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ECON2010 Intermediate Microeconomics

Semester One Mid-Semester Examinations, 2025

Question  1  (10  marks). Consider the utility function U (x, y) = xa yb, where a > 0 and b > 0. The optimal bundle associated to this utility function is given by

Suppose that a = 2, b = 3 and the consumer’s income is M = 100.

(a) Derive the inverse demad function for x and the inverse demand function for y. (1 mark)

(b) Consider the inverse demand function for x that you obtained in part (a).  Obtain the first and second derivative of this function with respect to x.  Determine the sign of these derivatives.  Explain the implications of these signs for the graph of the inverse demand function.  Determine whether this function intersects the vertical and/or horizontal axis. Draw the graph of this inverse demand function.  (8 marks)

(c) Derive the percentage of the consumer’s income that is allocated to buying x  and the percentage of the consumer’s income that is allocated to buying y. (1 mark)

Question  2   (10  marks). Refer to the file Q&A1problems2025 on Blackboard under Learning Resources / Lecture 1 :  Preference and Utility.

(a) Refer to slide 1 (the one starting with “Binary  relation:”). Change the entries  in the preference table in that slide, so that the table represents the preferences of a consumer with complete and transitive preferences, whose strictly preferred food is sushi, followed by seafood, which, in turn, she strictly prefers to steak and pasta; and between these last two, she strictly prefers steak to pasta. (4 marks)

(b) Redo your table from part (a), changing only one of the entries, so that the preferences fail to satisfy completeness.  (Hint:  there are many ways to do this, you just pick one of them.)  Explain in words why the change you made makes the preferences fail to satisfy completeness. (2 marks)

(c) Redo your table from part (a), changing entries as needed, so that the preferences fail to satisfy transitivity, while still satisfying completeness.  (Hint:  there may be many ways to do this, you just pick one of them.)  Explain in words why the change(s) you made makes the preferences fail to satisfy transitivity. Explain why these modified preferences still satisfy completeness. (4 marks)

Question  3 (10  marks). In the FederaI Budget announced in March of 2025, the AustraIian Gvovernment aIIocated M = 10.3 biIIions to heaIth. This amount was aIIocated to two main items: incentives for doctors (x*  = 8.5 biIIions) and pubIic hospitaI investment (y*  = 1.8 biIIions). Suppose that the budget aIIocation between these two items relects the government1s preferences U (x, y) over expenditure bundIes (x, y) subject to the budget constraint M = xpx +ypy , where px  = py  = 1 (which simpIy relects that each doIIar aIIocated to x costs one doIIar of the budget and, simiIarIy, each doIIar aIIocated to y costs one doIIar of the budget).

(a) Refer to the sIides in the fiIe Lecture 3 - slides. CouId the budget aIIocation (x* , y* ) decided by the government correspond to the case iIIustrated in sIide 22 (the one starting with “ Non-interior solutions”)? What about the case iIIustrated in sIide 7 (the one starting with “ Monotonic preferences”)? Justify your answers. (4 marks)

(b) Given the government1s budget aIIocation (x* , y* )  =  (8.5, 1.8), couId the government1s preferences be represented by the utiIity function U (x, y) = ax + by where a > 0 and b > 0 are two diferent constants? Justify your answer in detaiI. (3 marks)

(c) Given the government1s budget aIIocation (x* , y* )  =  (8.5, 1.8), couId the government1s preferences be represented by the utiIity function U (x, y) = min {ax, by} where a > 0 and b > 0 are two diferent constants? If your answer is NO, expIain why this is the case.  If your answer is YES, then provide vaIues of a and b that wouId be consistent with the current soIution (x* , y* ) = (8.5, 1.8). Justify your answer in detaiI. (3 marks)

Question 4  (10  marks). Refer to the problem in slide 17 (the one with title  “Problem”) in the file Q&A 6 2025 S1 - slides on Blackboard under Learning Resources / Lecture 6:  Welfare.  Instead of the utility function in that slide, consider the utility function U (x, y) = min {x, y} and suppose that M = 6.  Other than that, consider the same setup, and the same initial prices Px   =  1 and Py  = 1.

(a) Follow the three  steps in  slide  18  (the one starting with “Solution”) to compute the compensating variation for an increase in the price of X from Px  = 1 to Px  = 2. Additionally, draw one graph to explain and illustrate the three steps to solve this problem.  Interpret the compensating variation obtained. (5 marks)

(b) Follow the three steps in slide 24  (the one starting with “b.   Compute  the equivalent variation”) to compute the equivalent variation for an increase in the price of X from Px  = 1 to Px  = 2. Additionally, draw one grap to explain and illustrate the three steps to solve this problem. Interpret the equivalent variation obtained.  (5 marks)