Yoo-Mi Chin and Scott Cunningham (PubEc, 2019). “Revisiting the effect of warrantless domestic violence arrest laws on intimate partner homicides.”

I haven’t posted in a while, mainly because I started work last week and was working on a few other side projects prior to that and things have just been a little hectic. Today, I post about Chin and Cunningham’s (2019) work on the effect of domestic violence arrest legislation (whether arrest is discretionary, preferred, or mandatory) on spousal homicides in five slides.

The full paper may be found here. I’ll try to update more in the coming days!

On graduation

A few days ago I took my last final online and, without ado, graduated. I thought I would celebrate that evening with something more than sleep and Netflix, and also very naively thought I would be able to get back on track on all my personal projects the very next day, but my Netflix binge has lasted two whole days (I watched the entire season of Ryan Murphy’s Hollywood, a series which was very entertaining, even if not at all critically acclaimed).

I am now (maybe, sort of) back in the right headspace, well-rested and ready to work again, and desperately trying to finish everything I planned to complete in the limbo between graduation and the start of work. I had hoped to use this time to travel (to Australia, to visit my brother), but that, of course, is not possible now. I do not mean to sound bitter: I know that I am extraordinarily fortunate just to have found employment prior to graduation, so I’m at the very least not adrift in the ensuing economic storm. But back on topic: graduation.

Earlier this year, upon learning that I’m graduating this semester, my friend said, “Finally.” I retorted with “What do you mean by finally? I took just the right amount of time.” But right now, especially having spent the last weeks of my undergraduate education in isolation, it does feel like finally. It also feels like I matriculated last year, because all my memories of that semester are so vivid, and most of the intermediate semesters are a blur. I guess rites of passage draw up emotional clichés.

I remember being set on studying Economics when I entered NUS. Back then it was mostly because I had done relatively well in H2 Economics, and wanted to work in a bank (I swiftly realized some time after matriculating that I had very little passion for finance). It was secondly because I had not done well enough to be admitted to law school, and most of everything else did not interest me. It was thirdly because at that time I expected that I would be able to muddle through most other majors I had a vague interest in – Political Science, Global Studies – because I had a natural affinity for these subjects, and wanted to challenge myself doing something more mathematical, having never been too good at math. I suppose this reason should be labelled youthful hubris.

Anyway, I began studying Economics, and it was serendipitous: if I had done better in the A Levels, I would have chosen Law; if I had not taken a gap year to work and chosen to enter university a little earlier, I might have been keener to tread the path of less resistance, and chosen to study something like Political Science instead. But I ended up studying Economics, and found that I love empirical work, and I love applied micro, and I love experimental work, and I love all the cool theories in papers I have and haven’t read. I remain not quite literate and certainly inarticulate in Economics, and struggle through papers sometimes (most of the time), and when I attend theory seminars sometimes (most of the time) I have no idea what’s going on. But I do love Economics, and want to get better at reading Economics, and I guess this may be read as a commitment to regular updates on this blog, as work permits.

Competitively debating and coaching debate (alongside completing other RA duties) while trying to take as many courses as I can to maximize learning in 4 years, approximately SGD 32,000, has not been easy. I think if I could go back I would have chosen to do many things differently. Coaching at two schools was definitely a bad idea, and I should have limited myself to one from the beginning. Maybe I should not have coached at all, but that would have left me financially stranded, so I don’t know. I wish I had obtained better grades in my time at NUS, but they are what they are, and the past is past. I hope there will be subsequent opportunities to study Economics again, and I hope that in these opportunities I will be able to devote myself full-time to my education.

Melissa Dell and Benjamin Olken (REStud, 2020). The development effects of the extractive colonial economy: The Dutch Cultivation System in Java.

I didn’t want to leave this space untouched for too long, so I thought I would squeeze in a quick post before my last final of this semester (and my undergraduate candidature in NUS!)

I summarize a paper by Melissa Dell, 2020 JBC medal winner, in this post. Since this is a summary, I do not cover everything in the paper, and do not describe the robustness checks performed. If you are curious about the details, the paper can be found here.

