I Teach Economics, Not Algebra and Calculus
John D Hey. Journal of Economic Education. Washington: Summer 2005.Vol.36, Iss. 3; pg. 292, 13 pgs
Abstract (Document Summary)
Most people learn to drive without knowing how the engine works. In a similar vein, the author believes that students can learn economics without knowing the algebra and calculus underlying the results. If instructors follow the philosophy of other economics courses in using graphs to illustrate the results, and draw the graphs accurately, then they can teach economics with virtually no algebra or calculus. The author’s intermediate micro course is taught using mathematical software that does the mathematics and that draws accurate graphs from which students can see the key results. He backs up this no-algebra no-calculus approach with tutorial exercises in which students do economics and with exams that require no knowledge of algebra and calculus. The students end up feeling the economics, rather than fearing the algebra and the calculus. [PUBLICATION ABSTRACT]
Full Text (6313 words)
Copyright HELDREF PUBLICATIONS Summer 2005[Headnote]
Abstract: Most people learn to drive without knowing how the engine works. In a similar vein, the author believes that students can learn economics without knowing the algebra and calculus underlying the results. If instructors follow the philosophy of other economics courses in using graphs to illustrate the results, and draw the graphs accurately, then they can teach economics with virtually no algebra or calculus. The author’s intermediate micro course is taught using mathematical software that does the mathematics and that draws accurate graphs from which students can see the key results. He backs up this no-algebra no-calculus approach with tutorial exercises in which students do economics and with exams that require no knowledge of algebra and calculus. The students end up feeling the economics, rather than fearing the algebra and the calculus.
Key words: economics, graphs, mathematical software, mathematics
JEL codes: A22, D0
The basic point I strive to make in this article is that instructors can teach intermediate microeconomics using accurately drawn graphs rather than using algebra and calculus. Recent advances in computing software make this possible in a way that it was not before. More important, adopting this approach means that students can get a feel for economics. Moreover, given that life constantly involves tradeoff problems and given that there is now software available that does the mathematics, my proposal is that instructors just teach the economics and let the software do the mathematics. A demonstration of the way that I teach intermediate microeconomics can be accessed at http://www-users.york.ac.uk/~jdl/jee.htm. I recommend that readers access this site.
HOW TO TEACH INTERMEDIATE MICROECONOMICS
In any intermediate microeconomics textbook, one thing is immediately obvious: it is full of graphs. However, it seems that the graphs are not used to their full potential. They are usually there to illustrate the general principles of a result being discussed in the text; instead, they could be used to convince the students of the truth of the propositions of the text.
Consider demand theory. What is the important thing that students should learn? The crucial point is that demands depend on preferences. The instructor can, of course, show this mathematically, but another way (and a potentially better one for the typical intermediate microeconomics audience) is to show it graphically. The preferences determine indifference curves, and the indifference curves determine the demand. If the instructor draws the indifference curves accurately for a particular type of preferences, then the student can derive the demand curves accurately for those particular preferences. Some cases are trivial. With perfect I to a substitutes, it is clear from the graph that the individual demands only good 1 when p^sub 1^/p^sub 2^ < a and that the individual demands only good 2 when p^sub 1^/p^sub 2^ > a. With perfect 1 with a complements, it is clear from the graph that the individual’s demands satisfy q^sub 2^/q^sub 1^ = a (and the budget constraint). Some cases are not so obvious mathematically but become clear graphically. Take Cobb-Douglas preferences with weights a and 1 – a on goods 1 and 2. From an accurate graph,1 it is apparent, for example, that variations in p^sub 1^ do not affect the demand for good 2. The graph shows that the demand for good 2 is constant, and it can also be seen that the amount spent on good 2 is a proportion (1-a) of income. The result is that individuals with these preferences spend a proportion (1-a) of their income on good 2-and hence spend a proportion a on good 1. The graph shows it all. In my lectures and in my Web page, I use an animated graph which shows the result in stages. This kind of animation is possible (but cumbersome) if an instructor uses PowerPoint or overhead projector slides, but with mathematical software such as Maple, the animations are an important part of the software.2
The code that generates this graph is not difficult to follow, but students do not need to see this code. However, it is something that they could use. To demonstrate the robustness of the result, the instructor (or student) could change an item (the income, the fixed price of good 2, the parameter a, or the number and values of the prices of good 1), and students could immediately see the implication. The students have to be sure that the software is correctly drawing the Cobb-Douglas indifference curves, but it is not too difficult to convince them of that.
