Microeconomics II 2016 17 Mid term 2016 11 02 version A solution

Prévia do material em texto

A 
UNDERGRADUATE DEGREE IN ECONOMICS 
MICROECONOMICS II 
 
MID-TERM 
 
02 November 2016 
Time allowed for the exam: 1h20m 
 
Teacher: Henrique Monteiro 
 
Write on the top of your answer sheet the letter A 
 
Group I – 5 pts. (20m) 
Correct answers: 0.5 pts.; 
Wrong answers: -0.25 pts. (T/F) or -0.125 pts. (multiple choice). 
 
True-False Questions: 
 
 
1. The weak axiom of revealed preference is a sufficient condition to ensure 
consistency between the consumer’s behavior and the economic theory of 
optimizing consumer. 
 
 
2. An increase in the unit wage can make the labor supply decrease. 
 
 
3. In Laspeyres indexes, we use the weights from the period (t) for which we are 
computing the index. 
 
 4. Hotelling’ rule is a no arbitrage condition. 
 
 
Complete the following sentences: (wrong answers have no point deduction penalty) 
 
5. For a Giffen good, quantity demanded ______________________ when the price 
decreases. 
 
6. The present value of €630 received in a year and €441 received two years from now, 
discounted at a 5% interest rate, adds up to ______________________. 
 
 
Multiple-choice questions: 
 
7. We know that the change in consumer surplus and the equivalent and compensating 
variations when the price of a good changes will all be the same if the utility function if 
of the following form: 
a) Cobb-Douglas � 
b) Linear � 
c) Exponential � 
d) Quasilinear � 
 
A 
 
8. Which effect can never be positive as a response to a price increase? 
a) Total effect � 
b) Substitution effect � 
c) Ordinary income effect � 
d) Endowment income effect � 
 
 
9. If a consumer prefers to concentrate consumption in a single period of time instead of 
distributing it by two periods, we say this consumer’s preferences are: 
a) Concave � 
b) Linear � 
c) Convex � 
d) Quasilinear � 
 
 
10. The set of optimal consumption bundles for all price levels of one of the goods, 
holding all other prices and income constant is called: 
a) The demand curve � 
b) The income offer curve � 
c) The Engel curve � 
d) The price offer curve � 
 
 
Group II – 5 pts. (20m) 
 
 
1. (2 pts.) Vasco, the painter usually buys his brushes and ink buckets from “The 
Artshop”, using his loyalty card to get discounts. The registries of his purchases 
from the last two years are the following: 
 
Year Brushes Ink buckets PB (€/brush) PI (€/can) 
t-1 5 21 2 20 
t-2 2 20 3 15 
 
Use the Paasche quantity index to find out whether he was better or worse off in 
the last year compared to two years ago 
 
 
2. (3 pts.) Will a lender be better off or worse off with an interest rate decrease? 
Justify your answer using the principle of revealed preferences. 
 
 
 
 
A 
Group III – 10 pts. (40m) 
 
 
1. (10 pts.) Virgílio is a regular user of ijingles, a website where users can buy or 
sell songs in mp3 files. His utility function is ����� ��� � 	
����, where �� 
denotes the number of Adele songs and �� stands for the number of songs by 
Boss AC he possesses. Assume that his endowment bundle includes 5 songs by 
Adele and 3 by Boss AC. Both these artists’ songs sell at ijingles for €0.5. 
 
a) (2 pts.) Write the expression of the budget constraint and sketch it in a graph. 
 
b) (2 pts.) Determine Virgílio’s demand functions for songs of each of these artists. 
 
c) (2 pts.) Find Virgílio’s optimal consumption bundle. 
 
d) (2 pts.) If the price of songs by Adele was €1.5 instead how would your answer 
to the previous question differ? Compute the Hicks substitution effect and both 
the ordinary and endowment income effects for this price change. 
 
e) (2 pts.) Calculate the compensating variation. 
 
 
 
A 
Solution 
 
Group I 
 
True-False Questions 
1) F 
2) T 
3) F 
4) T 
Complete the sentences 
5) Decreases 
6) €1,000 
Multiple-choice questions 
7) d 
8) b 
9) a 
10) d 
 
Group II 
 
1) (2 pts.) 
 
