Elasticity
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left#1\right}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Agricultural Economics: Elasticity
Food Market [1]
1.0 Introduction
Perhaps one of the most useful concepts in economics is the notion of elasticity. Taken from engineering, economists adopted this term into their toolset to be able to provide an intuitive mechanism for examining the marginal changes in economic processes, such as changes in the prices of goods on the quantity demanded (consumed) or supplied (produced) of that good. Elasticity can help us answer questions like (i) how much should a firm produce?; (ii) how will a change in demand impact a firm's sales?; (iii) how much can a firm charge for a product given a competitor's position in the market?; and (iv) should the government tax a particular good to raise more revenue for public programs? All of these decisions revolve around a marginal change in one variable in response to a marginal change in another variable. A marginal change is defined as the addition of one or more units of a given economic variable (e.g. goods produced) to the current level of that economic variable We call this relative or marginal response an elasticity.
This chapter will introduce you to the concepts of elasticity, present different forms of elasticities found in economics, and take you through a set of detailed case studies about how elasticity can be calculated, interpreted, and used. The case studies will show you how elasticity is one of the most important concepts in economics.
1.1 Learning Objectives
By the end of this chapter, you should be able to:
(1) Define the concept of an elasticity;
(2) Calculate and interpret different types of elasticities, including own price elasticities, price elasticities, and income elasticities; and
(3) Be able to use the concept of elasticity in different economic problem situations.
A few case studies are provided that help illustrate the calculation and usefulness of elasticities. These case studies will help you to examine elasticities in different decisionmaking contexts and include questions to help you review and go through these examples on your own. In addition, the case studies will present you with some additional concepts related to elasticities and extend how they are useful in different economic situations and decisionmaking.
2.0 What is Elasticity?
The concept of elasticity originally arose in the fields of engineering and physics. Elasticity refers to the capability of a body that has been strained to recover its size and shape after deformation. Another perspective is how easily an object can be stretched [2]. It is this latter definition that gives rise to the notion of elasticity in economics. From an economic perspective, it is how much an economic variable can be "stretched" or how responsive that variable is to a given change in another economic variable.
By definition, elasticity is the responsiveness of one economic variable to a marginal change in another related economic variable. Mathematically, elasticity can be defined as the percentage change in one variable in response to the percentage change in another variable. When used as a single term, elasticity usually refers to "ownprice elasticity", which is a measure of the percentage change in the quantity demanded or supplied of a good in response to a percentage change in the "own" price of that same good. By "own" price, we mean the price of the good or service we are examining. The ability of an elasticity to capture this responsiveness or marginal change in economics makes the concept of elasticity especially important in conducting marginal analysis, one of the main tools used in economics. For example, elasticity is very useful in measuring the responsiveness of consumers' demand to different output prices, income, and other economic variables, which can help provide a basis for firms or businesses to make more profitable production and marketing decisions. These decisions include how much to spend on advertising, offering discounts, increasing/decreasing production, which markets to sell in, and many more [3,4].
Elasticity can be represented mathematically. It is the mathematical formula and calculation that illustrates the usefulness of this concept in economics.
To begin, elasticity is the relationship between two economic variables. Let \(Y\) represent the economic variable of interest, say the quantity demanded of a good. We are interested in how responsive \(Y\) is to a change in another related economic variable \(X\), say the price of that same good. To be able to calculate an elasticity using data, we need, at a minimum, two points of data. Let these two points be (\(Y_1,X_1\)) and (\(Y_2,X_2\)). We are interested in the marginal change (△) in each variable. That is:
Marginal Change in \(Y\): \(△Y=Y_2Y_1\); and
Marginal Change in \(X\): \(△X=X_2X_1\)
We want to make the changes unitless. That is, to compare the marginal change in one variable to a marginal change in another variable in more general terms, we can use the percentage changes of each variable. As the two variables are measured in percentage changes, the units cancel out [3]. In addition, using percentage changes helps to avoid problems of scale or differences in the units of the variables being examined. To do this, we convert the above marginal changes into percentage changes (%△) by:
Percentage change in \(Y\): \(\%△Y=\frac{△Y}{Y_1}\)
Percentage change in \(X\): \(\%△X=\frac{△X}{X_1}\)
We can put these together to calculate an elasticity. Using the two variables defined here, the elasticity is the percentage change in \(Y\) given a percentage change in \(X\). The elasticity (\(E\)) can be calculated as: \[E=\frac{\%△Y}{\%△X}=\frac{△Y}{Y_1}\times\frac{X_1}{△X}=\frac{△Y}{△X}\times\frac{X_1}{Y_1}\] The formula given by equation (1) is known as a point elasticity, as it is calculated relative to the point (\(Y_1,X_1\)). It should be noted here that the point (\(Y_2,X_2\)) could have been used instead.
