Social Security with Growth:

Households live for two periods, so at every time t there are two generations alive, young (y) and old (o). Households preferences are given by

U=2 cyt +2 cot+1.

Each generation born at time t has size Nt, where Nt+1 = (1 + n) Nt, i.e. n is the population growth rate. In each period t, young households receive an endowment of income yt, consume cyt and save for retirement an amount st+1. Young workers have no initial wealth. The old households are retired and consume cot+1. They consume what they have saved as young plus the interests cumulated at the fixed rate r. Income grows over time at rate g, i.e. yt+1 = (1 + g) yt, so every generation has larger income than the previous generation by a factor (1 + g) .

a) Solve for the optimal consumption choices {cy,co} as a function of (y,r) in this economy without social security.

Suppose the government introduces a pay-as-you-go social security system that works as follows. At every date t, the government taxes young workers' income at rate τ and redistributes these tax revenues equally (i.e., lump-sum) to the old at time t as pension transfers.

b) Solve for the optimal consumption choices {cy,co} as a function of (y,r,τ.g,n) in the economy with social security.

c) Find a parametric condition under which introducing social security makes every agent better off. Explain this condition.