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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 x-axis). 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 Cobb-Douglas utility function – its indifference curves, like that of the Cobb-Douglas 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: 17 ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. [ The FOCs are: ( ) Re-arrange 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: 18 ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 2012-2013) 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, cross-price 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: 19 ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 2012-2013) 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 2012-2013) University of Toronto, Department of Economics (STG). ECO...
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