6c. Marginal Rate of Substitution and Monotonic Transformations of Utility

This video uses calculus.

In this video, I use a simple example to motivate the result that monotonic transformations of utility functions do not change the preferences that are described by the resulting (transformed) utility function. After using the example to motivate this result, I prove it by taking all the right derivatives (using the chain rule).

FYI: The result in this video is why preferences depend on the utility function's ordinal properties (rather than its cardinal properties).

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[amsalorio]

how can you be such a great teacher? you are 10 times better than my proffesor at university! thank you very much!

[intromediateecon]

Most of the time. If U is a negative number, however, that's going to cause problems. It won't cause problems in the Cobb Douglas Utility case though because Cobb Douglas Utility is X^a * Y^b where (X, Y) are both positive... so we commonly transform Cobb Douglas by logging it.

[davidpark]

OMG... thank you so much for this lecture!! your lecture is really organized and easy to understand!

[intromediateecon]

If the two utility functions have the same slope everywhere (at every bundle, not just one), then yes, I believe that's correct. You do, however need to worry about weird things like flipping the sign (so that both goods become bads), but by and large (for most circumstances), that won't get you into trouble.

[LorenzoWake]

Is this the homogeneity/homotheticity concept? My microeconomics course does not even try to explain in plain language, so maybe you can shed some light on this for me.

I understand that homogeneity implies homotheticity (latter is monotonic transformation of the former), but when is a function homothetic but not homogenous? And do you maybe have a calculus example of this?

[miguelpereira]

If U = U(X, Y) can I always transform that utility function by rewriting V=log((U)^5)? Is that universally true for all utility functions? I mean, can that formula always be applied to any utility function? Thanks for the videos, you're pretty awesome!

[DavidAbel]

Oh muchas gracias, realmente me sirvio de mucho el ejemplo... desde Perú...

[davidm]

If one utility function is a monotonic transformation of the other, it is very clear that they have the same MRS.
But does the fact that the MRS of two different utility functions are same mean that they are monotonic transformations?
For example
U(x, y)= (x^2)*(y^2)
V(x, y)= -x*y

Clearly, the MRS is exactly the same, but I think that they are not monotonic transformation because one is an increasing function and the other is not.

Correct me if I'm wrong. Thanks!!

[jomonavi]

when you take the MUx, why isn't the y raised to the .6? since it's treated like a constant, shouldn't you multiply the whole thing, y^.6 times the derivative of x?

[nickdizzle]

thanks what are all the possible monotonic transformations?

[howardchan]

Can I say in a reversed way that if 2 utility functions have the same marginal rates of substitution then one of them MUST be a monotonic transformation of the other?

[gabspen]

Does this mean that if I am solving the consumers problem the optimal x and y will be the same after any monotonic transformation of the utility function?

[brianlibgober]

Aren't we taking the derivative of g with respect to different functions? g'(x, y) on the top should be dg/dx, while g'(x, y) on the bottom is dg/dy

[intromediateecon]

No. I used a rule of logs here: A = log(B^5) = 5log(B)