Class 09 (answers) CS, CV and EV

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Microeconomics II 
Undergraduate degree in Economics 
 
 
Class nr. 09 
 
Subject: 1. Consumer Theory 
1.3. Consumer surplus, compensating variation and equivalent variation 
(practice class) 
 
 
 
Exercise 14.3. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, pp. 179-180 
 
Quasimodo consumes earplugs and other things. His utility function for earplugs x and 
money to spend on other goods y is given by: 
( ) yxxyxu +−=
2
100,
2
 
 
a) What kind of utility function does Quasimodo have? 
b) What is his inverse demand curve for earplugs? 
c) If the price of earplugs is $50, how many earplugs will he consume? 
d) If the price of earplugs is $80, how many earplugs will he consume? 
e) Suppose that Quasimodo has $4,000 in total to spend a month. What is his total 
utility for earplugs and money to spend on other things if the price of earplugs is 
$50? 
f) What is his total utility for earplugs and other things if the price of earplugs is 
$80? 
g) What is the change in utility when the price changes from $50 to $80. 
h) What is the change in (net) consumer surplus when the price changes from $50 
to $80? 
 
Answers: 
 
a) Quasilinear utility. 
b) When we have a quasilinear utility function, the inverse demand function can be 
found directly by deriving the utility function in order to the interest variable: 
( ) yxxyxu +−=
2
100,
2
 
Microeconomics II 
Undergraduate degree in Economics 
 
 
( ) ( ) xxpxyxx
xx
yxup −=⇔−=





+−
∂
∂
=
∂
∂
= 100100
2
100,
2
 
 
or 
 
Optimization problem: 
( )
( )





−=
=
=−
⇔








+=
=
=
−
⇔





=−−
=−
=−−
⇔









=
=
=
−−++−=
=−⇔=





−+−⇔=
−+−⇔





=+
+−
pxmy
px
ypxm
p
x
ypxm
px
d
dL
dy
dL
dx
dL
ypxmyxxL
or
pxpxmxx
dx
d
dx
yxdu
pxmxx
mypxts
yxx
x
yx
1
100
1
100
0
01
0100
0
0
0
2
100
1000
2
1000,
2
100max
..
2
100max
2
2
2
2
,
λλ
λ
λ
λ
λ
λ
 
 
c) 50 = 100 – x fl x = 50 
 
d) 80 = 100 – x fl x = 20 
 
e) 
px = 50 fl x = 50 
y = 4000 - px x = 4000 – 50*50 = 1500 
( ) 52501500
2
50501001500,50
2
=+−×=u 
 
f) 
px = 80 fl x = 20 
y = 4000 - px x = 4000 – 80*20 = 2400 
Microeconomics II 
Undergraduate degree in Economics 
 
 
( ) 42002400
2
20201002400,20
2
=+−×=u 
 
g) u(20,2400) – u(50,1500) = 4200 – 5250 = -1050 
 
h) In the presence of quasilinear utility function the consumer surplus variation, the 
equivalent variation and the compensating variation are all equal. Also, the consumer 
surplus variation measures the exact change in utility that occurs after a price change, 
thereby: 
DCS = -1050 
 
Let us check this theoretical result by making use of the consumer surplus formulas for 
a rise in prices. (Note that the formulas that are used for a rise in prices are suitably 
different from the ones that would be adequate if we were considering a fall in prices!) 
 
