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Ida Landgärds-Tarvoll, Daniel Göller, Bridging mathematical and microeconomic perspectives: a praxeological analysis of the Lagrange multiplier method, Teaching Mathematics and its Applications: An International Journal of the IMA, Volume 43, Issue 4, December 2024, Pages 315–338, https://doi.org/10.1093/teamat/hrae020
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Abstract
This article investigates the challenges that economics students face when they make the transition from service mathematics course (s) to microeconomics courses with a focus on how the concept of the Lagrange multiplier method is used in constrained utility maximization problems. The study aims to identify and understand discrepancies in the application of the Lagrange multiplier method as introduced in the mathematics course and subsequently applied within the microeconomics course. For this purpose, we conducted a comparative praxeological analysis of textbooks used in the two courses. The analysis reveals significant differences in the techniques and technologies used in both textbooks, which may cause difficulties for students in their transition between the courses. Additionally, the analysis revealed total mismatches in praxeologies within the textbooks. The article concludes with suggestions on topics for service mathematics teachers to address for aligning their instruction with the microeconomics application of the Lagrange multiplier method.
1 Introduction
Calculus plays an important part in the economics discipline in higher education and has a key role in developing understanding of key economic principles and concepts. Multivariable functions have significant applications in foundational economics modules such as microeconomics, and therefore, analysis of such functions is crucial in the first-year undergraduate mathematics course(s) for economics education (Voßkamp, 2023). However, the literature shows that many students struggle to make sense of mathematics learned in their service mathematics course in other economics courses (Ariza et al., 2015; Mkhatshwa & Doerr, 2015; Feudel & Biehler, 2020, 2021). Discrepancies between how mathematical concepts are understood and taught in mathematics compared to the way they are used and understood in other economics courses cause learning problems for many students (Alpers, 2020). Therefore, Biza et al. (2022) call for research concerning the intersection of mathematics and other disciplines courses. They write: ‘We would like to encourage researchers to consider this intersection and to question content and approaches in calculus courses, as well as implicit assumptions about their role in teaching’ (Biza et al., 2022, p. 220).
As authors and lecturers of the mathematics-for-economists and microeconomics courses at the University of Agder (UiA), we have identified a significant discrepancy in the communication of mathematical concepts between the mathematics and economics departments. This situation mirrors findings from Willcox & Bounova (2004) in engineering education, where faculty members lacked knowledge of how mathematics is applied in adjacent fields. Inspired by these insights and Rønning’s (2022) research, which emphasizes the need for in-depth communication between the disciplines, we initiated discussions to enhance the alignment of our courses at UiA’s business school. We realized we were missing understanding of each other’s disciplines’ approaches to derivatives, integrals and the Lagrange multiplier method (Landgärds, 2023). Additionally, we noted from the frequently asked questions during microeconomics lectures that many students faced difficulties applying the Lagrange multiplier method in microeconomics, despite having learned it in the mathematics course the preceding semester.
Educational research on mathematics in economics is limited and whereas Mkhatshwa & Doerr (2018) and Feudel & Biehler (2021) investigate the derivative concept, there is little research on how the concept of the Lagrange multiplier method is taught and applied. The concept of Lagrange multiplier method is standard content of service mathematics courses for economics (Voßkamp, 2023) and is crucial in the microeconomics theory on consumer utility maximization (Perloff, 2022). Given the importance of this concept, the broader international challenge of students struggling to apply mathematical concepts from service mathematics courses to their economics studies and influenced by specific observations at UiA, this article reports on a comparative analysis of the textbooks used in the two courses at UiA. The study aims to identify and understand discrepancies in how the Lagrange multiplier method is introduced in the mathematics-for-economists course and then applied in the microeconomics course. We believe such knowledge is essential for practitioners to provide pertinent mathematical education tailored for economics students to support them in their transition from the service mathematics to the microeconomics course.
For this purpose, we take an institutional perspective and draw on the framework of Anthropological Theory of the Didactic (ATD) (Chevallard, 2006, 2019) to investigate how the practices (the praxeologies) in the two courses compare. The study is guided by the following research question:
How do the mathematical praxeologies compare between the mathematics-for-economists’ textbook and the microeconomics textbook?
2 The history of the mathematization of the economics discipline
Unlike other service mathematics fields such as engineering and natural sciences, where the symbiotic relationship between mathematics and the application field has driven advancements in both fields with each discipline enriching the other over centuries (Dugger Jr, 1993; Pospiech, 2019; Hafni et al., 2020; Pepin et al., 2021), the field of economics was gradually mathematized starting in the 19th century (Hodgson, 2013). The integration of mathematical tools, techniques and concepts into economics has been driven by the desire to provide more rigour, precision and clarity to economic theories and to better understand and predict economic phenomena (Tarasov, 2019). An example of the mathematization of economics is the introduction of the concept of the Lagrange multiplier method into the field during the 1880s. Creedy (1980) describes how many of the past century’s most innovative economists, who were known to endorse the use of mathematics, and particularly the concept of the Lagrange multiplier method, might not have fully understood the intricacies of the method. Furthermore, Creedy (1980) elucidates how Edgeworth began incorporating mathematical techniques from other scientific disciplines into economics from 1881. Creedy writes about Edgeworth’s work on the use of Lagrangian multipliers (p. 375):
He was not simply translating existing results into a succinct notation, but made genuinely original and lasting contributions to economic theory. It is not surprising that Edgeworth’s early work was not widely appreciated; indeed, there must have been very few economists who were capable of understanding his discussion, and Edgeworth was the first to admit that the theory, ‘could be presented by a professed mathematician more elegantly and scientifically’ (1881, p. 24).
