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Unformatted text preview: , ⏟, ⏟ ⏟ Why do monotone preferences simplify the UMP? Because with 2 inequality constraints we go from having to check 8 cases to 4 cases: ( , ) ..⏟ ,(,) ⏟ , , , , (, , ) [ ⏟ ⏟ [ , , , , Each “animal” implies 2 possibilities and we get: 19 ECO 204 CHAPTER 3 Utility Maximization Problems (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. , , , , , , , Case A , , , , , Case B , , , Case C , Case D According to this we have to check for 4 cases but in fact as we show below, with monotone preferences, Case A (as shown above) is always impossible. For the time being here is a summary of the 4 cases (items with ? mark are what we have to check and/or verify): Case A Case B Check Case C Case D ? Check Check Check Check Check ? Check Here’s what’s happening: the math tells you that there are 4 possible solutions. Sometimes, one of these cases wi ll definitely be the solution; at other times, depending on the circumstances, each of the cases can be a solution and we would then have to derive conditions for each case to be a solution. Look at the table above: some items have to be “checked” – it is these checks that tell you when a case may be a solution. There is no fixed, set, procedure for solving these types of problems. Only through practice, repetition, experience and intuition will you learn how to solve such UMPs. Math is like learning how to ride a bike, salsa dancing, spinning, cooking, etc. – the only way to get better is through practice and sadly there is no “magic bullet”. Let’s do some examples. 20 ECO 204 CHAPTER 3 Utility Maximization Problems (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Do NOT memorize. Instead try to understand the process. Can you do it on your own? Explain to someone else? _________________________________________________________________________________________________ Example: A specific linear UMP. We have seen that the linear utility function models a consumer who perceives goods as perfect substitutes (review chapter 2). Suppose that the consumer tells us that she perceives the two goods as perfect | is: substitutes and that she is willing to give up 2 units of good 2 for a unit of good 1. That is, her | | ⏟ | | | Recall that for linear utility model | Thus we can choose any value of , such that: | , For example: | | . With this the general UMP is: , , The consumer has monotone e same “price” (the price comparison adjusts for the fact she has a greater affinity for good 2). Note: the linear utility function model is a special case of the quasi-linear utility function: () Where ( ) is any function of good 1. For example, each of the following is a quasi-linear utility function of the form: () : √ On the other hand, these are not quasi-linear utility functions (can you see why?): √ √ The fact that the linear utility function model is a special case of the quasi-linear utility function that if you solve the problem: , () ..⏟ ,(, ⏟ () means ) And then set ( ) you’ll get the answers for the linear UMP _________________________________________________________________________________________________ 36 ECO 204 CHAPTER 3 Utility Maximization Problems (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. 4. Strategies for Possibly Further Simplifying UMP with Monotone Preferences Let’s summarize what we done so far. If we solve the general UMP: , (, ).. , ⏟, ⏟ ⏟ This UMP requires us to check 8 different cases (i.e. explore 8 possible solutions to the UMP and outline conditions for each case to be a solution). Fortunately we saw that if the consumer has (or we assume that she has) monotone preferences then we can solve the simpler UMP with monotone preferences: , (, ).. , ⏟, ⏟ ⏟ This problem entails checking for 4 possible cases. But in fact, there’s even more good news (well maybe). With monotone preferences we may be able to drop one or both non-negativity constraints: if we manage to drop both nonnegativity constraints we don’t have to check for any cases (do you know why? Pray such a problem comes on the test) while if we manage to drop one non-negativity constraints we have to check for 2 cases. How do we possibly drop one or both of non-negativity constraints? The “trick...
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