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 yaxis (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 xaxis (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 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. _________________________________________________________________________________________________
Example: Suppose we are asked to solve the CobbDouglas 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 nonnegativity constraints? Do the 3 step check above:
❶ Check if
utility function: (or indeterminate) at any bundle on the yaxis (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 xaxis (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 CobbDouglas monotone preferences UMP simplifies to:
.. , , , , ..
38 ECO 204 CHAPTER 3 Utility Maximization Problems (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. _________________________________________________________________________________________________
Example: Suppose we are asked to solve this (quasilinear) 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 nonnegativity constraints? Do the 3 step check above:
❶ Check if
utility function: (or indeterminate) at any bundle on the yaxis (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 xaxis (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 quasilinear UMP simplifies to:
, .. , , 39
ECO 204 CHAPTER 3 Utility Maximization Problems (this version 20122013) 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 nonnegativity constraints? Do the 3 step check above:
❶ Check if
utility function: (or indeterminate) at any bundle on the yaxis (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 xaxis (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 20122013)...
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 Fall '09
 AJAZHUSSAIN
 Microeconomics, Utility, S. Ajaz Hussain, Sayed Ajaz Hussain

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