Research Question

Acemoglu and Robinson’s (2012) Why Nations Fail (a debater starter pack book) drew vivid contrasts between different case studies, reifying that economic and political institutions are instrumental determinants of economic growth. Across almost all countries in Southeast Asia, many of these key institutions are transplants, kept intact even as the sun set on the age of empire. Have these institutions hurt or helped economic development? In Singapore, the Bicentennial last year reinvigorated heated debate about this.

Dell and Olken’s (2020) paper answers this question in the context of Java, Indonesia. Java came under Dutch rule in 1800. In 1830, as Dutch historian Cees Fasseur (1986) writes, Dutch Governor-General Johannes van den Bosch established the cultuurstelsel, or Cultivation System. This was a system that coerced the Javanese into agricultural activity (most predominantly, sugar cultivation) to enrich the Dutch government.

Perhaps useful TL;DR and disclaimer here: the Cultivation System instituted by the Dutch did bring about a net economic gain, but Olken highlights that these results should not be taken as representing that Dutch colonial rule was a net positive for Indonesia. Indeed, there are several problems with this interpretation of the study. A comprehensive cost-benefit analysis must also take into account the costs suffered by subjugated locals, the social costs of enduring myths about the native population, the use of anti-colonial rhetoric to mobilize the masses to vote for economic policies that actually work against them, alongside other factors. All of these fall outside the scope of this study.

The Cultivation System, as the paper lays out, could have affected the development trajectory of Java, and specifically the sugar cultivation villages, through four causal mechanisms, summarized below.

Sources of data

To study whether a positive or negative effect on economic growth dominates, the paper uses historical data on the Cultivation System from manuscript archival records, and 1900 infrastructure maps published by the Dutch Topographic Bureau. The manuscript archival records contain information on which villages contributed to each sugar factory and the contribution (in terms of land and labor) of each village. In total, by geographical coordinate matching, 6,383 historical villages were able to be mapped to the 2,519 modern villages that now occupy these territories. Factories were mapped in this way as well. Data on modern growth and growth-related outcomes were obtained from the Indonesian government’s Central Bureau of Statistics (BPS) datasets.

Empirical strategy

Effects of proximity to a sugar processing plant

To determine whether being in close proximity to a sugar processing factory had an effect on economic development, Dell and Olken (2020) compared the outcomes of villages near actual old sugar processing factories to the outcomes of villages near counterfactual sugar processing factories (i.e. locations that would have been suitable for sugar processing factories to be built, but where sugar processing factories were not built because they would have been located too near another sugar processing factory and ate into its catchment area).

They constructed counterfactual factory locations based on three criteria:

  1. Counterfactual location must be within 5 to 20 kilometres upstream or downstream from the actual factory location
  2. Counterfactual location must have as much land suitable for sugar cultivation (determined by slope, elevation) within a 5 kilometre radius as the 10th percentile of the distribution of actual locations
  3. Counterfactual locations must be spaced as far apart as actual factories within the 10th percentile of the distribution

The specification for the regression ran was as such:

out_{v} = \alpha + \sum_{i=1}^{20} \gamma_{i} dfact_{v}^{i} + \beta X_{v} + \sum_{j=1}^{n}fact_{j}^{v} + \epsilon_{v}

  • out_{v} is the outcome variable of interest for the village
  • \sum_{i=1}^{20}\gamma_{i} dfact_{v}^{i} = \gamma_{1}(dfact_{v}^{1} + \gamma_{2}dfact_{v}^{2} + \gamma_{3}dfact_{v}^{3} + ... \gamma_{20}dfact_{v}^{20}, where dfact_{v}^{1} is a dummy variable indicating whether the village is located within a 0-1 kilometre radius of the nearest factory (and \gamma_{1} is obviously the coefficient on this term), dfact_{v}^{2} is a dummy variable indicating whether the village is located within a 1-2 kilometre radius of the nearest factory, and so on
  • X_{v} is a set of controls, including variables like elevation, slope, etc.
  • \sum_{j=1}^{n}fact_{j}^{v} are nearest factory fixed effects, to compare each village to villages near the same sugar processing factory (shown in the below diagram)

