On the Misteaching of Consumer Theory

A debate was taken up by econobloggers on the usefulness of mathematical economics a few months ago [1]. While their discussion focused on the descriptive and instructional power of mathematics, I’d like to make a note on mathematical formalism at the introductory level of economics – that is, the method of teaching economic principles to undergraduates through mathematical models.

One of the things that have stuck with me from my time studying economics at the University of Maryland is a question a classmate asked in the discussion section in Intermediate Microeconomics: “Is a price floor binding when it’s above or below the market price?” Now, this was in the beginning of the year, and I don’t know how long of a break the student had had since taking the principles level course, yet this led me to wonder where the educational process had gone wrong (remember – these are people who are in a 300-level class in a university which is ranked in the top 25 in the world for economics). The question can easily be restated so that any non-econ major could have answered it: “When the market price for an apple is $10, would a government law that forces buyers pay at least $8 affect the market? What about $12?” [2]

While this specific example doesn’t demonstrate any explicit failure of math in economic pedagogy, it does raise the question of why our future economists can’t answer some of the easiest questions in their field. My contention is that economics professors have failed to provide students with a solid foundation for individual choice theory – that is, how individuals make choices given their incentives and their real-world options.

A problem in consumer theory

One of the basic ideas behind economics is that individuals receive utility, or satisfaction, from the consumption of various goods. Given two types of goods (A and B), different combinations of both of these goods yield different levels of utility for consumers. This is generally recognized by various schools of thought. Mainstream consumer behavior theory then introduces a concept called “indifference curves.” The idea behind this model is that for a given level of utility, there are various possible combinations of amounts of A and B that give a person the same level of utility. For example, 3 apples and 5 bananas may give me as much utility as 4 apples and 4 bananas, and also as much utility as 6 apples and 3 bananas. Taking all of these points (combinations) together, we can draw a curve which graphically shows all the bundles of goods which yield a certain level of utility. We say that a person is indifferent between all of these bundles – she has no preference of one over another. Figure 1 shows an example indifference curve, labeled U1. Another indifference curve shown in Figure 1, U2, represents a higher utility level. That is, all bundles on U2 are preferred over all bundles on U1.

What is the problem with these models? Indifference curves must have a very specific shape to actually describe what is going on (either consciously or subconsciously) in the minds of consumers. The curves have to be bowed in toward the origin, as seen in Figure 1 (like the bottom left half of a circle). Why is this so, however? Couldn’t we draw a bowed out indifference curve (like the top right half of a circle)? The shape of the curve is justified in neoclassical theory by the idea that people like balance – or that averages are preferred to extremes. For example, consider bundles X and Y on curve U1 in Figure 2. If we take the average of these two bundles, we get bundle Z, which is on curve U2. This tells us that Z is preferred to both X and Y, since its indifference curve is higher up.

But what is the justification for this? In the presentation slides for consumer theory in my class, “balance” is listed as an assumption. Underneath, as a bullet, it says “also called convex preference.” What this means is that we are essentially told that the curves are shaped that way – we are told to assume it, without a basis in reality or an explanation why. And the entirety of economic theory is based off of this assumption.

In my Intermediate Macro class we were told a similar version of the assumption – that people like diversity. The professor justified this by waving his hands and saying that people usually like having some of both goods rather than all of one good or all of another. When a student challenged him by arguing that sometimes a person may indeed choose much more of one good and just a little bit of the other, the professor emphatically agreed and stated that we’re just using an approximate assumption that works most of the time. Naturally, he smiled, this is just an assumption.

Before beginning to pick this apart, I’ll note that another (even more mathematical) way of expressing the bowed in shape is to state that the second derivative of the curve is negative. I would definitely not be surprised if there are some professors who, believing that they are being especially rigorous, teach it in this way. Unfortunately, this is no more of a justification of why the curves are shaped that way, but merely a description of the fact that they are.

