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 
As a choice condition, they are clearly equivalent.
I suppose the equations themselves are saying different things, but this is a pretty pedantic/pointless thing to ask, imo. After all, the following expressions are clearly equivalent, but are ‘saying’ different things:
1 + 1
2 * 1
I’d have said true as well. I don’t see how the ordinal/cardinal point enters into it. If you redefine preferences by V(X,Y) = g(U(X,Y)) where g(U) is monotonic increasing, then you’d still get the same expressions when the g'(U) terms cancel.
Well, I voted “true”. I have seen it taught both ways, and have taught it both ways myself. I can’t figure out any devious reasoning why it should be false. And we don’t (at least not deliberately) ask “trick” questions in ECON 1000.
Every so often, in every multiple choice test bank, and in study guides, I find the occasional mistake. I always do the test myself, then compare my answers to the “official” answers. I think this was a simple typo.
And the cardinal/ordinal thing doesn’t work. As soon as you write “MUx” you are assuming cardinality. So both versions assume cardinality.
Gotta be a typo. Stuff happens. (Or I’m totally out to lunch, I suppose.)
I think I would contact the prof, and ask what we are all missing: I would certainly have answered “true”. (b) can be generalized to more than two goods more easily.
And i didn’t peek at Stephen’s or Jared’s answer first, honest teecher!
I would also add that I too can’t see why anyone would see the point in asking this question.
Nick “I think this was a simple typo.”
I don’t think so, I was quite careful to try to get this right, and the solutions (which I don’t have in front of me) do contain an explanation of why the answer is supposed to be false. But I will double check. And the question does say “equilibrium equations” when they are, in fact, optimization conditions.
Linda, I’ve always preferred formulation (b), too, and am glad to have a reason for this.
Jared, Stephen, Linda, I’ll see if I can get some clarification on why false. Because I can’t see it either.
In physics torque and energy both have the same dimensions but are not exactly equivalent. Torque is measured in Newton-metres (Nm) where the force (newtons) is perpendicular to the distance (metres).
Energy (Joules) is where the force and the distance are in line, it is force acting over a distance.
Torque can be converted to Energy by multiplying by the angle in radians (a scalar).
Thus the distinct notation of Nm and J.
This is a little more slip-shod as there is no conceptual difference like perpendicularity.
It seems to me to simply be a difference of interpretation of something which is dimensionally equivalent. For instance in Newtonian physics force and energy are interrelated and provide an equivalent solution through different mathematical operations.
Hmm, is there a math philosopher in the house?
Determinant: “Thus the distinct notation of N*m and J.”
I think this is what the instructor was trying to get at.
I should also note that I never understood choice theory in terms of ‘equilibrium’, so the language used is strange to me anyway. Better wording: (?)
T/F: the following conditions are both utility-maximizing
i) MUx/MUy = Px/Py
ii) MUx/Px = MUy/Py
(but now maybe I’m the one being pedantic..)
Ah, Frances beat me to it.
Hmmm. If there are solutions, it’s unlikely to be a typo. Oh well. I always end up proving England will export wine to Portugal.
I wouldn’t worry much about the equilibrium/optimisation thing though. I would say it’s both. Optimisation is a necessary condition for equilibrium, and it’s expressed as an equation, so I don’t see the problem in describing it as an equilibrium equation. Do others see it differently?
Determinant: I follow what you are saying about torque and force being different, even though they have the same units, but you lost me after that.
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 that only works if we treat (MUx/MUy) as a whole, that you cannot unpack as one thing divided by another thing. But in that case, you would just write it as MRSxy instead.
Ok, then I didn’t get the lesson. I also didn’t take that lesson, I was busy in First Year Engineering and took two History courses for my electives.
Nick is also right when he says that if you asking a question about dimensionally identical units with different models/interpretations, then you ought to be very clear about the units and terminology you are using. Again first-year engineering profs are very clear about this.
This question is both obscure and insufficiently specified.
If you can directly derive the one from the other (which obviously you can) then they are equivalent. There really should be no debate here.
From 400 BC to 300 AD, greek mathematicians studied how to compute the surface of polygons through the method of exhaustion. But applying it to very complex shape was almost impossible and seeing only the geometrical application of that method made them completely miss out on inventing calculus.
It took 13 centuries for Newton and Leibnitz to reinvent calculus starting from new bases. And today we essentially use only Leibnitz approach.
Even though units may be the same, you need to know what you are looking for and understand whence you are coming when you build a model.
Even though Nxm and J are the same units, an engineer write them differently because it is very important to know whether a part is subject to torsionnal forces or not.
We should be as careful about what we ask our students to think and how to think about the what.
why did they ask this question in econ 1000? It is really not what matter…
I have been teaching first year class for some time now and would have pick A….
