In practice the general ump can be simplified by

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Unformatted text preview: for each animal can be checked in any order. Below, we do it in reverse order: , , , ,⏟ ,⏟ , ⏟[ 6 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. For the general soon. 6 UMP we would have to check eight separate cases6 (ouch)! Not to worry – we’ll see a short cut In the tree below: 7 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. , , , Propose: , , Propose: , , , Propose: , Propose: , , , , Propose: , Propose: , , , , , , Propose: Propose: Propose: Propose: Propose: Propose: Propose: Propose: Check Check Check Check Check Check Check Check 8 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. Very rarely in ECO 204 will we have to do a UMP which requires checking so many cases. In fact, as we show below, in almost all UMPs in ECO 204 the consumer has monotone preference which greatly simplifies the calculations. Before we show the simplification, let’s derive some general results about consumer choice and behavior (these results hold for any utility function even when the consumer perceives all goods to be “bad”). What are the parameters in the general UMP? Well: , ⏟ , , (, , [ ) ⏟] [ [ ⏟] Since we don’t know the actual utility function we can’t write down its parameters7. That said, the other parameters and variables of the general UMP are: Variables (What you’re solving for) Parameters (What you’re given as constants) Utility parameters (N/A for arbitrary function) , , , , , ,, , From chapter 1 you understand that a change in any parameter (most likely) changes the optimal solutions (, , , , ). From chapter 1 you know that from the KT conditions: ) and the optimal objective ( ⏟( , ) ⏟[ [ ⏟] ⏟ Thus, changes in get: are equivalent to changes in [ ⏟] ⏟ . We can now apply the envelope theorem on the function to By the 3 steps of the envelope theorem (review these; it’s helpful to compare and contrast with constrained revenue maximization problems in chapter 1): (, 7 For example, if we knew that ) or [ then the utility function parameters would be , . 9 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. Applying steps ❷ and ❸ of the envelope theorem to each of the parameters gives us the following general results (you really need to know how we go this -- reason this out for yourself): Impact on due to a small change in income: ⏟ From the KT conditions remember that cannot be negative (in what kind of problems can the be negative?). This result tells us that with a higher income (and after optimal readjustment) the consumer’s utility either increases or stays the same but can’t go down (see graphical example below). More money, more (or same) happiness . 2 1 In Wolfram Alpha type: plot u=2log(q1)+q2^0.5) from q1=0,10 q2=0,10 Impact on due to a small change in price of good 1 : ⏟⏟ From the KT conditions remember that cannot be negative and that (with the current consumption set) the quantities must be non-negative. This result tells us that with a higher price of good 1 (and after optimal readjustment), the consumer’s utility either decreases or stays the same but can’t go up. This means for example, an excise tax on good 1 can’t make you better off (see graphical example below): 10 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. 2 1 In Wolfram Alpha type: plot u=q1 + 0.5log(q2) from q1=0,10 q2=0,10 Impact on due to a small change in price of good 2: ⏟⏟ From the KT conditions remember that cannot be negative and that (with the current consumption set) the quantities must be non-negative. This result tells us that with a higher price of good 2 (and after optimal readjustment), the consumer’s utility either decreases or stays the same but can’t go up. This means for example, an excise tax on good 2 can’t make you better off (see graphical example below): 1 2 In Wolfram Alpha type: plot u=(q1^0.3...
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