At this point, the students have found the form of Cobb-Douglas (with parameter a) preference demand curves: Individuals with such preferences spend a fraction a of their income on good 1 and the rest on good 2. It is simple to generalize this result (either verbally or graphically) to Stone-Geary (with parameter a and subsistence levels s^sub 1^ and s^sub 2^) preferences: Individuals with such preferences first buy a quantity s^sub 1^ of good 1 and a quantity s^sub 2^ of good 2 and then spend a fraction a of their residual income on good 1 and the rest on good 2. By this point, it is clear in the students’ minds that demands depend upon preferences: If they know the preferences, they can predict the demands. I then introduce the opposite idea of inferring the preferences from the demands (by using their knowledge of how the preferences determine the demands). This scientific process of inference is crucial to the methods used in economics, and it is vital that students learn it at an early stage.3 How it is made clear to them is something that I discuss in the next section. So the method becomes clear: Economists make some assumptions (in this context about preferences); if those assumptions are correct, then the economist can predict demand; but first, the economist must test (using knowledge of the relationship between preferences and demands) whether those assumptions are true (or approximately true); if the assumptions are not true, then the assumptions need to be modified.
In many economics textbooks, the theory of demand usually comes from the theory of the individual or of the household, whereas the theory of supply usually comes from the theory of the firm. I find this a bit misleading, even though from one perspective, it is the norm; in practice, it is usually the case that individuals and households demand goods and services, and firms supply them. This is misleading because individuals and households also supply labor and capital, whereas firms demand labor and capital. So I start with an exchange economy in which each individual’s income is in the form of endowments of the goods. In this context, whether individuals are buyers or sellers in a particular market depends on their preferences and on the prices of the goods. Students see that whether individuals enter a market as buyers or as sellers is not exogenous but depends on the preferences and the endowments of the individuals and the market conditions. Both demand and supply (and whether individuals are buyers or sellers) depend on the preferences (and endowments) of individuals. Students can see how demand and supply curves depend on preferences (and endowments).
Another point is important. Throughout the course, I stress the gains from trade-the reason that people carry out exchanges-and ways of measuring the gains from trade. To avoid any ambiguities, I start with quasilinear preferences, for which all measures-change in the surplus, equivalent variation, and compensating variation-coincide. For students to be able to check the results arithmetically, I start with discrete goods and with preferences defined through reservation prices. This means that the demand and supply curves are step functions with steps at the reservation prices. Moreover, the results about surpluses (the buyer surplus is the area between the price paid and the demand curve, and the seller surplus is the area between the price received and the supply curve) can be verified with simple arithmetic. (The mathematics of the proof is messy and obscures what is going on.) I then extend the analysis to continuous goods but stay with quasilinear preferences, where the results about surpluses can be verified by simple arithmetic. This gives the students confidence in the surplus as a measure of the gains from trade. Obviously, this result has to be qualified for the case of nonquasilinear preferences, and this I do later in the course. This accurategraphs approach enables me to get across quite sophisticated ideas in a relatively simple way.
Having discussed demand and supply in this pure exchange economy, I look to see if exchange is possible between two individuals. At this point, early in the course, I introduce the Edgeworth box, in which many of the key ideas of economics can be represented. Its visual appeal is manifested by its use in many economics textbooks. I find it crucial to draw the boxes accurately. Students quickly learn that if the two individuals have the same (homothetic) preferences, then the contract curve is the line joining the two origins and therefore that if the endowments of the two individuals are the same, then the endowment point is on the contract curve, and exchange is not possible. From this, the students see that (except when the endowment point happens to fall on the contract curve) if the two individuals are different in either their preferences or in their endowments, mutually advantageous trade is possible. Moreover, students can quickly and accurately identify the area of mutually advantageous trade.
Visually, students can also see the importance of the contract curve and its value for eliminating many inefficient trades. From this figure, they also appreciate the meaning of the terms efficient and inefficient. This figure is also vital for showing that economists can go only so far and that they must enlist the help of those prepared to make value judgments to choose between the points on the contract curve. The Edgeworth box shows clearly how much economists can say and, at the same time, shows that they cannot say everything. The Edgeworth box is also important for showing the properties of different exchange mechanisms. The box shows that the price-offer curves of the two individuals intersect along the contract curve. This is such a crucial property that I spend time discussing it. Has the mathematical software made a mistake? Is it a coincidence? Is it always true? Students learn a lot from thinking about these issues, and they are led to them by the accurate-graphs approach. The Edgeworth box also shows where other exchange mechanisms end up and shows, for example, the inefficiency of an exchange mechanism in which one of the agents sets the price and the other takes it as given. Again, thinking about why this is so is crucial to the formation of students as economists.
