Unlike fluids, solids do not respond to outside force by flowing or easily compressing. The term elasticity refers to the manner in which solids respond to stress, or the application of force over a given unit area. An understanding of elasticity—a concept that carries with it a rather extensive vocabulary of key terms—helps to illuminate the properties of objects from steel bars to rubber bands to human bones.
HOW IT WORKS
Characteristics of a Solid
A number of parameters distinguish solids from fluids, a term that in physics includes both gases and liquids. Solids possess a definite volume and a definite shape, whereas gases have neither; liquids have no definite shape.
At the molecular level, particles of solids tend to be precise in their arrangement and close to one another. Liquid molecules are close in proximity (though not as much so as solid molecules), and their arrangement is random, while gas molecules are both random in arrangement and far removed in proximity. Gas molecules are extremely fast-moving, and exert little or no attraction toward one another. Liquid molecules move at moderate speeds and exert a moderate attraction, but solid particles are slow-moving, and have a strong attraction to one another.
One of several factors that distinguishes solids from fluids is their relative response to pressure. Gases tend to be highly compressible, meaning that they respond well to pressure. Liquids tend to be noncompressible, yet because of their fluid characteristics, they experience external pressure uniformly. If one applies pressure to a quantity of water in a closed container, the pressure is equal everywhere in the water. By contrast, if one places a champagne glass upright in a vise and applies pressure until it breaks, chances are that the stem or the base of the glass will be unaffected, because the pressure is not distributed equally throughout the glass.
If the surface of a solid is disturbed, it will resist, and if the force of the disturbance is sufficiently strong, it will deform—for instance, when a steel plate begins to bend under pressure. This deformation will be permanent if the force is powerful enough, as in the above example of the glass in a vise. By contrast, when the surface of a fluid is disturbed, it tends to flow.
Types of Stress
Deformation occurs as a result of stress, whether that stress be in the form of tension, compression, or shear. Tension occurs when equal and opposite forces are exerted along the ends of an object. These operate on the same line of action, but away from each other, thus stretching the object. A perfect example of an object under tension is a rope in the middle of a tug-of-war competition. The adjectival form of "tension" is "tensile": hence the term "tensile stress," which will be discussed later.
Earlier, stress was defined as the application of force over a given unit area, and in fact, the formula for stress can be written as F /A, where F is force and A area. This is also the formula for pressure, though in order for an object to be under pressure, the force must be applied in a direction perpendicular to—and in the same direction as—its surface. The one form of stress that clearly matches these parameters is compression, produced by the action of equal and opposite forces, whose effect is to reduce the length of a material. Thus compression (for example, crushing an aluminum can in one's hand) is both a form of stress and a form of pressure.
Note that compression was defined as reducing length, yet the example given involved a reduction in what most people would call the "width" or diameter of the aluminum can. In fact, width and height are the same as length, for the purposes of most discussions in physics. Length is, along with time, mass, and electric current, one of the fundamental units of measure used to express virtually all other physical quantities. Width and height are simply length expressed in terms of other planes, and within the subject of elasticity, it is not important to distinguish between these varieties of length. (By contrast, when discussing gravitational attraction—which is always vertical—it is obviously necessary to distinguish between "vertical length," or height, and horizontal length.)
The third variety of stress is shear, which occurs when a solid is subjected to equal and opposite forces that do not act along the same line, and which are parallel to the surface area of the object. If a thick hardbound book is lying flat, and a person places a finger on the spine and pushes the front cover away from the spine so that the covers and pages no longer constitute parallel planes, this is an example of shear. Stress resulting from shear is called shearing stress.
Hooke's Law and Elastic Limit
To sum up the three varieties of stress, tension stretches an object, compression shrinks it, and shear twists it. In each case, the object is deformed to some degree. This deformation is expressed in terms of strain, or the ratio between change in dimension and the original dimensions of the object. The formula for strain is δL /L o, where δL is the change in length (δ, the Greek letter delta, means "change" in scientific notation) and L o the original length.
Hooke's law, formulated by English physicist Robert Hooke (1635-1703), relates strain to stress. Hooke's law can be stated in simple terms as "the strain is proportional to the stress," and can also be expressed in a formula, F = ks, where F is the applied force, s, the resulting change in dimension, and k, a constant whose value is related to the nature and size of the object under stress. The harder the material, the higher the value of k ; furthermore, the value of k is directly proportional to the object's cross-sectional area or thickness.
The elastic limit of a given solid is the maximum stress to which it can be subjected without experiencing permanent deformation. Elastic limit will be discussed in the context of several examples below; for now, it is important merely to know that Hooke's law is applicable only as long as the material in question has not reached its elastic limit. The same is true for any modulus of elasticity, or the ratio between a particular type of applied stress and the strain that results. (The term "modulus," whose plural is "modu li," is Latin for "small measure.")
Moduli of Elasticity
In cases of tension or compression, the modulus of elasticity is Young's modulus. Named after English physicist Thomas Young (1773-1829), Young's modulus is simply the ratio between F /A and δL /L o—in other words, stress divided by strain. There are also modu li describing the behavior of objects exposed to shearing stress (shear modulus), and of objects exposed to compressive stress from all sides (bulk modulus).
Shear modulus is the relationship of shearing stress to shearing strain. This can be expressed as the ratio between F /A and φ. The latter symbol, the Greek letter phi, stands for the angle of shear—that is, the angle of deformation along the sides of an object exposed to shearing stress. The greater the amount of surface area A, the less that surface will be displaced by the force F. On the other hand, the greater the amount of force in proportion to A, the greater the value of φ, which measures the strain of an object exposed to shearing stress. (The value of φ, however, will usually be well below 90°, and certainly cannot exceed that magnitude.)
