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ajaz_eco204_2009_chapter_2.4.1

ajaz_eco204_2009_chapter_2.4.1 - 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 2.4.1: Modeling Consumer Choice, Applications 1 Please help improve the course by sending me an e mail about typos or suggestions for improvements Section #s continue from chapter 2.3 2. UMP: Perfect Substitutes (Linear) Utility Function Do you perceive differences in gasoline brands? Do you think Shell and Esso gasoline are imperfect substitutes? If so, your preferences may be modeled by a Cobb Douglas utility function (review chapter 2.3 ). But what if you perceive gasoline brands as a “commodity”, i.e. you perceive all gasoline brands to be the same? In this case, your optimal choice may be modeled by the perfect substitutes (linear) utility function. We will set up and solve the perfect substitutes UMP followed by an analysis of optimal choices and business applications. 2.1 Optimal Solution and Utility A utility function with a constant MRS is labeled a “perfect substitutes” utility function. A common misconception is that “perfect substitutes” means the consumer perceives good to be identical . Here are two examples: Example Suppose a consumer has the utility function ܷ ൌ ܳ ൅ 2ܳ . The ܯܴܵ ൌ െ1/2 means the consumer perceives a ½ unit of good 2 to be perfe able w good 1. ctly substitut ith a unit of Example Suppose a consumer has the utility function ܷ ൌ ܳ ൅ ܳ . The ܯܴܵ ൌ െ1 means the consumer perceives a unit of good 2 to be perfectly substitutable with a unit of good 1. That said, let’s see how to do the UMP for perfect substitutes (linear) utility function. For ܰ ൌ 2 commodities, the UMP is: 1 Thanks: Ksenija Stupar 1 ECO 204 (Draft) Chapter 2.4.1
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University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain max ,ொ ߙܳ ject to ܲ ܳ ൅ ܲ ܳ ൑ ܻ ߚܳ sub ߙ, ߚ can be any arbitrary numbers. If ߙ, ߚ ൐ 0 , the consumer has monotonic preferences 2 and will spend her entire inco U me. Hence, the MP is: max ,ொ ߙܳ ൅ ߚܳ subject to ܲ ܳ ൅ ܲ ܳ ൌ ܻ This is the same as: max ,ொ ߙܳ ൅ ߚܳ subject to ܲ ܳ ൅ ܲ ܳ െ ܻ ൌ 0 Optimization problems with equality constraints are solved by the Lagrangean technique. Form the Lagrangean: max ,ொ ,ఒ ܮ ൌ ߙܳ ൅ ߚܳ െ ߣሾܲ ܳ ൅ ܲ ܳ െ ܻሿ The FOCs are 3 : ߲ܮ ߲ ൌ ߙ െ ߣܲ ൌ 0 ܳ ߲ܮ ൌ ߚ െ ߣܲ ൌ 0 ߲ܳ ߲ܮ ߲ߣ ൌ ܲ ܳ ൅ ܲ ܳ െ ܻ ൌ 0 Now observe that if you divide the first and second FOCs, you get: ߣܲ ܲ ߙ ܲ ߚ ߣܲ 2 ௬௜௘௟ௗ௦ ሱۛۛۛሮ ߙ ܲ ߚ ܲ ௬௜௘௟ௗ௦ ሱۛۛۛሮ െ ߙ ߚ ൌ െ ܲ ܲ 2 This is because her marginal utili ill be positive.
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