Arrow, Chenery, Minhas and Solow in their new famous paper of 1961 developed the Constant Elasticity of Substitution (CES) function. This function consists of three variables Q, С and L, and three parameters A, and.

CES (Constant Elasticity of Substitution) production functions are commonly used in economics to model the relationship between inputs (typically labor and capital) and output in the production process. They have several properties and economic interpretations:

**General Form**: The CES production function is expressed as Q = A [αK^ρ + (1-α)L^ρ]^(1/ρ), where Q is the quantity of output, A is a scale parameter, K is the quantity of capital, L is the quantity of labor, α is the output elasticity of capital, and ρ is the elasticity of substitution between capital and labor.**Homogeneity**: CES production functions exhibit constant returns to scale. This means that if you double the amount of capital and labor, output will also double. It’s a linear homogenous production function.**Substitution Elasticity**: The parameter ρ represents the elasticity of substitution between labor and capital. If ρ is equal to 1, it implies perfect substitutability, while if ρ approaches infinity, it implies perfect complementarity.**Returns to Factors**: Depending on the values of α and ρ, CES production functions can exhibit different returns to factors. When α + ρ = 1, the production function will show constant returns to scale, and when α + ρ > 1, there will be increasing returns to scale. Conversely, if α + ρ < 1, there will be decreasing returns to scale.**Cobb-Douglas and Special Cases**: The Cobb-Douglas production function is a special case of the CES function where ρ is equal to 1. In this case, it assumes constant returns to scale and constant factor shares.**Elasticity of Substitution Interpretation**: The elasticity of substitution (ρ) measures the ease with which one input can be substituted for another. A higher ρ implies that inputs are more substitutable, while a lower ρ indicates they are less substitutable. This has important implications for firms’ decisions on how to allocate capital and labor.**Empirical Applications**: CES production functions are widely used in empirical research to estimate the parameters of production functions, especially in the context of growth theory and labor economics.

In summary, CES production functions are a flexible tool in economics for modeling production relationships, capturing different degrees of substitutability between inputs, and allowing for various returns to factors. Their properties and parameters have economic interpretations that are valuable for understanding the behavior of firms and industries.