We will show another related formula known as the arc elasticity in section 2.1.1, as it provides a more accurate estimate of the marginal changes being captured by the elasticity in some cases.
The elasticity formula is a general formula, which can be used to look at the responsiveness of a large number of economic variables to marginal changes in other economic variables. Substituting different economic variables into equation (1) gives rise to a multitude of different elasticity measures, as will be seen in this and the next section.
The interpretation follows from this mathematical formula. The formula in equation (1) tells us by how much \(Y\) will change as a percentage, given a 1% change in \(X\). Consider the case where \(Y\) is the quantity supplied of shoes, and \(X\) is the price of shoes, then \(E\) here represents the own price elasticity of supply for shoes (to be discussed in more detail later). If say \(E=1.5\), then a 1% increase in the price of shoes (\(X\)), will increase the quantity supplied of shoes (\(Y\)) by 1.5 %. Now consider the following question.
Question #1
In general, elasticities are classified into three separate categories based on the responsiveness of the economic variable of interest \(Y\) to a change in another economic variable \(X\. The three categories, which are based on the absolute value of the elasticity, are:
1. Elastic: \(∣E∣ > 1.0\)
2. Unitary Elastic: \(∣E∣ = 1.0\)
3. Inelastic: \(∣E∣ < 1.0\)
Question #2
We now turn to looking at different types of elasticities commonly used in economics, how to compute them, and how to interpret them. The following video provides a review of this material and focuses on the own price elasticity of demand, which you will encounter in the next section.
Own Price Elasticity of Demand [5]
Question #3
2.1 Types of elasticities
As mentioned earlier, numerous types of elasticities are useful in economics. We want to examine a number of these in this section. All of these elasticities are calculated using the same formula, the only difference being the economic variables involved. For now, we will focus on 3 fundamental elasticities: own price elasticity of demand and supply, cross price elasticities, and income elasticities.
2.1.1 Arc vs. Point Elasticities
We presented equation (1) initially as the formula for calculating elasticity. While point elasticity can be useful, this measure is highly dependent on the point being chosen and does not take into consideration the information from other points (or data) used to calculate the elasticity.
An alternative formula advocated in economics is the midpoint arc elasticity formula or, more commonly known, the arc elasticity formula. This measure uses the average of the points in the formula, rather than one point, as in the point elasticity formula. That is, an arc elasticity is calculated using the following formula: \[E=\frac{△Y}{△X}\times\frac{\overline{X}}{\overline{Y}}\](2)
where \(\overline{Y}=\frac{Y_1+Y_2}{2}\) and \(\overline{X}=\frac{X_1+X_2}{2}\). That is, instead of using the point (\(Y_1,X_1\)), we use the averages of the two points, (\(\overline{Y},\overline{X}\)), to calculate the elasticity.
The use of the arc elasticity formula was advocated by R.G. D. Allen, especially for the case when using elasticities to look at comparative statics for market demand and supply. The arc elasticity has the following useful properties: (1) it is symmetric with respect to the two economic variables being examined, (2) it is not dependent on units of measure, and (3) it is equal to one for the own price elasticity of demand and supply if total revenues or expenditures (price times quantity) at the two points being examined are equal. [6].
2.1.2 Own Price Elasticity
The ownprice elasticity (more often referred to as just "elasticity") measures the change in quantity (\(Q\)) demanded or supplied as a result of a change in the price (\(P\)) of the good. The own price elasticity can be represented as a mathematical function, based on equation (2), as: \[E=\frac{\%△Q}{\%△P}=\frac{△Q}{△P}\times\frac{\overline{P}}{\overline{Q}}\]
More specifically, there are two own price elasticities of interest. These are:
Own price elasticity of demand: This elasticity is represented by the Greek letter \(η\) and it measures the responsiveness of consumers' consumption or quantity demanded of a good or service to a change in the market (or own) price of that good or service. These elasticities are usually negative, following the Law of Demand. The own price elasticity of demand (\(η\)) is calculated as: \[η=\frac{\%△Q^D}{\%△P}=\frac{△Q^D}{△P}\times\frac{\overline{P}}{\overline{Q^D}}\] (3)
where \(Q^D\) is the quantity demanded of the good or service being examined.
Own Price elasticity of supply: This elasticity is represented by the Greek letter ϵ and it measures the responsiveness of a firm's production or quantity supplied of a good or service to a change in the market price of the good or service being produced. These elasticities are usually positive, following the Law of Supply. The own price elasticity of supply (\(ϵ\)) is calculated as: \[ϵ = \frac{\%△Q^S}{\%△P}=\frac{△Q^S}{△P}\times\frac{\overline{P}}{\overline{Q^S}}\] (4)
where \(Q^S\) is the quantity supplied of the good or service being examined.
