An investigation of business calculus students’ covariational reasoning, procedural knowledge and conceptual knowledge in the context of price elasticity of demand

Abstract

Motivated by the paucity of research that has examined students’ covariational reasoning in economic contexts, the study reported in this article investigated business calculus students’ covariational reasoning about the economic concept of price elasticity of demand. Furthermore, the study examined students’ procedural knowledge and conceptual knowledge in the context of explaining what it means for demand to be inelastic, elastic or unit elastic, respectively. Additionally, the study examined students’ calculational knowledge of price elasticity of demand. The analysis of 10 students’ responses to three tasks used in the study revealed that most of the students struggled making sense of what it means for demand to be inelastic, elastic or unit elastic, respectively. A majority of the students only engaged at the lowest levels of covariational reasoning when prompted to reason about the relationship between the concept of price elasticity of demand and its relationship to the economic quantities of price, demand and revenue. Furthermore, showing that maximum revenue occurs when price elasticity of demand is equal to one was particularly challenging for all the students. Overall, findings from this study suggest that the study participants demonstrated a weak understanding of the concept of price elasticity of demand. Directions for future research and implications for instruction are included.

1 Introduction

Price elasticity of demand (elasticity of demand) is an economic measure of the sensitivity of the quantity of demand in response to changes in the quantity of price (cf. Hughes-Hallett et al., 2018; Haeussler et al., 2019). According to Haeussler et al. (2019), elasticity of demand ‘…can be defined as the ratio of the resulting percentage change in quantity demanded to a given percentage change in price…’ (p. 543). These researchers went on to remark that ‘…when elasticity [of demand] is evaluated, no units [of measure] are attached to it-it is nothing more than a real number’ (p. 544). When measured, quantities have units of measure (cf. Thompson, 1993). Examples of quantities in this study include price, demand and revenue. We note that in as much as elasticity of demand can be measured (i.e. can be calculated), elasticity of demand is not a quantity by virtue of not having a unit of measure. Stated another way, elasticity of demand is a unitless measure of the sensitivity of the demand of a product in response to changes in the price of the product.

Much research has reported on how students make sense of quantities, especially in physics and engineering contexts (cf. Beichner, 1994; Bingolbali & Monaghan, 2008; Prince et al., 2012; Ärlebäck et al., 2013; Flynn et al., 2018; Panorkou & Germia, 2021; English, 2022; Zimmerman, 2023). A substantial body of research has reported on students’ thinking about quantities in other natural or life sciences contexts (cf., Rasmussen & King, 2000; Rasmussen & Marrongelle, 2006; Jones, 2017; Johnson et al., 2017; Mkhatshwa, 2023). A number of researchers have lamented the scarcity of research that has examined how students make sense of quantities or concepts in economic contexts, let alone how students reason about quantities or concepts in these contexts (cf. Mkhatshwa & Doerr, 2018; Feudel & Biehler, 2022). Of the few available studies, Mkhatshwa & Doerr (2015, 2018) found that students tend to confuse marginal cost with total cost when tasked with solving application problems in business calculus. Similar results were reported by Feudel (2017). Contributing toward addressing this knowledge gap, this study investigated business calculus students’ reasoning about the economic idea or concept of elasticity of demand.

Undoubtedly, the aforenoted studies have provided beneficial information as it pertains to students’ thinking about economic quantities or concepts, especially at the undergraduate level. However, there is still much to be explored about how engaging students in different modes of reasoning might reveal about their understanding of economic quantities or concepts. Covariational reasoning is one mode of reasoning that has been particularly identified by leading mathematics educators and scholars as important (cf. Carlson et al., 2002; Thompson, 2011; Moore, 2014). They concluded that this form of reasoning helps to make sense of how variables or quantities are related to each other. This study investigated the following research question: What do students’ reasoning about the economic concept of elasticity of demand reveal about their covariational reasoning, procedural knowledge and conceptual knowledge? In using the terms, procedural knowledge and conceptual knowledge, we are referring to knowledge associated with determining, by way of calculating using a formula, numerical values of the measure of elasticity of demand (in the case of the former type of knowledge) or knowledge associated with interpreting the concept of elasticity of demand (in the case of the latter type of knowledge). More details about the aforementioned knowledge types, and most importantly how they are operationalized in this study, are discussed in the next section, followed by a section that provides context on the research reported in this study, which is followed by a section on the methods of data collection and analysis used in the study.

2 Theoretical background

The research reported in this study is rooted in constructivism (cf. Piaget, 1970; von Glasersfeld, 1995). Following is a description of the theoretical constructs used in this study, namely covariational reasoning, procedural knowledge, calculational knowledge and conceptual knowledge.

2.1 Covariational reasoning

According to Carlson et al. (2002), covariational reasoning describes how two or more related quantities change in tandem in a dynamic situation. These researchers went on to propose five increasingly sophisticated mental actions of covariational reasoning (see Table 1). Moreover, Carlson et al. (2002) described five levels of covariational reasoning (herein referred to as L1–L5) associated with the mental actions. Students who are at L1 (i.e. the coordination level) demonstrate behaviours associated with MA1. In other words, these students are aware that two or more quantities are changing in tandem. Students who are at L2 (i.e. the direction level) demonstrate behaviours associated with MA1 and MA2. That is, these students are able to attend to how at least one quantity changes (increases/decreases) with changes in other quantities. Students who are at L3 (i.e. the quantitative coordination level) demonstrate behaviours associated with MA1–MA3. Specifically, these students demonstrate evidence of quantifying the amount of change of at least of the changing quantities. Students who are at L4 (i.e. the average rate level) demonstrate behaviours associated with MA1–MA4. In particular, these students are able to attend to the average rate of change of at least one quantity with uniform increments in another quantity. Students who are at L5 (i.e. the instantaneous rate level) demonstrate behaviours associated with MA1–MA5. Put another way, students at this level are able to attend to the instantaneous rate of change of at least one quantity with continuous changes in another quantity. We note that it is possible for students to exhibit what Carlson et al. refer to as a pseudo-analytical behaviour, especially in connection with MA5. What this means is that students are sometimes able ‘…to exhibit MA5 without applying L5 covariational reasoning [i.e. MA1 to MA 4]’ (p. 359).

