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ECON6001/6701: Answers to Midterm S1 2022
发布时间:2023-04-05
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ECON6001/6701: Answers to Midterm S1 2022
1 Short-Answer Questions (45 points)
Group Q1 (3 points)
1. Consider u(x, y) = Which of the following statements are correct? (There might be several correct statements or none.)
I. 从 represented by u is not continuous.
II. Utility u(x, y) represents the same preference as v(x, y) = xy . (***)
III. 从 represented by u is not monotone.
2. The same questions for u(x, y) = {
10 .
Group Q2 (3 points)
1. Consider u(x, y) = Which of the following statements are correct? (There might be several correct statements or none.)
I. 从 represented by u is not continuous. (***)
II. Utility u(x, y) represents the same preference as v(x, y) = xy .
III. 从 represented by u implies that x and y are perfect substitutes.
2. The same questions for u(x, y) = {
10 .
Group Q3 (5 points)
1. A consumer has utility u(x, y) = <x + 2(y + 1). Then MRS at x = and y = 7 is __. (Answer: 1)
2. A consumer has utility u(x, y) = <xy + y . Then MRS at x = 4 and y = 1 is __. (Answer: 0.125)
Group Q4 (4 points)
1. Suppose S = {s1 , s2 } and a consumer evaluates AA acts as follows
V (f) = max{z : f(s)(z) ≥ 0.5}.
s
Consider the following act g:
Then the value of act g is __ (50).
2. Suppose S = {s1 , s2 } and a consumer evaluates AA acts as follows
V (f) = min{z : f(s)(z) ≥ 1 }.
Consider the following act g:
Then the value of act g is __ (40).
Group Q5 (6 points)
1. Suppose S = {s1 , s2 } and a consumer evaluates AA acts as follows
V (f) = max{z : f(s)(z) ≥ 0.4}.
s
Consider the following acts g and h:
Then the value of act 0.5g + 0.5h is __ (60).
2. Suppose S = {s1 , s2 } and a consumer evaluates AA acts as follows
V (f) = min{z : f(s)(z) ≥ 1 }.
Consider the following acts g and h:
Then the value of act 0.5g + 0.5h is __ (20).
Group Q6 (10 points)
1. Suppose S = {s1 , s2 , s3 } and there are 3 assets available in the market with the following return vectors: x1 = (1, 2, 0) that costs q1 = 4, x2 = (2, 1, 0) that costs q2 = 5, and x3 = (3, 3, 1) that costs q3 = 12. The price of the riskless bond y = (1, 1, 1) is __ (Answer: 6). Consumer wants to purchase a consumption vector (4, 1, 2), then she will buy __ (-2.667) of x1 , __ (0.333) of x2 and __ (2) of x3 . You should round all numbers up to the second digit, i.e., 28.8864 should be entered
as 28.89.
2. Suppose S = {s1 , s2 , s3 } and there are 3 assets available in the market with the following return vectors: x1 = (0, 2, 1) that costs q1 = 8, x2 = (3, 4, 5) that costs q2 = 30, and x3 = (2, 2, 4) that costs q3 = 20. The price of the riskless bond y = (1, 1, 1) is __ (Answer: 7.333). Consumer wants to purchase a consumption vector (1, 5, 7), then she will buy __ (3) of x1 , __ (-2) of x2 and __ (3.5) of x3 . You should round all numbers up to the second digit, i.e., 28.8864 should be entered as
28.89.
Group Q7 (7 points)
1. Suppose consumer’s initial wealth is w0 = $40 and she is facing a lottery (−$40, ; +$60,
). If consumer’s utility is u(w) = ^w, then the certainty equivalent is __ (44.444) and the risk-premium is __ (22.222). 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 = $80 and she is facing a lottery (−$80, 0.25; +$60, 0.75). If consumer’s utility is u(w) = ^w, then the certainty equivalent is __ (78.75) and the risk-premium is __ (26.25). You should round all numbers up to the second digit, i.e., 28.8864 should be entered as 28.89.
