Chris Papageorgiou

Kaz Miyagiwa

English

Research Papers in Economics

Elasticity of Substitution and Growth:
Normalized CES in the Diamond Model
Kaz Miyagiwa
Department of Economics
Louisiana State University
Baton Rouge, LA 70803
kmiyagi@lsu.edu
Chris Papageorgiou
Department of Economics
Louisiana State University
Baton Rouge, LA 70803
cpapa@lsu.edu
February, 2001
Abstract
Klump and de La Grandville (2000) used the \normalized" Constant Elasticity of Substitu-
tion (CES) speci¯cation to prove that the Solow growth model exhibits a positive relationship
between per capita output and the elasticity of substitution both in transition and in steady
state. This paper shows that their result does not extend tothe Diamond overlapping generations
model. In particular, their result is reversed when capital and labor are relatively substitutable;
countries with a higher elasticity of substitution have lower per capita output and growth.
JEL Classi¯cation Numbers: E13, E23, O40.
Keywords: CES, Diamond Overlapping Generations Model, Economic Growth.1 Introduction
In a recent paper, Klump and deLa Grandville (2000) utilized the\normalized"Constant Elasticity
of Substitution (CES) production function in the Solow (1956) growth model and found that a
country endowed with a greater elasticity of substitution experiences greater capital and output
perworker both in transition and in steady state. Theobjectiveof this paper isto examinewhether
their result carries over to the Diamond (1965) overlapping-generations model. Such examination is
warranted becausetheDiamond model has increasingly been used in recent yearsto study economic
growth as an alternative to theSolow model. Our main ¯nding isthat the Klump-de La Grandville
result does not hold in the Diamond model; in particular, their result is reversed if the elasticity of
substitution is su±ciently large.
2 The Normalized CES Production Function in the Solow Model
Oliver de La Grandville (1989) suggested that a meaningful examination of the properties of di®er-
ent members of the same family of CES production functions requires the following normalization.
Given the standard intensive-form CES production function f(kt) = A[±k
½
t +(1 ¡±)]
1
½, where kt
is the capital per worker at time t, choose arbitrary baseline values for capital per worker (¹ k),
output per worker (¹ y) and the marginal rate of substitution between capital and labor de¯ned by
¹ m = [f(¹ k) ¡¹ kf0(¹ k)]=f0(¹ k) (primes denote derivatives with respect to k). Then, use those baseline
values to solve for the normalized e±ciency parameter A(¾) = ¹ y
³
¹ k1¡½+¹ m
¹ k+¹ m
´1=½
, and the normalized
distribution parameter ±(¾) =
¹ k1¡½
¹ k1¡½+¹ m as a function of ¾ = 1
1¡½, the elasticity of substitution.
Substituting these normalized parameters into the initial equation yields the normalized CES pro-
duction function:1
f¾(kt) =A(¾)f±(¾)k
½
t +[1 ¡±(¾)]g
1
½ : (1)
Figure 1 illustrates the de La Grandville normalization. Despite disparate values for ¾, all the
isoquants for a given initial level of output (¹ y) are shown to go through the common point (point
A) de¯ned by ¹ k (given by ray OA) and ¹ m (given by line BAC). As shown by Pitchford (1960),
an increase in ¾ without the normalization causes not only an increase in the curvature of the
1For extensive discussions on the normalized CES function see de La Grandville (1989, p.476), and Klump and
Preissler (2000, pp.44-45).
1Figure 1: Illustration of de La Grandville's normalized CES production function
K K
L
? = ?
? = 1
? = 0 A
    0
    B
   C
isoquant for a given level of output; it also causes the isoquant to shift inward by making factors
more e±cient. The de La Grandville normalization prevents such dispersions.
The de La Grandville normalization generates a family of dynamical paths in the Solow growth
model that depend only on the value of ¾. Paths of capital per worker for three values of ¾ are
shown in Figure 2. Figure 2 di®ers from Figure 1 in Klump and de la Grandville (2000 p.284)
because here the Solow model is recast in a discrete-time setting to facilitate comparison with
the Diamond model below. More speci¯cally, the paths shown in the ¯gure are generated by the
dynamical equation
kt+1 =
°
1 +n
f¾(kt);
where ° is the exogenous saving rate out of output per worker, n is the exogenous labor growth
rate and where for simplicity capital is assumed to depreciate fully at the end of each period.