Paasche quantity index: 
 
��
���
�� �
��
����
�� � ��
����
��
��
����
�� � ��
����
��
�
� � � � �
 � �	
� � � � �
 � �
�
��
�
�
� 	�
������ � 	 � 
The consumer is better-off. 
 
Note 1 (not required): 
Proof: 
��
���
�� � 	 �
��
����
�� � ��
����
��
��
����
�� � ��
����
��
� 	 � 
� ��
����
�� � ��
����
�� � ��
����
�� � ��
����
��
 
The bundle from �� was affordable in ��! and was not chosen, therefore the bundle 
from ��! ha been revealed preferred to the one from �� . 
 
Note 2: The question explicitly required that the Paasche quantity index was used to 
find out whether Vasco, the painter, was better or worse off. However, that could be 
seen directly from the data table. Notice that Vasco’s consumption bundle in ��! is 
strictly greater than the one he had in �� . 
�"
�!� #
�!� � ����	� � ����
� � �"
� � #
� � 
Using the monotonicity property of preferences, we conclude that Vasco prefers the 
bundle which has strictly greater quantities of both goods. 
�"
�!� #
�!� � ����	� $ ����
� � �"
� � #
� � 
 
 
 
 
A 
 
2) (3 pts.) 
 
The principle of revealed preference states that when two consumption bundles are 
affordable at the same time and one is chosen while the other is not, then the former has 
been revealed preferred to the latter. Let’s use this insight to see what happens to a 
lender when the interest rate decreases. 
 
If he remained a lender, he will be worse off. 
His previous consumption would no longer be affordable and he would have to settle 
for a previously affordable worse bundle which remained affordable. 
 
Baseline situation Final situation ∆& ' 
 
Lender �(! ' )!� Remains a lender 
 
 
 
 
 
 
 
He’s worse off (proof by the 
principle of revealed preferences) 
 
If he became a borrower, we wouldn’t be able to tell whether he was better or worse off. 
There is no preference revelation because he would be choosing a consumption bundle 
which was previously not affordable. The previous optimal bundle would also no longer 
be affordable. There wouldn’t be a situation in which both bundles were affordable 
simultaneously. 
 
Baseline situation Final situation ∆& ' 
 
Lender �(! ' )!� Becomes a borrower 
 
 
 
 
 
 
 
No conclusion 
 
The answer depends on the agent’s choice of whether to remain a lender or become a 
borrower. If he remained a lender he would be worse off than in the baseline situation, 
but it may happen that becoming a borrower is better or even worse. 
 
 
 
 
C’ 
c2 
c1 
M 
C 
c2 
c1 
M 
C 
C’ 
 
A 
Group III 
 
1) (10 pts.) 
 
a. (2 pts.) 
 
Mathematical expression of the budget constraint: 
 
Endowment: ( )3,5 ==≡Ω BA ωω 
Prices: ( )0.5, 0.5A Bp p= = 
Value of the endowment: 435,055,0BAA =×+×=+≡ ωω Bppm 
Budget constraint equation: BABAABAA 5,05,04 xxxpxppp BB +=⇔+=+ ωω 
 
Graphical representation of the budget constraint: 
ABBA 85,05,04 xxxx −=⇔+= 
 
 
 
 
 
 
 
 
 
 
 
 
b. (2 pts.) 
 
Optimization problem: 
( )




+=
=
BBAA
BABA
xx
xpxpmts
xxxxU
BA
..
10,max
,
 
 
Solution using the Lagrangian method: 
[ ]10 A B A A B BL x x m p x p xλ= + − − 
Bx
 
8 
8 
3=Bω
 
Ω 
Ax
 
5=Bω
 
 
A 
10 10 100
10 0
100 10 0
0
0
2
5
2
B
B A
AA
B A AB
A
A B
B B
A A B B
A A B B
A B
B A B
B
A B
A
A A B A
B A
A
xdL
x x
pdx
x p p p
xdL
x p
dx p
m p x p x
dL m p x p x
d
mp x
x x pp
m
p p
p
m p x p x mp x
p
λ
λ
λ λ
λ
λ

