Microeconomic theory
Class 6
A. Elasticity of substitution in consumption
Elasticity of substitution in consumption, 𝑒21 , tells what is a % change of the product
quantity ratio when MRS changes by 1%.
𝑒21 =
𝑞
𝑑 𝑙𝑛 (𝑞2 )
1
𝑑𝑀𝑅𝑆12
The indifference curve curvature depends on the elasticity of substitution:
Slika 1 Elastičnost supstitucije u potrošnji
𝜷
Problem 1. The utility function is 𝒖 = 𝒒𝜶𝟏 𝒒𝟐 where α+β = 1. Find elasticity of
substitution.
′
′
𝑞2
𝑞2
(𝑙𝑛 (𝑞 ))
(𝑙𝑛 (𝑞 ))
1
1
𝑒21 =
⇒ 𝑒21 =
= 1
′
𝑞2 ′
𝑙𝑛𝑀𝑅𝑆12
𝑙𝑛 (𝑞 )
1
Note: Cobb-Douglas utility function always has e21 = 1.
Problem 2. Find elasticity of substitution for the following functions?
𝑞12 𝑞22
9
)
(𝑞
𝑢(𝑞1 , 𝑞2 = 12
a) 𝑢(𝑞1 , 𝑞2 ) =
b)
+ 𝑞22 )0.5
c) 𝑢(𝑞1 , 𝑞2 ) = 𝑞1 + 𝑞2 .
d) u(q1,q2) = min{ q1,q2}.
1
𝜌
𝜌
e) 𝑢(𝑞1 , 𝑞2 ) = (𝑎1 𝑞1 + 𝑎2 𝑞2 )𝜌 .
f) 𝑢 = 𝑞1 + ln 𝑞2
Solutions::
a) 𝑒21 = 1
b) 𝑒21 = 2
c) 𝑒21 = ∞
d) 𝑒21 = 0
e) 𝑒21 =
1
1−𝜌
𝑞
f) 𝑒21 = 1 − 𝑞2
1
B. Indirect utility function and Roy identity
Solving the utility function for the utility maximizing quantities 𝑞1∗ and 𝑞2∗ we get:
𝑢(𝑞1∗ , 𝑞2∗ ) = 𝑢[𝑞1𝑀 (𝑝1 , 𝑝2 , 𝐼), 𝑞1𝑀 (𝑝1 , 𝑝2 , 𝐼)] = 𝑣(𝑝1 , 𝑝2 , 𝐼) (3)
where 𝑣(𝑝1 , 𝑝2 , 𝐼) is indirect utility function which is a maximand of the utility
maximization problem subject to prices and income.
According to the envelope theorem a derivatitive of a maximand (here 𝑣(𝑝1 , 𝑝2 , 𝐼)) with
respect to a variable is equal to derivative of Lagragean function with respect to the same
variable. Hence:
𝜕𝑣
𝜕ℒ
= 𝜕𝑝 = −𝜆𝑞1
(4)
𝜕𝑝
1
𝜕𝑣
𝜕𝑝2
𝜕𝑣
1
𝜕ℒ
= 𝜕𝑝 = −𝜆𝑞2
𝜕ℒ
2
(5)
= 𝜕𝐼 = −𝜆
(6)
If we divide (4) with (6) and (5) with (6) we get:
𝜕𝐼
𝜕𝑣
𝜕𝑝1
𝜕𝑣
𝜕𝐼
𝜕𝑣
𝜕𝑝2
𝜕𝑣
𝜕𝐼
=−
𝜆𝑞1
=−
−𝜆
𝜆𝑞2
−𝜆
= 𝑞1
(7)
= 𝑞2
(8)
Since v is the maximum utility function then 𝑞1 and 𝑞2 are optimal values of the utility
function 𝑞1𝑊 (𝑝1 , 𝑝2 , 𝐼) and 𝑞1𝑊 (𝑝1 , 𝑝2 , 𝐼). Therefore:
𝜕𝑣
𝜕𝑝1
𝜕𝑣
𝜕𝐼
𝜕𝑣
𝜕𝑝2
𝜕𝑣
𝜕𝐼
= 𝑞1𝑊
(9)
= 𝑞2𝑊
(10)
where pri čemu se rezultati (9) and (10) is called Roy identitity. This is a method for
obtaining Walrasian (uncompensated) demand functions.