Despite the translation into the discrete-time setting, the Klump-de la Grandville result is
evident; a country having a greater value of ¾ clearly has morecapital per worker in transition and
in steady state than a country endowed with a lower value of ¾. It follows that, the greater the
value of ¾, the greater income per worker is both in transition and in steady state.
2Figure 2: Transitional paths of per capita capital for di®erent ES in the Solow model
3 The Normalized CES Production Function in the Diamond Model
In the Diamond (1965) overlapping-generations model a new generation is born at thebeginning of
every period. Agents are identical and live for two periods. In the ¯rst period each agent supplies
a unit of labor inelastically and receives a competitive wage
w¾;t =f¾(kt) ¡ktf0
¾(kt) =[1 ¡±(¾)][A(¾)]½[f¾(kt)]1¡½:
To makethemodel consistent with the Solow model, assumethat agents savea ¯xed proportion
° of the wage income to ¯nance consumption in the second period of their lives. All savings are
invested as capital to be used in the next period's production; that is
kt+1 =
°
1 +n
w¾(kt) =
°
1 +n
[1¡±(¾)][A(¾)]½[f¾(kt)]1¡½ ´h¾(kt); (2)
where n is the exogenous labor growth rate and where capital depreciates fully.2 Equations (2)
2Alternatively, we could assume that agents have preferences over consumption in the two periods of their lives
3determines the dynamical path of capital per worker. Then, the dynamical path of output per
worker is obtained from (1).
Steady states for k (denoted by ¤) are solutions to the polynomial equation
k ¡h¾(k) = 0: (3)
If ¾ ¸ 1 (½ 2 [0; 1]), there always exists one unique positive steady state for k¤, since lim
k!0
h0
¾(k) > 1 and lim
k!+1
h0
¾(k) =0. If ¾ <1 (½ <0), there are either zero or two positive and distinct
steady-state values for k¤; depending on the value of the scale factor A(¾).3
We now turn to our two main ¯ndings. (All proofs are in the Appendix.)
Theorem 1 Suppose that a country is represented by the one-sector Diamond model with a nor-
malized CES aggregate production function. If ½ ¸ ¹ m
¹ k , for any kt > ¹ k,
(A) the higher the elasticity of substitution the lower the level of capital and output per worker at
any stage of the transition path, and
(B) the higher the elasticity of substitution the lower the growth rates of capital and output per
worker along the transitional path.
Theorem 2 Suppose that a stable steady state exists in the one-sector Diamond model with a
normalized CES aggregate production function. If ½ ¸ ¹ m
¹ k , the higher the elasticity of substitution,
the lower the steady-state level of capital and output per worker.
Figure 3 illustrates thedynamical pathsof capital per worker in the Diamond model for ½ ¸ ¹ m
¹ k ,
where we set ¹ m = 1 and ¹ k = 5. As ¾ increases from 1:25 to 5 and to 1, the level of capital per
worker falls both in transition and in steady state for any kt > ¹ k; thereby reversing the Klump-de
La Grandville result.4
given by U(c
1
t; c
2
t+1) = (1 ¡ °)lnc
1
t + ° lnc
2
t+1, where c
i
t+j is period i consumption by the representative agent in
period t +j, j = 0; 1. The representative agent maximizes U(c
1
t; c
2
t+1) subject to the constraint, c
1
t +
c2
t+1
R¾;t+1 · w¾;t,
where w¾;t and R¾;t+1 represent the returns to labor and capital, respectively. Maximization yields the transition
equation, kt+1 =
°
1+nw¾(kt), which is equivalent to equation (2).