With tensile and compressive stress, A is a surface perpendicular to the direction of applied force, but with shearing stress, A is parallel to F. Consider again the illustration used above, of a thick hardbound book lying flat. As noted, when one pushes the front cover from the side so that the covers and pages no longer constitute parallel planes, this is an example of shear. If one pulled the spine and the long end of the pages away from one another, that would be tensile stress, whereas if one pushed in on the sides of the pages and spine, that would be compressive stress. Shearing stress, by contrast, would stress only the front cover, which is analogous to A for any object under shearing stress.
The third type of elastic modulus is bulk modulus, which occurs when an object is subjected to compression from all sides—that is, volume stress. Bulk modulus is the relationship of volume stress to volume strain, expressed as the ratio between F /A and δV /V o, where δV is the change in volume and V o is the original volume.
Elastic and Plastic Deformation
As noted earlier, the elastic limit is the maximum stress to which a given solid can be subjected without experiencing permanent deformation, referred to as plastic deformation. Plastic deformation describes a permanent change in shape or size as a result of stress; by contrast, elastic deformation is only a temporary change in dimension.
A classic example of elastic deformation, and indeed, of highly elastic behavior, is a rubber band: it can be deformed to a length many times its original size, but upon release, it returns to its original shape. Examples of plastic deformation, on the other hand, include the bending of a steel rod under tension or the breaking of a glass under compression. Note that in the case of the steel rod, the object is deformed without rupturing—that is, without breaking or reducing to pieces. The breaking of the glass, however, is obviously an instance of rupturing.
Metals and Elasticity
Metals, in fact, exhibit a number of interesting characteristics with regard to elasticity. With the notable exception of cast iron, metals tend to possess a high degree of ductility, or the ability to be deformed beyond their elastic limits without experiencing rupture. Up to a certain point, the ratio of tension to elongation for metals is high: in other words, a high amount of tension produces only a small amount of elongation. Beyond the elastic limit, however, the ratio is much lower: that is, a relatively small amount of tension produces a high degree of elongation.
Because of their ductility, metals are highly malleable, and, therefore, capable of experiencing mechanical deformation through metallurgical processes, such as forging, rolling, and extrusion. Cold extrusion involves the application of high pressure—that is, a high bulk modulus—to a metal without heating it, and is used on materials such as tin, zinc, and copper to change their shape. Hot extrusion, on the other hand, involves heating a metal to a point of extremely high malleability, and then reshaping it. Metals may also be melted for the purposes of casting, or pouring the molten material into a mold.
The tension that a material can with stand is called its ultimate strength, and due to their ductile properties, most metals possess a high value of ultimate strength. It is possible, however, for a metal to break down due to repeated cycles of stress that are well below the level necessary to rupture it. This occurs, for instance, in metal machines such as automobile engines that experience a high frequency of stress cycles during operation.
The high ultimate strength of metals, both in tension and compression, makes them useful in a number of structural capacities. Steel has an ultimate compressive strength 25 times as great as concrete, and an ultimate tensile strength 250 times as great. For this reason, when concrete is poured for building bridges or other large structures, steel rods are inserted in the concrete. Called "rebar" (for "reinforced bars"), the steel rods have ridges along them in order to bond more firmly with the concrete as it dries. As a result, reinforced concrete has a much greater ability than plain concrete to with stand tension and compression.
Steel Bars and Rubber Bands Under Stress
Metals are crystalline materials, meaning that they are composed of solids called crystals. Particles of crystals are highly ordered, with a definite geometric arrangement repeated in all directions, rather like a honeycomb. (It should be noted, however, that the crystals are not necessarily as uniform in size as the "cells" of the honeycomb.) The atoms of a crystal are arranged in orderly rows, bound to one another by strongly attractive forces that act like microscopic springs.
Just as a spring tends to return to its original length, the highly attractive atoms in a steel bar, when it is stretched, tend to restore it to its original dimensions. Likewise, it takes a great deal of force to pull apart the atoms. When the metal is subjected to plastic deformation, the atoms move to new positions and form new bonds. The atoms are incapable of forming bonds; however, when the metal has been subjected to stress exceeding its ultimate strength, at that point, the metal breaks.
The crystalline structure of metal influences its behavior under high temperatures. Heat causes atoms to vibrate, and in the case of metals, this means that the "springs" are stretching and compressing. As temperature increases, so do the vibrations, thus increasing the average distance between atoms. For this reason, under extremely high temperature, the elastic modulus of the metal decreases, and the metal becomes less resistant to stress.
POLYMERS AND ELASTOMERS.
Rubber is so elastic in behavior that in everyday life, the term "elastic" is most often used for objects containing rubber: the waistband on a pair of underwear, for instance. The long, thin molecules of rubber, which are arranged side-by-side, are called "polymers," and the super-elastic polymers in rubber are called "elastomers." The chemical bonds between the atoms in a polymer are flexible, and tend to rotate, producing kinks along the length of the molecule.
When a piece of rubber is subjected to tension, as, for instance, if one pulls a rubber band by the ends, the kinks and loops in the elastomers straighten. Once the stress is released, however, the elastomers immediately return to their original shape. The more "kinky" the polymers, the higher the elastic modulus, and hence, the more capable the item is of stretching and rebounding.
