# Elasticity (economics)

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In economics, elasticity is the measurement of the proportional change of an economic variable in response to a change in another. It shows how easy it is for the supplier and consumer to change their behavior and substitute another good, the strength of an incentive over choices per the relative opportunity cost. Economics is the social science that studies the production, distribution, and consumption of goods and services.

## Contents

It gives answers to questions such as:

• "If I lower the price of a product, how much more I will sell?"
• "If I raise the price of one good, how will that affect the sales of this other good?"
• "If the market price of a product goes down, how much will that affect the amount that firms will be willing to supply to the market?"

An elastic variable (with an absolute elasticity value greater than 1) is one which responds more than proportionally to changes in other variables. In contrast, an inelastic variable (with an absolute elasticity value less than 1) is one which changes less than proportionally in response to changes in other variables. A variable can have different values of its elasticity at different starting points: for example, the quantity of a good supplied by producers might be elastic at low prices but inelastic at higher prices, so that a rise from an initially low price might bring on a more-than-proportionate increase in quantity supplied while a rise from an initially high price might bring on a less-than-proportionate rise in quantity supplied.

Elasticity can be quantified as the ratio of the percentage change in one variable to the percentage change in another variable, when the latter variable has a causal influence on the former. A more precise definition is given in terms of differential calculus. It is a tool for measuring the responsiveness of one variable to changes in another, causative variable. Elasticity has the advantage of being a unitless ratio, independent of the type of quantities being varied. 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.

In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable at point a is defined as

Price elasticity of demand is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price when nothing but the price changes. More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price. The price elasticity of supply is a measure used in economics to show the responsiveness, or elasticity, of the quantity supplied of a good or service to a change in its price.

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 economics, marginal concepts are associated with a specific change in the quantity used of a good or service, as opposed to some notion of the over-all significance of that class of good or service, or of some total quantity thereof.

The theory of the firm consists of a number of economic theories that explain and predict the nature of the firm, company, or corporation, including its existence, behaviour, structure, and relationship to the market.

The distribution of wealth is a comparison of the wealth of various members or groups in a society. It shows one aspect of economic inequality or economic heterogeneity.

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. In statistics, linear regression is a linear approach to modeling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression. For more than one explanatory variable, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x) or log(x). This is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity.

A major study of the price elasticity of supply and the price elasticity of demand for US products was undertaken by Joshua Levy and Trevor Pollock in the late 1960s. 

## Mathematical construct

Elasticity indicates responsiveness. In mathematics, x-elasticity of y measures the responsiveness/fractional change of y with respect to x, i.e. how much y changes fractionally when x changes fractionally.

x-elasticity of y: $\varepsilon ={\frac {\partial \ln {y}}{\partial \ln {x}}}={\frac {\partial y}{\partial x}}{\frac {x}{y}}$ In economics, the common elasticities are price-elasticity of quantity-demanded (elasticity of demand), price-elasticity of quantity-supplied (elasticity of supply) and price-of-a-different-good-elasticity of quantity-demanded (cross-price elasticity). They all have the same form:

P-elasticity of Q: $\varepsilon =-{\frac {\mathrm {d} q(p)}{\mathrm {d} p}}*{\frac {p}{q(p)}}$ if continuous, or $\varepsilon ={\frac {Q_{2}-Q_{1}}{P_{2}-P_{1}}}\times {\frac {P_{1}+P_{2}}{Q_{1}+Q_{2}}}={\frac {\%\ {\mbox{change in quantity Q}}}{\%\ {\mbox{change in price P}}}}$ if discrete.
 $|\varepsilon |>1$ elastic Q changes more than P $|\varepsilon |=1$ unit elastic Q changes like P $|\varepsilon |<1$ inelastic Q changes less than P

Special cases:

perfect P-elasticity of Q, $\varepsilon \rightarrow \infty$ , Q changes while P = constant
perfect P-inelasticity of Q, $\varepsilon =0$ , P changes while Q = constant

