# Overlapping Generations Model Pension System Question

Part 1 Pension System

OLG Model with pension system:

• Each individual lives up to two periods.
• The surviving probability at period 2 is p.
• At period 1, the young household consumes c1, saves s1, and works for wage w.
• In period 2, conditional on survival, the old household retires, consumes c2, and receives returns from savings.
• The interest rate is fixed at r.
• The household derives the utility from a natural logarithm utility function, and discounts the future at the rate of β = 1.

Now we incorporate a pay-as-you-go pension system into the economy. The income tax rate levied on a young household’s wage income is τ, and the amount of pension paid to an old household is b.

􏰀 (f) Compare the present value of lifetime earnings for a household (survive at period 2) living with and without the pension system. Under what condition she prefers the economy with the PAYG pension?

􏰀 (g) Re-solve the optimal consumption and saving decisions of the household.

􏰀 (h) What is the savings rate when pension is available? Show that the availability of pension reduces the savings rate of the household.

• I solved this question without the PAYG system, but I'm just not sure how to do it with the PAYG system. Like what's the math that shows that the pension reduces the savings rate of the household?. Mostly, IDK how to do part (h) – user24609 Oct 19 '19 at 17:00