Andor et al. (2019). How effective is the European Union energy label? Evidence from a real-stakes experiment.

Very quick post: I try a new way of summarizing papers today. I think I’m going to alternate between this and text for future posts on empirical papers!

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.

Out of the five articles, three perpetuate that the consumption of wildlife is prevalent among Chinese people (including diasporas). I couldn’t find any sales data for bats as food in China, or in Chinatowns across the world, so I won’t try to claim that this prevalence is a myth, but it certainly is an allegation that requires more empirical validation.

Another observation I made is that while Statesman News Network disputed the attribution of blame to Chinese culture, this showed up in the search results snippet text, which I think is rather misleading about the content of the actual article.

Other observations from the data: While COVID-19 did not originate in dogs, it appears that people living in the US are once again interested in the Chinese tradition of eating dogs! Donald Trump’s term for the virus has also caught on, with “Chinese virus” becoming a popular search term. People also increasingly think that the Chinese are dirty, and are retreating to the territory of racial slurs.

Of course, I know not everyone in the US is searching these terms. In fact, it is impossible to determine what people’s intentions are when they enter these search terms into Google, so I won’t even say that all of those who are making these searches are sinophobes. But what people search for tells us a lot about the current climate, while official COVID-19 related hate crime statistics have not been consolidated (and are likely to be underreported anyway). I’ll end on this gloomy note.

Andrei Hagiu and Julian Wright (MS, 2015). Marketplace or Reseller?

I write about the basic formulation in Hagiu and Wright (2015)’s paper modelling the choice an intermediary makes between being a marketplace and a reseller to revise for my Industrial Organization II final.

Professor Julian Wright taught the model in this paper to us this semester, and I’m going to try my best to do it justice here (hope I haven’t misunderstood anything!) Sidenote: Prof Wright is very clear and very, very, patient, and if you’re an NUS Economics student reading this, I highly recommend EC4324 Economics of Competition Policy and EC4322 Industrial Organization II! I learned a lot, and these modules got me really interested in micro theory.

Introduction

Hagiu and Wright (2015) model the conditions under which an intermediary chooses to be a marketplace, and those under which it chooses to be a reseller. Basically, if it is better for the supplier to retain “control rights over a non-contractible decision variable,” then the intermediary will choose to be a marketplace, and if it is better for the intermediary to hold these rights, then it will choose to be a reseller. The authors give the example of marketing, and they use Best Buy, so I’ll give you a different example for further illustration.

Let’s look at, for instance, knick knack stores in Singapore. We have Naiise, a multi-brand reseller which stocks toys, novelty gifts, and more.

We also have a marketplace like Boutique Fairs Singapore, a flea market for crafts, with product offerings in domains similar to the range of items stocked by Naiise.

How do these intermediaries decide whether to set up in reseller or marketplace mode?

An intermediary in this market may have more updated information about the general trends in the market, so they can drive up buyer demand through advertising that products are handcrafted, for example. But the individual artisans (suppliers) have more information about their specific target clientele or loyal customers, who they may interact with, and think that it is more important to advertise that the product was made from recycled fabric. Assume that it is extremely difficult or costly for the intermediary and supplier to convey their private information to each other, perhaps because this information is constantly being updated as trends change in the market. Whether the intermediary or the supplier has relatively more important information will determine whether the intermediary chooses to be a reseller or a marketplace.