3When there are two positive steady states, the larger of the two is locally asymptotically stable. In this case,
the trivial steady state (k
¤ = 0) is also locally asymptotically stable. The domains of attraction of the two stable
steady states are distinct, and depend on whether the initial capital stock lies above or below the locally unstable
equilibrium. The conditions for and characterization of multiple equilibria in the Diamond (1965) model (see e.g.
Azariadis 1993, pp.198-204) remain una®ected by the normalization.
4Parametric examples of the dynamic relationship between yt+1 and yt are available upon request.
4Figure 3: Transitional paths of per capita capital in the Diamond model when ½ ¸ ¹ m
¹ k
Moreover, if ½ < ¹ m
¹ k the relationship between the ¾ and the level of capital per worker is not
unique because, as shown in Figure 4, the dynamical paths of k for di®erent values of ¾ cross each
other at some kt >¹ k.5
Why do our results contrast with those of Klump and de La Grandville? The Diamond model
di®ers from the Solow model in oneimportant respect: individual savings comeout of wage income
in the former and out of total (wage and rental) income in the latter. A useful way to demonstrate
the di®erence is o®ered by Galor (1996). Suppose that the fraction saved out of wage income, °w,
di®ersfromthefraction saved out of rental income, °r, possibly becauseof di®erencesin preferences
or endowments among agents. Then the law of motion for capital per worker in the normalized
CES production function is
kt+1 =
°w
1 +n
£
f¾(kt) ¡f0
¾(kt)kt
¤
+
°r
1 +n
f0
¾(kt)kt: (4)
5In constructing Figure 4, we set ¹ m = 3, ¹ k = 5 to keep the diagram from getting cluttered.
5Figure 4: Transitional paths of per capita capital in the Diamond model when ½ < ¹ m
¹ k
Since °w = °r = ° in the Solow model while °w = ° and °r = 0 in the Diamond model, the
dynamical path in the former contains the additional term,
°
1+nf0
¾(kt)kt, that represents savings
out of rental income.
For example, when ¾ =1 (capital and labor are perfect substitutes), equation (4) reduces to
kt+1 =
°¹ y¹ m
(1 +n)(¹ k + ¹ m)
+
°¹ y
(1 +n)(¹ k + ¹ m)
kt;
in the Solow model. Thus, kt+1 is a linear positive function of kt with the vertical intercept at
°¹ y¹ m
(1+n)(¹ k+¹ m) and the slope
°¹ y
(1+n)(¹ k+¹ m). On the other hand, in the Diamond model equation (4)
reduces to
kt+1 =
°¹ m¹ y
(1 +n)(¹ k + ¹ m)
:
Thus, kt+1 is a horizontal line at °¹ m¹ y
(1+n)(¹ k+¹ m).6 Then, as k grows from the common initial value ¹ k,
the entire capital intensity path of the Solow model lies above the path of the Diamond model.
6The former line is depicted by the parametric curve ¾ ¼ 1 in Figure 2, while the latter line is depicted by the
parametric curve ¾ = 1 in Figure 3.
6Now, to get an intuition of our result di®erentiate equation (4) with respect to ¾ to obtain:
@kt+1
@¾
=
°w
1+n
·
(1¡¼t)
@f¾(kt)
@¾
¡f¾(kt)
@¼t
@¾
¸
| {z }
@wt
@¾ (?)
+
°r
1 +n
·
¼t
@f¾(kt)
@¾
+f¾(kt)
@¼t
@¾
¸
| {z }
@rkt
@¾ (+)
; (5)
where¼t ´
f0
¾(kt)kt
f¾(kt) istherental incomeshare. The¯rst and thesecond term on theRHS of equation
(5) show the change in wage and rental incomes, respectively, due to a change in ¾. The second
expression is clearly positive while the ¯rst is generally ambiguous in sign. No matter what, the
second expression must dominate the¯rst in the Solow model (where °w = °r =°) since
@kt+1
@¾ > 0
as shown by Klump and de La Grandville (2000).