Maple enables me to draw the Edgeworth box for any pair of preferences and for any initial endowment. The competitive equilibrium depends on both these things. I think that seeing these properties graphically helps students understand them much better than if the properties are demonstrated mathematically.
The software can also be used to calculate the utility possibility frontier determined by any allocation problem. This is not something that I can show in an article (it requires animation and color), but the relationship between the preferences of the individuals and the shape of the frontier is something that is apparent from an animated graphics presentation. It is not apparent when presented algebraically.
Up to this point, I have been considering a pure-exchange economy and must now generalize the analysis to a production economy. I follow convention in using the expression firms to refer to those agents who carry out production. To minimize the amount of effort required by students and to enable economiesof-scale to be reaped in learning, I draw an analogy. The individuals that I have been considering so far have bought goods and services and have converted them into utility. Firms, in contrast, buy inputs and convert them into one or more outputs. If I restrict attention to firms that produce one output, the analogy is complete: Households buy goods and services and turn them into utility; firms buy inputs and convert them into output. For households, the transformation into utility depends on the preferences of the individuals; for firms, the transformation into output depends on the technology of the firm. I represent individuals’ preferences through indifference curves; I represent firms’ technologies through isoquant maps. The analogy is complete, except for one point: Utility is not measurable but output is. This means that I can carry over all the indifferencecurve analysis to the study of the firm, with the added dimension given by the fact that output is measurable. Thus, I can derive demand curves for the factors of production of the firm in just the same way as I did before, and the students can see that the demand for inputs depends on the technology of the firm. I can also exploit the fact that output is measurable to talk about different returns to scale. I then get on to cost functions, starting with total cost functions and relate the (increasing/constant/decreasing) returns to scale to the concavity/linearity/ convexity of the total cost function. Students can see this graphically. The properties of, and relationships between, the other types of cost functions, if treated graphically, now fall into place. An additional advantage of starting with total cost curves is that, when I get to the conditions for the maximization of profit for both the competitive firm and the monopoly, it is clear from looking at the graphs of total costs and total revenue that the profit is maximized when the total cost and total revenue are farthest apart-which happens when the slopes of the two curves are equal-that is, when marginal cost equals marginal revenue. The graphs show the key results.
Students are now ready to consider production possibility frontiers for a society. The analogy with utility possibility frontiers is immediate, and they can use the software to illustrate the derivation of the ppf by moving along the contract curve in an Edgeworth box for production.
Now that the concept of production possibility frontiers for society has been covered, students can begin to explore the nature of general (production and exchange) equilibrium in a society. If treated with algebra and calculus, general equilibrium is beyond the reach of most of our audience. If treated graphically and in color, the important concepts and results become approachable. Color is needed to make this point, but the idea is simple: I take the production possibility frontier; I take one point on it (which determines the initial endowment point of the members of society); I draw an Edgeworth box within the frontier at that point; and I do the competitive analysis. The point on the frontier is clear and so is the competitive equilibrium. I can then move the point on the frontier until I have found the best point. From the graph, it is clear that if the slope of the frontier at the point chosen is not equal to the slopes of the indifference curves of the two individuals at the competitive equilibrium, then the point chosen can be improved on and is therefore not an equilibrium. Students immediately see the result that in competitive equilibrium the marginal technical rate of substitution has to be equal to the marginal rates of substitution for both individuals. The graphs, the accuracy of the graphs, and the animation of the graphs make clear a result that is algebraically quite remote.
I have now covered demand and supply and exchange. In a sense, I can claim to have done everything-albeit in a very simple context. At this point, I think it is crucial to show students that what I have done has empirical relevance. I therefore insert a lecture, which many students tell me afterward is the most influential that they have ever attended. It is, however, a difficult lecture as it brings in econometrics and also some ideas about scientific methodology that the majority of students have never thought about before. One of these ideas is that no theories (in economics or elsewhere) are exactly correct. Econometrics specifically captures this point through its inclusion of an error term in almost all equations. Many intermediate microeconomic students think that econometrics and its obsession with the error term is a bore, and it is difficult to convince them otherwise: that the error is, in some ways, more important than the deterministic component. If the students have done no econometrics at all, then my empirical lecture is difficult to teach. At the same time, it is vital because it shows the students that the theory that they have learned is useful, is important, and is crucial for guiding the empirical work. Moreover, if they want to do any economics in practice, they have to do some empirical work. I must admit that my accurategraphs approach has less impact in this lecture than in the others.
