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Why do mainstreamers care about monotonic transformations?

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abskebabs posted on Tue, Jan 4 2011 6:49 PM

I've been revising lately and reading Varian's Intermediate Microeconomics, and found something a little puzzling. The relevant chapter that piqued me was chapter 4, on utillity. He introduces and makes a big fuss about monotonic utillity functions and how they make the ordering of the transformed utilities remain the same as that of the "original" utillity functions. He then goes on to say, and somewhat assure the reader that these are not used in anything apart from establishing the "Marginal Rate of Substition"(MRS), and the idea that all they care about are relative preferences (I have objections to this too, but shall lay them to rest for now).

 

What puzzles me, is that if their only purpose is ultimately establishing the identity whereby the ratio of the partial derivatives of the utillity function equate to the MRS, dx2/dx1, which is later given some meaning by equating it with the slope (for some reason economists don't like the word gradient...) of the budget line, then why do economists bother with making them "monotonic" at all? I mean just from the chain rule, I know any trivial transformation would satisfy that identity, since if the new utillity is f, and the old one is u, so f=f(u), then:

 

(df/du*du/dx1)/(df/du*du/dx2)=(du/dx1)/(du/dx2)=dx2/dx1

 

and this is true regardless of whether the tranformation is montonic or not, for instance, if f(u)=sin(u). So why do economists care about montonic utillity functions when they wouldn't make the blindest bit of difference to the results of the mathematical treatment?

 

I can't think of any apart from the fact they may arbitrarily seem "intuitively sensible", or possibly to backup crackpot welfare analysis making interpersonal comparisons of utillity. Are there any others?

 

Or have I misunderstood something very simple here and come across very foolish? blush

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The error is not a simple one.  The point of the positive monotonic transform argument is that the optimal choice of the individual is invariant to any such transform.  That is, given some utility function u, the individual chooses an optimal bundle (x*,y*) (for the 2 good case).  The choice (x*,y*) will not change given any positive montone transform of u, call it h.

However, any non-positive monotonic transform will generally change the optimal bundle from (x*,y*) to (x**,y**).  This is despite the fact that the MRS remains the same for any transform; that is, regardless of whether it is positive or not.  This is only violated when the implicit function theorem is violated.  And this occurs when you get the 0 in the dominator (as discussed above). 

For example, suppose u(x,y)=xy.  Then the solution is x*=w/2p_x and y*=w/2p_y (w is wealth). 

Now, if h(u)=-xy, then the solution would be  {(x, y): xy=0, x>=0, y>=0, p_xx+p_yy<=w}-i.e. there would be set of solutions, not just one.

The MRS for the 2 functions are the same.  But the solutions are not.  The mathematical treatment is therefore distinctly different.  When solving any of these problems the general method is to use what's called the Kuhn-Tucker Conditions.  You would find that under the new utility function h(u), the Kuhn-Tucker conditions would be violated at the old solution. 

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I think the worry is that if they admit non-montonic transformations into their analysis they have given up on ordinal utility and have adopted cardinal utility (which sophisticated neoclassicals do not want to do). On the other hand, I am very sleepy and might be talking nonsense.

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Any other thoughts?

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They use monotonic transformations because they realize that cardinal utility is theoretically untenable. It's how they side-step (rather than resolve) a major theoretical problem with their analysis, that's all.

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"(df/du*du/dx1)/(df/du*du/dx2)=(du/dx1)/(du/dx2)=dx2/dx1

 and this is true regardless of whether the tranformation is montonic or not, for instance, if f(u)=sin(u)."

Not true; if f(u) = sinu, and u = pi/2, then df /du = cos(u) = 0 and you cannot divide by df/du.

In other words, I'm saying maybe monotonically increasing transformations are required here merely to avoid minor yet annoying mathematical problems?

Edit: another theoretical benefit is that a monotonically increasing function everywhere differentiable has a unique inverse that is a one-to-one function. If you admit functions like sine then I can say an apple gives me utility 0.8; an orange, -0.6; and if I've applied the sine function you have absolutely no clue which piece of fruit I prefer.

Edit 2: If I apply the monotonically non-decreasing function f(u) = 42, so that df/du = 0,  then the equations will blow up everywhere. I've destroyed all information regarding values with this transformation.

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Edit 2: If I apply the monotonically non-decreasing function f(u) = 42, so that df/du = 0,  then the equations will blow up everywhere. I've destroyed all information regarding values with this transformation.

They define utility functions in such a way as to avoid this problem. I think they are probably right for doing so.

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Not true; if f(u) = sinu, and u = pi/2, then df /du = cos(u) = 0 and you cannot divide by df/du.

Fair point. So you would need to make sure the first derivative didn't zero. In many cases montonic functions would produce the same problem too, if they have a stationary point of inflection. I'm guessing those would be disallowed too?

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>In many cases montonic functions would produce the same problem too, if they have a stationary point of inflection. I'm guessing those would be disallowed too?

Yeah, a monotonic non-decreasing function can stay constant for a while. A monotonic increasing function can't do this - it will always have a postiive derivative (or an infinite/undefined derivative if it "jumps", which is probably best to forbid as well).

Note that I know nothing about mainstream economics and anything I say is based just on math knowledge.

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The error is not a simple one.  The point of the positive monotonic transform argument is that the optimal choice of the individual is invariant to any such transform.  That is, given some utility function u, the individual chooses an optimal bundle (x*,y*) (for the 2 good case).  The choice (x*,y*) will not change given any positive montone transform of u, call it h.

However, any non-positive monotonic transform will generally change the optimal bundle from (x*,y*) to (x**,y**).  This is despite the fact that the MRS remains the same for any transform; that is, regardless of whether it is positive or not.  This is only violated when the implicit function theorem is violated.  And this occurs when you get the 0 in the dominator (as discussed above). 