𝜶
Problem 3. The utility function is 𝒖(𝒒𝟏 , 𝒒𝟐 ) = 𝒒𝜶𝟏 𝒒𝟏−
, prices are p1 and p1, and the
𝟐
income is I. Check if the Roy identity for good 1 holds.
𝐼(1 − 𝛼)
𝐼𝛼
, 𝑞1 =
𝑝2
𝑝1
1−𝛼
𝛼
𝐼𝛼
𝐼(1 − 𝛼)
𝛼 𝛼 1 − 𝛼 1−𝛼
𝑢 = 𝑣(𝑝1 , 𝑝2 , 𝐼) = ( ) (
)
= 𝐼( ) (
)
𝑝1
𝑝2
𝑝1
𝑝2
𝑞2 =
𝜕𝑣
𝜕𝑝1
𝜕𝑣
𝜕𝐼
=
1−𝛼 1−𝛼
)
𝑝2
𝛼
1−𝛼
𝛼
1−𝛼
( ) (
)
𝑝1
𝑝2
𝛼𝛼+1 𝐼 𝑝1−𝛼−1 (
𝛼𝐼
vrijedi.
= 𝑝 = 𝑞1𝑊
1
C. Minimum expenditure function and Shephard lemma
Solving the expenditure function for the optimal values 𝑞1∗ and 𝑞2∗ which minimize the
expendituresfor obtaining the fixed level of utility one gets:
𝐸(𝑞1 , 𝑞2 ) = 𝐸[𝑞1𝐻 (𝑝1 , 𝑝2 , 𝑢), 𝑞2𝐻 (𝑝1 , 𝑝2 , 𝑢)] = 𝑒(𝑝1 , 𝑝2 , 𝑢)
(13)
𝑒(𝑝1 , 𝑝2 , 𝑢) is called minimum expenditure function which is a minimand of the
expenditure minimization problem subject to the prices and level of utility.
According to the envelope theorem a minimand derivative (here 𝑒(𝑝1 , 𝑝2 , 𝑢)) with respect
to a variable is equal to the derivative of Lagrangean with respect to the same variable,
Hence:
𝜕𝑒
𝜕ℒ
= 𝜕𝑝 = 𝑞1
(14)
𝜕𝑝
1
𝜕𝑒
𝜕𝑝2
1
𝜕ℒ
(15)
= 𝜕𝑝 = 𝑞2
2
Since 𝑒(𝑝1 , 𝑝2 , 𝑢) is a minimum expenditure function then 𝑞1 and 𝑞2 are equal to the
Hicksian demand functions 𝑞1𝐻 (𝑝1 , 𝑝2 , 𝑢) and 𝑞1𝐻 (𝑝1 , 𝑝2 , 𝑢) which assume expenditure
minimization. Hence one obtains:
𝜕𝑒
= 𝑞1𝐻
(16)
𝜕𝑝
1
𝜕𝑒
𝜕𝑝2
(17)
= 𝑞2𝐻
Results (16) and (17) are called Shephard lemma.
𝜶
Problem 4. The utility function is 𝒖(𝒒𝟏 , 𝒒𝟐 ) = 𝒒𝜶𝟏 𝒒𝟏−
, prices are p1 and p1, and
𝟐
̅
desired level of utility is 𝒖. Check if the Roy identity for good 1 holds.