It was not until the 1920s that the work of Edgeworth was understood and further developed by economists with strong mathematical abilities (Creedy, 1980). However, the role of mathematics in economics has always been and is still a matter of debate. Hodgson (2013, p. 36) writes: ‘Mathematics plays a useful and sometimes vital complementary role, aiding conceptual clarification and providing thought-stretching heuristics. But historically much important theory in economics is verbal.’ Therefore, today both economists and students of economics recognize mathematical knowledge as important. Nonetheless, the field of economics adopts a distinctive approach to utilizing it. Consequently, mathematical concepts are being ‘reshaped’ to better align with their application in economics. For students, it is crucial to understand the connection between the use of the concepts in mathematics and economics to effectively comprehend and engage with economic texts, models, and theories. However, as we discussed above, the literature suggests that students struggle with this crucial transition (Ariza et al., 2015; Feudel, 2018; Mkhatshwa & Doerr, 2018; Feudel & Biehler, 2021).
3 Embedding the research
The transition from studying service mathematics to applying the mathematical concepts in the economics courses is a multifaceted transition and in particular the issue of relevance is addressed in the literature (Landgärds-Tarvoll, 2024). Similar to the case of engineering education, the transition from service mathematics to application courses in economics encompasses the problem of students not seeing the relevance of the mathematics taught (e.g. Flegg et al., 2012; Faulkner et al., 2019, 2020), the relevance issues of students not managing to apply the mathematics in their engineering courses (e.g. Harris et al., 2015; Faulkner et al., 2019) and the relevance issue of mathematical concept and notation discrepancies (González-Martín, 2018; Hochmuth & Peters, 2021; Peters & Hochmuth, 2021). Alpers (2017, p. 1) writes ‘In order to recognize potential cognitive barriers, the use of mathematics in application subjects must be investigated and compared with the treatment provided in mathematics education.’ In particular, it is crucial for mathematics instructors to understand how the concepts they teach are applied in application courses to help students transition from the former to the latter (González-Martín, 2018; Peters & Hochmuth, 2021; Rønning, 2022). While the fields of economics and engineering education differ in knowledge to be taught and course structure, they face similar challenges in terms of the relevance of the educational content. This is why we found it relevant to draw on methodologies and results from the engineering discipline for investigating the concept of the Lagrange multiplier method in the economics education.
As outlined in the introduction, the research in this article addresses the institutional perspective and concerns the knowledge to be taught, that is, we address the relevance issue of mathematical concepts in economics courses. Most research concerning application of mathematical concepts concerns the field of engineering (Hochmuth, 2020). Alpers (2017) focuses on the discrepancy in notation that often exists between mathematics and engineering by elaborating on the example of vectors. Additionally, he points out that engineers often employ shortcuts in mathematics by relying on assumptions. González-Martín (2021) explores how integrals are used and taught in engineering courses and service mathematics calculus courses, respectively. He points to a mismatch between the technical skills taught in the calculus course and the more conceptual grasp and only basic mathematical calculations of the integrals which are needed in the engineering courses. Similar curricular results were found by Hitier & González-Martín (2022). They found that textbooks of physics kinematics mostly require students to use ‘ready-to-use equations.’ They noticed a shift toward algebraic methods and task: after introducing the concept of motion in kinematics, the application of derivatives disappears. Hochmuth & Peters (2021) and Peters & Hochmuth (2021) investigate how the institutional discourse of mathematics for engineers relates to the discourse of electrical engineering. In particular, Peters & Hochmuth (2021) identify ways in which the two techniques and discourses are intertwined utilizing the ‘extended praxeological model’. Based on the analysis of engineering problems, they suggest different forms of embedding the mathematics discourse in the mathematical discourse of electrical engineering. Similarly, Rønning (2022) investigates the interplay between the engineering approach and the mathematics approach needed to solve problems in engineering context. He highlights the need for deep knowledge in both fields to master the interplay.
Within the sparse body of educational research at the intersection of economics and mathematics, certain investigations have delved into the concept of the derivative. An early study on students’ understanding of the economic interpretation of the derivative was made by Mkhatshwa & Doerr (2015), who find that students reason about marginal change as an amount of change (a difference) rather than a rate of change (in economics a rate of change over a subinterval of unit length). Ariza et al. (2015) investigate the relationship between a function and its derivative, which is essential in marginal analysis in microeconomics. They explain that in economics students need to study derivatives through different systems of representations (both algebraically and graphically) and students’ need for an ability to convert functions from graphical to algebraic form and vice versa. Through textbook analysis, Feudel (2019) investigates what knowledge of derivatives economics students need in their microeconomics and business administration courses. He concludes that students need a thorough understanding of the economic interpretation and its connection to the mathematical concept via linear approximation, the geometric representation, monotonicity/convexity understanding, optimization and the differentiation rules. Therefore, students need a deep understanding of derivatives beyond just computational methods (Feudel, 2019). Feudel & Biehler (2021) explore students’ economic understanding of derivatives in the context of marginal cost. Feudel (2019a) develops a framework that explains the link between the mathematical and the economical concept of the derivative. This framework was used by Feudel & Biehler (2021) for examining students’ exam answers. They discovered that students have difficulties connecting the mathematical and the economic aspects of derivatives and often miss the linear approximation method used in economics.
