Prove that a diminishing marginal rate of substitution neither implies
nor is implied by diminishing marginal utility.

Hi dime365!!


At McGraw-Hill Higher Education website, in the 'Online Training
Center' I found the requested proof.
It is in the subsite of the book "Macroeconomics, 15/e" (Campbell R.
McConnell, University of Nebraska, Emeritus and Stanley L. Brue,
Pacific Lutheran University) at the Student Center:
http://www.mhhe.com/economics/mcconnellmacro15e/student_index.mhtml

The lecture is called "The Marginal Rate of Substitution", and it
starts as follows:
"Following the explanation in the text, you might expect that if two
goods each exhibit diminishing marginal utility, then the marginal
rate of substitution between them will also be diminishing. As we will
show below, the marginal rate of substitution is equal to the ratio of
the marginal utilities of the two goods, so this hypothesis seems
reasonable. Unfortunately, this hypothesis is not generally true if
the amount consumed of one good influences the marginal utility of the
other good. For example, an increase in your butter consumption may
decrease your marginal utility of margarine, or an increase in your
coffee consumption may increase your marginal utility of cream. This
note investigates the conditions under which the marginal rate of
substitution is decreasing, as hypothesized in the text."

Then shows work using partial derivatives utility function for two
goods X and Y: U = U(X,Y) to conclude "...with the uneasy conclusion
that diminishing marginal utility is neither a sufficient nor a
necessary condition for a diminishing marginal rate of substitution
(convex indifference curves) and we must therefore simply state this
as an assumption, albeit a reasonable one: The MRS between any pair of
goods X and Y is diminishing."

You can read the article following this link:
http://www.mhhe.com/economics/mcconnellmacro15e/graphics/mcconnell15eco/common/dothemath/marginalrateofsubstitution.html

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The following lectures will be useful for you:

"The Utility Surface":
http://www.digitaleconomist.com/utility.html

"Consumer Theory" by Shizuka Nishikawa:
http://econ.la.psu.edu/~sun106/spring2004/302consumer.pdf

"THE BEHAVIOR OF CONSUMERS":
http://www.ksu.edu/economics/malik/lect6f00.html

"Diminishing Marginal Utility":
http://www.mhhe.com/economics/mcconnellmacro15e/graphics/mcconnell15eco/common/dothemath/diminishingmarginalutility.html

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Search strategy:
"marginal rate of substitution" "marginal utility" diminishing
"marginal rate of substitution" "marginal utility"
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I hope this helps you. 
Note that this answer is not finished until you feel satisfied with
it. If you find anything unclear, or if a link doesn't work for you,
please request clarification; I'll be glad to offer further
assistance.

Best regards
livioflores-ga

Exactly what i was looking for!