Model Setup

• There are $N > 1$ independent suppliers, each incurring a marginal cost of $c = 0$ in producing their products.
• Each buyer is willing to pay $v$ for each product she is interested in. The buyer must make the purchase from/through the intermediary.
• The number of buyers for product $i$ is $m - (a_{i} - a_{i}^*)^{2}$. What this says is basically that the marketing activity chosen for the product affects buyer demand.
• $a_{i}$ is the choice of marketing activities made by the intermediary (when it is in reseller mode) or the supplier (when the intermediary is in marketplace mode and the supplier is marketing its own product on the marketplace.
• $a_{i}^{*}$ is the optimal marketing activity for the product $i$.
• There are two things we should realize from the use of the squared term $(a_{i} - a_{i}^{*})^{2}$ to denote how the buyer demand is affected by the choice of marketing activity here.
• The first thing we immediately observe is that $(a_{i} - a_{i}^{*})^{2}$ is always positive (and so obviously $(a_{i} - a_{i}^{*})^{2}$ is always negative), so any difference between $a_{i}$ and $a_{i}^{*}$, whether that difference is in the positive direction $a_{i} - a_{i}^{*} > 0$ or the negative direction $a_{i} - a_{i}^{*} < 0$ will reduce buyer demand (i.e. consumers want what they want, no more, no less).
• The second thing we observe is that $(a_{i} - a_{i}^{*})^{2}$ punishes larger deviations from the optimal marketing activity. Say the total number of buyers in the market is 20, and the optimal marketing activity $a_{i}^{x}$ is 3. If I choose $a_{i} = 4$, the number of buyers for product $i$ is $20 - (4 - 3)^{2} = 19$, i.e. I lose one buyer. If I choose $a_{i} = 5$, the number of buyers for product $i$ is $20 - (5 - 3)^{2} = 16$, i.e. I lose three more buyers from the same magnitude of increase in my choice of marketing activity. If I choose $a_{i} = 6$, the number of buyers for product $i$ is $20 - (6 - 3)^{2} = 11$… You get my drift.
• $a_{i}^{*} = \theta + \gamma_{i} + \delta_{i}$
• The optimal marketing choice can be additively separated into what both the supplier and the intermediary know ($\theta$), what the supplier knows ($\gamma_{i}$), and what the intermediary knows ($\delta_{i}$). Further exposition is in the bullet points below.
• $\theta$ is known to both the supplier and the intermediary.
• $\gamma_{i}$ represents the private information about the marketing of product $i$ known only to the intermediary, and is a random variable that is independently and identically distributed. $E[\gamma_{i}] = 0$ and $Var[\gamma_{i}] = V_{\gamma}$.
• $\delta_{i}$ represents the private information about the marketing of product $i$ known only to the supplier, and is a random variable that is independently and identically distributed. $E[\delta_{i}] = 0$ and $Var[\delta_{i}] = V_{\delta}$.
• The entity holding control over marketing activity (reseller or supplier) will incur a fixed cost for each product, $F$, and a variable cost for each sale, $f$. Later we discuss different cost structures.

Marketplace or reseller?

Under the assumption that each entity learns their private information before deciding on the marketing activity $a_{i}$ (the full set of timing assumptions may be found in the paper), and that the intermediary will make positive profits whether in marketplace or reseller mode, we have

Reseller

The reseller offers to buy each supplier’s product for zero to maximise its own profit. We assume that suppliers who are indifferent between selling to the reseller and not selling to the reseller will sell to the reseller. Since suppliers’ outside option (alternative) gives zero profit, and their marginal costs are normalized to zero, they are indifferent when presented with an offer of zero, and therefore choose to sell to the reseller. The cost the reseller incurs on buying each unit to resell is therefore zero.

The reseller’s expected profit is therefore given by the following expression

$\pi_{R}(n) = (v - f)\sum_{i=1}^{n}E_{R}[m-(a_{i} - (\theta + \gamma_{i} + \delta_{i}))^{2}] - nF$

from

$\pi_{R}(n) = (v - f)\sum_{i=1}^{n}E_{R}[m-(a_{i} - a_{i}^{*})^{2}] - nF$

Expanding a little, we get

$\pi_{R}(n) = (v - f)\sum_{i=1}^{n}E_{R}[m - a_{i}^{2} + 2a_{i}(\theta + \gamma_{i} + \delta_{i}) - (\theta + \gamma_{i} + \delta_{i}))^{2}] - nF$

The price is fixed, and the costs are fixed. But the reseller can still choose its marketing activity, $a_{i}$, to maximise the number of buyers. How does the reseller choose $a_{i}$? We first “take out” all the terms in $a_{i}$.