A thorough understanding of derivatives is a prerequisite for the topic of optimization. Of particular interest in microeconomics is the concept of the Lagrange multiplier method (Voßkamp, 2023). The Lagrange multiplier method provides a strategy for converting constrained optimization problems (finding the maximum or minimum of a function of several variables subject to equality constraints) into unconstrained optimization problems, by introducing a new variable
4 Theoretical perspective
Our aim is to identify and understand discrepancies in the application of the Lagrange multiplier method as introduced in the mathematics course and subsequently applied within the microeconomics course. For this purpose, our research draws on the Anthropological Theory of the Didactic (ATD) (Chevallard, 2006, 2019). Our primary rationale for using this theoretical framework is that it provides tools for analyzing the dynamic interactions between human practices and institutions. It particularly focuses on how human practices are shaped and restricted by institutions through the relations to practices that institutions either mandate or promote. Accordingly, also knowledge about these practises is institutionally situated (Castela, 2017). In ATD, an institution is anything instituted such as a class or a school or a family etc. (Chevallard & Bosch, 2020). The knowledge to be taught in an institution can be accessed through didactic materials such as: official programs, textbooks, recommendations to teachers etc. (Bosch & Gascón, 2014).
The ATD framework models human knowledge and practice as institutional praxeologies. Specifically, our analysis focuses on the mathematical praxeologies in the mathematics and microeconomics textbooks to identify how the concept of Lagrange multiplier method is applied in each of the two institutions. According to the notion of praxeology, an activity can be dissected into its elementary components, which are known as tasks. Defined by the quadruplet
We take inspiration from the second part of Klein’s double discontinuity as framed by Winsløw & Grønbæk (2014) within the ATD framework to elaborate on our research perspective taken in this article. The second part of Klein’s double discontinuity consider the institutional transition student teachers encounter when trying to apply their university mathematics studies as teachers in schools. While the economics students do not undertake a new institutional position such as the student teachers, the transition between the mathematical objects is similar. The notation
Hence, examining the kinds of tasks, techniques and technologies used in the mathematics and the microeconomics textbooks is crucial for understanding the difficulties that economics students might meet when transitioning from the mathematics to the microeconomics course. In this regard, we have taken an institutional perspective focusing on the knowledge to be taught to explore how the praxeologies concerning the concept of Lagrange multiplier method compare in the two courses. Castela (2017) also highlights that such praxeological analysis is a pertinent instrument for tackling the challenge of selecting the most suitable mathematics curriculum for application-oriented programs.
5 Methodology
Bosch & Gascón (2014) highlight that an ATD analysis always starts by approaching institutional praxeologies and then referring individual behaviour to them. As explained earlier, in this article, we restrict the focus to the first step, which is of synchronic praxeological analysis as described by Strømskag & Chevallard (2024). This involves investigating and comparing the institutional praxeologies of the two courses. Institutional praxeologies can be considered on different granularities: ‘a distinction is made between a point praxeology (containing a single type of task), a local praxeology (containing a set of types of tasks organized around a common technological discourse) and a regional praxeology (which contains all point and local praxeologies sharing a common theory)’ (Bosch & Gascón, 2014, p. 69). Following González-Martín (2021), Hitier & González-Martín (2022) and Rønning (2022) who did similar research on concept discrepancies in the engineering and physics field (concerning the concepts of integrals, derivatives and differential equations, respectively) and Strømskag & Chevallard (2024) on concave/convex functions we view the two courses as institutions. According to Bosch & Gascón (2014), the knowledge to be taught in these institutions can be accessed through didactic materials such as official programs, course textbooks and recommendations to teachers etc.
Therefore, our data consists of textbooks of both mathematics and microeconomics. Textbooks are generally acknowledged as essential resources in mathematics courses (Remillard, 2005; Hadar, 2017; Mkhatshwa, 2022) and economics courses alike (Feudel, 2019). The importance of the textbooks stems from their capacity to offer an organized framework of concepts, facilitate the teaching and learning process and enable critical thinking and comprehension of the subject matter (Hadar, 2017) for both teachers and students (Remillard, 2005). Therefore, textbooks have a significant impact on the implemented curriculum and in turn, influence students’ learning opportunities (Törnroos, 2005).
5.1. Study context
Our study is conducted at the University of Agder in Norway where both authors work. The considered programme is the bachelor’s programme in Business Administration, which from an international perspective is a hybrid of business studies and economics. The programme is scheduled over 3 years, each year divided into two semesters. In the first year’s second semester, students are required to take the mathematics course ‘Mathematics Applied in Business Administration’1, which ideally should provide students with the mathematical foundation for studies in economics and business administration. The course is taught in Norwegian and covers a wide range of topics from algebra and calculus. The primary reference for the course’s curriculum is the (2019) textbook ‘Matematikk for økonomistudenter’ by Dovland and Pettersen. The teacher (the first author of this article) derives both the content and teaching approach from this central resource. It is noteworthy that this book is the preferred choice in a majority2 of Business schools in Norway.