Own price elasticities are classified into three separate categories, as seen earlier, based on the responsiveness of the quantity demanded or supplied to a change in the market price of the good or service being examined. The three categories are:
1. Price Elastic: \(∣η∣>1.0\) or \(ϵ>1.0\)
2. Unitary Elastic: \(∣η∣=1.0\) or \(ϵ=1.0\)
3. Inelastic: \(∣η∣<1.0\) or \(ϵ<1.0\)
From a demand perspective, goods that are price elastic have changes in quantity demanded that are greater than 1% when the (own) price changes by 1%. These goods may have many substitutes in consumption, and as price increases, consumers will substitute these goods for cheaper goods. Goods that are price inelastic have changes in quantity demanded that are less than 1% when price changes by 1%. As such, these goods tend to have fewer substitutes and include products like food staples, cigarettes, and alcohol. The use of substitutes/complements in consumption/production will be more closely examined using cross price elasticities.
Based on this information, consider the following question to test your understanding.
Question #4
Now let's look at how to calculate and interpret own price elasticities of demand and supply by examining the market for bananas.
Bananas at Market [12]
For this example, assume the demand schedule for bananas for a small group of people is given in the table below.
Price of Bananas (P) 
Quantity Demanded of Bananas (Q) 

$1 
8 
$2 
6 
$3 
4 
$4 
2 
Along a linear demand curve, the own price elasticity of demand will change as you move down the curve (as the price decreases). Let's consider what the own price elasticity of demand will be going from $1 to $2 (a price increase). To calculate the elasticity, we will use the arc elasticity formula given by equation (3).
For this calculation, we are given the following two points, (\(P_1,Q_1^D\)) = (\(1,8\)) and (\(P_2,Q_2^D\)) = (\(2,6\)), from the demand schedule. To calculate the arc elasticity, we first need to find the change in the market price and quantity demanded. These are given by:
\(△Q^D= 6  8 = −2\) and \(△P =21=1\).
Second, we will need the average quantity demanded and price, which are given by:
\(\overline{Q^D}=\frac{6+8}{2}=7\) and \(\overline{P}=\frac{2+1}{2}=1.5\)
Plugging these values into the formula for the own price elasticity for demand given above gives: \[η=\frac{△Q^D}{△P}\times\frac{\overline{P}}{\overline{Q^D}}=\frac{2}{1}\times\frac{1.5}{7}=\frac{3}{7}=0.4\]
rounded to the first decimal place. The value of the elasticity is negative due to the Law of Demand and it indicates that demand for bananas is price inelastic between $1 and $2.
How do we interpret this elasticity? The calculated own price elasticity of demand for bananas tells us that for every 1% increase in the price of bananas (\(P\)), consumers in the small group will decrease their consumption or quantity demanded (\(Q^D\)) by 0.4%, and vice versa. Now let's consider what would happen if the price of bananas increased by 5% when the current price is between $1 and $2. Remember that for each 1% increase in price, the quantity demanded of bananas decreases by 0.4 %. If the price increases by 5%, then the quantity demanded will decrease by 5 x 0.4% = 2%.
Now let's have you calculate and interpret the ownprice elasticity of demand using the above example.
First, let's calculate the elasticity of demand for bananas between $2 and $3
Question #5
Question #6
On your own, you can now calculate the average quantity and average price that will be needed to calculate the arc elasticity of demand for bananas between $2 and $3 using equation (3). Once you have these values answer the next question.
Question #7
Now try the calculation fully on your own and interpret some of the results obtained.
Question #8
For the next two questions, refer back to your answer to Question #7.
Question #9
Question #10
Calculating and interpreting the own price elasticity of supply will follow the same procedures as in the above example and text.
2.1.3 Income Elasticity of Demand
We now begin to look at factors that impact demand (and supply) other than the own price. Remember these are referred to as determinants of demand (and supply). The following video provides a review of the income elasticity of demand. Watching this will help understand how elasticity can be used to look at the impact of changes in determinants of demand on the demand for a good or service.
Income Elasticity of Demand [8]
As a review for the coming discussion, answer the following question:
Question #11
We focus on income as a determinant of demand in this subsection to introduce the income elasticity of demand or income elasticity for short. This elasticity measures the percentage change in quantity demanded in response to a percentage change in consumers' income. We will designate the income elasticity of demand as \(E^I\). Income elasticity can be calculated using the following mathematical formula: \[E^I=\frac{\%△Q^D}{\%△I}=\frac{△Q^D}{△I}\times\frac{\overline{I}}{\overline{Q^D}}\] (5)
where \(I\) is income, \(△I=I_2I_1\), \(\overline{I}=\frac{I_1+I_2}{2}\) and \(\overline{Q^D}=\frac{Q_1^D+Q_2^D}{2}\). The formula is calculated using two points (\(I_1,Q_1^D\)) and (\(I_2,Q_2^D\)).