It is interesting to note that steel and rubber, materials that are obviously quite different, are both useful in part for the same reason: their high elastic modulus when subjected to tension, and their strength under stress. But a rubber band exhibits behaviors under high temperatures that are quite different from that of a metal: when heated, rubber contracts. It does so quite suddenly, in fact, suggesting that the added energy of the heat allows the bonds in the elastomers to begin rotating again, thus restoring the kinked shape of the molecules.
The tensile strength in bone fibers comes from the protein collagen, while the compressive strength is largely due to the presence of inorganic (non-living) salt crystals. It may be hard to believe, but bone actually has an ultimate strength—both in tension and compression—greater than that of concrete!
The ultimate strength of most materials is rendered in factors of 108 N/m2—that is, 100,000,000 newtons (the metric unit of force) per square meter. For concrete under tensile stress, the ultimate strength is 0.02, whereas for bone, it is 1.3. Under compressive stress, the values are 0.2 and 1.7, respectively. In fact, the ultimate tensile strength of bone is close to that of cast iron (1.7), though the ultimate compressive strength of cast iron (5.5) is much higher than for bone.
Even with these figures, it may be hard to understand how bone can be stronger than concrete, but that is largely because the volume of concrete used in most situations is much greater than the volume of any bone in the body of a human being. By way of explanation, consider a piece of concrete no bigger than a typical bone: under relatively small amounts of stress, it would crumble.
WHERE TO LEARN MORE
Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.
"Dictionary of Metallurgy" Steelmill.com: The Polish Steel Industry Directory (Web site). <http://www.steelmill.com/DICTIONARY/Diction-ary.htm> (April 9, 2001).
"Engineering Processes." eFunda.com (Web site). <http://www.efunda.com/processes/processes_home/process.cfm> (April 9, 2001).
Gibson, Gary. Making Shapes. Illustrated by Tony Kenyon. Brookfield, CT: Copper Beech Books, 1996.
"Glossary of Materials Testing Terms" (Web site). <http://www.instron.com/apps/glossary> (April 9, 2001).
Goodwin, Peter H. Engineering Projects for Young Scientists. New York: Franklin Watts, 1987.
Johnston, Tom. The Forces with You! Illustrated by Sarah Pooley. Milwaukee, WI: Gareth Stevens Publishing, 1988.
ANGLE OF SHEAR:
The angle of deformation on the sides of an object exposed to shearing stress. Its symbol is φ (the Greek letter phi), and its value will usually be well below 90°.
The modulus of elasticity for a material subjected to compression on all surfaces—that is, volume stress. Bulk modulus is the relationship of volume stress to volume strain, expressed as the ratio between F /A and dV/Vo, wheredV is the change in volume and Vo is the original volume.
A form of stress produced by the action of equal and opposite forces, whose effect is to reduce the length of a material. Compression is a form of pressure. When compressive stress is applied to all surfaces of a material, this is known as volume stress.
A property whereby a material is capable of being deformed far beyond its elastic limit without experiencing rupture—that is, without breaking. Most metals other than cast iron are highly ductile.
A temporary change in shape or size experienced by a solid subjected to stress. Elastic deformation is thus less severe than plastic deformation.
The maximum stress to which a given solid can be subjected without experiencing plastic deformation—that is, without being permanently deformed.
The response of solids to stress.
A principle of elasticity formulated by English physicist Robert Hooke (1635-1703), who discovered that strain is proportional to stress. Hooke's lawcan be written as a formula, F = ks, where F is the applied force, s the resulting change in dimension, and k a constant whose value is related to the nature and size of the object being subjected to stress. Hooke's law applies only when the elastic limit has not been exceeded.
In discussions of elasticity, "length" refers to an object's dimensions on any given plane, thus, it can be used not only to refer to what is called length in everyday language, but also to width or height.
MODULUS OF ELASTICITY:
The ratio between a type of applied stress (that is, tension, compression, and shear) and the strain that results in the object to which stress has been applied. Elastic modu li—including Young's modulus, shearing modulus, and bulk modulus—are applicable only as long as the object's elastic limit has not been reached.
A permanent change in shape or size experienced by a solid subjected to stress. Plastic deformation is thus more severe than elasticdeformation.
The ratio of force to surface area, when force is applied in a direction perpendicular to, and in the same direction as, that surface.
A form of stress resulting from equal and opposite forces that do not act along the same line. If a thick hard-bound book is lying flat, and one pushes the front cover from the side so that the covers and pages no longer constitute parallel planes, this is an example of shear.
The modulus of elasticity for an object exposed to shearing stress. It is expressed as the ratio between F /A and φ, where φ (the Greek letter phi) stands for the angle of shear.
The ratio between the change in dimension experienced by an object that has been subjected to stress, and the original dimensions of the object. The formula for strain is dL /L o, where dL is the change in length and L o the original length. Hooke's law, as well as the various modu li of elasticity, relates strain to stress.
In general terms, stress is any attempt to deform a solid. Types of stress include tension, compression, and shear. More specifically, stress is the ratio of force to unit area, F /A, where F is force and A area. Thus, it is similar to pressure, and indeed, compression is a form of pressure.
A form of stress produced by a force which acts to stretch a material. The adjectival form of "tension" is "tensile": hence the terms "tensile stress" and "tensile strain."
The tension that a material can with stand without rupturing. Due to their high levels of ductility, most metals have a high value of ultimate strength.
The stress that occurs in a material when it is subjected to compression from all sides. The modus of elasticity for volume stress is the bulk modulus.
A modulus of elasticity describing the relationship between stress to strain for objects under either tension or compression. Named after English physicist Thomas Young (1773-1829), Young's modulus is simply the ratio between F /A and δL /L o—in other words, stress divided by strain.