2.2 Procedural knowledge, calculational knowledge and conceptual knowledge

When it comes to making sense of mathematical concepts, there is a consensus among several researchers that there are generally two types of mathematical knowledge, namely procedural knowledge and conceptual knowledge (cf. Skemp, 1978; Hiebert & Carpenter, 1992; Osana & Pelczer, 2015; Rittle-Johnson et al., 2015; Lenz et al., 2019). This study uses Lenz et al.’s (2019) distinction between these two types of knowledge:

…The term procedural knowledge is used to describe knowledge of operations in the sense of a sequence of steps or partial actions, which are performed to achieve a specific goal…Conceptual knowledge is commonly defined as knowledge of concepts and relations which are fundamental in a certain domain. (p. 811).

It is worth noting that Lenz et al.’s definition of conceptual knowledge and procedural knowledge is consistent with how these types of knowledge have been defined in other studies (cf. Hiebert & Lefevre, 1986; Byrnes, 1992; Rittle-Johnson et al., 2001). Lenz argued that the aforementioned types of knowledge are both useful and ‘mutually interdependent’ (p. 811), an argument that has been echoed by other researchers (cf. Byrnes & Wasik, 1991; Rittle-Johnson et al., 2001).

In this study, procedural knowledge entails knowledge of how to calculate elasticity of demand [|$\eta$|], herein referred to as calculational knowledge. In other words, we define calculational knowledge as students’ facility (or lack thereof) with using the formula/equation, |$\eta =\mid \frac{p}{q}\ast \frac{dq}{dp}\mid$|⁠, where |$p$| is the price of a product and |$q$| is the demand of the product to determine numerical values of elasticity of demand. Procedural knowledge also entails an understanding of what it means for demand to be inelastic, elastic or unit elastic, respectively, that is limited to comparing the numerical value of |$\eta$| to the numerical value one. In particular, this means an understanding of inelastic demand that is limited to |$\eta <1$|⁠, an understanding of unit elastic demand that is limited to |$\eta =1$| and an understanding of elastic demand that is limited to |$\eta >1$|⁠. We note that students’ explanations of the ideas of elastic, inelastic or unit elastic demand that is limited to comparing the numerical value of |$\eta$| to the numerical value of one was classified (in this study) as procedural knowledge because students could reply on rote memorization when explaining the aforementioned ideas in this manner.

In this study, conceptual knowledge entails understanding what it means for demand to be inelastic, elastic or unit elastic, respectively, that takes into consideration the relationship between the quantities of price and demand in determining elasticity of demand. Specifically, this means understanding inelastic demand to mean that a 1% increase in the price of a product will cause the demand of the product to decrease by less than 1% and understanding unit elastic demand to mean that a 1% increase in the price of a product will cause the demand of the product to decrease by 1%. Additionally, it means understanding elastic demand to mean that a 1% increase in the price of a product will cause the demand of the product to decrease by more than 1%.

3 Context of the study: business calculus and elasticity of demand

Business calculus is an undergraduate service mathematics course that is typically taken by business majors (e.g. finance, supply chain management, accounting and business administration) or economics majors in the USA. It is rarely taken by other students pursuing other majors (e.g. computer science) as an elective. We remark that more than 300,000 students enroll in business calculus every year in the USA (Gordon, 2008). Topics included in the business calculus course where this study was conducted include limits and continuity, derivatives, graphing and optimization, exponential and logarithmic functions, integration and applications to problems arising in business. Business calculus is usually the only mathematics course where students are first introduced to the concept of elasticity of demand, among other economic concepts such as marginal change (marginal cost, marginal revenue and marginal profit). Additionally, business calculus is often a prerequisite course for business or economics courses where the economic concept of elasticity of demand and other economic concepts are covered in greater detail and from a business or economics perspective.

At the institution where this study was conducted, elasticity of demand was covered after students had learned about various rules of differentiation, including the power rule, quotient rule, product rule and chain rule. Elasticity of demand was defined (during classroom instruction) the same way as in the textbook (Haeussler et al., 2019) used in the course:

Elasticity of demand is a means by which economists measure how a change in the price of a product will affect the quantity demanded. That is, it measures consumer response to price changes. More precisely, it can be defined as the ratio of the resulting percentage change in quantity demanded to a given percentage change in price. (p. 543).

Students in this course were introduced (via course lectures) to the formula for calculating elasticity of demand, denoted by |$\eta$| or the Letter |$E$|⁠:

where |$q=f(p)$| is a differentiable demand function and |$p$| is the price of a product.

Students were taught that |$\eta <1$| means that demand is inelastic, |$\eta =1$| means that demand is unit elastic and that |$\eta >1$| means that demand is elastic via course lectures. Furthermore, the ideas of inelastic demand, unit elastic demand and an elastic demand were explained to students in a manner that is consistent with how these concepts are explained in the textbook (Haeussler et al., 2019) used in the course:

If demand is elastic, then for a given percentage change in price there is a greater percentage change in quantity demanded. If demand is inelastic, then for a given percentage change in price there is a smaller percentage change in quantity demanded. Unit elasticity means that for a given percentage change in price there is an equal percentage change in quantity demanded. To better understand elasticity, it is helpful to think of typical examples. Demand for an essential utility such as electricity tends to be inelastic through a wide range of prices. If electricity prices are increased by 10%, consumers can be expected to reduce their consumption somewhat, but a full 10% decrease may not be possible if most of their electricity usage is for essentials of life, such as heating and food preparation. On the other hand, demand for luxury goods tends to be elastic. A 10% increase in the price of jewelry, for example, may result in a 50% decrease in demand. (p. 545).

