Returns to Scale vs Experience
How many wristwatches are cheaper than Big Ben?
Big machines are sometimes more efficient. But they cost more, so fewer can be produced with a finite budget. Small machines are cheaper and may benefit from improvement over time, driven by experience in building more units. When does this experience lead to greater overall efficiency? We derive an approximation which, given a learning rate, tells how much smaller a machine must be to overcome an initial efficiency disadvantage.
Learning curves were characterized in the context of industrial production in the 1930s by Wright. The production cost of a machine follows a power law in the number of units made so far
Cn = C1 n^(-a) < 1 >
where C1 is the cost of the first unit, Cn is the cost of the nth unit, and a is the learning rate.
Across diverse industries, Wright and others have documented cost reductions of around 20% per doubling in the number of units. This corresponds to a learning rate of nearly
a = -log2(0.8) = 0.32 < 2 >
and in the following we assume 0 < a < 1.
Notably, Moore's law may be a special case of Wright learning when unit production increases exponentially.
A machine is a device that emits some commodity at a constant rate. We call this rate the machine's size. For any commodity of interest, there may be machines of different sizes
Ub - rate of output for a big machine Us - rate of output for a small machine Ub > Us
Each machine must itself be manufactured at a one-time cost in dollars
Cb - cost for first unit of a big machine Cs - cost for first unit of a small machine Cb > Cs
and its size is fixed at time of manufacture. Its efficiency can be defined by a cost per rate of commodity output
Eb = Cb/Ub - efficiency for first unit of a big machine Es = Cs/Us - efficiency for first unit of a small machine
and we may assume larger machines have an initial advantage
Eb < Es
For any finite budget B and learning rate a, we can afford to manufacture Nb units of a big machine or Ns units of a small one, Nb < Ns
Nb Ns B = Sum Cb n^-a = Sum Cs m^-a < 3 > n=1 m=1
Nb Ns Cb Sum n^-a = Cs Sum m^-a < 4 > n=1 m=1 Cb/Cs = H(Ns, a) / H(Nb, a) < 5 >
where H(x, y) is the generalized harmonic number of order y of x.
In the limit as Nb goes to infinity, the right side of equation < 5 > admits the following approximation
H(Ns, a) / H(Nb, a) ~ (Ns/Nb)^(1-a) < 6 >
This approximation seems quite good even at small orders, and is fine for our purposes since the learning rate a is statistical in nature anyway
Substituting it into equation < 5 > gives
Cb / Ns \ (1-a) ---- ~ ( ---- ) < 7 > Cs \ Nb /
From our definitions, we can further substitute the product of size and efficiency for initial cost
UbEb / Ns \ (1-a) ------ ~ ( ---- ) < 8 > UsEs \ Nb /
The total output of a fleet of machines is the product of the number built and their size, NsUs for small machines and NbUb for big ones. Setting these equal, we get
Ns/Nb = Ub/Us < 9 >
Using this to substitute in < 8 > gives
UbEb / Ub \ (1-a) ------ ~ ( ---- ) < 10 > UsEs \ Us / / Ub \ a Es ( ---- ) ~ ---- < 11 > \ Us / Eb
a log Ub/Us ~ log Es/Eb < 12 >
which relates the ratio of sizes to the ratio of initial efficiencies in the case where fleets of each machine type produce the same total output for the same total cost.
This framework can be applied to the question of reactor size in the nuclear energy industry. The United States operates a modest fleet of large reactors, which produce about Ub = 1 GW of electricity per unit. In recent years, it has been claimed that mass production of much smaller reactors may be cheaper in the long run.
Though the question of capital cost for reactors is murky, it appears large reactors can be built in some markets for nearly Eb = $4/W.
On the other hand, a company called Oklo plans to build Us = 1.5 MW reactors at an initial cost Cs = $10M, corresponding to an initial efficiency Es = $6.7/W.
Which type of reactor will produce more electricity when built into fleets of equal cost? We'll assume the learning rate from < 2 > for both types and solve < 12 > for Us. If the solution is larger than Oklo's 1.5 MW, we may conclude the Oklo fleet will produce more electricity
0.32 ln(1000 MW/Us) ~ ln(6.7/4) ln(1000 MW/Us) ~ 0.52/0.32 Us ~ 1000 MW / exp(0.52/0.32) Us ~ 200 MW
An initial efficiency advantage of a big machine must be due to efficiencies of scale, so it is not independent of size. Can this dependency be added here, to allow a closed-form solution for optimal machine size?
Thanks to Ed Pheil (@EdPheil) for many engaging discussions on Twitter.
 Wright (1936) Factors Affecting the Cost of Airplanes
 Nagy et al (2013) Statistical Basis for Predicting Technological Progress
 Alexandre et al (2020) Limit of a ratio of harmonic numbers?
 Energy Technologies Institute (2018) The ETI Nuclear Cost Drivers Project: Summary Report
 Adams (2020) Oklo has filed first COLA with the NRC since 2009