S = d + ad + a²d + a³d + a⁴d + ... forever with the power of a increasing by 1 each term

if 0 < a < 1, then S = kd where k = 1/[1-a] loosely speaking the multiplier is the factor that sums the series.

Previously
DY= DI + bDI + b²DI + b³DI + b⁴DI ...
DY = k DI where in this case k = 1/[1-b]
where DY=change GDP, DI = change investments, and b = MPC

Money expansion multiplier.
DM = DG + (1-RR)DG + (1-RR)²DG + (1-RR)³DG + (1-RR)⁴DG + (1-RR)⁵DG ...
DM = kDG where in this case k = 1/[1- (1-RR)] = 1/RR
where DM =change money supply, DG = change government securities, and RR is the required reserve ratio.

DG is + when FED buys => DM + and
DG is - when FED sells => DM -
we now work out the first 2 terms of the + geometric series

Standard bank for how the Fed expands and contracts the money supply.

Where A = assets L = Liabilities, R = reserves, L = loans, G = government securities, and D= deposits. D is a liability to the banks because it is owed to the depositors (At the same time it is an asset to the depositors)

In the real world there are many ways the mechanism works. For class we stick to one scenario to keep it simple. The action moves from one bank to the next. It could all take place within one bank.

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