$E_{R}[- a_{i}^{2} + 2a_{i}(\theta + \gamma_{i} + \delta_{i})]$

The reseller has to guess at $\delta_{i}$, which is the private information about optimal marketing activity only known by suppliers, as mentioned above. Recall that $E_{R}[\delta_{i}] = 0$ (so we set all terms in $\delta_{i}$ to be zero). We then get

$E_{R}[- a_{i}^{2} + 2a_{i}(\theta + \gamma_{i})]$

With all the information the reseller has, they now choose $a_{i}$ to maximise the number of buyers, so then we have the first order condition

$\frac{\partial m - a_{i}^{2} + 2a_{i}(\theta + \gamma_{i})}{\partial a_{i}} = 2a_{i} - 2\theta - 2\gamma = 0$

From this, we get

$a_{i} = \theta + \gamma_{i}$

So with $a_{i} = \theta + \gamma_{i}$, what do we have for the expected number of buyers?

$\pi_{R}(n) = (v - f)\sum_{i=1}^{n}E_{R}[m - ((\theta + \gamma_{i}) - (\theta + \gamma_{i} + \delta_{i}))^{2}] - nF$

Simplifying, we observe that this reduces to

$\pi_{R}(n) = (v - f)\sum_{i=1}^{n}E_{R}[m - \delta_{i}^{2}] - nF$

Since $E_{R}[m] = m$ and $E_{R}[\delta_{i}^{2}] = E_{R}[\delta_{i} - 0]^{2} = E_{R}[\delta_{i} - E_{R}[\delta_{i}]]^{2} = Var[\delta_{i}] = V_{\delta}$, we have that the expected profit of selling all $N$ products is

$\pi_{R} = N(v - f)(m - V_{\delta}) - NF$

Marketplace

The marketplace charges each supplier $P$ to be on the platform. We first see that each supplier $i$‘s profit-maximizing $a_{i}$ will be $a_{i} = \theta + \delta_{i}$, by the same solving process as above. So each supplier makes

$\pi_{i} = (v-f)(m - V_{\gamma}) - F - P$

The marketplace may set $P = (v-f)(m - V_{\gamma}) - F$ in order to extract all surplus from suppliers. At this point suppliers are indifferent between joining and not joining, and we assume, as above, that all suppliers who are indifferent will join the marketplace.

The monopolist therefore makes

$\pi_{M} = N(v-f)(m - V_{\gamma}) - NF$

Which makes larger profit: the marketplace or the reseller? For your convenience, I’m going to put both profit functions here for the comparison.

$\pi_{R} = N(v - f)(m - V_{\delta}) - NF$, $\pi_{M} = N(v-f)(m - V_{\gamma}) - NF$

It is easy to see that which mode allows the intermediary to make larger profit depends on whether $V_{\delta}$ or $V_{\gamma}$ is higher. As mentioned in the summary, whether the intermediary or the supplier has relatively more important information will determine whether the intermediary chooses to be a reseller or a marketplace.

Now we add different cost structures and heterogeneous product demand.

Heterogeneous products and costs

We now have that the reseller and the supplier (in marketplace-mode) incur variable costs $f_{R}$ and $f_{M}$ respectively, and fixed costs $F_{R}$ and $F_{M}$ respectively.

We now substitute these new parameters into the profit expressions we solved for previously.

For each product $i$, the reseller gets

$(v-f_{R})(m - V_{\delta}) - F_{R}$

For each product \$latex i, the marketplace gets

$(v-f_{M})(m - V_{\gamma}) - F_{M}$

Which mode will the intermediary choose? Now, we see that this also depends on what $f_{R}$, $f_{M}$, $F_{R}$, $F_{M}$, and $m$ are.

For the intermediary to be indifferent between operating in the two modes,

$(v-f_{R})(m - V_{\delta}) - F_{R} = (v-f_{M})(m - V_{\gamma}) - F_{M}$

$vm - f{M}m - vV_{\gamma} +f_{M}V_{\gamma} - F_{M} = vm - f_{R}m - vV_{\delta} +f_{R}V_{\delta} - F_{R}$

We remove the common terms, such that we have

$(f{R} - f_{M})m = V_{\gamma}(v-f_{M}) - V_{\delta}(v-f_{R}) + F_{M} - F_{R}$R

If $f_{M} > f_{R}$, $m \geq m^{*}$,

$m \geq m^{*} = \frac{V_{\gamma}(v-f_{M}) - V_{\delta}(v-f_{R}) + F_{M} - F_{R}}{f_{R} - f_{M}}$

