Consider a Diamond economy where production is Cobb-Douglas (F (K, Lt) = KgLta) and utility is logarithmic (u (ct, C2t+1) = In ct + In c2t+1). There is no capital depreciation 1. Solve for the planner's allocation in steady state 2. Solve for the competitive equilibrium. Is the steady state in the competitive equilibrium the same as the planner's allocation? 3. Pay-as-you-go social security. Suppose the government taxes each young individual an amount d and uses the proceeds to pay benefits to old individuals; thus each old person receives (1n) d. (a) How, if at all, does this change affect the equilibrium equation giving kt41 as a function of k,? (b) How, if at all, does this change affect the steady state value of k? (c) If the economy is initially in a steady state that is efficient (i.e., the same as the planner's allocation), how does a marginal increase in d affect the welfare of current and future generations? What happens if the initial steady state is inefficient? 4. Fully funded social security. Suppose the government taxes each young person an amount d and uses the proceeds to purchase capital. Individuals born at t therefore receive (1Tt+1) d when they are old. (a) How, if at all, does this change affect the equilibrium equation giving kt41 as a function of kt? (b) How, if at all, does this change affect the steady state value of k?

How to solve the above questions?