Optimal Growth Models
One Sector Ramsey Model
Role of the Financial Sector
Cass (1965) Koopman(1965) type Optimal Growth Model

Ut 

 t ln Ct 
0   1
t 0
Yt  AKt
0  1
K t 1  I t
C t  I t  Yt
K t 1  I t
C t  Yt  I t
K0  K0
K t 1  K t 1     I t
Ct  AK t   K t 1  K t 1   
0  1
Steady State in an Optimal Growth Model

Ut 




 t ln AK t  K t 1
t 0

 1


U t    t ln AKt  K t 1   t 1 ln AKt1  K t 2  .. 
C t 1  t 1
 t AK t11
Ct

U t




AK t11  0
K t 1
Ct Ct 1
t
t 1

U t  ln AK
C t 1
 AK t11
Ct
 
t
C t 1 C

 AK  1
Ct
C
1
  1
..  Ct 1  Ct  Ct 1  ..  C
 1

K  

A



Y  AA1

1
C  Y  I  AK  K  AA

K
t 0
..  K t 1  K t  K t 1  ..  K
Y  AK 


1
1
 A

1
A 1
 A
1
1
 
1
1
  1

Optimal Growth Model with less than 100% depreciation
K t 1  K t 1     I t

  ln AK   K
t
t
K t 1  K t 1     I t
Ct  AKt  K t 1  K t 1   
C t  Yt  I t
Ut 
0   1
t 1

 K t 1   
t 0




U t    t ln AKt  K t 1  K t 1      t 1 ln AKt1  K t 2  K t 1 1     .. 
U t Ct Ct 1
 t  t 1


AK t11  1      0
Ct Ct 1 K t 1
Ct Ct 1



C t 1
  AK t11  1   
Ct
..  Ct 1  Ct  Ct 1  ..  C

..  K t 1  K t  K t 1  ..  K

U t  ln AK


 
 K  K 1   
t 0
t
Steady State in an Optimal Growth Model
with less than 100% depreciation


AK 
C t 1 C

  AK  1  1   
Ct
C
K    1A  1  1   
1

 1   1    

K  
 A

Y  AK
I  K  1   K
C Y I

1

1
 1      

K    1A  1  1    
1


1
 1

0   1
 A


K  


1  1  
Y A
I  K
 A


C  
 1   1    
2 
1







1


1




1
1 

1 
 A


I  K   


1


1





1 
 A


  
 1   1    
1
1 
1
1 
Optimal Growth Model with Financial Intermediation
S t  I t
Ct  AK t   K t 1  K t 1   



U t    t ln AK t   K t 1  K t 1   
t 0





Ct 1 
 AK t11   1   
Ct

U t    t ln AK t   K t 1  K t 1      t 1 ln AK t1   K t  2  K t 1 1     .. 

U t
 t  t 1


AKt11   1      0
Ct
Ct
Ct 1
..  K t 1  K t  K t 1  ..  K

..  Ct 1  Ct  Ct 1  ..  C
U t  ln AK  K  K  1     t


t 0

Steady State in Optimal Growth Model with Financial Intermediation


Ct 1 C


 AK  1   1   
Ct

C
AK 
1


  1     

    1    

K  

A



Y  AK

1

1


1
 1


A

K  
    1    


A

Y  





1






A

I  K   





1




C Y I
K    1A    1    
K    1A     1   





1
1
I  K  1   K
1
1
1


A

C  





1





1


A

  





