Indifference Curves for Beginners - Jens Peter Siebel PDF

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Australasian Journal of Economics Education Volume 7, Number 2, 2010, pp.1-12

INDIFFERENCE CURVES FOR BEGINNERS * Jens Peter Siebel International College, University of Applied Sciences Kaiserslautern ABSTRACT This paper presents a rather simple method to make students familiar with indifference curves. It shows with the help of a numerical example how indifference curves can be derived without advanced mathematical skills. Hence the method is especially suitable for introductory microeconomics lectures. Anecdotal evidence from the author’s own lectures suggests that the intuitive way of approaching indifference curves presented in this paper is attractive to students and reduces antipathy to economic modelling. Keywords: utility functions, indifference curves, introductory microeconomics. JEL classifications: D01, D03.

1. INTRODUCTION For many undergraduate students utility functions in the two-good case are their first experiences with multivariate functions and indifference curves are, therefore, their first examples of contour lines. Despite this underdevelopment of mathematical practice among students, many introductory microeconomics lectures open with continuous utility functions with two arguments. As a consequence, deriving indifference curves graphically (by means of a threedimensional graph or a four-quadrant scheme) and analytically (by means of equation-solving or total derivatives) often tends to overstrain students. In order to remedy these problems, I present an alternative approach to derive indifference curves that I have successfully applied in my *

Correspondence: Jens Peter Siebel, University of Applied Sciences Kaiserslautern, International College, Schoenstraße 9, D-67659 Kaiserslautern, Germany, E-mail: [email protected]. Thanks to Athanassios Pitsoulis and two anonymous referees for helpful comments. ISSN 1448-448X © 2010 Australasian Journal of Economics Education

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J.P. Siebel

introductory microeconomics lectures at the International College Kaiserslautern (ICK). These lectures are mainly targeted at foreign students preparing for their B.A. studies in Germany. The International College (“Internationales Studienkolleg“) is a special institution offering preparatory courses for foreign students in Germany. Depending on country of origin and school degree, attending the International College is compulsory. Students striving for economics-related studies are mainly taught German and Mathematics, supplemented by basic lectures in economic principles. About 50% of these students come from Morocco, whereas the remainder mainly originate from Indonesia, Nepal and China. Generally, my students have little or no experience in economics and business lectures. Furthermore, our entrance tests reveal that very few students have knowledge about differentiation.1 Normally, students stay at ICK for two terms. My microeconomics lecture is in the first term, when students’ knowledge of functions and their graphs is still rather limited. The approach here is based on a numerical example of two discrete utility functions and an additional table showing the overall utility of all possible consumption bundles. Drawing lines between all bundles of a certain utility level gives the students a first impression of indifference curves. This approach requires none of the skills and techniques mentioned above. The paper is organized as follows: Section two presents my numerical example to derive indifference curves. In section three I report on two evaluation polls at ICK and Brandenburg Technical University Cottbus (BTUC), in which students had to compare my approach to a graphical approach with a four-quadrant scheme. Section four concludes. 2. NUMERICAL EXAMPLE Before dealing with indifference curves, students should understand the following properties of a utility function in the single good case: • A utility function describes the utility level depending on the quantity consumed; 1

The entrance test in mathematics comprises basic themes such as quotients, logarithms, univariate functions and simple linear equation systems. Students revealing further knowledge in sets of numbers, differentiation and matrices may proceed straight to the second term if their German score also exceeds a certain threshold value.

Indifference Curves 3

• In reality there exists no unit to express utility with cardinal numbers. Using cardinal numbers as in the following example serves for expository purposes only. However, one can express utility ordinally by comparing different goods or different quantities of the same good; • Gossen’s Law: Marginal utility of any additional unit consumed is positive but decreasing. Table 1: Partial Utilities from Consuming Coffee and Donuts Cups of Coffee (x)

0

1

2

3

4

5

6

7

8

9

10

Utility v(x)

0

240

276

308

336

360

380

396

408

416

420

Donuts (y)

0

1

2

3

4

5

6

7

8

9

10

Utility w(y)

