A representative household (HH) lives for 2 periods, working and earning income y1 in period 1 and only consuming in period 2 . The market real interest rate is r and the HH discount factor is . The one-period (instantaneous) utility function of the HH is as follows: U(c)=ln(c). a. Find the distribution of HH consumption over time. What is the function? b. Introduce to the model the PAYG-type retirement system, financed from income taxes in period 1 (according to the rate), and paid out in period 2. The value of the pension system PS increases with the increase of income (g) and population (n), and it is balanced, i.e.: PS=(1+g)(1+n)y1. What is the in such case? What is the allocation of consumption over time? For which n,g, and r consumers have higher utility (i.e. their consumption is higher in both periods) after the introduction of the PAYG system