In the first semester of the students’ second year, hence the semester after having completed their service mathematics course, students take a microeconomics3 course taught by the second author of this article based on the (2022) international textbook ‘Microeconomics: Theory and Applications with calculus 5th edition’ by Perloff.
We recognize that our research is a case study focussed on two textbooks utilized at a single Norwegian university. However, these textbooks adhere to standard approaches in presenting constrained utility maximization within the two fields. Moreover, given the prevalence of similar courses internationally (Voßkamp, 2023), our findings should be of interest to a wider audience.
Furthermore, we consider it a strength of our study that the authors have expertise in the two distinct disciplines where the investigated concept is applied, enhancing both the utilization and interpretation of the textbooks. Interdisciplinary research collaboration for aligning curricula is emphasized as significant for improving students learning opportunities (Biza et al., 2022; Hitier & González-Martín, 2022). Our approach is in line with Castela (2017) who writes about researching concept discrepancies in mathematics and application studies (p. 7):
Mathematics researchers and lecturers are too often not aware of the necessity and complexity of such an investigation; they are not necessarily prepared for it by their mathematics education. This should be accomplished collectively with researchers and professionals of the domains using the mathematics at stake in the program.
5.2. Analysis
We conducted a five-step analysis inspired by Bittar’s (2022) methodological model for analyzing mathematical praxeologies in textbooks. The model comprises the following steps: 1) Selection of textbooks and sections to be analyzed; 2) Modellation of mathematical praxeologies present in the course part of the selected sections. Modellation here means that the researcher interprets the textbook content in terms of praxeologies; 3) Mathematical analysis and modellation of the proposed activities in the books as well as the solution manual; 4) Modellation of how mathematical praxeologies are presented within the textbooks, encompassing both the course part and the proposed activities; 5) Data triangulation. In particular, step 2–4 considers the different levels of granularities as described by Bosch & Gascón (2014). The second and the third step involved identifying all point praxeologies of the textbook while the fourth step involved grouping point praxeologies with a common technological discourse into local praxeologies which served as reference praxeologies for the comparison of the textbooks praxeologies. Lastly, the reference mathematical praxeologies found in the two textbooks were compared in accordance with our research question. It is worth noting that the steps of the model are not necessarily sequential.
5.3. Selection of textbooks and sections
Considering the first step in our study, we selected sections that introduce constrained maximization using a graphical approach in both the mathematics and the microeconomics textbooks. We also focus on the sections that discuss the Lagrange multiplier method using calculus in both texts. These sections are relevant for our study as they represent the sections between which students need to do the transition, that is, establish the relation
Matematikk for økonomistudenter by Dovland & Pettersen (2019). Pages: 507–519, 528 and the corresponding solution manual.
Microeconomics: Theory and Applications with Calculus by (Perloff, 2022), Pages: 86–116, 126–127 and the corresponding solution manual.
5.4. Praxeological analysis
The second step in our analysis is modellation of the mathematical praxeologies present in the course part (information, examples, solved problems, proofs, etc.) of the books (Bittar, 2022). The course part of the books provides opportunities for students to familiarize themselves with task, techniques, and rationales behind the techniques, while also providing opportunities for educators to introduce these in their teaching (Thompson et al., 2012; Bittar, 2022). By investigating both the course part and the proposed activities, a more comprehensive understanding of potential learning opportunities provided by the textbooks could be achieved compared to analyzing only one of them in isolation (cf. Thompson et al., 2012; González-Martín, 2021; Bittar, 2022). Therefore, as a third step, we examined the proposed exercises in each of the books. The mathematical praxeologies established in the previous step served as support for the analysis of the exercises and the solutions present in the solution manuals.
Each task, along with its associated technique and technology, was documented, leading to the discovery of 34 mathematical point praxeologies within the mathematics textbook and 15 mathematical point praxeologies within the microeconomics textbook. The microeconomics book features fewer point praxeologies because it presents activities as sample tasks that students can repeatedly practice with varying numbers via the accompanying digital resource. The fourth step analysis resulted in that five distinct local praxeologies were established for the mathematics book and three for the microeconomics book.
The two authors collaborated on the analysis. The first author did the coding and the detailed analysis. The second author reviewed the entire work, made improvements, and discussed the questionable parts with the first author. The distinct backgrounds of the authors in the disciplines under consideration enriched the interpretation of the various techniques and technologies.
6 Results
This section discusses the results. First, the result from the praxeological analysis of the mathematics book is presented in Section 6.1, followed by the presentation of the analysis of the microeconomics textbook in Section 6.2. The result of the praxeological analysis of each book is summarized in Table 1 and Table 2, respectively. These results are compared in Section 6.3 to answer the research question which was outlined in Section 1.
