Economic growth and economic development 237

Introduction to Modern Economic Growth
The utility function (5.4) is referred to as constant elasticity of substitution (CES),
since if we define the level of consumption of each good as xˆij = xij −ξ ij , the elasticity

of substitution between any two xˆij and xˆij 0 would be equal to σ.

Each consumer faces a vector of prices p= (p1 , ..., pN ), and we assume that for
all i,
N
X

pj ¯ξ < y i ,

j=1

so that the household can afford a bundle such that xˆij ≥ 0 for all j. In Exercise
5.6, you will be asked to derive the optimal consumption levels for each household
and show that their indirect utility function is given by
i
h P
N
i
i
p
ξ
+
y
−
j
¡
¢

j
j=1
,
(5.5)
vi p,y i = h
1
PN 1−σ i 1−σ
j=1 pj

which satisfies the Gorman form (and is also homogeneous of degree 0 in p and y).

Therefore, this economy admits a representative household with indirect utility:
h P
i
− N
p
ξ
+
y
j=1 j j
v (p,y) = h
1
PN 1−σ i 1−σ
j=1 pj
R
R
where y is aggregate income given by y ≡ i∈H y i di and ξ j ≡ i∈H ξ ij di. It is also

straightforward to verify that the utility function leading to this indirect utility
function is

(5.6)

σ
# σ−1
"N
X¡
¢ σ−1
.
xj − ξ j σ
U (x1 , ..., xN ) =

j=1

We will see below that preferences closely related to the CES preferences will
play a special role not only in aggregation but also in ensuring balanced growth in
neoclassical growth models.
It is also possible to prove the converse to Theorem 5.2. Since this is not central
to our focus, we state this result in the text rather than stating and proving it
formally. The essence of this converse is that unless we put some restrictions on the
distribution of income across households, Gorman preferences are not only sufficient
for the economy to admit a representative household, but they are also necessary.
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