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 It = (1+n)', production is Cobb-Douglas so Y, = Kel, " with a E (0, 1), the wage rate and interest rate at time t are w, and 7:1, and lifetime utility for an agent born at time t is given by U(Cut, C21 1) = In(cut) + B In(c2141) where cu and ca are consumption when the agent is young and old, and Be (0, 1) is the discount factor. Also, there is goods-market clearing and asset-market clearing Y = Gith where C := Cult + Cal 1 is aggregate consumption, , is investment, K, is the capital stock, and s, is the saving an agent born in t 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 Ci, C2 ,1, and s,1.) b) Find the wage rate w, and the rental rate of capital R, as a function of the capital- labor ratio ki = It At from firm optimization. (Hint: there's no need to solve the firm 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 K, = Style.)