Task ( | Technique ( | Technology ( |
---|---|---|
Maximize and/or minimize | 1. Plot curves of | 1. Analogy of a landscape ( |
Maximize and/or minimize | 1. Define $\left\{ 4. Compare values: - Interior points from step 3 - Boundary points | 1. Utilizes graphical analysis from |
Give an approximation of the increase/decrease in maximum/minimum value of the objective function from a change in | 1. Perform steps 1 to 3 from | 1. The solution obtained by Lagrange’s method depends on the value |
Investigate the existence of max/min for | 1. Check whether the constraint set defined by | 1. The technologies |
Find the slope of the level curve at some point | 1. Calculate the partial derivatives | 1. The partial derivatives show how the function |
Task ( | Technique ( | Technology ( |
---|---|---|
Maximize and/or minimize | 1. Plot curves of | 1. Analogy of a landscape ( |
Maximize and/or minimize | 1. Define $\left\{ 4. Compare values: - Interior points from step 3 - Boundary points | 1. Utilizes graphical analysis from |
Give an approximation of the increase/decrease in maximum/minimum value of the objective function from a change in | 1. Perform steps 1 to 3 from | 1. The solution obtained by Lagrange’s method depends on the value |
Investigate the existence of max/min for | 1. Check whether the constraint set defined by | 1. The technologies |
Find the slope of the level curve at some point | 1. Calculate the partial derivatives | 1. The partial derivatives show how the function |
Task ( | Technique ( | Technology ( |
---|---|---|
Maximize and/or minimize | 1. Plot curves of | 1. Analogy of a landscape ( |
Maximize and/or minimize | 1. Define $\left\{ 4. Compare values: - Interior points from step 3 - Boundary points | 1. Utilizes graphical analysis from |
Give an approximation of the increase/decrease in maximum/minimum value of the objective function from a change in | 1. Perform steps 1 to 3 from | 1. The solution obtained by Lagrange’s method depends on the value |
Investigate the existence of max/min for | 1. Check whether the constraint set defined by | 1. The technologies |
Find the slope of the level curve at some point | 1. Calculate the partial derivatives | 1. The partial derivatives show how the function |
Task ( | Technique ( | Technology ( |
---|---|---|
Maximize and/or minimize | 1. Plot curves of | 1. Analogy of a landscape ( |
Maximize and/or minimize | 1. Define $\left\{ 4. Compare values: - Interior points from step 3 - Boundary points | 1. Utilizes graphical analysis from |
Give an approximation of the increase/decrease in maximum/minimum value of the objective function from a change in | 1. Perform steps 1 to 3 from | 1. The solution obtained by Lagrange’s method depends on the value |
Investigate the existence of max/min for | 1. Check whether the constraint set defined by | 1. The technologies |
Find the slope of the level curve at some point | 1. Calculate the partial derivatives | 1. The partial derivatives show how the function |
Task ( | Technique ( | Technology ( | |
---|---|---|---|
Exemplary tasks outlined in Section 6.2 | 1. Draw the budget line | Assumption of the model (A1)–(A4) 1. As more is preferred to less a consumer (weakly) prefers bundle ( | |
1. Set up the function | |||
1. If the IC are strictly convex and differentiable, calculate the slope of the IC, |
Task ( | Technique ( | Technology ( | |
---|---|---|---|
Exemplary tasks outlined in Section 6.2 | 1. Draw the budget line | Assumption of the model (A1)–(A4) 1. As more is preferred to less a consumer (weakly) prefers bundle ( | |
1. Set up the function | |||
1. If the IC are strictly convex and differentiable, calculate the slope of the IC, |
(Continued)
Task ( | Technique ( | Technology ( | |
---|---|---|---|
Exemplary tasks outlined in Section 6.2 | 1. Draw the budget line | Assumption of the model (A1)–(A4) 1. As more is preferred to less a consumer (weakly) prefers bundle ( | |
1. Set up the function | |||
1. If the IC are strictly convex and differentiable, calculate the slope of the IC, |
Task ( | Technique ( | Technology ( | |
---|---|---|---|
Exemplary tasks outlined in Section 6.2 | 1. Draw the budget line | Assumption of the model (A1)–(A4) 1. As more is preferred to less a consumer (weakly) prefers bundle ( | |
1. Set up the function | |||
1. If the IC are strictly convex and differentiable, calculate the slope of the IC, |
(Continued)
Task ( | Technique ( | Technology ( | |
---|---|---|---|
Exemplary tasks: ‘Spenser has a quasilinear utility function | 1. Perform 2. Check if the bundle obtained is non-negative. If yes, it is the optimal bundle. If not, the optimal bundle is a corner solution: | ||
Derive the exponents of the Cobb Douglas utility function Exemplary task: Suppose that a consumer has a Cobb–Douglas utility function and buys two goods, | 1. Calculate the budget shares | 1. The MRS = MRT condition implies |
Task ( | Technique ( | Technology ( | |
---|---|---|---|
Exemplary tasks: ‘Spenser has a quasilinear utility function | 1. Perform 2. Check if the bundle obtained is non-negative. If yes, it is the optimal bundle. If not, the optimal bundle is a corner solution: | ||
Derive the exponents of the Cobb Douglas utility function Exemplary task: Suppose that a consumer has a Cobb–Douglas utility function and buys two goods, | 1. Calculate the budget shares | 1. The MRS = MRT condition implies |
Task ( | Technique ( | Technology ( | |
---|---|---|---|
Exemplary tasks: ‘Spenser has a quasilinear utility function | 1. Perform 2. Check if the bundle obtained is non-negative. If yes, it is the optimal bundle. If not, the optimal bundle is a corner solution: | ||
