Economic growth and economic development 251

Introduction to Modern Economic Growth and for all i ∈ H0 , p∗ ·ˆ xi > p∗ · (5.14) à 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 th component of x Then construct xˆi (ε) such that xˆij (ε) = xˆij and xˆij (ε) = xˆij + 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 à ! X X X (5.15) ωi + xˆi > p∗ · θif y f ∗ , p∗ · i∈H i∈H = p∗ · à 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 = for all f Finally, since y∗ is profit-maximizing at prices p∗ , we have that X X â ê (5.16) p · 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