Effects on villages made to grow sugar cane

To study the effect of the Cultivation System on the villages coerced into sugar cultivation (“subjected villages”), Dell and Olken (2020) further used a regression discontinuity design, exploiting the discontinuity at the boundaries of catchment areas made to grow sugar cane. Within these boundaries, villages were made to grow sugar cane; outside them, villages were not. The sample under analysis comprises only villages that had arable land suitable for the cultivation of sugar then.

out_{v} = \alpha + \gamma cultivation_{v} + f(geographic location_{v}) + g(dfact_{v}) + \beta X_{v} + \sum_{i=1}^{n}seg_{v}^{i} + \epsilon_{v}

  • out_{v} is, as mentioned above, the outcome variable for each village
  • cultivation_{v} is a dummy variable taking the value of 1 if the village grew sugar cane under the Cultivation System (“subjected”), and 0 otherwise
  • f(geographic location_{v}) is the regression discontinuity polynomial, which is estimated separately for each catchment area
  • g(dfact_{v}) controls for distance from a sugar processing factory, in order to isolate the effect of being made to cultivate sugar from that of being located near a sugar processing factory
  • seg_{v}^{i} are the fixed effects such that villages are compared to other villages nearest to them (i.e. in the same segment of the catchment area)

Main findings

Effects of proximity to sugar processing factory

Economic structure

Living in a village within a few kilometres of a historical factory is associated with a 20 to 25 percentage point decrease in the likelihood of working in agriculture, relative to living in a village 10 to 20 kilometres away from such a historical factory. Moreover, living near an actual historical factory is associated with a 17 percentage point decrease in the likelihood of working in agriculture, relative to living near a counterfactual historical factory.

Sugar-related industrial activity

Even after limiting the sample to historical sugar processing factories that are not near modern sugar processing factories, being within 0 to 1 kilometre of a historical factory is associated with an increase in employment in manufacturing industries downstream from sugar (food processing plants that use sugar as an ingredient, etc.). This may reveal that agglomeration effects are an important contributor to the continuity of industrial activity in these areas: manufacturing companies downstream from sugar still have an incentive to locate near historical factories because of potential cost savings arising from many factories in the same or related industries located there.

Public good provision

Being located in the immediate vicinity of a historical factory is associated with an increase in the likelihood of having a local high school as well as having electricity. This may be due to the greater accessibility of these places (being located in the immediate vicinity of a historical factory is associated with higher road and rail density), greater village lobbying power due to being more industrialized, or local governments having a greater incentive to invest in distributing public goods to these areas as the returns on such investment in industrialized areas are higher.

Household consumption

Being located in the immediate vicinity of a historical factory is associated with an increase in household consumption. This increase may be attributed to an average 1.25 years increase in schooling from being located near a historical factory.

Effects on villages made to grow sugar cane

Economic structure

Individuals in subjected villages have a 15% decreased likelihood of being employed in agriculture, 14% increased likelihood of being employed in manufacturing, and 7% increased likelihood of being employed in retail.

Education

Based on data in the 2000 Population Census, individuals in subjected villages have approximately 0.24 years more schooling, relative to a sample mean of 5 years. They are also more likely to complete primary school and junior high.

Land ownership

In 1980 and 2003, village census collected information on village-owned land. In both years, it was found that subjected villages owned more land (approximately 1.4 percentage points more in 2003, relative to a sample mean of 9%, and approximately 1.2% more in 1980, relative to a sample mean of 11%).

Conclusion

The Dutch Cultivation System improved economic growth prospects for the areas near sugar processing factories and villages that were subjected. There are persistent changes in economic structure, public good provision, years of education, and land ownership in these places.

I have four more days before my final paper, after which I can get back to more frequent posts. I have a few posts in draft, so do visit this space again soon for my next update!

