Answer all three questions keeping these notes in mind -

For question #1a, should either use the Lagrangian or recognize that MRS=MRT for each set of relationships, i.e. the MRS for Clothes/Shelter = MRT for Clothes/Shelter and the MRS for Clothe/Food = MRT for Clothes/Food.

Yes you can use the idea of equalizing the marginal utility per dollar... however, that's not what you've got written as an answer. The marginal utility per dollar for food is MU_F/P_F = (q_S*(1+q_C))^(0.5)/(P_F*2*(q_F)^0.5)). That's very different from what you have.

You will also get a solution for all the variables of interest: q_F, q_S, and q_C (should have 4 equations and 4 unknowns). Need to show q_C

For Question #1b, need compensating and equivalent variation, it's similar to the change in consumer surplus and all of those are measured in dollar values. Your solutions should give a dollar value. I'm also not certain why you're equating utilities in the way that you are as that's not something shown in lecture, tutorials, or the book. That just makes life more challenging as now you have 4 variables and 2 (maybe 3) equations. Will not be able to solve that system as infinite set of possible outcomes.

For Question #1c, check you math. Particularly the MU_T as you seem to have lots q_T... Also, your answer should SHOW that these do not in general violate the 3 big assumptions we made about preferences. For example, can you show that "More is Better" holds with same level of good S, but one additional unit of good T.

1. (28 points) This question will feature a variety of utility functions to give you some time/ practice with deriving demand functions and some aspects of interest in the class. (a) (10 points) Suppose a consumer who is a nudist has the following utility function: U(QFaQSaQC) = qF X 9'3 X (1 + go) where qF = the quantity of food, qs = quantity of shelter, and go = quantity of clothing. Let income be Y = $10, the price of food is PF 2 $1, the price of shelter is P3 : $20, and the price of clothing is P0 : $6. Show the resulting quantity demanded of clothing is negative, implying qg} : 0. Is this result consistent with the idea of a nudist (i.e. someone who does not like to wear clothes)? Explain. What are q}; and q; given income and prices (hint: if qg : 0, then what is the maximization problem)? (10 points) Suppose a video game player has the following utility function: U(QA:QB) = (x/qA + QTBf where (M is the quantity of Axes used in the game and (13 is the quantity of bombs. The video game player has $50 per month to allocate to the game. Initially the video game seller priced axes and bombs identically: PA = PB = $1. After several months, the game seller adjusts the prices to PA : $0.50 and PB : $2 since not enough people were buying axes and too many people were buying bombs. Find BOTH the compensating variation and equivalent variation for the video game player from the price change. (8 points) Suppose an individual has the following utility function: 1 4 Warsaw) : (IS QT where gs is good 8 and qT is good T. Explain why the utility being negative does not violate out main assumptions about preferences. Assuming price of good 8 is PS, price of good T is PT, and income of Y; nd the demand functions q; and (1}. Finally, explain what the relationship between the goods would represent (hint: only concepts discussed are complements, substitutes, or unrelated goods)