Bridging mathematical and microeconomic perspectives: a praxeological analysis of the Lagrange multiplier method

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 19^th 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 |$\lambda$|⁠, called the Lagrange multiplier (Baxley & Moorhouse, 1984). In economics, the concept is often used to solve problems of utility maximization, cost minimization and other optimization scenarios where constraints (budget, resources etc.) are present (Baxley & Moorhouse, 1984). Mkhatshwa (2023) discusses that research on students’ opportunities to learn optimization on undergraduate level is scarce. One exception is Xhonneux & Henry (2011) who examined how the Lagrange multiplier method is presented in textbooks specifically targeting economics students and those specializing in mathematics. They observed that the calculus textbook for economics students is predominantly focused on task, techniques and justification for the techniques concerning tasks like ‘find candidates to be optimal solutions for a constrained optimization problem subject to equality constraints.’ In contrast, calculus textbooks for mathematics students are focussed on tasks considering proving and developing the theory of Lagrange’s theorem. Our study investigates the presentation and use of the concept in two courses, with the same audience, that is, the economics students. It draws on the engineering educational studies’ approach to investigating concepts. This article extends the preliminary findings concerning graphical approach to constrained maximization in Landgärds (2023). Our study hence broadens existing literature on how the use of mathematical concepts in economics education compares to the mathematical use. It serves as a resource for educators teaching mathematics for economists, as well as those teaching microeconomics, by raising awareness of discrepancies on how the Lagrange multiplier method is used and interpreted in both fields.

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 |$\mathfrak{p}=\left[T,\tau, \theta, \varTheta \right]$|⁠, a praxeology consists of four key elements. Type of task T, which in our research are mathematical tasks presented in the two textbooks, and the technique τ, which is the method employed in the textbooks to execute the task. These two elements together constitute the praxis block or the ‘know how’ block of the praxeology. The other part of the praxeology, the logos block or the ‘knowledge block’, consists of the technology θ and the theory|$\varTheta .$|The technology θ is a rationale, by which the technique applied is justified and explained (Chevallard, 2019). Castela & Romo-Vázquez (2011) detail six categories of technological functions. These categories serve as useful references for elaborating the different institutions’ technologies: Describing the technique; Validating it (i.e. proving that this technique actually does produce what is expected); Explaining why it is efficient (concerning causes); Motivating its different stages (regarding objectives); Facilitating its use; and, finally, Appraising it (with respect to the field of efficiency and to the comfort of use relative to other techniques available) (Castela & Romo-Vazquez, 2022, p. 623). The theory|$\varTheta$| is a discourse (in a broad sense) that generates, controls and justifies the technology. In Section 2, we outlined how economics was mathematized in the 19^th century and concepts were changed to be applicable for the field of economics. This is consistent with the ATD’s epistemological hypothesis that transformation of knowledge (transpositional effects) occurs when social knowledge moves between institutions (Chevallard, 2019). However, for this article, we do not focus on these transpositional aspects on the evolution of the concept in the two disciplines’ scholarly knowledge and their relation to the knowledge to be taught. Instead, we investigate the mathematical praxeologies concerning the concept of Lagrange multiplier method as they appear in the knowledge to be taught in the mathematics course |${\mathfrak{p}}_M$| and the mathematical praxeology in the microeconomics course |${\mathfrak{p}}_E,$|that is, synchronic comparative praxeological analysis as described by Strømskag & Chevallard (2024).

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 |${R}_I\left(x,o\right)$| represents a compact abbreviation indicating the relationship of a position|$x$| to a praxeology|$o$| within the institution|$I$| (Winsløw & Grønbæk, 2014; Chevallard & Bosch, 2020). The purpose of the mathematics-for-economics course is to enable students to establish relationship of |${R}_M(s,\mathfrak{p})\to{R}_E(s,\mathfrak{p}).$| The arrow refers to the transition of a single person occupying the position as a student |$s$| of the mathematics course and as a student of the microeconomics course undertaking the relationship to the concept of Lagrange multiplier method (the praxeology |${\mathfrak{p}}_M$| and |${\mathfrak{p}}_E$|⁠, respectively) in the two institutions.

