Let N be the population of an economy inhabited only by active and retired worke

Question

Let N be the population of an economy inhabited only by active and retired workers. Let pi 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 sigma be the mandatory savings per dollar of income, and suppose the government sets sigma so that each retired worker may consume a quantity c of the good produced in the economy. Find a formula for sigma as a function of pi, y, and c. (Hint: think of the amount collected from active workers and the amount transferred to retired workers as supply and demand for savings, respectively.)

Explanation / Answer

Total population of the economy = N

No. of active workers = *N

No. of retired workers = N - *N = (1-)*N

Income received by one active worker = y

Savings made by each active worker = times the worker's income = *y

Total savings made the active workers = (*y)*(*N)

These savings made by the active workers should be sufficient enough to distribute amongst (1-)*N retired workers so that each of them can get c quantity of goods.

Total cost of quantity c of goods = c*(1-)*N

Hence, (*y)*(*N) =  c*(1-)*N which is the required relation.

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