What was the problem with my Micro professor? She never explained why the curves are shaped as they are. What was the problem with my Macro professor? His explanation left students believing that economists make ad-hoc pseudo-scientific assumptions that may easily be challenged at will. [3]

The alternative?

It’s true that the students in class, if they can remember the assumptions, will likely be just fine on their exams and in higher level classes. However, they will not be able to have a deeper understanding that is the seed of thinking that germinates into the highest level of economics – including Nobel Prize-level economics.

The Austrian school of economics gives a concrete explanation of consumer behavior that is both easily remembered and is a necessary basis for understanding how people actually function. It also gives a clean explanation of the shape of indifference curves, which neoclassicals would do well to assimilate.

Austrians begin by stating that humans act. They act to achieve ends. To achieve these ends, they have to use various means – their time, their labor, their property, etc. The means in the world that we can use to achieve our ends are scarce – we don’t have enough “stuff” to achieve every single thing we might want to achieve. People have preferences, which determine which ends they want to achieve first. Then, in accordance with these preferred ends, people economize, or make the best use of, their means to achieve their chosen ends. An implication of action aimed at achieving ends (as opposed to arbitrary action for no purpose) is that more valued ends (some say more urgent ends) are acted upon before less valued (less urgent) ends.

Taking this information, what can we say about people’s satisfaction (due to achieving ends) relative to the amount of goods they have at their disposal? First off, ceteris paribus – that is, assuming everything else remains the same – the more of a useful good that a person has, the more ends he can achieve. With a loaf of bread I can prevent myself from starving. With two, I can feel pretty full. With three, I can afford to keep a dog as well. And so on. Hence, the more of a good, the higher a person’s level of utility.

Secondly, there is another very useful law of economics we can derive. Notice that, as was mentioned, people act to achieve their most urgent ends first, followed by a little less urgent, then even less urgent, and so on. Well, what does this mean when we have a limited amount of resources? It means that given a certain amount of goods, we use it to satisfy our most valued end first. Next, given some more of the good, we satisfy our next most-valued end. Given yet another increment of the good, we satisfy the third-most valued end. And so on.

Note that with each addition to our stock (or quantity) of the good, we achieve an end that gives us less utility than the previous addition (since it was lower on our value scale). That is, the utility from each additional unit of the good decreases (since we achieve a less-valued end). This utility of the next unit of a good is called the marginal utility. The additional unit of the good is called the marginal unit of the good. The law we just derived in easy-to-follow steps was the Law of Diminishing Marginal Utility. It’s a law in that it necessarily describes how humans work, starting from the simple and uncontroversial empirical fact of humans acting.

How does this means-ends framework help us to understand indifference curves? Let’s begin with a graph of two points that we are told lie on a single indifference curve in Figure 3. This graph will eventually be built up into a whole indifference curve, but for now, we have just two of the points (M and N) from the eventual indifference curve.

Start at point M, which we will say corresponds to using 2 of good A and 10 of good B (the graph is not to scale). We are told that point N is on the same indifference curve, and consists of 3 of good A and 7 of good B. Since they are on the same indifference curve, they provide the same satisfaction, by definition (the consumer is indifferent between them). Moving from M to N, in order to gain 1 of A (while still being as satisfied as before), we were willing to give up 3 of B. What does this mean in terms of preferences and ends? It means that, moving from M to N, the end we can now achieve with 1 more unit of A is valued so highly that it can balance out the ends we will not be able to achieve after giving up 3 of B. The ends we now achieve with one extra A are the most valuable ones remaining that can be achieved with A. The ends that are not any more achieved with the B we lost are the least valued ones that were being achieved with B. That is, when we lose means, we don’t stop doing the thing we value most with them – we stop doing the thing we value least (at the margin). (This is fairly important, so let me restate it – when we gain a unit of a good, we satisfy our next-most valuable end, which, compared to the ends we are already achieving, is the least-valued one overall; when we lose a unit of a good, we stop doing the least valuable thing we were doing using that type of good; these two facts can be understood intuitively if we consider a quick example. If I have 10 units of a good and I am given an 11th one, that 11th one will achieve an end that is valued less than every other end my previous 10 units were achieving. Now, if that 11th one is taken away, I will lose, once again, the least valued end that I was able to achieve.)

