Teste Intercalar 2013 2014 Resolução

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A 
UNDERGRADUATE DEGREE IN ECONOMICS 
MICROECONOMICS II 
 
MID-TERM 
 
29 October 2013 
Time allowed for the exam: 1h20m 
 
Teachers: Henrique Monteiro; Telma Gonçalves 
 
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 convexity of preferences implies there is a single optimal consumption 
bundle for each income level. 
 
 2. The inverse demand function gives us the consumer’s maximum willingness to 
accept compensation for each unit of the good that is consumed. 
 
 3. For Giffen goods, quantity demanded decreases when price falls. 
 4. Gross demand is obtained by subtracting the endowment from the net demand. 
 
 
 
Complete the following sentences: (wrong answers have no point deduction penalty) 
 
5. Discounting enables us to calculate ______________________ of a series of cash-
flows earned in different moments in time. 
 
6. We call the preferences ___________ when any two bundles can be compared. 
 
 
 
 
Multiple-choice questions: 
 
7. To calculate the Hicks substitution effect, we must hold fixed the following variable: 
a) Income □ 
b) Purchasing power □ 
c) Utility □ 
d) Quantity consumed □ 
 
 
 
 
A 
8. How do you call the price for which the consumer is indifferent between consuming 
or not consuming a good: 
a) Equilibrium price □ 
b) Market price □ 
c) Reservation price □ 
d) Indifference price □ 
 
 
9. A net buyer, when faced with a price increase: 
a) becomes a net seller and is better-off □ 
b) remains a net buyer and is better-off □ 
c) becomes a seller and is worse-off □ 
d) we cannot conclude what he will do □ 
 
 
10. With a 10% interest rate, what is the value today of €1000 received 3 years from 
now? 
a) €578 □ 
b) €751 □ 
c) €1300 □ 
d) €1331 □ 
 
 
Group II – 5 pts. (20m) 
 
 
1. (3 pts.) Consider the following table of observations for the prices (p) of goods 1 
and 2 and for the quantities demanded (x) by a consumer. 
Observation/Consumption bundle p1 p2 x1 x2 
A 4 4 6 6 
B 5 3 3 9 
C 3 5 7 1 
Sketch in a graph the set of bundles worse than A. 
 
2. (2 pts.) Explain the concept of compensating variation. Give an example of an 
application of this welfare measure. 
 
 
Group III – 10 pts. (40m) 
 
 
1. (6 pts.) Zacarias is a civil servant with a net income of €1500 in period 1. In 
period 2 he expects a wage cut that will reduce his income to €1100. Assume 
that the price of consumption goods is €1, there is no inflation and Zacarias’ 
 
 
A 
utility function is ( ) 2121, CCCCU ×= , where 1C is Zacarias’ period 1 
consumption and 2C is his consumption in period 2. 
 
a) (3 pts.) Assume there is no financial system that enables him to borrow Money 
or get his savings remunerated. Sketch in a graph Zacarias’ budget set and find 
his demand functions for consumption in each period. 
 
b) (2 pts.) Calculate the Slutsky substitution effect for an increase in the price of 
consumption in period 1 from 1 €1P = to 1 €2P = . 
 
c) (1 pts.) Assume again that 1 €1P = , but that now it is possible to borrow Money 
and get savings remunerated at a 10% interest rate. Write down the equation for 
the new budget constraint and sketch it in a graph. What happened to Zacarias’ 
consumption possibilities? 
 
 
2. (4 pts.) Martinho is a boy who likes to eat seasonal fruits. His consumption was 
observed in the months of August and November and is presented in the following 
table together with the unit prices of the fruits he consumed. 
Month Melon (M) (kg) Chestnuts (C) (kg) PM (€/kg) PC (€/kg) 
August (A) 3 1 0.5 5 
November (N) 1 12 3 1 
 
a) (2 pts.) Explain what is an index number. Calculate the Paasche quantity index 
for the period between August and November. 
 
b) (2 pts.) How has Martinho’s satisfaction changed? Justify your answer. 
 
 
 
A 
Solution 
 
Group I 
 
True-False Questions 
1) T 
2) F 
3) T 
4) F 
Complete the sentences 
5) Present value 
6) Complete 
Multiple-choice questions 
7) c 
8) c 
9) d 
10) b 
 
Group II 
 
1) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2) (2 pts.) 
 
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. 
Possible example: calculating the impact on welfare of an increase in user charges in 
public hospital or in the price of the social pass for public transportation, etc. 
 
B 
A 
C 2 
4 
6 
8 
4 3 2 1 x1 
x2 
5 
10 
12 
6 10 9 8 7 11 12 
14 Set of bundles worse 
than A 
 
 
A 
Group III 
 
1) (6 pts.) 
 
a) (3 pts.) 
 
Budget constraint 
2 2 2
1 1 1 1 2
2 1
11001500 2600
1 1 1 0 1 0
2600
m C Cm C C C C
r r
C C
+ = + ⇔ + = + ⇔ = + ⇔+ + + +
⇔ = −
 
Graphical representation: 
 
 
 
 
 
 
 
 
 
 
 
 
 
( )
1 2
1 2 1 2,
1 2 1 1 2 2
max ,
. .
C C
U C C C C
s t m m p C p C
⎧ = ×⎪⎨ + = +⎪⎩
 
Solution using the Lagrange method: 
[ ]1 2 1 2 1 1 2 2L C C m m p C p Cλ= × + + − − 
2
11
2 1 1 1 2 2
1
1 2
2 2
1 2 2 2 2 21 2 1 1 2 2
0
0
0 0
00
0
CdL
pdC C p p C p C
CdL C p
dC p
m m p C p Cm m p C p C
dL
d
λ
λ
λ λ
λ
⎧⎧ == ⎪⎪ ⎪⎪ − = =⎧ ⎧⎪⎪ ⎪ ⎪= ⇔ − = ⇔ = ⇔ −⎨ ⎨ ⎨ ⎨⎪ ⎪ ⎪ ⎪ + − − =+ − − =⎩ ⎩⎪ ⎪−=⎪ ⎪⎩ ⎩
 