1




1
1
Table 5
Capital Stock, Output, Consumption and Investment in the Steady State
Technology
Capital share:
alpha
Beta
Initial capital K0
Delta
Intermediation
cost
Capital Stock
Output
Consumption
Investment
I
44.025
0.4
0.9
100
1
1
100
278
178
100
Parameters of the Infinite Horizon Model
II
II
IV
V
VI
44.025 44.025
44.025
44.025
44.025
100
VIII
100
0.6
0.9
100
0.05
0.4
0.9
100
1
0.4
0.9
100
0.05
0.6
0.9
100
0.05
1
1
1
1.2
1.05
Infinite Horizon Economy in the Steady State
2,499
149
344,202 218,202 304,677
9,750
2,472
420,017 319,518 390,376
7,251
2,323
75,815 101,315
85,699
2,499
149
344,202 218,202 304,677
1
1
1.05
392
1,090
698
392
9,807
62,607
52,800
9,807
2,369,142
4,214,584
1,845,441
2,369,142
0.4
0.9
100
0.05
0.2
0.9
100
0.05
0.6
0.9
100
0.05
0.6
0.9
100
0.05
VII
IX
100
Computations for Optimal Growth
Excel based model
GAMS based Models
Basics of an Optimal growth Model
Yt  K t Lt1    Ct  I t
Capacity constraint
Accumulation
K t 1  1   K t  I t
Terminal Capital
First order condition
For investment
Lt  1  g  L0
t 1
K0  K0
I T  g   KT
Pt  PKt 1
First order condition for consumption
First order condition of capital market
The marginal revenue product of capital
PT  PTCT
1 g 
Pt Ct  C0  C0 

 1 r 
Pt k  RK t  1   Pt k 1
t
PTCTk  RK t  1   Pt k 1
RK t  Pt K t Lt1   K t
t
Steps for Implementing a Dynamic Optimal Growth Model
• *1. declare the time and define the first and last periods
• *2. declare and assign the values for the benchmark
parameters
• *3. calibrate the model
– discount factors and assign the value for initial capital stock
• *4. declare variables and equations
• *5. derive the first order conditions for optimisation and
market clearing conditions
• *6. write all equations
• *7. declare the model
• *8. Solve the model
• *9. produce the results in readable format
• *10. Interpret them using economic theory
• *11. modify the model to incorporate new issues.
set t /t1*t30/
set tfirst(t)
set tlast(t)
;
tfirst(t) = yes$(ord(t) eq 1);
tlast(t) = yes$(ord(t) eq card(t));
*declare the Key parameters to benckmark the economy
scalar
g
/0.02/
R
/0.05/
K0
/3/
kstock /1/
delta /0.07/
I0
c0
kvs
;
I0 = (delta+g)*K0;
C0 = 1-I0;
*declare reference prices and quantities
parameters
qref(T)
pref(t)
alpha
;
QREF(T) = (1+g)**(ord(t)-1);
pref(t) = (1/(1+r))**(ord(t)-1);
kvs = (delta+r)*K0;
alpha(t) = ((1+g)/(1+r))**(ord(t)-1);
alpha(tlast) = alpha(tlast)/(1-((1+g)/(1+r)));
*declare variables
Variables
C(t)
I(t)
K(t)
P(t)
PK(t)
RK(t)
PTC(t)
;
*Declare equations
Equations
capacity(t)
capital(t)
Terminal(t)
Foc_con(t)
Foc_K(t)
Foc_I(t)
rent_k(t)
*declaration of model equations
capacity(t)..
(K(t)/K0)**kvs*qref(t)**(1-kvs) =e= C(t)+I(t);
capital(t)..
(1-delta)*K(t-1) +I(t-1) +K0*Kstock$tfirst(t) =G= K(t);
Terminal(tlast)..
I(tlast) =e= (g+delta)*K(tlast);
Foc_con(t)..
P(t)*C(t)=e=C0*Alpha(t);
Foc_K(t)..
Pk(t) +PTC(t)*(g+delta)$tlast(t)=e=
P(t)*((kvs*K(t)**kvs*qref(t)**(1-kvs))/K(t)) +(1-delta)*pk(t+1);
Foc_I(t)..
P(t) =e= PK(t+1) +PTC(t)$tlast(t);
rent_k(t)..
rk(t) =e= P(t)*((kvs*K(t)**kvs*qref(t)**(1-kvs))/K(t));
model ramsey1/
capacity.p
capital.pK
Foc_con.c
Foc_K.k
Foc_I.I
Terminal.ptc
rent_k
/;
C.l(t)=c0*qref(T);
I.l(t)=i0*qref(T);
K.l(t)=k0*qref(T);
P.l(t)=Pref(T);
PK.l(t)=(1+R)*Pref(T);
*RK.lo(t)=Pref(T);
PTC.l(T)=PREF(T);
pk.lo(t) =1e-6;
pk.up(t) =+inf;
solve ramsey1 using mcp;
parameter base, report;
base(t, "cons")= C.l(t);
base(t, "inv") = I.l(t);
base(t, "cap") = K.l(t);
base(t, "labour") = qref(t);
base(t, "price-y")= P.l(t);
base(t, "price-k")= PK.l(t);
base(t, "rent") = RK.l(t);
base(t, "term-K")= PTC.l(T);
Display base, report;
Interpretation of Model Equations in GAMS
Capacity constraint
Yt  K t Lt1    Ct  I t
(K(t)/K0)**kvs*qref(t)**(1-kvs) =e= C(t)+I(t);
Accumulation
K t 1  1   K t  I t
K0  K0
(1-delta)*K(t-1) +I(t-1) +K0*Kstock$tfirst(t) =G= K(t);
Terminal Capital
First order condition
For investment
I(tlast) =e= (g+delta)*K(tlast);
I T  g   KT
P(t) =e= PK(t+1) +PTC(t)$tlast(t);
Pt  PKt 1
PT  PTCT
First order condition for consumption
1 g 
t
P(t)*C(t)=e=C0*Alpha(t); Pt Ct  C0  C0 