"Elasticity." Science of Everyday Things. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/science/news-wires-white-papers-and-books/elasticity
"Elasticity." Science of Everyday Things. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/science/news-wires-white-papers-and-books/elasticity
The quantity of a commodity that people want to purchase (demand) per unit of time or want to sell (supply) per unit of time depends in part upon the price of the commodity. Economists have sought ways to describe and measure the relationship between changes in the rate of quantity demanded or supplied and changes in price.
Alfred Marshall (1885, p. 260) first used the term elasticity to refer to the percentage change in quantity demanded (Δq/q) divided by the percentage change in price (Δp/p), and indicated that elasticity at C, on the demand curve ABCDE (see Figure 1), is CG divided by CF. He elaborated these ideas in his Principles (1890, book 3, chapter 3) and the concept has since been adopted by the profession.
Prior to Marshall, there had been no generally accepted terminology or measure of the relationship between price and quantity, although concepts similar to elasticity had been suggested. For example, Mill (1848, book 3, chapter 2, sec. 4) compared percentage changes in price and in quantity, but he did not divide the two to get what we now call elasticity. Cournot (1838, chapter 4, sec. 24) showed the relationship between elasticity and total revenue (see below) by comparing the ratio of the change in quantity demanded to the change in price (Δq/Δp) and the ratio of quantity demanded to price (q/p), but he did not suggest dividing the two ratios to measure elasticity.
Elasticity of demand. Elasticity of demand is defined as the percentage change in the quantity demanded, divided by the percentage change in price. It is derived from the demand curve, which shows the absolute quantity demanded as a function of price. Since the rate of demand rises if the price falls, and vice versa, the elasticity of demand is actually negative, except that it is common to ignore the algebraic sign and use the absolute value. By definition, demand is inelastic if the elasticity is less than one and is elastic if the elasticity is greater than one. An elasticity of one is called unit, or unitary, elasticity.
Elastic demand. If demand is elastic, a given percentage increase in price causes a larger percentage decline in the quantity demanded, or a given percentage reduction in price causes a larger percentage increase in quantity demanded. In the limiting case of perfectly (infinitely) elastic demand, which is illustrated by the horizontal line segment DE in Figure 1 (if it is visualized as extended horizontally so as to constitute the entire demand curve), any increase in price, no matter how small, will cause the quantity demanded to decline to zero; and any reduction in price, again no matter how small, will cause the quantity demanded to increase without limit.
Demand is usually elastic for commodities for which there are good substitutes, so that the de-
mand facing a firm is more elastic than the demand facing an industry, since a buyer may substitute the products of other firms in the industry. It is part of the definition of perfect competition that there must be so many firms in the industry producing such excellent substitutes that the demand facing the firm is perfectly (infinitely) elastic, even though the demand facing the industry may be inelastic. [SeeCompetition.]
Inelastic demand. Conversely, if demand is inelastic the percentage change in quantity is smaller than the percentage change in price. An increase in price therefore causes a proportionately smaller decline in the quantity demanded, and a reduction in price causes a proportionately smaller increase in the quantity demanded. In the limiting case of zero elasticity (perfectly inelastic demand), which is illustrated by the vertical line segment AB in Figure 1 (if it is visualized as extended vertically so as to constitute the entire demand curve), neither an increase nor a reduction in price has any effect upon the quantity demanded.
Products that have no good substitutes, like tobacco, or that require a very small proportion of the consumer’s income, like salt, frequently have inelastic demands.
Relation to total revenue. Elasticity of demand also shows how the total dollar value of sales of the commodity responds to a change in its price. For example, if the price of a commodity with inelastic demand is increased, the total revenue of the seller will increase even though the quantity demanded declines somewhat, since the increase in price more than offsets the reduction in volume. This creates a presumption that a seller faced with an inelastic demand will raise his price and will continue to do so until demand becomes elastic. Similar reasoning shows that a reduction in price reduces total revenue if demand is inelastic but increases total revenue if demand is elastic, since the increase in volume more than offsets the reduction in price. Finally, an increase in price reduces total revenue if demand is elastic and does not change total revenue if demand has unit elasticity.
Relation to marginal revenue. There is also a logical relationship between the elasticity of demand and marginal revenue. Marginal revenue is defined as the change in the total revenue per unit change in quantity sold, if the relationship between the price and the quantity sold is given by the demand curve. Positive marginal revenue therefore implies that total revenue increases if the quantity sold increases, or declines if the quantity sold declines. Since the relation between price and quantity is given by the demand curve, an increase in sales volume must be accompanied by a decline in price. Thus, a positive marginal revenue means that total revenue increases as volume increases and price declines. Since a decline in price increases total revenue only if demand is elastic, positive marginal revenue must be associated with elastic demand. Furthermore, since profit maximization requires that marginal cost equal marginal revenue, and marginal cost is always positive, it follows that the marginal revenue of the profit-maximizing firm must be positive and the demand elastic.
Similar reasoning shows that negative marginal revenue is associated with inelastic demand: if demand is inelastic, an increase in price reduces volume and increases total revenue, so that marginal revenue is negative. The intermediate case of unit elasticity of demand implies zero marginal revenue, since any change in price is just balanced by a change in volume, and total revenue is the same at any price or at any volume.
Relation to price. Elasticity of demand usually changes with the price of the product. Thus, it is incorrect to say that the demand for cigarettes is inelastic or that the demand for butter is elastic. The correct statement is that these demands are elastic or inelastic at the prices that are currently charged. At higher prices the demand for cigarettes might well become elastic, while the demand for butter might well become inelastic if its price fell far enough below the price of oleomargarine.