Group Q8 (7 points)
1. Suppose indirect utility is v(px , py , I) = 4(I27p北(+p北)p(+)p2yy )3 . If u = 27, px = 1, py = 2, then Hicksian demand for good x is __ (2).
2. Suppose indirect utility is v(px , py , I) = 4(I27p2北(+p北) If u = 4, px = 5, py = 5, then Hicksian demand for good x is __ (1).
2 Long-Answer Questions (55 points)
Group Q9 (30 points)
1. Suppose u(x, y) = x(y + 1)2 . Note that the problem has corner solutions, if you do not consider the corner solutions case, you will be able to obtain not more than 75% of the available points.
(a) Derive Hicksian demand for both goods. (16 points)
Solution: We have to solve EMP. For the interior solution, we have MRS=price ratio, hence
(y + 1)2 y + 1 px
MRS = 2x(y + 1) = 2x = py ,
implying y = 2xp(p)y(北) − 1. By plugging this back into utility constraint, we have
x(y + 1)2 = 4x3 = u ⇒ hx = 2 −
u
px(−)
p
hy = 2hx
− 1 = 2
u
p
py(−)
− 1.
Note that it is possible for hy to be negative, which means that we have corner solutions when
1 1
hx = {u(2)23o
1u3ther(p)23ise(p)
, if 2
u3(1)p
py(−)3(1) ≥ 1
and hy = {0(2) o(u)13the(p)13xrw(p)
13e − 1, if 2
u
p
py(−)
≥ 1 .
(b) Derive the indirect utility. (14 points)
Solution: The expenditure function for the interior solution is
e(p, u) = 2− u
p
p
+ 2
u
p
p
− py = 2
u
p
p
∗ 1.5 − py .
The expenditure function for the corner solution is
e(p, u) = upx .
Then the indirect utility for the interior solution is
I = 2 v
p
p
∗ 1.5 − py ⇒ v(px , py , I) =
and the indirect utility for the corner solution is
I = vpx ⇒ v(px , py , I) = I
Finally, we have to translate the condition on px , py , u into a condition on px , py , I :
2 u
p
py(−)
≥ 1 ⇔ 2u
≥ 1 ⇔
≥ 1 ⇔
≥ 1 ⇔ I ≥ 0.5py .
Hence, the indirect utility is
v(px , py , I) = {
2. Suppose u(x, y) = x(y + 1)3 . Note that the problem has corner solutions, if you do not consider
the corner solutions case, you will be able to obtain not more than 75% of the available points.
(a) Derive Hicksian demand for both goods. (16 points)
Solution: We have to solve EMP. For the interior solution, we have MRS=price ratio, hence
(y + 1)3 y + 1 px
MRS = 3x(y + 1)2 = 3x = py ,
implying y = 3x − 1. By plugging this back into utility constraint, we have
x(y + 1)3 = 27x4 = u ⇒ hx = 3 −
u
px(−)
p
hy = 3hx
− 1 = 3
u
p
py(−)
− 1.
Note that it is possible for hy to be negative, which means that we have corner solutions when
1 1
hx = {u(3)34o
1u4ther(p)34ise(p)
, if 3
u
p
py(−)
≥ 1
and hy = {0(3) o(u)14the(p)14xrw(p)
14e − 1, if 3
u
p
py(−)
≥ 1 .
(b) Derive the indirect utility. (14 points)
Solution: The expenditure function for the interior solution is
e(p, u) = 3− u
p
p
+ 3
u
p
p
− py = 3
u
p
p
∗
− py .
The expenditure function for the corner solution is
e(p, u) = upx .
Then the indirect utility for the interior solution is
I = 3 v
p
p
∗
− py ⇒ v(px , py , I) =
and the indirect utility for the corner solution is
I = vpx ⇒ v(px , py , I) = I
Finally, we have to translate the condition on px , py , u into a condition on px , py , I :
3 u
p
py(−)
≥ 1 ⇔ 3u
≥ 1 ⇔
≥ 1 ⇔
≥ 1 ⇔ 3I ≥ py .