So what happens next? How do we put another point of the indifference curve? Well, consider two more possible points – points S and T in Figure 4:

Let’s try to predict which one of the two is likely to be the next point. We know that moving from M to N, we gained 1 A. We know that if we were to gain one more A, we would be able to satisfy another end with A. But this one would be valued less than the previous ones satisfied with A. Furthermore, remember that from M to N we lost 3 of B. We lost the ability to achieve the least-valued things we were able to before with B. If we were to lose even more of B, we would have to start giving up ends that are more and more valuable. So A is becoming less and less valuable, while B is becoming more and more valuable (given the ends we can achieve with each of them). Therefore, if we were offered 1 more of A, we would be willing to give up less of B than we were before.

As a real world example, imagine that A and B are bread and water, respectively. At point M, I was able to eat just a little bit of bread, but I had a lot of water – enough to use water to drink, to flush the toilet, to clean the house, to water my garden, to wash my car, and to use as sprinklers for my little cousins to play in. If someone offered to give me one more loaf bread, I would be willing to give up a bunch of water to gain it, even though I would have to give up, say, the sprinklers and the car wash. If someone were to give me yet another loaf, I would value it less, since I’m less hungry at this point, so I would be willing to give up less water than before. The water I am now giving up is more valuable than the water I gave up before, since now I have to give up washing my house, and washing my house was decently valuable to me. If someone wanted to once again give me one more loaf of bread, I would be even less willing to part with my water, because I am beginning to eat into the water that I would be using to flush the toilet. We see, then, that when I have to trade off bread for water, the more bread I acquire, the less willing I am to part with my remaining water for more bread. Why? Because the bread is satisfying ends that are less and less urgent, and the remaining water I have is satisfying the very most important things to me.

The graph of the indifference curve has to reflect this. So we know that from M to N I got 1 A for 3 B. For the next 1 of A that I gain, I will be willing to give up less than 3 of B. Maybe this time I will give up only 2 of B. Which point – S or T – satisfies this fact?

Here are the points once again in Figure 5, this time with the distances in question emphasized:

If I move from N to T, it is apparent that I will be gaining 1 of A at the expense of more than 3 of B. If, on the other hand, I move from N to S, I will be giving up less than 3 of B. S satisfies the deduction we went through earlier, so S is the next point on the indifference curve. If we connect the points on the same curve, we get the bowed-in shape. Using the same logic, we can deduce that all the points on the indifference curve will form a bowed-in curve, seen in Figure 6:

Now, whether it is very bowed in or not very much bowed in depends on the specific preferences of the consumers, yet the bowed in shape is retained. And this concludes our definitive explanation of why indifference curves are shaped the way they are.

Other critiques

Other critiques of the undergraduate approach to economics include the lack of emphasis on mutually-beneficial terms of trade and on the fact that trade involves goods that are not of equal value.

A voluntary trade being mutually-beneficial (that is, beneficial for both sides) stems from the fact that if it weren’t, then it would never take place. Given that a trade occurs, it can be inferred that both parties expect to benefit.

The second point is that for a trade to take place, a person must value what the other person has more than he (the same person) values what he himself has (this is called an inverse valuation of goods). As such, he is giving away something of lesser value and gaining something of greater value. The same holds for the person with whom he is trading – lesser value for greater value. If this sounds paradoxical, it’s because the subjective theory of value has not been internalized: value is subjective – it is entirely dependent on the individual and what her desires are. As such, it is entirely possible for both people in a trade to be giving away something they want less and gaining something they value more. In fact, this is exactly what happens when a trade is voluntary. As such, trade creates wealth for both sides.

