The system is based on the false notion that for the depositor, the irregular deposit contract is a true deposit contract, while for the banker it is a loan or mutuum contract.
For example, supppose that Mr X makes a demand deposit of $1,000,000 in Bank A, and Bank A receives the money not as a deposit, but as a loan - it can (almost) freely use them. While keeping only a portion of this deposit on hand as a security reserve (reserve ratio, suppose - 10 percent, in our example) to satisfy possible demands for deposit withdrawals, the banker would be available to to use the other part of this demand deposit to his own benefit.
First of all, why is it necessary for the banker to maintain any reserve ratio? Does he not realize that when he acts legitimately as true intermediary between lenders and borrowers he need not maintain any?
It is important to emphasize that to all intents and purposes demand deposits are like physical units; that is, they are perfect money substitutes. The depositor can use them to make payments at any time by issuing a check on which he writes the sum he wishes to pay and giving instructions to the bank to make the payment. The portion of these perfect money substitutes, or demand deposits, which is not fully backed by physical monetary units in the bank's vault (the $900,000 not backed by reserves in the present example) is called fiduciary money.
Therefore, there has been an increase in the amount of money in circulation in the market, due to beliefs held simultaneously and with good reason by two different economic agents:
In other words, the bank's appropriation of $900,000 from a demand deposit results in an increase equal to $900,000 in the aggregate balances of money existing in the market.
Demand deposits backed by cash reserves at the bank ($100,000 in our example) are also called primary deposits, while the portion of demand deposits not backed by the bank's reserves (fiduciary money) is also called a derivative deposit.
We will now consider the limits to an isolated bank's capacity to create loans and expand deposits from nothing.
The following variables are involved:
d1 = (1 - k) * x
But d1 is equal to:
d1 = d - (c * d + c * k * x)
(1 - k) * x = d - (c * d + c * k * x) x * (1 - k + c * k) = d * (1 - c)  x = d * (1 - c) / (1 - k * (1 - c))
As formula  makes clear, the lower c is and the higher k is, the higher x will be.
Up until now we have assumed k to be the average percentage of deposit money unused by borrowers. However k can include other phenomena which have the same ultimate effect.
For instance, for the very great likelihood in a market where few banks operate, a borrower will make payments to some other customers of his own bank. When this happens, these customers will deposit their checks in their own accounts at the same bank, thus keeping money from leaving the bank. This phenomenon has the same ultimate effect as an increase in the average percentage of loans unused by borrowers.
The fewer the banks operating in the market, the higher k will be. The value of k is also increased when deposits are made in other banks, which in turn expand their loans, and their borrowers ultimately deposit in the original bank a significant portion of the new money they receive.
Let us now consider a particular type of isolated bank: a very small bank; that is, one in which k=0. This means borrowers immediately withdraw the entire amount of their loans, and those to whom they make payments are not customers of the same bank as the borrowers.
If k=0, then by substituting this value into formula  we obtain x=$900,000
Even if k is only slightly larger than 0, an isolated bank can create a considerably larger amount of fiduciary money, and the sum created may even exceed the total of original deposits in the isolated bank.
If we suppose that the proportion of loans which remain unused k=0.2 then, by formula  we obtain x=$1,097,561
Therefore we see that a bank will be able to grant loans not only for the sum of $900,000, but for a considerably larger amount, $1,097,560. Hence, the capacity for credit expansion and ex nihilo deposit creation is 22 percent greater than we initially supposed.
With this in mind, it is easy to understand why banks compete as fiercely as they do to attract the largest possible number of deposits and customers. Bankers try to attract as many customers as they can, because the more customers they have, the larger k will be; and the larger k is, the greater their capacity to expand loans and generate deposits.
In any case, k is a crucial factor in determining a bank's earning power. Competition between banks keeps k significantly below 1, however each bank fights to continually raise the value of its k factor.
Let us now suppose that k=1. We are dealing with a monopolistic bank: all final recipients of payments made by borrowers of the bank are also clients of the bank. When we substitute the value k=1 into formula , we obtain:
 x = d * (1 - c) / cReturning to our example in which d=$1,000,000 and c=0.1, we obtain x=$9,000,000
In this case, the bank could create ex nihilo loans and deposits (fiduciary money) for the sum of $9,000,000, which means it could multiply its total money supply by 10.
Supporse that the bank's practice of repetitively granting its own clients loans for an amount equal to 90 percent of the funds it receives in cash. The clients withdraw the full amount of the loan, but because they have no account in any other bank, they ultimately deposit the money they receive back into the same bank. This permits the bank, in turn, to grant new loans and generate new deposits, and the process is repeated again and again. Thus, the total volume of deposits in a monopolistic bank would be equal to the value of the original deposits, d, divided by the reserve ratio, c. Formula  is the simplest version of the so-called "bank multiplier"
And we could use the following formula to calculate the net credit expansion:
x = d/c - d
It is clear that the assets generated by the banking system do not actually belong to anyone. To a large extent, however, they could be considered the property of banks' shareholders, directors and administrators, the people who actually take advantage of many of the economic benefits of this wealth, with the additional advantage of not appearing as the owners, since the account books indicate that the depositors own the wealth.
