A NOTE ON CES FUNCTIONS
Drago Bergholt, BI Norwegian Business School 2011
Functions featuring constant elasticity of substitution (CES) are widely used in applied
economics and finance. In this note, I do two things. First, I derive a number of conditions
(such as the optimal demand schedule) when aggregation technology is CES. Second, I
show how the CES function nests some particular functional forms as special cases.
1
P RELIMINARIES
Consider a general CES function:
Y =A
J
X
1
η
αj Yj
η−1
η
η
! η−1
,
(1)
j=1
PJ
where A, η > 0, αj , Yj > 0 ∀ j, and j=1 αj = 1. In models with monopolistic competition between firms, Y is interpreted as aggregate output and {Yj }Jj=1 as output by
intermediate firms. In turn, the production technology of firms can also have a CES structure, with each Yj being referred to as an input factor such as labor or capital or land.
Y can also be aggregate demand or consumption, a composite of individual consumer
classes or goods. In open economy macroeconomics, one uses the CES structure to distinguish between domestically produced goods and imports. The scaling parameter A is
often normalized to 1, but in the context of production theory, it is typically interpreted as
a measure of total factor productivity (which can be stochastic in a time series context).
2
I MPLICATIONS OF CES TECHNOLOGY
What are the implications of assuming a CES technology for aggregation in economic
models? To shed light on this question, we start with a maximization problem. The problem is to choose the vector {Yj }Jj=1 that maximizes Y subject to some budget constraint
J
X
Pj Yj ≤ Z,
(2)
j=1
where Z is total money spent. Let us state the Lagrangian (where Λ is the multiplier for
the constraint):
η
! η−1
!
J
J
η−1
1
X
X
−Λ
Pj Yj − Z
L=A
αjη Yj η
j=1
j=1
η−1
1
1
η
−1
The necessary first order condition is A η Y η αj Yj η = ΛPj . This must hold for all
j ∈ J , so we can write
−η
α j Pj
Yj =
Yi ∀ i, j
(3)
αi Pi
1
2.1
A NATURAL PRICE INDEX
Next, we derive an exact price index for a unit increase in Y . Plug (3) into the budget
constraint and solve for Yi :
J
X
J
Yi Piη X
Z=
P j Yj =
αj Pj1−η
αi j=1
j=1
Zαi Pi−η
⇒ Yi = PJ
1−η
j=1 αj Pj
Insert this result into Y (now with indexing i):
η
! η−1
J
η−1
1
X
Y =A
αiη Yi η
i=1

= A
J
X
Zαi Pi−η
PJ
1−η
j=1 αj Pj
1
η
αi
i=1
= AZ
J
X
!−1
αj Pj1−η
j=1
= AZ
J
X
J
X
η
 η−1
! η−1
η

η
! η−1
αi Pi1−η
i=1
1
! η−1
αj Pj1−η
j=1
Finally, define P as the expenditure needed to buy one unit of Y , i.e. P ≡ Z|Y =1 . From
the expression above, it naturally follows that
J
X
1
P =
A
1
! 1−η
αj Pj1−η
.
(4)
j=1
Thus, P is the natural price index for Y .
2.2
T HE BUDGET CONSTRAINT REVISITED
Next, note that inserting (3) into (2) gives
Z=
J
X
P j Yj
j=1
J
X
αj
=
Pj
αi
j=1
Pj
Pi
−η
Yi
Yi
= Piη A1−η P 1−η
αi
−η
Pi
Z
⇒ Yi = αi
.
P
A1−η P
2
(5)
Substitute into (1) and solve for Z. The result is
η

 η−1
! η−1
−η
J
η
1
X
Pj
Z

Y = A
αjη αj
= ZP −1 .
1−η
P
A
P
j=1
Thus, Z = P Y , and we can write the budget constraint as
PY =
J
X
Pj Yj .
j=1
2.3
T HE OPTIMAL DEMAND SCHEDULE
Finally, substitution of Z = P Y into (5) (which holds ∀ i, j ∈ J ) gives the optimal
demand for Yj :
−η
Pj
Y
(6)
Yj = αj
P
A1−η
One can also get rid of A and end up with