 
 
 
 
 
 
 
 
 
 
 
Using formulas for the areas of geometric figures: 
( ) ( ) ( ) [ ] 1050450600
2
50802050508020 −=+−=


 −×−
+−×−=∆CS 
 
 
Using generic formulas with integrals: 
( )[ ] ( )[ ] ( )[ ] ( )[ ]
( ) ( )
10501250200
2
0050
2
505050
2
0020
2
202020
2
50
2
205020
5010080100ˆ*
2222
50
0
220
0
250
0
20
0
50
0
20
0
ˆ
0
*
0
−=−=
=











−×−





−×−











−×−





−×=
=





−−





−=−−−=
=−−−−−=−−−=∆
∫∫
∫∫∫∫
x
x
x
xdxxdxx
dxxdxxdxPxPdxPxPCS
xx
 
 
or 
P 
x 
MB(x) = D 
p* = 80 
∆CS1 
 
p^ = 50 
∆CS2 
x* = 20 x^ = 50 
100 
100 
Microeconomics II 
Undergraduate degree in Economics 
 
 
[ ] ( )[ ] [ ] ( )[ ]
( ) [ ]
( ) ( )[ ] ( ) 10508001250600
2
202050
2
5050500302030
2
50305030
501005080ˆˆ*
22
50
20
2
20
0
50
20
20
0
50
20
20
0
ˆ
*
*
0
−=−−−=











−×−





−×−×−×−=
=





−−−=−−−=
=−−−−−=





−+−−=∆
∫∫
∫∫∫∫
x
xxdxxdx
dxxdxdxPxPdxPPCS
x
x
x
 
or 
 
( ) ( )
( )[ ] ( )
105048003750
2
8080100
2
5050100
2
100100
100100
2250
80
50
80
2ˆ
*
−=−=
=











−×−





−×=





−=−==∆
−=⇔−=
∫∫
PPdPPdPPxCS
ppxxxP
P
P
 
 
Exercise 14.5. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, pp. 181-182 
Bernice’s preferences can be represented by ( ) { }yxyxu ,min, = , where x is pairs of 
earrings and y is dollars to spend on other things. She faces prices ( ) ( )1,2, =yx pp and 
her income is 12. 
 
 
a) Determine her optimal consumption bundle. Sketch, in a graph, her optimal 
consumption bundle, her budget constraint and the indifference curve that goes 
through her optimal consumption bundle. 
b) The price of a pair of earrings rises to $3 and Bernice’s income stays the same. 
Determine her new optimal consumption bundle. Sketch, in a graph, her new 
optimal consumption bundle together with her budget constraint and the 
indifference curve that passes through her new optimal consumption bundle. 
c) What bundle would Bernice choose if she faced the original prices and had just 
enough income to reach the new indifference curve? Sketch (in a graph) the 
budget constraint that passes through this bundle at the original prices. How 
much income would Bernice need at the original prices to have the new budget 
constraint? 
 
d) What is the maximum amount that Bernice would pay to avoid the price 
increase? How would you call this amount? 
 
e) What bundle would Bernice choose if she faced the new prices and had just 
enough income to reach her original indifference curve? Sketch, in a graph, the 
budget constraint that passes through this bundle at the new prices. How much 
income would Bernice have with this new budget constraint? 
 
Microeconomics II 
Undergraduate degree in Economics 
 
 
f) By how much would Bernice’s original income have to increase in order to be as 
well-off as she was with her original bundle, even after the price change? How 
would you call this variation in income? 
 
Answers: 
 
a) 
From the utility function one can conclude that the goods in question are perfect 
complements, in this case in proportion of 1 to 1. Therefore they will be bought and 
consumed together, in pairs, which means that for each pair of earrings we will have in 
the optimal consumption bundle $1 to spend on other things. 
The price of each complementary set of goods (earrings and money to spend on other 
things) is: px + py = 2 + 1 = 3. 
The maximum amount of these sets of goods that she can buy with an income of 12 is 4. 
Budget constraint: xyyxypxpm yx 212212 −=⇔+=⇔+= 
44212 =×−=y 
 
 
 
 
 
 
 
 
 
 
 