Complexity derives from the fact that in reality a certain percentage of the money supply "filters" out of the system and is kept by individuals who do not wish to deposit it in a bank. The larger the this percentage at each stage, the smaller the bank's expansive capacity to generate new loans.
The credit expansion brought about by a banking system out of which some money filters would be equal to:
x = d / (c + f) - d
Let us see to what value credit expansion is reduced if, as before, d=$1,000,000 and c=0.1, while in addition f=0.15
The total sum of deposits would be $3,617,029 instead of $10,000,000 as is the case when f=0. Therefore, when the percentage of money which filters out f is greater than zero, the capacity of the banking system to create loans and generate deposits ex nihilo decreases noticeably.
Another complication takes place when banks hold cash reserves exceeding the minimum requirement. This tends to occur at certain stages in the economic cycle in which banks behave relatively more prudently, or they are obliged to increase their reserves due to difficulties in finding enough creditworthy borrowers willing to request loans, or both. The maintenance of cash reserves exceeding the necessary level reduces the system's capacity for credit expansion in the same way as f, a percentage of the money supply which filters out of the banking system.
Let us imagine that banking is just beginning to emerge, and banks act as true depositaries of money as stipulated in an irregular deposit contract.
As long as the general legal principles are upheld, banks will accept moneys and keep them in their vaults, and in return they will give certificates for the entire sum deposited. If the bank fulfills its commitments for a lengthy period of time and people completely trust it, it is certain that the public will gradually begin to use the certificates as if they were the units of commodity money themselves, thus converting the certificates into monetary units (they would not need to go to the bank and withdraw the monetary units they originally deposited).
When this situation arises, bankers may start to feel tempted to issue deposit receipts for an amount exceeding the sum of monetary units actually deposited.
Clearly if bankers succumb to this temptation, they violate universal legal principles and commit not only the crime of counterfeiting (by issuing a false receipt unbacked by a corresponding deposit), but the crime of fraud as well, by presenting as a means of payment a document that in reality lacks all backing.
Nevertheless, if people place enough trust in the bank and the banker knows from experience that a reserve ratio of 0.1 will permit him to honor his commitments under ordinary circumstances, he will be able to issue up to nine times more in new false deposit receipts. If people trust the bank, borrowers will agree to receive their loans in bills, which will circulate as if they were money.
We can conclude that once the certificates have acquired the nature of monetary units, no one will ever return them to the bank to withdraw the money deposited, since the bills circulate freely and are considered money themselves. Under these conditions the banker may believe, with good reason, that no one will ever return these bills to the bank to withdraw the original money deposited. The moment the banker decides this is the case, his judgment may manifest itself as an accounting entry identifying the $9,000,000 false bills put into circulation by the bank as part of the year's profit, which the banker may freely appropriate.
In fact the nature of banknotes is identical to that of secondary deposits and both produce the same economic effects. Both activities generate considerable assets for banks, who gradually take this wealth from all economic agents in the market through a process the agents cannot understand or identify, one which leads to small decreases in the purchasing power of the monetary units all use in society.
In this way banks appropriate a large volume of wealth, which from an accounting standpoint they guarantee with deposits that permit them to disguise the fact that economically speaking they are the only beneficiaries who completely take advantage de facto of these assets. Thus they have found a perennial source of financing which will probably not be demanded from them, a "loan" they will never have to return.
From an economic point of view, bankers and other related economic agents are the ones who take advantage of these extraordinary circumstances. They possess the enormous power to create money, and they use this power continually to expand their assets. Furthermore they have managed to keep their activities relatively hidden from most of the public, including economists, by backing their created loans with liability accounts (deposit accounts or banknote accounts) that do not coincide with their actual equity.
Banknotes take the form of bearer bonds and each has a particular face value, allowing the notes to be transferred from one person to another without it being necessary for the bank to make any accounting entry in its books.
In contrast deposits offer customers the advantage of being able to write an exact figure on a check without needing to hand over a specific number of bills of a set value. However the fact that the banker must follow the transactions conducted and record them in his books constitutes a disadvantage.
Still, apart from these legal differences and differences in form, from an economic standpoint the two operations are essentially identical and produce the same effects.
However, when the theory of money was first being developed, theorists only recognized the immorality of the creation of unbacked banknotes and the serious harm it causes. They did not initially realize nor respond to the fact that the expansive creation of loans backed by deposits generated from nothing has exactly the same effects.
This explains why the Peel Act of July 19, 1844, the foundation of all modern banking systems, prohibited the issuance of unbacked bills yet failed miserably to achieve its objectives of monetary stability and an adequate definition and defense of citizens' property rights with respect to banking.
