Unformatted text preview: versity of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. As
we see that
(so that the indifference curves don’t touch the xaxis). Thus, the optimal choice
must entail
and the UMP becomes: (3.9) Given utility function of the form ( ) { defined over }. (a) Solve the UMP and derive the demand functions for goods 1 and 2 and (any) Lagrange multipliers for a consumer
()
()
Show all calculations and state all assumptions. [Hint:
Answer
First, simplify the utility function by performing a positive monotonic transformation:
( ) ( ( [( ) ) )] This is the log CobbDouglas utility function – its indifference curves, like that of the CobbDouglas utility function don’t touch either axis and if we assume that
then the consumer has monotone preferences since for
: With monotone preferences and the UMP becomes: The UMP is:
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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. [
The FOCs are: ( ) Rearrange equations ❶and ❷: Equating ❶and ❷: Multiplying both sides by 1 tells us that the optimal bundle is where i.e. the slope of the indifference curve = slope of the budget line: We can solve this equation simultaneously with equation
( :
) Now, from: Substitute in budget constraint:
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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. ( ) ( ) Substitute this in: We now solve for . From the 1st FOC: But: so that: (b) Calculate the price, crossprice and income elasticity of goods 1 and 2. Show all calculations and state all
assumptions.
Answer
The demand functions are: Let’s use the “log” method for calculating elaticities where:
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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. First: Now: That is, a 1% rise in dampens demand for good 1 by 1%; a 1% rise in
1% rise in boosts demand for good 1 by 1%. has no impact on demand for good 1; and, a Next: Now: That is, a 1% rise in dampens demand for good 2 by 1%; a 1% rise in
1% rise in boosts demand for good 2 by 1%. has no impact on demand for good 2; and, a (c) Calculate the marginal optimal utility of income by the envelope theorem and the “value function” approach, i.e.
calculate . Show all calculations and state all assumptions. Answer
We want . By the envelope theorem, we:
❶ Express the objective function in terms of parameters:
[
❷ Differentiate with respect to the parameter of interest (in this case, income ): 20
ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO...
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 Fall '09
 AJAZHUSSAIN
 Microeconomics, Utility, Department of Economics, S. Ajaz Hussain, Sayed Ajaz Hussain

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