$x_t=a_0w_t +a_1w_{t-1}$ $\text{Var}(x_t)$=$\text{Var}(a_0w_t +a_1w_{t-1})$ Assume $w_t, w_{t-1}$ are independent. Then variances of sum is sum of variances. $\text{Var}(x_t) = \text{Var}(a_0 w_t) + \text{Var}(a_1 w_{t-1})$ $Var(x_t)$ = $a_0^2\text{Var}(w_t)+a_1^2\text{Var}(w_{t-1})$ $w_t, w_{t-1}$ are both standard normal distributions. So their variance is 1: $Var(x_t)$ = $a_0^2\text{Var}(w_t)+a_1^2\text{Var}(w_{t-1})$ $Var(x_t)$ = $a_0^2+a_1^2$ $E[x_t] =0$

Analysis We do a simple reverse engineering exercise. Assets, income $a,y$ are the states and $a'$, savings for tomorrow, is the choice variable. The most straightforward way to visualize extrapolation and its impact on MPCs out of stimulus payments (modeled as wealth transfers for the moment) is to model MPCs in the following functional form: [name=Brandon Kaplowitz]