My best friend's son is taking Eco 100. While studying for his midterm exam, he encountered this question:
True or false: There is no difference between these two equilibrium equations in Eco 100 consumer theory as one equation can be transformed mathematically into the other (a) MUx/MUy=Px/Py (b) MUx/Px=MUy/Py.
What do you think? Is this statement true or false? Please vote here (only the first 200 answers will be recorded.)
I think this statement is true. Clearly the statements are mathematically equivalent.
However the "right" answer is false. I'm not sure exactly why. Perhaps because the interpretation of the conditions is different. MUx/MUy=Px/Py is generally interpreted as "the slope of the indifference curve is equal to the slope of the budget constraint." MUx/Px=MUy/Py, on the other hand, is generally interpreted as "the utility gained from the last dollar spent is the same for each commodity." But this seems a little obscure for an Eco 100 class. My friend's son's explanation was a little garbled, but seemed to have something to do with whether utility was ordinal or cardinal.
Currently the Ontario government is planning a more differentiated university system. Each university must define for itself a clear role within the Ontario system (see The Benefits of Greater Differentiation of Ontario's University Sector.)
My friend's son is at a university that regards itself as a leading educational institution. I suspect they will define themselves as a research intensive school. Which begs the question: how will they ensure their first year students receive a quality education? If they can't guarantee a quality education, should they be educating first year students in the first place?
But without thousands of first year students, how will they be able to pay for the graduate student teaching assistantships and "internationally competitive" salaries that are the basic prerequisites of a "research intensive" school?
Update: of the 27 people who have responded to the survey so far, 24, or 89 percent, have answered "True." The best justification for a false answer is given by ECON 1000 guru Nick Rowe, "If x is leisure, and y is milk, then MUx/MUy has the units litres/hour, so we don't need to talk about utils." But the units of MUx/Px are utils per dollar. The problem with this response is, as Nick points out, that it's the MRS that is measured in units of litres per hour, MUx/MUy is measured as utils per litre of milk/utils per hour.
Update #2: I've managed to obtain the "official" answer, which is: There is a difference! First line above [MUx/MUy=Px/Py] refers to Indifference Theory that does not require measureable utility; there are no units of measurement on the LHS or RHS of the equation. Second line [MUx/Px=MUy/Py] refers to Utility Theory which does require measurement of satisfaction.
There have now been 72 responses to the survey. 83.3% of respondents answered "true."
Worthwhile Canadian Initiative 
Stephen – perhaps he was confusing you and Mike? Or perhaps he means that your posts are understandable and relevant to the real world?
Frances: “Stephen; “I think we need the person who wrote that ‘answer’ to explain why it’s not completely stupid, because that solution is – to say the least – unsatisfying.”
I’m imagining the email “Dear Professor ____, One of your students has brought this question to the attention of this blog, and we were wondering if you could explain yourself.”
Any volunteers?”
Who is/are the author(s)?
Its ironic how the confusion here exactly matches the confusion in Nick’s Volunteer army post, and for the same reasons people answer ‘true’ to that question and don’t see that the volunteer army analysis depends on an unsubstantiated assumption, which is precisely the point of this question.
The notion of a unit “utils” has been discredited for fifty years in the US. Is it really true that the only professor in Canada who knows this is teaching your nephew’s eco100 class?
Well you did say it was a top school!
Again, no. The ordinal/cardinal point doesn’t apply in this context. It does apply when you use the expected utility criterion.
Jon: “The notion of a unit “utils” has been discredited for fifty years in the US.”
Writing down MUx presupposes the existence of a continually differentiable utility function U=f(x). A one unit change in x will cause a change in U. We can use the word “util” to refer to the change in U with respect to a change in x without supposing cardinal utility, scalar measurement, or all sorts of other things.
Yes, utility theory not a literal description of human choice, it is only a model. People who find the assumptions of this model obnoxious can derive just about everything they need from revealed preference theory, which removes the need to talk about utility, marginal utility, or utils.
As Nick said earlier, if you rewrote the question so that MUx/MUy was replaced with MRS (marginal rate of substitution) then the professor’s answer might make sense, because you can have a MRS without a utility function – just use revealed preference theory. And it would, in this case, be a really great question – if the students had in fact been taught revealed preference theory.