It is worth mentioning that the presentation of the concept of elasticity of demand during classroom instruction closely followed the textbook presentation of the same concept. We remark that the researcher in this study was not the instructor of the course.

4 Related literature

4.1 A survey of students’ covariational reasoning

A reoccurring theme from studies that have investigated students’ covariational reasoning is that engaging in the lower levels of covariational reasoning, namely coordination, direction and quantitative coordination, is often straightforward for students (cf. Carlson et al., 2002; Jones, 2017; Nagle et al., 2013; Mkhatshwa, 2023). However, engaging in higher levels of covariational reasoning, namely average rate and instantaneous rate, tend to be problematic for students (cf. Jones, 2017; Mkhatshwa, 2023). Findings of a related line of research indicate that students who are able to perform physical enactments of dynamic situations when solving application problems in calculus are often able to exhibit stronger covariational reasoning skills compared to students who do not make such enactments (cf. Carlson et al., 2002). On another note, findings by Carlson et al. (2003) suggest that using a models and modelling perspective (Lesh & Doerr, 2003; Lesh & Zawojewski, 2007) in mathematics teaching has the potential to help students develop robust covariational reasoning skills.

4.2 A survey of students’ thinking about rates of change and amounts of change in economics

To reiterate, research on students’ thinking about economic quantities is scarce. A common finding that emerges from the few available studies indicate that students’ interpretations of the concept of the derivative in mathematics is generally not consistent with students’ interpretations of the same concept in economics or business contexts (cf. Mkhatshwa & Doerr, 2015; Feudel, 2017, 2018; Feudel & Biehler, 2022). Specifically, students often demonstrate an understanding of derivatives as amounts of change and not as rates of change when working in economic contexts. For instance, a number of studies have reported on students’ conflations of marginal cost with total cost or marginal revenue with total revenue (cf. Mkhatshwa & Doerr, 2015, 2018; Mkhatshwa, 2016; Feudel, 2018). It can be argued that, to some extent, this stems from the fact that rates of change are often interpreted as amounts of change in economics such as when marginal cost is interpreted as the amount of change in cost when production increases by one unit (cf. Mkhatshwa, 2016; Feudel & Biehler, 2022). In fact, Feudel & Biehler (2022) posited that the derivative is ‘… commonly used [interpreted] in economics as the amount of change when increasing the production by one unit’ (p. 437). As Ariza et al. (2015) eloquently put it, ‘…understanding the relationship between a function [total cost, an amount of change] and its derivative [marginal cost, a rate of change] is essential to make sense of the marginal analysis on which these economic concepts are based’ (p. 616).

It is worth noting that students’ tendencies with regard to interpreting rates of change as if they were amounts of change are widespread and extend to other contexts, including the natural sciences (cf. Prince et al., 2012; Rasmussen & Marrongelle, 2006; Flynn et al., 2014, 2018; Lobato et al., 2012; Mkhatshwa, 2020). For instance, Lobato et al. (2012) reported on a student who interpreted the quantity of speed as if it were the quantity of distance, Rasmussen & Marrongelle (2006) reported on ‘students were not making a conceptual distinction between rate of change in the amount of salt and amount of salt’ (p. 408), and Flynn et al. (2014) reported on students who confused the total amount of water accumulated over time with the flow rate of water into a roof drain. On a related note, other studies have reported on students’ predisposition to confuse rates of change with other rates of change (Mkhatshwa, 2020, 2023). Mkhatshwa (2020), for instance, reported on a student who confused the rate of change of the radius of a circular puddle with the rate of change of the area for the same puddle. Findings from another line of research have reported on students’ inclination to confuse amount quantities with other amount quantities (Mkhatshwa, 2019). Specifically, Mkhatshwa (2019) reported on students who confused the amount quantity of total profit with the number of items that must be produced and sold in order to generate the greatest amount of total profit possible.

4.3 A survey of students’ calculational knowledge of derivatives

Although the focus of this study is on students’ covariational reasoning in the context of elasticity of demand, it is important to survey the existing literature on students’ calculational knowledge of derivatives. This is because one of the tasks used in this study (i.e., Task 2 in the Methods section) prompts students to calculate numerical values of elasticity of demand. We note that success in finding the aforementioned values involves calculating the derivative of the demand function |$q=1000-2{p}^2$| given in the same task. A common theme that emerges from research that has examined students’ calculational knowledge of derivatives when solving application problems in calculus is that lack of facility with different rules of differentiation often serves as a stumbling block for students when solving such problems. Particular rules of differentiation students struggle with include the product rule, quotient rule and chain rule (cf. Clark et al., 1997; Maharaj, 2013; Maharaj & Ntuli, 2018; Jeppson, 2019; Mkhatshwa, 2020). On a positive note, some studies have found that using the power rule of differentiation (i.e., differentiating polynomial functions) is often straightforward for most students (cf. Orton, 1983; Mkhatshwa, 2023).

5 Methods

This qualitative study used task-based interviews (Goldin, 2000) to assess students’ covariational reasoning in the context of working with the concept of elasticity of demand. The interviews, which lasted for an average of 45 minutes per interview, covered three tasks (see Appendices B, C and D). The tasks were administered sequentially i.e. students completed Task 1 before completing Task 2, and so on. The interviewer asked clarifying questions as students worked through each task. Additionally, the interviewer asked the following questions at the conclusion of each task:

• What was the easiest part when solving this problem? Explain.

• What was the difficult part when solving this problem? Explain.

Since the evaluation of the adequacy (or lack thereof) of students’ answers to prompt a) in Task 1 requires a careful analysis of the concept of elasticity of demand from both a mathematical and an economics point of view, something that is beyond the scope of this article, students’ responses to this prompt were excluded from the analysis.

