Unformatted text preview: pace (good 1) and horsepower (good 2). Which of the utility is/are not suitable? Give explanations, using graphs if
{
}.
necessary. In all cases, the utility function is defined over the consumption set
(a)
(b) (c) () where
where ( ) is any function of where such that () and Show all calculations and state all assumptions.
Answer
A car must have strictly positive interior space and HP. As such, indifference curves cannot touch either axis. This rules
out:
The utility model
where
because its indifference curves do touch both axes (implying that
consumers may choose a car with lots of space and no engine or a car with no space and just an engine).
()
The utility model
where ( ) is any function of such that ( )
because its indifference
curves do most likely touch both axes (again, implying that consumers may choose a car with lots of space and no engine
or a car with no space and just an engine).
This leaves the “CobbDouglas” model
a “bad good” and obviously implausible. where and This too is unsuitable because good 2 (HP) is (3.8) Consider the following utility function defined over the consumption set { }: (a) Does this consumer has monotone preferences over every bundle in the consumption set? If not, are there any
bundles in the consumption set where she does have monotone preferences? Show all calculations and state all
assumptions.
Answer
To see if the consumer has monotone preferences, calculate the marginal utilities: Now for any bundle ( ) notice that and that for any bundle where , . Next, for any
15 ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. bundle in the consumption set,
. Therefore, the consumer has monotone preferences over bundles in the
interior of the consumption set, and perceives good 2 as a “good” good and good 1 as a “neutral” good along any bundle
where
.
(b) State  but do not solve – a “simplified” Utility Maximization Problem (UMP). That is, state the UMP by simplifying
and/or eliminating some constraints from the general UMP. Show all calculations and state all assumptions. Assume all
pecuniary parameters are
.
Answer
In general, the UMP is:
( ) Can we simplify this UMP? Now: Notice that when
and when
. This means that the consumer will always have zero
utility by choosing a bundle where
and positive utility from a bundle with
. Thus, the optimal choice can
never be a bundle with
so that we can drop the constraint
. With this, the UMP becomes: The consumer has monotone preferences over all bundles where so that the UMP becomes: The other way to simplify the UMP is by graphing indifference curves to see if they touch either axis. The equation of the
indifference for an arbitrary level of utility is: ( As we see that ) (so that the indifference curves touch the yaxis). Next, notice that: 16
ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 20122013) Uni...
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 Fall '09
 AJAZHUSSAIN
 Microeconomics, Utility, Department of Economics, S. Ajaz Hussain, Sayed Ajaz Hussain

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