Frances: I don’t follow the distinction you’re drawing. If the preference relation cannot be represented by a continuous utility function, then the ratio of marginal utilities isn’t defined, but neither is MRS. Similarly, if behavior satisfies the weak axiom but is not rational, neither the ratio of marginal utilities nor MRS is defined. What am I missing?
The answer seems very wrong to me. There’s a distinction drawn between “utility theory” and “indifference theory” which doesn’t exist (the latter is a pedagogical approach to teaching the former), the word “measurable” is abused, the claim that price and marginal utility ratios are unitless is wrong (presumably what is meant is “the units don’t involve `utils'”), and, as many have pointed out, neither way of expressing this equation implies that utility is cardinal. I think we can even dispense with the technical problem that arises when p_x=0 because the question tells us to invoke Econ 101 assumptions, which include non-satiation.
Chris: “I don’t follow the distinction you’re drawing. If the preference relation cannot be represented by a continuous utility function, then the ratio of marginal utilities isn’t defined, but neither is MRS.”
It’s between (a) postulating the existence of a utility function and from that deriving predictions about consumer behaviour and (b) postulating certain assumptions about consumer behaviour and from that deriving a utility function. If you start with the MRS and then go on from that to derive the existence of a utility function that’s approach (b), and that’s the distinction I’m drawing.
I agree completely with what you say in your second paragraph.
Francis, I don’t like the professor’s wording either, but it seems to me that the point of the question is that two theories can have different epistemology even when they appear to have some mathematical equivalence.
That’s a VERY important point when discussing models from which we intend to draw inferences. For instance, we might agree on the meaning of probability–such that we can compute probabilities in a finite space by enumeration, but then interpret probabilities differently: i.e., frequentist vs baynesian interpretation. (Are there any frequentists still !?!?)
“True or false: There is no difference between these two equilibrium equations in Eco 100 consumer theory as one equation can be transformed mathematically into the other (a) MUx/MUy=Px/Py (b) MUx/Px=MUy/Py.”
What does “no difference” mean that’s what this matter hinges on. If you insist on thinking about “no difference” in purely mathematical terms, you get one answer, but as Bill Wooley explained earlier MUy/MUx is a notation, not a quotient. In particular the theory in question states that the only the relative prices reveal the relative utility). MUy/MUx is a direct rejection of that calculus of utility implicit in MUy/Py; MUy and MUx cannot be known as individual quantities. Only their ratio can be observed by the manifestation of Py/Px. That’s the whole point the notation is making…
The question appears to take as given the terms are mathematically convertible, so it somewhat hints that it isn’t asking about mathematics. Its asking about the epistemology.
Your best friend’s son is going to a good school.. but I find the the solution in the “Update #2” to be quite rather muddled in making the point.
Jon: “two theories can have different epistemology even when they appear to have some mathematical equivalence.”
Yes, that would be true of revealed preference theory and utility theory.
That is not true of “utility theory” and this so-called “indifference theory”.
You write: “MUy/MUx is a direct rejection of that calculus of utility implicit in MUy/Py”
How could “MUy” be a direct rejection of the calculus of utility when MUy is defined in terms of utility calculus, i.e. dU/dy?
“Your best friend’s son is going to a good school..”
That is totally and utterly irrelevant. The variance in the quality of instruction within universities is much greater than the variance between universities.
Min:
I learned the basis of calculus using limits. Besides, most of the interesting questions are actually related to integration or the solving of differential equations where you use Laplace Transformations or weirder things like the Method of Froebenius. And you have to use limits when you integrate to infinity anyway so it serves a double purpose.
Then there is complex number calculus, in which we get interesting results like any integral over a closed path being zero (Cauchy’s Theorem).
Speaking of limits, a certain economist on this blog got caught short by not invoking L’Hosptial’s Theorem when he should have.
Francis,
The the point is that you can define a concept of MU, and a concept of MUy/MUx but MUi is unobservable, and not computable (neither is MUi/Pi). That’s what there being no calculus of utility means. In this line of thinking, MUy/MUx cannot be calculated directly, but happens to be observable by inspection of the relative prices, i.e., from Py/Px.
