Equation 1 · What the Machine Breakers Actually Wanted
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol Δ
Δ is part of the quantity the equation computes from the expression on the right.
Symbol w
the real wage, Y/L output per worker, and labour’s share of value added.
Symbol Y
Y occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol L
L occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol s_L
is one of the signed contributions combined to compute the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →change
Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
Allen’s analysis of the same period gives the mechanism a name. Across 1760–1913 he finds a two-stage evolution of inequality in which, over the first half of the nineteenth century, “the real wage stagnated while output per worker expanded”, the profit rate roughly doubled, and profits’ share of national income rose at the expense of labour and land; only after mid-century did real wages resume growing in line with productivity, with factor shares stabilising [ 8 ] . The gap between output per worker and the real wage is an accounting identity once the labour share is admitted: . where w is the real wage, Y/L output per worker, and labour’s share of value added.…
Read the full surrounding passage
Allen’s analysis of the same period gives the mechanism a name. Across 1760–1913 he finds a two-stage evolution of inequality in which, over the first half of the nineteenth century, “the real wage stagnated while output per worker expanded”, the profit rate roughly doubled, and profits’ share of national income rose at the expense of labour and land; only after mid-century did real wages resume growing in line with productivity, with factor shares stabilising [ 8 ] . The gap between output per worker and the real wage is an accounting identity once the labour share is admitted: . where w is the real wage, Y/L output per worker, and labour’s share of value added. Productivity growth is necessary for durable wage growth but not sufficient for it: the second term can absorb the first for decades. Allen’s contribution is to show that in Britain it did.
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.
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