The optimum basket:
𝑝1 (1−𝛼) 𝛼
𝑞2 = 𝑢̅ (
𝑝2 𝛼
) i 𝑞1 = 𝑢̅ (𝑝
𝑝2 𝛼
1
1−𝛼
)
(1−𝛼)
̅ , p1 and p2 are no longer considered as constants but as variables instead then q1 and q2
If 𝒖
become Hicksian demand functions:
1−𝛼
𝑝1 (1 − 𝛼) 𝛼 𝐻
𝑝2 𝛼
𝑞2𝐻 (𝑝1 , 𝑝2 , 𝐼) = 𝑢̅ (
) , 𝑞1 (𝑝1 , 𝑝2 , 𝐼) = 𝑢̅ (
)
𝑝2 𝛼
𝑝1 (1 − 𝛼)
A minimum expenditure function is:
1−𝛼
𝑝2 𝛼
𝑝1 (1 − 𝛼) 𝛼
𝐸(𝑞1 , 𝑞2 ) = 𝑒(𝑝1 , 𝑝2 , 𝑢) = 𝑝1 𝑢̅ (
)
+ 𝑝2 𝑢̅ (
)
𝑝1 (1 − 𝛼)
𝑝2 𝛼
𝑝1 𝛼 𝑝2 1−𝛼
𝑒(𝑝1 , 𝑝2 , 𝑢) = 𝑢̅ ( ) (
)
𝛼
1−𝛼
Shephard lemma:
𝜕𝑒
𝜕𝑝1
1−𝛼
𝑝
2
= 𝛼1−𝛼 𝑢̅𝑝1𝛼−1 (1−𝛼
)
𝑝2 𝛼
= 𝑢̅ (𝑝
1
Shephard lemma holds.
1−𝛼
)
(1−𝛼)
= 𝑞1𝐻
D. Hicks and Walras demand equations relation and Slutsky equation
If one substitutes 𝑢̅ in the expenditure minimization problem with indirect utility function
𝑣(𝑝1 , 𝑝2 , 𝐼) then:
𝑢̅ = 𝑣(𝑝1 , 𝑝2 , 𝐼)
(18)
In that case:
𝑞1𝐻 = 𝑞1𝑊 and 𝑞2𝐻 = 𝑞2𝑊
(19)
Also, if one substitutes I in the utility maximization problem with a minimum expenditure
function 𝑒(𝑝1 , 𝑝2 , 𝑢) then:
𝐼 = 𝑒(𝑝1 , 𝑝2 , 𝑢)
(20)
In that case:
𝑞1𝐻 (𝑝1 , 𝑝2 , 𝑢) = 𝑞1𝑀 [𝑝1 , 𝑝2 , 𝑒(𝑝1 . 𝑝2 , 𝑢)]
(21)
If it is differentiated with respect to p1 one gets:
𝜕𝑞1𝐻
𝜕𝑝1
𝜕𝑞1𝑀
=
𝜕𝑝1
+
𝜕𝑞1𝑀
𝜕𝑒
Shephard lemma states that
that
𝑞1𝐻
=
𝑞1𝑀 .
𝜕𝑒
(22)
∙ 𝜕𝑝
1
𝜕𝑒
𝜕𝑝1
= 𝑞1𝐻 , and (20) says that:
𝜕𝑞1𝑀
𝜕𝑒
=
𝜕𝑞1𝑀
𝜕𝐼
. From (19) we know
By putting it in (22) one gets:
𝜕𝑞1𝐻
𝜕𝑝1
𝜕𝑞1𝑊
=
𝜕𝑝1
+
𝜕𝑞1𝑊
𝜕𝐼
∙ 𝑞1𝑊
(23)
∙ 𝑞1𝑊
(24)
Rearranging (23) one gets:
𝜕𝑞1𝑊
𝜕𝑝1
=
𝜕𝑞1𝐻
𝜕𝑝1
−
𝜕𝑞1𝑊
𝜕𝐼
Result (24) is called Slutsky equation.
Slutsky equation states that the total effect of a price p1 price change on Walrasian demand
is equal to the derivative of Walrasian demand which is equal to the sum of income effect
and substitution effect::
𝑆𝐸 =
𝜕𝑞1𝐻
𝜕𝑝1
, 𝐼𝐸 = −
𝜕𝑞1𝑊
𝜕𝐼
∙ 𝑞1𝑊
(25)
̅ is equal to the indirect utility
Problem 5. Deduct Hicksian demands for problem 4 if 𝒖
function.
min 𝐸(𝑞1 , 𝑞2 ) = 𝑝1 𝑞1 + 𝑝2 𝑞2
s.t.
