ECON7030 MICROECONOMIC ANALYSIS PROBLEM SET 1

发布时间：2024-06-03

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ECON7030 MICROECONOMIC ANALYSIS

PROBLEM SET 1: 100 MARKS

DUE: 28 MARCH 2024 10:00 AM

You have 24 hours to complete AND submit your answers by the due time. Hand- written as well as typed answers are welcome. Make sure that you upload a single PDF file containing multiple pages. Submit on time, ideally at least 1 hour in advance of the deadline. It shows competence and professionalism, and it will ensure that you avoid late penalties.

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.

Question 1 (30 marks)

True, False, or Uncertain? Justify your answers.

(a) (6 marks) Suppose you visit a clothing store and find that all the clothing items, with the exception of a pair of socks, have their prices properly labelled. Hence, the binary relation “as expensive as” defined over the set of all clothing items is complete and transitive. (Hint: To check completeness you need to check reflexivity.)

(b) (6 marks) Lexicographic preferences are monotone and complete but not transitive.

(c) (6 marks) Allan’s preferences ⪰ over the set of bundles that contains chicken and pork are such for any two arbitrary bundles A = (cA , pA ) and B = (cB , pB ),

A ⪰ B if and only if cA ≥ cB .

That is, Allan only cares about his consumption of chicken and does not care about his consumption of pork. His preference relation ⪰ is complete and strongly mono- tone but not strictly convex.

(d) (6 marks) Suppose the utility function U : R → R+ , given by U(x,y) = √xy , represents Matt’s preferences. Hence, for any a > 0, the utility functions (i) V (x,y) =2ln(xy) + a; (ii) V (x, y) = xya; and (iii) a xy also represent the same preferences.

(e) (6 marks) Angela’s preferences over x and y can be represented by a differentiable function U(x,y). Consider the function V (x,y) = f(U(x,y)), where f : R → R is a strictly increasing function. Then the marginal utility of x computed from U(x,y) must be equivalent to the marginal utility of x computed from V (x,y), that is,

MUx = MVx. Similarly, MUy = MVy . As a result, MU(MU)y(x) = MV(MV)y(x) = −MRSxy .

Question 2 (21 marks)

Matt considers Butter (B) and Sour Cream (S) as substitutes. His utility function has the form u(B, S) = α 1 B + α2 S, where α 1 > 0 and α2 > 0. Suppose the price of Butter is PB = 3, the price of Sour Cream is PS = 2, and Matt has an income of M = 100 dollars to spend between Butter and Sour Cream.

(a) (3 marks) In a well labelled diagram, draw Matt’s indifference map, putting Butter in the x-axis and Sour Cream in the y-axis. Make sure to indicate the direction in which utility levels are increasing.

(b) (3 marks) In a well labelled diagram, draw Matt’s budget set indicating the intercepts with the axes and the slope of the budget line.

(c) (3 marks) Obtain the Marginal Rate of Substitution between Butter and Sour Cream. Indicate the MRS in the map drawn in part (a).

(d) (3 marks) Does Matt’s preferences comply with the property of diminishing marginal rate of substitution? Justify your answer.

(e) (9 marks) Obtain Matt’s optimal bundle (set of optimal bundles) when:

(i) α 31 > α 2 2

(iii) α 3 1 < α 2 2

(iii) α 3 1 = α 2 2

Question 3 (23 marks)

Jack consumes only Bread (good x) and Cheese (good y). His utility function is given by

U(x,y) = 3ln(x) + 2ln(y)

(a) (7 marks) The price of good x is $6 and the price of goody is $3. Jack has an income of $120. Find his utility maximising consumption bundle.

(b) (5 marks) Now the price of good y increases to $6 while the price of good x re- mains at $6. Jack’s income remains at $120. Calculate his new utility maximising consumption bundle.

(c) (7 marks) Find the minimal expenditure required for Jack to achieve his original utility level (i.e., the one that corresponds to the bundle you found in part (a)) under the new prices in (b). As you are finding this minimal expenditure, solve for the expenditure minimising bundle that achieves the original utility level under the new prices.

(d) (4 marks) In one clear, sufficiently large, and well-labelled graph, draw the original budget line in (a) and the budget line associated with the minimal expenditure you solved for in part (c). Additionally, draw the indifference curves where the bundle you found in (a) and the bundle you found in (c) lie.

Question 4 (26 marks)

Lucy consumes only scoops of ice-cream (x) and cones (y). Moreover, she insists on consuming these two goods in the combination of α 1 scoops of ice-cream and α2 cones, where α 1 and α2 are real numbers greater than 1. If there are more scoops of ice-cream than cones, she throws the extra ice-cream away. If there are more cones than scoops of ice-cream, she throws the extra cones away.

(a) (4 marks) Draw a couple of Lucy’s indifference curves when: (i) α 1 = α2 ; (ii) α 1 > α2 ; and (iii) α 1 < α2 . Put ice-cream on the horizontal axis.

(b) (8 marks) Suppose each scoop of ice-cream costs α2 , and each cone costs α 1 . Lucy has an income of (α12 +α22 ) dollars. Obtain Lucy’s utility maximising consumption bundle for general α 1 , α2 (i.e., without making any of the assumptions about α 1 and α2 in (a)).

(c) (4 marks) Draw a diagram showing Lucy’s optimal bundle, clearly depicting her budget set and the indifference curve where the optimal bundle lies.

(d) (10 marks) Lucy’s sister, Mary, also insists on a particular combination of ice-cream and cones. Unlike Lucy, however, Mary must consume these two goods in the com- bination of α2 scoops of ice-cream and α 1 cones. Mary is envious of Lucy and wants to achieve the same utility level that Lucy gets when she is consuming her optimal bundle in (b). What is the minimum income that Lucy’s and Mary’s parents must give to Mary so that she can achieve the utility level of Lucy at the current prices? Is this income greater or lower than the income of Lucy in (b).