The inventor of this theory is none other than one of the founders of marginal analysis in economics… Carl Menger, so please lets not call it a “so-called indifference theory”. Indeed, in fact this is what we know as Marginal Utility. “Utility” as a quanity is something entirely different–something that does not exist, and therefore there is no calculus of utility.
Or in Menger words:
Jon: “The inventor of this theory is none other than one of the founders of marginal analysis in economics… Carl Menger, so please lets not call it a “so-called indifference theory”.”
I’m afraid history of thought is not my area. This is from the UK edition of Dick Lipsey’s text:
http://www.oup.com/uk/orc/bin/9780199563388/01student/interactive/lipsey12e_extra_ch05/page_02.htm.
So perhaps this idea is out there some places. It’s certainly not one that’s in common currency, as you can see from the comments on the blog.
If you want to get some idea of how widely used the term is, this is what Wikipedia says on the subject:
“The page “Indifference theory” does not exist. You can ask for it to be created, but consider checking the search results below to see whether the topic is already covered.”
And again, it’s a restatement of the ordinal/cardinal utility point. And again, it’s a point that doesn’t matter here. A monotonic transformation of U(X,Y) will change units, but you still end up with both expressions.
Stephen,
not sure that the ordinal/cardinal point doesn’t matter. I think the issue is that for a given set of prices, which are observable, one of the expressions is invariant to choosing a different cardinal representation of the preference ordering, the other isn’t.
If you start with a choce of utility function that represents the preference ordering and is consistent with observed prices, then apply a monotonic transformation to the utility function in order for b to be true under the new utility function you need to multiply all prices by the derivative of the transformation.
On the other hand, a is true for all representations of the preference ordering.
I matters because prices are observed, thus should be invariant to the utility representation. Thus you want to work with the invariant expression, that’s (a), not (b).
Adam P “Thus you want to work with the invariant expression, that’s (a), not (b).”
The thing is, you never actually “work with” either expression. All you observe are prices and quantities demanded. Both (a) and (b) are ways of representing a consumer’s state of mind and, together with the budget constraint, define the consumer’s demand function.
I have to say, I find expression (b) much easier to think about – in fact the only way I can ever get to expression (a) is to derive it from expression (b).
The idea that expression (a) is a concrete expression of reality somehow divorced from utility theory makes no sense. When you ask people to think about their own personal marginal rates of substitution you’re asking them to set U=K i.e. keep their own utility constant. Indifference as a concept is meaningless without reference to a preference ordering. (If you want to draw a distinction between preference orderings and utility that’s fine, we’re back in the world of revealed preference, but in that case don’t talk about marginal utility in the first place).
I don’t like the MRS much because it involves this weird thought experiment – how much of this could you give up for how much of that and be just exactly as well off as you were before. It’s extraordinarily difficult for people to conceptualize because in real life no one ever makes these kinds of trade-offs.
Expression (b), however – allocate resources so that the value to you of the last dollar spent is the same for every purchase you make – just makes total sense.
And as Linda says about 60 comments ago, expression (b) is much easier to generalize to a multi-good case.
“True or false: There is no difference between these two equilibrium equations in Eco 100 consumer theory as one equation can be transformed mathematically into the other (a) MUx/MUy=Px/Py (b) MUx/Px=MUy/Py.”
Based solely on reading the question I would have said true. In the context of intro micro (bear in mind I’m not an economist, I only play one in blog comments) don’t you write down a budget constraint, you write down a utility function, you assume the functions are well behaved, maximize utility subject to the budget constraint, and presto? So by definition and assumption both expression are describing the same thing (given well behaved utility and budget constraint functions).
The cardinal/ordinal thing seems like a stretch, especially since the question wasn’t something like “which expression is still valid if we mess around with cardinal representation of the preference ordering”.
-1 to the Prof for asking an unfair question.
Frances Woolley: “I don’t like the MRS much because it involves this weird thought experiment – how much of this could you give up for how much of that and be just exactly as well off as you were before. It’s extraordinarily difficult for people to conceptualize because in real life no one ever makes these kinds of trade-offs.”
Gee, I thought I made that kind of trade-off while grocery shopping.
OC, I have a different equation in mind:
PxΔx = PyΔy = Δ$
where Δ$ is small.