By this point, I have covered most of the important concepts in economics. It is now important to check them, generalize them, and apply them in other contexts. The accurate-graphs no-algebra-or-calculus approach helps me to do this. To reinforce what has already been taught, I make the point of using the same method of analysis if at all possible. Indeed, this is often the case-in many ways, I have already done everything. I have three lectures on implications, three on applications to intertemporal problems, three on applications to risky problems, and one on an application to the labor market.
The three lectures on implications also benefit from the accurate-graphs noalgebra no-calculus approach. I discuss here only two of them-aggregation and measures of the gains from trade. These pick up a theme repeated throughout the textbook that the reason economics exists is that people want to trade because they are better off after trade. Such considerations lead naturally to thinking about ways of measuring the gains from trade. In the first part of the course, where I assumed quasilinear preferences, students were led naturally to the surplus as the obvious (and, in that context, unique) measure. The question remains, seeing that the analysis of the surplus was done at the level of the individual, what happens to this measure if there are only aggregate demand and supply curves, which is often the case in practice? Aggregation becomes an issue: How are demand and supply curves aggregated? Are the measures of the surplus valid at the aggregate level?4 These are important questions. They are probably also questions the students have never thought about and find it difficult to get excited about. But they should. They are also questions with trivial answers for anyone who knows some elementary calculus (the sum of integrals is the integral of the sum), but they are rather mystical questions for someone who does not know any calculus. If the instructor draws accurate graphs, then students can verify arithmetically that the surpluses aggregate in the manner expected.
In the early part of the course, the use of the surplus as the measure of the gains from trade was justified through its uniqueness and obvious interpretation if the preferences of the individual are quasilinear. Having started at this point and then having introduced other kinds of preferences that are rather obviously not quasilinear and having encouraged students to realize that quasilinear preferences are very much a special case, instructors are obliged to explore the question of measuring gains from trade with nonquasilinear preferences. This is naturally done through asking the question, How much better or worse off is an individual if a price changes and hence the quantity bought or sold changes? This leads to the ideas of compensating and equivalent variations. If portrayed graphically and accurately, it is immediately obvious that these two things coincide when the preferences are quasilinear but not necessarily otherwise. Students can see from the graph that it is because the indifference curves are parallel (in the money direction) that the two variations are the same. Moreover, with all the nonquasilinear preferences that they have so far come across, they can see immediately that the compensating variation is bigger than the equivalent variation. They can also begin to understand under what circumstances it may be the converse.
I now have three lectures on intertemporal choice (using the discounted utility model) and three lectures on risky choice (using the expected utility model). As much as possible, I use the same framework as before. So, when I am talking about intertemporal exchange or about the exchange of risk, I use an Edgeworth box and, in essence, repeat the discussion I did before when talking about the exchange of goods in general. To do this, and to do it graphically, I have to work in two dimensions. In intertemporal choice, I have two periods, the present and the future, periods 1 and 2. In risky choice, I have two possibilities, two ex, ante risky states of the world, states 1 and 2.
In the intertemporal choice lectures, I draw Edgeworth boxes with consumption in periods 1 and 2 on the two axes. As the relative slopes of the indifference curves are crucial to determining what exchanges take place, I derive an important implication of the discounted utility model (DUM): that the slopes of all indifference curves along the equal-consumption line are equal to -(1 + ρ), where ρ is the individual’s discount rate. This is something that students can see from accurately drawn indifference curves (although they have to believe that the software has drawn the curves correctly). It is also one of the few formulas that I ask my students to remember.5 This property enables me to derive many of the results connected with intertemporal choice and exchange.
In the three lectures on exchange under risk, I draw Edgeworth boxes with consumption in states 1 and 2 on the two axes. Because once again, the relative slopes of the indifference curves are crucial to determining what exchanges take place, I derive an important implication of the expected utility model (EUM): that the slopes of all indifference curves along the certainty line (c^sub 1^ = c^sub 2^) are equal to -π^sub 1^ / π^sub 2^, where π^sub 1^ and π^sub 2^ are the probabilities of the two states of the world. This property is crucial to many of the results that students can derive in the theory of choice and of exchange under risk. The Edgeworth box analysis of the exchange of risk shows clearly how the risk is shared in the competitive equilibrium. For example, students can see that, in an exchange between a risk-neutral individual and a risk-averse individual, the former will take all the risk, leaving the risk-averse individual completely insured (although, of course, the individual pays for the privilege!).
