# Uzawa-Lucas Growth Model

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LECTURE NOTES 8
Uzawa-Lucas Growth Model

Sources
Our discussion of the Uzawa-Lucas growth model builds on foundations presented by
Robert Barro and Xavier Sala-i-Martin on pages 182-198 and 204-205 of their text Economic
Growth. The objective of these lecture notes is to add detail to the presentation found in this
source, showing every step taken and explaining all aspects of the model.

Uzawa-Lucas Growth Model
In the Uzawa-Lucas growth model, we work with the same objective functional we worked
with in the Romer growth model:


C1   t
M ax      e dt
0
1 

Once more, we seek to maximize the present value of this future utility stream. And while
we also continue to work with a Cobb-Douglas production function that features constant
returns to scale, here it takes a slightly different form:

Y = K(uH)1-
In this expression, K represents the stock of physical capital and H represents the stock of
human capital. The coefficient u is the portion of human capital devoted to the production
of output Y. Suppose that 0  u  1, and let 1 – u be the portion of human capital devoted
to the production of more human capital. This leads to an equation of motion for the
human capital stock:


H  (1  u )H

To interpret , note that


H  (1  u )H 
H
 (1  u ) 
H
 H  (1  u )
and

101
u0
H  

Thus,  represents the growth rate of human capital if u = 0. But note also that u = 0
implies that 100% of human capital (1 – u = 1 – 0 = 1) would be devoted to producing more
human capital. Consequently,  indicates the maximum possible growth rate of human
capital. And note finally that this upper bound on the growth rate of human capital is
unlikely to be attained, since u = 0 implies that 0% of human capital would be devoted to
producing output, and Y = K(uH)1- would imply Y = K(0H)1- = 0. Thus, the human
capital stock could only attain its maximum growth rate if no output were produced. With
regard to the physical capital stock, its equation of motion can be obtained in the usual way
(assuming for simplicity once more that physical capital doesn’t depreciate ( = 0):


K I

K S

K  Y C 
K  K  ( uH)1  C


At this point we can fully specify the optimal control problem emerging in the Uzawa-Lucas
model:


C 1 t
M ax      e dt
0
1 
subject to


H  (1  u )H
and

K  K ( uH )1  C


Given this problem, we set up the Hamiltonian function reproduced below. (Since H
represents the stock of human capital in this model, we will use J to denote the
Hamiltonian.)

C 1 t
J      e   H (1  u )H   K ( K  ( uH)1  C )
1 

102
In this problem, the control variables are C and u, the state variables are H and K, and the costate
variables affiliated with these state variables are H and K, respectively. To solve this
dynamic optimization problem, it is necessary to satisfy the following six Maximum Principle
conditions:

J
(1)        C   e t   K  0
C
J
( 2)        H H  (1   ) K K  u   H 1   0
u
J                                                    
( 3)        H (1  u )  (1   ) K K  u 1  H     H
H
J                               
(4 )         K K  1 ( uH )1    K
K
J                      
(5)          (1  u )H  H
 H
J
(6)          K  ( uH )1   C  K 
 K

Before continuing, recall first that in all of the growth models that we have examined in this
course steady-state equilibrium has featured aggregate consumption growing at the same rate
as the physical capital stock: C = K. Here in the Uzawa-Lucas growth model, this will be a
feature of steady state equilibrium as well. To ease the search for this characteristic of
steady-state equilibrium, we follow the example of Barro and Sala-i-Martin in their text
C
Economic Growth by letting   . If  represents the ratio of consumption to the
K
physical capital stock, observe that

C
    
K
K  C 
 
K  K  C 

K K C
          
       
K K C
 
 K C     
  
 K C
  K  C 

103
  C  K

Thus, the steady-state equilibrium condition involving equal growth rates in consumption
and the physical capital stock is satisfied when the ratio  is at rest: C = K   = 0.

Observe next that like the Mankiw-Romer-Weil growth model, the Uzawa-Lucas growth
model features two types of capital: (1) physical capital K, and (2) human capital H. In the
Mankiw-Romer Weil growth model, and here again in the Uzawa-Lucas growth model,
steady-state equilibrium features the physical capital stock growing at the same rate as the
human capital stock: K = H. Once more, to ease the search for this characteristic of
steady-state equilibrium, we take the suggestion of Barro and Sala-i-Martin in their text
K
Economic Growth by letting   . If  represents the ratio of the physical capital stock
H
to the human capital stock, note that

K
    
H
H  K 
   
H  H  K 

H H K
           
        
H H K
 
 H K     
  
 H K
  H  K 
  K  H

Consequently, the steady-state equilibrium condition that requires the growth rates of the
physical capital stock and the human capital stock to be equal is satisfied when the ratio  is
at rest: K = H   = 0.