For example, suppose u(x,y)=xy.  Then the solution is x*=w/2p_x and y*=w/2p_y (w is wealth). 

Now, if h(u)=-xy, then the solution would be  {(x, y): xy=0, x>=0, y>=0, p_xx+p_yy<=w}-i.e. there would be set of solutions, not just one.

The MRS for the 2 functions are the same.  But the solutions are not.  The mathematical treatment is therefore distinctly different.  When solving any of these problems the general method is to use what's called the Kuhn-Tucker Conditions.  You would find that under the new utility function h(u), the Kuhn-Tucker conditions would be violated at the old solution. 

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> suppose u(x,y)=xy.  Then the solution is x*=w/2p_x and y*=w/2p_y (w is wealth). 

This makes sense; I calculated the same result.

But really, unless the individual wants to blow all their money on goods X and Y, shouldn't it be u(x,y,w)? So the individual could transition from some state (0,0,w) to (x*,y*,w*)?

To do this, also won't there have to exist complementary individuals who transition from (x,y,0) to (x*,y*,w*)?

Also, is it assumed the individual will pick the global maximum? Even for a computer, finding the global maximum in a multi-dimensional problem involving integer solutions can be unbelievably expensive. When real values are allowed, I'm guessing it can be impossible. This poses two problems:

1. The model has another level of unrealism in that it endows actors with god-like mental capabilities.

2. Actually solving your model can be impractical since you yourself need the same god-like computing abilities to do so.

Note: the problems can be alleviated somewhat if you allow only linear utility functions. Though even these problems are quite nasty:  http://en.wikipedia.org/wiki/Knapsack_problem

Finally, what happens if there are multiple global maxima?

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But really, unless the individual wants to blow all their money on goods X and Y, shouldn't it be u(x,y,w)? So the individual could transition from some state (0,0,w) to (x*,y*,w*)?

I was just stating the most basic type of problem.  Usually the problem is given as follows. 

There is an individual with a utility function u.  This individual is endowed with, or awakes with, or is spontaneously given, an income; commonly denoted as w, I, or m.   This endowmend w can be thought of as something like money in that the individual can use it to purchase goods but it does not itself directly contribute to utility.  That is, w does not enter in the utility function.  Only goods enter into the utility function; hence u(x,y) instead of u(x,y,w) (of course, there are problems where this is not the case).  Moreover, w is given to the individual, or put differently, it is exogenous to the model, or a paramater of the model.  The indiviual cannot affect it.  The individual can choose x and y of course-commonly called endogenous or choice variables. 

Therefore the problem boils down to:

An Individual maximizes u(x,y) by choosing x and y given that his choice must satisfy p_x*x+p_y*y<=w.  That is, he cannot consume more than his given endowment w.

To do this, also won't there have to exist complementary individuals who transition from (x,y,0) to (x*,y*,w*)?

In this simple problem there is just 1 individual.  We only want to find what he would choose given an endowment w, without thinking about other individuals and so forth.  It is just a very basic choice problem. 

Of course, one way to enhance the model, is to introduce multiple individuals.  The most basic problem is to consider an endowment-exchange economy.  In this case there are M individuals, each with their own god-given endowments (however, now the endowment is in the form of goods, not some blob like w).  As before, each individual maximizes his utility given his endowment.  However, the problem changes in the sense that not only must each individual maximize his utility, it must also be that the optimal bundles chosen by each individual are in "harmony" with one another.  In other words, there is the additional requirement that the supply and demand for each good must be equal.  And this will only occur if there is a set of positive prices for which it is true.  This is commonly called a Walrasian equilibrium.  A Walrasian General Equilibrium is nothing more than a set of prices, such that each individual has chosen a bundle that maximizes his utility and the demand for and supply of those bundles are equal (markets clear).  

 

Also, is it assumed the individual will pick the global maximum? Even for a computer, finding the global maximum in a multi-dimensional problem involving integer solutions can be unbelievably expensive. When real values are allowed, I'm guessing it can be impossible. This poses two problems:

For the most part, at least in the first year of a standard PhD program, you're mainly concerned with finding global maxima-this is consisten with the idea of a "rational agent".  And it is true that this is not easy.  A lot of time is taken to first: describe and understand when global maxima exist (Weierstrass Theorem); and second; to know whether we can say a local maximum is a global maximum.  In many cases neither of these conditions is fulfilled.  That is why much of the literature restricts itself to only certain types of utility functions.  These types of functions are referred to as quasi-concave utility functions.  Certain theorems show that when an individual has this shape of a utility function, we can be certain that if he does find a "solution bundle", that this bundle will be his global maximum (there are other parts to the theorem as well-restrictions on his "budget set" and so forth). 

And usually the literature does not restrict the solutions to be integers.  This would certainly be more realistic but I imagine it would cause some headaches.  That's not to say it hasn't been done.  I imagine there's a large amount of literature devoted to economic problems only involving integer solutions.  It's just not generally taught in the core of a PhD program.

And yes, the compuational cost of calculating solutions quickly becomes very large for even fairly basic problems.   

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Dr. Acula:

2. Actually solving your model can be impractical since you yourself need the same god-like computing abilities to do so.

Note: the problems can be alleviated somewhat if you allow only linear utility functions. Though even these problems are quite nasty:  http://en.wikipedia.org/wiki/Knapsack_problem

 

You appear to be better versed than I in mathematics. I have had an interest in researching Knapsack Problems with regard to the complexities of the economic calculation problem. Would you be able to elaborate a bit more on how they could relate to an economic context?

 

Thanks.

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For Alexander Zinoviev and the free market there is a shared delight:

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