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Unformatted text preview: a 1% excise tax on good 1? Show all
calculations.
Answer
Notice that: Is a log positive monotonic transformation of the CobbDouglas utility function
The “log CobbDouglas” UMP is:
⏟ To solve this problem we’d need the Lagrangian function:
[
We are told
and
and are asked for the impact on optimal utility due to a 1% excise on good
1. First, let’s use the envelope theorem to see the impact on
due to 1% excise on good 1 (review the three steps of
the envelope theorem):
❶ Express the objective function in terms of parameters:
[
❷ To get change in objective with respect to the parameter differentiate with respect to the parameter that is
changing: here it is (since the excise tax is on good 1): ❸ To get change in the optimal objective evaluate the derivative in ❷ at the optimal solution:
12
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. ( )( ) Now, at the optimum (why?):
[⏟ ⏟ Therefore: Therefore: Thus: We are told there has been a 1% excise tax on good 1. Thus
( Change in price of good 1 = () and so: ) (b) Without solving the UMP, what is impact on the consumer’s utility due to a 1% excise tax on good 2? Show all
calculations.
The “log CobbDouglas” UMP is:
⏟ To solve this problem we’d need the Lagrangian function:
[
We are told
and
and are asked for the impact on optimal utility due to a 1% excise on good
2. First, let’s use the envelope theorem to see the impact on
due to 1% excise on good 2 (review the three steps of
the envelope theorem):
❶ Express the objective function in terms of parameters:
[ 13
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. ❷ To get change in objective with respect to the parameter differentiate with respect to the parameter that is
changing: here it is (since the excise tax is on good 2): ❸ To get change in the optimal objective evaluate the derivative in ❷ at the optimal solution: ( )( ) Now, at the optimum (why?):
[⏟ ⏟ Therefore: Therefore: Thus: We are told there has been a 1% excise tax on good 2. Thus
( Change in price of good 2 = () and so: ) (3.6) A consumer’s preferences are representable by the following utility function over goods 1 and 2 (defined over the
{
}):
consumption set
√( ) Does this consumer have rational preferences? A one or two sentence answer suffices.
Answer
Yes. In ECO 204, we assumed that if a consumer has rational preferences then her preferences can be represented by a
utility function.
14
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. (3.7) The following utility functions are given to model automobile consumers’ preferences over cubic feet of interior
s...
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 Fall '09
 AJAZHUSSAIN
 Microeconomics, Utility

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