Economic growth and economic development 71

Introduction to Modern Economic Growth k(t+1) 45° sf(k(t))+(1–δ)k(t) k(t+1) 45° k(t+1) sf(k(t))+(1–δ)k(t) 45° sf(k(t))+(1–δ)k(t) k(t) k(t) Panel B Panel A k(t) Panel C Figure 2.5 Examples of nonexistence and nonuniqueness of steady states when Assumptions and are not satisfied Equation (2.18) and (2.19) then follow by definition Ô Figure 2.5 shows through a series of examples why Assumptions and cannot be dispensed with for the existence and uniqueness results in Proposition 2.2 In the first two panels, the failure of Assumption leads to a situation in which there is no steady state equilibrium with positive activity, while in the third panel, the failure of Assumption leads to non-uniqueness of steady states So far the model is very parsimonious: it does not have many parameters and abstracts from many features of the real world in order to focus on the question of interest Recall that an understanding of how cross-country differences in certain parameters translate into differences in growth rates or output levels is essential for our focus This will be done in the next proposition But before doing so, let us generalize the production function in one simple way, and assume that f (k) = af˜ (k) , where a > 0, so that a is a shift parameter, with greater values corresponding to greater productivity of factors This type of productivity is referred to as “Hicksneutral” as we will see below, but for now it is just a convenient way of looking at the impact of productivity differences across countries Since f (k) satisfies the regularity conditions imposed above, so does f˜ (k) 57