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 5^th 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 |${R}_M(s,\mathfrak{p})\to{R}_E(s,\mathfrak{p})$| as explained in Section 4. They represent the practices, that is, the mathematical praxeologies |${\mathfrak{p}}_M$| present in the mathematics course and in the mathematical praxeologies |${\mathfrak{p}}_E$| present in the microeconomics course concerning the concept of Lagrange multiplier method. The following books and pages were analyzed:

• 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.

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 |$xy$|-plane by drawing contour lines (i.e. level curves) for different |$z$|-values of the objective function. The maximum or minimum point being at the tangency point between the level curves and the constraint is explained using the map-analogy and drawing the graphs in the |$xy-$|plane. This approach was discussed in Landgärds (2023). Following this, Dovland & Pettersen (2019) introduce the technique of solving the Lagrange equation as a more efficient method for solving constrained maximization problems compared to the graphical method.

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 |$f\left(x,y\right)$| when |$g\left(x,y\right)=c.$|

We assume that 𝑓 and g have continuous partial derivatives. We construct the Lagrangian function

$$ \mathrm{L}\left(\mathrm{x},\mathrm{y},\mathrm{\lambda} \right)=\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)-\mathrm{\lambda} \left(\mathrm{g}\left(\mathrm{x},\mathrm{y}\right)-\mathrm{c}\right) $$

Then, if |$\left({x}^{\ast },{y}^{\ast}\right)$| solves the given optimization problem with the constraint, there exists a value for 𝜆 such that the following equations are satisfied:

$$ \begin{align*} \genfrac{}{}{0pt}{}{{\mathrm{L}}_{\mathrm{x}}^{\prime}\left({\mathrm{x}}^{\ast },{\mathrm{y}}^{\ast },\mathrm{\lambda} \right)=0}{\begin{array}{c}{\mathrm{L}}_{\mathrm{y}}^{\prime}\left({\mathrm{x}}^{\ast },{\mathrm{y}}^{\ast },\mathrm{\lambda} \right)=0\\{}\mathrm{g}\left({\mathrm{x}}^{\ast },{\mathrm{y}}^{\ast}\right)=\mathrm{c}\end{array}} \end{align*} $$

Lagrange’s method provides a necessary but not sufficient condition for solving the optimization problem. To identify potential solution points, we follow this procedure:

1. Define the Lagrangian function.

2. Calculate the partial derivatives |${L}_x^{\prime }$| and |${L}_y^{\prime }$|⁠, and set the expressions equal to 0.

3. You now have three equations |${L}_x^{\prime }=0$| and |${L}_y^{\prime }=0$| and |$g\left(x,y\right)=0$| to find |$x,y$| and possibly |$\lambda .$| (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 |$g\left(x,y\right)=c$| that touches a level curve of |$f\left(x,y\right).$| In this point, the slope of the curve |$g\left(x,y\right)=c$| equals the slope of the level curve of |$f\left(x,y\right).$| Or (2), it must be a point on the curve |$g\left(x,y\right)=c$| which also is a stationary point of |$f\left(x,y\right).$| Then second, Dovland & Pettersen (2019) demonstrate how points of type (1) and (2) satisfy the requirement in Lagrange’s multiplier method. The section validates the technique in the sense of Castela & Romo-Vázquez (2011) categorization of the technological functions as it establishes that the technique produces what it claims to produce:

We start by demonstrating that type (1) points satisfy the requirement in Lagrange’s method. Therefore, we assume that we have a point |$\left({x}^{\ast },{y}^{\ast}\right)$| lying on the curve |$g\left(x,y\right)=c$|⁠, which satisfies

$$ -\frac{{\mathrm{f}}_{\mathrm{x}}^{\prime }}{{\mathrm{f}}_{\mathrm{y}}^{\prime }}=-\frac{{\mathrm{g}}_{\mathrm{x}}^{\prime }}{{\mathrm{g}}_{\mathrm{y}}^{\prime }} $$

This equation can be rearranged to

$$ \frac{{\mathrm{f}}_{\mathrm{x}}^{\prime }}{{\mathrm{g}}_{\mathrm{x}}^{\prime }}=\frac{{\mathrm{f}}_{\mathrm{y}}^{\prime }}{{\mathrm{g}}_{\mathrm{y}}^{\prime }} $$

