Demand function, Engel curve and Slutsky
equation
27 de octubre de 2011
4.1
1. Optimal conditions:
(
2x1 = x2
p1 x1 + p2 x2 = m
p1 x1 + p2 2x1 = m
x1 (p1 + 2p2 ) = m
x1 (m, p1 , p2 ) =
m
p1 + 2p2
x2 (m, p1 , p2 ) =
2m
p1 + 2p2
2. Engel curve for good 2:
1
m = x2 · p2 + p1
2
|
{z
}
slope
1
4.2
1.
|M RS| =
p1
p2
p1
αxα−1
x1−α
1
2
−α =
p
(1 − α)xα
x
2
1 2
Simplify:
p1
αx2
=
(1 − α)x1
p2
(1 − α)x1 p1 = αx2 p2
x1 =
αx2 p2
(1 − α)p1
2
Budget constraint:
p1 α x2 p2
+ p2 x2 = m
p1 (1 − α)
α
p 2 x2 1 +
=m
(1 − α)
m
x2 (m, p2 ) =
p2 1 +
α
(1−α)
=
m
(1 − α)
p2 [(1 − α) + α]
Then:
α
x1 (m, p1 ) =
(1−α)
m
p2 [(1−α)+α]
(1 − α)p1
p2
=
αm(1−α)
[(1−α)+α]
(1 − α)p1
=
m
αm(1 − α)
α
=
p1 [(1 − α) + α] (1 − α)
p1 (1 − α) + α
2.
m(x2 , p2 ) = x2 p2
|
3
[(1 − α) + α]
(1 − α)
{z
}
slope
3. The parameter
α
indicates what percentage of income the consumer will
be spent in good 1; therefore
(1 − α)
tell us what percentage of income devoted
to good 2.
4.3
1. The general shape of a utility fonction representing these preferences is:
u(x1 , x2 ) = ax1 + bx2 ; a, b ∈ (−∞, +∞)
Examples:
u(x1 , x2 ) = 5x1 + 3x2 ,
MRS is
u(x1 , x2 ) = 200x1 + 100x2 ,
u(x1 , x2 ) = 10x1 + 10x2 ,
u(x1 , x2 ) = x1 − 2x2 ,
− 35
MRS is
MRS is
MRS is
−2
−1
(perfect substitutes)
1
2 (good 1 is a good while good 2 is
a bad)
u(x1 , x2 ) = −6x1 − 2x2 ,
MRS is
3
4
(both goods are bads)
2. Engel curves for both goods when
p1 < p2 :
3.
5
The law of demand says that prices and quantities move in opposite directions. So if the price increases, demand decreases and vice versa.
Both goods aren´t Gien, the law of demand is satised.
4.4
1. An indierence map homothetic can be represented by perfect substitutes,
perfect complementary, Cobb-Douglas preferences...
6
2.
Optimal conditions:
(
x1 = x2
p1 x1 + p2 x2 = m
x2 (p1 + p2 ) = m
m(x2 , p1 , p2 ) = x2 (p1 + p2 )
m(x1 , p1 , p2 ) = x1 (p1 + p2 )
Engel curves are straight lines with slopes
(p1 + p2 ).
3.
In the case of good 1 if
If
p2
x1 (m, p1 , p2 ) =
m
p1 + p2
x2 (m, p1 , p2 ) =
m
p1 + p2
p1
down, its demand increase: so good 1 is not Gien.
low, demand of good 2 also increase: good 2 is not Gien. Remember that
to see if a good is Gien is necessary to analyse the variation of the good about
their own price.
4.5
1. Optimal conditions:
|M RS| =
p1
p2
x2
p1
=
x1
p2
p1 x1 = p2 x2
7
Budget constraint:
2p1 x1 = m
If
m = 10 , p1 = 1
y
x1 (m, p1 ) =
m
2p1
x2 (m, p2 ) =
m
2p2
p2 = 1:
(
x1 = 5
x2 = 5
2.
With the new prices, how much income do I need to buy the original conssumption?
2 · 5 + 1 · 5 = m0
m0 = 15
Now we calculate the articial bundle with the new prices and new rent:
00
x1 =
00
x2 =
m0
15
=
= 3,75
2p1
4
m0
15
=
= 7,5
2p2
2
Then
(
SE
00
good1 : x1 − x1 = 3,75 − 5 = −1,25
00
good2 : x2 − x2 = 7,5 − 5 = 2,5
3.
8
u(5, 5) = u
25 =
m0 m0
,
2·2 2
m0 m0
4 2
m02 = 200
m = 200 /2 ' 14,14
1
(
00
x1 =
00
x2 =
1
200 /2
2·2
1
200 /2
2
= 3,5355
= 7,071
(
00
good1 : x1 − x1 = 3,5355 − 5 = −1,4645
SE
00
good2 : x2 − x2 = 7,071 − 5 = 2,071
9
4. Slutsky descomposition:
10
Hicks descomposition:
4.6
1), 2) and 3)
Optimal conditions:
(
x1 = x22
p1 x1 + p2 x2 = m
p1
x2
+ p 2 x2 = m
2
11
x2
x2 =
1
p1 + p2
2
=m
m
12
=
= 4,8
2,5
+ p2
(
x∗1 = 2,4
x∗2 = 4,8
1
2 p1
12
4).
(
x2 = 2x1
p1 x1 + p2 x2 = m
x1 (p1 + 2p2 ) = m
(
For
p2 = 2
and
m
x1 (p1 , p2 , m) = p1 +2p
2
x2 (, p1 , p2 , m) = p12m
+2p2
m = 12:
x1 (p1 , 2, 12) =
13
12
p1 + 4
5).
∆p1 = 3 ⇒ p1 + 3 = 4
u(2,4, 4,8) = u
2,4 =
2m0
m0
,
4+2·2 4+2·2
m0
4+2·2
m0 = 19,2
4m = 19,2 − 12 = 7,2
14
4.7
1.
2.
15
3.