$m(f_{R} - f_{M}) \geq V_{\gamma}(v-f_{M}) - V_{\delta}(v-f_{R}) + F_{M} - F_{R}$

$m(f_{R} - f_{M}) \geq V_{\gamma}(v-f_{M}) - V_{\delta}(v-f_{R}) + F_{M} - F_{R}$

$(v-f_{R})m - V_{\delta}(v-f_{R}) - F_{R} \geq (v-f_{M})m - V_{\gamma}(v-f_{M})-F_{M}$

So we know that when $f_{M} > f_{R}$, for large enough $m$, product $i$ should be offered in reseller-mode, because the profits made by a reseller are higher than those for a marketplace. For small $m$, product $i$ should be offered in marketplace-mode. The converse applies for $f_{M} < f_{R}$.

This is an intuitive result: if the intermediary incurs a higher $f_{M}$ than $f_{R}$, they should sell more popular products (with large $m$) in reseller mode (in order to pay a lower variable cost on each product sold), and if they incur a higher $f_{R}$ than $f_{M}$, they should sell more products (with small $m$) in marketplace mode for the same reason.

Network externalities

If the number of buyers $m$ is increasing in $n$ (more buyers want to join the intermediary when there are more products being sold on it/by it, because it’s more convenient to browse, etc.). Then we have the number of buyers joining the intermediary = $m(n)$, which is increasing in $n$, and $m(N) = m$. What this means is that if more suppliers join, more buyers join, which attracts more suppliers, which attracts more buyers, and so on. We examine what happens in marketplace mode.

Favorable beliefs

Favorable beliefs refer to the situation where suppliers believe that all other suppliers will join the intermediary whenever it is an equilibrium (they make non-negative profits). Their optimism means that their expected profit from joining an intermediary is $\pi_{i}^{e} = (v-f)(m - V_{\gamma}) - F - P$ in equilibrium. In this case, the previous results hold.

Unfavorable beliefs

This refers to the situation where suppliers believe that no other suppliers will join the marketplace in equilibrium, like in the case in which a new marketplace has just been set up. Their expected profit from joining an intermediary is $\pi_{i}^{e} = (v-f)(m(1) - V_{\gamma}) - F - P$ in equilibrium.

What this means is that the monopolist can only charge the supplier $(v-f)(m(1) - V_{\gamma}) - F$.

In this case, the monopolist makes profit

$\pi_{M} = n((v-f)(m(1) - V_{\gamma}) - F)$

What does the reseller make? Since the reseller is purchasing the products and selling them, it gets

$\pi_{R} = n((v-f)(m(N) - V_{\delta}) - F)$

So in this case, reseller-mode is always preferred to marketplace-mode if

$V_{\delta} > V_{\gamma} + m(N) - m(1)$

This should be easy to observe from the comparison of the two profits above.

More extensions and the proofs are available in the paper! I sincerely hope I didn’t get anything wrong, and if I did, you can let me know in the comments. I’m going to write up an extension on this that I attempt in preparation for my final in a later post.

The full paper is available here.

Keisuke Hattori and Yusuke Zennyo (2018). Heterogeneous consumer expectations and monopoly pricing for durables with network externalities.

This is an explanation of the model and some of the basic results in Hattori and Zennyo’s (2018) working paper “Heterogeneous Consumer Expectations and Monopoly Pricing for Durables with Network Externalities”. I do not go into all the results and proofs, but the paper is linked at the end of this post.

Some background first: while I was studying for my Industrial Organization final and looking for alternative environments in which the concepts I was taught may be applied, I came across this paper and decided to write about the model here for fun! I sacrifice brevity here and end up repeating explanations many times, because I personally find that while reading papers, sometimes it’s difficult to remember all the characteristics of the setup or explanations that were set forth in previous sections of the paper, and I frequently have to go back and refer to those parts before I can carry on. Therefore, to save you the trouble, I try to make my explanation at every point as clear and as self-contained as possible.

