Household Social Security Consumption-Only Optimization

Motivation. We pay for social security by taking money from the young and giving it to the old. The payment framework for a three-period social security is: let

• population $N_{\tau }$ grows by rate $n$
• young will pay lump sum $t$ per capita
• old will receive lump sum $b$ per capita then consider period $\tau =t$:

$$\begin{array}{rl}\stackrel{\text{ distribution to old }}{\overbrace{N_{t-1}{b}_t}}& =\stackrel{\text{ payment by young }}{\overbrace{N_t{t}_t}}\\ t& =\frac{b}{1+n}\end{array}$$

Household Intertemporal Pension Optimization §

Variation of Household Intertemporal Consumption Only Optimization. We have

• $y_{\tau }$: income at time $\tau$
• $r$: interest rate. ass. perfect Financial Markets.
• $W_{\tau }$: lifetime wealth, value at time $\tau$ Then we have budget constraint for a houshold young at time $t$:

$$\begin{array}{rl}{W}_t& =y_t-t_t+\frac{y_{t+1}+b_{t+1}}{1+r}\\ & =y_t-\frac{b}{1+n}+\frac{y_{t+1}}{1+r}+\frac{b_{t+1}}{1+r}\\ & =\underset{\text{NPV gross income }}{\underbrace{y_t+\frac{y_{t+1}}{1+r}}}+\underset{\text{ NPV retirement income }}{\underbrace{\frac{b_{t+1}(n-r)}{(1+n)(1+r)}}}\end{array}$$

The last term, NPV of retirement income is positive when $n-r>0$, i.e. when population growth outpaces interest rates. if not, then instead of social security one should invest in the capital markets (i.e. 401(k) policies, etc.) to use $r$ to grow their money.