tutorial 1 : lecture 1 and 2
☐
1) 9 ( ✗ ( t ) ) = ✗ Lt ) -
✗ (t -
l )
✗ Ct
-
1)
Ict ) = ict ) / ✗ Ct ) ict ) =-D ✗ Idt
yet )=y◦e9ᵗ
) show that g. ( ylt ) )
☐
i if is small
≈
g g
91+-1)
yoe9ᵗ
☐ 9ᵗ
)) yo @ Yoe
g Cylt
= -
= -
I
91+-1 ) 91+-1)
Yoe Yo @
9T -
g. Lt
-
1)
e. ≈ + I
g
9+1 I ≈ g
-
ii ) take the Inc ) of ylt )
lnlylt )) 1h ( est )
=
Inly ) ◦
+
lnlylt)) =
lnlyo ) + gt
take the time derivative of Cii )
'
iii ) and show that ight ) ±
ylt) / yet ) =
g
dlnlylt ))
=
9
dt
y^Lt)=g
^ Inly )
iv )
↳ Slope :
g
I
,2)
logs and derivatives
Lt)=_ yit) / ylt )
i ) y.lt
) = kct )
lnlylt) )
=
lhlklt ) )
dlnlylt) ) =
dlnlklt))
dt dt
K' I hence j=k
f- ¥ is
=
= . = .
=
ii ) ) Kitt
ylt
=
1 It )
In / klt ) ) lnlllt ))
lnlylt ))
= -
D= i i
-
✗
=/ )
kit )
iii ) ylt ) =
Ict)
lnlyltn.in/lY 1)
dt :
y^=aci -
I]
3) growth rates : within
country comparison
1900 : $1433 2000 : $23.971
""
i ) yiqoo ( I + 9) =
92000
.
'◦°
1433 ( It g) = 23971
-
' ◦◦
( I +
g) = 16.7278436846
"' ◦ °
(1+9)=(16.7-28)
Itg
=
I. 02857
0.029
g
=
" "
ii ) 23971 (11-0.029) yz
' =
, ◦◦
418045.57
Yzioo
=
"
"◦ ◦
e×=y
iii ) 1433 .
e =
23971 ✗ Inte )= Inly ) ✗ =
Inly )
,4) growth rates cross
country comparison
:
1970
-
200s
China :
7.3% 5955
V. S :
2.2 % 36806
exponential
geometric
" "
◦ ◦ 73£ ◦ ◦" t
(1+9) (1+9)
Yzoosc
'
Yzoos
'
'
=
'
us sqgs .
e. =
36806 @
.
0.073T
5955 '
( 1.073 )ᵗ =
36806 '
( 1.022 )ᵗ e =
36806
0-0226
5955
( )t
e
1.073 36806
=
1. 022
5955
073
1h14
1-073
I. 022 ) In (
3680°
5955 ) t.tn/e0' / e.
0.022
= In
/ 368061
595s
)
36806
t.tn (1.073/1.022) = In (36806/5955) t =
In
( Sass
≈ 35-3
+ In (36806/5955) 37.4 ◦ 073
)
= ≈ '
e
In (1.073/1.022) In / e.
0.022
38 years
20051-35.3 ≈ 2041
2005+38=2043
, 5) Solow
✗
)=klt)✗Llt ]
' -
Ylt
" "
dYltl.aklti-tl.lt ) "=a( ¥7 )
'
a)
dklt )
[¥ )✗
-
✗
b) dYlt)=( I -
a) Kal =L , -
a) (
dllt )
" ✗
kltit.lt)
" " ✗
-
-
c) hilt )=dYlt ) / dt =
dklt ) 'kLCt ) + ( I -
a)
d) Vict )=Y.lt/Ylt)=dkt( I -
a) I
6) Solow
'
Ltt )=Loehᵗ
-
✗
Ylt )=Klt)•Llt ) K' (f) SYLT ) 8klt )
-
=
'
( 0=1 h=0 ✗ = 13 5=0.20 8=0.05
,
a) steady states
)*=(g+sn )
"
output :
Ylt Loe "
)
ylt )=klt
.
"
Into )klt )
Ent
/
skit )
:( ) gin
K' (f)
-
consumption l -
s .
(◦
=
kit ¥
)=(n /
capital
( ! ) iloent
stock : > ,
n ,
y☒=(n! )%
a
y*:( nio )
' "
,
( nˢ+o)%
s)
Cct )=( l -
s )ylt)*
( l
)%
-
.
-11
sin!
-
÷ ,
E)
.