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. In a later post, I will further extend this model to look at the case of screening gunowners so that a “trustworthy” proportion of the population owns a gun.

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).

At this symmetric equilibrium, the two resellers split the market. Each gets \frac{n}{2} product types to sell from latex \frac{n}{2}$ suppliers. In this equilibrium, we have

\pi_{R}^{1}(n_{1}) = (v-f)\sum_{i=1}^{n_{r}}E_{R}[m-(a_{i}-a_{i})^{2}] - n_{r}(F + \tau_{r})

We know (from the original model) that resellers will set

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

So we have that

\pi_{R}^{1}(n_{r}) = n_{r}(v - f)(m - V_{\delta}) - n_{r}(F + \tau_{r})

From the previous analysis, we have that

\tau_{r} = (v - f)(m - V_{\delta}) - F

So the supplier makes (v - f)(m - V_{\delta}) - F profit and each reseller makes zero profit.

Marketplace

We now look at the case in which both intermediaries choose marketplace-mode.

From the analysis above, it also follows that if either marketplace z \in \{1,2\} charges suppliers any P_{z} > P_{-z}, suppliers will not join the marketplace. So in Bertrand pricing, we have p = 0. Here, the supplier makes (v - f)(m - V_{\gamma}) - F and each marketplace makes zero profit.

Marketplace and reseller

What if one intermediary chooses marketplace-mode and the other chooses reseller-mode?

Suppliers will choose to join the intermediary that offers them higher profit. The reseller can offer any \tau up to (v - f)(m - V_{\delta}) - F and the marketplace can offer the supplier any profit up to (v - f)(m - V_{\gamma}) - F. If (v - f)(m - V_{\delta}) - F > (v - f)(m - V_{\gamma}) - F, i.e. V_{\delta} < V_{\gamma}, basically we have something like Bertrand competition with asymmetric costs.

The reseller will offer \tau_{r} = (v-f)(m - V_{\gamma}).

The marketplace will randomise P over [(v-f)(m - V_{\gamma}), (v-f)(m - V_{\gamma}) + \epsilon] (offering anything within this interval will give the marketplace zero profit).

At this point, the reseller makes positive profit either \frac{N}{2}(v-f)(V_{\gamma} - V_{\delta}) - \frac{N}{2}F or N(v-f)(V_{\gamma} - V_{\delta}) - NF.

  • If (v - f)(m - V_{\gamma}) - F > (v - f)(m - V_{\delta}) - F, i.e. V_{\gamma} < V_{\delta}, the marketplace will offer \tau_{r} = (v-f)(m - V_{\delta}).
  • The reseller will randomise \tau_{r} over [(v-f)(m - V_{\delta}), (v-f)(m - V_{\delta}) + \epsilon (offering anything over this interval will give the reseller zero profit).
  • We assume that in both cases, the intermediary that is offering lower profit always charges the minimum price.
  • At this point, the marketplace makes positive profit either \frac{N}{2}(v-f)(V_{\delta} - V_{\gamma}) - \frac{N}{2}F or N(v-f)(V_{\delta} - V_{\gamma}) - NF.

If V_{\delta} < V_{\gamma}, the Nash equilibrium is (Reseller-mode,Reseller-mode).

Marketplace-modeReseller-mode
Marketplace-mode0,0 0,>0 (see above)
Reseller-mode>0 (see above),00,0

If V_{\gamma} < V_{\delta}, the Nash equilibrium is (Marketplace-mode, Marketplace-mode).

Marketplace-modeReseller-mode
Marketplace-mode0,0>0 (see above),0
Reseller-mode0, >0 (see above)0,0

These are the results derived in the original model as well.

Sequential entry

Say Intermediary 1 enters the market first. We assume that suppliers can costlessly switch between Intermediary 1 and Intermediary 2. We also assume that Intermediary 1 has to commit to the same price throughout period 1 and period 2. There are m - (a_{i} - a_{i})^{2} buyers for each product in both period 1 and period 2. We represent the discount factor Intermediary 1 applies to profits in period 2 with \eta (I know the usual notation is \delta but it’s already representative of the private information known to the supplier).