Introduction

This paper examines firms’ pricing strategies for a durable good that has network externalities, but which consumers get bored of after a while. Think of something like Words with Friends – the more people there are playing the game, the shorter the wait time to match with another online consumer to play a game, so your utility is increasing in the number of consumers on the network. At the same time, the gameplay gets kind of tired after a while, since there are (to my knowledge) no new levels, just a leaderboard that records each consumer’s prowess, so consumers may get bored and stop playing. Another feature of the model set forth in this paper is that it examines consumers’ utilities for infinite periods with discounting. What this means is that if you’re making the decision about whether or not to download Words with Friends right when the app was first released and you have some foresight, you anticipate that more consumers are going to hop on the network later on, so you take the future discounted expected utility from a larger network into account when making your decision. (If you’re naive, you may not take this into account, and you only make your decision based on the utility from the small network of consumers on the app right now.)

First, some basic setup.

Model setup

• Each consumer derives benefit $v(n_{t})$ from using the good in each period, where $n_{t}$ refers to the number of consumers joining the network of consumers using the good in period $t$ (it should be trivial to observe that $n_{t} \in \{1, ..., n\}$, since $n$ is the total population of potential consumers). $v(1)$ here refers to the inherent utility the good provides (intuitively, if I am the only person using the good, there are no network benefits, so the amount of utility I derive is just what I get from using the good). $v^{\prime} > 0$ (i.e. the utility is strictly increasing in the number of consumers using the good).
• We represent the discount rate with $\delta_{C} \in [0,1)$, and the depreciation rate of the durable good (or basically, how quickly consumers get bored of it) with $\beta \in [0,1)$. What this means is that, for example, if I am the only consumer on the network and no other consumers join ever, and assuming $v(1) = 1$, $\delta_{C} = 0.8$, $\beta = 0.25$, in the first period, I derive utility of $v(1) = 1$; in the second period, I derive utility of $\delta_{C}(1 - \beta) v(1) = 0.6$; in the third period, I derive utility of $\delta_{C}^{2}(1 - \beta)^{2} v(1) = 0.36$; and so on. Therefore, we have that the combined discount factor = $\delta_{C}(1-\beta)$. For simplicity, we represent this with $\lambda_{C}$. We know that the combined discount factor in period $k$ is $\lambda_{C}^{k-1}$.
• The monopolist incurs marginal cost $c = 0$ in offering the product.

Let’s take a look at what happens on the consumer side.

Two types of consumers: Naive and sophisticated

First assume that there are two consumer types: naive consumers (total number = $n_{N}$) and sophisticated consumers (total number = $n_{S}$). We start by looking at the naive consumers (although the paper begins with the sophisticated consumers).

Naive consumers have no foresight when it comes to future expected utility, and therefore only take into account the number of consumers on the network in the immediately preceding period $t-1$ ($= n_{t-1}$) for their purchase decision in period $t$. So for them, the expected number of consumers on the network if they choose to purchase will be $n_{t-1} + 1$. (The $+1$ here comes from the consumers himself being added to the network.) What utility will the naive consumer expect?

$u_{N}^{e}(t) = \sum_{k=t}^{\infty}[\lambda_{C}^{k-1}v(n_{t-1}+1)]$

The expected utility is given by the amount of utility derived from $n_{t-1} + 1$ consumers on the network summed over an infinite number of periods, starting from when the consumer joins the network. Of course, we know that the naive consumer’s expected utility does not take into account all consumers that will join the network in the future. What the naive consumer actually gets should be

$u_{N}(t) = \sum_{k=t}^{\infty}[\lambda_{C}^{k-1}v(n_{k})]$

The equation above properly accounts for all of the consumers who will join the network in each period $t, t+1, t+2, ..., \infty$. We notice immediately that $u_{N}(t) \geq u_{N}^{e}(t)$. This is because more consumers may join in the later periods, while all consumers on the network stay on the network, so $n_{k} \geq n_{t-1} + 1$.

What about sophisticated consumers? Sophisticated consumers have perfect foresight and can project into the future and derive a correct expectation of the number of users that will join the network. Therefore, the sophisticated consumer’s expected utility will be the actual utility that he will derive, which, as we know, is given by the function above. So, we have

$u_{S}^{e}(t) = u_{S}(t) = \sum_{k=t}^{\infty}[\lambda_{C}^{k-1}v(n_{k})]$

We know that both consumer types will make the decision to join the network as long as their expected utility is greater than the price of the good (consumer $i \in \{N,S\}$ with $u_{i}^{e}(t) = p$ is indifferent between joining and not joining, and all consumers with $u_{i}^{e}(t) > p$ will join the network). Therefore, for consumers who join, we have the expected and actual indirect utility functions expressed below.

