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

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

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

Introduction

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

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

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

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

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

Model Setup

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

Marketplace or reseller?

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

Reseller

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

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

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

from

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

Expanding a little, we get

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

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

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

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

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

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

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

From this, we get

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

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

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

Simplifying, we observe that this reduces to

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

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

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

Marketplace

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

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

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

The monopolist therefore makes

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

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

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

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

Now we add different cost structures and heterogeneous product demand.

Heterogeneous products and costs

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

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

For each product $i$, the reseller gets

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

For each product \$latex i, the marketplace gets

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

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

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

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

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

We remove the common terms, such that we have

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

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

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

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

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

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

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

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

Network externalities

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

Favorable beliefs

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

Unfavorable beliefs

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

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

In this case, the monopolist makes profit

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

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

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

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

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

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

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

The full paper is available here.

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

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

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

Introduction

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

First, some basic setup.

Model setup

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

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

Two types of consumers: Naive and sophisticated

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Concluding remarks

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

The full paper I read may be found here.