Ex 2.2: Used stuff always goes for cheaper Sonos PLAY:1 speakers are really awesome. I bought eight of them on sale; they usually go for $200. After I plugged them in, I realized I might not need all eight. However, if I want to sell them, since I took them out of the box, I could only get about $100 for each one. If I wanted to buy more, additional speakers would cost $200 apiece. I'm trying to decide whether I should sell some of my speakers and use the proceeds to buy more Christmas presents for my family. As of now I have $800 budgeted for Christmas presents. My preferences over speakers and money to spend on other Christmas presents may be represented by the Cobb-Douglas utility function u(x1, 12) = xqx]-a, where speakers are good 1 and money for other Christmas presents is good 2. For simplicity, assume speakers can be bought and sold as if they were a continuous good (i.e., not discrete). (a) What is my endowment point (e1, ez)? 1 (b) What is my MRS at the endowment point? The expression you derive will be a function of a; based on the information given in the question, what can you say about the value of a? (c) Draw my budget constraint. Explain why it is kinked, and what the slopes on the two sections of the budget line represent. (d) Suppose I decide to sell four of my speakers for $100 each to my friend Jeff. Draw the indifference curves passing through my endowment point (e1,e2) and my optimal choice (31,37). What value of a would lead me to this being my optimal behavior? (e) Assuming this is my true value of a, derive my "supply curve" for speakers: that is, how many speakers would I sell as a function of the market price for used speakers, p? Draw this curve in a diagram, labeling the appropriate points. (Hint: you know the point (4 speakers, p = 100) must be on it...) Ex 2.2: Used stuff always goes for cheaper Sonos PLAY:1 speakers are really awesome. I bought eight of them on sale; they usually go for $200. After I plugged them in, I realized I might not need all eight. However, if I want to sell them, since I took them out of the box, I could only get about $100 for each one. If I wanted to buy more, additional speakers would cost $200 apiece. I'm trying to decide whether I should sell some of my speakers and use the proceeds to buy more Christmas presents for my family. As of now I have $800 budgeted for Christmas presents. My preferences over speakers and money to spend on other Christmas presents may be represented by the Cobb-Douglas utility function u(x1, 12) = xqx]-a, where speakers are good 1 and money for other Christmas presents is good 2. For simplicity, assume speakers can be bought and sold as if they were a continuous good (i.e., not discrete). (a) What is my endowment point (e1, ez)? 1 (b) What is my MRS at the endowment point? The expression you derive will be a function of a; based on the information given in the question, what can you say about the value of a? (c) Draw my budget constraint. Explain why it is kinked, and what the slopes on the two sections of the budget line represent. (d) Suppose I decide to sell four of my speakers for $100 each to my friend Jeff. Draw the indifference curves passing through my endowment point (e1,e2) and my optimal choice (31,37). What value of a would lead me to this being my optimal behavior? (e) Assuming this is my true value of a, derive my "supply curve" for speakers: that is, how many speakers would I sell as a function of the market price for used speakers, p? Draw this curve in a diagram, labeling the appropriate points. (Hint: you know the point (4 speakers, p = 100) must be on it...)