Yj = αj  −η
Pj
PJ
1−η
i=1 αi Pi
1
1−η


J
X
1
η
αi Yi
η−1
η
η
! η−1
.
i=1
Thus, optimal demand for Yj in (6) is not determined by A, but rather by the price Pj
relative to all other prices, as well as by total factor demand. Equation (6) is widely used
as demand function in theoretical models of optimizing economic behavior.
2.4
E LASTICITIES
To understand why the functional form in (1) exhibits CES between any pair Yi , Yj , we
Y
define the elasticity of substitution as the percentage rise in Yji following a one percent
η
Y
α
rise in the relative price PPji . It is clear from (3) that Yji = αji PPji , so we can write
Y
η−1
∂ Yji
αj P i
=η
.
αi Pj
∂ Pi
Pj
Y
By dividing by Yji / PPji , a constant elasticity of substitution emerges:
η−1
Y
α
Y
∂ Yji / Yji
η αji PPji
Elasticity of substitution ≡ =
η−1 = η
αj
Pi
∂ PPji / PPji
αi
Pj
What about the income elasticity? Earlier we defined Z as the income spent on Y . The
income elasticity follows from (5):
−η 1
αi PPi
∂Yj /Yj
1−η
Elasticity of income ≡
=
−η A 1 P = 1
P
∂Z/Z
αi i
1−η
P
3
A
P
3
S PECIAL CASES OF THE CES FUNCTION
Next, we show how the CES function nests as special cases some particular functional
forms. Before we start, it is useful to note that equation (1) is equivalent to
Y =A
J
X
! γ1
αj Ỹjγ
,
(7)
j=1
Y
where Ỹj ≡ αjj and γ ≡ η−1
. Our task is to find an expression for Y when i) η →
η
∞ or γ → 1 (linear), ii) η → 1 or γ → 0 (Cobb-Douglas), and iii) η → 0 or γ →
−∞ (Leontief). For reasons that will be clear below, it is convenient to work with a log
transformation of equation (7):
!
J
X
F (γ)
1
αj Ỹjγ = ln (A) +
,
(8)
ln (Y ) = ln (A) + ln
γ
G (γ)
j=1
P
J
γ
α
Ỹ
and G (γ) ≡ γ. Finally, in order to derive the Cobbwhere F (γ) ≡ ln
j
j
j=1
Douglas and Leontief functions we need L’Hôpital’s rule:
The L’Hôpital’s rule:
Consider two functions F (γ) and G (γ), which are differentiable on an interval (a, b),
except possibly at a point c ∈ (a, b). Suppose limγ→c F (γ) = limγ→c G (γ) = 0 or
F 0 (γ)
0
±∞, limγ→c G
0 (γ) exists, and G (γ) 6= 0 ∀ γ 6= c ∈ (a, b). Then
F (γ)
F 0 (γ)
= lim 0
.
γ→c G (γ)
γ→c G (γ)
lim
In our case the first derivatives are F 0 (γ) =
0
F (γ)
=
G0 (γ)
αj Ỹjγ ln(Ỹj )
PJ
γ
j=1 αj Ỹj
PJ
j=1
and G0 (γ) = 1, implying
γ
Ỹj
α
Ỹ
ln
j=1 j j
.
PJ
γ
α
Ỹ
j
j
j=1
PJ
(9)
Next, we consider each of the three special cases of the CES function.
3.1
T HE LINEAR CASE (γ → 1)
This one is trivial as it follows immediately from equation (7) that
lim Y = A
γ→1
J
X
j=1
4
Yj .
(10)
T HE C OBB -D OUGLAS CASE (γ → 0)
3.2
Here we want to evaluate equation (8) when γ → 0. However, the result is
P
J
γ
ln
α
Ỹ
j=1 j j
0
= ln (A) + “
”.
lim ln (Y ) = ln (A) + lim
γ→0
γ→0
γ
0
P
The last term holds as long as Jj=1 αj = 1.1 To proceed, we apply L’Hôpital’s rule.
Evaluating equation (9) in the limit as γ → 0, we can write (8) as
PJ
γ
j=1 αj Ỹj ln Ỹj
lim ln (Y ) = ln (A) + lim
PJ
γ
γ→0
γ→0
j=1 αj Ỹj
J
X
= ln (A) +
αj ln Ỹj
j=1
= ln A
J
Y
!
αj
Ỹj
,
1
J
Y
j=1
or
α
Yj j .
αj
j=1 αj j=1
lim Y = A QJ
γ→0
(11)
This is the widely used Cobb-Douglas function. The scaling parameter A is often normalized so that A QJ 1 αj = 1.
j=1
αj
3.3
T HE L EONTIEF CASE (γ → −∞)
When γ → −∞, then equation (8) becomes
P
J
γ
α
Ỹ
ln
j
j
j=1
−∞
lim ln (Y ) = ln (A) + lim
= ln (A) + “
”.
γ→−∞
γ→−∞
γ
−∞
Again, using L’Hôpital’s rule, we can write
γ
α
Ỹ
ln
Ỹj
j
j
j=1
.
PJ
γ
j=1 αj Ỹj
PJ
lim ln (Y ) = ln (A) + lim
γ→−∞
γ→−∞
αj Ỹjγ ln(Ỹj )
PJ
γ
j=1 αj Ỹj
2
PJ
However, a new problem arises because limγ→−∞
j=1
=“
0
0
”, which can-
not be solved
n by multiple
o uses of L’Hôpital’s rule. In order to proceed, we define
Ỹm ≡ min Ỹ1 , . . . , ỸJ . Dividing the nominator and denominator by Ỹmγ , we get
γ PJ
Ỹj
ln Ỹj
α
j=1 j Ỹm
γ
lim ln (Y ) = ln (A) + lim
PJ
Ỹj
γ→−∞
γ→−∞
j=1 αj Ỹ
m
1
2
PJ
If instead j=1 αj > 1 (< 1), then the last term goes to ∞ (−∞) as γ → 0.
Ỹjγ will always show up both in the nominator and in the denominator.
5
n
o
= ln (A) + ln min Ỹ1 , . . . , ỸJ
n
o
= ln A min Ỹ1 , . . . , ỸJ
,
or
lim Y = A min
γ→−∞
YJ
Y1
,...,
α1
αJ
.
(12)
This is the Leontief function.
3.4
F ROM ISOELASTIC UTILITY TO LOG UTILITY
Consider the isoelastic utility function
U=
C 1−σ − 1
,
1−σ
with C > 0. Apparently a “ 00 ”-issue arises when one evaluates U as σ → 1. In that case
one can define F (σ) ≡ C 1−σ − 1 and G (σ) ≡ 1 − σ. Because F 0 (σ) = −C 1−σ ln (C)
and G0 (σ) = −1, we get
F 0 (σ)
= ln (C) .
σ→1 G0 (σ)
lim U = lim
σ→1
6
(13)