5.1 Setting, participants and data collection

The interviewees were 10 undergraduate students (Pseudonyms Pat, Eve, Laura, Ava, Caleb, Emily, James, Adam, Brian and Nancy) who were enrolled in a business calculus course at a research university in the USA. The students took the course in the fall semester of 2022 and were recruited via email using an official class roster that was solicited from the course instructor. Final course grades solicited from the course instructor, with written consent from students, revealed that six students passed the course with an A grade, two students passed the course with a B grade and two other students passed the course with a C grade. The course instructor was an experienced professor who had previously taught multiple sections of business calculus. Even though students’ participation in the study was voluntary, and they could withdraw at any point in time if they wanted to or felt the need to, none of the students withdrew from the study. The study participants consisted of eight business majors, one economics major and one student who had not declared a major at the time of the study. Business calculus is generally a required course for business or economics majors at the institution were the study participants were recruited. Nine students were freshman, i.e., first-year students and one student was a sophomore, i.e., a second-year student. The data for the study consist of transcriptions of the task-based interviews and work written by the students during each interview session.

5.2 Data analysis

The data were analyzed in four phases. In the first phase of the analysis, we examined students’ procedural and conceptual knowledge of the concept of elasticity of demand, i.e., how students reasoned about what it means for demand to be inelastic, elastic or unit elastic in Task 1. Additionally, we examined how students reasoned about the ease and difficulty of solving the task. In the second phase of the analysis, we examined students’ calculational knowledge of elasticity of demand, i.e., how students calculated values of elasticity of demand in Task 2, and how they interpreted the results of their calculations. Furthermore, we examined how they reasoned about the ease and difficulty of solving the task. In the third phase of the analysis, we searched for evidence (or lack thereof) of students’ engagement in the five levels of covariational reasoning identified in the theoretical background section, namely coordination, direction, quantitative coordination, average rate and instantaneous rate, while working on Task 3. Moreover, we examined how students reasoned about the ease and difficulty of solving the task. In the fourth phase of the analysis, we examined students’ ability to verify, algebraically, that maximum revenue occurs when the numerical value of elasticity of demand is equal to one.

6 Results

In this section, we present results on students’ knowledge (procedural, calculational or conceptual) of elasticity of demand exhibited while working on Tasks 1 and 2, how they engaged in covariational reasoning while working on Task 3 and how they attempted to verify that maximum revenue occurs when demand is unit elastic while working on Task 3.

6.1 Students’ procedural and conceptual knowledge of elasticity of demand

Only one student (Caleb) provided a correct response that demonstrated conceptual knowledge of what it means for demand to be elastic while working on Task 1. In addition, Caleb exhibited procedural knowledge of what it means for demand to be unit elastic. Following is an excerpt illustrating how Caleb reasoned about several economic terms, namely elastic demand, inelastic demand and unit elastic demand.

• Researcher: What does it mean to say that demand is elastic? Explain.

• Caleb: Elastic to me means that if you increase the price, your demand will change (it may increase or decrease).

• Researcher: What does it mean to say that demand is inelastic? Explain.

• Caleb: Inelastic means that regardless what you change, your price to the demand doesn’t change. This may happen for a product like insulin, since it is a necessity.

• Researcher: What does it mean to say that demand is unit elastic? Explain.

• Caleb: Unit elastic means that your price is equal to the demand and it is not elastic or inelastic. It also means that when you are finding the demand of elasticity [i.e. in the sense of calculating a numerical value of the elasticity of demand], it equals one.

• Researcher: What was the easiest part when solving this problem? Explain.

• Caleb: I think the easiest part of solving the problem was identifying the definitions [i.e. explaining what elasticity of demand, elastic demand, inelastic demand and unit elastic demand means, respectively] since we go over these a ton in MATH 150 [business calculus course] and you either know the topic or you don’t.

• Researcher: What was the difficult part when solving this problem? Explain.

• Caleb: I think the hardest part of solving this problem was coming up with real-world applications to show someone who might not understand these terms and describing these terms in everyday language. It was also hard as we are not used to give a real-world example.

We interpreted Caleb’s explanation of what it means for demand to be elastic to be correct in that he correctly stated that when demand is elastic, increasing the price of a product would result in changes in the demand of the product. However, his claim that demand could increase or decrease when demand is elastic is only partially correct in the sense that when demand is elastic, the demand of a product decreases by more than 1% if the price of the product increases by 1%. Three other students (Emily, James and Adam) only demonstrated correct procedural knowledge of what it means for demand to be elastic. This occurred when they stated that this means that the numerical value of the elasticity of demand is greater than one. Another student (Pat) remarked that the numerical value of the quantity of demand is less than one when demand is elastic. This demonstrates incorrect procedural knowledge of what it means for demand to be elastic. Other students either provided vague or incorrect definitions of what elastic demand means, such as Laura who claimed that elastic demand means that ‘…the demand of a project can change.’

We interpreted Caleb’s explanation of what it means for demand to be inelastic to be imprecise with respect to what has been taught in the student’s business calculus course, as it shows no evidence of understanding that when demand is inelastic, the demand of a product shrinks by less than 1% if the price of the product is increased by 1%. Three other students (Emily, James and Adam) only demonstrated correct procedural knowledge of what it means for demand to be inelastic when they stated that this means that the numerical value of the elasticity of demand is less than one. As with students’ explanations of what elastic demand means, the other students either provided vague or incorrect responses when prompted to explain what inelastic demand means. An example of this is Brian who stated that inelastic demand means that ‘…the demand is unable to be stretched, or it is either decreasing or losing attraction.’

While Caleb’s claim that unit elastic demand is when price equals demand is incorrect, this student provided two other correct claims regarding the meaning of demand being unit elastic. Specifically, Caleb correctly claimed that when demand is unit elastic it is neither elastic nor inelastic and that the elasticity of demand ‘…equals one.’ Because of the latter two claims, we considered Caleb to have some understanding of what it means for demand to be unit elastic. We note, however, that Caleb’s understanding of unit elasticity as it relates to the value of the elasticity of demand equalling one only demonstrates correct procedural knowledge of the idea of unit elasticity. A demonstration of conceptual knowledge would include remarking on the fact that when demand is unit elastic, one would expect that the demand of a product would decrease by 1% when the price of the product is increased by 1% as well. Four other students (Pat, Emily, James and Adam) demonstrated correct procedural knowledge of unit elasticity when they claimed that demand is unit elastic when the value of the elasticity of demand is equal to one. Other students provided incorrect definitions of unit elasticity such as Laura who remarked that unit elastic means that demand ‘…either changes or doesn’t change.’