Both of these points do in fact appear in intermediate neoclassical theory. However, they are only minor features of the models presented and can be missed entirely if not pointed out – they are not at all obvious when a student is given two equations and told to solve them. These concepts should instead be front and center in introductory and intermediate courses, and not a mere byproduct of the model. They are, indeed, what drives the model. In favor of appearing to be rigorous by using high-level math, we are neglecting the central ideas in economics. Indeed, when economists Ferraro and Taylor went to a major economics conference and asked a large number of economics students and PhDs an introductory-level question about a concept (opportunity cost) instead of an equation, almost eighty percent of them got it wrong [4].

Conclusion

I have tried to show in this article the importance of the means-ends framework in economic analysis. The mainstream approach of indifference curves deemphasizes important aspects of human action that should instead be underlined. Humans constantly use various means to achieve ends – thereby becoming better off. The occurrence of a voluntary trade implies that both parties are better off, and both parties trade away something they value relatively less than what they gain. Understanding these core principles of economics is vital to grasping the importance and relevance of markets.

References and Notes

[1] “A few words about math” (http://noahpinionblog.blogspot.jp/2013/08/a-few-words-about-math.html), “The Point of Economath” (http://krugman.blogs.nytimes.com/2013/08/21/the-point-of-economath/?_r=0), “Economath Fails the Cost-Benefit Test” (http://econlog.econlib.org/archives/2013/08/economath_fails.html), “My Thoughts on Formalism in Economics” (http://consultingbyrpm.com/blog/2013/08/my-thoughts-on-formalism-in-economics.html)

[2] To be fair, there is a way to interpret the question so that it actually turns out to be a relatively valid question. Namely, “could non-binding price floors have a general equilibrium effect on equilibrium prices?” It’s easiest to explain what this means through an example. Imagine that the government set the minimum wage at $50/hour tomorrow. If enforced, there would be massive unemployment of every wage earner who produces below that level – since their labor would literally be made illegal. Now, the question is whether this minimum wage would have an effect on wages above $50; for example, on the wages of people who earn $60/hour. The simplest analysis (partial equilibrium) would answer in the negative, pointing out that $60 is already above $50. However, general equilibrium (taking into account all the different markets in the economy) might yield a different result. Since the people who get paid $60/hour rely on the products produced by lower-paid workers as inputs for their own production processes, they would no longer be able to sustain the same level of productivity. As a result, their wages would drop accordingly. For example, a person who cuts and polishes jewels would likely lose his entire wage if all the people who mine jewels were suddenly put out of work. As such, the minimum wage imposed on the other workers lowered his wage, despite it being above the equilibrium price level for his labor market. This is an interesting question to consider, yet, unfortunately, it’s likely not what the student had in mind.

[3] Here I have to give credit to my labor economics professor, who did insist on deriving indifference curves from diminishing marginal utility, although she then didn’t justify DMU – which is an important part of the issue.

[4] “Do Economists Recognize an Opportunity Cost When They See One? A Dismal Performance from the Dismal Science” – http://www2.gsu.edu/~wwwcec/docs/ferrarotaylorbep.pdf

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12 thoughts on “On the Misteaching of Consumer Theory

  1. I wasn’t able to read the whole post, but indifference curve are convex because of diminishing marginal utility. As you move down the curve, for example, it takes more of Good A to make sacrificing an additional unit of Good B desirable.

  2. I’m not necessarily convinced that we can say diminishing marginal utility is a law without implicit supplementary axioms to the action axiom. For instance, it must be assumed that each discrete unit must be used separately to achieve different ends, whereas it may very well be the case that in reality, some highly valued end can only be achieved by consuming quite a number of units of a good. Is it not conceivable that under certain conditions marginal utility can increase? One can certainly define the law of diminishing marginal utility so that it is necessarily true but then the scope of its application is narrowed substantially. None of this is to say that marginal utility often increases or that it is useful to consider cases in which it increases. I also do not mean to imply that neither model is more intuitive than the other. However, I am saying that it seems to me that neither framework will, a priori, generate a generalizable convex indifference curve and that both require assumptions which are indeed, to some extent, ad hoc.

