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 from using the good in each period, where refers to the number of consumers joining the network of consumers using the good in period (it should be trivial to observe that , since is the total population of potential consumers). 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). (i.e. the utility is strictly increasing in the number of consumers using the good).
• We represent the discount rate with , and the depreciation rate of the durable good (or basically, how quickly consumers get bored of it) with . What this means is that, for example, if I am the only consumer on the network and no other consumers join ever, and assuming , , , in the first period, I derive utility of ; in the second period, I derive utility of ; in the third period, I derive utility of ; and so on. Therefore, we have that the combined discount factor = . For simplicity, we represent this with . We know that the combined discount factor in period is .
• The monopolist incurs marginal cost 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 = ) and sophisticated consumers (total number = ). 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 () for their purchase decision in period . So for them, the expected number of consumers on the network if they choose to purchase will be . (The here comes from the consumers himself being added to the network.) What utility will the naive consumer expect?

The expected utility is given by the amount of utility derived from 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

The equation above properly accounts for all of the consumers who will join the network in each period . We notice immediately that . This is because more consumers may join in the later periods, while all consumers on the network stay on the network, so .

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

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 with is indifferent between joining and not joining, and all consumers with will join the network). Therefore, for consumers who join, we have the expected and actual indirect utility functions expressed below.

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 , all naive consumers, since their expected utility is the same as that of that one naive consumer, will join in period . 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 less than each consumer type’s expected utility from joining the network. We know that naive consumers have expected utility given by

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

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

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

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

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

If ,

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

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

If the monopolist sets , then the sophisticated consumers will predict that naive consumers will not join in period 2, and therefore, they will not purchase if

Notice that , since is monotonically increasing. Therefore, we know that if the monopolist sets , 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 . So,

By comparison, we can see that which is lower will depend on the and the difference between and . If is sufficiently small (), then is bigger, and the firm will charge . Otherwise, the converse will apply.

Therefore, we have for ,

where is the discount factor for the monopolist

(Sophisticated consumers are being charged their maximum willingness to pay)

For ,

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.