 1 r 
First order condition of capital market
Pt k  RK t  1   Pt k 1
t
(t) +PTC(t)*(g+delta)$tlast(t)=e= P(t)*((kvs*K(t)**kvs*qref(t)**(1-kvs))/K(t)) +(1-delta)*pk(t+1)
PTCTk  RK t  1   Pt k 1
RK t
Marginal product of capital
 Pt K t Lt1   K t rk(t) =e= P(t)*((kvs*K(t)**kvs*qref(t)**(1-kvs))/K(t));
Ramsey Model: Benevolent Social Planner’s Problem
U t  ln Ct
Instantaneous utility
T
Max U    t ln C t
1 g 
t  

1

r


t
Subject to

1  
Yt  K t Lt
or
K t 1  1   K t  I t
I T  g   KT
Yt  Ct  I t
Lt  1  g  L0
t

Yt    K t  1   Lt

t

1
 
Ramsey Model: Decentralised Market
T
Max U   1    ln C t
t
t
Yt  K t1  Lt
K t 1  1   K t  I t
I T  g   KT
Ct 1  Rt 1   Ct 1 1   
Rt 1    1   K t Lt
Lt  1  g  L0
t 1
Yt  Ct  I t
Optimal Consumption-Saving Model while Young, Adult and Old
Max U  C , C , C   ln C   ln C   ln C
1 2
2
3
3
 1 2 3
Subject to:
1.
2.
C
W
C
W
C  2  3 W  2  3
1 1 r 
1 r 2 1 1 r  1 r 2
( W , W , W ) = (120, 1200, -120)
1
3. ( C1 0,
2
3
C2 0, C 3 0)
What is the optimal consumption and saving in each period ?
Derivation of the Marginal Productivity = User Cost of Capital Condition

F K 
1   P2K K
k

P K 
1  r  1
1 r
Producer’s Problem:
Optimality Condition:
Implication:
K


1


P
 F ' K 
2 0

 Pk 
K 1  r  1
1 r
MPK  1  r P1k  1   P2K



MPK  1  r   1    1   K P1


MPK  r     K P1
Assumptions:
K
P2K
 K 1
P1
k
k
 K  0
References
•
•
Bhattarai (2003) Role of Financial Markets in an Economy, memio University of Hull.
Cass, D. (1965): Optimum Growth in Aggregative Model of Capital Accumulation,
Review of Economic Studies, 32:233-240.
•
GAMS User Manual, GAMS Development Corporation, 1217 Potomac Street, Washington D.C.
•
Stokey, N. L. and R.E. Lucas (1989) Recursive Methods in Economic Dynamics,
Harvard UP, Cambridge, MA.