We denote the value of these fractions as λ, and we obtain

$$ \frac{{\mathrm{f}}_{\mathrm{x}}^{\prime }}{{\mathrm{f}}_{\mathrm{y}}^{\prime }}=\mathrm{\lambda} \ \frac{{\mathrm{g}}_{\mathrm{x}}^{\prime }}{{\mathrm{g}}_{\mathrm{y}}^{\prime }}=\mathrm{\lambda} $$

This is then transformed into

$$ {\mathrm{f}}_{\mathrm{x}}^{\prime }-\mathrm{\lambda} {\mathrm{g}}_{\mathrm{x}}^{\prime }=0\ {\mathrm{f}}_{\mathrm{y}}^{\prime }-\mathrm{\lambda} {\mathrm{g}}_{\mathrm{y}}^{\prime }=0 $$

But this precisely demonstrate that |${L}_x^{\prime }=0$| and |${L}_y^{\prime }=0$|⁠.

Finally, we need to show that type (2) points above satisfy the requirement in Lagrange’s multiplier method. Let |$\left({x}^{\ast },{y}^{\ast}\right)$| be a point lying on the curve |$g\left(x,y\right)=c$| and which is a stationary point for |$f\left(x,y\right).$| We aim to demonstrate that there exists a λ such that the following equations are satisfied

$$ {\mathrm{L}}_{\mathrm{x}}^{\prime }={\mathrm{f}}_{\mathrm{x}}^{\prime }-\mathrm{\lambda} {\mathrm{g}}_{\mathrm{x}}^{\prime }=0 $$

$$ {\mathrm{L}}_{\mathrm{y}}^{\prime }={\mathrm{f}}_{\mathrm{y}}^{\prime }-\mathrm{\lambda} {\mathrm{g}}_{\mathrm{y}}^{\prime }=0 $$

But since we have a stationary point, then |${f}_x^{\prime }={f}_y^{\prime }=0.$| Then, we see that we satisfy the equation by setting |$\mathrm{\lambda} =0$| (Dovland & Pettersen, 2019, p. 513).

After this, a short ‘Remark’ (Norwegian: ‘Merknad’) about the technique is provided:  

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. |$x,y\ge 0$|⁠. After the note, the chapter provides several example tasks and ends the chapter with proposed activities to be solved.

In the analysis, we grouped point task of the type: ‘maximize |$f\left(x,y\right)= xy$| with the constraint |$2x+3y=24$| ‘and ‘we consume |$x$| units of A which costs |$2$| per unit, and |$y$| units of B which costs |$1$| per unit. We have a budget of 50. Find the combination of A and B that maximizes utility when our utility function is given by |$U\left(x,y\right)={x}^{0.4}{y}^{0.6}$| under the reference task: ‘Maximize and/or minimize |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$|using calculus’, see (Task |${T}_2$|in Table 1). This reference task is solved using the 4-step technique outlined above. The technology encompasses the functions of description, validation and facilitating the use as described by Castela & Romo-Vázquez (2011). The task and the technique are described in a pictorial analogy of a 3d map which helps the reader to understand the graphical 2d representation and solution. The technique is validated trough mathematical argumentation using mathematical concepts such as stationary points, partial derivatives and tangency point for deriving the results obtained by the graphical approach. Furthermore, assuming both |$f$| and |$g$| have continuous partial derivatives facilitates the technique working effectively and avoid errors.

All together there were 19 tasks which were grouped into the reference praxeology|${\mathfrak{p}}_{2M}=\left[{T}_{2M},{\tau}_{2M},{\theta}_{2M},{\varTheta}_M\right]$|described above. Similarly, |${\mathfrak{p}}_{1M}$| corresponds to the praxeology with the reference task|${T}_{1M}$| of ‘Maximize and/or minimize |$f\left(x,y\right)$| s.t |$g\left(x,y\right)=c$| graphically.’ These two local praxeologies were the most common and accounted for 70% of all point praxeologies. In addition to these two, we identified three other praxeologies, briefly outlined in Table 1. The identified praxeologies align closely with those identified in the economics calculus books by Xhonneux & Henry (2011). This alignment not only validates our coding approach but also supports our belief that our findings have relevance beyond just the Norwegian context.

