Economic growth and economic development 251

Introduction to Modern Economic Growth
and for all i ∈ H0 ,
p∗ ·ˆ
xi > p∗ ·

(5.14)

Ã

ωi +

X

!

θif y f ∗ .

f ∈F

The second inequality follows immediately in view of the fact that xi∗ is the
utility maximizing choice for household i, thus if xˆi is strictly preferred, then it
cannot be in the budget set. The first inequality follows with a similar reasoning.
Suppose that it did not hold. Then by the hypothesis of local-satiation, ui must be
strictly increasing in at least one of its arguments, let us say the j 0 th component
of x. Then construct xˆi (ε) such that xˆij (ε) = xˆij and xˆij 0 (ε) = xˆij 0 + ε. For ε ↓

0, xˆi (ε) is in household i’s budget set and yields strictly greater utility than the

original consumption bundle xi , contradicting the hypothesis that household i was
maximizing utility.
Also note that local non-satiation implies that ui (xi ) < ∞, and thus the right-

hand sides of (5.13) and (5.14) are finite (otherwise, the income of household i

would be infinite, and the household would either reach a point of satiation or
infinite utility, contradicting the local non-satiation hypothesis).
Now summing over (5.13) and (5.14), we have
Ã
!
X
X
X
(5.15)
ωi +
xˆi > p∗ ·
θif y f ∗ ,
p∗ ·
i∈H

i∈H

= p∗ ·

Ã
X
i∈H

f ∈F

ωi +

X

f ∈F

!

yf ∗ ,

where the second line uses the fact that the summations are finite, so that we can
P
change the order of summation, and that by definition of shares i∈H θif = 1 for all
f . Finally, since y∗ is profit-maximizing at prices p∗ , we have that
X
X
â ê
(5.16)
p Ã
y f p ·
y f for any y f f ∈F with y f ∈ Y f for all f ∈ F.
f ∈F

f ∈F

However, by feasibility of xˆi (Definition 5.1, part 1), we have
X
X
X f
xˆij ≤
ω ij +
yˆj ,

i∈H

i∈H

237

f ∈F