Its failure was due to legislators' inability to comprehend that bank deposits with a fractional reserve have exactly the same nature and economic effects as unbacked banknotes. As a result, the Act did not outlaw fractional-reserve banking and allowed the age-old practice of "issuing" unbacked (secondary) deposits to continue.
In reality secondary deposits predated the fiduciary issue of banknotes, but because the former proved much more complex, only the latter was (very belatedly) prohibited. The monetary bank-deposit contract with a fractional reserve is still legal today, even though it has exactly the same economic nature and produces the same damaging effects as the issuance of unbacked banknotes prohibited in 1844 by the Peel Act.
One of the central problems posed by the process of credit expansion and ex nihilo deposit creation, and thus by the bank deposit contract involving a fractional reserve, is that just as this process inevitably unleashes forces that reverse the effects of credit expansion on the real economy, it also looses forces which lead to a dual process of credit tightening or contraction.
Any of the following events serve to establish that such a process has been set in motion:
First, it is clear that if a certain sum in original deposits is withdrawn from a bank, all created loans and deposits would disappear in a chain reaction, implying a significant drop in the money supply. The result is severe deflation, which, in the short and medium term, further aggravates the recession.
Second, a desire of the public to keep more money outside the banking system produces the same effects. It provokes an increase in f and a decline in banks' capacity for credit expansion, which in turn brings about a recession and a monetary squeeze.
Third, a decision by banks to be more "prudent" and to increase their reserve ratio c leads to a contraction as well.
Fourth, the repayment of loans produces equally deflationary effects when enough new loans are not granted to at least offset the ones returned.
Fifth, if the loans lose their value due to the failure of the economic activity for which they were employed, the corresponding bank must record this fact as a loss.
The crucial problem posed by credit tightening consists of the fact that the very process of credit expansion based on a fractional reserve inevitably triggers the granting of loans unsupported by voluntary saving, resulting in a process of intertemporal discoordination, which in turn stems from the distorted information the banking system imparts to businessmen who receive loans generated ex nihilo by the system. Hence businessmen rush out to launch investment projects as if society's real saving had increased, when in fact this has not happened. The result is artificial economic expansion or a "boom," which inevitably provokes an adjustment in the form of a crisis and economic recession.
The crisis and economic recession reveal that a highly significant number of investment projects financed under new loans created by banks are not profitable because they do not correspond to the true desires of consumers. Therefore many investment processes fail, which ultimately has a profound effect on the banking system. The harmful consequences are evidenced by a widespread repayment of loans by many demoralized businessmen assessing their losses and liquidating unsound investment projects (thus provoking deflation and the tightening of credit); they are also demonstrated by an alarming and atypical rise in payment arrears on loans (adversely affecting the banks' solvency).
Just as the money supply was expanded according to the bank multiplier, artificial economic expansion fostered by the ex nihilo creation of loans eventually triggers an endogenous recession, which in the form of a widespread repayment of loans and an increase in arrears, reduces the money supply substantially.
Therefore the fractional-reserve banking system generates an extremely elastic money supply, which "stretches" with ease but then must contract just as effortlessly, producing the corresponding effects on economic activity, which is repeatedly buffeted by successive stages of boom and recession.
In short, bank customers' economic difficulties, one of the inevitable consequences of all credit expansion, render many loans irrecoverable, accelerating even more the credit tightening process (the inverse of the expansion process). In fact the bank may completely fail as a result, in which case the bills and deposits issued by it (which we know are economically equivalent) will lose all value, further aggravating the monetary squeeze (instead of the $9,000,000 decrease in the money supply caused by the return of a loan, here the money supply would drop by $10,000,000 that is, including the $1,000,000 in primary deposits held by the bank).
Furthermore, one bank's solvency problems are enough to sow panic among the customers of all other banks, leading them to suspend payments one by one, with tragic economic and financial consequences.
Moreover we must point out that, even if the public continues to trust banks (despite their insolvency), and even if a central bank created ad hoc for such situations provides all the liquidity necessary to assure depositors their deposits are fully protected, the inability to recover loans initiates a process of credit tightening that is spontaneously set off when loans are repaid and cannot be replaced by new ones at the same rate.
This phenomenon is typical of periods of recession. When customers default on their loans, banks become more cautious about granting more. Hence the natural reluctance of the demoralized public to request loans is reinforced by banks' greater prudence and rigor when it comes to giving them. In addition, as bankers see their profitability fall along with the value of their assets as a result of irrecoverable loans, they will attempt to be more careful, and other things being equal, to increase their cash on hand by raising their reserve ratio, which will have an even greater tightening effect.
Finally business failures and frustration arising from the inability to honor commitments to banks will contribute even more to the demoralization of economic agents and to their determination to avoid new investment projects financed with bank loans.