There is one final application (to the labor market) and then the final part of the course concentrates on market inefficiencies of various types. As might be expected, this part of the course is a bit of a mixture. I start with taxation, then monopoly and duopoly (the latter preceeded by a bit of game theory) and natural monopoly and discrimination, then externalities, public goods, and asymmetric information. I cannot claim that my accurate-graphs no-algebra no-calculus approach has particular advantages in all these lectures, but I can show things about externalities in Edgeworth boxes that are difficult to demonstrate mathematically, whereas separating equilibria in problems of asymmetric information are best shown (and their limitations exposed) graphically.
HOW TO REINFORCE THE TEACHING
Students do not learn solely by passively listening to lectures and passively reading the textbook. This listening and reading has to be reinforced in some way. Many courses try to do this through exercises of various kinds. Many exercises in some textbooks are more akin to nothing more than algebraic manipulation. I do not think that this is appropriate. I try to reinforce the understanding through getting the students to actually do economics. Reading about economics is not enough. They have to experience economics. To a certain extent, they do this in real life, but instructors have to reinforce the scientific approach that they are advocating. This I do in a variety of ways-what is feasible depends upon the number of students and the resources that are available. Different countries have different levels of support. Certain things instructors can do wherever they are as long as they get the support from the students. Some students object to what I do, saying that it is game playing, but these students will complain about any challenge.
The first exercise I give the students is a problem of allocation. In one sense, the whole course is all about allocation, so it is good to get them thinking about this early. The exercise is very simple. I split a group of students into two halves in some way. One half will be potential buyers of some hypothetical good, and the other half will be potential sellers. To keep life simple, I restrict each of them to buying or selling at most one unit of the good; I then induce values. Because I am an economist who does experiments, the way to do this is natural to me. Each potential buyer is given a reservation value r; if they buy at some price p, the instructor promises6 to pay them in Euros euro(Y – p). Each potential seller is given a reservation value r; if they sell at some price p, the instructor promises to pay them euro(p – r). Note the differences: It obviously does not pay a buyer to buy at a price higher than their reservation value, nor a seller to sell at a price below their reservation value. Having induced values, the instructor then discusses the issue of exchange. Is it possible to have exchanges that make both parties to the exchange better off? Well, this depends on how reservation values were allocated. If such exchanges are possible, the instructor then can enter into a discussion of the best way of organizing the exchanges. This is not the place to go into detail, but I promise that the resulting discussion is usually interesting and stimulating. The students might be encouraged to organize the exchange in such a way that the instructor pays out as much as possible. This will amuse-and educate-the students.
I have already hinted at another kind of reinforcement when talking about inferring preferences from demands, and I now give more detail. In a sense, it is a kind of game, but I would prefer to think of it as a sort of detection process. Instructors can run it as a game if they want-students do respond to competition-but it is not necessary. The idea is that students see a sequence of observations on an individual’s demands, and they have to infer the preferences of the individual from these observations-the fewer observations they need, the better. For the kind of students that I am teaching, I need to give them a little help.7 So I restrict the set of possible preferences and the number of goods. Two goods is enough, and I think it is enough to restrict them to the following preferences: (1) perfect 1 to a substitutes, (2) perfect 7 with a complements, and (3) Cobb-Douglas with parameter a. It would probably be sensible to restrict the parameter a to be positive and finite in the first two cases and strictly between O and 7 in the third. I^sup 8^ then give them a first observation: at income m and with prices p^sub 1^ and p^sub 2^, the individual demanded a quantity q^sub 1^ of good 1 and a quantity q^sub 2^ of good 2. (Obviously, I give them specific numbers.) Can they infer the preferences of the individual from this one observation? Well, it depends and that is the fun of the game. If one of the q^sub i^ is zero, they know immediately that the preferences are perfect substitute preferences, and they know something about the parameter a (it is either less than P^sub 1^/p^sub 2^-, or greater than it, depending on which q^sub i^ is zero). If both q^sub 1^ are nonzero, they can infer nothing because the preferences could be perfect substitutes with parameter a = p^sub 1^/p^sub 2^, or they could be perfect complement preferences with parameter a = q^sub 1^/q^sub 2^, or they could be Cobb-Douglas preferences with parameter a = p^sub 1^q^sub 1^/m. I then give them a second observation. With the two together, can they detect the preferences and the parameter a? It depends on the nature of the prices and