I would however never have asked such silly question though…
Nick is right that (MUx/MUy) does not require utils, which is the point of the question. Some principles texts–including Krugman’s–use (MUx/MUy) as the notation for MRSxy, and in the context of a particular course this would not be confusing to students who had been paying attention and reading carefully. The choice of notation, although not precisely correct, is used to highlight that the condition is mathematically equivalent to MUx/Px = MUy/Py, but that economically the interpretations of the two equations are slightly different (ordinal vs. cardinal). I don’t know of any books that clearly explain the distinction between diminishing marginal utility and diminishing MRS until intermediate micro; There are, however, professors making this distinction as part of the principles coursework.
Given Economics recent complete failure and given Steve Keen has shown it to be nonsense why would anyone want to pass Eco 101.
Wearn_and_worn,
Given that you’re probably the only person on this thread who believes that, why would you ask such a rhetorical question?
Nick Rowe,
“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?”
To avoid this problem for universities, perhaps Canada should move onto the Scottish system: universities get ample home undergraduate students at the taxpayers’ expense (and those students can’t get free higher education elsewhere) so they can focus on getting extra money through government research funds rather on that old-hat “educating” rubbish.
Nick: ” 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.”
This is what the prof was getting at, and this is the reason that the answer to the question was supposed to be false.
John: “why did they ask this question in econ 1000?”
I can think of four explanations:
1. It’s in the test bank, and someone is assigning test bank questions without thinking about them
2. To manipulate the grade distribution; without questions like this there would be too many As
3. To reduce the size of upper year classes; without questions like this people might major in Econ
4. The prof was trying to measure whether or not students really understood the units used in different types of economic measures, but picked a lousy way to do it.
Rather that focusing on the cardinal/ordinal business, suppose that the two ways of showing the expression reflect two different approaches to the problem described in class–allocating money income to different goods vs. indifference curves. The result immediately appears in one of those two forms. Each approach has advantages and disadvantages. The results are quite similar. Mathematically, the results can be expressed in a way that looks identical. But they come out of different traditions and different ways of looking at the problem.
By the way, my answer was “True.” But you have to admit that with a true answer, it seems like an odd question.
You can say that the ratio of marginal utilities approach comes out of an approach that focuses on preference orderings (for combinations of products) and the ratio of marginal utilities to prices come from allocating income to different goods until the utils per dollar of income are equal, which suggests utils are measurable units of pleasures and was developed by people who thought along those lines, but I the mengerian approach is subjective and doesn’t use indiffenence analysis.
If you were in class that day and paid attention to the discussion of the two different approaches brought to question by economists (and maybe cost of production theory too,) then the answer is obvious.
Bill “suppose that the two ways of showing the expression reflect two different approaches to the problem described in class–allocating money income to different goods vs. indifference curves….Mathematically, the results can be expressed in a way that looks identical. But they come out of different traditions and different ways of looking at the problem.”
And I think that’s why the phrase “in Eco 100 consumer theory” is crucial. Because in any other kind of consumer theory, the consumer’s choice problem is max U(X,Y) subject to PxX+PyY=M, and then the two equations given are simply alternative ways of writing the first order conditions. We’re all saying true because we’re not thinking in terms of Eco 100 consumer theory.
But it seems a bit tough to expect first year students to get that.
And also you have to think “in a typical exam situation, who is going to choose the answer false.” It’ll be three types of students
– people who are totally lost so just guess
– people who choose their answers strategically “i.e. the answer is obviously true therefore the answer must be false.”
– people who reason as you suggest here
I would guess that the third group will be much smaller than the first two. So the question won’t do a good job of differentiating between students who know the material and students who don’t.
I’d be willing to bet that the question also skews in favour of males educated in Canada, because they’re more likely to try to out-psych the prof.
Sorry, for “Nick Rowe” read “Frances Woolley”. The similar writing styles and the fact that the name isn’t at the top of the article fools me time after time.
W. Peden “The similar writing styles” Really?
“the fact that the name isn’t at the top of the article”
We’re actually in the process of doing a minor blog tweak. Would you like to have the names of the authors more prominently displayed at the top of the article?
This reminds me of how calculus used to be taught. You would see the notation, dx/dy, but the student was cautioned not to treat it as a regular quotient, because it represented the limit of Īx/Īy as both go to 0. Later calculus texts dispensed with that notation.
In these days of non-standard analysis where infinitesimals are first class objects, it really does not matter anymore. And besides, IIUC, marginal utilities are not infinitesimals, anyway, right?
Now the term, “equilibrium”, makes me wonder about psyching out the prof. In general, equilibrium requires continuity. Otherwise, we might get oscillation, for instance. As a practical matter the assumption of continuity is a mathematical convenience.
Now it may be that the prof is treating marginal utilities as infinitesimals in one equation, but not the other. In which case, he should be shot.
In any event, if I were grading the test, I would flunk the prof. š
Frances Woolley,
If there are differences in the writing styles, I haven’t noticed them.
Yes, I think it would help a little to have the author name prominently displayed. Since neither of you always stick to a narrow topic or agree about everything, it would help to get a sense of who the author was before reading the blog.
If there are differences in the writing styles, I haven’t noticed them.