Intermediary 1 enters in reseller mode

The condition on \tau_{1} is that

N[(v-f)(m - V_{\delta}) - F - \tau_{1}] + \eta n_{1}[(v - f)(m-V_{\delta}) - F - \tau_{1}] \geq N[(v-f)(m - V_{\delta}) - F]

i.e. the lower bound on \tau_{1} is given by

\tau_{1} \geq \frac{\eta n_{1}(v-f)(m-V_{\delta}) - F}{N - \eta n_{1}u}

Intuitively, this is because if Intermediary 1 charges any \tau_{1} < \frac{\eta n_{1}(v-f)(m-V_{\delta}) - F}{N - \eta n}, it gets lower profit than it would have if it had charged \tau_{1} = 0 and gotten N[(v-f)(m - V_{\delta}) - F] in period 1, and nothing in period 2 (from Intermediary 2 undercutting and offering suppliers a higher surplus than 0 in period 2).

Intermediary 2 enters in period 2. Note that if Intermediary 1 has entered in reseller-mode, it indicates that entering in reseller-mode is the dominant strategy (V_{\delta} < V_{\gamma}),, and Intermediary 2 will never enter in marketplace-mode.

If Intermediary 2 offers \tau_{2} higher than \tau_{1}, it gets the full market in period 2, and its expected profit is

N[(v-f)(m - V_{\delta}) - F - \tau_{2}]

Intermediary 1’s profit is

N[(v-f)(m - V_{\delta}) - F - \tau_{1}]

If Intermediary 2 offers suppliers the same profit as Intermediary 1 does, it gets half the market in period 2

Intermediary 2’s profit is

\frac{N}{2}[(v-f)(m - V_{\delta}) - F - \tau_{2}]

And Intermediary 2’s profit is

N[(v-f)(m - V_{\delta}) - F - \tau_{1}] + \eta \frac{N}{2}[(v-f)(m - V_{\delta}) - F - \tau_{1}]

If Intermediary 2 offers suppliers a higher profit than Intermediary 1 does, it gets no market share in period 2. Therefore,

Intermediary 2’s profit is 0 and Intermediary 1’s profit is

(1+\eta)N[(v-f)(m - V_{\delta}) - F - \tau_{1}]

So we know that Intermediary 1, in seeking to maximize profits, must charge \tau_{2} \leq \tau_{1}. Within the interval \tau_{1} \geq \frac{\eta n_{1}(v-f)(m-V_{\delta}) - F}{N - \eta n_{1}}, Intermediary 1 makes more profit from lowering its \tau_{1} to match \tau_{2}. Therefore, it will lower its price all the way down to \frac{\eta n_{1}(v-f)(m-V_{\delta}) - F}{N - \eta n_{1}}. We know that n_{1} can only take the values 0, \frac{1}{2} or 1. However, since Intermediary 2 will always set \tau_{2} \leq \tau_{1} down to \tau_{2} = 0, where it makes zero profit as well, we have that n_{1} can only take the values of 0 or \frac{1}{2}. So in equilibrium, we have that

\tau_{2} = \frac{\eta \frac{N}{2}(v-f)(m-V_{\delta}) - F}{N - \eta \frac{N}{2}}

and

Intermediary 1 randomizes over setting

\tau_{1} = \frac{\eta \frac{N}{2}(v-f)(m-V_{\delta}) - F}{N - \eta \frac{N}{2}}

or

\tau_{1} = 0

At this point, suppliers make either (1+\eta)\frac{\eta \frac{N}{2}(v-f)(m-V_{\delta}) - F}{N - \eta \frac{N}{2}} or \eta\frac{\eta \frac{N}{2}(v-f)(m-V_{\delta}) - F}{N - \eta \frac{N}{2}} in profit.

