CESfunktionen

The CES function, or Constant Elasticity of Substitution function, is a flexible production (or utility) form in economics that allows the substitutability between inputs to vary, while maintaining a constant elasticity of substitution. It generalizes traditional forms such as Cobb-Douglas and Leontief and nests them as special cases.

A common two-input CES production function for capital K and labor L is written as

F(K,L) = [α K^{ρ} + (1 − α) L^{ρ}]^{1/ρ},

where α is a share parameter between 0 and 1, and ρ ≤ 1 with ρ ≠ 0. The elasticity of substitution between inputs is σ = 1/(1 − ρ). Equivalently, using the substitution parameter σ > 0, the function can be written as

F(K,L) = [α K^{(σ−1)/σ} + (1 − α) L^{(σ−1)/σ}]^{σ/(σ−1)}.

Special cases and limits. As σ → 1, the CES reduces to the Cobb-Douglas form, F(K,L) = K^{α} L^{1−α}. As σ → ∞, the inputs become perfect substitutes, yielding a linear form in K and L. As σ → 0, substitutability vanishes and the function approaches Leontief-type behavior, with capital and labor used in fixed proportions.

Properties. The CES function is homogeneous of degree one and is concave in inputs under standard parameter ranges. It yields isoquants that can be smoother or kinked depending on σ, reflecting varying ease of substitution.

Applications. It is widely used in microeconomics and macroeconomics to model production or utility where the ease of substituting inputs (or goods) is uncertain or policy-sensitive, such as capital-labor substitution in growth models or energy-labor trade-offs in sector analyses.

History. The CES form was developed in the 1960s, commonly attributed to Arrow, Chenery, Minhas, and Solow, and has since become a standard tool for comparative statics and empirical modeling.