Economic growth and economic development 694

Introduction to Modern Economic Growth contrast equilibrium allocation to the Pareto optimal allocation We will start with this latter comparison in the next subsection 15.3.4 Pareto Optimal Allocations The analysis of Pareto optimal allocation is very similar to the analysis of optimal growth in Chapter 13 For this reason, we will present only a sketch of the argument As in that analysis, it is straightforward to see that the social planner would not charge a markup on machines, thus we have xSL (ν, t) = pL (t)1/β L and xSH (ν, t) = pH (t)1/β H (1 − β)1/β (1 − β)1/β Combining these with the production function and some algebra establish that net output, which can be used for consumption or research, is equal to (see Exercise 15.6): S −1/β Y (t) = (1 − β) (15.31) i h ¡ ¢ σ−1 ¢ σ−1 ε¡ S ε S σ σ β γ NL (t) L + (1 − γ) NH (t) H In view of this, the current-value Hamiltonian for the social planner can be written as ¢ C S (t)1−θ − ¡ S S S S S S +µL (t) η L ZLS (t)+µH (t) η H ZH (t) , H NL , NH , ZL , ZH , C , µL , µH = 1−θ subject to h ¡ i ¢ σ−1 ¢ σ−1 −1/β ε¡ S S ε S S σ σ + (1 − γ) NH (t) H (t) γ NL (t) L − ZLS (t) − ZH C (t) = (1 − β) The necessary conditions for this problem give the following characterization of the Pareto optimal allocation in this economy Proposition 15.5 The stationary solution of the Pareto optimal allocation involves relative technologies given by (15.27) as in the decentralized equilibrium The stationary growth rate is higher than the equilibrium growth rate and is given by ´ £ 1/ 1 Ô (1 β) β (1 − γ) (η H H) + γ (η L L) − ρ > g∗, g = θ ∗ where g the BGP a clue brim growth rate given in (15.29) S Ô Proof See Exercise 15.7 680