Intermediary 1 makes profit

\pi_{R}^{2} = \frac{N}{2}[(v-f)(m - V_{\delta}) - F - \frac{\eta \frac{N}{2}(v-f)(m-V_{\delta}) - F}{N - \eta \frac{N}{2}}]

or

\pi_{M}^{2} = N[(v-f)(m - V_{\gamma}) - F - \frac{\eta \frac{N}{2}(v-f)(m-V_{\gamma}) - F}{N - \eta \frac{N}{2}}]

Intermediary 2 makes profit

\pi_{R}^{1} = N[(v-f)(m - V_{\delta}) - F]

If, instead, Intermediary 1 enters in marketplace-mode (because V_{\delta} > V_{\gamma} then it is quite easy to observe that

Suppliers make either (1+\eta)\frac{\eta \frac{N}{2}(v-f)(m-V_{\gamma}) - F}{N - \eta \frac{N}{2}} or \eta\frac{\eta \frac{N}{2}(v-f)(m-V_{\gamma}) - F}{N - \eta \frac{N}{2}} in profit.

Intermediary 2 makes profit

\pi_{M}^{2} = \frac{N}{2}[(v-f)(m - V_{\gamma}) - F - \frac{\eta \frac{N}{2}(v-f)(m-V_{\gamma}) - F}{N - \eta \frac{N}{2}}]

or

\pi_{M}^{2} = N[(v-f)(m - V_{\gamma}) - F - \frac{\eta \frac{N}{2}(v-f)(m-V_{\gamma}) - F}{N - \eta \frac{N}{2}}]

Intermediary 1 makes profit

\pi_{M}^{1} = N[(v-f)(m - V_{\gamma}) - F]

Basically, we’re just substituting V_{\gamma} for V_{\delta}. With sequential entry in a two-period model, it is easy to observe that both Intermediary 1 and Intermediary 2 make higher profits.

Multihoming

Assume that suppliers will multihome whenever the increase in their profits is non-negative. Instead of a case in which all products have identical buyer demand 2[m - (a_{i} - a_{i}^{*})^{2}], where each intermediary has its own captive potential buyers m - |a_{i} - a_{i}^{*}| ,we have [m - (a_{i} - a_{i}^{*})^{2}] (for simplicity, let m = \gamma_{i}^{2} + \delta_{i}^{2}) split between the two suppliers if a_{i} is the same, but otherwise with buyers choosing which intermediary to purchase from.

From this, we know that if Intermediary 1 enters as a reseller and sets a_{i} = \theta + \gamma_{i}, it gets m - \delta_{i}^{2} buyers. There are \delta_{i}^{2} buyers left over for each product i.

The intermediary, entering in Stage 2, can choose to enter as a reseller or as a marketplace.

If it enters as a reseller, both intermediaries will pay the supplier i the minimum they can = its marginal cost = 0 (recall that if the supplier is multihoming whenever the increase in its profits is non-negative, the two suppliers are no longer in competition). At this point, each reseller will sell product i to half the buyers who buy at a_{i} = \theta_{i} + \gamma_{i}. We have the following profit expressions.

\pi_{R}^{1} = N[(v-f)(m - V_{\delta}) - F + \eta N[(v-f)\frac{1}{2}(m - V_{\delta}) - F]

\pi_{R}^{2} = N[(v-f)\frac{1}{2}(m - V_{\delta}) - F)]

If Intermediary 2 chooses to enter as a marketplace,

Supplier i‘s optimal choice is to set a_{i} = \theta_{i} + \delta_{i}, and the buyer demand for its product here is m - \gamma_{i}^{2}.