Min: “Gee, I thought I made that kind of trade-off while grocery shopping.”
This comment just illustrates how few people really understand the notion of indifference – the “equilibrium equation” described in (a) above is not something you would have experienced while grocery shopping, unless you are the type of person who takes time out from your grocery shopping to ponder the nature of choice.
In the grocery store hardly anyone, ever, thinks to themselves “I could take an apple out of my shopping basket and replace it with a mango and be just as well off as I am right now because the relative price of apples and mangoes is just exactly equal to my willingness to substitute the two goods.”
People think “I’m going to buy apples instead of mangoes because they’re equally delicious and apples are cheaper.” Or people think “I’m going to buy mangoes instead of apples because even though mangoes are more expensive, they taste better.” (Unless they’re in a tropical climate, where mangoes are cheaper than apples).
In other words, people in terms of how to make themselves better off, how to move from one indifference curve to another. People rarely if ever think in terms of indifference, i.e. “I could do this, or not, and it would make absolutely no difference to me.” When they do think in those terms, often they’re talking about a good that gives them no utility at all, i.e. MUx=0.
Frances Woolley: “People rarely if ever think in terms of indifference,”
Who said they have to do so to make trade-offs? Thinking in terms of indifference is for theorists.
Frances:”In the grocery store hardly anyone, ever, thinks to themselves “I could take an apple out of my shopping basket and replace it with a mango and be just as well off as I am right now because the relative price of apples and mangoes is just exactly equal to my willingness to substitute the two goods.”
It simply means that ,in our socio-economic class, have no budget constraint in the grocery store,only satiation. The process is so fast and automatic we don’t ever realize it took place.But a welfare mom will have to judge both how to attain her highest indifference curve and how to stay on it.
As a did last time I bought a car. And my ex,on a disability pension, who took to year to choose her new car.
If you are in the 1% though( better yet the 0.1%), I agree with you. Except that the MU of buying a bigger megayacht than your neighbour is priceless.
Min: “Who said they have to do so to make trade-offs? Thinking in terms of indifference is for theorists.”
All of the alleged advantages of the MUx/MUy formulation rely upon the notion of indifference – only by keeping the consumer on a single indifference curve can we eliminate any need to talk about the gain in utility from moving from one indifference curve to another.
If you’re coming around to the point that the difference between “indifference theory” and utility theory is an obscure distinction unsuitable for Eco 100 then, good, I agree.
If all we’re interested in is the conditions which define consumers trade-offs then, fine, expressions (a) and (b) above are equivalent – which is what I’ve been maintaining all along.
Jacques Rene – “It simply means that ,in our socio-economic class, have no budget constraint in the grocery store,only satiation.”
Not at all. If you are a welfare mom, you are of course thinking about how to get onto the highest possible indifference curve.
What you don’t do is waste time thinking about various combinations of goods that are alternative ways of obtaining that highest possible level of indifference. Imagine saying to a welfare mom “right now you have 2 boxes of pasta and 1 bag of potatoes in your shopping cart. I’m going to take away 1 box of pasta. How many more potatoes would you need to have to be just exactly as well off as you were before.”
She would – rightly – say to you – “Give me back that box of pasta and quit asking me stupid questions.”
We think in terms of making ourselves as well off as possible. We do not think in terms of indifference. It’s a really really unnatural way for animals focussed on survival to think.
AdamP: “. I think the issue is that for a given set of prices, which are observable, one of the expressions is invariant to choosing a different cardinal representation of the preference ordering, the other isn’t.”
Start with MU_X/P_X = MU_Y/P_Y. Apply a monotonic transformation g() to the utility function and you get:
g'()MU_X/P_X = g'()MU_Y/P_Y,
which is the same expression since the g'() terms cancel. There is no sense in which one of those equations requires cardinality and the other does not.
I actually use the formulation: MUx/(Muy/Py) = Px. The LHS is willingness to pay; the denominator is Marshall’s Marginal utility of money in a 2-good world. This is how I get from the consumer equalizing MU per dollar to demand for x.
Jacques René Giguère: “Except that the MU of buying a bigger megayacht than your neighbour is priceless.”