    • You might be more convinced by this article: http://mises.org/daily/3100

      I am not quite sure whether I agree with his analysis (haven’t thought about it enough), but I find it difficult to see how marginal utility can increase. If it were to increase, why wasn’t the marginal unit allocated to the most valued end in the first place? Could you give an example?

      Plus, I don’t think that the analysis assumes that each discrete unit must be used separately.

      • I think Carden’s analysis is absolutely correct in what he says and he makes very explicit his extraneous assumption. He writes: “one might be tempted to look at the situation and exclaim, ‘Aha! With the fourth egg, Joe can feed his family with cake, which he clearly prefers to feeding them with scrambled eggs! Clearly, then, the marginal utility of the fourth egg is higher than the marginal utility of the third egg; therefore, marginal utility is increasing!’ This line of reasoning neglects a crucial point: the fourth egg can only be used to bake a cake in the presence of the first three eggs. Since ‘marginal utility’ is a concept that can only be applied to homogeneous units of a given supply, ‘one egg’ is no longer the relevant unit of analysis. The homogeneity of units is determined by the set of wants that can be satisfied with a unit of a good.” Now, this is all fine and good; he has carefully and accurately defined the problem out of existence. I had previously said that “[o]ne can certainly define the law of diminishing marginal utility so that it is necessarily true.”

        However, I also warned with such definitions, “then the scope of [the law’s] application is narrowed substantially.” Of what use is the pure law of diminishing marginal utility with Carden’s formulation (or Rothbard’s, et al.) when economists want to talk about real people and real goods? Real people do not make decisions based only on the wants that can be satisfied with one unit of a good that an economist happens to be analyzing; they can make decisions based on wants that can be satisfied with parts of units and by combinations of units as well. It follows that real people act all of the time in cases in which, according to Carden, the concept of marginal utility does not apply.The assumption required to give diminishing marginal utility any application becomes as ad hoc as the assumption of convex preferences. Indeed, it should be as the two assumptions appear to be logically equivalent.

  3. Mises addresses the “sometimes you need a lot of it” argument explicitly in HA, Chapter 7, sec 1.

    May as well quote it here:
    The law of marginal utility does not refer to objective use-value, but to subjective use-value. It does not deal with the physical or chemical capacity of things to bring about a definite effect in general, but with their relevance for the well-being of a man as he himself sees it under the prevailing momentary state of his affairs. It does not deal primarily with the value of things, but with the value of the services a man expects to get from them.

    If we were to believe that marginal utility is about things and their objective use-value, we would be forced to assume that marginal utility can as well increase as decrease with an increase in the quantity of units available. It can happen that the employment of a certain minimum quantity—n units—of a good a can provide a satisfaction which is deemed more valuable than the services expected from one unit of a good b. But if the supply of a available is smaller than n, a can only be employed for another service which is considered less valuable than that of b. Then an increase in the quantity of a from n — 1 units to n units results in an increase of the value attached to one unit of a. The owner of 100 logs may build a cabin which protects him against rain better than a raincoat. But if fewer than 30 logs are available, he can only use them for a berth that protects him against the dampness of the soil. As the owner of 95 logs he would be prepared to forsake the raincoat in order to get 5 logs more. As the owner of 10 logs he would not abandon the raincoat even for 10 logs. A man whose savings amount to $100 may not be willing to carry out some work for a remuneration of $200. But if his savings were $2,000 and he were extremely anxious to acquire an indivisible good which cannot be bought for less than $2,100, he would be ready to perform this work for $100. All this is in perfect agreement with the rightly formulated law of marginal utility according to which value depends on the utility of the services expected. There is no question of any such thing as a law of increasing utility.

