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:

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