0

120

156

188

216

240

260

276

288

296

300

Similar to most lecturers and textbooks, I introduce the two-good case with a food and beverage example – in my case cups of coffee (good x) and donuts (good y). Table 1 and Figure 1 show the partial utilities v(x) and w(y), that the hypothetical consumer derives from the consumption of coffee and donuts, respectively. 450 400

360

408

416

420

288

296

300

8

9

v (x)

336

350

308 276

300

Utility

380

396

240

240

250

260

276

w (y)

216 188

200

156

150

120

100 50 0 0 0

1

2

3

4

5

6

7

10

Units of good

Figure 1: Utility Functions from Consuming Coffee and Donuts

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For the sake of simplicity and in order to obtain a sufficient number of matching pairs in the later analysis, I chose values such that: 𝑣(𝑥 + 1) − 𝑣(𝑥) = 𝑤(𝑦 + 1) − 𝑤(𝑦)

for any pair 𝑥 = 𝑦 ≥ 1, i.e. the second (third, fourth etc.) cup of coffee delivers the same marginal utility as the second (third, fourth etc.) donut. The basic idea is to write the overall utility u(x, y) = v(x) + w(y) from all possible consumption bundles into another table, which resembles an x-y-diagram. This is done in Table 2 below. Cups of coffee (x) are on the horizontal axis and donuts (y) are on the vertical axis. The table starts in the lower left corner with 0 cups of coffee and 0 donuts. Any cell contains the sum of the relevant partial utilities. An Excel spreadsheet might serve best here. Depending on the time available, one can also make the students fill out the table independently. Table 2: Overall Utility from Consuming Coffee and Donuts y: Donuts 10

300

540

576

608

636

660

680

696

708

716

720

9

296

536

572

604

632

656

676

692

704

712

716

8

288

528

564

596

624

648

668

684

696

704

708

7

276

516

552

584

612

636

656

672

684

692

696

6

260

500

536

568

596

620

640

656

668

676

680

5

240

480

516

548

576

600

620

636

648

656

660

4

216

456

492

524

552

576

596

612

624

632

636

3

188

428

464

496

524

548

568

584

596

604

608

2

156

396

432

464

492

516

536

552

564

572

576

1

120

360

396

428

456

480

500

516

528

536

540

0

0

240

276

308

336

360

380

396

408

416

420

0

1

2

3

4

5

6

7

8

9

10

x: Cups of Coffee

Indifference Curves 5

Obviously, there are some consumption bundles of coffee and donuts which yield the same utility level ū to the consumer. In this context, one introduces the definition of “indifference” to the students, e.g. by showing them that the consumer is indifferent between two consumption bundles (5 cups of coffee, 2 donuts) and (7 cups of coffee, 1 donut), as both bundles give the same utility level, ū=516. In the next step, let the students highlight all utility levels that appear four times. Then ask them to connect all highlighted consumption bundles of the same utility level with a line, as shown in Figure 2. Finally, one should explain that such lines are called indifference curves. ū=576

ū=636

ū=656

ū=596

ū=516

ū=536 x: Cups of Coffee

Figure 2: Indifference Curves for Different Utility Levels

Additionally, this numerical example reveals three basic properties of indifference curves to the students: 1) The utility level increases with the distance from the origin; 2) Indifference curves must not intersect anywhere. In the example we have no consumption bundle that represents more than one utility level. The mathematical intuition behind this result is

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straight forward, any sum of utility from any combination of coffee and donuts can only have one value; 3) Usually, indifference curves are strictly convex shaped (except for the perfect substitutes case). Table 3: Best Affordable Consumption Bundle for Given Prices and Income y: Donuts 10

300

540

576

608

636

660

680

696

708

716

720

9

296

536

572

604

632

656

676

692

704

712

716

8

288

528

564

596

624

648

668

684

696

704

708

7

276

516

552

584

612

636

656

672

684

692

696

6

260

500

536

568

596

620

640

656

668

676

680

5

240

480

516

548

576

600

620

636

648

656

660

4

216

456

492

524

552

576

596

612

624

632

636

3

188

428

464

496

524

548

568

584

596

604

608

2

156

396

432

464

492

516

536

552

564

572

576

1

120

360

396

428

456

480

500

516

528

536

540

0

0

240

276

308

336

360

380

396

408

416

420

0

1

2

3

4

5

6

7

8

9

10

x: Cups of Coffee

One can easily modify and extend the approach to search for the best affordable consumption bundle. Reconsider Table 2 and assume that both one cup of coffee and one donut cost $1 each and that disposable income is $10. Table 3 shows all maximum affordable combinations of coffee and donuts in the grey cells. The best affordable consumption bundle is (5 cups of coffee, 5 donuts), which yields the highest utility of all cells highlighted in grey. 3. EVALUATION In December 2010 I ran a poll with the students in my economics course at ICK, asking them to compare and evaluate my approach

