For the time being here is a summary of the 4 cases

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Unformatted text preview: for dropping one or both of the nonnegativity constraints is that in a monotone preferences UMP at least one of the goods has to be strictly positive. Thus, with monotone preferences, the bundle , cannot be a solution and the only types of bundles that can be solutions are: ⏟ ⏟ ⏟ ⏟ ⏟ , , , Next in ECO 204, with monotone preferences, the utility at the optimal bundle must be positive, i.e. any bundles where can be ruled out as solutions. Here is the practical procedure: . This means ❶ Check if (or indeterminate) at any bundle on the y-axis (except the origin): substitute , utility function. If it means cannot be a solution and in which case we can drop the constraint because the optimal bundle must have (and this “constraint” doesn’t require the KT method). ❷ Check if (or indeterminate) at any bundle on the x-axis (except the origin): substitute , utility function. If it means cannot be a solution and in which case we can drop the constraint because the optimal bundle must have (and this “constraint” doesn’t require the KT method). ❸ Check if check that (or indeterminate) at any bundle in the interior substitute (with monotone preferences this will be always true). , into the into the into the utility function and 37 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. _________________________________________________________________________________________________ Example: Suppose we are asked to solve the Cobb-Douglas general UMP: .. , , , Check/impose conditions on parameters which ensure she has monotone preferences: ⏟ She will have monotone preferences when , ⏟ . The UMP simplifies to: .. , , , Can we drop any of the non-negativity constraints? Do the 3 step check above: ❶ Check if utility function: (or indeterminate) at any bundle on the y-axis (except the origin). Substitute This means cannot be a solution and in which case we can drop the constraint bundle must have (and this “constraint” doesn’t require the KT method). ❷ Check if utility function: (or indeterminate) at any bundle on the x-axis (except the origin). Substitute , into the because the optimal , (or indeterminate) at any bundle in the interior. Substitute into the because the optimal This means cannot be a solution and in which case we can drop the constraint bundle must have (and this “constraint” doesn’t require the KT method). ❸ Check if , into the utility function: The Cobb-Douglas monotone preferences UMP simplifies to: .. , , , , .. 38 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. _________________________________________________________________________________________________ Example: Suppose we are asked to solve this (quasi-linear) general UMP: , .. , , Check/impose conditions on parameters which ensure she has monotone preferences: ⏟ As long as ⏟ is finite she has monotone preferences. The UMP simplifies to: , .. , , Can we drop any of the non-negativity constraints? Do the 3 step check above: ❶ Check if utility function: (or indeterminate) at any bundle on the y-axis (except the origin). Substitute , into the ⏟ This means cannot be a solution and in which case we can drop the constraint bundle must have (and this “constraint” doesn’t require the KT method). ❷ Check if utility function: because the optimal (or indeterminate) at any bundle on the x-axis (except the origin). Substitute , into the ⏟ This means can be a solution and we cannot drop the constraint ❸ Check if (or indeterminate) at any bundle in the interior. Substitute ⏟ . , into the utility function: ⏟ This specific quasi-linear UMP simplifies to: , .. , , 39 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. _________________________________________________________________________________________________ Example: Suppose we are asked to solve the linear general UMP: , .. , , Check/impose conditions on parameters which ensure she has monotone preferences: ⏟ She will have monotone preferences when , , ⏟ . The UMP simplifies to: .. , , Can we drop any of the non-negativity constraints? Do the 3 step check above: ❶ Check if utility function: (or indeterminate) at any bundle on the y-axis (except the origin). Substitute , into the , into the ⏟ This means can be a solution and we cannot drop the constraint . ❷ Check if utility function: (or indeterminate) at any bundle on the x-axis (except the origin). Substitute ⏟ This means can be a solution and we cannot drop the constraint ❸ Check if (or indeterminate) at any bundle in the interior. Substitute ⏟ . , into the utility function: ⏟ Unfortunately, we cannot further simplify the linear UMP and we must solve: , .. , , We did in fact solve this UMP earlier and we will examine the economics of the linear and other utility models in chapter 4. 40 ECO 204 CHAPTER 3 Utility Maximization Problems (this version 2012-2013)...
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This note was uploaded on 03/20/2014 for the course ECON 204 taught by Professor Ajazhussain during the Fall '09 term at University of Toronto- Toronto.

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