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 (⁠|${\mathrm{q}}_1$|⁠) and burritos (⁠|${q}_2$|⁠)), uses up the whole budget and maximizes her/his utility. All points (bundles of goods) on an indifference curve make the consumer equally satisfied but shifting between indifference curves changes the level of satisfaction (called utility). Graphically, the maximum point is where the budget line touches the highest attainable indifference curve. Intuitively, at the optimal point the consumer gets as much utility from spending the last dollar on good 1 as on good 2. The graphical technique|${\tau}_{1 EG}$| (see Table 2) to find (the interior) maximum was described in Landgärds (2023) and constitutes the two rules of ‘highest indifference curve rule’ and the ‘tangency rule of MRS=MRT.’

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(⁠|${q}_1,{q}_2)$|= |${Aq}_1^a{q}_2^{1-a},$|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 |${\mathfrak{p}}_{1E}$| has the reference task |$\underset{q_1,{q}_2}{\max }U\left({q}_1,{q}_2\right)\ s.t.Y={p}_1{q}_1+{p}_2{q}_2$| where |$Y,{p}_1,{p}_2$| are positive parameters when there exists an interior solution. |${\mathfrak{p}}_{1E}$| constitutes point tasks of type:

1. ‘Show how to find the optimal bundle graphically’ (Perloff, 2022, p. 106)

2. ‘Julia has a Cobb–Douglas utility function |$U={q}_1^a{q}_2^{1-a}.$| Use the Lagrangian method to find her optimal values of |${q}_1$| and |${q}_2$| in terms of her income and the prices’ (Perloff, 2022, p. 111–112)

3. ‘Diogo’s utility function is |$U\left({q}_1,{q}_2\right)={q}_1^{0.75}{q}_2^{0.25},$| where |${q}_1$| is chocolate candy and |${q}_2$| is slices of pie. If the price of a chocolate bar, |${p}_1,$| is $1, the price of a slice of pie, |${p}_2$|⁠, is $2 and |$Y$| is $80, what is Diogo’s optimal bundle.’ (Perloff, 2022, p. 127)

4. ‘Baki likes kofta, K, and Falafel F. His utility function is |$U={\big(\sqrt{K}+\sqrt{F}\big)}^2.$|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, |${e}_1$|⁠, 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 |${e}_2$|⁠.’ (Perloff, 2022, 127)

The technique|${\tau}_{1E}$|to solve the tasks of this type is divided into three different possible techniques which are used interchangeable in the book: |${\tau}_{1 EG}$| is graphical reasoning, |${\tau}_{1 EL}$| utilizes the Lagrangian equation and |${\tau}_{1 ES}$| represents the short cut method. Next, we illustrate by an example how these techniques are introduced and outline how they all rest on the same technology|${\theta}_{1E}$|⁠.Perloff (2022, p. 109) sets up the following task|${T}_{1E}$| to be solved: ‘Lisa’s objective is to maximize her utility, |$U\left({q}_1,{q}_2\right)$| s.t. |$Y={p}_1{q}_1+{p}_2{q}_2$|’. To solve the task, Perloff (2022, p. 111) introduces the Lagrangian method by first stating the Lagrangian expression that corresponds to the task: |${}^{\prime}\mathcal{L}=U\left({q}_1,{q}_2\right)+\lambda \left(Y-{p}_1{q}_1-{p}_2{q}_2\right),$|where |$\lambda$| (the Greek letter lambda) is the Lagrange multiplier.’ Then he continues:

For values of |${q}_1$| and |${q}_2$| such that the constraint holds, |$Y-{p}_1{q}_1-{p}_2{q}_2=0$|⁠, so the functions |$\mathcal{L}$|and |$U$| have the same values. Thus, if we look only at values of |${q}_1$| and |${q}_2$| for which the constraint holds, finding the constrained maximum value of |$U$| is the same as finding the critical value of |$\mathcal{L}.$|

Equations 3.21, 3.22, and 3.23 are first-order conditions that determine the critical values |${q}_1,{q}_2$| and |$\lambda$| for an interior maximization:

$$\begin{equation} \frac{\partial \mathcal{L}}{\partial{q}_1}=\frac{\partial U}{\partial{q}_1}-\lambda{p}_1={U}_1-\lambda{p}_1=0, \end{equation}$$

(3.21)

$$\begin{equation} \frac{\partial \mathcal{L}}{\partial{q}_2}={U}_2-\lambda{p}_2=0, \end{equation}$$

(3.22)

$$ \begin{equation} \frac{\partial \mathcal{L}}{\partial \lambda }=Y-{p}_1{q}_1-{p}_2{q}_2=0. \end{equation} $$

(3.23)

At the optimal levels of |${q}_1,{q}_2$| and λ, Equation 3.21 shows that the marginal utility of pizza, |${U}_1=\frac{\partial U}{\partial{q}_1}$|⁠, 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 |${q}_1,{q}_2$| and |$\lambda$| (Perloff, 2022, p. 111).