Ouch. You sure do know how to hurt a blogger’s feelings…
Laughing! Brett says he can usually spot who’s writing the post in the first few lines, but then writing’s his profession.
One bit of advice I tell my students: on a multiple choice (or T/F) exam, always choose the answer you think the prof thinks is right, not the answer you think is right. Essays are the place where you get the chance to argue what you think is right and against what the prof thinks is right. MC questions just aren’t any good at letting you do this. MC questions have their role, but this isn’t it.
Some other economics prof (John Palmer?) said he always gives the students the option to write him a short note afterwards arguing their case on an MC question he has graded wrong. Not a bad idea in principle. But few take him up on the offer.
Perhaps if one of you started off with a split infinitve, it would be easy to swiftly and unambigiously identify who is blogging…
For the record, I answered “FALSE” to the quiz, but only because of the technicality that the two expressions are not mathematically equivalent when MUx and Px are both zero. I am evangelical in teaching that utility is only ordinal, and that the assumption of diminishing marginal utility when teaching indifference curves is meaningless, but I still teach why optimisation at interior solutions requires MUx/pX=MUy/py and why that formulation is useful for constructing arbitrage arguments for the tangency condition.
Maybe in second year. But in Eco 100?
Update: I managed to obtain the “official” answer to the question:
“There is no difference between these two equilibrium equations in Eco 100 consumer theory as one equation can be transformed mathematically into the other
MUx/MUy=Px/Py
MUx/Px=MUy/Py
Answer: No.
There is a difference! First line above 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 refers to Utility Theory which does require measurement of satisfaction.
I don’t get it. As I noted above, you get the exact same result if you apply a monotonic, increasing transformation to the original utility function.
“There is a difference! First line above 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 refers to Utility Theory which does require measurement of satisfaction.”
So the variables in the different equations have different meanings. In which case summary execution is the merciful path to final equilibrium. š
Now see here Min I was taught that dy/dx should not be treated as a regular quotient and I’m under 30!
Though you can split them if you are differentiating or integrating you only do it under certain circumstances which require it. Otherwise you leave it alone.
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. Any change in units of measurement on one side of the second equation will be cancelled out by the same change on the other.
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?
It’s time for someone to summon up the courage to set up an anonymous e-mail account!
Determinant: “Now see here Min I was taught that dy/dx should not be treated as a regular quotient and I’m under 30!
“Though you can split them if you are differentiating or integrating you only do it under certain circumstances which require it. Otherwise you leave it alone.”
Perhaps we are seeing the fruits of non-standard analysis. š My college text, IIRC, did mention dy/dx, but only as an alternative way of writing d/dx(y), where d/dx(.) represents a function. Splitting was not allowed.
I now have sympathy for students complaining about marks.
d/dx(y)??? That’s nonsense. Neither my OAC Calc textbook or my first-year calc book used that notation.
First, y=f(x).
Differentiating: dy = f'(x)dx The variable is reduced to infinitesimals.
So dy/dx = f'(x).
On more splitting, however else are you going to solve a 5% growth problem to project population:
If the growth rate is 5%, Population is P and t is time, then dP/dt = 0.05P
dt = dP/0.05P
=20dP/P
Integrating, or anti-differentiating,
T = 20lnP
P = exp(0.05t)+C, the initial population.
“We’re actually in the process of doing a minor blog tweak. Would you like to have the names of the authors more prominently displayed at the top of the article?”
Most emphatically, yes. Seriously. As an amateur observer the name tells me alot. At the risk of overgeneralizing Nick Rowe = “trying to understand a grand model while arguing with Delong and Krugman and/or explaining better left wing ideas than what is in the NDP policy book”. Frances = “applying the theory to a small-scale real-life event and/or musings on role of higher-ed in Canada”. Livio = “nuts and bolts and data about Ontario”. Stephen: “more businessy”. The guy from Haute etudes who guest posts = “smart stuff on finance”
Now that I have offended all of you I will sign off by saying I enjoy it all even when I only understand to the third paragraph.
Chris J – thanks for the feedback. The only person you’re likely to have offended is Mike Moffatt š
My self-description on twitter is “I theorize about life” so your perception of me, anyways, is pretty much spot on.
“businessy”?!?
Determinant: “First, y=f(x).
Differentiating: dy = f'(x)dx The variable is reduced to infinitesimals.
So dy/dx = f'(x).”
Eminently sensible. And that’s pretty much how people thought about infinitesimals in the beginnings of calculus. Later on, people began to worry about dividing by dx. In some ways, we treat infinitesimals as equal to zero, so dividing by an infinitesimal is like dividing by zero. In this case it gives us the right answer, but. . . . One solution to the logical enigma was to do the division to produce Īy/Īx, and then take the limit as they approach 0.
But then non-standard analysis provided infinitesimals with a sound logical basis. I suppose that some time later you had textbooks return to the previous style, as indicated in your post. š
As for d/dx, check this out.
y = f(x)
dy/dx = d/dx(y) = d/dx(f(x)) = f'(x)
š