$V_{i}^{e}(t) = u_{i}^{e}(t) - p$

$V_{i}(t) = u_{i}(t) - p$

One thing to note at this point is that there is no heterogeneity among consumers beyond the naive/sophisticated distinction. As seen above, all naive consumers have the same expected utility, and all sophisticated consumers have the same expected utility. So, what we’ll see here is that if one naive consumer joins in period $t$, all naive consumers, since their expected utility is the same as that of that one naive consumer, will join in period $t$. The same applies for sophisticated consumers.

Two different pricing strategies: Simultaneous-diffusion and sequential-diffusion

Now, let’s examine two different pricing strategies: simultaneous-diffusion and sequential-diffusion. In simultaneous-diffusion, the monopolist sets a price to have all consumers join in period 1. In sequential-diffusion, the monopolist sets a price such that sophisticated consumers join in period 1, and naive consumers join in period 2 (we observe that sophisticated consumers will never join in the period after naive consumers, as their expected utility is higher than that of the naive consumers).

In the simultaneous-diffusion strategy, the monopolist must set a price $p$ less than each consumer type’s expected utility from joining the network. We know that naive consumers have expected utility given by

$u_{N}^{e}(1) = \sum_{k=t}^{\infty}[\lambda_{C}^{k-1}v(1)] = \frac{v(1)}{1-\lambda_{C}}$

The expected utility of sophisticated consumers at the point at the simultaneous-diffusion equilibrium price is

$u_{S}^{e}(1) = \sum_{k=t}^{\infty}[\lambda_{C}^{k-1}v(n)] = \frac{v(n)}{1-\lambda_{C}}$

We have that the naive consumer’s expected utility is the maximum price at which all consumers will purchase in the same period, so

$p^{sim} = \frac{v(1)}{1-\lambda_{C}}$

At this point, the profit made by the monopolist (recall the assumption that the marginal cost $= c = 0$) is

$\pi^{sim} = n\frac{v(1)}{1-\lambda_{C}}$

Since the actual utility function for both naive and sophisticated consumers are the same, and the price charged to each consumer type i the same as well, the surplus for each consumer is given by

$V_{i}^{sim} = \frac{v(n)}{1-\lambda_{C}} - \frac{v(1)}{1-\lambda_{C}} = \frac{v(n) - v(1)}{1-\lambda_{C}}$

Now, we go on to examine the sequential-diffusion strategy.

If $p^{sim} > \frac{v(1)}{1-\lambda_{C}}$,

In order to get all sophisticated consumers to join in period 1, the monopolist must set

$p^{seq} \leq v(n_{S}) + \frac{\lambda_{C}v(n)}{1-\lambda_{C}}$

In order to get all naive consumers to join in period 2, the monopolist is further constrained by

$p^{seq} \leq \frac{v(n_{S} + 1)}{1-\lambda_{C}}$

If the monopolist sets $p^{seq} > \frac{v(n_{S} + 1)}{1-\lambda_{C}}$, then the sophisticated consumers will predict that naive consumers will not join in period 2, and therefore, they will not purchase if

$p^{seq} > \frac{v(n_{S})}{1-\lambda_{C}}$

Notice that $p^{seq} \leq \frac{v(n_{S})}{1-\lambda_{C}} < \frac{v(n_{S} + 1)}{1-\lambda_{C}}$, since $v$ is monotonically increasing. Therefore, we know that if the monopolist sets $p^{seq} > \frac{v(n_{S} + 1)}{1-\lambda_{C}}$, no consumer will join. Also, we know that one of the other conditions on the price in sequential-diffusion is that the monopolist must set $p^{seq} \leq v(n_{S}) + \frac{\lambda_{C}v(n)}{1-\lambda_{C}}$. So,