Finally, steady-state equilibrium in the Uzawa-Lucas model features an optimal choice for
the portion of human capital u to be devoted to the production of output. Since this
optimal choice implies a particular value for u (and hence for 1 – u, the portion of human
capital to be devoted to producing more human capital), u should come to rest at this
optimal value, and therefore u should display zero growth in steady-state equilibrium as well:
u = 0.

In sum, steady-state equilibrium in the Uzawa-Lucas growth model is characterized by the
three homogeneous equations shown below:

 = 0

104
 = 0
u = 0
Thus, the search for this steady-state equilibrium in the Uzawa-Lucas growth model starts
with a search for , , and u, the left-hand sides of these homogeneous equations. But
before starting this search, note one more implication of steady-state equilibrium in this
model:

Y  K  ( uH )1  
Y  K  1( uH )1  K  (1   )K  u   H1  u  (1   )K  u 1  H   H 
                                                                            
Y K  1( uH )1  K (1   )K  u   H1  u (1   )K  u 1  H   H
                                                                             
                                                                            
Y        K  ( uH )1           K  ( uH )1               K  ( uH )1 
      
Y K (1   )u (1   )H
           
                               
Y     K           u            H
 Y   K  (1   ) u  (1   ) H

Since u = 0 in steady-state equilibrium, the preceding result simplifies as follows in steady-
state equilibrium:

 Y   K  (1   ) u  (1   ) H 
 Y   K  (1   ) H

And since K = H ( = 0) in steady-state equilibrium, the preceding result implies

 Y   K  (1   ) H 
 Y   H  (1   ) H 
Y  H

And finally since steady state equilibrium features C = K ( = 0) as well, we can fully

 =  = u = 0
and
Y = H = K = C

Turning our attention back to the six Maximum Principle conditions necessary to maximize
the present value of the future utility stream accruing to workers in the Uzawa-Lucas
economy, we recall once more that  = C – K, and we work initially with condition (1) to
move in the direction of obtaining an expression for the growth rate of aggregate
consumption C:

105
C   e  t   K  0 
C   e t   K 
                 
 C   1 e t C  C   e t   K 
 C   1 e t C C   e t  K
                   
   t           
C   e t       C e            K

C          
   K 
C          K

K
  C   
K

Looking for opportunities to combine and simplify, from condition (4) observe that we can


obtain a second expression for K :
K

 K K  1 ( uH )1   K  

 1        1 

K
K ( uH )                      
K
 1 1      1 

K
K u H                         
K
K  1 1            

  1 u   K 
H                     K
 1              
K                      
  u 1   K 
H                      K

  1 u 1   K 
K
 1 1 

K
  u 
K


K                                 
Settting this result for      equal to the previous result for K , we find
K                                 K

106
  C      1 u 1  
 C      1 u 1  
 C    1 u 1    
1
 C  (   1 u 1    )


Having obtained this result for C, we continue along the road to  by looking next for K.
Note that the growth rate of the physical capital stock emerges readily from condition (6):

K  ( uH )1  C  K 

K  ( uH )1 C K     
  
K         K K
 1      1 
K ( uH )     K 
K  1 1
u    K 
H  1
 1
K         1 
  u    K 
H
 1 u 1     K

Equipped with this result, we can obtain an expression for :

  C  K 
1
   (   1 u 1    )  (   1 u 1    ) 

                
     1 u 1      1 u 1    
                
                    
     1  1 u 1    
                     

Having obtained an expression for , we look next for . Recalling that  = K – H and
noting that we have already found K, we turn our attention to the growth rate of the human
capital stock H. Note that this result can be obtained quickly from condition (5):

107
(1  u )H  H 
H
(1  u )  
H
(1  u )   H

Given this result, we can obtain an expression for :

  K  H 
    1 u 1    (1  u )

Having found a result now for  as well as , we turn our attention to u. Note also to this
point that we have employed Maximum Principle conditions (1), (4), (5), and (6). To start
our search for u, we work next with condition (2):

  H H  (1   ) K K  u   H 1   0 
(1   ) K K  u   H 1    H H 
(1   ) K K  u   H     H  
K  
(1   ) K  u   H  
H

 K  
(1   ) K   u   H  
H
(1   ) K   u     H  
                                                                        
(1   ) K   u    (1   ) K   1 u     (1   ) K   u   1 u   H  

                                                                         
(1   ) K   u   (1   ) K   1 u    (1   ) K   u   1 u  H 

                                                                  
(1   ) K   u        (1   ) K   u           (1   ) K   u         H

K                
u H
                   
K                u H

K                       

      u  H
K                       H

Before continuing, note that since steady-state equilibrium features  = 0 and u = 0,

  

steady-state equilibrium also features the costate variables growing at equal rates: K  H .
K H
Recall that we showed that the costate variables in the Romer model displayed equal growth
rates in steady-state equilibrium as well.

