# Elasticity of substitution

The elasticity of substitution is a measure used in microeconomics to calculate the ease of substituting one good for another.

The elasticity of substitution measures how much the quantity of a good or service must adjust to maintain a constant level of profit or production. It is an indicator free of measurement units, since it is expressed as a percentage of variation.

The elasticity of substitution can be applied both to the consumption of final goods and to the factors of production. In the first case, the substitution between two consumer goods or services is measured, keeping the utility level constant. While in the second case, the substitution between production factors is measured, keeping the level of production constant.

## The relationship between TMS and elasticity of substitution

The Marginal Replacement Rate (TMS) tells us how much the quantity of one good should be modified when we increase or decrease the quantity of another, all with the aim of maintaining constant utility or production.

The TMS measures the slope of the utility curve (in the case of consumption) or isoquant (in the case of production) and is affected by the unit of measurement that we use: kilos, units, tons, etc.

The elasticity of substitution measures the curvature of the utility or isoquant curve. That is, the percentage of change in the ratio of use or consumption of two goods, divided by the percentage of change in the TMS.

## Formula of elasticity of substitution

The formula for the elasticity of substitution is as follows:

Where:

- X1, X2 = goods or services.
- TMS: Marginal rate of replacement.

## Example of elasticity of substitution of factors

Below we see how the concept is applied in the field of production. In production, the isoquant is the curve that shows us the different combinations of productive factors (suppose Capital (K) and Labor (L)) that allow us to obtain the same amount of production. The elasticity of substitution, meanwhile, refers to the ease with which one productive factor (let's say K) can be substituted for another (L). The formula for elasticity in this case is as follows:

Where:

- K, L = Capital, Labor.
- TMS: Marginal rate of replacement.

Another closer example is the substitution between two consumer goods such as pizza and hamburgers. People, depending on their preferences, might be willing to substitute hamburgers for pizza. The rate at which these two goods must be exchanged for the consumer to be equally happy (same level of utility) is the Marginal Rate of Substitution.

To obtain a free measure of units (pieces of pizza or hamburger buns) we resort to the concept of elasticity that will give us a percentage value. The higher that value, the easier it is to substitute one good for another.

## Graph of the elasticity of substitution of factors

The elasticity of substitution is related to the curvature of the Isoquant and the production function. In the following graph we see an example of an isoquant curve.

The elasticity of this isoquant curve is calculated as:

= Proportional change in the slope of 2 rays (OA and OB) from the origin to the two points on the isoquant / Proportional change in the slopes of the isoquants (the tangents drawn) at the two points (A and B)

## Extreme values of the elasticity of substitution

Elasticity can take extreme values in the following cases:

a) When the substitution is perfect, the isoquants are straight lines and the elasticity is infinite.

b) When the substitution allows only fixed proportions, the isoquants are right angles and the elasticity is zero.

c) There are production functions that have a constant elasticity. This means that the elasticity is not affected by the relative variations of the production factors or, in other words, the substitutability is the same at all points of the isoquant. A widely used example of a production function that meets these characteristics is the Cobb-Douglas production function.

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