Hence, the indirect utility is
v(px , py , I) = {
Group Q10 (25 points)
1. Suppose there are two states of the world, s1 and s2 . We observe the following data about con- sumption in these states: x1 = (3, 2) with prices p1 = (1, 2); x2 = (3, 5) with prices p2 = (3, 1); x3 = (2, 4) with prices p3 = (1, 1).
(a) (15 points) Do these data satisfy WARP and GARP?
Solution: To identify which bundles were affordable in which budget, we construct the cost table:
|
(1,2) |
(3,1) |
(1,1) |
x1 = (3, 2) |
7 |
11 |
5 |
x2 = (3, 5) |
13 |
14 |
8 |
x3 = (2, 4) |
10 |
10 |
6 |
We can conclude that x1 ∈ B 1 , B2 , B3 ; x2 ∈ B2 ; x3 ∈ B2 , B3 . So we do not have two budgets, in which two of our bundles could be affordable and each of them chosen at least in one of
those budgets. This fact implies that WARP is trivially satisfied.
For GARP, we have the following relationships:
B2 : x2 〉x1 , x3 ; and B3 : x3 〉x1
and B 1 is not informative. Hence, we get x2 〉 x3 〉 x1 , which is consistent with rational preference, implying that GARP is satisfied.
(b) (10 points) Do these data satisfy SARSEU?
We have the following order of consumptions:
x2(2) > x2(3) > x 1(1) = x1(2) > x2(1) = x1(3) .
We need to construct all possible sequences of pairs that satisfy conditions (2) and (3) in SARSEU. By checking all possible combinations, we have the following three sequences:
• x2(2) > x1(2) , x 1(1) > x2(1), implying the corresponding price ratio is =
< 1.
• x2(3) > x2(1) , x 1(1) > x1(3), implying the corresponding price ratio is =
< 1.
• x2(3) > x1(3) , x 1(1) > x2(1), implying the corresponding price ratio is =
< 1.
Thus, these data satisfy SARSEU.
2. Suppose there are two states of the world, s1 and s2 . We observe the following data about con- sumption in these states: x1 = (1, 5) with prices p1 = (3, 2); x2 = (2, 2) with prices p2 = (4, 4); x3 = (5, 2) with prices p3 = (1, 3).
(a) (15 points) Do these data satisfy WARP and GARP?
Solution: To identify which bundles were affordable in which budget, we construct the cost table:
|
(3,2) |
(4,4) |
(1,3) |
x1 = (1, 5) |
13 |
24 |
16 |
x2 = (2, 2) |
10 |
16 |
8 |
x3 = (5, 2) |
19 |
28 |
11 |
We can conclude that x1 ∈ B 1 ; x2 ∈ B 1 , B2 , B3 ; x3 ∈ B3 . So we do not have two budgets, in which two of our bundles could be affordable. This fact implies that WARP is trivially satisfied.
For GARP, we have the following relationships:
B 1 : x1 〉x2 ; and B3 : x3 〉x2
and B2 is not informative. Hence, we get x1 , x3 〉 x2 , which is consistent with rational
preference, implying that GARP is satisfied.
(b) (10 points) Do these data satisfy SARSEU?
We have the following order of consumptions:
x2(1) = x1(3) > x1(2) = x2(2) = x2(3) > x 1(1) .
We need to construct all possible sequences of pairs that satisfy conditions (2) and (3) in SARSEU. By checking all possible combinations, we have the following three sequences:
• x2(1) > x2(2) , x1(2) > x 1(1), implying the corresponding price ratio is =
< 1.
• x2(1) > x2(3) , x1(3) > x 1(1), implying the corresponding price ratio is =
< 1.
• x2(1) > x 1(1) , x1(3) > x2(3), implying the corresponding price ratio is =
< 1.
Thus, these data satisfy SARSEU.