In this short note I clarify two features of Figure 2.3 in Barro and Sala-i-Martin (2004). The figure, as it appeared in the first and second editions of the book, is confusing if not wrong. I hope this note will serve as a corrigendum to the figure.Transition dynamics; Saving rate; Neo-classical growth model

Hashmi, Aamir Rafique

English

Research Papers in Economics

A Note on
the Transitional Behavior of the Saving Rate in
the Neo-Classical Growth Model (the Cobb-Douglas Case)
Aamir Raque Hashmi
March 17, 2009
Abstract
In this short note I clarify two features of Figure 2.3 in Barro and Sala-i-Martin [2004]. The
gure, as it appeared in the rst and second editions of the book, is confusing if not wrong. I
hope this note will serve as a corrigendum to the gure.
Key words: Transition Dynamics; Saving Rate; Neo-classical Growth Model
Department of Economics, National University of Singapore. E-mail: aamir@nus.edu.sg
1Barro and Sala-i-Martin [2004] (BS from here on) discuss the behavior of the saving rate during
transition to the steady state in the neo-classical growth model in Section 2.6.4 of their book. They
summarize the behavior in gure 2.3 (p.109). I reproduce the gure below.
Figure 1: Phase diagram for the behavior of the saving rate (in the Cobb-Douglas case)
There are two features of this gure which are confusing. First, in each part of the gure,
the curve showing the transitional behavior of the saving rate (from here on, the stable saddle
path) starts from point 1= on the y-axis. It implies that y intercept of these curves is 1=. This
is confusing because when capital is zero, given the assumptions of the model, the saving rate is
undened. It is also confusing because, in parts (a) and (c) of the gure, if the system begins at
point 1= it will diverge instead of converging to the steady-state saving given by s.
Second, in parts (a) and (c) of the gure, it appears that as ^ k grows large, the saving rate
asymptotically approaches s. This is confusing too. Since a unique nite steady state exists,
there is a nite ^ k at which the stable saddle paths in parts (a) and (c) of the gure will cross the
horizontal line representing the steady state saving rate. In what follows I attempt to clarify these
confusions.1
I use the same notation as in BS except that I replace ^ c=^ y with ^ z to ease the notation a little
bit. With this notation, equations (2.37) and (2.38) in BS become
1There is another problem with the gure as it appears in the second edition of the book: the caption does
not correspond to the gure. According to Robert Barro, the caption was prepared for another gure which was
inadvertently deleted in production.
2^ z =

1  
1


+   
^ k1 
A
; (2:37)
^ z = 1  
(x + n + )
A
 ^ k1 ; (2:38)
where in (2.37),    [( +  + x)=   (x + n + )]. Depending on the values of model
parameters,   could be greater than, equal to or less than zero. Hence the plot of (2.37) could be
upward sloping, horizontal or downward sloping. The plot of (2.38) is always downward sloping.
I plot these equations in Figure 2. The three cases correspond to three possible scenarios about
 . The directional arrows follow from equations (2.35) and (2.36) in BS. The curves with arrows
represent the stable saddle paths.
q q
y y
6 -
?

6 
?
-
1
1-1

_ ^ k=0
_ ^ z=0
s*
^ k* ^ k
^ z
- -  
6 -
?

6 
?
-
1
_ ^ z=0
_ ^ k=0
1-1
=s*
^ k* ^ k
^ z
) 9
1
3
6 -
?

6 
?
-
1
1-1

_ ^ z=0
_ ^ k=0
s*
^ k* ^ k
^ z
Figure 2: Stable Saddle Path in ^ z and ^ k space
The stable saddle path represents the equilibrium ratio ^ z for any given ^ k. Since the saving rate
is given by 1   ^ z, it is represented by the vertical distance between the stable saddle path and the
horizontal dashed line at ^ z = 1 in Figure 2. I plot this vertical distance in Figure 3, which is the
same as Figure 1 (i.e. Figure 2.3 in BS) except for two dierences.
3
3
9 9
1
-
s*
^ k* ^ k
s
- -   s*=1

^ k* ^ k
s
R
j
i
y
1
-
s*
^ k* ^ k
s
Figure 3: Saving Rate during transition to Steady State
3The rst dierence is that these curves do not start from 1= on y-axis. The reason is that the
saving rate is undened when ^ k = 0. This is because when capital is zero, output and consumption
are both zero and their ratio ^ z is undened. Even if we ignore this indeterminacy of the saving rate
when ^ k = 0, the stable saddle path cannot start from point 1= on y-axis in parts (a) and (c) of
the gure. To see this rst consider part (a). If s = 1= in Figure 3(a) then ^ z = 1   1= in Figure
2(a). At this point, although _ ^ z = 0, _ ^ k > 0 and hence the system will move eastwards. When the
system moves eastwards it will be above _ ^ z = 0 line and diverge. The system will eventually reach
point (0,1) in Figure 2(a) and the Euler's equation will be violated. Using a similar argument one
can show that if the system starts at point ^ z = 1   1= in Figure 2(c), it will diverge to a point
(not shown) where ^ z = 0 and the economy will save everything it produces. This will violate the
transversality condition. Hence the only possible starting position for the stable saddle path in
Figure 2(a) is somewhere below point 1  1=. Similarly the only possible starting position for the
stable saddle path in Figure 2(c) is somewhere above the point 1   1=.
The second dierence is that these stable saddle paths intersect the horizontal line representing
s when ^ k = ^ k. This is clear in parts (a) and (c) of Figures 2 and 3. Hence in Figure 3(a),
for example, if initial period capital stock is less than ^ k the saving rate will increase during the
transition. But if the initial capital stock is greater than ^ k the saving rate will decrease during
the transition (a possibility not shown in Figure 1).
To sum up, the transitional behavior of the saving rate in the Neo-classical growth model (in
the Cobb-Douglas case) is more accurately described by Figure 3 than by Figure 1.
References
R. J. Barro and X. Sala-i-Martin. Economic Growth. The MIT Press, Cambridge, MA, 2nd edition,
2004.
4

A note on the transitional behavior of the saving rate in the neo-classical growth model (the Cobb-Douglas case)

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