In response to a prompt on the easiest part when working on Task 1, Caleb remarked on the ease of defining the economic terms, namely elastic demand, inelastic demand and unit elastic demand. Two other students (Ava, Laura and Nancy) echoed Caleb’s response regarding the easy part of solving Task 1. That is, these students remarked that defining/explaining the meaning of elastic demand, inelastic demand and unit elastic demand was straightforward. Other students’ responses to the same prompt were different and did not point to a particular theme. For instance, Pat stated that ‘…remembering that elasticity of demand is an absolute value…’. On the other hand, Adam remarked on applying previous knowledge from his economics background to define elastic demand, inelastic demand and unit elastic demand as the easiest part when solving Task 2. In response to a prompt on the challenging part when working on Task 1, Caleb remarked on the difficulty of explaining what elastic demand, inelastic demand and unit elastic demand mean in non-technical or economic terms, i.e., using everyday language. Taken together, Caleb’s remark about the difficulty of explaining what it means for demand to be inelastic, elastic or unit elastic in non-technical terms suggests that this student struggled to make sense of the notion of elasticity of demand conceptually. In general, there was no particular theme regarding what students found to be particularly challenging when working on Task 1. For instance, two students (Pat and Brian) remarked on forgetting how to distinguish elastic demand from inelastic demand. Eve noted that ‘these terms [elastic demand, inelastic demand and unit elastic demand] were not defined in class,’ while Nancy commented on not having an economics background.

6.2 Students’ calculational knowledge of elasticity of demand

All the students determined numerical values of the demand function |$q=1000-2{p}^2$| at different price values in the process of calculating numerical values of the elasticity of demand in response to prompts a), c) and e) in Task 2. However, only three students (Adam, Eve and Caleb) used the correct formula in their attempt to calculate numerical values of the of elasticity of demand in Task 2. Following is an excerpt illustrating how one of these students, Eve, reasoned about the ease and difficulty of working on Task 2.

• Researcher: What was the easiest part when solving this problem? Explain.

• Eve: It isn’t very hard math [to calculate the elasticity of demand] once you figure out the formula [the elasticity of demand formula denoted as |$\frac{p}{q}\ast \frac{dq}{dp}$| in Fig. 1] out.

• Researcher: What was the difficult part when solving this problem? Explain.

• Eve: Remembering the formulas and which meant elastic and inelastic [i.e. deciding how to determine whether demand is elastic or inelastic based on the value of the elasticity of demand].

With the exception of part e) where Eve made a calculational error, she correctly calculated the numerical values of the elasticity of demand in parts a) and c). Eve incorrectly stated that demand would be elastic in part a) instead of inelastic. Additionally, she incorrectly claimed that demand would be inelastic in part c) instead of elastic. Adam and Caleb are two other students who, despite having correctly determined when demand would elastic, inelastic or unit elastic in Task 2, made calculational errors when evaluating the elasticity of demand formula to determine numeric values of the elasticity of demand.

None of the other students correctly determined if demand was elastic, inelastic or unit elastic in Task 2. These students either incorrectly recalled the formula for calculating the elasticity of demand, or they simply did not know the aforementioned formula. For instance, one of the students (Emily) who incorrectly determined the elasticity of demand values in Task 2 only evaluated the demand equation |$q=1000-2{p}^2$| given in Task 2 at the different price values [|$p=10$|⁠, |$p=12.95$| and |$p=15$|] provided in the same task. Surprisingly, a majority of the students either claimed that calculating numerical values of the elasticity of demand or recalling the formula for calculating the elasticity of demand was the easiest part when working on Task 2. One other student (James) noted that finding the derivative of the demand function,|$q=1000-2{p}^2$|⁠, was the easiest part when working on Task 2. Another student (Emily) remarked that evaluating the demand function at the different price values given in Task 2 was the easiest part when working on Task 2. Strangely, the same students who claimed that calculating elasticity of demand was the easiest part when working on Task 2 also either indicated that remembering the formula for calculating the elasticity of demand or determining when demand is elastic, inelastic or unit elastic was the challenging part when prompted to talk about the difficult part when working on Task 2.

6.3 Students’ covariational reasoning in the context of elasticity of demand

All the students in the study engaged in the lowest levels of covariational reasoning in response to prompt a) in Task 3 that we used to elicit students’ ability to engage in covariational reasoning in the context of elasticity of demand. Specifically, three students (Eve, Laura and Ava) engaged in the first three levels of covariational reasoning, namely coordination, direction and quantitative coordination. The following excerpt illustrates how Laura, whose engagement in covariational reasoning in Task 3 is representative of these students, responded to prompt a) in Task 3.

• Researcher: What conclusion can you draw, based on the information presented in the table [Task 3], about the effect of price change on demand and revenue? Explain.

• Laura: The effect of price change as it increases revenue up to 13 then decreases. Demand continues to decrease as price increases.

Laura’s recognition of the quantities of price and demand changing in tandem, as indicated by her claim that, ‘the effect of price change as it increases revenue…’ demonstrates evidence of engaging in the first level of covariational reasoning (i.e. coordination). Her recognition that with changes in price, revenue increases constitutes evidence of engaging in the second level of covariational reasoning (i.e. direction). Lastly, her attention to the increase of revenue as prices increases up to 13 is evidence of engaging in the third level of covariational reasoning (i.e. quantitative coordination).

Seven other students only engaged in the first two levels of covariational reasoning. The following excerpt illustrates how James, who is representative of how these students engaged in the first two levels of covariational reasoning, responded to prompt a) in Task 3.