If








If






If



α
β
>
p1
p2
⇒ (x∗1 , x∗2 ) = ( pm1 , 0)
α
β
<
p1
p2
⇒ (x∗1 , x∗2 ) = (0, pm2 )
α
β
=
p1
p2
⇒ (x∗1 , x∗2 ) = any point on budget constraint
4.
p1
p2
|M RS| =
α
p1
=
β
p2
p1 = p2
α
β
Substituting in budget constraint:
α
p2
x1 + p2 x2 = m
β
m
β
∗
x1 =
− x2
p2
α
Substitute
p2 = 1,m = 10
y
α=β=1
:
x∗1 = 10 − x2
The demand function is found in the previous section:
 ∗
x1 = pm1 ,
if







x∗1 = 0,
if






x∗ = m−(p2 x2 ) , if
1
p1
α
β
>
p1
p2
α
β
<
p1
p2
α
β
=
p1
p2
16
h
i
∀ x2 ∈ 0, pm2
4.8
1.
Optimal condition:
|M RS| =
p1
p2
p1
x2
=
x1
p2
p1 x1 = p2 x2
Substituting:
2p1 x1 = m
If
m = 10 , p1 = 1
and
x1 (m, p1 ) =
m
2p1
x2 (m, p2 ) =
m
2p2
p2 = 1:
(
x1 = 5
x2 = 5
With a tax of 25 % an good 2 (wine)
(
p2
will be equal to
1,25:
0
x1 = 5
0
x2 = 4
(
0
good1 : x1 − x1 = 5 − 5 = 0
TE
0
good2 : x2 − x2 = 4 − 5 = −1
2.
With the new prices, how much income do I need to buy the original conssumption?
1 · 5 + 1,25 · 5 = m0
m0 = 11,25
Now we calculate the articial bundle with the new prices and new rent:
00
x1 =
11,25
m0
=
= 5,625
2p1
2
17
00
x2 =
m0
11,25
=
= 4,5
2p2
2,5
Then:
(
SE
(
RE
00
good1 : x1 − x1 = 5,625 − 5 = 0,625
00
good2 : x2 − x2 = 4,5 − 5 = −0,5
0
00
good1 : x1 − x1 = 5 − 5,625 = −0,625
0
00
good2 : x2 − x2 = 4 − 4,5 = −0,5
3.
u(5, 5) = u
m0 m0
,
2 2,5
m0 m0
2 2,5
25 =
m02 = 125
m = 125 /2 ' 11,18
1
( 00
x1 =
00
x2 =
(
good1 :
SE
good2 :
(
good1 :
RE
good2 :
1
125 /2
2
1
125 /2
2,5
= 5,590
= 4,472
00
x1 − x1 = 5,590 − 5 = 0,590
00
x2 − x2 = 4,472 − 5 = −0,528
0
00
x1 − x1 = 5 − 5,590 = −0,590
0
00
x2 − x2 = 4 − 4,472 = −0,472
4.
4m? =⇒ 4m = m0 − m
Slutsky : 4m = 11,25 − 10 = 1,25
Hicks : 4m =
√
125 − 10 = 1,18
18
5.
x1 p1 + x2 p2 (1 + 0,25) = m
x2 p2 · 0,25
| {z }
x1 p1 + x2 p2 +
=m
money f or governement
Governement will get:
4 · 0,25 = 1.
6.
(
p1 = 1
p2 = 1,25
|M RS| =