Four different types of goods are relatively important concerning income elasticity of demand: normal, necessity, luxury, and inferior goods. The definition and presentation of these types of goods should be reviewed. The income elasticity can indicate the type of good being examined as follows:
· Normal Goods: \(E^I>0\)
o Necessity Goods: \(0<E^I<1\)
o Luxury Goods: \(E^I>1\)
· Inferior Goods: \(E^I<0\)
Normal goods are those goods with a positive income elasticity. That is, an increase in income will increase the consumption of a normal good. Normal goods can be broken down into two subcategories: necessity and luxury goods.
Necessity goods are goods considered to have a positive income elasticity but are less than one in value. A 1% increase in income results in a less than 1% increase in the consumption of the good. Examples of necessity goods may include rice, beans, and corn.
Luxury goods are goods that consumers consume more of as income increases at an increasing rate. That is a one percent increase in income results in a larger than one percent increase in consumption of the good. Examples of luxury goods for consumers could include yachts, expensive wines, art, and jewelry.
Inferior goods are goods with a negative income elasticity. That is, an increase in income will result in a decrease in consumption of an inferior good. Possible inferior goods for consumers include public transportation, generic foods, and offbrand electronics.
Let's now consider an example of how to calculate and interpret income elasticity. Consider a farmer whose farm income increases from $75,000 to $90,000 per year. Earning $75,000 a year, the farmer would go and play 10 rounds of golf. Earning $90,000 a year, the farmer would now go and play 15 rounds of golf.
What is the income elasticity of demand for golf for the farmer between these two levels of income? To answer this question, we will use equation (5). Our two data points are \((I_1,Q_1^D)=(75,000,10)\) and \((I_2,Q_2^D)=(90,000, 15)\). Using these two points we can compute:
\(△I= 90,000 − 75,000 = 15,000\)
\(△Q^D= 15 – 10 = 5\)
\(\overline{I}=\frac{75,000+90,000}{2}= 82,500\) and
\(\overline{Q^D}=\frac{15+10}{2}=12.5\)
Plugging all of this into equation (5) gives: \[E^I=\frac{△Q^D}{△I}\times\frac{\overline{I}}{\overline{Q^D}}\times\frac{5}{15,000}\times\frac{82,500}{12.5}=2.2\]
The income elasticity calculated tells us that a one percent increase in income for the farmer increases their playing or consumption of golf by 2.2 percent. This makes golf a luxury good for the farmer.
Now consider what happens when the farmer's income decreases from $75,000 to $50,000. In this case, the farmer will reduce their playing of golf from 10 games to 7 games. Use this information to answer the following questions.
Question #12
Question #13
Income elasticities are important to consider because as income grows for consumers in the economy, the demand for goods and services will grow, as well. For elastic goods, as income increases, consumption of these goods will increase at an increasing rate. For inelastic goods, as income increases, consumption of these goods will increase at a decreasing rate. This type of information can be useful for firms planning marketing and production decisions, as well as governments looking to plan fiscal policy. For example, if a government wants to determine recessionproof goods or consumer staple stocks that will be produced in enough quantities for consumers in times of economic hardships, the government may want to look for highly income inelastic goods [14].
2.1.4 Cross Price Elasticity
Another important elasticity that captures the effects of changes in determinants of supply and demand is cross price elasticity. Cross price elasticity measures the quantity change of one good in response to a change in the price of another good. For example, butter and margarine, which are not produced together, impact the price and quantity demanded/ supplied by one another inversely. That is, they are substitutes. This relationship can be captured by the cross price elasticity. What we will find is that the cross price elasticity can be applied to both demand and supply.
Cross price elasticity of demand: This measure captures the responsiveness of consumers' consumption (quantity demanded) of a good or service in response to a change in the price of a related good or service. As with other elasticities, this can be captured using percentage changes. The cross price elasticity of demand here is represented by the symbol \(σ_X^D\). The formula for the cross price elasticity of demand for a good concerning the price of good \(X\) is given by the following function: \[=σ_X^D=\frac{\%△Q^D}{\%△P_X}=\frac{△Q^D}{△P_X}\times\frac{\overline{P_X}}{\overline{Q^D}}\] (5)
where \(P_X\) is the price of the related good \(\overline{P}_X=\frac{P_{X,1}=P_{X,2}}{2}\).
Cross price elasticity of supply: This measure captures the responsiveness of firms' production (quantity supplied) of a good or service in response to a change in the price of a related good or service. As with other elasticities, this can be captured using percentage changes. The cross price elasticity of supply here is represented by the symbol \(σ_X^S\). The formula for the cross price elasticity of supply for a good concerning the price of good \(X\) is given by the following function: \[=σ_X^S=\frac{\%△Q^S}{\%△P_X}=\frac{△Q^S}{△P_X}\times\frac{\overline{P_X}}{\overline{Q^S}}\]
where \(P_X\) is the price of the related good \(\overline{P}_X=\frac{P_{X,1}=P_{X,2}}{2}\)
Depending if one is looking at demand or supply, the cross price elasticities can be categorized into two different categories: substitutes or compliments, based on their sign, as follows:
(i) Substitutes:
· Substitutes in Consumption: \(σ_X^D>0\)
· Substitutes in Production: \(σ_X^S<0\)
(ii) Complements:
· Complements in Consumption: \(σ_X^D<0\)
· Complements in Production: \(σ_X^S>0\)
Recall, that substitutes are goods that are produced or consumed as an alternative to the original good.
