Questions for Quiz 1 ECON6001/ECON6701, Semester 1 2022
– INSTRUCTIONS
1. Unlike the usual MCQ you may be accustomed to, in the following questions, there is not necessarily a unique correct choice multiple options may be correct. For questions with multiple correct answers, total points for a question are divided equally between each correct selection. I deduct points if you are over-selecting answers. For example: To illustrate, suppose (a) and (b) are the only two correct answers to a 10 point question.
2. This Quiz is testing you on how well you understand the material covered in Lec 1-3. Review these lectures well, make sure that you understand what all the Theorems are saying and the assumptions they rely upon. Work through the Problem Sets. In particular, Condition , Condition , WARP, Roy’s Identity, Sheppard’s Lemma, properties of the Slutsky Matrix (second last slide, Lec 3) all play a role.
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For the most part, there are no complicated derivations, often simple examples / counter- examples to the given statements are adequate.
3. Each of the TEN questions is worth 10 points.
Questions.
Note. There was a typo in the definition of WARP in the original slides, which I then corrected in class. (Slide 23, Lec 1) For convenience, I repeat the corrected definition here: WARP is said to hold if for any given pair of alternatives a; b: a is revealed preferred to b implies b is not strictly revealed preferred to a.
1) DMhasa well-definedstrictrankingoverthefinitesetofalternativesX=fa;b;c;d;e;f;gg. Withoutlossofgenerality,assumea b c d e f g. AtanyBX,C(B)isthe medianrankedalternative. MedianisdefinedasthemiddlerankedelementifBhasanodd number of elements and the middle two elements if B has an even number of elements. (So, for example, C(fa;b;cg)=fbg since a b c and C(fc;d;f;gg)=fd;fg since c d f g.)
a) Condition is satisfied. b) Condition is satisfied.
c) Not enough information to conclude if Condition holds. d) None of the above.
your choice mark a,b 10
a,b,c 5 a,d 5
Explanation
(5 deducted for choosing (c), since )
you choose three options when only two are correct no deduction for choosing the incorrect (d),
since only two options are chosen
2) In a two good economy, a consumer chose the bundle A when given the blue budget con- straint (the relatively steep line) and B when given the brown budget constraint. Assume that choice in each case is unique. What property/condition/axiom does she necessarily violate? You will be able to write your answer directly on Canvas
3) Consider a choice structure hX ; B ; C i where X is finite and B includes all two and three element subsets of X. Which of the following are true statements?
a. Condition
b. Condition
c. Condition B2B.
and Condition imply the WARP .
and Condition are implied by the WARP.
implies Condition , provided C(B) has exactly one element for every
d. A preference relation (complete, reflexive and transitive relation) that rationalizes the agent’s choice may not exist even though the Weak Axiom is satisfied.
4) Fill in the blanks labelled A , B and C and D in the statement below. You will be able to enter your answer directly on Canvas Quiz portal.
A Theorem ensures that B preferences on R+n have such a utility representation which allows us to apply C Theorem to gaurantee the existence of a solution to the Utility Maximization Problem for any feasible set is D .
5) Let X=fx;y;zg and B=ffx;yg;fx;y;zgg and C(fx;yg)=fxg. Which of the following are consistent with WARP?
a. C(fx;y;zg)=fyg b. C(fx;y;zg)=fxg c. C(fx;y;zg)=fzg
d. C(fx;y;zg)=fx;zg
6) We used A to connect preferences with observable demand and then made an assumption that preferences are B to deduce that the area under the curve shown in Figure 2 below is a measure of a consumer’s utility when price of the good changes .
xi(pi;pj;y)
Figure 2. Marshallian Demand for Good i.
7) Consider a typical utility maximizing consumer, with a continuous utility function defined over two goods. So X = R2+ as usual but with a small twist on the budget sets:
The cost of a unit of Good 1 consumed above 50 units is one and a half times the cost of a unit that good consumed up to 50 units. So depending on the income level, a typical budget line may look either the thick blue line or the thick black line, depending on whether the consumer has sufficient income to consume 50 units or not.
50 x1 Figure 3.
Assume that Assumption 3 on preferences is met (so, in particular, there is a continuous utility function). Which of the following statements are true?
a. The Extreme Value Theorem ensures that the utility maximization problem has a well-defined solution for all (p; y) 0.
b. For every (p; y) 0, there must be a unique utility maximizing choice.
c. Part (b) but only if preferences are also strictly convex and locally non-satiated.
Price of Good i
Qty of Good i
d. Since the budget sets are not convex, Extreme Value Theorem does not apply here.
8) Repeat the previous question, but this time assume that Every unit of Good 1 consumed above 50 units costs oneandahalftimesthecost half the cost a unit of consumption consumed up to 50 units. Using a figure of the appropriate budget sets, answer which of the folowing of the following statements are true .
a. The Extreme Value Theorem ensures that the utility maximization problem has a well-defined solution for all (p; y) 0.
b. For every (p; y) 0, there must be a unique utility maximizing choice.
c. Part (b) but only if preferences are also strictly convex and locally non-satiated.
d. Since the budget sets are not convex, Extreme Value Theorem does not apply here.
9) The following is a Slutsky Matrix for some rational consumer in a three good economy at
some given(andfixed)(p;u):
0@ ¡ 8 a b 1A S= c¡2d 3ef
Using the properties of the Slutsky Matrix, we may conslude that a. b=a and d=e.
d. None of the above.
10) Suppose the expenditure function is given by the following equation:
e(p;u) = 13p1+pp1p2+23p2u
At a target utility of u = 1, which of the following statements are true, where xh is the
Hicksian demand function?
a. xh1(p;u)>13 andxh2(p;u)>23
b. xh1(p;u)13 andxh2(p;u)23
c. x1(p;u)= 13 and x2(p;u)= 23 d. None of the above.
Hint: What Lemma connects Expenditure function with the Hicksian Demand? See Lec 3.
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