Introduction to Modern Economic Growth where η ≡ η H /η L and the *’s denote that this expression refers to the BGP value The notable feature here is that relative productivities are determined by the in- novation possibilities frontier and the relative supply of the two factors In this sense, this model totally endogenizes technology Equation (15.27) contains most of the economics of directed technology However, before discussing this, it is useful to characterize the BGP growth rate of the economy This is done in the next proposition: Proposition 15.1 Consider the directed technological change model described above Suppose that (15.28) Ô Ê (1 ) ( H H)σ−1 + γ ε (η L L)σ−1 σ−1 > and Ê Ô (1 ) (1 − γ)ε (η H H)σ−1 + γ ε (η L L)σ−1 σ−1 < ρ Then there exists a unique BGP equilibrium in which the relative technologies are given by (15.27), and consumption and output grow at the rate Ô 1³ £ (15.29) g∗ = β (1 − γ)ε (η H H)σ−1 + γ ε (η L L)σ−1 σ−1 − ρ θ Proof The derivation of (15.29) is provided by the argument preceding the proposition Exercise 15.2 asks you to check that condition (15.28) ensures that free entry conditions (15.20) and (15.21) must hold, to verify that this is the unique relative equilibrium technology, to calculate the BGP equilibrium growth rate and to verify that the transversality condition is satisfied Ô It can also be verified that there are simple transitional dynamics in this economy whereby starting with technology levels NH (0) and NL (0), there always exists a unique equilibrium path and it involves the economy monotonically converging to the BGP equilibrium of Proposition 15.1 This is stated in the next proposition: Proposition 15.2 Consider the directed technological change model described above Starting with any NH (0) > and NL (0) > 0, there exists a unique equilibrium path If NH (0) /NL (0) < (NH /NL )∗ as given by (15.27), then we have ZH (t) > and ZL (t) = until NH (t) /NL (t) = (NH /NL )∗ If NH (0) /NL (0) < (NH /NL )∗ , then ZH (t) = and ZL (t) > until NH (t) /NL (t) = (NH /NL )∗ 671