Substitutes in consumption are goods that impact the price and quantity demanded of one another in the same direction. That is, as the price of a substitute good increases, the consumption of the good being examined increases, and vice versa. An example of substitutes in consumption are tortilla chips and potato chips. A consumer will tend to buy more of the cheaper product to minimize their expenses on food.
Substitutes in production are goods that impact the price and quantity supplied of one another inversely. That is, as the price of a substitute in production increases, the production of the good being examined decreases, and vice versa. An increase in the price of the substitute will increase revenue from that good for the producer. Assuming costs are unchanged, it is now more profitable for the producer to produce more of the substitute good, as they now earn more revenue relative to the other good. For example, farmers must usually decide how many different crops to plant each year to maximize their profit. These crops, e.g. corn, sorghum, soybean, wheat, and cotton are often substitutes in production.
Complements in consumption are goods that impact the price of a related good and quantity demanded of the good of interest inversely. That is, as the price of a complement in consumption increases, the consumption of the good of interest decreases, and vice versa. An example of goods that are complements in consumption include cake and frosting. As the price of cake increases, you will likely purchase less frosting, given you consume less cake.
Complements in production are goods that are produced together. Therefore, if the price of a complement in production increases, the production of the good of interest will increase, and vice versa. For example, peanut oil and peanut meal must be produced together, as peanut meal is a byproduct from the production of peanut oil. In fact, peanut meal has its own market and can be used for a number of different uses, including as a feed source for livestock. This makes peanut oil and peanut meal complements in production.
Let's examine the use of cross price elasticity by looking at the production of ethanol and a byproduct of the production process, distiller's dried grains (DDGs). DDGs are a good source of protein for livestock feed rations.
Question #14
Ethanol Plant [10]
Assume that 5 billion gallons of ethanol are produced and the price of DDGs is $90 per dry ton. A change occurs in the market over time increasing the production of ethanol to 6 billion gallons and the price of DDGs changes to $100 per ton. Use this information and the above material on cross price elasticities to answer the following questions [11].
Question #15
Question #16
2.2 Important Aspects of Elasticity
Much of the importance, calculation, and interpretation of different types of elasticities was discussed in the prior subsections of this section of the chapter. In this short subsection, we want to highlight briefly two points to keep in mind about elasticities.
(1) While several different elasticities were presented above, there are many more used in the field of economics. As stated earlier the relationship of many economic variables can be summarized using an elasticity. That is, elasticities can be used in general to compare many different relationships between two economic variables. It is a widely applicable and used concept in the field of economics and other related disciplines. You will encounter some additional elasticity measures in the case studies below.
(2) Elasticities provide a generalizable and intuitive way to not only capture the sign (e.g. positive or negative) of the relationship between two economic variables, but elasticities provide a measure that captures the magnitude of this relationship (i.e. if it is elastic or inelastic). These properties allow the use of elasticities to do "back of the envelope" types of calculations for a quick assessment of changes in the market or of policy changes that are accessible and understandable by more lay groups (who may not fully understand economics). These quick assessments though should be placed in context, as the use of an elasticity usually assumes that all other variables are held constant. That is, when using elasticities we are implicitly invoking the notorious ceteris paribus clause often encountered in economic analysis.
The remainder of the chapter provides some indepth case studies to provide you with more practice in using, calculating, and interpreting different elasticity measures.
3.0 Case Studies
In this section of the chapter, we examine three indepth case studies on beef demand, parking demand on campus, and agricultural land use.
3.1 Beef Demand and Cattle Producer Competitiveness
The own price elasticity of demand may be one of the most useful elasticity measures in economics. It is a crucial and highly important measure for agribusiness to take account of and try to determine. Knowing or having an accurate prediction of the own price elasticity of demand can help a firm figure out how much product to produce; the effect of sales, discounts, or other promotions on their product; allow them to be more competitive in the market; determine the impact of government taxing policies; among other benefits [3,4]. The purpose of this case study is to examine the impact of knowing the own price elasticity of demand for beef on agribusiness firm decisionmaking.
