Geometric Series
(a) In a Fractional Reserve system of banking, each bank need only keep a fraction $\alpha$, say $5\%$, of its deposits and can lend out $1-\alpha$. The loan recipient will then deposit $1-\alpha$ at its own bank (or it could be the same bank) and use the proceeds as/when needed. The second bank in turn can lend out $1-\alpha$ of this amount $(1-\alpha {)}^2$, and so on. Compute the Velocity of Money: The total amount of currency generated by each new unit of currency

$$1+(1-\alpha )+(1-\alpha {)}^2+\dots$$

What is the velocity of money when $\alpha =5\%$ ?

(b) A simple model for a company's stock value is the Dividend Discount Model. It is assumed that the company pays periodic dividends with growth rate $g$, that is each new periodic dividend $D_{n+1}=$ $(1+g)D_n$. Discounting each future dividend using the same periodic discount rate $r$, the stock value $S$ is computed as the discounted value of future dividends

$$S=\sum _{i=1}^{\infty }{D}_0{\left(\frac{1+g}{1+r}\right)}^i$$

where $D_0$ is the current dividend amount. Using the above model, compute $S$ as a function of $D_0,r,g$, and provide conditions to ensure the sum converges.