𝛼 𝛼 1 − 𝛼 1−𝛼
𝐼( ) (
)
= 𝑞1𝛼 𝑞21−𝛼
𝑝1
𝑝2
Solution:
𝛼 𝛼 1 − 𝛼 1−𝛼
𝑝2 𝑞2 𝛼 𝛼
𝐼( ) (
)
=(
) (𝑞2 )1−𝛼
𝑝1
𝑝2
𝑝1 (1 − 𝛼)
𝐼(1 − 𝛼)
𝑞2𝐻 =
𝑝2
which is equal to 𝑞2𝑊 (The same result is obtained when in Problem 3 instead of I one puts a
minimum expenditure function e.
Problem 6. Extract income and substitution effect in Problem 5.
𝑞1𝐻 = 𝑞1𝑊 :
𝑞1𝐻 (𝑝1 , 𝑝2 , 𝑢) = 𝑞1𝑊 (𝑝1 , 𝑝2 , 𝐼) when 𝐼 = 𝑒(𝑝1 , 𝑝2 , 𝑢) hence:
𝑞1𝐻 (𝑝1 , 𝑝2 , 𝑢) = 𝑞1𝑊 (𝑝1 , 𝑝2 , 𝑒(𝑝1 , 𝑝2 , 𝑢))
Derivative with respect to p1 is:
𝜕𝑞1𝐻 𝜕𝑞1𝑊 𝜕𝑞1𝑊 𝜕𝑒
=
+
∙
𝜕𝑝1
𝜕𝑝1
𝜕𝑒 𝜕𝑝1
𝜕𝑒
Apply Shephard lemma: 𝜕𝑝 = 𝑞1𝐻 as well as the fact that 𝑞1𝐻 = 𝑞1𝑊 . Since 𝐼 = 𝑒(𝑝1, 𝑝2 , 𝑢)
then
𝜕𝑞1𝑊
𝜕𝑒
=
𝜕𝑞1𝑊
𝜕𝐼
1
. We get:
𝜕𝑞1𝐻 𝜕𝑞1𝑊 𝜕𝑞1𝑊 𝑊
=
+
∙ 𝑞1
𝜕𝑝1
𝜕𝑝1
𝜕𝐼
By rearranging we get:
Income effect is: IE= −
𝜕𝑞1𝑊
𝜕𝐼
𝜕𝑞1𝑊 𝜕𝑞1𝐻 𝜕𝑞1𝑊 𝑊
=
−
∙ 𝑞1
𝜕𝑝1
𝜕𝑝1
𝜕𝐼
∙ 𝑞1𝑊 , And substitution effect is SE =
𝜕𝑞1𝐻
𝜕𝑝1
2
:
𝜕𝑞1𝑊 𝑊
𝛼 𝐼𝛼
𝐼𝛼
∙ 𝑞1 = − ∙
=− 2
𝜕𝐼
𝑝1 𝑝1
𝑝1
𝐻
𝜕𝑞1
𝑆𝐸 =
= (1 − 𝛼)𝑢̅𝑝1𝛼−2 (𝑝2 𝛼)1−𝛼
𝜕𝑝1
𝐼𝐸 = −
𝛼 𝛼
Utilitiy level 𝑢̅ is equal to = 𝐼 (𝑝 ) (
1
1−𝛼 1−𝛼
𝑝2
𝛼 𝛼
)
:
(1 − 𝛼)𝛼𝐼
𝜕𝑞1𝐻
1 − 𝛼 1−𝛼 𝛼−2
(1
𝑆𝐸 =
= − 𝛼)𝐼 ( ) (
)
𝑝1 (𝑝2 𝛼)1−𝛼 = −
𝜕𝑝1
𝑝1
𝑝2
𝑝12
Total effect is:
(1 − 𝛼)𝛼𝐼 𝐼𝛼 2
𝛼𝐼
𝑇𝐸 = 𝑆𝐸 + 𝐼𝐸 = −
− 2 =− 2
2
𝑝1
𝑝1
𝑝1
Shares of IE and SE in TE are:
(1 − 𝛼)𝛼𝐼
𝐼𝛼 2
−
𝐸𝑆
𝐸𝐷
𝑝12
𝑝12
=−
= 1 − 𝛼,
=−
=𝛼
𝛼𝐼
𝛼𝐼
𝑈𝐸
𝑈𝐸
− 2
− 2
𝑝1
𝑝1
Hence if α = 0.5 then IE and SE are the same.