamma)$ $\kappa_{i} = \frac{0.5(\beta_{i} + a_{i}) - 0.25\gamma}{1-\gamma}$

What we see is that for sufficiently large $\gamma$ ( $\gamma > 0.5$), $\frac{0.5(\beta_{i} + a_{i}) - 0.5}{1-\gamma} > 0.5(\beta_{i} + a_{i})$, no matter what $\beta_{i}$ and $a_{i}$ are.

What is the implication of this?

Let’s take a $\bar{\kappa}$ of $0.7$ and $\gamma$ of $0.5$. I previously had $\kappa_{i} = 0.5(0.6 + 0.6) = 0.6 < \bar{\kappa}$

Now, I have $\kappa_{i} = \frac{0.5(0.6 + 0.6) - 0.5(0.5)}{1-0.5} = 0.7 = \bar{\kappa}$

So, given $\gamma > 0.5$, we will see an increase in the number of people committing violent crimes.

Ban on guns, but individuals can obtain them illegally

Next, we introduce a situation with a ban on guns but the ability to obtain weapons illegally. We assume the probability of an individual obtaining a gun is dependent on $\beta_{i}$ (the intuition behind this is that the more dogged they are in their pursuit of a gun to act out their violent tendencies, the more likely it is that they get the gun), with some cutoff: $\beta_{i} > \bar{\beta} > 0.5$ means that the individual has a gun. What this means is that the average person does not have a gun, and the expectation that someone is carrying a gun is $1 - \bar{\beta}$.

For an individual with $\beta_{i} > \bar{\beta}$, and with the same expectations as laid out before, it’s easy to observe that $\kappa_{i} = 0.5(\beta_{i} + a_{i} + \gamma\kappa_{i} - (1-\bar{\beta})(0.5\gamma+ \bar{\beta}c)$ $\kappa_{i} = \frac{0.5(\beta_{i} + a_{i} + \bar{\beta}c - 0.5\gamma(1-\bar{\beta}))}{1-\gamma}$

Since $1 - \gamma < 1$ and $c > 0$, $\kappa_{i}$ here is obviously higher than in the case with no guns at all. It is also higher than $\frac{0.5(\beta_{i} + a_{i}) - 0.25\gamma}{1-\gamma}$, derived from the case where everyone carries a gun.

For an individual with $\beta_{i} < \bar{\beta}$, and with the same expectations as laid out before, we observe that $\kappa_{i} = 0.5(\beta_{i} + a_{i} - (1-\bar{\beta})(0.5\gamma + c))$

The individual has a lower $\kappa_{i}$ than in the case with no guns, as well as a lower $\kappa_{i}$ than in the case with everyone carrying a gun.

What this tells us is that allowing everyone to carry a gun (relative to a situation in which we ban guns but there is still illegal gun ownership) decreases the probability that someone who has high pre-existing violent desire ( $\beta_{i} < \bar{\beta}$) commits a crime, but increases the probability that someone who has lower pre-existing violent desire ( $\beta_{i} > \bar{\beta}$) commits a crime. What this means for the overall violent crime rate depends on what $\bar{\beta}$ is: the implication being that if there’s a very low proportion of people who manage to obtain guns in the presence of a gun ban, then we should ban guns to increase general safety.

I’ll stop here for now and just highlight some simplifying assumptions made in this model. There are, of course, assumptions made in this model. One is that the ability is exogenously determined, and independent of desire to be violent. Another important one is that the decision to commit a violent crime is made based on an expectation of the average ability of all victims to be violent and retaliate, where in reality, violent criminals are choosing their victims because these victims are perceived to be of lower ability to retaliate.

# A duopolistic setting: Marketplace or reseller?

I attempt to model the choice between marketplace and reseller in a duopolistic setting, with each intermediary deciding whether to be a marketplace or a reseller, while revising for my Industrial Organization final. Please read to the end to see my very trivial results and let me know if there are any mistakes!

It is based on the model I’ve posted about here (I employ the same notation as in the paper in this post, so if you haven’t read the paper or the post, or taken EC4322, you probably should take a look). I try this extension for my own learning in this post.

Assume we have two intermediaries (Intermediary $1$, Intermediary $2$) deciding on whether to operate in reseller-mode or in marketplace-mode. The timing of events is the same as in the paper. We first examine the case of simultaneous entry, then look at sequential entry.

Simultaneous entry

We first assume that supplier $i$ will only sell their products to or through one intermediary (otherwise this will be no different from the monopolist case). All products have identical buyer demand $2[m - (a_{i} - a_{i}^{*})^{2}]$, where each intermediary gets $m - (a_{i} - a_{i}^{*})^{2}$ buyer demand. As with the previous paper, $a_{i}^{*} = \theta + \gamma_{i} + \delta_{i}$. Suppliers only care about the profit they make, and have no preference for either intermediary.

Reseller

We first look at the situation in which both intermediaries choose to sell in reseller mode.

It is easy to see here that when a reseller $r \in \{1,2\}$ raises the fee $\tau_{r}$ at which it purchases goods from the suppliers to sell, more suppliers will be willing to sell to it, and when a reseller lowers the fee at which it purchases goods from the suppliers to sell, fewer suppliers will be willing to sell to it. We have, in essence, Bertrand competition here, where the two intermediaries $1$ and $2$ bid up the fee to the point at which they make zero profit. This is an equilibrium: if Reseller $1$ lowers its fee below that which Reseller $2$ is paying it will also gain zero profit (since no suppliers are selling to it, it does not have any products to sell). If Reseller $1$ increases its fee above that which Reseller $2$ is paying at this point it will make a loss on every good sold (its marginal cost is larger than the revenue it makes on each good sold here).

# Hate in the time of corona

There has been an increase in the number of hate crimes against those of East Asian descent since the pandemic hit Western countries (you can read about it here, here, here, here, and here). Living as part of the Singapore Chinese majority, I obviously have no personal encounters with COVID-19 related racism to add to the conversation (for clarity, since a friend pointed it out, what I mean is that people have not been racist towards ME). I’ll just say that I’m now looking at postponing graduate study because I’m afraid to live overseas in the near future, and I’m worried about the safety of my friends who are still studying abroad. I was in California just last summer, and Italy, France and Greece last December. It’s fortunate that I traveled so much last year, because I don’t know when I’ll be able to visit all these places again.

What I can contribute is what I’ll post today: a visualization of the frequency at which sinophobic search terms have been keyed into Google recently (very rough, hastily thrown together, but I just wanted to quickly share it). Graph data is taken from weekly Google Trends search data and limited to searches in the US. It goes back to one year ago (i.e. it spans the week of 21 April 2019 to the week starting on 12 April 2020); the x-axis represents Week 1, Week 2, …, Week 52. We see an explosion in the number of searches in the topics I looked at in the 40th week (the week of 19 January 2020).

For each topic, I collect data on a number of related popular search terms. It should be noted that the data is incomplete, because there may be a variety of search terms I have not tried since they didn’t pop up on the list of relevant keywords Google Trends recommended. Actual numbers may be significantly higher. I have a table of the search terms on which I collected Google frequency data on below, and the topic they fall under.

As you can observe from the graph above, why Chinese people eat bats is something people in the US are very curious about. I tried the search term “why do chinese eat bats” as well and the first five search results (outside of a Wikipedia page about bats as food) are below.