In the Diamond model (where °w = ° and °r = 0) the second expression in equation (5) is
absent. Within the ¯rst expression, (1¡¼t)
@f¾(kt)
@¾ represents a positive e®ect of ¾ on wage income
due to an increase in labor productivity for a given wage income share (1 ¡¼t). The second term
¡f¾(kt)@¼t
@¾ represents a negative e®ect of ¾ on wage income due a decrease in the wage income
share triggered by substitution of capital for labor. If the wage incomeshare (1¡¼t) is su±ciently
small, then thenegativee®ect dominatesand ourresult follows. Indeed ½ ¸ ¹ m
¹ k implies(1¡¹ ¼) <1=2
which is su±cient to obtain our result.7
4 Conclusion
In this paper we have shown that the positive relationship between the elasticity of substitution
and economic growth discovered recently by Klump and de la Grandville does not carry over to
the Diamond model. Thus, whether the elasticity of substitution has a positive or negative e®ect
on economicgrowth depends on our view of the world, that is, on the particular framework (Solow
vs. Diamond) we believe as a better representation of the world.
Both our work and that of Klump and de la Grandville take the elasticity of substitution as
exogenous. However, as pointed out by Hicks (1932), the aggregate elasticity of substitution itself
is likely to be in°uenced by factors that also a®ect economic growth. Thus, endogenizing the
elasticity of substitution in the context of a growth model seems like a natural next step in this
line of research.
7Since ¹ ¼ =
¹ k
¹ k+ ¹ m implies that
¹ m
¹ k ´
1¡¹ ¼
¹ ¼ ; substitution into ½ ¸
¹ m
¹ k yields ¹ ¼ ¸
1
1+½. Given that ½ 2 (0; 1] under
½ ¸
¹ m
¹ k , ½ ¸
¹ m
¹ k ¯nally implies ¹ ¼ ¸ 1=2.
7Appendix
Proof of Theorem 1
Rewrite equation (2) as
kt+1 =
°
1 +n
[f¾(kt) ¡ktf0
¾(kt)]
=
°
1 +n
[f¾(kt)(1 ¡¼t)];
where ¼t ´
f0
¾(kt)kt
f¾(kt) is the rental income share. Di®erentiating with respect to ¾ yields
@kt+1
@¾
=
°
1+n
·
(1¡¼t)@f¾(kt)
@¾
¡f¾(kt)@¼t
@¾
¸
:
Substituting
@f¾(kt)
@¾ = ¡ 1
¾2
1
½2yt
h
¼tln ¹ ¼
¼t +(1 ¡¼t)ln 1¡¹ ¼
1¡¼t
i
and @¼t
@¾ = 1
¾2(1¡¼t)¼tln kt
¹ k from Klump
and de La Grandville (2000, pp.284-285) yields
@kt+1
@¾
= ¡
°
1+n
½(1¡¼t)yt