What this tells us is that if the supplier i multihomes when Intermediary 1 is a reseller and Intermediary 2 is a marketplace, it gets buyer demand m - \delta^{2} + m - \gamma^{2}. Since we know m = \delta_{i}^{2} + \gamma_{i}^{2}, this simplifies to buyer demand = m, i.e. supplier i is able to sell to the entire market. At this point, Intermediary 1 (the reseller) pays \tau_{1} = 0 and its profit is

\pi_{R}^{1} = (1+\eta)N[(v-f)(m - V_{\delta}) - F]

Intermediary 2 (the marketplace) is able to charge (1+\eta)N[(v-f)(m - \gamma_{i}^{2}) - F, i.e. its profit is

\pi_{M}^{2} = N[(v-f)(m - V_{\gamma}) - F]

Therefore, even if V_{\delta} < V_{\gamma}, under the conditions laid out above, if \pi_{M}^{1} = V_{\delta} > \frac{1}{2}\gamma, Intermediary 2 has an incentive to enter in marketplace-mode when Intermediary 1 is in reseller mode. The converse applies for the case in which Intermediary 1 has an incentive to set up in marketplace-mode (V_{\gamma} < V_{\delta}).

You may ask whether these results may be inferred from other papers. The answer is yes. But the question you should be asking is whether I have too much time on my hands, and, well:

COVID-19: Migrant worker-related searches in Singapore



Popular Youtuber Preetipls’ campaign for donations to migrant workers in Singapore

After I posted about US search trends in sinophobic terms after the COVID-19 outbreak, a friend told me it would be interesting to look at trends in Singapore. As the COVID-19 pandemic disproportionately affects migrant workers in Singapore, many have taken to social media platforms to encourage donations to migrant worker NGOs, raising awareness of migrant workers’ living conditions and quarantine environment. Those keeping up to date on COVID-19 statistics will also know that infection rates among migrant workers are significantly higher than those among locals, and perhaps be curious about migrant workers.

I therefore visualized migrant worker-related search trends in Singapore.

Data from Google trends

Weekly data was collected on the past year in searches, from the week in 21 April 2019 to the week in 12 April 2020 (1st to 52nd week on the x-axis in the graph). I visualized some data about search results on migrant worker issues and migrant worker NGOs in Singapore, with search terms binned into broad topics (categorization described below).

TopicSearch terms
Migrant“construction worker”; “migrant workers”, “migrant”
Dormitory“dormitories”; “dormitory”
Migrant Workers’ Centre (MWC)“mwc”; “mwc singapore”; “migrant workers centre”
Transient Workers Count Too (TWC2)“twc2”; “transient”; “transient workers”

As shown from the data above, there has been increased interest in migrant workers in Singapore, their living conditions, and NGOs that promote their welfare. Has this interest translated into action to help migrant workers? There are currently various COVID-19 specific campaigns set up on Giving.sg to raise funds to provide food and other essentials to migrant workers in isolation facilities. I tried to compare differences in amount donated in COVID-19 related migrant worker welfare campaigns and other migrant worker welfare campaigns, but it’s difficult to say what the difference is at this time, considering that many of the COVID-19 campaigns have not ended (also, a quick note: Giving.sg does not show campaigns that have ended, so it’s difficult to collect data – perhaps a data request is in order).

Hope this was interesting to you! If you’re Singaporean and would like to show some Solidarity through donating your S$600 cash payout to the ongoing campaigns raising funds for migrant workers on Giving.sg, here’s a link to the campaigns.

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).

Data from Google trends, visualized in R using the amazing and beautiful Wed Anderson palette library (Darjeeling1)

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.

TopicSearch terms
Chinese virus“chinese virus 2020”; “chinese virus us”; “chinese virus”; “china virus”
Dirty“dirty chinese”; “china dirty”
Eating bats“chinese people eat bats”; “why do chinese eat bats”; “do chinese people eat bats”; “why chinese eat bats”; “why do chinese eat bats” “do the chinese eat bats”; “did chinese eat bats”; “why do chinese people eat bats”; “why do the chinese eat bats”; “does chinese eat bats”; “do chinese eat bats and snakes”
Eating dogs“chinese eat dogs”; “why do chinese eat dogs”; “in china they eat dogs”; “do they eat dogs in china”; “do people in china eat dogs”
Pejoratives“chink virus”; “chink”; “yellow virus”

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.

Article

Article

Article

Article

Article

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.

Google search results

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.

Amazon.com: Amazon.com eGift Card: Gift Cards
Source

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.

Source

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.