In the particular case of the straight-line demand curve, which is so frequently used for illustrative purposes in economics, the demand is elastic and marginal revenue is positive throughout the upper half of the possible price range, while demand is inelastic and marginal revenue negative throughout the lower half of the price range.
Elasticity of supply. Defined as the percentage change in the quantity offered for sale divided by the percentage change in price, elasticity of supply is derived from the supply curve, which shows the absolute amount producers wish to sell as a function of price. Since sellers wish to sell more at a higher price, the elasticity of supply is positive. If the supply curve is inelastic, a relatively larger increase in price is required to produce a given (percentage) increase in quantity supplied. If the supply curve is elastic, the sellers are more responsive to a change in price, and the increase in the quantity supplied is larger, on a percentage basis, than the increase in price.
Other elasticities. Marshall originally introduced the concept of elasticity in connection with demand and supply curves, but economists soon started to use the ratio of the percentage changes to describe the relationship between any two variables, provided only that the variables are functionally related. By 1913, Johnson (p. 503) was able to write, “This form of expression [the ratio of percentage changes] corresponds to the general notion of elasticity.”
The most important elasticities, other than demand and supply, are income elasticity of demand and cross-elasticity of demand.
Income elasticity of demand. The demand for a commodity depends upon its price, consumers’ incomes, the prices of other commodities on which consumers might spend their incomes, and perhaps on other variables. Quantity demanded is therefore functionally related to price, if incomes and other prices do not change; and the (price) elasticity of demand is a property of this function. Quantity demanded is also functionally related to income, if the price of the commodity and other prices do not change. The income elasticity of demand is a property of this function and is defined as the percentage change in the quantity demanded divided by the percentage change in income.
If income elasticity of demand is negative, consumers purchase less of the commodity as their incomes rise, and the commodity is, by definition, an inferior good. If income elasticity is positive, but less than one, consumers buy more of the product as their incomes rise (if prices do not change), but spend a smaller proportion of their incomes on the commodity. For commodities with income elasticities greater than one, consumers spend a higher proportion of a higher income on the commodity, and the quantity demanded grows at a faster (percentage) rate than income grows.
Cross-elasticity of demand. Quantity demanded is also functionally related to the prices of other commodities, if the commodity’s own price and consumers’ incomes do not change. Cross-elasticity of demand is a property of this function and is defined as the percentage change in the quantity demand of commodity A divided by the percentage change in the price of commodity B.
The commodities A and B are, by definition, sub stitutes if the cross-elasticity is positive, since an increase in the price of B then causes a reduction in the demand for B and an increase in the demand for A. Conversely, if cross-elasticity is negative, the commodities are complements, since an increase in the price of B causes consumers to demand less of both commodities.
Mathematical definitions. Point elasticity. If the variables q, p, and y are related by the function q = f (p, y ), the p elasticity of q, or the elasticity of q with respect to p, is customarily defined as ηqp = (∂q/∂p)(P/q), no matter what variables the letters represent.
If, as is customary, q represents the quantity demanded, p the price, and y the consumers’ incomes, then ηqp is the (price) elasticity of demand and is negative. Marginal revenue, M, is defined as M = ∂(pq)/∂q, and it can be shown that M = p( l + l/ηqp).
Arc elasticity. Sometimes the functional relationship between p and q is unknown, but two points, B and D, with coordinates (q1,p1) and (q2, p2) as in Figure 1, are known; and the arc elasticity between the two points is desired. The usual practice is to assume that the demand curve passing through these points is a straight line, HBIDJ, and to calculate the point elasticity at I, the point midway between B and D. The formula for the arc elasticity, so defined, is
While this procedure is arbitrary, it has two advantages over calculating the elasticity at either B or D. First, the elasticity has the same value whether the change is from B to D or from D to B. More important, however, is the fact that the relationships between elasticity, total revenue, and marginal revenue, which were discussed above, are valid if elasticity is calculated at I, but not if it is calculated at either B or D.
Cournot, Antoine Augustin (1838) 1960 Researches Into the Mathematical Principles of the Theory of Wealth. New York: Kelley. → First published in French.
Johnson, W. E. 1913 The Pure Theory of Utility Curves.Economic Journal 23:483–513.
Marshall, Alfred 1885 On the Graphic Method of Statistics. Pages 251–260 in Royal Statistical Society, London, Jubilee Volume of the Statistical Society. London: Stanford.
Marshall, Alfred (1890) 1961 Principles of Economics. 9th ed., 2 vols. New York and London: Macmillan. → A variorum edition. The eighth edition is more convenient for normal use.
Mill, John Stuart (1848) 1961 Principles of Political Economy, With Some of Their Applications to Social Philosophy. 7th ed. Edited by W. J. Ashley. New York: Kelley.
"Elasticity." International Encyclopedia of the Social Sciences. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/social-sciences/applied-and-social-sciences-magazines/elasticity
"Elasticity." International Encyclopedia of the Social Sciences. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/social-sciences/applied-and-social-sciences-magazines/elasticity
Elasticity is a measure of the responsiveness of one economic variable to another. For example, advertising elasticity is the relationship between a change in a firm's advertising budget and the resulting change in product sales. Economists are often interested in the price elasticity of demand, which measures the response of the quantity of an item purchased to a change in the item's price. A good or service is considered to be highly elastic if a slight change in price leads to a sharp change in demand for the product or service. Products and services that are highly elastic are usually more discretionary in nature—readily available in the market and something that a consumer may not necessarily need in his or her daily life. On the other hand, an inelastic good or service is one for which changes in price result in only modest changes to demand. These goods and services tend to be necessities.
