ajaz_eco204_2009_chapter_2.4.2

ajaz_eco204_2009_chapter_2.4.2 - University of Toronto,...

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University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain ECO 204 2009 2010 S. Ajaz Hussain (Draft) Chapter 2.4.2: Complements Utility Function UMP Please help improve the course by sending me an e mail about typos or suggestions for improvements Section #s continue from chapter 2.3 and chapter 2.4.1 In chapter 2.3 we discussed that it was possible to model consumer preferences by utility functions. In chapter 2.4.1 we modeled optimal choice for consumers with Cobb Douglas (imperfect substitutes) and linear (perfect substitutes) utility functions. In this chapter we will model optimal choice for consumer with the “complements” utility function 1 and some interesting utility functions. 3. UMP: Complements Utility Function By definition, some commodities are complements ( Xbox and Xbox games , cars and gasoline) while other commodities ‐‐ depending on the individual ‐‐ are perceived to be complements (milk and cookies, gin and tonic). The utility function for two commodities that are, or perceived to be, complements is: ܷൌ m inሺߙܳ ,ߚܳ Here ߙ,ߚ are parameters. We will now solve for the optimal choice of goods 1 and 2. 3.1 Optimal Solution and Utility The consumer makes her optimal choices by maximizing utility subject to her budget constraint. For ܰൌ2 commodities the UMP is: max ,ொ ܷ ൌ minሺߙܳ ሻsu j c ܲ ܳ ൅ ܲ ܳ ൑ܻ be t to The parameters can be any numbers; but if ߙ,ߚ ൐ 0 , the consumer has monotonic preferences and will always spend her income. Thus, the UMP becomes: 1 The complements utility function will be repeatedly used in RSM 332. 1 ECO 204 (Draft) Chapter 2.4.2
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University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain max ,ொ ܷൌm inሺߙܳ ,ߚܳ ሻsubject to ܲ ܳ ൅ ܲ ܳ ൌܻ This being an equality constrained optimization problem, it can be solved by the Lagrangean method. First, set up the Lagrangean: max ,ொ ,ఒ ܮ ൌ minሺߙܳ ሻ െ ܳ ߣሾܲ ܳ െܻ From chapter 2.2 we know that the utility function ܷ ൌ minሺߙܳ is not differentiable everywhere. Thus, we cannot solve the UMP by taking FOCs. Instead, let’s resort to insights from logic and/or graphical analysis. Logically, if the consumer’s utility is the minimum of ߙܳ or ߚܳ she won’t “waste” resources by having an excess of either or (that is, at the optimal choice, it cannot be that ൐ߚܳ or ൏ߚܳ ). After all, her utility is determined by the smaller of or ; thus, at the optimal choice , it must be that: ൌߚܳ Alternatively, we can exploit insights from graphical analysis. From chapter 2.1 you know that ܷ ൌ minሺߙܳ has “L” shaped indifference curves, where the corners are always on the line: ௬௜௘௟ௗ௦ ሱۛۛۛሮ ܳ ߙ ߚ ܳ Some indifference curves are plotted below (remember, there are an infinite number of
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto- Toronto.

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ajaz_eco204_2009_chapter_2.4.2 - University of Toronto,...

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