    All that from Mises. I think the gist of his answer is that we have to talk about utility for a particular person in a particular situation. You can’t mix two cases. When A has 95 logs that’s one situation with it’s utility scheme. When he has 5 logs that’s a different situation. Only when we mistakenly talk about utility of logs in a vacuum, thinking there is such a thing as their objective utility, do we get the illusion of rising marginal utility.

    And why do we have to talk about a given person in a given situation? Because that’s where value comes from. It’s subjective, meaning it springs from the feelings of the person. And he feels different things in different situations, obviously.

    Best of luck,
    Smiling Dave

    • Smiling Dave, I think you will agree that Mises is essentially taking the same approach that Carden takes in the post to which Michael linked above. And, as I said, I believe that analysis is absolutely correct in what it says. The way in which both formulate the law of marginal utility does in fact give it the status of law. However, I think you will also agree that economists (and their students) want to talk about objectively defined goods (i.e. physical “things”). In fact, I suspect that Michael may have intended for “Good A” and “Good B” to refer to objectively defined “things”. If in fact he did, then, by Mises’s own admission, the law of diminishing marginal utility cannot be applied a priori and consequently, Michael cannot establish the convexity of indifference curves a priori. If not, then I think he is using the term “good” for something quite different (and far less intuitive) than what is intended for analysis in introductory economics courses.

      • I’m using good in the sense you think I’m not, haha. I’m using it as in “item with a specific serviceability” – that is, that may satisfy a certain end/want. Goods are homogeneous as long as units of them may satisfy the same wants.

        I am not sure, however, that your critique of using the more powerful Misesian law is good simply because, as you point out, applying it is more difficult. As economists, we know that goods exist subjectively for actors. If a person legitimately thinks thinks that after he buys 10 bananas all further bananas bought will turn into oranges (or at least have the same exact serviceability as oranges), he will act as if all bananas after the 10th are in fact oranges. Yes, this does put a wrinkle in our plans, but we must learn how to deal with it.

        One way is to make the argument that even though such things are possible, they are so rare that they do not change the overall shape of market demand curves.

      • ” economists (and their students) want to talk about objectively defined goods (i.e. physical “things”).”
        We are all talking about physical things. However, the value these physical things have do not exist in them. A thing can weigh a pound, or two pounds, or whatever. But it cannot be said to be “worth” two dollars, or to have a “value” of two dollars in and of itself. Worth and value are not like weight, or color. Worth and value are relationships between a person and an object. Just as a human cannot be married without a wife, because it takes two to tango, so too, a thing cannot have value inherently. Value is a thought in the mind of a person when contemplating an object. It is never inherent in an object.

        So that, much as economists may want to talk about a married bachelor, it is impossible to talk about. The things may be clearly defined without a person involved, but their worth and value cannot.

        BTW, Mises and Carden are different. Mises says having 5 logs or 95 puts the person in a different situation. Carden says having 5 logs or 95 turns them into different goods. I think that’s fudging the issue.

      • Nobody here ever claimed that value exists objectively in the material world. What I did say, and still do say, is that economists want to talk about objectively defined goods at various quantities, not only quantities where the “situation” is the same (i.e. the units of the good under consideration are homogeneous). Economists want to talk about what happens when someone gets an additional log, regardless of whether the “situation” changes or not at the new quantity of logs. Indifference curves, the subject of this post, are used without paying heed to whether at certain units, the situation changes. In short, they want to talk about homogeneous units of a good, even though people act without valuing the good in homogeneous units. Note well that this is not internally inconsistent or contradictory. However, it does force us to conclude that for the purposes for which economists employ indifference curves, convexity is a condition because it describes how people do act, rather than how they must act.

        BTW, Carden does not say having 5 logs or 95 logs makes them different goods, but only that if 95 logs are used together for one end, the unit of analysis is not homogeneous and consequently not subject to the law of diminishing marginal utility.

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