¾2½2
·
¼tln
¹ ¼
¼t
+(1 ¡¼t)ln
1 ¡¹ ¼
1 ¡¼t
¸
+
y
¾2(1¡¼t)¼tln
kt
¹ k
¾
= ¡
°
1+n
(1 ¡¼t)yt
¾2½2
·
¼tln
¹ ¼
¼t
+(1¡¼t)ln
1 ¡¹ ¼
1 ¡¼t
+½2¼tln
kt
¹ k
¸
; (A1)
where ¹ ¼ =
¹ k
¹ k+¹ m. From
¹ ¼
¼t
=
Ã
¹ k
kt
yt
¹ y
!½
;
1¡¹ ¼
1 ¡¼t
=
µyt
¹ y
¶½
;
we obtainkt
¹ k = ¹ ¼
¼t
1¡¼t
1¡¹ ¼. Substituting this into the last term in the brackets of equation (A1) gives
@kt+1
@¾
=¡
°
1 +n
(1¡¼t)yt
¾2½2
·
¼tln
¹ ¼
¼t
+(1¡¼t)ln
1 ¡¹ ¼
1 ¡¼t
¡½¼tln
µ
¹ ¼
¼t
1¡¼t
1¡ ¹ ¼
¶¸
:
Since the logarithmic function is strictly concave, we have that
ln
¼t
¹ ¼
<
¼t
¹ ¼
¡1 ) ¡ln
¼t
¹ ¼
>1 ¡
¼t
¹ ¼
) ln
¹ ¼
¼t
>1 ¡
¼t
¹ ¼
; (A2)
ln
1 ¡¼t
1 ¡¹ ¼
<
1¡¼t
1 ¡¹ ¼
¡1 ) ¡ln
1 ¡¼t
1 ¡¹ ¼
>1 ¡
1¡¼t
1 ¡¹ ¼
) ln
1 ¡¹ ¼
1 ¡¼t
>1 ¡
1 ¡¼t
1 ¡¹ ¼
; (A3)
ln
µ
¹ ¼
¼t
1 ¡¼t
1¡¹ ¼
¶
<
¹ ¼
¼t
1 ¡¼t
1 ¡¹ ¼
¡1 ) ¡ln
µ
¹ ¼
¼t
1 ¡¼t
1 ¡¹ ¼
¶
>1 ¡
¹ ¼
¼t
1 ¡¼t
1 ¡¹ ¼
: (A4)
8Assume that ¾ > 1 (½ 2 (0;1]) and kt >¹ k: Multiplying both sides of the ¯nal inequalities in (A2),
(A3) and (A4) by ¼t; (1¡¼t) and ½¼t, respectively, yields the following inequality:
¼tln
¹ ¼
¼t
+(1 ¡¼t)ln
1 ¡¹ ¼
1 ¡¼t
¡½¼tln
µ
¹ ¼
¼t
1¡¼t
1 ¡¹ ¼
¶
> ¼t
µ
1¡
¼t
¹ ¼
¶
+(1 ¡¼t)
µ
1 ¡
1¡¼t
1¡ ¹ ¼
¶
+½¼t
µ
1¡
¹ ¼
¼t
1¡¼t
1¡ ¹ ¼
¶
=
¼t
¹ ¼
(¹ ¼¡¼t) +
1 ¡¼t
1 ¡¹ ¼
(¼t ¡ ¹ ¼) +
½
1¡ ¹ ¼
(¼t ¡¹ ¼)
= (¼t ¡ ¹ ¼)
µ1 ¡¼t
1 ¡¹ ¼
+
½
1 ¡¹ ¼
¡
¼t
¹ ¼
¶
=
µ¼t ¡¹ ¼
1¡¹ ¼
¶µ
1¡¼t +½ ¡
¼t(1 ¡¹ ¼)
¹ ¼
¶
=
µ¼t ¡¹ ¼
1¡¹ ¼
¶µ
1+½ ¡
¼t
¹ ¼
¶
=
µ
¼t ¡¹ ¼
1¡¹ ¼
¶"
1+½ ¡
Ã
¹ k + ¹ m
¹ k
!
k
½
t¹ k1¡½
k
½
t¹ k1¡½ + ¹ m
#
; (A5)
where the last equality comes from the fact that ¼t =
k
½
t¹ k1¡½
k
½
t ¹ k1¡½+¹ m. The function
Á(kt) ´
"
1 +½ ¡
Ã
¹ k + ¹ m
¹ k
!
k
½
t ¹ k1¡½
k
½
t¹ k1¡½ + ¹ m
#
;
is monotonically decreasing with thehorizontal asymptote at ½¡ ¹ m
¹ k : Therefore, if ½ ¸ ¹ m
¹ k , Á(kt) ¸ 0.
Then since ¼t¡¹ ¼
1¡¹ ¼ ¸ 0, the last expression in (A5) is non-negative. Consequently,
@kt+1
@¾ < 0.
To prove that output per worker is a decreasing function of the ¾ when ½ ¸ ¹ m
¹ k and k > ¹ k;
rewrite
@yt+1
@¾ as
@yt+1
@¾
=
@yt+1
@kt+1
@kt+1
@¾
:
We have just shown that
@kt+1
@¾ < 0 for ½ ¸ ¹ m
¹ k and k > ¹ k. Given that
@yt+1
@kt+1 is positive for all
kt+1 >0, then
@yt+1
@¾ <0. This completes the proof of Theorem 1A.
