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