Learning Curves

394. Learning Curves

The empirical observation that cost or time per unit decreases by a constant percentage every time cumulative production doubles. Discovered by T. P. Wright (1936) studying aircraft production at Curtiss-Wright.

394.1. Wright’s formula

Time (or cost) for the $𝑛$ -th unit:

𝑇_{𝑛} = 𝑇_{1} 𝑛^{- 𝑏}

where:

$𝑇_{1}$ : time for the first unit
$𝑏$ : learning exponent
$𝑛$ : cumulative unit number

The learning rate $𝐿$ — the proportion of time the $2 𝑛$ -th unit takes relative to the $𝑛$ -th:

𝐿 = \frac{𝑇_{2 𝑛}}{𝑇_{𝑛}} = \frac{{(2 𝑛)}^{- 𝑏}}{𝑛^{- 𝑏}} = 2^{- 𝑏}

So $𝑏 = - {log}_{2} (𝐿) = - ln \frac{𝐿}{ln} 2$ .

394.2. Common learning rates

Learning rate $𝐿$	Exponent $𝑏$	Example industry
80%	0.322	Aircraft assembly (Wright’s original)
85%	0.234	Automotive
90%	0.152	Electronics, software
95%	0.074	Highly automated; little learning
70%	0.515	Complex assembly, lots of skill

Lower $𝐿$ → faster learning. $𝐿 = 100 %$ means no learning.

394.3. Crawford’s unit model vs cumulative average

Two conventions:

Wright (unit model): $𝑇_{𝑛}$ is the time for the $𝑛$ -th unit specifically.

Crawford (cumulative average): ${| 𝑇 |}_{𝑛}$ is the average time across all units $1$ to $𝑛$ . Same $𝐿$ , but different curves and different interpretations.

	Wright unit	Crawford cumulative average
Formula	$𝑇_{𝑛} = 𝑇_{1} 𝑛^{- 𝑏}$	${\| 𝑇 \|}_{𝑛} = 𝑇_{1} 𝑛^{- 𝑏}$
Interpretation	time of the $𝑛$ -th unit	average time over first $𝑛$ units

Crawford’s average always overstates per-unit improvement vs Wright (averaging includes early high-time units). Industries vary in which they use.

394.4. Cumulative cost

Total time / cost for the first $𝑛$ units:

𝐶_{𝑛} = \sum_{𝑘 = 1}^{𝑛} 𝑇_{1} 𝑘^{- 𝑏} \approx 𝑇_{1} \cdot \frac{𝑛^{1 - 𝑏}}{1 - 𝑏} (continuous approximation for large 𝑛)

For Crawford model: $𝐶_{𝑛} = 𝑛 {| 𝑇 |}_{𝑛} = 𝑇_{1} 𝑛^{1 - 𝑏}$ .

394.5. Strategic implications (BCG, 1960s)

Boston Consulting Group famously generalized learning curves to experience curves — all costs (not just labor) decline with cumulative volume.

Strategic implications:

Cost leadership through scale: the highest-volume producer has the lowest unit cost
Predatory pricing: price below current cost to grab share, ride the curve to profitability
First-mover advantage: head start on the curve is hard to overtake

Famous critiques (Henderson, BCG): treat learning as a strategic asset and a barrier to entry.

394.6. Limitations

Empirical, not causal: explains pattern, doesn’t explain why learning happens
Discontinuities: new product / process resets the curve
Diminishing returns: long-run learning saturates
Stanford-B / DeJong adjustments: include an “incompressibility factor” — costs can’t fall below some floor

DeJong’s model:

𝑇_{𝑛} = 𝑇_{1} (𝑀 + (1 - 𝑀) 𝑛^{- 𝑏})

where $𝑀 \in [0, 1]$ is the incompressibility floor.

394.7. See also

Factory Physics
Takt Time — performance target as learning advances