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ECON7030 MICROECONOMIC ANALYSIS
创建日期：2024-07-10 16:17:56
所属分类：经济Economics
标签： ECON7030
MICROECONOMIC ANALYSIS

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ECON7030 MICROECONOMIC ANALYSIS


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).

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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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