Optimal Pay$-$As$-$You$-$Go Social Security $$ Consider the Overlapping Generations economy with non$-$storable

incomes. Each household cohort lives for two periods: Young and Old. There is no population growth. When

Young, each produces output by working with the production function Yyoung $=$ Lyoung. When Old, each

receives exogenous income of Yold $=$ W $<1.$ Each household solves

max log

$$

Cyoung

$$

$$ Lyoung $+$ log $($Cold$)$ subject to Cyoung $+$ S $=$ Lyoung

Cold $=$ W $+(1+$ r$)$ S

$($a$)(4$ points$)$ Solve for the optimal consumption$-$labor choice for each household under the competitive

equilibrium absence of government. $($Hint: In OLG, saving is$?)$