Beside notational differences, the technique|${\tau}_{1 EL}$| described in this example is similar to the technique|${\tau}_{2M}$| in the mathematics textbook. However, differently from the mathematics textbook, Perloff (2022) continues the introduction by addressing |$\lambda$| and elaborating on the microeconomic interpretation of the optimum. He writes:

What is λ? If we solve both Equation 3.21 and 3.22 for |$\lambda$|and then equate these expressions, we find that

$$ \begin{equation} \lambda =\frac{U_1}{p_1}=\frac{U_2}{p_2}. \end{equation} $$

(3.24)

That is, the optimal value of the Lagrangian multiplier |$\lambda,$| equals the marginal utility of each good divided by its price, |$\frac{U_i}{p_i}$|⁠, 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, |$U\left({q}_1,{q}_2\right)={q}_1^a{q}_2^{1-a}$| is addressed. Perloff (2022, p. 112) writes:

Solve these three first-order equations for |${q}_1$| and |${q}_2.$| By solving the right sides of the first two conditions for |$\lambda$| and equating the results, we obtain an equation that depends on |${q}_1$| and |${q}_2$|but not on λ:

$$ \begin{equation} \left(1-a\right){p}_1{q}_1=a{p}_2{q}_2 \end{equation} $$

(3.28)

The budget constraint, and the optimality condition, Equation 3.28, are two equations in |${q}_1$| and |${q}_2.$| Rearranging the budget constraint, we know that |${p}_2{q}_2=Y-{p}_1{q}_1$|⁠. By subtracting this expression for |${p}_2{q}_2$| into Equation 3.28, we can write the expression as |$\left(1-a\right){p}_1{q}_1=a\left(Y-{p}_1{q}_1\right).$| By rearranging terms, we find that

$$ \begin{equation} {q}_1=a\frac{Y}{p_1}. \end{equation} $$

(3.29)

Similarly by substituting |${p}_1{q}_1=Y-{p}_2{q}_2$| into Equation 3.28 and rearranging, we find that

$$ \begin{equation} {q}_2=\left(1-a\right)\frac{Y}{p_2}. \end{equation} $$

(3.30)

Thus, we can use our knowledge of the form of the utility function to solve the expression for |${q}_1$| and |${q}_2$| that maximize utility in terms of income, prices, and the utility function parameter |$a$|⁠.

Equations 3.28–3.30 derived above becomes a new technique|${\tau}_{1 ES}$| for solving constrained maximization problems which is called a short-cut method. Perloff highlights that by rearranging Equation 3.28, the Cobb Douglas optimum is achieved, as elaborated in the graphical approach, where |$\mathrm{MRS}=-\big(\frac{a}{\left[1-a\right]}\big)\big(\frac{q_2}{q_1}\big)=\frac{p_1}{p_2}=\mathrm{MRT}.$| Hence, using Equations 3.29 and 3.30 one can get the equilibrium quantities quickly.

This technique|${\tau}_{1 ES}$| represents an algebraization of the other two techniques introduced for solving the maximization problem. Using the |${\tau}_{1 EL}$| technique for different kinds of utility functions, standard results of the MRS = MRT condition are tabulated. This allows students to algebraically insert values into the formulas (like 3.29 and 3.30 for the Cobb–Douglas function) to determine the maximum quantities.

The three techniques outlined above all rest on the same logos block. The technology|${\theta}_{1E}$| blends mathematical arguments and microeconomics theory. In terms of the technological functions by Castela & Romo-Vázquez (2011), the technology facilitates the implementation of the techniques by establishing four assumptions on which the consumer behaviour model rests:

(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 |$\mathrm{MRS}=\frac{d{q}_2}{d{q}_1}.$| This slope, the tangent at a specific point (for a specific bundle) on the curve, reflects the trade-off between |${q}_1$| and |${q}_2$| that keeps utility constant. It is rooted in the concept of marginal utility, which is the additional satisfaction from consuming one more unit of a good, indicating that MRS equates to the ratio of the marginal utilities of the two goods, represented as |$\mathrm{MRS}=-\frac{U_1}{U_2}.$| The slope of the budget constraint is the marginal rate of transformation (MRT), which is the amount of one good that a consumer must give up to obtain more of the other good. Hence, the rate at which the consumer is able to trade goods (when prices and income are fixed). As the budget line is represented as: |${q}_2=\frac{Y-{p}_1{q}_1}{p_2}$|⁠, the MRT is given by: |$\mathrm{MRT}=\frac{d{q}_2}{d{q}_1}=-\frac{p_1}{p_2}.$| The tangency condition then results in |$\mathrm{MRS}=-\frac{U_1}{U_2}=-\frac{p_1}{p_2}=\mathrm{MRT}$|⁠. Equivalently, solving for |$\lambda$|in the Lagrangian equation one get: |$\lambda =\frac{U_1}{p_1}=\frac{U_2}{p_2}.$| This represents the condition that, the extra utility one gets from one more of good 1 per dollar (or other currency) spent on that good equals the extra utility one gets from one more of good 2 per dollar spent on that good. The utility is hence maximized when the consumer gets as much utility from spending the last dollar on good 1 as on good 2. Rearranging the terms yields |$\mathrm{MRS}=\mathrm{MRT}$|⁠.

The microeconomics praxeology |${\mathfrak{p}}_{2E}$| concerns tasks where students are asked to find the maximum, as for |${\mathfrak{p}}_{1E}$| but without the condition of an interior solution. The technique includes either of the techniques for the first praxeology, |${\tau}_{1M}$|⁠, but is extended as the tasks do not assume an interior solution. The technique is then to consider whether there is an interior or corner solution. For instance, in the case of quasilinear utility functions, for low incomes the only possible point of tangency involves a negative quantity of the second good. In this case, the consumer spends all her income on the first good, and the marginal utility per dollar spent on the two goods is not equal (for an example, see Fig. 3.10 in Perloff (2022, p. 116)). Other possible situations discussed are when |$MRS\ne MRT$| and the goods are not perfect complements⁴. The technology for this praxeology builds on technology|${\theta}_{1M}$| but excludes the assumption that the solution derived with the technique|${\tau}_{1M}$| is the optimal interior bundle. The knowledge considered hence pertains to the scope, conditions, and limits of the technique relative to this task. Hence, the technology for this praxeology has the additional function of appraising the technique as described by Castela & Romo-Vázquez (2011) in that it guides the use of the technique relative to other techniques available.

Due to space constraints, a comprehensive praxeological analysis of |${\mathfrak{p}}_{2E}$| and |${\mathfrak{p}}_{3E}$| cannot be provided herein. However, a brief overview of the result is included in Table 2.

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 |${\mathfrak{p}}_{1E}$|with the reference task |${T}_{1E}$| can be solved using different techniques |${\tau}_{1 EG},{\tau}_{1 EL}$| and |${\tau}_{1 ES}$| which all rest on the same technology |${\theta}_{1E}$|⁠. What we recognize, is an algebraization of the technique, where short-cut methods are derived from the results obtained using the other two methods in the microeconomics book. This finding aligns with the findings by González-Martín (2021) and Hitier & González-Martín (2022) who found similar algebraization in the engineering and physics context respectively. However, we also recognize that the microeconomics technique using the Lagrangian equation |${\tau}_{1 EL}$|⁠, is very similar to the technique presented in the mathematics book |${\tau}_{2M}.$| In Microeconomics, this technique is needed to derive the results for the short cut method |${\tau}_{1 ES}$|⁠. Whenever the students do not have a table of standard results at hand, she/he needs to master the Lagrangian technique |${\tau}_{1 EL}$|⁠.Not only do the praxeologies in the two books differ in terms of the phenomenon of algebraization of techniques, but they also differ in terms of technology. As illustrated by the examples provided in this article, in both cases, the technology facilitates the technique by assumptions that must be satisfied for the technique to work. However, these assumptions are different in that they concern the mathematical features of functions in |${\theta}_{2M}$| compared to assumptions concerning the consumer behaviour model in the microeconomics |${\theta}_{1E}.$| Furthermore, the mathematics technology’s function of description and validation using a pictorial map analogy and a mathematical proof for the technique finding interior solutions differ from the microeconomics technology which explains and validates the technique using microeconomics theory and concepts. In particular, the microeconomics technologies are a mix of graphical discussion, calculations and the microeconomic interpretation of the derived expressions (such as referring to marginal utility and its interpretation in terms of the model, which is the extra utility from the last good consumed, instead of referring to the mathematical partial derivative notion when referring to |$\frac{\partial U}{\partial{q}_1}$|⁠). These findings align with the findings of Feudel (2019) and Ariza et al. (2015) who found economics discourse assuming a flexible understanding of mathematical concepts such as the derivative in terms of geometric representation, economic interpretation and algebraic form and a mathematical understanding for how the different representations can be converted to one another. Above all, the praxeological analysis highlights how the mathematical discourse in the microeconomics textbook intertwines mathematical approaches with microeconomic concepts, while the mathematics technology present in the mathematics textbook relies exclusively on mathematical reasoning, without incorporating any microeconomic elements. Our results of the comparative analysis demonstrate notable similarities to the work by Peters & Hochmuth (2021) and Rønning (2022) on the expanded praxeological model in the engineering discipline. In particular, the praxeological analysis of the microeconomics textbook indicate that the approach by Peters & Hochmuth (2021) is a promising approach for future research with the goal of designing teaching resources on the concept.

