(25 pts) Consider a payasyougo pension system. Workers are facing two periods in their life: lPeriod t when they works and receives 11) as the wage and pay 7' as the tax. 2Period t+l when they are retired and receives the pension benets ,8. In period t, workers spend ct on consumption while in period t+1 they con- sume Ct+1. The lifetime utility function for each worker is U = (1 / 4)log(ct2)+ 2log(ct+12). Let us take 5' = [1 + k]1', where k: > O is a constant which its value is decided by the social security system based on the population growth. 2' is the discount rate (one Dollar that will be Spent or earned in future is worth (1/(1+i)) today). (a) What is the equilibrium value of ct. (hint: dene the lifetime budget constraint and use the Lagrangian method) (9pts) (b) How much each worker is able to save personally in period 1;? What effect does an change in A: have on her saving? (8pts) (e) Let us assume we are in period 2 and there has been no p0pulation growth, so the true value of the pension which each retired worker receives is 3 = 7' while they expected to receive 5 = [1 + k]T. How much they can Spend now if the interest rate is also equal to i? How their welfare in period 2 changes with change in expected k? (Spts)