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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: ܷ ൌ minሺߙܳ , 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 , ߚܳ ሻ su j c ܲ ܳ ൅ ܲ ܳ ൑ ܻ b e t to The parameters ߙ, ߚ 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 ,ொ ܷ ൌ minሺߙܳ , ߚܳ ሻ subject 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 , ߚܳ ሻ െ ܳ ߣሾܲ ܳ ൅ ܲ െ ܻ From chapter 2.2 we know that the utility function ܷ ൌ minሺߙܳ , ߚܳ 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 ߙܳ or ߚܳ (that is, at the optimal choice, it cannot be that ߙܳ ൐ ߚܳ or ߙܳ ൏ ߚܳ ). After all, her utility is determined by the smaller of ߙܳ or ߚܳ ; thus, at the optimal choice , it must be that: ߙܳ ൌ ߚܳ Alternatively, we can exploit insights from graphical analysis. From chapter 2.1 you know that ܷ ൌ minሺߙܳ , ߚܳ has “L” shaped indifference curves, where the corners are always on the line: ߙܳ ൌ ߚܳ ௬௜௘௟ௗ௦ ሱۛۛۛሮ ܳ ߙ ߚ ܳ
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