 
 
b) 
From the utility function one can conclude that the goods in question are perfect 
complements, in this case in proportion of 1 to 1. Therefore they will be bought and 
consumed together, in pairs, which means that for each pair of earrings we will have in 
the optimal consumption bundle $1 to spend on otherthings. 
The price of each complementary set of goods (earrings and money to spend on other 
things) is: px + py = 3 + 1 = 4. 
The maximum amount of these sets of goods that she can buy with an income of 12 is 3. 
 2 
 4 
 6 
 8 
4 3 2 1 
x 
y 
10 
12 
5 6 
X 
Indifference Curves 
U = 2 
U = 4 
U = 6 
Microeconomics II 
Undergraduate degree in Economics 
 
 
Budget constraint: xyyxypxpm yx 312312 −=⇔+=⇔+= 
33312 =×−=y 
 
 
 
 
 
 
 
 
 
 
 
 
c) 
Once again, since the goods in question are perfect complements in proportion of 1 to 1 
(in this case), in order to achieve the new indifference curve Bernice would have to 
consume at the point (3,3). At the original prices this consumption bundle would cost: 
91323 =×+× . 
 
 
 
 
 
 
 
 
 
 
 
 
d) 
Bernice is willing to pay 12 – 9 = 3 monetary units in order to avoid the price increase. 
The symmetric result of this income difference is called equivalent variation 
( 3129'' −=−=−= mmEV ), because is equivalent to the price variation in what 
concerns the impact on Bernice’s utility. 
X1 
X2 
 2 
 4 
 6 
 8 
4 3 2 1 
10 
12 
5 6 
U = 4 
U = 3 
 2 
 4 
 6 
 8 
4 3 2 1 x 
y 
10 
12 
5 6 
X1 
Indifference Curves 
X2 
U = 4 
U = 3 
x 
y 
Microeconomics II 
Undergraduate degree in Economics 
 
 
 
Note: We follow Mas-Colell et al. (1995)1 and Varian (1992)2 instead of the course’s 
textbook by Varian in what concerns the concepts of compensating and equivalent 
variation. These advanced books are not mandatory (or recommended at this 
intermediate level) reading. Refer to the class slides regarding these concepts; you only 
need to check the book if you have serious difficulties in understanding the course 
materials. 
 
e) 
Once again, since the goods in question are perfect complements in proportion of 1 to 1 
(in this case), in order to achieve the original indifference curve Bernice would have to 
consume on the point (4,4). At the new prices this consumption bundle would cost: 
161434 =×+× . 
 
 
 
 
 
 
 
 
 
 
 
 
 
f) 
In order to be as well-off as she was with her original consumption bundle even after the 
price change, Bernice’s income would have to rise 16 – 12 = 4 monetary units. Only 
this rise in her income compensates the loss in her purchasing power. 
The symmetric result of this income difference is called compensating variation 
( 41612' −=−=−= mmCV ), because it compensates the loss in her purchasing power 
caused by the increase in the price of earrings in what concerns the impact on Bernice’s 
utility. 
 
 
1
 Mas, Colell, Andreu, Michael D. Whinston and Jerry R. Green (1995), Microeconomic Theory. Oxford 
University Press. 
2
 Varian, Hal R: (1992), Microeconomic Analysis. W.W. Norton & Co. 
X1 
X2 
 2 
 4 
 6 
 8 
4 3 2 1 
10 
12 
5 6 
U = 4 
U = 3 
x 
y 
Microeconomics II 
Undergraduate degree in Economics 
 
 
Note: We follow Mas-Colell et al. (1995)3 and Varian (1992)4 instead of the course’s 
textbook by Varian in what concerns the concepts of compensating and equivalent 
variation. These advanced books are not mandatory (or recommended at this 
intermediate level) reading. Refer to the class slides regarding these concepts; you only 
need to check the book if you have serious difficulties in understanding the course 
materials. 
 
 
 
3
 Mas, Colell, Andreu, Michael D. Whinston and Jerry R. Green (1995), Microeconomic Theory. Oxford 
University Press. 
4
 Varian, Hal R: (1992), Microeconomic Analysis. W.W. Norton & Co.