To prove Theorem 1B, de¯ne the growth rate of capital per worker by gk =
kt+1
kt ¡1 and the
growth rate of output per worker by gy =
yt+1
yt ¡1. Di®erentiation yields
@gk
@¾
=
1
kt
@kt+1
@¾
<0; 8 ½ ¸
¹ m
¹ k
; and k > ¹ k;
@gy
@¾
=
1
yt
@yt+1
@¾
< 0; 8 ½ ¸
¹ m
¹ k
; and k >¹ k:
This completes the proof.
9Proof of Theorem 2
At steady state, kt =kt+1 = k¤ and therefore equation (2) reduces to the polynomial equation
(3). Di®erentiating k¤ with respect to ¾ yields
@k¤
¾
@¾
=
°
1+n(w¤
¾)
0
1¡ °
1+n(w¤
k)
0 ; (B1)
where(w¤
¾)
0
=
h
(1¡¼¤)@f¤
@¾ ¡f¤@¼¤
@¾
i
and (w¤
k)
0
=
h
(1 ¡¼¤)(f¤)0 ¡f¤@¼¤
@k¤
i
. By Theorem 2, (w¤
¾)
0
<
0 for ½ ¸ ¹ m
¹ k and k¤ > k, so the numerator is negative. To show that the denominator is positive,
solvethede¯nition of ¼¤ =
(f¤)0k¤
f¤ and thesteady-statepolynomial equation k¤ = °
1+n [f¤ ¡(f¤)0k¤]
simultaneously for f¤and (f¤)0 to obtain
f¤ =
(1+n)
°
k¤
1 ¡¼¤; (B2)
(f¤)0 =
(1 +n)
°
¼¤
1¡¼¤: (B3)
Substituting equations (B2), (B3) and @¼¤
@k¤ =
½¼¤(1¡¼¤)
k¤ into (w¤
k)
0
gives
(w¤
k)
0
= (1 ¡¼¤)(1 +n)
°
¼¤
1¡¼¤ ¡
(1+n)
°
k¤
1 ¡¼¤
½¼¤(1¡¼¤)
k¤
=
(1+n)
°
(1 ¡½)¼¤:
Then
°
1+n(w¤
k)
0
= (1 ¡½)¼¤ and hence 1 ¡
°
1+n(w¤
k)
0
=1¡(1¡½)¼¤ >0. Therefore @k¤
@¾ <0.
To prove that thesteady-state output per worker is a decreasing function of the ¾ when ½ ¸ ¹ m
¹ k
and k > ¹ k; once again rewrite
@y¤
@¾ as
@y¤
@¾
=
@y¤
@k¤
@k¤
@¾
:
Given that @y¤
@k¤ > 0 for all k¤ > 0; and @k¤
@¾ < 0 for ½ ¸ ¹ m
¹ k and k > ¹ k as shown above, @y¤
@¾ < 0 as
desired.
10References
Azariadis, C., Intertemporal Macroeconomics. Cambridge, MA: Blackwell Publishers, 1993.
Diamond, P.A., \National Debt in a Neoclassical Growth Model," American Economic Review,
55:1126-1150, 1965.
deLa Grandville, O., \In Quest of theSlutsky Diamond," American EconomicReview, 79:468-481,
1989.
Galor, O., \Convergence? Inferences from Theoretical Models," EconomicJournal, 106:1056-1069,
1996.
Hicks, J.R., The Theory of Wages, London: MacMillan, 1932.
Klump, R. and de La Grandville, O., \Economic Growth and Elasticity of Substitution: Two
Theorems and Some Suggestions," American Economic Review, 90:282-291, 2000.
Klump, R. and Preissler, H., \CES Production Functions and Economic Growth," Scandinavian
Journal of Economics, 102:41-56, 2000.
Pitchford, J.D, \Growth and the Elasticity of Substitution," Economic Record, 36:491-504, 1960.
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nomics, 70:65-94, 1956.
11

Elasticity of Substitution and Growth: Normalized CES in the Diamond Model

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Elasticity of Substitution and Growth: Normalized CES in the Diamond Model

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