Market for network services
Although the first paper on network issues was written by Rohlfs in 1974, network goods (or services) have become a very important part of most advanced economies .
What follows is my first lecture on “Network Services”, used frequently in Industrial Organization Courses
What is the difference between an ordinary service, say a haircut and a network service, for instance telephone service?
Haircut (only I look ugly as before!!)
The consumer who is getting a haircut gains utility from it, hopefully, and no one else is affected. But the utility one gets from having a phone depends on how many other people have telephones. So one consumer’s utility depends on the total number of buyers of the service.
Networking!!
We will discuss two market structures, monopoly and duopoly in the network market, also compare this with the social optimum solution. As we will see, the nature of equilibrium is substantially different from non-network goods.
First we derive the demand curve of a network service. Unlike non-network goods and services, the demand curve is not negatively sloped! This changes the nature of equilibrium.
Assume that consumers pay a single price p for accessing the network , but there is no charge for subsequent pay per use.
N is potentially the maximum number of consumers that may want to subscribe to the network.
Let vi be the value that consumer “i” places on the network when everyone subscribes to the network. In other words, vi is the maximum amount that “i” will pay for the network.
If all consumers are identical, then this is an easy model to analyze, but that is not a realistic assumption. So we assume that customers are different. They are distributed uniformly over the interval [0,100].
So, if the consumer knows that f is the fraction of total consumers subscribing to the network, the maximum that consumer “i” would pay is a function of f and vi. For simplicity, we assume that the consumer “i” will pay f.vi.
Therefore given a price p, there will be a consumer whose willingness to pay is vi^ such that
The price p = f vi^.
By our assumption of uniform distribution, the fraction of consumers who want to subscribe to this service is 1 – f = vi^/100
Therefore p = 100f(1-f) after a little algebra
This is the demand curve for network services, note that this is inverse demand showing p as a function of f, when f is the fraction of the total subscribing to the network.
| “f” is fraction of people in the network | p is demand price |
| 0 | 0 |
| 0.1 | 9 |
| 0.2 | 16 |
| 0.3 | 21 |
| 0.4 | 24 |
| 0.5 | 25 |
| 0.6 | 24 |
| 0.7 | 21 |
| 0.8 | 16 |
This demand curve is not negatively sloped.
It is positively sloped for f < ½ , reaching a maximum at p = 25.
Over the positively sloped range, some consumers quit (f falls), there are two effects:
Due to the price effect, demand price goes up (the people who would want to pay higher price would remain in the market)
Due to the network effect, the value of the network falls to existing customers, so some others quit as well.
As the network effect dominates the price effect over this range, as f falls, demand price p also falls
Over the negatively sloped range, the network effect is small because f is large, and the demand curve is negatively sloped.
Assuming Q = fN, we get the total revenue curve as
Q = pfN = 100f(1-f)fN = 100f2N – 100f3N
Network monopoly with marginal cost of 11.11 (no fixed costs)
My numbers are a little different from the ones in text , pages 642-3.
Π = pfN – (11.11)fN
= 100f(1-f).fN -11.11fN
Differentiating with respect to f, and setting to zero
100f(2-3f)N-11.11N = 0
Or
2f – 3f2 – 11.11/100 = 0
Or 3f2 -2f + (1/9) = 0
The solution is
Optimal f* = 1/6 {2 ±( 4 – 4x2x(1/9)}1/2
Which comes to f* = 0.6, p* = 24, and profit per-unit as 13.49
Total profit is 13.49N
We try zero marginal cost next
Π = pfN
= 100f(1-f).fN
Check the solution is f = 2/3, p = 100(2/3)(1/3) = 22.22
Profit per-unit is 22.22 (2/3) N = 14.8N
If price is zero, then everyone subscribes, total demand is N
The total social surplus is 1/6 N
The government may supply this for free and charge a tax per user of 1/6 ?
The other problem is how a network is built over time. In regular market as a price is announced, there could be a fraction of the total number of buyers at first, but the rest would come over time.
In a network market, a fraction of buyers may come initially, but if f is less than the breakeven point, the network may collapse if some decide to leave.
Network duopoly with Marginal cost 11.11
There is a Bertrand solution with price = 11.11.
But there is also another Nash equilibrium where one sells at the monopoly price and the other sells at 11.11 with zero customers. Because of the network externality, the firm with price 11.11 will not get any customers, if the monopoly is there first.
Interestingly, entry is also not possible here if there is a monopoly. One way to enter would be a dynamic strategy where the entrant offers it at a zero price, at a loss. Then the incumbent will either offer a zero price or exit. If the incumbent offers a zero price, they can both increase their price to 11.11 and make normal profits.
Alternately, because of network externality, any initial provider should provide the service at well below cost or for free and charge a one-time sign up fee. This would establish a large network, and new entry would be blocked since cost per period for the consumers is zero.
Lastly we generalize the monopoly model with and without fixed cost with the assumption that the consumers’ willingness to pay is uniformly distributed on an interval [0,K]. Here
P = Kf(1-f) and
Π = pfN
= Kf(1-f).fN
Here, the monopoly price would be a function of K
Notice that the breakeven price is P = K/2