Elasticity is usually expressed as a positive number when the sign is already clear from context. Elasticity measures are reported as a proportional or percent change in the variable being studied. The general formula for elasticity, represented by the letter "E" in the equation below, is:
E = percent change in x / percent change in y.
Elasticity can be zero, one, greater than one, less than one, or infinite. When elasticity is equal to one there is unit elasticity. This means the proportional change in one variable is equal to the proportional change in another variable, or in other words, the two variables are directly related and move together. When elasticity is greater than one, the proportional change in x is greater than the proportional change in y and the situation is said to be elastic.
Inelastic situations result when the proportional change in x is less than the proportional change in y. Perfectly inelastic situations result when any change in y will have an infinite effect on x. Finally, perfectly elastic situations result when any change in y will result in no change in x. A special case known as unitary elasticity of demand occurs if total revenue stays the same when prices change.
ELASTICITY FOR MANAGERIAL DECISION MAKING
Economists compute several different elasticity measures, including the price elasticity of demand, the price elasticity of supply, and the income elasticity of demand. Elasticity is typically defined in terms of changes in total revenue since that is of primary importance to managers, CEOs, and marketers. For managers, a key point in the discussions of demand is what happens when they raise prices for their products and services. It is important to know the extent to which a percentage increase in unit price will affect the demand for a product. With elastic demand, total revenue will decrease if the price is raised. With inelastic demand, however, total revenue will increase if the price is raised.
The possibility of raising prices and increasing dollar sales (total revenue) at the same time is very attractive to managers. This occurs only if the demand curve is inelastic. Here total revenue will increase if the price is raised, but total costs probably will not increase and, in fact, could go down. Since profit is equal to total revenue minus total costs, profit will increase as price is increased when demand for a product is inelastic. It is important to note that an entire demand curve is neither elastic nor inelastic; it only has the particular condition for a change in total revenue between two points on the curve (and not along the whole curve).
Demand elasticity is affected by three things: 1) availability of substitutes; 2) the urgency of need, and 3) the importance of the item in the customer's budget. Substitutes are products that offer the buyer a choice. For example, many consumers see corn chips as a good or homogeneous substitute for potato chips, or see sliced ham as a substitute for sliced turkey. The more substitutes available, the greater will be the elasticity of demand. If consumers see products as extremely different or heterogeneous, however, then a particular need cannot easily be satisfied by substitutes. In contrast to a product with many substitutes, a product with few or no substitutes—like gasoline—will have an inelastic demand curve. Similarly, demand for products that are urgently needed or are very important to a person's budget will tend to be inelastic. It is important for managers to understand the price elasticity of their products and services in order to set prices appropriately to maximize firm profits and revenues.
see also Financial Analysis; Pricing
Haines, Leslie. "Elasticity is Back" Oil and Gas Investor. November 2005.
Hodrick, Laurie Simon. "Does Price Elasticity Affect Corporate Financial Decisions?" Journal of Financial Economics. May 1999.
Montgomery, Alan L., and Peter E. Rossi. "Estimating Price Elasticity with Theory-Based Priors." Journal of Marketing Research. November 1999.
Perreault, William E. Jr., and E. Jerome McCarthy. Basic Marketing: A Global-Managerial Approach. McGraw-Hill, 1997.
Hillstrom, Northern Lights
updated by Magee, ECDI
"Elasticity." Encyclopedia of Small Business. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/entrepreneurs/encyclopedias-almanacs-transcripts-and-maps/elasticity
"Elasticity." Encyclopedia of Small Business. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/entrepreneurs/encyclopedias-almanacs-transcripts-and-maps/elasticity
In economics, elasticity measures a response of one variable to changes in the other variable. The concept of elasticity can be applied to any two variables, but the most commonly used are price elasticity of demand and elasticity of substitution between factors of production, consumer goods, or bundles of consumption in different periods of time (elasticity of intertemporal substitution).
Elasticity measures the percentage change in variable Y in response to a 1 percent change in variable X. Formally, the elasticity of Y with respect to X is defined as
or, for continuous changes,
The concept of the price elasticity of demand, which was introduced in 1890 by Alfred Marshall, measures the percentage change in the quantity of a good demanded when the price of this good changes by 1 percent. The demand is elastic if price changes lead to large changes in quantity demanded. The demand is inelastic if the quantity demanded does not respond much to changes in price. If demand is perfectly elastic, even a small increase in price will send the quantity demanded to zero. If demand is perfectly inelastic, the quantity demanded will be constant regardless of the price.
Price elasticity of demand for goods depends on the characteristics of the goods—that is, whether or not they are necessities and whether or not there are close substitutes for these goods. Price elasticity of demand also varies with the time horizon in consideration. For example, demand for gasoline is very inelastic in the short run, because people have to fill up their gas tanks, but in the long run, if prices remain high, people will switch to more gasoline-efficient cars and will demand less gasoline. By the estimates of the Mackinac Center for Public Policy, a 10 percent increase in the price of gasoline will lower the demand for it by 2 percent in the short run (elasticity of -0.2), and by 7 percent in the long run (elasticity of -0.7).
Price elasticity of demand also depends on how narrowly a good is defined. For example, demand for milk in general is quite inelastic, because there are no close substitutes. However, demand for milk of a specific brand is very elastic because there are many close substitutes (i.e., other brands of milk).
