CESfunktio

CESfunktio, short for Constant Elasticity of Substitution function, is a production or utility function used in economics to model how easily one input can be substituted for another while keeping output at a desired level. Its defining feature is that the elasticity of substitution between inputs remains constant.

For a two-input production function with capital K and labor L, the CES form is

F(K,L) = [ α K^ρ + (1-α) L^ρ ]^(1/ρ),

where α ∈ (0,1) and ρ ≤ 1, ρ ≠ 0. The elasticity of substitution between K and L is σ = 1/(1-ρ). This parameter determines how readily inputs can be swapped as relative prices change.

Limit cases and intuition: as ρ → 0, the CES function converges to the Cobb-Douglas form F(K,L) = K^α L^(1-α), which has elasticity σ = 1. As ρ → -∞, the elasticity approaches zero, corresponding to Leontief-like behavior with strong complementarity (approximately min{K,L}). As ρ → 1, the elasticity tends toward infinity, implying very easy substitution between inputs (approaching a linear, perfect-substitutes form).

Applications and interpretation: CESfunktio is used in production analysis, macroeconomic models, and trade theory to study how input mixes respond to price changes. The parameter α captures input shares, while ρ (or σ) governs substitutability. The function can be extended to more than two inputs as F(X1,...,Xn) = [ ∑ α_i X_i^ρ ]^(1/ρ).

History: The CES form was introduced in the early 1960s by Arrow, Chenery, Minhas, and Solow, and has since become a standard tool in theoretical and applied economics.