Consider the Diamond model, with the following information: The individual's utility is logarithmic, with discount factor p The effectiveness of labour is constant across time, such that A = At+1 = A Each individual supplies one unit of labour inelastically in each period of their life The savings of the young, St, becomes the capital stock in the following period, K++1 a) What are the new budget constraints for each period of the life cycle? Combine these two constraints to form the intertemporal budget constraint. b) Write the Lagrangian that describes the individual's constrained optimization problem. c) Solve for the optimal consumption for each young person, C,t? Solve for the optimal consumption for each old person, C2,t+1 d) Using your answer from part c), what is the amount of saving done by each young person, s? e) What is the savings function s(r), represented by the fraction of young income (Awt) saved, in this case? f) Compare your answers for c) and e) to the case where the individual only earns income when young.