incomes in the two observations and on the nature of the demands. Some reflection will reveal that, if the relative prices in the two observations are the same, but the incomes are different, the student can learn nothing (given the restricted preferences that I am allowing). Realizing this fact is important for the students. With all three preferences in this exercise, a change in the income (with the relative price held constant) causes the demands for the two goods to change by the same proportion. However, if I vary the relative price, the student can learn something. Indeed, if the preferences are either perfect complements or Cobb-Douglas, then two observations are not only sufficient to decide which of the two preferences is the true one, but are also sufficient to determine the value of the parameter a. If, however, the preferences are perfect substitutes, the student will detect this after two observations (with different relative prices) but may need a lot of observations to detect the value of the parameter a. This exercise is instructive as it not only teaches the students about inference from data, it also tells them about efficient inference from data. The instructor can also do a similar kind of exercise to try and detect the technology of firms from information about their minimum costs for producing certain outputs at particular input prices. This can be structured to be a much more complicated exercise.
Instructors can also adapt experiments into teaching tools from virtually any area of economics. For many years (until I got frustrated by the sheer weight of student numbers), I used to run double-auction market experiments to show students that the Invisible Hand does indeed often take the market to the competitive equilibrium. Showing this is useful to students who are skeptical about markets actually attaining the competitive equilibrium (as instructors never actually prove to them that they do). If an instructor has a sufficient number of computer terminals, the students can run the market experiments on the computer; there is plenty of software available for doing this. Indeed, the availability of hardware opens up a whole new world of tutorial reinforcement. I have not the space to go into detail here, but I strongly recommend a visit to Charlie Holt’s Web site9 (Holt n.d.). I find that experiments are an excellent way to reinforce knowledge; students really do learn economics best by actually doing economics.
HOW TO EXAMINE STUDENT LEARNING
If an instructor has spent the whole course saying (and showing) that algebra and calculus are not important to the understanding of economics, it is crucial that this philosophy is carried through to the examination. Many microeconomics examinations seem, however, to be tests of mathematical ability rather than tests of economic understanding. The problem is that it is very difficult to set examination questions that test understanding. I tell the students that the exams are not tests of mathematics and that they are not required to remember any formulas other than those listed in note 5. If any other formulas are required, then I will tell them the formulas in the question. Remembering formulas is not one of the skills required of an economist.
I attach two sample questions in the appendix. My questions are usually in four parts, and I specify the breakdown of the marks between the four parts (the numbers in square brackets at the beginning of each part). There is no algebra or calculus needed in any part, and the structure of the questions builds up throughout the question. In question 1, the first part requires the student to derive demand and supply functions for a discrete good where the preferences are quasilinear and expressed through reservation prices. The demand and supply curves are thus step functions, with steps at the reservation prices. The second part simply asks them to find the competitive equilibrium, which they can do directly from the graph. The surplus can be calculated either from the demand and supply curves or by comparing the posttrade position with the pretrade. Students should get used to the idea of trying to find independent checks of results that they have found. The final part asks them to consider different trading mechanisms. This question contains a lot of economics but no algebra and no calculus.
The second sample question looks more difficult, but I guide the students through it. They have to build an Edgeworth box. To help them, I insert indifference curves for one of the two individuals because these would be difficult for them to draw accurately. The indifference curves of the other individual are simpler, being straight lines (which I tell them but which they could work out as this individual is risk-neutral), and I ask the students to insert them. For the student who understands what is going on, the contract curve leaps out of the page, and the price-offer curves can be constructed easily and graphically because the indifference curves for B are accurately drawn. This completes the second part. The third part asks them to find the competitive equilibria and explain the reasons for their choice. Again, no algebra or calculus is needed to answer the question, but economic understanding is needed.
Students do not need to understand algebra and calculus for them to understand economics. If they have accurately drawn graphs, that is sufficient. The students can see and feel the economics.
1. One such graph is contained in the presentation at www-users.york.ac.uk/~jdhl/im-pad.htm. Readers are recommended to consult this site while reading this article.
