Question 1. Consider an overlapping generations (OLG) economy which lasts forever in discrete time. Agents only live for two periods and supply a single unit of labor inelastically when they are young (not working when they are old) and make consumption and savings decisions. The number of young people born in t is L, = (l+n)*, production is CobbDouglas so Y: = Kng'\" with a E (D, 1), the wage rate and interest rate at time t are wt and n+1, and h'fetime utility for an agent born at time t is given by U(Clt1 c2t+1) = 10(Clt} + lnfc2t+1i where 01c and C2t+1 are consumption when the agent is young and old, and 13' E (0, 1) is the discount factor. Also, there is goods-market clearing and assetmarket clearing YE UriIt Kt+1 = 3t+1Lt where C; := CltLt + ngLt_1 is aggregate consumption, It is investment, K; is the capital stock, and 3H1 is the saving an agent born in it makes when they are young. a) Solve the consumer optimization problem. (Hint: maximize lifetime utility subject to the lifetime budget constraint and use the Euler equation to get c;,,c;,+1, and 8:14.) 1)) Find the wage rate wt and the rental rate of capital R: as a function of the capital- labor ratio is; := if from rm optimization. (Hint: there's no need to solve the rm problem, just use the results you derived in homework 1.) c) Find the law of motion for the capital-labor ratio and it's value in the steady-state. (Hint: use the asset-market clearing condition Kt\" = 5t+1Lt-)