• Researcher: What conclusion can you draw, based on the information presented in the table [Task 3], about the effect of price change on demand and revenue? Explain.

• James: I can conclude that as the price increases, the demand will decrease. The revenue doesn’t have a direct relationship with either the price or demand in this instance.

James’s recognition of the quantities of price and demand changing simultaneously provides evidence of engaging in the first level of covariational reasoning (i.e. coordination). His attention to the fact that ‘…as price increases, the demand will increase…’ provides evidence of engaging in the second level of covariational reasoning (i.e. direction).

6.4 Students’ ability to show that maximum revue is achieved when demand is unit elastic

None of the students were able to show, algebraically, that maximum revenue occurs when the elasticity of demand equals one (i.e. when demand is unit elastic) in their response to prompt b) in Task 3. Four students (Eve, Ava, Adam and Nancy) stated that they were ‘unsure’ of how they might go about showing that maximum revenue occurs when elasticity of demand equals one. Two other students (Emily and Brian) attempted to use a numerical or graphical approach to illustrate that maximum revenue occurs when elasticity of demand equals one. Figure 2 illustrates Emily’s attempt to use a numerical approach, while Fig. 3 illustrates Brian’s attempt to use a graphical approach to show that maximum revenue occurs when the elasticity of demand equals one, respectively. Emily correctly stated that the quantity of revenue (⁠|$R$|⁠) can be found by multiplying the quantities of price (⁠|$p$|⁠) and demand (q) at the price levels of $10, $13 and $15. She concluded that maximum revenue would occur at a price level of $13, since the elasticity of demand is closest to one at this price level. This is compared to the price levels of $10 and $15 where the values of the elasticity of demand are 0.5 and 1.64, respectively. Brian, on the other hand, created a graph depicting the relationship between the elasticity of demand and revenue based on his understanding of the information shown in the table that appears in Task 3. Brian then concluded that maximum revenue would occur when the elasticity of demand equals one-this is indicated by the point (1, |$x$|⁠) in Fig. 3, where the number 1 represents the value of the elasticity of demand and the variable |$x$| represents an unknown value of the quantity of revenue.

The other students did not make any meaningful progress towards showing that maximum revenue occurs when the elasticity of demand equals one. For instance, Caleb attempted summing all the revenue values provided in the accompanying table in Task 3 and dividing this sum by the count of the revenue values (i.e. six) and got a result of 869 as can be seen in Fig. 4. This student then concluded that this meant that maximum revenue must occur when the elasticity of demand equals one as can be seen in his work that appears in Fig. 4.

Another example of a student who did not make meaningful progress in his attempt to respond to prompt b) in Task 3 is Pat. This student correctly wrote (Fig. 5) that revenue can be calculated by multiplying the quantities of price (⁠|$p$|⁠) and demand (⁠|$q$|⁠)-this is indicated by the equation |$R=q\ast p$| in Fig. 5. This student then went on to calculate the difference between the quantities of revenue and demand at a price level of $12 and determined that the aforenoted difference, which he labelled as the elasticity of demand as indicated by his use of the letter |$E$| in Fig. 5, would be greater than 1. The student calculated another difference between the quantities of revenue and demand; this time at the price level of $14 and determined that this difference would be greater than one. Based on these two calculations, the student concluded that maximum revenue would occur when the elasticity of demand equals one as can be seen in Fig. 5. Pat’s labelling of the difference between the quantities of revenue and demand as elasticity of demand shows that this student confused the elasticity of demand with the meaningless quantity calculated by the student as the difference between the quantities of revenue and demand. Another student (Leah) confused the elasticity of demand with another meaningless quantity, whose units of measure would be dollars per item, when she used the formula |$E=\frac{R-p}{10q}$| where |$E$| is the elasticity of demand, |$R$| is the revenue in dollars, |$p$| is the price in dollars and |$q$| is the demand measured in number of items, to calculate the elasticity of demand.

We remark that nine students indicated that responding to prompt b) in Task 3 was particularly challenging when asked to talk about the difficult part while working on Task 3. Another student (Brian) identified being able to explain why his reasoning [in prompt a) of Task 3] works as the difficult part while working on Task 3. On another note, all the students noted that interpreting or analyzing the information presented in the accompanying table to Task 3 was the easiest part of working on the task. In fact, two students (Caleb and Nancy) went on to remark on using either their statistical or economics background to their advantage when responding to prompt a) in Task 3.

7 Discussion and conclusions

This study investigated business calculus students’ covariational reasoning, procedural knowledge and conceptual knowledge in the context of the economic concept of elasticity of demand. Following is a discussion of the key findings of the study. First, none of the students were able to provide explanations that demonstrate correct conceptual knowledge of what it means for demand to be elastic, inelastic or unit elastic, respectively. To reiterate, correct conceptual knowledge of what it means for demand to be elastic entails remarking on the fact that a 1% increase in the price of a product would cause a more than 1% decrease in the demand of the product. On a positive note, half of the students at one point provided responses that demonstrated correct procedural knowledge of what it means for demand to be elastic, inelastic, or unit elastic, respectively. Specifically, these students correctly indicated that demand would be elastic, inelastic, or unit elastic when the values of the elasticity of demand is greater than one, less than one or equal one, respectively. We recommend that business calculus instructors provide more opportunities (e.g. assigning graded problems that require calculating and interpreting elasticity of demand in the context of inelastic, unit or elastic demand, respectively), both during course lectures and in formal assessments such as in exams. This would enable students to make sense of the concept of elasticity of demand, which is clearly not well understood by the participants of this study. We argue that students’ lack of a good understanding of the concept of elasticity of demand may be widespread and not just limited to the participants of this study. This is because definitions of concepts typically do not receive much attention in students’ assessment of mathematical ideas. This is especially true in calculus courses in the United States that are generally taught using a traditional lecture format.