2. I use Maple, but other software, such as MATLAB® or Mathematica, does a similar job.
3. The idea that there is always noise in data is rather difficult to incorporate at this early stage. This coincides with the notion that no theories are precisely true and therefore that all theories are literally wrong. Economists are looking for theories that explain the data well enough in relation to their parsimony. These complicated notions are difficult for students to understand. Usually they are introduced by the econometrician in an introductory course in econometrics. This is far from perfect because it is the economist who should introduce these ideas.
4. By which I mean whether the surplus measured with respect to the aggregate curve (the area between the price paid and the demand curve for the buyer surplus and the area between the price received and the supply curve for the seller surplus) is equal to the aggregate of the surpluses measured with respect to the individual curves.
5. The list is very short because I encourage students to work things out from first principles if at all possible. If students cannot do the mathematics, they cannot work out the following problems and must commit them to memory.
* The slope of a budget line with prices p^sub 1^ and p^sub 2^ is – p^sub 1^/p^sub 2^
* Individuals with Cobb-Douglas preferences (with parameter a) spend a fraction a of their income on good 1 and the rest on good 2.
* Individuals with Stone-Geary preferences (with parameter a and subsistence levels s^sub 1^ and s^sub 2^) buy s^sub 1^ of good 1, s^sub 2^ of good 2, and then spend a fraction a of their residual income on good 1 and the rest on good 2.
* The slope of a budget constraint in an intertemporal choice problem with a constant interest rate ris -(I + r).
* Individuals with discounted utility model preferences with discount rate ρ have indifference curves in (c^sub 1^, c^sub 2^) space with slope -(I + p) along the equal-consumption line.
* The slope of the budget constraint with fair insurance where π^sub 1^ and π^sub 2^ are the probabilities of the two states of the world is -π^sub 1^/π^sub 2^.
* Individuals with expected utility model preferences have indifference curves in (c^sub 1^, c^sub 2^) space with slope -π^sub 1^ / π2 along the certainty line (where π^sub 1^ and π^sub 2^ are the probabilities of the two states of the world).
6. Obviously, it is more fun for them but more expensive for the instructor with real money. Otherwise use pretend money.
7. Indeed, at any level, they need help but rather more at this level than when they have completed a Masters in Economics. But that is irrelevant because it is a method that the instructor is teaching.
8. Well, it depends-it is more fun to play it as a competition. I split the tutorial group into two teams. Each team assumes particular preferences and a particular value of the parameter a, and the other team has to discover the preferences and the value of the parameter by asking them questions about their demands at particular income and prices.
9. This is Holt’s teaching Web site. It contains references to the vast literature on the use of experiments in teaching.
Hey, J. D. 2003. Intermediate microeconomics: People are different, UK: McGraw-Hill.
Holt, C. R. Teaching Web site, http://www.people.virginia.edu/~cah2k/teaching.html.
John D. Hey is a professor of economics and statistics at the University of York, UK (e-mail: firstname.lastname@example.org).
On My Web Site, I Teach Economics, Not Algebra and Calculus
John D Hey. Journal of Economic Education. Washington: Summer 2005.Vol.36, Iss. 3; pg. 305, 1 pgs
Full Text (278 words)
Copyright HELDREF PUBLICATIONS Summer 2005
The presentation on the University of York Web site is designed to show instructors the power of a graphical approach to the teaching of intermediate microeconomics, in a way that is not possible in the author’s Journal of Economic Education article that described his course. The Web site contains 11 examples of the type of animated graphics that can be used to teach economics, and could usefully be viewed in conjunction with the journal article.
The philosophy behind this course is related to the observation that most people learn to drive without knowing how the engine works. The author believes that students can learn economics without knowing the algebra and calculus underlying the results. If instructors follow the philosophy of other economics courses in using graphs to illustrate the results and the graphs are drawn accurately, then they can teach economics with virtually no algebra or calculus. The author’s intermediate micro course is taught using mathematical software that does the mathematics and that draws accurate (often animated) graphs from which students can see the key results. Eleven of these graphs are available on the Web site. Instructors can show them in class as an aid to teaching the concepts and ask students to use them to review the course material. Instructors using the Web site might usefully back up this no-algebra, no-calculus approach with tutorial exercises in which students do economics and with exams that require no knowledge of algebra and calculus. The students end up feeling the economics, rather than fearing the algebra and the calculus.
John D. Hey is a professor of economics and statistics at the University of York (e-mail: email@example.com).