$p^{seq} = \min\{\frac{v(n_{S} + 1)}{1-\lambda_{C}}, v(n_{S}) + \frac{\lambda_{C}v(n)}{1-\lambda_{C}}\}$

By comparison, we can see that which is lower will depend on the $\lambda_{C}$ and the difference between $v(n_{S})$ and $v(n)$. If $\lambda_{C}$ is sufficiently small ($\lambda_{C} \leq \frac{v(n_{S}+1) - v(n_{S})}{v(n) - v(n_{S})}$), then $\frac{v(n_{S} + 1)}{1-\lambda_{C}}$ is bigger, and the firm will charge $v(n_{S}) + \frac{\lambda_{C}v(n)}{1-\lambda_{C}}$. Otherwise, the converse will apply.

Therefore, we have for $\lambda_{C} \leq \frac{v(n_{S}+1) - v(n_{S})}{v(n) - v(n_{S})}$,

$\pi^{seq} = n_{S}[v(n_{S}) + \frac{\lambda_{C}v(n)}{1-\lambda_{C}}] + \delta_{F}n_{N}[v(n_{S}) + \frac{\lambda_{C}v(n)}{1-\lambda_{C}}]$ where $\delta_{F}$ is the discount factor for the monopolist

$V_{S}^{seq} = 0$ (Sophisticated consumers are being charged their maximum willingness to pay)

$V_{N}^{seq} = \frac{v(n)}{1-\lambda_{C}} - v(n_{S}) - \frac{\lambda_{C}v(n)}{1-\lambda_{C}} = \frac{(1-\lambda_{C})v(n)}{1-\lambda_{C}} - v(n_{S}) = v(n) - v(n_{S})$

For $\lambda_{C} > \frac{v(n_{S}+1) - v(n_{S})}{v(n) - v(n_{S})}$,

$\pi^{seq} = n_{S}[\frac{v(n_{S} + 1)}{1-\lambda_{C}}] + \delta_{F}n_{N}[\frac{v(n_{S} + 1)}{1-\lambda_{C}}]$

$V_{S}^{seq} = v(n_{S}) + \frac{\lambda_{C}v(n) - v(n_{S}+1)}{1-\lambda_{C}} = \frac{\lambda_{C}(v(n) - v(n_{S})) - (v(n_{S}+1)-v(n_{S})}{1-\lambda_{C}}$

$V_{S}^{seq} = \frac{v(n)}{1-\lambda_{C}} - \frac{\lambda_{C}v(n_{S} + 1)}{1-\lambda_{C}} = \frac{v(n) - v(n_{S} + 1)}{1-\lambda_{C}}$

We can make two observations from the above. The first is that naive consumers always derive a higher consumer surplus than sophisticated consumers in sequential-diffusion pricing. This is because sophisticated consumers only receive utility from a smaller network when they join the network at first, and receive a discounted utility from the full network in period 2 onwards when naive consumers join, but naive consumers receive utility from the full population being on the network as soon as they join. The second observation is that whether simultaneous-diffusion or sequential-diffusion offers higher profit to the monopolist depends on the discount factor the monopolist applies to its profits in period 2 under sequential-diffusion. Some proofs and further exposition are available in the paper.

Concluding remarks

I’ll end with one concluding thought. While the authors include a term for depreciation of the product, the inclusion of this term in the model does not seem to do anything beyond increase the combined discount rate. When I started reading this paper, I thought that it might be used to analyse the case in which consumers in preceding periods drop out of the network. This is something they mention might be interesting to research in the Discussion section of the paper, but they do not include it in the scope of this paper. Perhaps I’ll try to see if I can model network departures in a future post.

The full paper I read may be found here.

Online seminars

I’ve been noting dates and planning to attend numerous online Economics seminars. They’re hosted by different organizations, so I’ve compiled a list (available below, with links). This is maybe one of the better things to have come out of this virus.

On a side note, I’ve penciled in some time to study for my Industrial Organization and Behavioral Public Policy finals, and probably will be posting notes on the papers covered in the syllabus or related papers so I can revise and prepare. Excited for my exams to end so I can focus entirely on reading more new papers!

(Update: I’ve been told that the formatting for this table is a bit wonky on mobile platforms. It’s best viewed on desktop!)