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ECON7030 MICROECONOMIC ANALYSIS
MOCK PROBLEM SET 1
Answer all questions. Justify all your answers. Every graph (figure or diagram) in your answers has to be well-labelled. For functions that intersect the axis, the points where they intersect them must be identified (providing the corresponding numbers). All quantities of goods can be treated as continuous variables unless explicitly stated otherwise. Show your work. Marks are as indicated.
Question 1 (20 marks) True, False, or Uncertain? Justify your answers.
(a) (4 marks) Suppose there are 3 bundles in a consumption set: bundles A, B, and C. Assume that A ≿ B, B ≿ A, C ≿ B, and A ≿ C. This information is sufficient to establish that ≿ is complete.
(b) (4 marks) Suppose there are 3 bundles in a consumption set: bundles A, B, and C. Assume that A ≿ B, B ≿ A, C ≿ B, and A ≿ C. This information is sufficient to establish that ≿ is transitive.
(c) (4 marks) An indifference curve cannot be thick.
(d) (4 marks) For a 2-good world, a utility-maximising consumer with a monotone prefer ence who faces a budget constraint must choose the optimal bundle in which
where MU1 and MU2 are the marginal utilities for good 1 and good 2 respectively. P1 and P2 are the prices for good 1 and good 2 respectively. X1 and X2 are the quantities for good 1 and good 2 respectively. M is the consumer’s budget.
(e) (4 marks) Giving a consumer enough income to afford their original bundle after a price increase is the correct way to measure the compensation required to restore their original utility level.
Question 2 (20 marks) Suppose there are only two goods: Beer and Milk. Jos´e’s preference over bundles of beer and milk is as follows: for any two bundles A = (bA, mA) and B = (bB, mB) (where b and m denotes the amount of beer and milk, respectively), A ≿ B if and only if:
Either: bA > bB;
Or: bA = bB and mA ≥ mB.
In other words, Jos´e cares first and foremost the amount of beer, but if the two bundles contain the same amount of beer, then he prefers having more milk to less.
(a) (4 marks) Is Jos´e’s preference strongly monotone? Explain.
(b) (4 marks) Is Jos´e’s preference convex? Explain.
(c) (4 marks) Recall that A ∼ B iff A ≿ B and B ≿ A. Define the relation A ∼ B in terms of bA, bB, mA and mB. (Hint: The relation A ≿ B in terms of bA, bB, mA and mB is defined by Either bA > bB; Or bA = bB and mA ≥ mB.)
(d) (4 marks) Recall that A ≻ B iff A ≿ B and B ≿ A. Define the relation A ≻ B in terms of bA, bB, mA and mB. (Hint: The relation A ≿ B in terms of bA, bB, mA and mB is defined by Either bA > bB; Or bA = bB and mA ≥ mB.)
(e) (4 marks) Explain how the shape of the indifference curve for Jos´e’s preference looks like. Draw three examples of indifference curves. Choose one bundle in the graph, label it E, and indicate the set that is strictly preferred to the bundle E. [Hint: Use the results from part (d).]
Question 3 (25 marks) Consider a consumer with Cobb-Douglas utility:
The consumer faces prices p1, p2 > 0 and has income m > 0.
(a) (5 marks) Derive the Marshallian (uncompensated) demand functions x
∗
1
(p1, p2, m) and x
∗
2
(p1, p2, m).
(b) (5 marks) What does the parameter α tell us economically about how the consumer allocates income across the 2 goods?
(c) (5 marks) Derive the Hicksian (compensated) demand functions x
h
1
(p1, p2, u¯) and x
h
2
(p1, p2, u¯), and the expenditure function e(p1, p2, u¯).
(d) (5 marks) Using your results from Parts A and B, decompose the effect of a change in p1 on the Marshallian demand for good 1 via the Slutsky equation:
Compute each term explicitly and verify that the equation holds algebraically.
(e) (5 marks) Is good 1 a normal good, inferior good, or Giffen good under this utility function? Justify your answer using the decomposition.
Question 4 (35 marks) Mary consumes only scoops of ice-cream (x) and cones (y). She insists on a particular combination of ice-cream and cones. Mary must have 2 scoops of ice-cream with every cone. Assume that both quantities of ice-cream scoops and cones are continuous variables. Her preference is complete, transitive, and monotone.
(a) (5 marks) Propose 2 different utility functions that represents Mary’s preference.
(b) Pick one of the utility functions you proposed in part (a).
(i) (5 marks) Draw a couple of indifference curves, indicate two bundles in each of them and the corresponding utility level. Make sure to label the axes appropri ately.
(ii) (5 marks) What is the Marginal Rate of Substitution for these preferences? Ex plain.
(c) Suppose each scoop of ice-cream costs $2, and each cone costs $1. Mary has an ice cream budget of $10.
(i) (5 marks) Draw her budget line in the diagram you have drawn in part (b). Show your work.
(ii) (5 marks) Find Mary’s utility maximising consumption bundle. Show your work.
(d) Suppose that now there is a promotion that each scoop of ice-cream is only $1. Everything else remains the same.
(i) (5 marks) Redraw the diagram from part (c) AND draw the new budget line as a result of this promotion. Label all the intercepts appropriately.
(ii) (5 marks) Assume now that she can only consume integer quantities of x and y.
Find Mary’s new utility maximising consumption bundle. Show your work.