The praxeological analysis also uncovered a disparity in the kinds of praxeologies featured within the textbooks. The mathematics book’s praxeologies |${\mathfrak{p}}_{3M}$| and |${\mathfrak{p}}_{4M}$|have no direct similarities in the microeconomics book. This is somehow surprising, as |${\mathfrak{p}}_{3M}$| which has the reference task ‘Give an approximation of the increase/decrease in maximum/minimum value of the objective function from a change in |$c$| of the constraint’ is of economic interest as it is the value of |$\lambda$| in the Lagrange multiplier method. However, the |$\lambda$| variable (commonly referred to as the shadow price) is frequently considered in master-level microeconomic courses. Furthermore, the microeconomic praxeology |${\mathfrak{p}}_{3E}$|⁠, which deals with the commonly used Cobb–Douglas utility function, lacks a corresponding praxeology in the mathematics textbook. The microeconomics text frequently addresses this type of utility function in the point praxeologies. In contrast, a mere 4 out of 34 point praxeologies in the mathematics text refer to the Cobb–Douglas function, without explicitly acknowledging it as such a function.

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 |$xy$|-plane) and the microeconomic indifference curve and utility curve understanding is a fundamental discrepancy that needs to be addressed in the teaching of the Lagrange multiplier concept. Concerning the maximum point, teaching needs to establish the relationship between the microeconomic ‘highest indifference curve rule’ which encompasses the theorization of the interior maximum point as the point where the consumer’s marginal rate of substitution equals marginal rate of transformation, and the mathematical terms of the slope of the curves being equal.

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 |${\tau}_{1 EL}$| disappears after the introduction of the short-cut methods. However, it is important for students to understand the concept of the Lagrange multiplier method to succeed in the transition between the institutions. The microeconomics discourse integrates the mathematical Lagrangian technique with the microeconomic interpretation of the derived partial derivative conditions and graphical representation in terms of slope of the indifference curves (level curves) and the tangency condition of MRS = MRT as outlined in Section 6.2. Hochmuth & Peters (2021) highlight the need for teaching to address mathematics in engineering instead of focusing on the applicationist view of mathematics for engineering. Following that view, we argue that service mathematics teaching in economics should not only focus on the distinct techniques and their technologies, instead an approach discussing the relationship between the graphical and the calculus-based techniques for different functions (such as the Cobb–Douglas function) could help students in their transition to the microeconomic praxeology.

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 |${\tau}_{2M}$| and |${\tau}_{1 EL}$| are ‘the same’ but the notation both in terms of establishing the Lagrangian function and the three partial derivative conditions differ between the books. This recognition and the discrepancies between the praxeologies set the agenda for future research opportunities. We see the possibility to further investigate these issues by adding another data layer to the analysis. For example, by interviewing service mathematics and microeconomics teachers from other universities about the issues students face in their transition between the courses, as well as interviewing or examining students working with the concept in the two courses. Further investigations would be beneficial to determine which of our findings are most critical and to potentially identify additional issues.

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

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: [email protected].

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: [email protected].