*
MPK : ✗ ( nts )
☐
1) 9 ( ✗ ( t ) ) = ✗ Lt ) -
✗ (t -
l )
✗ Ct
-
1)
Ict ) = ict ) / ✗ Ct ) ict ) =-D ✗ Idt
yet )=y◦e9ᵗ
) show that g. ( ylt ) )
☐
i if is small
≈
g g
91+-1)
yoe9ᵗ
☐ 9ᵗ
)) yo @ Yoe
g Cylt
= -
= -
I
91+-1 ) 91+-1)
Yoe Yo @
9T -
g. Lt
-
1)
e. ≈ + I
g
9+1 I ≈ g
-
ii ) take the Inc ) of ylt )
lnlylt )) 1h ( est )
=
Inly ) ◦
+
lnlylt)) =
lnlyo ) + gt
take the time derivative of Cii )
'
iii ) and show that ight ) ±
ylt) / yet ) =
g
dlnlylt ))
=
9
dt
y^Lt)=g
^ Inly )
iv )
↳ Slope :
g
I
,2)
logs and derivatives
Lt)=_ yit) / ylt )
i ) y.lt
) = kct )
lnlylt) )
=
lhlklt ) )
dlnlylt) ) =
dlnlklt))
dt dt
K' I hence j=k
f- ¥ is
=
= . = .
=
ii ) ) Kitt
ylt
=
1 It )
In / klt ) ) lnlllt ))
lnlylt ))
= -
D= i i
-
✗
=/ )
kit )
iii ) ylt ) =
Ict)
lnlyltn.in/lY 1)
dt :
y^=aci -
I]
3) growth rates : within
country comparison
1900 : $1433 2000 : $23.971
""
i ) yiqoo ( I + 9) =
92000
.
'◦°
1433 ( It g) = 23971
-
' ◦◦
( I +
g) = 16.7278436846
"' ◦ °
(1+9)=(16.7-28)
Itg
=
I. 02857
0.029
g
=
" "
ii ) 23971 (11-0.029) yz
' =
, ◦◦
418045.57
Yzioo
=
"
"◦ ◦
e×=y
iii ) 1433 .
e =
23971 ✗ Inte )= Inly ) ✗ =
Inly )
,4) growth rates cross
country comparison
:
1970
-
200s
China :
7.3% 5955
V. S :
2.2 % 36806
exponential
geometric
" "
◦ ◦ 73£ ◦ ◦" t
(1+9) (1+9)
Yzoosc
'
Yzoos
'
'
=
'
us sqgs .
e. =
36806 @
.
0.073T
5955 '
( 1.073 )ᵗ =
36806 '
( 1.022 )ᵗ e =
36806
0-0226
5955
( )t
e
1.073 36806
=
1. 022
5955
073
1h14
1-073
I. 022 ) In (
3680°
5955 ) t.tn/e0' / e.
0.022
= In
/ 368061
595s
)
36806
t.tn (1.073/1.022) = In (36806/5955) t =
In
( Sass
≈ 35-3
+ In (36806/5955) 37.4 ◦ 073
)
= ≈ '
e
In (1.073/1.022) In / e.
0.022
38 years
20051-35.3 ≈ 2041
2005+38=2043
, 5) Solow
✗
)=klt)✗Llt ]
' -
Ylt
" "
dYltl.aklti-tl.lt ) "=a( ¥7 )
'
a)
dklt )
[¥ )✗
-
✗
b) dYlt)=( I -
a) Kal =L , -
a) (
dllt )
" ✗
kltit.lt)
" " ✗
-
-
c) hilt )=dYlt ) / dt =
dklt ) 'kLCt ) + ( I -
a)
d) Vict )=Y.lt/Ylt)=dkt( I -
a) I
6) Solow
'
Ltt )=Loehᵗ
-
✗
Ylt )=Klt)•Llt ) K' (f) SYLT ) 8klt )
-
=
'
( 0=1 h=0 ✗ = 13 5=0.20 8=0.05
,
a) steady states
)*=(g+sn )
"
output :
Ylt Loe "
)
ylt )=klt
.
"
Into )klt )
Ent
/
skit )
:( ) gin
K' (f)
-
consumption l -
s .
(◦
=
kit ¥
)=(n /
capital
( ! ) iloent
stock : > ,
n ,
y☒=(n! )%
a
y*:( nio )
' "
,
( nˢ+o)%
s)
Cct )=( l -
s )ylt)*
( l
)%
-
.
-11
sin!
-
÷ ,
E)
.
*
MPK : ✗ ( nts )