p1
p2
x1 = 1,25x2
Substituting in budget constraint:
x1 + 1,25x2 = 10 + 0,25x2
2,25x2 = 10
Then:
(
x∗1 =
x∗2 =
11
2 = 5,5
11
2,5 = 4,4
Wine conssumption is reduced.
7.
The measure is neutral from the point of view of the estate, however, is not
neutral from the point of view of private welfare.
u(5, 5) = 25
u(5,5, 4,4) = 24,2
19
8.
New prices:
(
p1 = (1 − s)
p2 = 1,25
⇒ |M RS| =
x2
(1 − s)
p1
⇒
=
p2
x1
1,25
Budget constraints:


consumer ⇒ (1 − s)x1 + 1,25x2 = 10


governement ⇒ s x1 = 0,25x2
x2
(1 − s)
⇒ 1,25x2 = (1 − s)x1
=
x1
1,25
Consumer constraint:
(1 − s)x1 + 1,25x2 = 10
1,25x2 + 1,25x2 = 10
x∗2 = 4
Governement constraint:
s x1 = 0,25 · 4 = 1
Consumer constraint:
x1 − s x1 + 1,25x2 = 10
x1 − 1 + 1,25(4) = 10
x∗1 = 6
20
Then s is equal to:
s x∗1 = 1
s·6=1
s=
1
6
Check:
(1 − s)x∗1 + 1,25x∗2 = 10
5
· 6 + 1,25 · 4 = 5 + 5 = 10
6
Utility in this case is:
u(x∗1 , x∗2 ) = u(6, 4) = 24
9.
The reason for wich the distorsion of the prices makes the consumer be
worst o, is the kind of preferences he has. Notice that on both 4.8.6 and 4.8.8
the consumer chooses a bundle that was aordable under the initial conditions
but he rejected. Notice also that if, for example preferences were of the kind
u(x1 , x2 ) = x1 +x2 in all cases utility remains constant (and equal to 10 assuming
unit prices).
Finally, notice that the budget set doesn't enlarge eectively, since the "enlargement " depends on the consumer's choice (that's it, is variable depending on
the consumer's choice). If we work out the expressions for th budget constraint,
we would nd that in both cases is equivalent to the original one (P=(1,1)).
In 4,8,6
{(x, y) : x + (1 + t)y ≤ 10 + yt}
{(x, y) : x + y ≤ 10} ⇒ original B.C.
21
In 4,8,8
yt
(x, y) : x 1 −
x
+ y(1 + t) ≤ 10
{(x, y) : x − yt + y(1 + t) ≤ 10}
{(x, y) : x + y ≤ 10} ⇒ original B.C.
22