1. This question should guide you through the single period utility set up.
  1. Write down a single period utility function that consists of \(H_{t}\) and \(Z_{t}\) as given in the lecture.
  2. Give some practical examples of \(H_{t}\) and \(Z_{t}\).
  3. The assumption imposed for the utility function is \(U'(H)>0\) and \(U''(H)>0\). Explain.

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  1. This question pertains to time constraint.
  1. Write down the time constraint when time is broken down into 4 discrete blocks: \(T^{W}\), \(T^{Z}\), \(T^{H}\), \(T^{S}\).
  2. Given one practical example of \(T^{W}\), \(T^{Z}\), \(T^{H}\), \(T^{S}\). Just one is fine, but please be specific.
  3. Draw the labor using the following conditions: i) total time available = 24 hours, ii) \(T^{H}=6 \; hours\), and \(T^{S}=4 \; hours\). Denote the y and x-intercepts correctly, including the labels for axis.
  4. Draw an indiffference curve (for a person who values both work and leisure) on the same graph (for part c.) to determine the optimal \(T^{W}\) and \(T^{Z}\).
  5. Now using the following condition: i) total time available = 24 hours, ii) \(T^{H}=7 \; hours\), and \(T^{S}=1 \; hour\), draw a new labor-leisure trade off graph.
  6. What can you deduce from the newer labor-leisure graph compared to the older one? Please be brief.

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  1. This question is about production of health stock and home goods.
  1. In equation form, write down the production of health stock and home goods as shown in the lecture.
  2. Give an example of \(J_{t}\) and explain the difference between \(J_{t}\) and \(Z_{t}\).
  3. Explain the role of depreciation in production of health stock.
  4. Draw the health production frontier. Clearly label the important segments of the frontier.
  5. Think of the frontier in part d. as a constraint. Explain your thought succinctly.
  6. Clearly point the optimal health stock \((H^{*})\) and home goods \((Z^{*})\) by using an indifference curve for a person who values both health stock and home goods.

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  1. Repeat part 3 (f) but this time consider that the person only values health stock. Using a graph show how the optimal points of health stock \((H^{*})\) and home goods \((Z^{*})\) changes.

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  1. This question pertains to the multi period utility function.
  1. Write dpwn the multi-period utility function as an additive form (of today’s, tomorrow’s and so on utility). Label the inputs in the utility function.
  2. Explain the role of discount factor \((\delta)\) in the multi period utility function.
  3. What does is mean to have a discount factor \((\delta)\) of zero.
  4. Derive a single period utility using a multi period utility function using \(\delta=0\).