Derive the exponents of the Cobb Douglas utility function Exemplary task: Suppose that a consumer has a Cobb–Douglas utility function and buys two goods, | 1. Calculate the budget shares | 1. The MRS = MRT condition implies |
Task ( | Technique ( | Technology ( | |
---|---|---|---|
Exemplary tasks: ‘Spenser has a quasilinear utility function | 1. Perform 2. Check if the bundle obtained is non-negative. If yes, it is the optimal bundle. If not, the optimal bundle is a corner solution: | ||
Derive the exponents of the Cobb Douglas utility function Exemplary task: Suppose that a consumer has a Cobb–Douglas utility function and buys two goods, | 1. Calculate the budget shares | 1. The MRS = MRT condition implies |
6.1. Praxeological analysis of constrained maximization in the mathematics-for-economists textbook
In the mathematics book, constrained optimization problems are informally introduced through the analogy of a landscape with a mountain where the constraint is the road you are restricted to move along when moving by car in the landscape. It is discussed how landscapes can be plotted in the
The type of task and the corresponding technique for solving the Lagrangian equation is presented as ‘rule 8.20’ on page 512. The introduction is not the ‘standard mathematics’ notation using the gradients that a mathematician would expect, therefore, to illustrate how it is introduced, a translated version is provided here:
Rule 8.20. We shall solve the following optimization problem with the constraint:
Maximize/minimize
when We assume that 𝑓 and g have continuous partial derivatives. We construct the Lagrangian function
Then, if
solves the given optimization problem with the constraint, there exists a value for 𝜆 such that the following equations are satisfied: Lagrange’s method provides a necessary but not sufficient condition for solving the optimization problem. To identify potential solution points, we follow this procedure:
Define the Lagrangian function.
Calculate the partial derivatives
and , and set the expressions equal to 0. You now have three equations
and and to find and possibly (Dovland & Pettersen, 2019, p. 512)
The presentation of the task and the technique is followed by a technological section named ‘Justification’ (Norwegian: ‘Begrunnelse’) which first refers back to the geometrical approach. Using the geometrical approach, it was established that the solution for the task must (1) be a point on the curve
We start by demonstrating that type (1) points satisfy the requirement in Lagrange’s method. Therefore, we assume that we have a point
lying on the curve , which satisfies This equation can be rearranged to
We denote the value of these fractions as λ, and we obtain
This is then transformed into
But this precisely demonstrate that
and . Finally, we need to show that type (2) points above satisfy the requirement in Lagrange’s multiplier method. Let
be a point lying on the curve and which is a stationary point for We aim to demonstrate that there exists a λ such that the following equations are satisfied But since we have a stationary point, then
Then, we see that we satisfy the equation by setting (Dovland & Pettersen, 2019, p. 513).
After this, a short ‘Remark’ (Norwegian: ‘Merknad’) about the technique is provided:
Lagrange’s method identifies candidates to be solutions. In addition to the points provided by Lagrange’s method, any endpoints on the curve describing the constraint are also possible solutions (Dovland & Pettersen, 2019, p. 513).
This note enlarges the technique with a fourth step without adding a technological discourse. From the graphical approach one can deduce that endpoints for the curve which defines the constraint is derived from a third condition of e.g.
In the analysis, we grouped point task of the type: ‘maximize
All together there were 19 tasks which were grouped into the reference praxeology
6.2. Praxeological analysis of constrained optimization in the microeconomics textbook
In the microeconomics textbook, constrained optimization problems are contextualized by optimal consumer choices subject to a budget constraint. Consumer preferences are represented by indifference curves with the underlying assumptions that the consumer chooses between two goods only (e.g., pizzas (
Following the graphical introduction, the Lagrange multiplier method concept is introduced in the microeconomics book as a method using calculus, which converts constrained maximization problems into unconstrained ones that can be solved using familiar maximization strategies. The introduction of the concept utilizes microeconomics concepts which might not be familiar to the reader, therefore, we briefly introduce them here:
Utility function: A function, for example U(
= that represents a consumer’s preferences, i.e., how a consumer’s well-being depends on the consumed bundle of goods, here q1 and q2. Utility functions can only be used for comparing different bundles as the exact utility values do not matter because a consumer’s preferences can be represented by many different utility functions. For example, any positive monotonic transformation of the original utility function preserves the consumer’s preferences.Indifference curve: level curves on which utility is constant between the bundles it represents.
Budget line: the constraint that defines the consumer’s opportunity set (the area between the line and the two axes).
Marginal rate of substitution (MRS) represents the willingness to substitute between goods. Holding utility level fixed, this is the slope of the indifference curve.
Marginal rate of transformation (MRT) represents the rate at which the consumer can trade goods (given prices and income are fixed). This is the slope of the budget line: the amount of one good that a consumer must give up to obtain more of the other good.
