Consider water pipes. Suppose all water pipes of every size are made from a very strong material of effectively zero thickness (this is to make the math much easier). The output of a water pipe is the number of gallons of water that can flow through it in a month. This is proportional to the area of a cross section. Water pipes are cylinders. The cost of constructing a water pipe is proportional to the amount of material used.

Suppose we’re thinking about building a ten-mile-long water pipe from a specific reservoir to a specific town.

a. Are there increasing returns to scale in the construction of this water pipe?

b. Does this process obey the two-thirds rule?

c. What is the mathematical relationship between cost and output?