1 2
2 1 2 1
1 1
1 2
1 2
1 2
1 22
22
2
22
2
2 2
m mp m m CC pp p
m m
m m m mC Cp p
λ
+⎧+ =⎧ ⎪=⎪ ⎪⎪ +⎪⎪⇔ − ⇔ =⎨ ⎨⎪ ⎪+ +⎪ ⎪= =⎪ ⎪⎩ ⎩
 
or 
Solution combining the optimum rule with the budget constraint. 
C1 
C2 
2600
2600
1100
1500
Ω
 
 
A 
1 2
1 2
21 12 11 1
, 2 1 2
2 21 22
1 2
2 11 2 1 1 2 2 1 2 1 1
1
2
2
2
C C
U m mCp pC pC pTMS C C pUp pC pp
m mC Cm m p C p C m m p C
p
∂⎧ +⎧⎪ =⎧ ⎧∂ ⎧ ⎪= − =− = −⎪− = −⎪ ⎪ ⎪ ⎪∂⇔ ⇔ ⇔ ⇔⎨ ⎨ ⎨ ⎨ ⎨ +⎪ ⎪ ⎪ ⎪ ⎪∂ =+ = + + =−⎩⎩ ⎩⎪ ⎪⎩−⎩
 
Or 
Solution substituting the budget constraint into the objective function: 
( ) ( ) ( )
( )
1 2
1 2
2
2
1 2 1 2,1 2 1 2 1 2 2 2,
2 21 2 2 2
111 2 1 1 2 2
1
21 2 2
2 2 2
1 1
max ,
max ,
max
. .. .
max
C C
C C
C
C
U C C C C
U C C C C m m p CU C Cm m p C ps t Cs t m m p C p C
p
m m pU C C C
p p
⎧ = ×⎧ = × ⎪ ⎛ ⎞+ −⎪ ⇔ ⇒ = × ⇔⎨ ⎨ ⎜ ⎟+ −= ⎝ ⎠+ = +⎪ ⎪⎩ ⎩
+⇔ = × −
 
( )2 21 2 2 1 2 2 1 2
2 2 2 2
2 2 1 1 1 1 2
1 2
1 2 2
2 1 2
1
1 1
0 0 2 0
2
2
2
U C m m p m m p m mC C C C
C C p p p p p
m mm m p
p m mC
p p
∂ ⎛ ⎞+ + +∂= ⇔ × − = ⇔ − = ⇔ =⎜ ⎟∂ ∂ ⎝ ⎠
++ − += =
 
b) (2 pts.) 
 
Original optimal consumption bundle: 
( ) ( )1 2 1 21 2
1 2
1500 1100 1500 1100, , , 1300,1300
2 2 2 1 2 1
m m m mC C
p p
⎛ ⎞+ + + +⎛ ⎞= = =⎜ ⎟ ⎜ ⎟× ×⎝ ⎠⎝ ⎠
 
With the change to 1 2P = , how much income would be necessary to keep the 
purchasing power constant? 
( )1 1 2 1 1300 1300m p x m mΔ = Δ ⇔ Δ = − × ⇔ Δ = 
Optimal consumption bundle with the price change and the income compensation: 
( ) ( )1 2 1 21 2
1 2
1500 1100 1300 1500 1100 1300, , , 975,1950
2 ' 2 2 2 2 1
m m m m m mC C
p p
⎛ ⎞+ + Δ + + Δ + + + +⎛ ⎞= = =⎜ ⎟ ⎜ ⎟× ×⎝ ⎠⎝ ⎠
Slutsky substitution effect: 1 1 1' 975 1300
sx x xΔ = − = − 
 
 
c) (1 pt.) 
 
Budget constraint 
2 2 2
1 1 1 1 2
2 1
11001500 2750 1,1
1 1 1 0,1 1 0,1
2750 1,1
m C Cm CC C C
r r
C C
+ = + ⇔ + = + ⇔ = + ⇔+ + + +
⇔ = −
 
Graphical representation: 
 
 
A 
 
 
 
 
 
 
 
 
 
 
 
 
Zacarias’ consumption possibilities have expanded. 
 
 
 
 
 
 
 
 
 
 
 
 
 
2) (4 pts.) 
 
a) (2 pts.) 
 
An index number summarizes the evolution of a set of phenomenons/variables (These k 
phenomenons/variables must have the same unit [ex.: monetary units; physical units]). 
Ex.: the evolution of the quantities consumed of a good; the evolution of a set of prices. 
3 1 1 12 1,5
3 3 1 1
N N N N
A N M M C C
q N A N A
M M C C
p x p xP
p x p x
+ × + ×= = =+ × + × 
 
b) (2 pts.) 
 
Martinho’s satisfaction increased. 
1A NqP > 
1 1
N N N N
A N N N N N N A N A N N A N AM M C C
q M M C C M M C C M M C CN A N A
M M C C
p x p xP p x p x p x p x m p x p x
p x p x
+> ⇔ > ⇔ + > + ⇔ > ++ 
This means that the consumption bundle chosen in the base year was affordable in year 
t, at those year’s prices, but was not chosen, therefore ( ) ( ), ,N N A AM C M Cx x x x; , which 
means that the consumer is better off. 
 
C1 
C2 
2750
2500 
1100
1500 
Ω
C1 
C2 
2750
2500 
1100
1500 
Ω
2600 
2600