This exercise asks you to construct and analyze the equivalent of the lab-equipment expanding variety model of Section 13.1 in discrete time. Suppose that the economy admits a representative household with preferences at time 0 given by
with β ∈ (0, 1) and θ ≥ 0. Production technology is the same as in the text, and the innovation possibilities frontier of the economy is given by N(t + 1) - N(t) = ηZ(t).
(a) Define the equilibrium in BGP allocations.
(b) Characterize the BGP and compare the structure of the equilibrium to that in Section 13.1.
(c) Show that there are no transitional dynamics, so that starting with any N(0) > 0, the economy grows at a constant rate.