In the analysis process, we identified three local reference praxeologies, presented in Table 2. The first praxeology
‘Show how to find the optimal bundle graphically’ (Perloff, 2022, p. 106)
‘Julia has a Cobb–Douglas utility function
Use the Lagrangian method to find her optimal values of and in terms of her income and the prices’ (Perloff, 2022, p. 111–112)‘Diogo’s utility function is
where is chocolate candy and is slices of pie. If the price of a chocolate bar, is $1, the price of a slice of pie, , is $2 and is $80, what is Diogo’s optimal bundle.’ (Perloff, 2022, p. 127)‘Baki likes kofta, K, and Falafel F. His utility function is
Baki has a weekly income of £108, which he spends entirely on kofta and falafel.
If he pays £6 for a falafel meal and £12 on a kofta meal, what is his optimal consumption bundle? Show Baki’s budget line, indifference curve, and optimal consumption bundle,
, in a diagram.Suppose that the price of a kofta meal decreases to £6. How does Baki’s optimal consumption of koftas and falafels change? On the same diagram as in a., show his new budget line and new optimal consumption bundle
.’ (Perloff, 2022, 127)
The technique
For values of
and such that the constraint holds, , so the functions and have the same values. Thus, if we look only at values of and for which the constraint holds, finding the constrained maximum value of is the same as finding the critical value of Equations 3.21, 3.22, and 3.23 are first-order conditions that determine the critical values
and for an interior maximization: (3.21) (3.22) (3.23)At the optimal levels of
and λ, Equation 3.21 shows that the marginal utility of pizza, , equals its price times λ. Equation 3.22 provides an analogous condition for burritos. Equation 3.23 restates the budget constraint. These three first-order conditions can be solved for the optimal values of and (Perloff, 2022, p. 111).
Beside notational differences, the technique
What is λ? If we solve both Equation 3.21 and 3.22 for
and then equate these expressions, we find that (3.24)That is, the optimal value of the Lagrangian multiplier
equals the marginal utility of each good divided by its price, , which is the extra utility one gets from the last dollar spent on that good. Equation 3.24 is the same as Equation 3.14 (and 3.13), which we derived using graphical argument (Perloff, 2022, p. 111).
Following this introduction, the special case of the Cobb–Douglas utility function, that is,
Solve these three first-order equations for
and By solving the right sides of the first two conditions for and equating the results, we obtain an equation that depends on and but not on λ: (3.28)The budget constraint, and the optimality condition, Equation 3.28, are two equations in
and Rearranging the budget constraint, we know that . By subtracting this expression for into Equation 3.28, we can write the expression as By rearranging terms, we find that (3.29)Similarly by substituting
into Equation 3.28 and rearranging, we find that (3.30)Thus, we can use our knowledge of the form of the utility function to solve the expression for
and that maximize utility in terms of income, prices, and the utility function parameter .
Equations 3.28–3.30 derived above becomes a new technique
This technique
The three techniques outlined above all rest on the same logos block. The technology
(A1) The consumer buys only two goods and spends the entire budget on them.
(A2) Completeness: The consumer can rank any bundles of goods.
(A3) Transitivity: If the consumer prefers bundle a to b and b to c, then she/he prefers a to c as well.
(A4) More is better than less: The consumer prefers more of any good to less.
Furthermore, the technology explains and validates the results obtained by the techniques through microeconomic theory. In particular, the tangency point is understood as the point where the consumer’s marginal rate of substitutions (MRS) equals the marginal rate of transformation (MRT). The slope of the indifference curve is the marginal rate of substitution (MRS) which is represented by the partial derivative of
The microeconomics praxeology
Due to space constraints, a comprehensive praxeological analysis of
6.3. Comparison of the reference praxeologies
In this section, we provide answers to the research questions on how the praxeologies in the two textbooks compare. The previous sections exemplified the praxeological analysis and the results for the typical task of finding maximum subject to a constraint. Tables 1 and 2 summarize the results of the full praxeological analysis conducted on the two textbooks.
It is apparent from these tables that the praxeologies featured in the mathematics book adhere to a pattern wherein each task is associated with a unique technique and its corresponding technology. This diverges from the praxeologies found in the microeconomics book where the first praxeology
The praxeological analysis also uncovered a disparity in the kinds of praxeologies featured within the textbooks. The mathematics book’s praxeologies
7 Discussion
The praxeological analysis shows that the knowledge to be taught in the different institutions differs both in term of context where Lagrange multiplier method is applied and in terms of techniques and technologies. The aim of the research was to explore discrepancies in the two courses’ praxeologies, and furthermore to address ways in which the praxeologies align and does not align with each other. Only with such insight, teaching in respective course can be changed to facilitate the transition between the courses for the students.
