Exam 13: Committing and Uncommitting

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Odysseus' inability to avoid temptation without a commitment mechanism means he can be best described as a naïf.

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Consider the example given on pages 3-12 and 3-13. Suppose kv=kc=1k_{v}=k_{c}=1 , that is both costs and rewards are immediate. Let the remaining parameters be unchanged, so that vt=v_{t}= {8,20,0},ct={0,9,1},β=12,δ=1\{8,20,0\}, c_{t}=\{0,9,1\}, \beta=\frac{1}{2}, \delta=1 . In which time period does the sophisticate act?

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At t=2t=2 the sophisticate solves the following problem, max{β(209),β}=\max \{\beta(20-9),-\beta\}= max{5.5,12}\max \left\{5.5,-\frac{1}{2}\right\} . And so the sophisticate chooses t=2.lnt=1t=2 . \ln t=1 , the sophisticate solves max{β8,β(209)}=max{4,5.5}\max \{\beta 8, \beta(20-9)\}=\max \{4,5.5\} and so the sophisticate chooses to act in t=2t=2 .

Time consistent individuals are always better off than either sophisticates or naïfs.

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Commitment mechanisms

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Erin has time consistent preferences must decide when to go to the doctor over the next 4 days. She knows she must go one of these four days. She knows that using backward induction to decide when to go will yield the optimal solution.

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Consider the example given on pages 3-12 and 3-13. Suppose kv=kc=0k_{v}=k_{c}=0 , that is both costs and rewards are immediate. Let the remaining parameters be unchanged, so that vt=v_{t}= {8,20,0},ct={0,9,1},β=12,δ=1\{8,20,0\}, c_{t}=\{0,9,1\}, \beta=\frac{1}{2}, \delta=1 . In which time period does the naïf act?

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Scenarios in which costs are experienced later than the action to incur the cost are called,

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Consider the example given on pages 3-12 and 3-13. Suppose kv=kc=0k_{v}=k_{c}=0 , that is both costs and rewards are immediate. Let the remaining parameters be unchanged, so that vt=v_{t}= {8,20,0},ct={0,9,1},β=12,δ=1\{8,20,0\}, c_{t}=\{0,9,1\}, \beta=\frac{1}{2}, \delta=1 . In this setting, the naïf obeys the Independence of Irrelevant Alternatives.

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Consider the example given on pages 3-12 and 3-13. Suppose kv=kc=0k_{v}=k_{c}=0 , that is both costs and rewards are immediate. Let the remaining parameters be unchanged, so that vt=v_{t}= {8,20,0},ct={0,9,1},β=12,δ=1\{8,20,0\}, c_{t}=\{0,9,1\}, \beta=\frac{1}{2}, \delta=1 . Suppose the decision-maker is partially naïve and believes β^>β\hat{\beta}>\beta . For which value of β^\hat{\beta} does the decision-maker's choice violate the dominance property?

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A naif cannot solve the inter-temporal problem recursively because

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Naifs and sophisticates always violate either the dominance property or the property of independence of irrelevant alternatives.

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Fiona must decide when to go to the doctor. There are four possible days she could go, each incurring a higher cost and the same benefit. Fiona must go to the doctor on one of these days. Fiona solves the problem using backward induction. This means she begins solving the problem from the point of view of

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Joy needs to clean the house before her parents come on Sunday. Today is Friday. This means Joy can clean either Friday, Saturday or Sunday. The reward for cleaning the house is that Joy gets to enjoy a clean home, but the cost is that she will have to give up planned activities with her friends. Let vt={10,0,0}v_{t}=\{10,0,0\} be the reward for cleaning the house on Friday, Saturday, Sunday, respectively. Let ct={0,2,1}c_{t}=\{0,2,1\} be the cost of cleaning the house on Friday, Saturday, Sunday, respectively. The Dominance Property requires Joy to clean the house on which day?

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Consider the example given on pages 3-12 and 3-13. Suppose kv=kc=0k_{v}=k_{c}=0 , that is both costs and rewards are immediate. Let the remaining parameters be unchanged, so that vt=v_{t}= {8,20,0},ct={0,9,1},β=12,δ=1\{8,20,0\}, c_{t}=\{0,9,1\}, \beta=\frac{1}{2}, \delta=1 . In this setting, the naif violates the dominance property.

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Joy needs to clean the house before her parents come on Sunday. Today is Friday. This means Joy can clean either Friday, Saturday or Sunday. The reward for cleaning the house is that Joy gets to enjoy a clean home, but the cost is that she will have to give up planned activities with her friends. Let vt={10,0,0}v_{t}=\{10,0,0\} be the reward for cleaning the house on Friday, Saturday, Sunday, respectively. Let ct={0,2,1}c_{t}=\{0,2,1\} be the cost of cleaning the house on Friday, Saturday, Sunday, respectively. Suppose her parents call and inform Joy that they are coming home on Saturday as opposed to Sunday. Does this change Joy's optimal action if she is time consistent and why?

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Consider the example given on pages 3-12 and 3-13. Suppose kv=kc=1k_{v}=k_{c}=1 , that is both costs and rewards are immediate. Let the remaining parameters be unchanged, so that vt=v_{t}= {8,20,0},ct={0,9,1},β=12,δ=1\{8,20,0\}, c_{t}=\{0,9,1\}, \beta=\frac{1}{2}, \delta=1 . In this setting, the sophisticate obeys the Independence of Irrelevant Alternatives.

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Fiona has time inconsistent preferences and must when to go to the doctor over the next 4 days. She is a sophisticate and so she uses backward induction to decide when to go because this method will yield the optimal solution.

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Sophisticates are always better off than naïfs.

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A sophisticate will never complete an identical task after a naïf.

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The optimal amount of savings for time consistent, naifs and sophisticates occurs where the marginal utility of savings is zero.

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