Let N be the population of an economy inhabited only by active and retired workers. Let π be the proportion of active workers in the total population. All active workers work the same number of ours to produce the only good available in the economy and receive labor income y. The economy has no financial markets, so that individuals are not allowed to privately save income. Active workers then split their income y between consumption (of the only good) and a mandatory "savings" imposed by the government (a benevolent social planner), who redistributes the total amount of savings equally among the retired workers. Retired workers have no other source of income.
Let σ be the mandatory savings per dollar of income, and suppose the government sets σ so that each retired worker may consume a quantity c of the good produced in the economy.
(a) Find a formula for σ as a function of π,y, and c. (Hint: think of the amount collected from active workers and the amount transfered to retired workers as supply and demand for savings, respectively.)
(b) Suppose the number of active workers decreases, while the number of retired workers increases (so that N does not change). If y remains constant, what should the government do in order to keep c at the same level.