# Elasticity (economics)

Elasticity (economics)
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In economics, elasticity is the measurement of how changing one economic variable affects others. For example:

• "If I lower the price of my product, how much more will I sell?"
• "If I raise the price, how much less will I sell?"
• "If we learn that a resource is becoming scarce, will people scramble to acquire it?"

In more technical terms, it is the ratio of the percentage change in one variable to the percentage change in another variable. It is a tool for measuring the responsiveness of a function to changes in parameters in a unitless way. Frequently used elasticities include price elasticity of demand, price elasticity of supply, income elasticity of demand, elasticity of substitution between factors of production and elasticity of intertemporal substitution.

Elasticity is one of the most important concepts in neoclassical economic theory. It is useful in understanding the incidence of indirect taxation, marginal concepts as they relate to the theory of the firm, and distribution of wealth and different types of goods as they relate to the theory of consumer choice. Elasticity is also crucially important in any discussion of welfare distribution, in particular consumer surplus, producer surplus, or government surplus.

In empirical work an elasticity is the estimated coefficient in a linear regression equation where both the dependent variable and the independent variable are in natural logs. Elasticity is a popular tool among empiricists because it is independent of units and thus simplifies data analysis.

Generally, an elastic variable is one which responds a lot to small changes in other parameters. Similarly, an inelastic variable describes one which does not change much in response to changes in other parameters. A major study of the price elasticity of supply and the price elasticity of demand for US products was undertaken by Hendrik S. Houthakker and Lester D. Taylor.

## Mathematical definition

The definition of elasticity is based on the mathematical notion of point elasticity.

In general, the "x-elasticity of y", also called the "elasticity of y with respect to x", is: $E_{y,x} = \left|\frac{\partial \ln y}{\partial \ln x}\right| = \left|\frac{\partial y}{\partial x}\cdot\frac{x}{y}\right| \approx \left|\frac {% \Delta y} {% \Delta x}\right|$

The approximation becomes exact in the limit as the changes become infinitesimal in size.

The absolute value operator is for simplicity – generally the sign of the elasticity is understood as being always positive or always negative. However, sometimes the elasticity is defined without the absolute value operator, when the sign may be either positive or negative or may change signs. A context where this use of a signed elasticity is necessary for clarity is the cross-price elasticity of demand — the responsiveness of the demand for one product to changes in the price of another product; since the products may be either substitutes or complements, this elasticity could be positive or negative.

## Specific elasticities

### Elasticities of demand

Price elasticity of demand
Price elasticity of demand measures the percentage change in quantity demanded caused by a percent change in price. As such, it measures the extent of movement along the demand curve. This elasticity is almost always negative and is usually expressed in terms of absolute value (i.e. as positive numbers) since the negative can be assumed. In these terms, then, if the elasticity is greater than 1 demand is said to be elastic; between zero and one demand is inelastic and if it equals one, demand is unit-elastic. A perfectly elastic demand curve is horizontal (with an elasticity of infinity) whereas a perfectly inelastic demand curve is vertical (with an elasticity of 0).
Income elasticity of demand
Income elasticity of demand measures the percentage change in demand caused by a percent change in income. A change in income causes the demand curve to shift reflecting the change in demand. IED is a measurement of how far the curve shifts horizontally along the X-axis. Income elasticity can be used to classify goods as normal or inferior. With a normal good demand varies in the same direction as income. With an inferior good demand and income move in opposite directions.
Cross price elasticity of demand
Cross price elasticity of demand measures the percentage change in demand for a particular good caused by a percent change in the price of another good. Goods can be complements, substitutes or unrelated. A change in the price of a related good causes the demand curve to shift reflecting a change in demand for the original good. Cross price elasticity is a measurement of how far, and in which direction, the curve shifts horizontally along the x-axis. A positive cross-price elasticity means that the goods are substitute goods.
Cross elasticity of demand between firms
Cross elasticity of demand for firms, sometimes referred to as conjectural variation, is a measure of the interdependence between firms. It captures the extent to which one firm reacts to changes in strategic variables (price, quantity, location, advertising, etc.) made by other firms.
Elasticity of intertemporal substitution
Combined Effects
It is possible to consider the combined effects of two or more determinant of demand. The steps are as follows: PED = (∆Q/∆P) x P/Q. Convert this to the predictive equation: ∆Q/Q = PED(∆P/P) if you wish to find the combined effect of changes in two or more determinants of demand you simply add the separate effects: ∆Q/Q = PED(∆P/P) + YED(∆Y/Y)
Remember you are still only considering the effect in demand of a change in two of the variables. All other variables must be held constant. Note also that graphically this problem would involve a shift of the curve and a movement along the shifted curve.

### Elasticities of supply

Price elasticity of supply
The price elasticity of supply measures how the amount of a good firms wish to supply changes in response to a change in price. In a manner analogous to the price elasticity of demand, it captures the extent of movement along the supply curve. If the price elasticity of supply is zero the supply of a good supplied is "inelastic" and the quantity supplied is fixed.
Elasticities of scale
Elasticity of scale or output elasticities measure the percentage change in output induced by a percent change in inputs. A production function or process is said to exhibit constant returns to scale if a percentage change in inputs results in an equal percentage in outputs (an elasticity equal to 1). It exhibits increasing returns to scale if a percentage change in inputs results in greater percentage change in output (an elasticity greater than 1). The definition of decreasing returns to scale is analogous.

## Applications

The concept of elasticity has an extraordinarily wide range of applications in economics. In particular, an understanding of elasticity is fundamental in understanding the response of supply and demand in a market.

Some common uses of elasticity include:

## Variants

In some cases the discrete (non-infinitesimal) arc elasticity is used instead. In other cases, such as modified duration in bond trading, a percentage change in output is divided by a unit (not percentage) change in input, yielding a semi-elasticity instead.

## Footnotes

1. ^ Hendrik S. Houthakker, Lester D. Taylor (1970).
2. ^ Negbennebor (2001).
3. ^ Perloff, J. (2008). p.36.
4. ^ Varian (1992). pp.16–17.
5. ^ Samuelson, W. & Marks, S. (2003). p.233.

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