Elasticity of complementarity¶
The percentage responsiveness of the ratio of two factors' marginal products or prices to a one-percent change in their relative input quantities, equal under the stated two-factor conditions to the inverse of elasticity of substitution.
Core Idea¶
The elasticity of complementarity measures how sensitively the relative marginal products—and, under the relevant price-taking interpretation, relative factor prices—respond to a proportional change in relative input quantities. For a differentiable two-input production function, it is a logarithmic derivative, so it is dimensionless and local to the evaluated input combination. Writing marginal products as f₁ and f₂, the displayed formulation compares the proportional change in f₁/f₂ with the proportional change in x₂/x₁.
Scope of Application¶
Use elasticity of complementarity with production function, factor pair, evaluation point, derivative convention, marginal-product or price interpretation, regularity assumptions, and comparison substitution elasticity stated. Use elasticity of complementarity with production function, factor pair, evaluation point, derivative convention, marginal-product or price interpretation, regularity assumptions, and comparison substitution elasticity stated.
- Production theory. Describes factor interaction.
- Labor demand. Analyzes relative wages and inputs.
- Agricultural economics. Compares productive factors.
- Energy economics. Studies capital-energy response.
- Functional-form analysis. Derives local curvature measures.
Clarity¶
A unit-free elasticity supports comparison across scales, but its value can change across input combinations and functional forms. Reporting one number as a global technology property can mislead. The closest near miss sets the boundary: Elasticity of substitution is closest: it reverses the response relation and is the reciprocal only under the stated formulation and regularity conditions.
Manages Complexity¶
Complementarity in this technical sense is derivative-defined. Sign conventions and reciprocal formulas depend on the orientation of input ratios, so algebraic definitions must precede verbal interpretation. The central local curvature–global label tradeoff is this: The derivative is precise at a point while verbal summaries suggest a constant technology.
Abstract Reasoning¶
Use three linked moves: specify differentiable two-factor production function; compute both marginal products; form relative marginal product and relative input ratios. As a collapse test, the case exits when marginal products or relative quantities are undefined, the derivative convention changes, or a multi-factor measure is substituted without reconciliation. A fourth check is to take the local logarithmic response with consistent orientation.
Knowledge Transfer¶
Relative-response elasticity transfers across economics, but production derivatives, factor ratios, and the complementarity convention delimit this measure. The nearest stopping boundary is explicit: Elasticity of substitution is closest: it reverses the response relation and is the reciprocal only under the stated formulation and regularity conditions. The inclusion test remains: A quantity is the elasticity of complementarity when it is the stated local log-derivative of relative marginal products or factor prices with respect to the relevant relative input ratio for a declared production function. The structure no longer applies when the case exits when marginal products or relative quantities are undefined, the derivative convention changes, or a multi-factor measure is substituted without reconciliation. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. It supplies the local percentage-response form.
Relationships to Other Abstractions¶
Current abstraction Elasticity of complementarity Domain-specific
Parents (1) — more general patterns this builds on
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Elasticity of complementarity is a kind of Elasticity Prime
It is explicitly a logarithmic-derivative, unit-free responsiveness ratio between two factor quantities.
Hierarchy path (1) — routes to 1 parentless root
- Elasticity of complementarity → Elasticity
Neighborhood in Abstraction Space¶
Elasticity of complementarity sits in a sparse region of the domain-specific corpus (61st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Financial & Economic Ratios (22 abstractions)
Nearest neighbors
- Total revenue test — 0.86
- Net domestic product — 0.85
- Factor cost — 0.85
- Shrinkflation — 0.84
- Money flow index — 0.84
Computed from structural-signature embeddings · 2026-10-08