Economic growth and economic development 267

Introduction to Modern Economic Growth (1) Rewrite the budget constraint of household h at time t, (5.17), including these bonds (2) Prove an equivalent of Theorem 5.8 in this environment with the extended set of bonds Exercise 5.11 Consider a two-period economy consisting of two types of households NA households have the utility function ¡ ¢ ¡ ¢ u ci1 + β A u ci2 , where ci1 and ci2 denotes the consumption of household i into two periods The remaining NB households have the utility function ¡ ¢ ¡ ¢ u ci1 + β B u ci2 , with β B < β A Each group, respectively, has income yA and yB at date 1, and can save this to the second date at some exogenously given gross interest rate R Show that for general u (·), this economy does not admit a representative household Exercise 5.12 Consider an economy consisting of N households each with utility function at time t = given by ∞ X t=0 i ¡ ¢ β t u ci (t) , with β ∈ (0, 1), where c (t) denotes the consumption of household i at time t The economy starts with an endowment of Y units of the final good and has access to no production technology This endowment can be saved without depreciating or gaining interest rate between periods (1) What are the Arrow-Debreu commodities in this economy? (2) Characterize the set of Pareto optimal allocations of this economy (3) Does Theorem 5.7 apply to this economy? N (4) Now consider an allocation of Y units to the households, {y i }i=1 , such that PN i i=1 y = Y Given this allocation, find the unique competitive equilibrium price vector and the corresponding consumption allocations (5) Are all competitive equilibria Pareto optimal? (6) Now derive a redistribution scheme for decentralizing the entire set of Pareto optimal allocations? 253