Economic growth and economic development 249

Introduction to Modern Economic Growth An economy E is described by preferences, endowments, production sets, consumption 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 ≥ for all j We can choose one of these prices as the numeraire and normalize it to We also define P 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, θ) ´ â ê is given by an allocation x = {xi∗ }i∈H , 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 235