108
Returning to the preceding analysis, we can obtain a preliminary expression for u:


K                   
H
      u      
K                   H

K H 
           u 
K H
     
1  K H 
             u
  K H 
          

K
From our earlier analysis, recall that we have obtained expressions for       and .
K
Consequently, to obtain a final expression for u, we still need to obtain an expression for

H                                   

. To obtain an expression for H , we turn to the Maximum Principle condition that
H                                   H
we have yet to use. In particular, from condition (3) note that


 H (1  u )  (1   ) K K  u 1 H     H 
(1   ) K K  u 1 H         
H
(1  u )                                       
H                   H
                         

(1  u )  (1   ) K K  u 1 H    H 
H                        H
 K K  1           

(1  u )  (1   )           u  H 
 H H                H
            
 K  K  1           H
(1  u )  (1   )       u                   
H  H                 H
 K  1           
H
(1  u )  (1   )      u 
H                 H

In order to substitute for H in the denominator on the left-hand side of this equation,
observe that we can obtain H from condition (2):

109
  H H  (1   ) K K  u   H 1  0 
(1   ) K K  u  H 1   H H 
(1   ) K K  u  H    H  
K  
(1   ) K      u  H 
H

 K  
(1   ) K   u   H  
H
(1   ) K   u    H  
1 
 K   u    H



H
Substituting this result into the preceding expression for       , we find
H

 K  1 

(1  u )  (1   )       u  H 
H        H
K                           
H
(1  u )  (1   )                       u 1         
1                                     H
 K   u 



(1  u )  u   H 
H

 H 
H


  H
H


H                                       
Equipped with this result for     to go along with earlier results for K and , we have
H                                       K
everything we need to obtain a final expression for u:

110

1  K H 
                u 
  K H 
             
1
   1 u 1      1 u 1    (1  u )   u 


   (1  u )   u


Finally, we can write down expressions for the three homogeneous equations that
characterize steady-state equilibrium in the Uzawa-Lucas model:

                    
   0    1  1 u 1      0
                    
   0    1 u 1    (1  u )  0

u  0           (1  u )  0

From the last two of these three homogeneous equations, note that

  1 u 1    (1  u )

   (1  u )

Thus,


 1 u 1 

And from the first of these three homogeneous equations, observe that

      1 1 
  1  u     0 
               
       
    1    1 u 1
        

Substituting back into the  = 0 equation,

111
  1 u 1     (1  u )  0 
          
   1    (1  u )  0 
         
    
             (1  u ) 
   
 
   (1  u ) 
 

 (1  u )

But recall from condition (5) that H = (1 – u), and recall further that in steady-state
equilibrium H = K = C = Y. Thus, the steady-steady equilibrium growth rates of the
human capital stock, the physical capital stock, aggregate consumption, and aggregate output
are given by

 
H  K  C   Y 

Note that these steady-state equilibrium growth rates vary positively with  but negatively
with  and . Observe further that we can also solve for the optimal portion of human
capital to devote to the production of output:


 ( 1  u ) 


u           


u 1


And of course, note finally that the optimal portion of human capital to devote to producing
more human capital emerges very easily:

 
u 1          

 
1 u


112
Problem Set
Consider a hypothetical economy that behaves in accordance with the Uzawa-Lucas growth
model. In particular, consider the dynamic optimization problem shown below:

          1       3
      t
M ax 2C e      2       25
dt
0
subject to
 1
H  (1  u )H
5
and
1             1

K  K ( uH ) 2  C
2

Note that the elasticity of utility with respect to consumption (1 – ) equals one-half
             1                                                  3 
 1       , the rate of time preference equals 12%     , the maximum possible
             2                                                 25 
    1
growth rate of human capital is 20%     , and the elasticities of output with respect to
    5
             1
physical capital and human capital are both one-half    1     .
             2

1. Set forth the Hamiltonian function pertinent to this optimal control problem.
2. Obtain six Maximum Principle conditions necessary to be satisfied to maximize the
present value of the future utility stream accruing to workers in this economy.
3. Obtain the steady-state equilibrium growth rates of human capital, physical capital,
consumption, and output in this economy.
4. Solve for the optimal fraction of human capital devoted to producing more human

113

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