Intermediate Microeconomics questions
Problem 1 [41 marks]
Consider the utility functions of the four Intermediate Microeconomics II tutors:
Nok: U_N (x_1,x_2 )=x_1^2 x_2^2
Xing: U_X (x_1,x_2 )=4log??x_1 ?+4log??x_2 ?
Kim: U_K (x_1,x_2 )=2/(x_1 x_2 )
Aaron: U_A (x_1,x_2 )=?4x?_1+?4x?_2
a) Explain the difference between a utility function and preferences. [2 marks]
b) For each of Aaron, Xing and Kim, SHOW whether he or she has the same preferences as Nok. [9 marks]
Hint: An appropriate method to show that preferences are the same is to show that one utility function is a monotonic transformation of the other. To show that preferences are not the same, it is appropriate either to show that it is not a monotonic transformation or to provide a pair of consumption bundles that the utility functions would rank differently.
Hint: log??(xy)=log?x+log?y ? and log??(x^a)?=a log?x
c) Find the equation of an indifference curve for an arbitrary utility level ‘k’ for each of Nok, Kim and Aaron, and write it in the form x_2=?. [3 marks]
d) Consider Nok’s preferences. Determine whether each commodity x_1 and x_2 is an economic good, an economic bad or an economic neutral. Do the same analysis for Kim’s and Aaron’s preferences. [6 marks]
e) Based on your answer from part d), does Nok have monotonic preferences? Does Kim? Does Aaron? Explain why. [3 marks]
f) If Nok’s and Kim’s indifference curves were drawn on a graph, their indifference curves would be represented by the same curves. Is this consistent with your result in part b) when Nok’s and Kim’s preferences were compared? Explain. [3 marks]
g) For each of Nok, Kim and Aaron, draw an indifference curve that gives a utility level of 36. Indicate three bundles on these indifference curves by specifying the coordinates of each bundle. [6 marks]
h) For each of Nok, Kim and Aaron, shade in the weakly preferred set of bundles you indicated on the indifference curve drawn in part g). [3 marks]
i) Based on your answer in part h), for each of Nok, Kim and Aaron, determine and explain whether the individual has convex, concave, or non-convex preferences. [4 marks]
j) Can you determine whether Xing’s preferences are convex or not convex without drawing her indifference curves? If yes, how? If no, why not? [2 marks]
Problem 2 [22 marks]
Kevin consumes two commodities, spaghetti (S) and homework (H). To motivate Kevin to study, Kevin’s parents pay him $2 for each hour spent doing homework. Assume Kevin can complete as much homework as he wishes at that price (i.e. he faces no time constraints and will never run out of homework). He has no other source of income. Spaghetti costs him $2/kg.
a) How much spaghetti can Kevin afford if he spends no time on homework? How much spaghetti can Kevin afford if he spends 15 hours on homework? [2 marks]
b) What is Kevin’s opportunity cost of one hour of homework? [1 mark]
c) Draw Kevin’s budget line with spaghetti on the horizontal axis and homework on the vertical axis. [2 marks]
Hint: Think about your answer to part a).
d) Write an equation for this budget line and shade in the budget set. [2 marks]
e) What assumptions are needed regarding Kevin’s preferences between spaghetti and homework to be able to express them with a continuous utility function? [2 marks]
f) Kevin has a utility function which represents well-behaved preferences given by U(S,H)= S^(1/2)-(3/28)H^(7/6). Provide an equation for Kevin’s marginal rate of substitution. What is the marginal rate of substitution for Kevin at his optimal bundle (8,8)? [3 marks]
g) We know that Kevin has convex preferences. Draw a typical indifference curve through Kevin’s optimal bundle. (You do not need to specify the utility of the indifference curve or other bundles on the indifference curve, just draw the typical shape of an indifference curve given that Kevin’s preferences are convex.) [3 marks]
h) Shade in the weakly preferred set of the optimal bundle. [2 marks]
i) In general, how do we determine whether preferences are convex? Based on your answers in part g) and h), confirm that Kevin has convex preferences. [3 marks]
j) Does Kevin have monotonic preferences? Justify your answer. [2 marks]
Problem 3 [18 marks]
Jess, Zoe and Terill are best friends and want to take a vacation to Queensland together. They are trying to decide which hotel to stay in.
Jess, Zoe and Terill rate their preferences over hotels in the same manner. If they are comparing two hotels and one is better in one or more characteristics that matter to them and no worse in any other, then they strictly prefer that hotel. If the hotels are identical in all characteristics that matter to them, then they are indifferent between the hotels. If the hotels are better in some characteristics but worse in others, then they cannot compare the hotels.
Jess prefers hotels that have more swimming pools, more restaurants and cost less. Zoe prefers more swimming pools, doesn’t care at all about the number of restaurants and prefers cheaper hotels. Terill prefers more swimming pools, more restaurants and does not care at all about the cost.
Hotel A has 2 swimming pools, 1 restaurant and costs $100 per night.
Hotel B has 1 swimming pool, 1 restaurant and costs $100 per night.
Hotel C has 1 swimming pool, 2 restaurants and costs $150 per night.
a) Using the symbols ?, ?, or ~, write down how Jess compares each possible pair of hotels (A vs. B, B vs. C, A vs. C). If no symbol can be used, you can state this. [3 marks]
b) Using the symbols ?, ?, or ~, write down how Zoe compares each possible pair of hotels (A vs. B, B vs. C, A vs. C). If no symbol can be used, you can state this. [3 marks]
c) Using the symbols ?, ?, or ~, write down how Terill compares each possible pair of hotels (A vs. B, B vs. C, A vs. C). If no symbol can be used, you can state this. [3 marks]
d) Explain what it means for preferences to be ‘complete’. [1 mark]
e) Are Jess’ preferences over the three hotels ‘complete’? Why or why not? [2 marks]
f) Explain what it means for preferences to be ‘transitive’. [1 mark]
f) Are Zoe’s preferences over the three hotels ‘transitive’? Why or why not? [2 marks]
h) What does it mean for preferences to be ‘reflexive’? [1 mark]
i) Are Terill’s preferences over the three hotels ‘reflexive’? Why or why not? [2 marks]
Problem 4 [19 marks]
Signe consumes only two commodities, apples (A) and bananas (B). She has an income of $64. The price of apples is $16/kg and the price of bananas is $3/kg. Her utility function is U(A,B)=4A^(1/4) B^(3/4).
a) Specify clearly what her objective is, what variable(s) she will choose in order to reach it and what constraint(s) she may face. Write out Signe’s maximization problem in mathematical terms. [3 marks]
b) Use the method of Lagrange to solve Signe’s optimization problem to find her optimal consumption of apples and bananas. [12 marks]
c) Write an equation for the indifference curve that passes through the optimal bundle. What amount of utility will Signe receive from her optimal consumption? [4 marks]
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