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Unformatted text preview: e: 
 ).
The consumer perceives goods 1 and 2 as “imperfect substitutes” with diminishing
(i.e. as
Along the yaxis (where
) we have
along the xaxis (where
) we have
; and
at any interior bundle
. Thus, the consumer may choose bundle on the
boundary and/or in the interior of
the consumption set. 7 University of Toronto, Department of Economics, ECO 204, 2013  2014 Question 3
(a) Show that the indifference curves of the utility function derived from a “log positive monotonic transformation” of
√ will have the exact same slope as the original utility function (albeit with different utility values). Answer:
The slope of an indifference curve of √ is: √ (√ The slope of an indifference curve of ) is: √ √
√ For all bundles other than ( ) we have:
√ (b) Show that the indifference curves of the utility function derived from a “raise to the power 3 positive monotonic
transformation” of
utility values). √ will have the exact same slope as the original utility function (albeit with different Answer:
The slope of an indifference curve of √ is: √ The slope of an indifference curve of (√ ) is:
(√ )
(√ For all bundles other than ( √
) ) we have: 8 University of Toronto, Department of Economics, ECO 204, 2013  2014 √ Question 4
Consider the following UMP for
“good/neutral” good: goods where at least one of the goods is a “good” good and the other good is a (
( )
) [ After you solve this UMP you will get
due to, all else equal, a small change in [ [ . Use the envelope theorem to derive expressions for the change in
. Answer
The UMP is:
(
( )
) [ [ [ If we knew the utility function then we’d solve this UMP as follows: ⏟ ⏟ [ [ [ From the FOCs and KT conditions we see that at the optimal solution (make sure you know why): ( ) [⏟ ⏟[ ⏟[ 9 University of Toronto, Department of Economics, ECO 204, 2013  2014 (
Thus, a change in
theorem: is equivalent to a change in (
Ceteris paribus, the change in Ceteris paribus, the change in Since ( ) )
due to due to [ ) . The change in [ can be found by applying the envelope [ is: is: we see that: In words: 10...
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