Second, all the students engaged at some level of covariational reasoning while working on Task 3. However, the students’ covariational reasoning was limited, to at most, the third level of covariational reasoning, i.e., the lower levels of covariational reasoning, namely coordination, direction and quantitative coordination. That is, none of the students engaged in the higher levels of covariational reasoning, namely average rate and instantaneous rate. This finding corroborates results previously reported by other studies that have examined students’ covariational reasoning when solving application problems in mathematics (cf. Carlson et al., 2002; Jones, 2017; Mkhatshwa, 2023; Nagle et al., 2013). It is worth noting that students’ failure to engage in the aforementioned higher levels of covariational reasoning may have been influenced by the fact that it is rather typical in economics to talk about rate of change as if it were an amount of change (cf. Feudel & Biehler, 2022). Findings by Carlson et al. (2003) suggest that using a models and modelling perspective (Lesh & Doerr, 2003; Lesh & Zawojewski, 2007) in mathematics could help students develop robust covariational reasoning skills. Thus, and in an effort to help students develop strong covariational reasoning skills, we recommend the integration of a models and a modelling perspective, among other teaching approaches, in the teaching of calculus at the undergraduate level, including in the teaching of business calculus. We remark that the business calculus course taken by the students in this study was taught using a traditional lecture format and thus did not incorporate the use of a models and modelling perspective. Furthermore, we call for future research to provide empirical evidence regarding the effectiveness (or lack thereof) of a models and modelling perspective in enhancing students’ covariational reasoning abilities.

Third, much research has reported on students’ propensity to confuse quantities when solving application problems in mathematics, including in business calculus. Specifically, students have a predisposition to confuse amount quantities with other amount quantities, amount quantities with rate quantities and rate quantities with other rate quantities (cf. Rasmussen & Marrongelle, 2006; Lobato et al., 2012; Prince et al., 2012; Flynn et al., 2014, 2018; Mkhatshwa & Doerr, 2018; Mkhatshwa, 2019, 2020, 2023). In this study, two students confused the concept of elasticity of demand with meaningless quantities. One student, for instance, incorrectly defined elasticity of demand as the meaningless quantity represented by the difference between the economic quantities of revenue and demand. To reiterate, elasticity of demand is an economic concept or idea and not a quantity in the sense that it does not have a unit of measure. We recommend that business calculus instructors provide more opportunities (e.g. a small graded research project where students are tasked with comparing and contrasting how the economic concept of elasticity of demand is defined and used in mathematics and economics textbooks) geared towards helping students not only make sense of the concept of elasticity of demand, but also how this economic idea/concept is related to the economic quantities of price, revenue and demand of a product, respectively.

Fourth, all the students were able to make sense of the information provided in the table provided in Task 3. Specifically, these students were able to keep track of how the quantity of revenue changes with changes in the elasticity of demand. Ultimately, most of the students correctly determined that maximum revenue would occur when the numerical value of the elasticity of demand is either one or very close to one. Disappointingly, none of the students were able to translate their understanding of the information presented in the table to algebra. In particular, none of the students were able to show, analytically, that maximum revenue occurs when the elasticity of demand equals one. In as much as helping students develop correct conceptual knowledge of economic ideas such as the concept of elasticity of demand is important, it is equally important to support them to hone their procedural knowledge of these ideas as well. Thus, and using the concept of the elasticity of demand investigated in this study as an example, we recommend that calculus instructors provide a balanced mix of opportunities for students to reason about economic ideas (that are covered in mathematics service courses) conceptually and procedurally.

Fifth, a number of studies have reported on students’ difficulties in connection with using different rules of differentiation when solving application problems in calculus, namely the product, chain and quotient rule, respectively (cf. Clark et al., 1997; Maharaj, 2013; Maharaj & Ntuli, 2018; Jeppson, 2019; Mkhatshwa, 2020). On the bright side, a few studies have found that using the power rule of differentiation to calculate derivatives of polynomial functions is often straightforward for students (cf. Orton, 1983; Mkhatshwa, 2023). Consistent with these findings, all the students in this study exhibited greater facility with using the power rule of differentiation to find the derivative of the demand function |$q=1000-2{p}^2$| in Task 2 in their attempt to calculate numerical values of the elasticity of demand in the same task.

In conclusion, this study provides valuable insight into students’ thinking about the economic concept of elasticity of demand. This is something we hope would help guide business or economics instructors in their teaching of this and other related concepts in their courses. One limitation of the study worth pointing out is that the study participants were all recruited from the same institution, and as such, the results reported in this study may or may not generalize to other settings. To some extent, we do not consider this to be a major limitation of the study as the overarching goal of this study was to explore students’ thinking about the concept of elasticity of demand, and not so much on generalizing the findings to other settings.

We posit that the concept of elasticity of demand has not received much attention in business calculus. On a related note, it is worth remarking that elasticity of demand is not only a concept that business calculus or economics students find difficult to understand (as was observed with the participants in this study), but also one that even principles of economics instructors find challenging to introduce during classroom instruction (cf. Andrews & Benzing, 2010; Dacass & Dilden, 2023). Even worse, Andrews & Benzing (2010) argued that principles of economics textbooks often present the concept of elasticity of demand in a way that confuses students:

Principles [of economics] textbooks have tended to confuse students with respect to price elasticity of demand in three ways. First, textbook authors often use graphs that lead students to confuse slope with the price elasticity of demand. Second, textbook authors over-rely on specific factors in determining price elasticity of demand. Lastly, some authors include tables of outdated price elasticity numbers that could lead students to believe that price elasticity of demand is constant over time regardless of changes in many significant factors. (p.2).

More research on students’ thinking about the concept of elasticity of demand and economic quantities such as the economic order quantity or points of diminishing returns is needed to better understand business calculus or economics students’ learning about calculus ideas that are covered in service mathematics courses that are intended for economics or business students. Additionally, and in light of Andrews & Benzing’s (2010) argument regarding the poor presentation of the concept of elasticity of demand in principles of economics textbooks, we recommend that future research examine the presentation of this concept in business calculus textbooks.

Disclosure statement

The author declares that there is no conflict of interest.

