I need help please!!! Thank you!! On Bilbo\'s quest, the rest of the group decid

Question

I need help please!!! Thank you!! On Bilbo's quest, the rest of the group decided that he was holding them back, so one morning, they left him stranded in the Mirkwook Forest. Bilbo is able to work 100 hours a week. He can split this time between gathering magical stones and gathering berries. His production functions are: And his utility function is: U = 9 ln(B) + 8 ln(S) How will Bilbo choose to allocate his labor? What will the optimal levels of Berries and Stones be? What will Bilbo's utility be? Suppose a group of elves encounters Bilbo in the woods. Since elves are magical, the magical stones have little value to them: they will trade with Bilbo at a price ratio of P_B/P_S = 16/7. If Bilbo continues his production from part a, what will he choose to consume now that he can trade? What will be his level of utility? Now, Bilbo realizes that, if he changes his production, he can increase his utility further. How docs Bilbo choose to change his production to maximize his utility? What is his consumption? What is his utility? (Sorry, but these aren't going to be pretty). Draw a graph illustrating your answers

Explanation / Answer

Given, 9 hours in berries and 8 hours in stones

Berries = underroot 72= 9 berries and 8 stones.

Hence Labour in berries = underoot 72= 36 hours

Labour in stones= 64 hours

(b) Optimal level of berries= 36 x 9= 324 berries

Stones = 64 x 8= 512 stones

(c) Bilbo's utility= 9x36+ 64x8= 836 utils

(d) 16/7 x 36= 64 berries

7/16x64= 28 stones

Utility= 9x64+8x28= 800

(e) Maximum = 9x64+8x36= 864 utils

(f) Graph=

Berries * * Supply

*

* * demand   

Stones

Hence solved,

By,

Nishant Bhatt

Stones

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