Dyskusja:Mnożnik kreacji pieniądza

message: Hope this is useful.
Let the monetary base be normalized to unity. Define the legal reserve ratio, \alpha \in\left(0, 1\right)\;, the excess reserves ratio, \beta \in\left(0, 1\right)\;, the currency drain ratio with respect to deposits, \gamma \in\left(0, 1\right)\;; suppose the demand for funds is unlimited; then the theoretical superior limit for deposits is defined by the following series:
Deposits = \sum_{n = 0}^{\infty}\left[\left(1 - \alpha - \beta - \gamma\right)\right]^{n} = \frac{1}{\alpha + \beta + \gamma}.
Analogously, the theoretical superior limit for the money held by public is defined by the following series:
Publicly Held Currency = \gamma \cdot Deposits = \frac{\gamma}{\alpha + \beta + \gamma}
and the theoretical superior limit for the totale loans lent in the market is defined by the following series:
Loans = \left(1 - \alpha - \beta\right) \cdot Deposits = \frac{1 - \alpha - \beta}{\alpha + \beta + \gamma}
By summing up the two quantities, the theoretical money multiplier is defined as
m = \frac{Money Stock}{Monetary Base} = \frac{Deposits + Publicly Held Currency}{Monetary Base} = \frac{1 + \gamma}{\alpha + \beta + \gamma}
where \alpha + \beta = Desired Reserve Ratio and \gamma = Currency Drain Ratio
The process described above by the geometric series can be represented in the following table, where
*loans at stage k\; are a function of the deposits at the precedent stage: L_{k} = \left(1 - \alpha - \beta\right) \cdot D_{k - 1}
*publicly held money at stage k\; is a function of the deposits at the precedent stage: PHM_{k} = \gamma \cdot D_{k - 1}
*deposits at stage k\; are the difference between additional loans and publicly held money relative to the same stage: D_{k} = L_{k} - PHM_{k}\;
 Process of money multiplication
n\;                 	Deposits                                          	Loans                                                                                   	Publicly Held Money
n = 0\;             	D_{0} = 1\;                                       	-                                                                                       	-
n = 1\;             	D_{1} = \left(1 - \alpha - \beta - \gamma\right)  	L_{1} = \left(1 - \alpha - \beta\right)                                                 	PHM_{1} = \gamma\;
n = 2\;             	D_{2} = \left(1 - \alpha - \beta - \gamma\right)^2	L_{2} = \left(1 - \alpha - \beta\right) \left(1 - \alpha - \beta - \gamma\right)        	PHM_{2} = \gamma \left(1 - \alpha - \beta - \gamma\right)
n = 3\;             	D_{3} = \left(1 - \alpha - \beta - \gamma\right)^3	L_{3} = \left(1 - \alpha - \beta\right) \left(1 - \alpha - \beta - \gamma\right)^2      	PHM_{3} = \gamma \left(1 - \alpha - \beta - \gamma\right)^2
…                   	…                                                 	…                                                                                       	…
n = k\;             	D_{k} = \left(1 - \alpha - \beta - \gamma\right)^k	L_{k} = \left(1 - \alpha - \beta\right) \left(1 - \alpha - \beta - \gamma\right)^{k - 1}	PHM_{k} = \gamma \left(1 - \alpha - \beta - \gamma\right)^{k - 1}
…                   	…                                                 	…                                                                                       	…
n \rightarrow \infty	D_{\infty} = 0                                    	L_{\infty} = 0                                                                          	PHM_{\infty} = 0
                    	                                                  	                                                                                        	
                    	Total Deposits:                                   	Total Loans:                                                                            	Total Publicly Held Money:
                    	D = \frac{1}{\alpha + \beta + \gamma}             	L = \frac{1 - \alpha - \beta}{\alpha + \beta + \gamma}                                  	PHM = \frac{\gamma}{\alpha + \beta + \gamma}
You just need to translate few words in Polish and you can insert it in your article. Cheers, Picodoro (dyskusja) 04:14, 11 cze 2010 (CEST)
thread_title:  Some formulas (sorry, I don't speak Polish) 

message: Brak informacji o przeciętnych i maksymalnych wartościach mnożnika (ew. skutkach, kryzysach etc.) Sagi2007 (dyskusja) 13:52, 13 paź 2012 (CEST)
thread_title:  Przeciętne i maksymalne wartości