Consider the following:
From the late 1940's to the late 1960's, leaders of the cattle industry urged producers to be conservative about their cattle herd sizes to improve cattlemans' market positions. At the same time, recommendations from academic livestock and range researchers focused on improving range management practices as a means of increasing range carrying capacity and in turn beef output. The motivation for this research was in part due to the rising demand for beef during the period following World War 2. Cattle producers were being pushed by industry leaders to decrease their output to increase the price received per pound (lb.) of beef, while simultaneously range researchers were looking to increase beef production, provided that beef prices would not decrease significantly as the quantity of beef increased [12]
Cattle Yard Circa 1950s [13]
From a beef producer's standpoint, the own price elasticity of demand is arguably a very important economic measure to estimate and understand. Unfortunately, it can also be one of the hardest measures to accurately determine. Many researchers and agribusinesses use historical data from the past to figure out or estimate the own price elasticity of demand for beef in the market. A cattle producer will be interested in deciding the best course of action for the future, but this decision can be enhanced by knowing the own price elasticity of demand in the market and past market trends. The table below provides historical data for the beef market, including the beginning period beef price, ending period beef price, the beginning quantity of beef consumed in the market for a period, the ending quantity of beef consumed in the market for a period, the average price over the period, and the average quantity of beef consumed over the period [12].
All the quantities in the table are in lbs per capita.

Beginning Price 
End Price 
Beginning Quantity 
Ending Quantity 
Average Price 
Average Quantity 

19471951 
$42.42 
$56.30 
69.6 
56.1 
$47.59 
63.22 
19521956 
$53.21 
$37.89 
62.2 
85.4 
$42.08 
77.46 
Type caption for table (optional)
We are interested in estimating the own price elasticity of demand for beef and determining how changes in the price of beef will impact the beef market and producers' decisions. The following questions ask you to use the arc elasticity formula given by equation (3) to answer the following questions. For these calculations use the average price and quantity provided in the above table instead of calculating \(\overline{P}\) and \(\overline{Q^D}\).
Question #17
Question #18
The own price elasticity of demand for beef is concerned with how the quantity demanded of beef changes as a percentage, because of a one percent change in price. Producers may likely be interested in the inverse relationship, as well. Given that price and quantity in the market are simultaneously determined, producers may want to know the effect a one percent change in the quantity demanded has on the price of beef.
Inversely related to the own price elasticity of demand, price flexibility shows the percentage change in price as a result of a one percent change in quantity demanded (or consumed). Price flexibility has the inverse interpretation of price elasticity. Each can be used separately to derive the percentage change in price or the percentage change in quantity demanded of a good. We introduce price flexibility as an alternative elasticity measure that may also be useful for market analysis. Price flexibility is denoted here by the symbol \(λ\) and is mathematically represented as: \[λ=\frac{\%△P}{\%△Q^D}=\frac{1}{η}\]
where \(η\) is the own price elasticity of demand. Thus, if you know the own price elasticity of demand, you can easily calculate the price flexibility and vice versa.
For example, assume the price elasticity of demand (\(η\)) for a good is 0.90. The elasticity can be interpreted as follows: a 1% increase in the price of the good would decrease the quantity demanded by 0.90%. The price flexibility for the good can then be calculated as (\(\frac{1}{η}\)) or \(\frac{1}{0.90}= 1.12\). The price flexibility can be interpreted as a 1% increase in the quantity demanded of the good will decrease the price of the good by 1.12%. Just as the own price elasticity of demand is negative, so will the corresponding price flexibility.
Question #19
Consider a 5% increase in total beef consumption following 1951. That is consumption is now equal to \(1.05\%\times56.1 = 58.91\) lbs per capita after the increase. The price flexibility can help us determine if we need to increase or decrease herd size by determining the corresponding percent change in price. In our case, the percent change in quantity is +5%.
To determine a % change in price given a percent change in the quantity demanded, it is useful to know the price flexibility (\(λ\)). The resulting percentage change in price from a given percentage change in the quantity demanded can be derived from the price flexibility equation. If one knows the price flexibility and the given percent change in \(Q^D\) then the percentage change in price can be found using the following formula: \[\%△P=λ\times\%△Q^D\]
Question #20
As seen in the discussion above, knowing the own price elasticity of demand and price flexibility can have a significant impact on a producer's bottom line, primarily through their total revenue (TR). TR is impacted because it is a function of both the market price and quantity sold. From the above example, if the quantity demanded of beef increases by 5% in the future, then we know the price will decrease due to the Law of Demand. The price flexibility in Question #19 gives us the expected percentage change in price.
Assume that it is the year 1957. We use the historical information from the period 1947 to 1951 to estimate the own price elasticity of demand, assuming the livestock industry improves in the near future similar to the average conditions in that period. The ending beef price in 1956 and the current price for us is $37.89. The current level of consumption (ending quantity demanded in 1956) is 85.4 lbs per capita. A 5% increase in consumption results in an increase in consumption from 85.4 to 85.4 x 1.05 = 89.67 lbs per capita.