The concept of the elasticity of substitution was introduced independently by John Hicks and Joan Robinson in 1932 and 1933, respectively. It is used to measure how easily factors of production can be substituted for one another. For example, how much will the ratio of capital input to labor input in production increase if the ratio of capital cost to labor cost falls by 1 percent? Samuel de Abreu Pessoa, Silvia Matos Pessoa, and Rafael Rob (2005) estimate this elasticity to be 0.7, on average.
The concept of the elasticity of substitution can also be applied to consumption of goods, to measure how relative consumption of two goods is affected by their relative price. Most frequently, the concept of the elasticity of substitution in consumption is used to construct the baskets of consumer goods in economic models and to measure the substitutability of consumption across different time periods—“elasticity of intertemporal substitution.” This measures how the ratio of future to current consumption reacts to the change in relative price of consumption tomorrow and consumption today, commonly measured by the interest rate. Elasticity of intertemporal substitution is estimated to be close to zero, indicating that consumers are not sensitive in their intertemporal consumption decisions to the changes in the interest rate.
SEE ALSO Demand; Hicks, John R.; Marshall, Alfred; Production Function; Robinson, Joan; Substitutability; Time
Anderson, Patrick L., Richard D. McLellan, Joseph P. Overton, and Gary L. Wolfram. 1997. Price Elasticity of Demand. Mackinac Center for Public Policy. http://www.mackinac.org/article.aspx?ID=1247.
Hicks, John. 1932. The Theory of Wages. London: Macmillan.
Marshall, Alfred.  1920. Principles of Economics. 8th ed. London: Macmillan.
Pessoa, Samuel de Abreu, Silvia Matos Pessoa, and Rafael Rob. 2005. Elasticity of Substitution between Capital and Labor and Its Applications to Growth and Development. PIER Working Paper Archive 05–012, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
Robinson, Joan. 1933. The Economics of Imperfect Competition. London: Macmillan.
Yogo, Motohiro. 2004. Estimating the Elasticity of Intertemporal Substitution When Instruments Are Weak. Review of Economics and Statistics 86(3): 797–810.
"Elasticity." International Encyclopedia of the Social Sciences. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/social-sciences/applied-and-social-sciences-magazines/elasticity-0
"Elasticity." International Encyclopedia of the Social Sciences. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/social-sciences/applied-and-social-sciences-magazines/elasticity-0
"The Elasticity of the Psychoanalytic Technique" is the title of a paper that Sándor Ferenczi gave to the Budapest Psychoanalytic Society, and which was first published in 1928. In essence he described the procedure he had introduced in his paper on the "contra-indications of the active technique" (1926), in which he recommended using relaxation to reduce tension in certain difficult cases. In two other articles from the same period ("Family Adaptation to the Child" and "The Problem of the End of Analysis") he dealt with difficulties in the educational environment. The question became one of how far the idea of elasticity could be taken. In 1967, Michael Balint would write on Ferenczi's problem, "His earlier experiences had familiarized him with two models: one was the classic technique with its objective and benevolent passivity, and apparently imperturbable and unlimited patience; the other was the active technique with its well-directed interventions founded on attentive observation and empathy."
In the 1928 paper, Ferenczi developed the technical importance of tact in deciding on the right moment to communicate to the patient any conjectures the analyst may have made, "based essentially on the dissection of our own Self." He stressed the notion of modesty, which should be "the expression of the acceptance of the limits to our knowledge," and to this end he preferred from the beginning of treatment to adopt a rather pessimistic attitude, in order to avoid creating enthusiastic confidence in the future patient, a confidence that often camouflaged "a healthy dose of distrust." Nothing could be more harmful, he continued, "than the attitude of a schoolmaster or an authoritarian doctor." He thus spoke of Einfühlung (feeling-with, empathy) as of a rule, from which he deduced the necessity, for the analyst, of developing "a rigorous control of his own narcissism and intense vigilance with regard to his own affective reactions." Analysts would have to "guess when the patient's esthetic sentiments have been offended by our own attitude" and, supporting this displeasure, behave like those little "culbutos" (small figures with lead ballast in their base that always return to a vertical position). Ferenczi proposed "a perpetual oscillation between feeling-with, self-observation and judgment activity."
He concluded this reflection on the counter-transference with a "metapsychology of the technique," denouncing the "fanaticism of interpretation as an infantile disease of analysis" because, in order for patients to become free of all emotional binds, they must "abandon, at least provisionally, all sorts of superegos, including that of the analyst." This position borders on "a demand for elasticity in the analysts themselves," a "metapsychology of the analysts." This then makes it absolutely essential to comply with the second rule of psychoanalysis, already problematic at the time, that analysts must themselves be analyzed.
See also: Active technique; Ferenczi, Sándor; Tact; Technique with adults, psychoanalytic.
Balint, Michael. (1967). Introduction. In Sándor Ferenczi, Oeuvres complètes (Vol. 4). Paris: Payot, 1982.
Ferenczi, Sándor. (1926). Contre-indications de la technique active. In his Oeuvres complètes (Vol. 3, pp. 389-428). Paris, Payot.
——. (1926). Le problème de l'affirmation du déplaisir. In his Oeuvres complètes (Vol. 3, pp. 389-428). Paris: Payot
——. (1928). The elasticity of psycho-analytic technique. In M. S. Bergmann and F. R. Hartman (Eds.), The evolution of psychoanalytic technique. New York: Basic Books.
"Elasticity." International Dictionary of Psychoanalysis. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/psychology/dictionaries-thesauruses-pictures-and-press-releases/elasticity
"Elasticity." International Dictionary of Psychoanalysis. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/psychology/dictionaries-thesauruses-pictures-and-press-releases/elasticity
Just about every solid material possesses some degree of elasticity, and so do most liquids. Some common highly elastic products are rubber bands, kitchen spatulas, and bicycle tires. Even buildings and bridges have some degree of elasticity (or give) so they can adjust to small shifts in Earth's surface.
