Economic growth and economic development 249

Introduction to Modern Economic GrowthAn 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.. Th

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

i∈Hxi

i∈Hωi

j +P

f ∈Fyfj 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· x as the inner product of p and x, i.e., p · x ≡P∞

j=0pjxj.2

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, θ)

is given by an allocation³

x∗ ={xi∗}i∈H, y∗ =©

yf ∗ª

f ∈F

´ and a price system p∗ such that

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

X

i∈H

xi∗j ≤X

i∈H

ωij +X

f ∈F

yjf ∗ for all j ∈ N

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

p∗· yf ∗ ≤ p∗· y for all y ∈ Yf (3) For every consumer i∈ H, xi∗ maximizes utility, i.e.,

ui¡

xi∗¢

≥ ui(x) for all x such that x∈ Xi and p∗· x ≤ p∗·

Ã

ωi+X

f ∈F

θifyf

! Finally, we also have the standard definition of Pareto optimality

2 You 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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