Economic growth and economic development 249

Introduction to Modern Economic Growth
An economy E is described by preferences, endowments, production sets, con-

sumption sets and allocation of shares, i.e., E ≡ (H, F, u, ω, Y, X, θ). An allocation

in this economy is (x, y) such that x and y are feasible, that is, x ∈ X, y ∈ Y, and
P
P
P
f
i
i
i∈H xj ≤
i∈H ω j +
f ∈F yj for all j ∈ N. The last requirement implies that
the total consumption of each commodity has to be less than the sum of its total
endowment and net production.
A price system is a sequence p≡ {pj }∞
j=0 , such that pj ≥ 0 for all j. We can

choose one of these prices as the numeraire and normalize it to 1. We also define
P
2
p · x as the inner product of p and x, i.e., p · x ≡ ∞
j=0 pj xj .

A competitive economy refers to an environment without any externalities and

where all commodities are traded competitively. In a competitive equilibrium, all
firms maximize profits, all consumers maximize their utility given their budget set
and all markets clear. More formally:

Definition 5.1. A competitive
equilibrium for the economy
E ≡ (H, F, u, ω, Y, X, )


â
ê
is given by an allocation x = {xi }iH , y∗ = y f ∗ f ∈F and a price system p∗ such
that

(1) The allocation (x∗ , y∗ ) is feasible, i.e., xi∗ ∈ X i for all i ∈ H, y f ∗ ∈ Y f for
all f ∈ F and
X
i∈H

xi∗
j ≤

X
i∈H

ω ij +

X

f ∈F

yjf ∗ for all j ∈ N.

(2) For every firm f ∈ F, y f ∗ maximizes profits, i.e.,

p∗ · y f ∗ ≤ p∗ · y for all y ∈ Y f .
(3) For every consumer i ∈ H, xi∗ maximizes utility, i.e.,
¡ ¢
ui xi∗ ≥ ui (x) for all x such that x ∈ X i and p∗ · x ≤ p∗ ·

Ã

ωi +

X

f ∈F

θif y f

!

.

Finally, we also have the standard definition of Pareto optimality.
2You may note that such an inner product may not always exist in infinite dimensional spaces.
But this technical detail does not concern us here, since whenever p corresponds to equilibrium
prices, this inner product representation will exist. Thus without loss of generality, we assume
that it does exist throughout.

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