# An Introduction to the Two-Factor CES Production Function

## Constant Elasticity of Substitution Property Defined

CES stands for constant elasticity of substitution. CES is a function describing production, usually at a macroeconomic level, with two inputs (which are usually capital and labor).

### Introduction to Elasticity of Substitution

The concept of elasticity of substitution in economics is the elasticity, or measurement of responsiveness, of the ratio of two inputs to a production function as it relates to the ratio of their marginal products.

Elasticity of substitution then measures the substitutability between production inputs or how easy (or difficult) it is to substitute one production input for another. In economics, constant elasticity of substitution (CES) is a property specific to some production functions as well as some utility functions.

### Definition of CES Production Function

The CES production function is a production function with roots in neoclassical economics, an approach to the study of economics that focuses on the study of prices, outputs, and income in a market in terms of supply and demand. The CES production function is an aggregator function that combines two or more types of productive inputs into an aggregate quantity or in other words, takes the values of multiple rows of production data and groups them together to form a single value that carries more significant meaning to the researcher.

The CES production function is a production function that exhibits just that, constant elasticity of substitution which holds that there is a constant percentage change in factor proportions as a result of a percentage change in the marginal rate of technical substitution (MRTS), which is the amount by which the quantity of one input is reduced when one additional unit of another input is utilized.

### The Two-Factor CES Production Function

As defined by Arrow, Chenery, Minhas, and Solow (1961), the two-factor CES production function is written this way:

V = (beta*K-rho + alpha*L-rho) -(1-rho)

Where:

• V = value-added (though y for output is more common)
• K is a measure of capital input
• L is a measure of labor input
• The Greek letters are constants. Normally alpha > 0 and beta > 0 and rho > -1.

In this production function, the elasticity of substitution between capital and labor is constant for any value of K and L. It is (1+rho)-1

### Relevance of CES Production Function in Economics

One key to economic efficiency is the substitution of limited production factors with a relatively more available one. This substitution of production factors is also a driving force of economic growth. One way to measure that growth force is through the elasticity of substitution between labor and capital, which is the primary focus of the CES production function. The relationship between economic growth and the size of the substitution elasticity has long been known and as such has been a primary area of study in dynamic macroeconomics.

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