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The Second Derivative

"Arguing in My Spare Time," No. 3.18

by Arnold Kling

July 13, 2000

If you know calculus, there is a relatively simple way to describe the contrast between believers and skeptics concerning the new economy. The dividing line concerns the sign of the second derivative, which believers take to be positive and skeptics take to be negative.

(If you do not know calculus, then I hope that the following explanation will help. Think of a moving object that you measure in relation to a fixed scale. The first derivative is the rate of change. The second derivative is rate of change OF the rate of change.

For example, imagine a ball shot from a cannon. You might measure the height of the cannonball relative to the earth. The first derivative is the rate at which its height is changing. Early in the flight, it might be rising at 20 feet per second. Late in the flight, it might be falling at a rate of 30 feet per second, or we could say it is rising at rate of -30 feet per second.

When we compare the rate at which the cannonball is rising at one point with its rate at a nearby point, we are taking the second derivative. Because the cannonball rises more slowly as it gets further from the cannon, the second derivative is negative.

In contrast, suppose that we had a company whose revenues were experiencing the mythical "hockey stick" growth. Revenues might increase 1 percent the first quarter, 2 percent the second quarter, 2.5 percent the third quarter, etc. Because each quarter's rate of increase is higher than the rate in the previous quarter, we say that the rate of increase is increasing. This means that the second derivative is positive.)

There are various ways in which new economy believers divide from skeptics on the issue of the second derivative.

1. Network economies. The theory of network economics (often called Metcalfe's Law by techies) is that as you add people to a network, the increase in the value of the network increases at an increasing rate. Belief in network economies underlies the strategy of sacrificing short-term profitability for market share. This McKinsey business plan, as I call it, has been used to justify the launching of many Internet "nonprofits."

2. Eyeball aggregation. A related strategy promoted by the new economy believers is eyeball aggregation. The theory is that as you increase your subscriber base, your sales will increase even more because of your ability to use data mining and cross-selling. According to this hypothesis, revenues increase at an increasing rate relative to the number of subscribers.

3. The long boom. The idea here is that better economic performance is so self-reinforcing that the rate of increase in economic growth is going to continue to increase. Therefore, even "old economy" stocks deserve high price-earnings ratios.

The opposite of the new economy view is the view that there remain important sources of diminishing returns in the economy. We skeptics therefore see negative second derivatives.

1. I have argued that there are reasons to believe that Metcalfe's Law holds only for a while, and then diminishing returns set in.

2. I have argued that there are diminishing returns to software development, particularly as systems increase in their complexity. I believe that the diseconomies of complexity far outweigh the economies of eyeball aggregation. Based on this view, I believe that's profits will be lower as a diversified retailer than would be the case had they stuck to books.

3. I have observed that the period of "hockey stick" growth in Internet usage is over. In my experience at, the period 1994-1996 was one in which usage increased at an increasing rate. Since then, usage has continued to increase, but at a slower rate.

Consider the following data for average daily page views on Yahoo:

MONTHAverage daily page views (millions)

Each quarter, page views increased. The first derivative clearly is positive.

However, look what has happened to the rate of increase. To abstract from seasonality, take the year over year increase each June. From June 1997 to June 1998, page views increased 203 percent. The following June, they were up 170 percent. The June after that, they were up 119 percent.

The second derivative is negative, and strongly so. It is -43 and then -51. If the second derivative remains in that range, then within three years Yahoo essentially will have stopped growing.

Suppose that Yahoo's growth continues to decelerate at the rate it has over the past two years, and then usage flattens out. Then its page views will asymptotically approach (top out at) 1.5 - 2.0 billion per day.

Yahoo's ability to monetize page views has consistently stayed within a range of $4 to $4.75 per 1000 page views. If we assume $5 per 1000 page views and 2 billion page views per day, then Yahoo's revenue will top out at $10 million per day, or $3.65 billion per year, or $6.72 per share.

As of July 13, Yahoo's stock price was around $125 per share. If you buy a share of Yahoo today, it will take them 20 years to bring in as much in gross revenues as you pay for that share of stock. Out of those revenues, they will have to pay expenses (currently running at $1.20 per share). I think it is safe to say that over the next twenty years you can get a better return in a money market fund than in Yahoo--probably far better.

My guess is that Yahoo's investors do not understand the concept of the second derivative.