Elasticity is commonly used because of its connection to revenue: revenue $PQ$ attains critical value (local max/min) when $\varepsilon =-1$ ${\frac {\partial (PQ)}{\partial P}}=0\Leftrightarrow \varepsilon ={\frac {\partial Q}{\partial P}}{\frac {P}{Q}}=-1\wedge Q\neq 0$ For conventional price-elasticity of quantity-demanded (downwards linear demand curve), revenue reaches global maximum when $\varepsilon =-1$ (unit elastic). Hence, to maximize revenues, firms must:

increase price if price-elasticity of quantity-demanded is inelastic, until reaching unit elastic: $\varepsilon =-1$ ,
decrease price if price-elasticity of quantity-demanded is elastic, until reaching unit elastic: $\varepsilon =-1$ ## Specific elasticities

### Elasticities of supply

Price elasticity of supply

The price elasticity of supply measures how the amount of a good that a supplier wishes 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 horizontal movement along the supply curve relative to the extent of vertical movement. If the price elasticity of supply is zero the supply of a good supplied is "totally inelastic" and the quantity supplied is fixed.

Elasticities of scale

Elasticity of scale or output elasticity measures the percentage change in output induced by a collective percent change in the usages of all 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.    

### Elasticities of demand

Price elasticity of demand

Price elasticity of demand is a measure used to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price. More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price (ceteris paribus, i.e. holding constant all the other determinants of demand, such as income).

Cross-price elasticity of demand

Cross-price elasticity of demand is a measure of the responsiveness of the demand for one product to changes in the price of a different product. It is the ratio of percentage change in the former to the percentage change in the latter. If it is positive, the goods are called substitutes because a rise in the price of the other good causes consumers to substitute away from buying as much of the other good as before and into buying more of this good. If it is negative, the goods are called complements.

## 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.

## Related Research Articles In microeconomics, the law of demand states that, "conditional on all else being equal, as the price of a good increases (↑), quantity demanded decreases (↓); conversely, as the price of a good decreases (↓), quantity demanded increases (↑)". In other words, the law of demand describes an inverse relationship between price and quantity demanded of a good. Alternatively, other things being constant, quantity demanded of a commodity is inversely related to the price of the commodity. For example, a consumer may demand 2 kilograms of apples at $70 per kg; he may, however, demand 1 kg if the price rises to$80 per kg. This has been the general human behaviour on relationship between the price of the commodity and the quantity demanded. The factors held constant refer to other determinants of demand, such as the prices of other goods and the consumer's income. There are, however, some possible exceptions to the law of demand, such as Giffen goods and Veblen goods. In economics, a demand curve is a graph depicting the relationship between the price of a certain commodity and the quantity of that commodity that is demanded at that price. Demand curves may be used to model the price-quantity relationship for an individual consumer, or more commonly for all consumers in a particular market. It is generally assumed that demand curves are downward-sloping, as shown in the adjacent image. This is because of the law of demand: for most goods, the quantity demanded will decrease in response to an increase in price, and will increase in response to a decrease in price. In microeconomics, marginal revenue (MR) is the additional revenue that will be generated by increasing product sales by one unit.

The Ramsey problem, or Ramsey–Boiteux pricing, is a Second best policy problem concerning what price a public monopolist or a firm faced with an irremovable revenue constraint should set, in order to maximize social welfare. A closely related problem arises in relation to optimal taxation of commodities.

In mathematics and economics, the arc elasticity is the elasticity of one variable with respect to another between two given points. It is the ratio of the percentage change of one of the variables between the two points to the percentage change of the other variable. It contrasts with the point elasticity, which is the limit of the arc elasticity as the distance between the two points approaches zero and which hence is defined at a single point rather than for a pair of points.

