ECON6001/6701: Midterm S1 2021

1    Short Questions (43 points)

Group Q1 (3 points)

1. Which of the following statements are correct? (There might be several correct statements or none.)

I. A consumer with convex preference who is indifferent between bundles (1,4) and (9,2) will prefer (5, 3)  (1, 4).

II. If preference is transitive, more is always preferred to less.

III. If preference is representable by a discontinuous utility, then this preference is not continuous.

2. Which of the following statements are correct? (There might be several correct statements or none.)

I. A consumer with convex preference who is indifferent between bundles (1,4) and (9,2) will prefer (1, 4)  (5, 3).

II. If preference is transitive, more is always preferred to less.

III. If preference is representable by a discontinuous utility, then this preference might be continuous.

Group Q2 (4 points)

1. Consumer’s indifference curve is y = 20 − 4 √x. When she buys bundle (25,9), her MRS is __.

2. Consumer’s indifference curve is y = 20 − 2 √x. When she buys bundle (25,9), her MRS is __.

3. Consumer’s indifference curve is y = 20 − 6 √x. When she buys bundle (4,9), her MRS is __.

Group Q3 (4 points)

1. Consider a consumer with u(x, y) = 4x + y . Which of the following statements are correct? (There might be several correct statements or none.)

I. Goods x and y are gross complements.

II. Utility v(x, y) = 10(16x2 + 8xy + y2 ) − 20 represents the same preference.

III. If px  = 5, py  = 2, the consumer will spend all of her money on good y .

2. Consider a consumer with u(x, y) = 2x + y . Which of the following statements are correct? (There might be several correct statements or none.)

I. Goods x and y are net complements.

II. Utility v(x, y) = 10(8x2 + 8xy + y2 ) − 20 represents the same preference.

III. If px  = 5, py  = 2, the consumer will spend all of her money on good y .

Group Q4 (4 points)

1. Suppose the consumer chooses bundle (1,6) with prices (2,10), (2,2) with prices (12,4) and (2,1) with prices (4,3). Which of the following statements are correct?  (There might be several correct statements or none.)

I. Consumer violates WARP.

II. Consumer violates GARP.

III. The revealed preference can be represented by a utility function.

2. Suppose the consumer chooses bundle  (1,5) with prices  (3,3),  (3,3) with prices  (6,4) and  (2,4) with prices (4,3). Which of the following statements are correct?  (There might be several correct statements or none.)

I. Consumer violates WARP.

II. Consumer does not violates GARP.

III. The revealed preference can be represented by a utility function.

Group Q5 (4 points)

1. Suppose S = {s1 , s2 } and a consumer evaluates AA acts as follows V (f) = f(s1 )(0) + f(s2 )(20).

Consider the following act g:

Then the value of act g is __ .

2. Suppose S = {s1 , s2 } and a consumer evaluates AA acts as follows

V (f) = f(s1 )(0) + f(s2 )(30).

Consider the following act g:

Then the value of act g is __ .

3. Suppose S = {s1 , s2 } and a consumer evaluates AA acts as follows

V (f) = f(s1 )(50) + f(s2 )(20).

Consider the following act g:

Then the value of act g is __ .

Group Q6 (6 points)

1. Suppose consumer’s demand is x = 0.2I − 4px , I = 600, px  = 30 and py  = 1. If px  increases by one cent, then the income effect on x is __ and the substitution effect on x is __.

2. Suppose consumer’s demand is x = I − 10px , I = 100, px  = 11 and py  = 6. If px  increases by one cent, then the income effect on x is __ and the substitution effect on x is __.

3. Suppose consumer’s demand is x = 0.5I − 4px , I = 50, px  = 5 and py  = 1. If px  increases by one cent, then the income effect on x is __ and the substitution effect on x is __.

Group Q7 (4 points)

1. Suppose there are two consumers with the following utilities:  u1 (w)  = lnw and u2 (w)  =  √w . Consumers’ wealth is w1   =  100 and w2   =  50.   Given these utilities and wealth, which of the consumers will be more risk-averse?

(a) Consumer 1

(b) Consumer 2

(c) Both consumers are identically risk-averse

(d) Consumers’ risk aversion is not comparable      (e) Not enough information to answer the question

2. Suppose there are two consumers with the following utilities:  u1 (w) = 2 lnw and u2 (w) = √w . Consumers’ wealth is w1   =  100 and w2   =  40.   Given these utilities and wealth, which of the consumers will be more risk-averse?

(a) Consumer 1

(b) Consumer 2

(c) Both consumers are identically risk-averse

(d) Consumers’ risk aversion is not comparable      (e) Not enough information to answer the question

Group Q8 (6 points)

1. Suppose consumer’s initial wealth is w0  = $50 and she is facing a negative shock (−$20, 0.5; −$10, 0.5). If consumer’s utility is u(w) = √w, then the certainty equivalent is __ and the risk-premium is  __. You should round all numbers up to the second digit, i.e., 28.8864 should be entered as 28.89.

2. Suppose consumer’s initial wealth is w0  = $20 and she is facing a negative shock (−$16, 0.5; −$10, 0.5). If consumer’s utility is u(w) = √w, then the certainty equivalent is __ and the risk-premium is  __. You should round all numbers up to the second digit, i.e., 28.8864 should be entered as 28.89.

3. Suppose consumer’s initial wealth is w0  = $40 and she is facing a negative shock (−$20, 0.5; −$10, 0.5). If consumer’s utility is u(w) = √w, then the certainty equivalent is __ and the risk-premium is  __. You should round all numbers up to the second digit, i.e., 28.8864 should be entered as 28.89.

Group Q9 (4 points)

1. Suppose expenditure function is e(px , py , u) = u √pxpy .  If u = 10, px  = 2, py  = 8, then Hicksian demand for good x is __.

2. Suppose expenditure function is e(px , py , u) = 2u√pxpy .  If u = 10, px  = 1, py  = 9, then Hicksian demand for good x is __.

3. Suppose expenditure function is e(px , py , u) = 4u√pxpy .  If u = 5, px  = 25, py  = 1, then Hicksian demand for good x is __.

Group Q10 (4 points)

1. Suppose expenditure function is  e(px , py , u)  =  u(3px  + py ).   If I  =  10, px   =  2, py   =  9, then Marshallian demand for good x is __.

2. Suppose expenditure function is  e(px , py , u)  =  u(px  + 2py ).   If I  =  80, px   =  2, py   =  9, then Marshallian demand for good x is __.

3. Suppose expenditure function is  e(px , py , u)  =  5u(px  + py ).   If I  =  10, px   =  2, py   =  8, then Marshallian demand for good x is __.

2   Long Answer Questions (57 points)

Group Q11 (22 points)

1. Suppose u(x, y) = x0 .75 + y0 .75 .

(a)  (12 points) Derive Hicksian demand for both goods.

(b)  (10 points) Derive the indirect utility.

2. Suppose u(x, y) = x  + y  .

(a)  (12 points) Derive Hicksian demand for both goods.

(b)  (10 points) Derive the indirect utility.

Group Q12 (35 points)

1. Suppose there are two states of the world S = {s1 , s2 } and the set of prizes Z = {0, 50, 100}.  A

consumer believes that state s2  is highly unlikely and evaluates AA acts as follows: V (f) = f(s1 )(50) + f(s1 )(100).

(a)  (15 points) Does this  satisfy monotonicity? If yes, prove it; if no, provide a counter-example. (b)  (15 points) Does this  satisfy independence? If yes, prove it; if no, provide a counter-example.

(c)  (5 points) Does this  have an SEU representation?  If yes, give an example; if no, explain why not.