Economic growth and economic development 571

Introduction to Modern Economic Growth This first-order condition for the consumption index immediately implies that (see Exercise 12.10): P ≡ (12.9) ÃN X i=1 p1−ε i ! 1−ε Since P is the price index corresponding to the consumption index C, it is typically referred to as the ideal price index, and in many circumstances, it will be convenient to choose this ideal price index as the numeraire Note, however, that we cannot set this as the price index in this particular instance, since we have already written the budget constraint in terms of money income, m, and also normalized the price of good y to In addition, the choice between C and y is also facilitated in this case, and boils down to the maximization of a semi-indirect utility function U (C, y) = C + v (y) , where we have use the definition of the consumption index C Similarly, combining (12.8) and (12.9) with the budget constraint, (12.7), we obtain a budget constraint expressed in terms of C and y, P C + y ≤ m Now the maximization of this semi-indirect utility function would respect to this budget constraint yields the following first-order condition: v0 (y) = , P which assumes that the solution is interior, an assumption we maintain throughout to simplify the discussion Since v (·) is strictly concave, v (·) is strictly decreasing and can be inverted, so that we obtain (12.10) ả y = v P µ ¶ 0−1 C = m−v P 0−1 Next, let us consider the production of the varieties Suppose that each variety can only be produced by a single firm, who is thus an effective monopolist for this 557