The Marshall–Lerner condition is the condition that an exchange rate devaluation or depreciation will only cause a balance of trade improvement if the absolute sum of the long-term export and import demand elasticities is greater than unity. If the domestic currency devalues, imports become more expensive and exports become cheaper due to the change in relative prices.

In economics, the total revenue test is a means for determining whether demand is elastic or inelastic. If an increase in price causes an increase in total revenue, then demand can be said to be inelastic, since the increase in price does not have a large impact on quantity demanded. If an increase in price causes a decrease in total revenue, then demand can be said to be elastic, since the increase in price has a large impact on quantity demanded.

In economics, tax incidence or tax burden is the effect of a particular tax on the distribution of economic welfare. Economists distinguish between the entities who ultimately bear the tax burden and those on whom tax is initially imposed. The tax burden measures the true economic weight of the tax, measured by the difference between real incomes or utilities before and after imposing the tax. An individuality on whom the tax is levied does not have to bear the true size of the tax. For the example of this difference, assume a firm, that contains employer and employees. The tax imposed on the employer is divided. The concept of tax incidence was initially brought to economists' attention by the French Physiocrats, in particular François Quesnay, who argued that the incidence of all taxation falls ultimately on landowners and is at the expense of land rent. Tax incidence is said to "fall" upon the group that ultimately bears the burden of, or ultimately suffers a loss from, the tax. The key concept of tax incidence is that the tax incidence or tax burden does not depend on where the revenue is collected, but on the price elasticity of demand and price elasticity of supply. As a general policy matter, the tax incidence should not violate the principles of a desirable tax system, especially fairness and transparency.

Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, stretched or generally deformed in any manner. Elasticity theory primarily develops formalisms for the mechanics of solid bodies and materials. The elastic potential energy equation is used in calculations of positions of mechanical equilibrium. The energy is potential as it will be converted into other forms of energy, such as kinetic energy and sound energy, when the object is allowed to return to its original shape (reformation) by its elasticity.

In economics, demand is the quantity of a good that consumers are willing and able to purchase at various prices during a given period of time. The relationship between price and quantity demanded is also known as the demand curve. Preferences which underlie demand, are influenced by cost, benefit, odds and other variables... In economics, supply is the amount of a resource that firms, producers, labourers, providers of financial assets, or other economic agents are willing and able to provide to the marketplace or directly to another agent in the marketplace. Supply can be in currency, time, raw materials, or any other scarce or valuable object that can be provided to another agent. This is often fairly abstract. For example in the case of time, supply is not transferred to one agent from another, but one agent may offer some other resource in exchange for the first spending time doing something. Supply is often plotted graphically as a supply curve, with the quantity provided plotted horizontally and the price plotted vertically.

In economics, income elasticity of demand measures the responsiveness of the quantity demanded for a good or service to a change in income. It is calculated as the ratio of the percentage change in quantity demanded to the percentage change in income. For example, if in response to a 10% increase in income, the quantity demanded for a good increased by 20%, the income elasticity of demand would be 20%/10% = 2.0.

Total revenue is the total receipts a seller can obtain from selling goods or services to buyers. It can be written as P × Q, which is the price of the goods multiplied by the quantity of the sold goods. Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body.

A markup rule is the pricing practice of a producer with market power, where a firm charges a fixed mark-up over its marginal cost.

In mathematical economics, an isoelastic function, sometimes constant elasticity function, is a function that exhibits a constant elasticity, i.e. has a constant elasticity coefficient. The elasticity is the ratio of the percentage change in the dependent variable to the percentage causative change in the independent variable, in the limit as the changes approach zero in magnitude. A Monopoly price is set by a Monopoly. A monopoly occurs when a firm lacks any viable competition, and is the sole producer of the industry's product. Because a monopoly faces no competition, it has absolute market power, and thereby has the ability to set a monopoly price that will be above the firm's marginal (economic) cost. Since marginal cost is the increment in total required to produce an additional unit of the product, the firm would be able to make a positive economic profit if it produced a greater quantity of the product and sold it at a lower price.

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