Question #21
Before the change in demand, total revenue for beef producers was TR = $37.89 x 85.4 = $3235.81. After the increase in the quantity demanded and the resulting change in price, TR = $3164.45. Thus, producers would know to expect their TR to fall and to lower their prices and increase their herd size marginally to be able to remain competitive in the market, assuming producers are in a perfectly competitive market. If a producer did not lower their price, they would likely make even less sales and even lower TR, reducing their profitability more than they may have expected. Thus, knowing the market and the associated demand elasticities can help a firm to remain competitive and profitable.
3.2 Parking Demand on University Campuses
Applications of elasticity may impact you even closer to home. Consider the issue of parking on college campuses. The own price elasticity of demand plays a role even here.
University Parking [14]
Consider the following:
Parking availability for dorms at many universities is relatively more substantial than for students who live off campus and pricing for parking permits may be relatively high. A reason for this is that students who live on campus have fewer options for parking their cars than students off campus. Thus, the demand for parking on campus by students living on campus is greater than for those living off campus. University parking services can take advantage of this when determining how much to price parking permits and how many parking spots to offer. These decisions will be impacted by the own price elasticity of demand for parking spaces by students living on campus.
Parking services at universities are in place to manage parking, but they can also generate revenue for the university. Thus, it may be reasonable to assume that their objective is revenue maximization. From this point of view, parking services would want to charge the highest possible price for a permit that maximizes the number of students parking in their lots. Considering there are limited viable parking substitutes (e.g. parking on side streets around campus) available to students living on campus and the close proximity of parking lots to the dorms (compared to other options); the own price elasticity of demand for oncampus parking can be thought of as "relatively inelastic". Let's assume the own price elasticity of demand is 0.25.
Assume parking services are attempting to maximize potential revenue for the coming school year and assume costs remain unchanged (due to no change in the size of the parking lot). Then the primary factor of importance for maximizing parking revenues is the total revenue to parking services.
Assume that there are 1000 parking spaces available and that parking services can charge whatever they see fit for each parking space. We know that parking services charged $200 per spot last year, and 850 of the spots were used. Use this information to answer the following question.
Question #22
When considering total revenue (TR), we are interested in how changes to the price of parking permits impact the quantity demanded of the permits (as this will impact how many are sold) and in turn, impacts TR. Remember from the previous case study that \(\%△P=λ\times\%△Q^D\), where \(λ\) is the price flexibility. This equation can be rewritten as a function of \(\%△P\) and the own price elasticity of demand, as well (\(η\)). That is \(\%△Q^D=η\times\%△P\).
Considering the information above and that the own price elasticity of demand for parking permits is 0.25, consider a 10% increase in the price of parking permits for the following question.
Question #23
For the following questions, refer to the original price of $200 and the original quantity demanded of 850 permits. Let's analyze how the 10% change in price would impact revenues earned by the parking services.
Question #24
Question #25
Question #26
Question #27
This case study shows how the own price elasticity of demand may provide insight into economic thinking and decisions in our everyday lives. University parking services usually deal with a demand for parking that is highly inelastic. In this situation, as the case study illustrates, by increasing the price of parking permits, parking services can substantially increase revenues earned, while still limiting the number of spots left unfilled.
3.3 Price Elasticities and Agricultural LandUse
While much work is done using own price and cross price elasticities to examine the effects of price changes on the quantity demanded of goods and services, these elasticities play a role in examining the supply side of the market, as well. This case study will examine the use and interpretation of own price elasticities of supply, cross price elasticities of supply, and input price elasticities of supply. To be able to put these into context, we will examine the supply of corn from a land use perspective.
Corn Field [15]
Corn is one of the primary crops grown in Kansas and around the rest of the nation. In 2017, there were 5,200,000 acres of corn harvested in Kansas. This area represents a production of over 680 million bushels of corn [16]. The supply of corn in the Kansas market is a function of the (own) price of corn, the prices of other crops, and the prices of inputs. The Law of Supply indicates that as the (own) price of corn increases, the quantity supplied of corn will increase. This could be interpreted from a land use perspective too: as the price of corn increases, the amount of land devoted to corn production increases. In this respect, we can talk about the supply of corn based on the number of acres planted for corn. We can examine the responsiveness of corn supply in terms of acreage to a change in the price of corn using our own price elasticity of (acreage) supply. This elasticity measures the percentage change in the acreage supplied or planted given a 1 percentage change in the own price of corn. For example, if the own price elasticity of acreage supply is 0.8, then a 1% increase in the price of corn will increase the acreage planted to corn by 0.8%.
While a change in the (own) price of corn directly impacts the quantity supplied, changes in prices of related crops and inputs will cause supply to change. That is, the supply curve will shift as changes in these other prices change, as they are determinants of supply. Elasticities can be used here to examine the responsiveness of supply to changes in one of these other prices or determinants. Consider the two following concepts:
(i) Cross price elasticity of acreage supply: This elasticity measures the percentage change in acreage devoted to a given crop to a 1% change in the price of another crop. The crops can be complements or substitutes in production. For example, if the cross price elasticity of acreage supply for corn for the price of soybeans is 0.5, then a 1% increase in the price of soybeans will increase the acreage devoted to corn production by 0.5%. When these elasticities are positive, the two crops being considered are complements in production (e.g. they may be grown in rotation). When these elasticities are negative, the two crops being considered are substitutes in production (i.e. they compete for the same area of land).