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versus the common graphical approach with a four-quadrant scheme. Since I designed the poll to obtain feedback from my students, I did not strive for statistical accuracy by running a regression and conducting inductive statistics. First, I split up students randomly into two groups. Group 1 (10 students) was taught my numerical approach first and then the standard graphical approach. Group 2 (11 students) was taught the graphical approach first and then the numerical approach. After the presentation of the first approach, students had to do a short multiple choice test about indifference curves (see Appendix 1). Students in group 1 scored an average of 75%, somewhat higher than those in group 2 at 67.5%. Additionally, I asked them to evaluate the first approach by answering the question “How much do you understand of the first approach?” using a Likert scale from 1 (Everything) to 4 (Nothing). The results are shown in Table 4. Table 4: Evaluation of the First Approach at ICK N = 21

Everything

Much

Little

Nothing

Numerical (Group 1)

5

5

0

0

Graphical (Group 2)

2

5

3

1

Table 5: Evaluation of the Second Approach at ICK N = 21

Everything

Much

Little

Nothing

Numerical (Group 1)

7

3

1

0

Graphical (Group 2)

3

5

1

1

After the presentation of the second approach, students had to evaluate this as well. The results of this survey are shown in Table 5. Then I asked them which approach they considered easier to understand and why. The results of this are shown in Table 6. Here, 18 of 21 students opted for my numerical approach. Two of the students, who preferred the graphical approach, came from group 2. Unfortunately, while only six students gave a reason for their choice, all of them were in favour

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J.P. Siebel Table 6: Evaluation of the Second Approach at ICK Question: Which approach do you consider easier to understand? N = 21

Group 1

Group 2

Numerical

9

9

Graphical

1

2

Table 7: Evaluation of the Second Approach at ICK Question: Which approach should the lecturer use in the future? N = 21

Group 1

Group 2

Numerical

9

9

Graphical

1

2

of the numerical approach (see Appendix 2). This could be due to the fact that my foreign students’ German skills are often limited at this early stage of their studies. Finally, I asked the students which approach should be used in future lectures. Table 7 shows that the answers agree with Table 6 to a high degree but not completely. Three students opted for both approaches here, although I did not offer that option. One of them wrote an extra comment that teaching the numerical approach first would make the graphical approach easier to understand. In February 2011 I ran another short evaluation during a teaching assignment at BTUC. The assignment was designed as a crash course for undergraduates of both Business Administration (11 students) and Culture and Technology (16 students) prior to their exams in maths and microeconomics. One participant was an undergraduate in engineering. All students came from Germany. In general, students at BTUC had a deeper knowledge of economics than students at ICK.

Indifference Curves 9 Table 8: Evaluation Results from BTUC (1) N = 28

Students

Group 1

Group 2

Numerical Business Administration Graphical

5

4

1

6

1

5

Numerical

12

6

6

Graphical

4

1

3

Numerical

1

1

0

Graphical

0

0

0

Culture and Technology Engineering

Approach

Table 9: Evaluation Results from BTUC (2) N = 28

Approach

Students

Group 1

Group 2

Numerical

5

4

1

Business Graphical Administration

6

1

5

Both

0

0

0

Numerical

10

6

4

Graphical

4

1

3

Both

2

0

2

Numerical

0

0

0

Graphical

0

0

0

Both

1

1

0

Culture and Technology

Engineering

Again, group 1 (13 students) was taught the numerical approach first and group 2 (15 students) the graphical approach first. Due to time constraints, I only asked students which approach they preferred and why. As Table 8 shows, the results were mixed: 18 of 28 students said that they considered the numerical approach easier to understand. However, there was a slight majority (six of eleven) fo...