😉 😉 😉
Frances Woolley: “If you’re coming around to the point that the difference between “indifference theory” and utility theory is an obscure distinction unsuitable for Eco 100 then, good, I agree.”
Well, that’s in part why I suggested summary execution. 😉 In part it is because if you want to make that distinction, comparing those equations is a lousy way to do so.
In response to you I was just thinking about experimentation. As I said, that actually led me to a different equation, not in terms of infinitesimals, but in terms of small differences. A supermarket does offer many choices involving small differences in both quantity and price. Some people find that paralyzing. 😉
As for the, ahem, utility of talking about utility, I doubt if that is an experimental question. Assuming that you can dispense with it, does that make things more complicated or less complicated?
Oh, this thread makes me so weary! I HATE the way we (well, many of us) teach this material in Principles, and I think the question in question is simply horrible. Students (in my experience) understand cardinality versus ordinality without much trouble at all (until we confuse them with “tricky questions” such as this, that is….). And the very simple idea that one will always spend an extra dollar on whatever good yields the greatest happiness is blindingly obvious to 99% of first year students. It is just a couple of simple steps to get them from this point to utility maximization, and you can do it without EVER using the dreaded words “marginal utility. Finally, its not that hard to convince them that diminishing MU is a total red herring: we give them too little credit to assume otherwise.
I think the way we teach consumer theory in Principles is the single biggest reason that many students flee econ in droves after first year. That and the fact that we drag them through weeks of cost-curve tedium.
All IMHO, of course
I blame Paul Samuelson and his incompetent attempt to create “science” — see Stanley Wong’s devastating analysis in his important book.
Did Hicks use “utils” in his original indifference curve work?
Hayek directed Hicks to Pareto and suggested he develop the indifference curve mechanics using Pareto’s work as a model.
Hayek would be appalled at the destruction to thinking competently these constructions have so often unintentionally produced.
“Utils” indeed.
No one has had this thought:
“If you are a welfare mom, you are of course thinking about how to get onto the highest possible indifference curve.”
Well said:
“We think in terms of making ourselves as well off as possible. We do not think in terms of indifference. It’s a really really unnatural way for animals focussed on survival to think.”
Emma “I HATE the way we (well, many of us) teach this material in Principles”
Agreed. It’s just so easy, though, to go on doing things the way we always have done and never step back and really think about what we’re doing. And coming up with a different way of doing things is so much like hard work…
Greg “I blame Paul Samuelson”
I wonder how the marginal value of time devoted to improving first year texts compares with the marginal value of time devoted to cutting edge research?
Frances: “I wonder how the marginal value of time devoted to improving first year texts compares with the marginal value of time devoted to cutting edge research?”
I think we need to distinguish between the social and private marginal values here: seems to me the social value is huge for the former, and small for the latter (on average). However, significant improvements to a first year text would not be recognized for a while, and would likely have little impact on salaries.
Frances: when I said “our socio-economic clas”, I meant the participants to this blog. We are in the 75-90 ( mostly). A welfare mom must think whether she will buy pears or apples.
Greg: Samuelson wrote for future professionnal economists when economics began to suffer from physics envy. Physicists were beginning to understand the weird implications of quantum physics. So we tried to stomach purely intellectual constructs like indifference curves.
Then we try to expand economics to people who had not intention nor interest in professional economics. No wonder they are confused.
There are very good discussions about “utils” ans ” indifference curves” and even “demand” and “supply curves” in
Alfred S. Eichner ed. “Why economics is not yet a science” from 1983, still relevant.
Linda: in the early ’80’s ,still young and foolish, I was explicitly told that no work I would do on authoring a textbook would be recognized by the econ. dept. But I could apply for credit at the Education Faculty ( is that a real faculty?). Professionnaly, I couls also put a bullet in my brain.
Linda: “I think we need to distinguish between the social and private marginal values here” True, so true!
As you say Jacques Rene, the professional equivalent of a bullet through the brain…
I think the issue here is that Econ 100 isn’t completely standardized; terminology, definitions and even assumptions vary, even though the analytical methods are relatively constant. The question in question isn’t analytical – it’s just a definition, one that someone who was studying from the textbook/lecture notes it was based on would have been able to answer.