The Last Inch and Metcalfe’s Law

"Arguing in My Spare Time" No. 2.12

by Arnold Kling

May not be redistributed commercially without the author's permission.
Index

Metcalfe’s Law states that the value of a network increases with the square of the number of people connected to it. Because Metcalfe’s law (due to Robert Metcalfe, inventor of Ethernet and now a journalist/pundit) implies increasing returns, it often is invoked to argue that traditional economics does not apply to the Internet. The point of this essay is that in fact there are diminishing returns, and it is Metcalfe’s Law that ought to be reconsidered.

On the homefair.com web site, one of my jobs is to read and respond to user comments. Let me paraphrase three of the comments that were received in the past week, along with the answers that I was tempted to write.

Question 1: I want to find out who owns the vacant lot next to my property. Where can I find that information on the Web?

Answer 1: Let me explain something. The information that is available on the Web is a subset of the information that is stored in electronic format. All of the information that is stored in electronic format in the United States is a subset of the information that is needed by credit card companies. Because a vacant lot typically does not have a mailbox which can be stuffed with credit card offers, chances are the ownership of the vacant lot next door is recorded only in paper format in the county recorder’s office. Did it ever occur to you to call the county?

Question 2: Can you help me find a job? I am looking to make $75,000 a year. My background is in cost accounting, with a specialty in compensation skeems.

Answer 2: We are not a placement service. Furthermore, there is no compensation "skeem" that is going to get you $75,000 a year.

Question 3: Your site keeps costing me job placements. The prospect will be all set to take a job offer, then they use your Salary Calculator, and it always over-estimates what they should be making by 10 percent or more. Whether I’m trying to move somebody from Akron to Milwaukee or the other way around, it always overstates the cost of living difference by 10 percent. How can I keep from having all my effort to place job candidates go to waste?

Answer 3: Tell your prospects right away that you are unable to grasp the concept of cost of living differentials. That way, nobody will be surprised when they found out later on in the process, and you won’t waste so much of your time.

Again, I was tempted to give these answers. In practice, I tried to be more polite.

What these questions illustrate, in my opinion, is the problem of the last inch. It is analogous to the problem of the last mile.

Pundits who talk about bandwidth (high-speed access to the Internet) tell us that fiber-optic cables have tremendous capacity to carry data. However, they point out, to enable fiber to reach homes ("the last mile") would require an effort to dig up and replace old copper wire. This would be prohibitively expensive.

The pundits’ concern may be misplaced. A co-worker lives in a new development that has fiber-optic cable connected to his home. But he has no high-speed access to the Internet. The faster alternatives of ISDN, ADSL, or cable modems are unavailable to him. He could obtain those services only by having someone dig up the fiber and replace it with copper, which would be prohibitively expensive.

What I call the problem of the last inch is the problem of getting information from the person’s eyeball into his or her brain. I would argue that all three questions listed above were symptomatic of the last inch problem.

I do not mean to imply that the problem of the last inch always is the fault of the user. We also receive comments about homefair.com that express frustrations that are due to inadequate on-screen documentation or a clumsy interface. However, I do not want to use homefair.com as a bad example.

Instead, let me suggest that about.com (the site formerly known as "The Mining Company") is a bad example. About.com is an Internet search tool where humans, called category experts, organize the information. This concept makes sense, and I would expect it to lead to the highest signal-to-noise ratio of any search tool on the Net. However, for me, about.com has one of the lowest signal-to-noise ratios. That is because the design of the site appears to totally negate the concept.

Some people venerate about.com—I’d love to hear the opinions of readers, just to place my views in perspective. However, for me as a user, about.com falls short of its potential in bridging the last inch.

In the larger scheme of things, the last inch is important because it is one of the factors that causes Metcalfe’s Law to break down. As more people become connected to a network, users become less uniform, and the challenge of crossing the last inch increases.

For example, suppose that you set up a nice email alert system to tell people when your web site has new information or offers available. In your email, there is a URL that looks something like "mywonderfulsite.com/personalnews/item.html?itemid=August15_1999&usertype=3A"

Now, imagine that this email is sent to a user of WebTV. This means that he or she cannot click on the link to get there. Nor do they have a mouse to drag the URL from one place to another. If the people for whom I’ve gotten WebTV are representative of the user demographics, your user also has failing eyesight and slight hand tremors. Do they have any prayer whatsoever of transcribing a 50-character URL from their email to their web browser?

Many of us are finding that user interfaces that were successful when the overwhelming majority of users were engineering-types are failing completely as Web usage broadens. Yet, it is nearly impossible to improve usability for late adopters without creating an interface that early adopters view as clumsy and insulting.

How does this affect Metcalfe’s Law? According to Metcalfe’s Law, if you build a network that costs you a fixed cost C, then if you get n people on the network, your profit will be something like (A)(n)(n) – C, where A is the rate at which you can monetize the value of your network. This is a world of increasing returns, where profits are always increasing as the number of eyeballs increases.

However, suppose that the cost of getting from eyeball to the brain—the last inch—increases as a function of the number of people that connect to your network. Then the profit function might look something like (A)(n)(n) – C – (B)(n)(n)(n) where B is some constant. In that case, the optimum network size has a finite limit.

There are other sources of diminishing returns. The ability to monetize the value of your network could be a decreasing function of size. For example, of all the types of products that consumers purchase, one of the most highly correlated with income is hard-back book purchases. Suppose that as the number of Internet users increases, the Internet user base starts to look more representative of the income distribution and less skewed toward affluence. In that case, the proportion of Internet users who purchase books frequently from Amazon.com probably will decline. In terms of our notation, A will be a declining function of n.

Finally, the value to one person of having other people on a network can reach a point of diminishing returns. Hal Varian has pointed out that Metcalfe’s Law assumes that the value to me of other people on the network is linear. That is, suppose that the value to me of being able to chat on AOL is proportional to the number of AOL users, n. If all n users have the same value function, then the total value is n times n, which gives Metcalfe’s Law.

Varian has pointed out the linearity assumption has the peculiar implication that AOL gets as much value out of connecting to a small network as the small network gets from connecting to AOL. If there are n users on the small network and N users on the big network, then the increase in the value of interconnection is nN for either network. Because multiplication is commutative both networks get the same benefit from interconnection.

For example, if one network has 36 members and it connects with a network that has 64 members, the small network has a new value of 36*100, or , 3600 compared with an old value of 36*36, or 1296, for a gain of 2304. The big network has a new value of 64*100, or 6400 compared with an old value of 4096 for an identical gain of 2304.

This result is counterintuitive. I believe that the reason it is counterintuitive is that it is wrong. I believe that for a typical individual, there are diminishing returns to having other people on a network. Even my teenage daughters reach a limit as to how many simultaneous conversations they can sustain.

For example, suppose that the value to me of other people on a network is equal to the square root of n. Now, when the network with 36 members connects to the network with 64 members, the value of the small network increases from 36*6 or 216 to 36*10, or 360. However, the value of the large network increases from 64*8, or 512, to 64*10, or 640. The smaller network’s value increases by 144, while the larger network’s value increases only by 128.

The result that interconnection is more beneficial to the small network conforms to intuition. That is because there are diminishing returns to adding people to a network. In other words, Metcalfe’s Law is false.

Blind faith in Metcalfe’s Law is motivating investors to pour literally billions of dollars into enterprises whose sole near-term objective is to accumulate eyeballs. What I have tried to do is inject a little bit of reality by pointing out that eyeballs bring diminishing marginal returns and increasing marginal costs. This means that conventional economics may be much more applicable in the Internet environment than is commonly assumed.