Acknowledgements

I would like to express my sincere gratitude to the anonymous reviewers whose thoughtful and insightful feedback has greatly improved the quality of the manuscript. Additionally, I am very grateful for the constructive feedback I received from participants at the 1^st International Network for Educational Research on Mathematics in Economics (INERME) Conference that was held at the University of Agder, Norway.

References

Appendix A. Carlson et al.’s (2002) Mental Actions

Appendix B. Solution to Task 1

Task 1

• a) What does the term, elasticity of demand, mean to you? Explain.

• b) What does it mean to say that demand is elastic? Explain.

• c) What does it mean to say that demand is inelastic? Explain.

• d) What does it mean to say that demand is unit elastic? Explain.

Solution.

The sensitivity of demand to changes in price varies with the product. For example, a change in the price of light bulbs may not affect the demand for light bulbs much, because people need light bulbs no matter what their price. However, a change in the price of a particular make of car may have a significant effect on the demand for that car, because people can switch to another make…Elasticity of demand [typically denoted by the letter |$E$| or the Greek symbol |$\eta$|] is a measure of the sensitivity of demand to changes in price. (Hughes-Hallett et al., 2018, p. 208)

• a) Demand is elastic when |$\eta >1$|⁠. This means that a 1% increase in the price of a product will cause the demand of the product to decrease by more than 1%.

• b) Demand is inelastic when |$\eta <1$|⁠. This means that a 1% increase in the price of a product will cause the demand of the product to decrease by less than 1%.

• c) Demand is elastic when |$\eta =1$|⁠. This means that a 1% increase in the price of a product will cause the demand of the product to decrease by 1%.

Appendix C. Solution to Task 2

Task 2 (Adapted from Hughes-Hallett et al., 2018)

The demand curve for a product is given by |$q=1000-2{p}^2$|⁠, where |$p$| is the price in dollars.

• a) Find the elasticity of demand at |$p=10$|⁠. Round your answer to one decimal place.

• b) Interpret you answer from part a) in the context of price and demand.

• c) Find the elasticity of demand at |$p=15$|⁠. Round your answer to one decimal place.

• d) Interpret you answer from part c) in the context of price and demand.

• e) Find the elasticity of demand at |$p=12.95$|⁠. Round your answer to one decimal place.

• f) Interpret your answer from part e) in the context of price and demand.

Solution

• a)

  |$\eta =\mid \frac{p}{q}\ast \frac{dq}{dp}\mid =\mid \frac{p}{1000-2{p}^2}\ast \left(-4p\right)\mid$|⁠. Thus, when |$p=10$|⁠, |$\eta =0.5.$|

• b)

  Since |$\eta <1$|⁠, this means that demand is inelastic. In other words, at a price |$p=\$10$|⁠, a 1% increase in the price of the product results in approximately a 0.5% decrease in the demand of the product.

• c)

  |$\eta =\mid \frac{p}{q}\ast \frac{dq}{dp}\mid =\mid \frac{p}{1000-2{p}^2}\ast \left(-4p\right)\mid$|⁠. Thus, when |$p=15$|⁠, |$\eta =1.6.$|

• d)

  Since |$\eta >1$|⁠, this means that demand is elastic. In other words, at a price |$p=\$15$|⁠, a 1% increase in the price of the product results in approximately a 1.6% decrease in the demand of the product.

• e)

  |$\eta =\mid \frac{p}{q}\ast \frac{dq}{dp}\mid =\mid \frac{p}{1000-2{p}^2}\ast \left(-4p\right)\mid$|⁠. Thus, when |$p=12.95$|⁠, |$\eta =1.0$|⁠.

• f)

  Since |$\eta =1$|⁠, this means that demand is unit elastic. In other words, at a price |$p=\$12.95$|⁠, a 1% increase in the price of the product results in approximately a 1.0% decrease in the demand of the product.

Appendix D. Solution to Task 3

Task 3 (Adapted from Hughes-Hallett et al., 2018) The table shows the price, |$p$|⁠, demand, |$q$|⁠, revenue, |$R$|⁠, and elasticity, |$E,$| for a product at several prices.

• a) What conclusion can you draw, based on the information presented in the table, about the effect of price change on demand and revenue? Explain.

• b) The information presented in the table suggests that maximum revenue occurs when elasticity is close to 1. Show analytically (i.e. using algebra and calculus) that maximum revenue occurs when |$E=1$|⁠.

Solution 

• a)

  The information presented in the table suggests that maximum revenue is achieved at a price of about $13, and at that price, |$E$| is approximately 1. At prices below $13, we have |$E<1$|⁠, so the reduction in demand caused by a price increase is small; thus, raising the price increases revenue. At prices above $13, we have |$E>1$|⁠, so the increase in demand caused by a price decrease is relatively large; thus, lowering the price increases revenue.

• b)
  We think of revenue as a function of price. Using the product rule of differentiation to differentiate |$R= pq$|⁠, we have

  $$ \frac{dR}{dp}=\frac{d}{dp}(pq)=p\frac{dq}{dp}+\frac{dp}{dp}q=p\frac{dq}{dp}+q $$

  At a critical point the derivative |$\frac{dR}{dp}$| equals zero, so we have

  $$ p\frac{dq}{dp}+q=0 $$

  $$ p\frac{dq}{dp}=-q $$

  $$ \frac{p}{q}\frac{dq}{dp}=-1 $$

  Thus, |$E=1$| since |$E=\mid \frac{p}{q}\ast \frac{dq}{dp}\mid$|⁠.

Appendix E. Author’s Biography

Thembinkosi Mkhatshwa is an Associate Professor of Mathematics at Miami University-Middletown, OH, USA. He teaches mathematics and statistics courses. His research spans three areas of study, namely students’ thinking about fundamental ideas in calculus or ordinary differential equations, experts’ views on effective instructional approaches in the teaching of fundamental ideas in undergraduate mathematics and opportunities to learn provided by undergraduate mathematics textbooks.