Elasticity is a chemical property that allows a solid body to return to its original shape after an outside force is removed. The key to determining whether a substance is elastic is to apply a force to it. With sufficient force, the substance should change its size, shape, or volume. If, when the force is removed, the sample returns to its original state, then it is elastic. If the substance returns only partially (or not at all) to its original state, it is called inelastic.
If too much force is applied, the material is in danger of reaching its elastic limit. The elastic limit is the point at which the material is bent beyond its ability to return to its original shape. Once the elastic limit is passed, the material will experience permanent reshaping, called plastic deformation, and will no longer act as an elastic substance.
This stretching/recoiling activity is easily seen by hanging a weight from a spring: if the weight is within the spring's elastic capacity, the spring will bounce back (in an elastic manner). However, if the weight is too heavy for the spring, the weight will pull the spring straight, making it inelastic. (Think of a Slinky™, the coiled wire toy that travels down stairs and then regains its original shape. If too much force is applied to it, the Slinky™ becomes bent out of shape or inelastic.)
Elasticity works because of two basic forces that operate at the molecular level: attracting force and repelling force. When at rest, these forces within the molecules balance each other. By adding a compressing force (say, by squeezing a spring), the repelling force increases in an attempt to once again balance the system. Likewise, by adding a stretching force (as in a weight pulling a spring), the attracting force increases, causing the elastic material to bounce back.
The first scientist to conduct in-depth research into the behavior of elastic materials was the famous English physicist Robert Hooke (1635–1703). Through experiments Hooke discovered that the relationship between tension (the force applied) and extension (the amount of bending that is produced) is directly proportional. For example, a weight will stretch a spring, and a weight twice as heavy will stretch it twice as much. Hooke's research has since been combined into a series of mathematical principles known as Hooke's law.
Words to Know
Strain: The amount by which a material stretches divided by its original length.
Stress: The force applied to an object divided by the area on which the force operates.
More than 100 years after Hooke's studies, another English scientist, Thomas Young, discovered that different elastic materials bend to different degrees when a force is applied. For example, brass bends more than lead, but less than rubber. The amount of elasticity of a particular material, Young found, can be expressed as a constant called Young's modulus. Knowledge of Young's modulus is essential to modern architects, who must be able to predict how construction materials will act when they are under stress.
"Elasticity." UXL Encyclopedia of Science. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/elasticity-1
"Elasticity." UXL Encyclopedia of Science. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/elasticity-1
e·las·tic / iˈlastik/ • adj. (of an object or material) able to resume its normal shape spontaneously after contraction, dilatation, or distortion. ∎ able to encompass variety and change; flexible and adaptable: the definition of nationality is elastic in this cosmopolitan country. ∎ springy and buoyant: Annie returned with beaming eyes and elastic step. ∎ Econ. (of demand or supply) sensitive to changes in price or income: the labor supply is very elastic. ∎ Physics (of a collision) involving no decrease of kinetic energy. • n. cord, tape, or fabric, typically woven with strips of rubber, that returns to its original length or shape after being stretched. DERIVATIVES: e·las·ti·cal·ly adv. e·las·tic·i·ty / iˌlaˈstisitē; ēˌla-/ n. e·las·ti·cize / iˈlastəˌsīz/ v.
"elastic." The Oxford Pocket Dictionary of Current English. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/humanities/dictionaries-thesauruses-pictures-and-press-releases/elastic-0
"elastic." The Oxford Pocket Dictionary of Current English. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/humanities/dictionaries-thesauruses-pictures-and-press-releases/elastic-0
elasticity, the ability of a body to resist a distorting influence or stress and to return to its original size and shape when the stress is removed. All solids are elastic for small enough deformations or strains, but if the stress exceeds a certain amount known as the elastic limit, a permanent deformation is produced. Both the resistance to stress and the elastic limit depend on the composition of the solid. Some different kinds of stresses are tension, compression, torsion, and shearing (see strength of materials). For each kind of stress and the corresponding strain there is a modulus, i.e., the ratio of the stress to the strain; the ratio of tensile stress to strain for a given material is called its Young's modulus. Hooke's law [for Robert Hooke] states that, within the elastic limit, strain is proportional to stress.
"elasticity." The Columbia Encyclopedia, 6th ed.. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/reference/encyclopedias-almanacs-transcripts-and-maps/elasticity
"elasticity." The Columbia Encyclopedia, 6th ed.. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/reference/encyclopedias-almanacs-transcripts-and-maps/elasticity
"elasticity." World Encyclopedia. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/elasticity
"elasticity." World Encyclopedia. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/elasticity
Hence elasticity XVII.
"elastic." The Concise Oxford Dictionary of English Etymology. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/humanities/dictionaries-thesauruses-pictures-and-press-releases/elastic-1
"elastic." The Concise Oxford Dictionary of English Etymology. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/humanities/dictionaries-thesauruses-pictures-and-press-releases/elastic-1
"elastic." Oxford Dictionary of Rhymes. . Encyclopedia.com. (May 30, 2017). http://www.encyclopedia.com/humanities/dictionaries-thesauruses-pictures-and-press-releases/elastic
"elastic." Oxford Dictionary of Rhymes. . Retrieved May 30, 2017 from Encyclopedia.com: http://www.encyclopedia.com/humanities/dictionaries-thesauruses-pictures-and-press-releases/elastic