==  = − =   
= ⇔ − = ⇔ = ⇔ − ⇔   
   
−− − =  = +
=  
 

= 
= 
 
⇔ − ⇔ = 
 
 = +
= 

 
 
Solution combining the optimum rule with the (budget) constraint: 
⇔







+=
=
⇔





−
−=−
⇔







−
−=
∂
∂
∂
−
⇔





+=
−=
A
B
A
BAA
A
B
A
B
B
A
A
B
B
A
E
A
BBAA
B
A
xx
x
p
p
pxpm
x
p
p
x
p
p
x
x
p
p
x
U
x
U
xpxpm
p
pTMS
BA 10
10
,
 







=
=
⇔
A
A
B
B
p
m
x
p
m
x
2
2
 
 
c. (2 pts.) 
 
( )
( )
40.5; 4 4
2 0.5
40.5; 4 4
2 0.5
B B
A A
x p m
x p m

= = = = ×


= = = =
 ×
 
 
d. (2 pts.) 
 
Calculating the total effect: 
New value of the endowment: A A B' 1.5 5 0.5 3 9Bm p pω ω≡ + = × + × = 
 
 
A 
( )
( )
90.5; 9 9
2 0.5
90.5; 9 3
2 1.5
B B
A A
x p m
x p m

= = = = ×


= = = =
 ×
 
 
Total effect: 143 −=−=∆ Ax 
 
Calculating the Hicks substitution effect: 
( ) 16044104,4 =××=== BA xxU 
How much income would the consumer need to maintain the utility level despite the 
price change? 
2' '10 160 ' 64 ' 64 1.5 0.5 ' 48
2 2
' 4 3 6.93
A B
A B
m m
m p p m m
p p
m
× × = ⇔ = × × ⇔ = × × ⇔ = ⇔
⇔ = ≈
 
 
Optimal level of consumption of Adele songs at the new prices with a compensated 
income: 
( ) 4 31.5; 4 3 2.312 1.5A Ax p m= = = ≈× 
Hicks substitution effect: 2.31 4 1.69sAx∆ ≈ − = − 
 
Calculating the ordinary income effect: 
Optimal level of consumption of Adele songs at the new prices with the new value of 
the endowment: 
( ) 41.5; 4 1.33
2 1.5A A
x p m= = = ≈
×
 
Ordinary income effect: 1.33 2.31 0.98nAx∆ ≈ − = − 
 
Calculating the endowment income effect: 
3 1.33 1.67Ax
ω∆ ≈ − = 
 
e. (2 pts.) 
 
In the previous question we calculated the income the consumer would need to maintain 
the initial utility level despite the price change. 
 
A 
93.634'
48'5,05,164'64'160
2
'
2
'10 2
≈=⇔
⇔=⇔××=⇔××=⇔=××
m
mmppm
p
m
p
m
BA
BA 
 
Compensating variation ≡ Amount of income we must take away from the consumer to 
leave him with the same utility level (in the same indifference curve), despite the price 
change () *)′). 
07.293.69349' =−≈−=− mm 
 
Notice we’re using income changes to compensate the impact of the change in price and 
keep the consumer with the same initial utility level. Without such compensation, the 
consumer would be better-off: 
( ) ( ) 16044104,427093109,3 =××===>=××=== BABA xxUxxU 
We have to take a positive amount of income to have him back with the original utility 
level. The positive sign of the compensating variation is what we should expect, 
because the consumer benefited (increased his utility) with the price change (because 
the affected good was very significant in the endowment) and we’re trying to measure 
the monetary value of that increase in utility. 
 
 
 
 
Ax
 
Bx
 
1 2 3 4 5 6 7 8 9 10 0 
0 
2 
4 
6 
8 
18 
12 
14 
16 
10 
20 
Ω 
The compensating variation measures in 
monetary units the improvement in the 
consumer’s well-being resulting from the 
increase of ,�from 0.5 to 1.5 represented as the 
difference between these two indifference 
curves (in this case, Virgilio benefited from the 
price increase because his endowment 
contained a significant amount of the good 
whose price changed). 
270=U
 
160=U