(ii) Input price elasticity of supply: This elasticity measures the percentage change in the acreage devoted to a crop to a 1% change in the price (or cost) of an input to produce that crop. For example, if the input price elasticity of acreage supply for corn for the price of labor is 1.5, then a 1% increase in the wage rate paid for labor on the farm will result in a 1.5% decrease in the amount of land devoted to corn production.
All of these elasticities are calculated in the same manner as presented in section 2.0. The only difference for an acreage supply elasticity is that the change in quantity is the change in the amount of land devoted to the production of the crop.
To be able to illustrate the use of these concepts, let's consider the production of irrigated corn in western Kansas. AlSudani [17] did a statistical analysis of the factors influencing the amount of land devoted to irrigated corn production in western Kansas between 2003 and 2015. The focus on irrigated corn production was due to its crop revenue generation and scarcity of water for corn production in this region of the state. We will focus on the estimated acreage supply elasticities for corn for different prices found in the study.
AlSudani [17] estimated the following acreage elasticities:
· Own price elasticity of acreage supply for corn = 0.66;
· Cross price elasticity of acreage supply for corn with respect to sorghum = 0.38;
· Cross price elasticity of acreage supply for corn with respect to soybean = 0.03; and
· Input price elasticity of acreage supply for corn with respect to fertilize price = 1.19.
These elasticities can be used to explain how the amount of acreage devoted to corn production changes when different commodity and input prices change in the market. This information would be useful to industry (e.g. cooperatives, elevators, traders) and farmers for marketing and planning their next growing season. Use this information to answer the following questions about the likely impact of different changes in the market.
Question #28
Question #29
Question #30
Question #31
4.0 Concluding Remarks
Elasticity is an important economic measure and can be a highly influential factor and aid when making economic decisions. Elasticities provide a readily accessible way to conduct marginal analysis for a multitude of problem situations. Elasticity has applications ranging across all reaches of economics as illustrated by the examples and case studies examined in this chapter. We have gone over a few of the essential elasticities, however, there are many more that assist in helping to examine different relationships between economic variables. Elasticity is a useful tool to utilize for producers, consumers, and policymakers and applies to our everyday lives.
Question #32
5.0 References
[1] Image courtesy of Luke Erickson through Pixabay, under license CC0 1.0 Universal.
[2] MerriamWebster Dictionary. 2018. Elasticity. Available at: https://www.merriamwebster.com/dictionary/elasticity.
[3] Barkley, A. and P.W. Barkley. "Principles of Agricultural Economics" 2nd Edition. New York, NY: Routledge, 2016
[4] Barkley, Andrew, "The Economics of Food and Agricultural Markets" (2016). NPP eBooks. 12.
[5] Video courtesy of mjmfoodie through Youtube, under license CC BY Attribution 3.0 Unported.
[6] Allen, R. G. D. 1933. "The Concept of Arc Elasticity of Demand". Review of Economic Studies 1 (3): 226–229.
[7] Image courtesy of Pexels, under their licensing agreement.
[8] Video courtesy of econom through Youtube, under license CC BY Attribution 3.0 Unported.
[9] InvestingAnswers, 2018. "Income Elasticity of Demand." Available at: http://www.investinganswers.com/inco...tydemand2906.
[10] Image courtesy of freddthompson through Wikimedia Commons, under license CC BYSA 2.0.
[11] Agricultural Marketing Service, USDA. 2018. "Weekly Distiller's Grain Summary." NW_GR115. Availalble at: https://www.ams.usda.gov/mnreports/nw_gr115.txt.
[12] Workman, J.P., S.L. King and J.F. Hooper. 1972. "Price Elasticity of Demand for Beef and Range Improvement Decisions." Journal of Range Management 25(5): 338  341.
[13] Image courtesy of U.S. National Archives and Records Administration through Wikimedia Commons, Public Domain photograph.
[14] Image courtesy of jalexartis through Flickr, under license CC BY 2.0
[15] Image courtesy of Petr Kratochvil through PublicDomainPictures, under license CC0 1.0 Universal.
[16] USDA, National Agricultural Statistics Service. 2018. 2017 State Agricultural Overview: Kansas. Available at: https://www.nass.usda.gov/Quick_Stats/Ag_Overview/stateOverview.php?state=kansas.
[17] AlSudani, A. 2018. Essays in land use and water scarcity in Kansas. Ph.D. dissertation, Department of Agricultural Economics, Kansas State University, Manhattan, KS.