Both books use the graphical approach to introduce the concept of the Lagrange multiplier method. The discrepancies of the graphical approach to solving constrained maximization were first discussed in Landgärds (2023). Building on the preliminary study, the praxeological analysis shows that although the two texts employ fundamentally similar graphs to introduce the concept, they differ in techniques and technology that is used to describe and explain them. In particular, the relationship between the level curve understanding of constrained optimization (interpreting the graph as a three-dimensional plot on the
As discussed previously, students not seeing the relevance of, or not being able to apply the mathematics taught in the service mathematics course is a common problem when it comes to students’ transition between service mathematics and their main study courses (Flegg et al., 2012; Harris et al., 2015; Faulkner et al., 2019, 2020; Landgärds-Tarvoll, 2024). This might certainly be the case when it comes to the transition discussed in this article. While in the service mathematics course, there is a distinct technique and technology discussing the Lagrange multiplier method, in the microeconomics textbook the explicit use of the Lagrangian technique
Finally, as outlined in the previous section, the full praxeological analysis revealed mismatches in praxeologies between the books. While the primary goal for the mathematics-for-economists course is to enable students to transition from the mathematics to the microeconomics praxeologies, it is important that the knowledge to be taught in the service mathematics course is the knowledge students subsequently need in their economics courses. This aligns with Castela (2017), who highlights the importance of praxeological analysis as a crucial instrument for selecting the most suitable mathematics curriculum for application-oriented programs.
8 Opportunities for further research
This synchronic praxeological analysis establishes a groundwork for future research, particularly in the form of meta studies aimed at transforming the institutional conditions and constraints affecting mathematics and microeconomics educators and curriculum designers (see archeorganisation in Strømskag & Chevallard (2024)). Such studies could specifically explore and potentially revise current pedagogical approaches to the Lagrange multiplier method. This presents a significant opportunity for further investigation and development in educational strategies and curriculum design.
In this article, we were not able to investigate whether the differences in notation pose difficulties for the students in the transition from mathematics to microeconomics, but as highlighted by Hochmuth (2020) they might do so: ‘because of the mismatch of practices, it is often not clear which activities and reasoning are allowed, required, or forbidden and, in particular, how students have to interpret symbols in view of a specific task in major-subject-courses’ (p. 771). Especially, we recognized that the techniques involving the Lagrangian equation
Furthermore, we see the potential to further investigate these research findings drawing on the ‘extended praxeological model’ (Castela & Romo-Vázquez, 2011) similar to the research conducted by Peters & Hochmuth (2021). This would involve taking the praxeological analysis one step further, that is, reconstruct the solutions for the microeconomics task by referring to the two different mathematical discourses in one extended praxeological model. Such an approach could serve as useful insights and knowledge base for producing teaching resources on the Lagrange multiplier method concept both for the service mathematics teaching and microeconomics teaching.
9 Potential topics for service mathematics teaching
The aim of this article was to inform practitioners teaching the service mathematics course for economists on how the concept of Lagrange multiplier method is applied in microeconomics and give some indications for what topics could be addressed in the service mathematics course to help students see the connection and make the transition between the mathematical and the microeconomic use of the concept. Therefore, from the analysis presented above, we conclude this article with a list of concrete aspects educators can consider in their service mathematics (and microeconomics) teaching to address the transition between the service mathematics and microeconomics.
Establish the link between the two books’ graphical approach. Especially discuss how the level curves in the mathematics book align with the indifference curve concept in microeconomics. Also approach the consumer model assumptions from the mathematics perspective. Address the discrete vs continuous case.
Establish the link between the mathematical ‘slope of the curves’ in terms of MRS and MRT.
Discuss the significance of the three partial derivative conditions and their mathematical, graphical and economic interpretation. Simultaneously discuss different notation.
Discuss what an interior maximum means in terms of mathematical approach and link to the microeconomic understanding of ‘the point where the consumer gets as much utility from spending the last dollar on good 1 as on good 2.’
Discuss what it means when Lagrange multiplier method does not yield a solution in the domain. Discuss and introduce the concept of corner solutions.
Discuss the Cobb Douglas function explicitly and derive the properties using Lagrange multiplier method to highlight the concept’s usefulness.
Acknowledgement(s)
The authors wish to express their gratitude to the anonymous reviewers for their insightful, thorough, and constructive feedback, which has significantly enhanced the quality of this article.
Footnotes
Course description to be found here: https://www.uia.no/en/studieplaner/topic/MA-138-1
Information received from internal statistics for 2023 by fagbokforlagt.no
Course description to be found here: https://www.uia.no/studieplaner/topic/SE-213-1
Perfect complements refer to goods that are consumed together in fixed proportions, where the consumption of one good is directly dependent on the consumption of the other. For instance, left and right shoes are perfect complements, as both are required to form a functional pair.
References
Ida Landgärds-Tarvoll is an Assistant Professor of mathematics education at University of Agder (UiA) in Kristiansand, Norway. She has been teaching the service mathematics course for economics students since 2017 and is doing research within that field as well. She is an active member of the Norwegian Centre for Excellence in Education – Centre for Research, Innovation and Coordination of Mathematics Teaching (MatRIC), based at UiA. E-mail: Ida.landgards@uia.no.
Dr. Daniel Göller is an Associate Professor at the Department of Economics and Finance at the School of Business and Law at the University of Agder. He has a Ph.D. in Economics from the University of Bonn (Germany) and has been teaching different courses in microeconomics and mathematics since over a decade. As a researcher, Göller’s main interests are law and economics, contract theory and bargaining theory. E-mail: daniel.goller@uia.no.